• Khan Academy
  • Math
    • K-8th grades
    • Eureka Math/EngageNY support
    • Remediation support for Eureka Math/EngageNY
    • On-grade support for Eureka Math/EngageNY
    • High school math
    • Early math
      • Counting
        • Counting
          • Counting with small numbers
          • Counting in order
        • Numbers 0 to 120
          • Number grid
          • Missing numbers between 0 and 120
        • Counting objects
          • Counting in pictures
          • Counting objects 1
          • Counting objects 2
        • Comparing small numbers
          • Comparing numbers of objects
          • Comparing numbers on the number line
          • Counting by category
      • Addition and subtraction intro
        • What is addition? What is subtraction?
          • Intro to addition
          • Intro to subtraction
        • Making small numbers
          • Making 5
        • Making 10
          • Getting to 10 by filling boxes
          • Adding to 10
        • Put together, take apart
          • Add and subtract: pieces of fruit
          • Addition and subtraction within 10
        • Relate addition and subtraction
          • Relating addition and subtraction
        • Addition and subtraction word problems
          • Addition word problems within 10
          • Subtraction word problems within 10
      • Place value (tens and hundreds)
        • Teens
          • Teens as sums with 10
          • Teen numbers: monkeys
        • Tens
          • Intro to place value
          • Place value example: 25
          • Place value example: 42
        • Comparing 2-digit numbers
          • Greater than and less than symbols
        • Hundreds
        • Comparing three-digit numbers
          • Comparing whole numbers
      • Addition and subtraction within 20
        • Addition within 20
          • Adding 7 + 6
          • Adding 8 + 7
          • Adding 5 + 3 + 6
        • Subtraction within 20
          • Subtracting 14 - 6
        • Equal sign
          • Equal sign
        • Missing number within 20
        • Word problems within 20
          • Addition and subtraction word problems: superheroes
          • Addition and subtraction word problems: gorillas
        • Word problems with "more" and "fewer"
          • Comparison word problems: marbles
          • Comparison word problems: roly-polies
        • Repeated addition
          • Repeated addition: haircuts
      • Addition and subtraction within 100
        • Adding 1s and 10s
          • Adding 1 vs. adding 10
          • Understanding place value when adding tens
          • Understanding place value when adding ones
        • Subtracting 1s and 10s
          • Subtracting 1 vs. subtracting 10
          • Subtracting 1s using place value
          • Subtracting 10s using place value
        • Intro to addition with 2-digit numbers
          • Adding 2-digit numbers without regrouping 1
          • Adding 2-digit numbers without regrouping 2
          • Breaking apart 2-digit addition problems
          • Regrouping to add 1-digit number
          • Adding by making a group of 10
        • Intro to subtraction with 2-digit numbers
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting two-digit numbers without regrouping (example 2)
          • Subtracting a 1-digit number with regrouping
        • Strategies for adding and subtracting within 100
          • Adding 53+17 by making a group of 10
          • Adding by making a group of 10
          • Strategies for adding 2-digit numbers
          • Addition and subtraction with number lines
        • Addition within 100
          • Adding 2-digit numbers without regrouping 2
          • Adding with regrouping
        • Subtraction within 100
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting with regrouping (borrowing)
        • Word problems within 100
          • Adding and subtracting on number line word problems
          • Subtraction word problem: tennis balls
          • Addition word problem: horses
          • Subtraction word problem: snow
          • Subtraction word problem: crayons
        • Word problems with "more" and "fewer" (within 100)
          • Subtraction word problem: basketball
          • Addition word problem: starfish
          • Addition word problem: spots
        • Skip-counting
          • Skip-counting by 5s
          • Counting by 10s
          • Skip-counting by 100s
        • Addition and subtraction missing value problems
          • Missing numbers in addition and subtraction
      • Addition and subtraction within 1000
        • Adding 1s, 10s, and 100s
          • Adding 10 or 100
          • Adding 1s, 10s, and 100s
          • Adding 3-digit numbers (no regrouping)
        • Subtracting 1s, 10s, and 100s
          • Subtracting 1, 10, or 100
          • Subtracting 1s, 10s, and 100s
          • Subtracting 3-digit numbers (no regrouping)
        • Strategies for adding 2- and 3-digit numbers
          • Breaking apart 3-digit addition problems
          • Solving 3-digit addition in your head
          • Addition using groups of 10 and 100
          • Adding and subtracting on number line 2
      • Measurement and data
        • Length and size
          • Ordering by length
        • Measuring length
          • Measuring lengths 2
          • Measuring lengths with different units
        • Comparing and estimating length
          • Comparing lengths
          • Estimating lengths
        • Length word problems
          • Length word problems
          • Measuring length: golden statue
        • Picture graphs
          • Picture graphs
          • Making picture graphs and line plots
        • Bar graphs
          • Reading bar graphs: dog bones
          • Reading bar graphs: bikes
          • Creating picture and bar graphs 1
        • Line plots
          • Intro to line plots
          • Making picture graphs and line plots
        • Time
          • Telling time (labeled clock)
          • Telling time (unlabeled clock)
        • Money
          • Counting dollars
          • Counting American coins
      • Geometry
        • Basic shapes
          • Cousin Fal's shape collection
          • Recognizing shapes
        • Properties of shapes
          • Cousin Fal's shape collection
          • Recognizing shapes
        • Composing shapes
          • Composing shapes
        • Fractions of shapes
          • Halves and fourths
          • Equal parts of circles and rectangles
    • Kindergarten
      • Counting and place value
        • Counting small numbers
          • Counting with small numbers
          • Counting in order
        • Numbers 0 to 100
          • Number grid
          • Missing numbers between 0 and 120
        • Counting objects
          • Counting in pictures
          • Counting objects 1
          • Counting objects 2
        • Comparing small numbers
          • Comparing numbers of objects
          • Comparing numbers on the number line
          • Counting by category
        • Teens
          • Teens as sums with 10
          • Teen numbers: monkeys
      • Addition and subtraction
        • What is addition? What is subtraction?
          • Intro to addition
          • Intro to subtraction
        • Making 10
          • Getting to 10 by filling boxes
          • Adding to 10
        • Making small numbers
          • Making 5
        • Put together, take apart
          • Add and subtract: pieces of fruit
          • Addition and subtraction within 10
        • Addition and subtraction word problems
          • Addition word problems within 10
          • Subtraction word problems within 10
      • Measurement and geometry
        • Comparing size
        • Basic shapes
          • Cousin Fal's shape collection
          • Recognizing shapes
        • Comparing shapes
        • Composing shapes
          • Composing shapes
    • 1st grade
      • Place value
        • Numbers 0 to 120
          • Number grid
          • Missing numbers between 0 and 120
        • Ones and tens
          • Intro to place value
          • Place value example: 25
          • Place value example: 42
        • Comparing 2-digit numbers
          • Greater than and less than symbols
      • Addition and subtraction
        • Relate addition and subtraction
          • Relating addition and subtraction
        • Addition within 20
          • Adding 7 + 6
          • Adding 8 + 7
          • Adding 5 + 3 + 6
        • Subtraction within 20
          • Subtracting 14 - 6
        • Equal sign
          • Equal sign
        • Missing number within 20
        • Word problems within 20
          • Addition and subtraction word problems: superheroes
          • Addition and subtraction word problems: gorillas
        • Word problems with "more" and "fewer"
          • Comparison word problems: marbles
          • Comparison word problems: roly-polies
        • Adding 1s and 10s
          • Adding 1 vs. adding 10
          • Understanding place value when adding tens
          • Understanding place value when adding ones
        • Intro to addition with 2-digit numbers
          • Adding 2-digit numbers without regrouping 1
          • Adding 2-digit numbers without regrouping 2
          • Breaking apart 2-digit addition problems
          • Regrouping to add 1-digit number
          • Adding by making a group of 10
      • Measurement and data
        • Length and size
          • Ordering by length
          • Measuring length: golden statue
        • Bar graphs
          • Reading bar graphs: dog bones
        • Time
          • Telling time (labeled clock)
      • Geometry
        • Shapes
          • Cousin Fal's shape collection
          • Recognizing shapes
        • Fractions of shapes
          • Halves and fourths
    • 2nd grade
      • Place value
        • Hundreds
        • Comparing 3-digit numbers
          • Comparing whole numbers
      • Addition and subtraction within 100
        • Repeated addition
          • Repeated addition: haircuts
        • Adding 1s and 10s
          • Adding 1 vs. adding 10
          • Understanding place value when adding tens
          • Understanding place value when adding ones
        • Subtracting 1s and 10s
          • Subtracting 1 vs. subtracting 10
          • Subtracting 1s using place value
          • Subtracting 10s using place value
        • Intro to addition with 2-digit numbers
          • Adding 2-digit numbers without regrouping 1
          • Adding 2-digit numbers without regrouping 2
          • Breaking apart 2-digit addition problems
          • Regrouping to add 1-digit number
          • Adding by making a group of 10
        • Intro to subtraction with 2-digit numbers
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting two-digit numbers without regrouping (example 2)
          • Subtracting a 1-digit number with regrouping
        • Strategies for adding and subtracting within 100
          • Adding 53+17 by making a group of 10
          • Adding by making a group of 10
          • Strategies for adding 2-digit numbers
          • Addition and subtraction with number lines
        • Addition within 100
          • Adding 2-digit numbers without regrouping 2
          • Adding with regrouping
        • Subtraction within 100
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting with regrouping (borrowing)
        • Word problems within 100
          • Adding and subtracting on number line word problems
          • Subtraction word problem: tennis balls
          • Addition word problem: horses
          • Subtraction word problem: snow
          • Subtraction word problem: crayons
        • Word problems with "more" and "fewer" (within 100)
          • Subtraction word problem: basketball
          • Addition word problem: starfish
          • Addition word problem: spots
        • Skip-counting
          • Skip-counting by 5s
          • Counting by 10s
          • Skip-counting by 100s
        • Addition and subtraction missing value problems
          • Missing numbers in addition and subtraction
      • Addition and subtraction within 1000
        • Adding 1s, 10s, and 100s
          • Adding 10 or 100
          • Adding 1s, 10s, and 100s
          • Adding 3-digit numbers (no regrouping)
        • Subtracting 1s, 10s, and 100s
          • Subtracting 1, 10, or 100
          • Subtracting 1s, 10s, and 100s
          • Subtracting 3-digit numbers (no regrouping)
        • Strategies for adding 2- and 3-digit numbers
          • Breaking apart 3-digit addition problems
          • Solving 3-digit addition in your head
          • Addition using groups of 10 and 100
          • Adding and subtracting on number line 2
      • Measurement and data
        • Measuring length
          • Measuring lengths 2
          • Measuring lengths with different units
        • Comparing and estimating length
          • Comparing lengths
          • Estimating lengths
        • Length word problems
          • Length word problems
          • Measuring length: golden statue
        • Picture graphs
          • Picture graphs
          • Making picture graphs and line plots
        • Bar graphs
          • Reading bar graphs: bikes
          • Creating picture and bar graphs 1
        • Line plots
          • Intro to line plots
          • Making picture graphs and line plots
        • Time
          • Telling time (labeled clock)
          • Telling time (unlabeled clock)
        • Money
          • Counting dollars
          • Counting American coins
      • Geometry
        • Shapes
          • Cousin Fal's shape collection
          • Recognizing shapes
        • Fractions of shapes
          • Equal parts of circles and rectangles
    • 3rd grade
      • Addition and subtraction
        • Strategies for adding two and three-digit numbers
          • Breaking apart 3-digit addition problems
          • Addition using groups of 10 and 100
          • Adding and subtracting on number line 2
          • Estimating adding and subtracting 3-digit numbers
        • Adding with regrouping within 1000
          • Intro to place value
          • Adding 3-digit numbers
          • Adding with regrouping
          • Adding 3-digit numbers with regrouping
        • Subtracting with regrouping within 1000
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Subtracting 3-digit numbers (regrouping) 2
          • Worked example: Subtracting 3-digit numbers (regrouping twice)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
          • Subtracting in your head (no regrouping)
        • Addition and subtraction missing value problems
          • Missing number for addition and subtraction within 1000
      • Multiplication and division
        • Multiplication intro
          • Intro to multiplication
          • Multiplication as groups of objects
          • More ways to multiply
        • Multiplication facts
        • Relating multiplication and division
          • Relating division to multiplication
          • Multiplication word problem: parking lot
          • Division word problem: school building
          • Multiplication word problem: soda party
          • Division word problem: blueberries
        • Division intro
          • The idea of division
        • Division facts
        • More with 1-digit multiplication and division
          • Unknowns with multiplication and division
        • Properties of multiplication
          • Properties of multiplication
          • Properties and patterns for multiplication
          • Associative property of multiplication
        • One and two-step word problems
          • 2-step estimation problem: marbles
          • 2-step word problem: truffles
          • 2-step word problem: running
          • 2-step word problem: theater
        • Multiplying by tens
          • Multiplying by multiples of 10
          • Multiplying by tens word problem
        • Patterns in arithmetic
          • Finding patterns in numbers
          • Intro to even and odd numbers
          • Patterns in multiplication tables
      • Fractions
        • Fractions intro
          • Intro to fractions
          • Cutting shapes into equal parts
          • Identifying unit fractions word problem
        • What fractions mean
          • More than one equal section
          • Recognizing fractions greater than 1
          • Identifying numerators and denominators
        • Fractions on the number line
          • Fractions on a number line
          • Fractions on number line widget
        • Fractions and whole numbers
          • Representing 1 as a fraction
          • Whole numbers as fractions
        • Equivalent fractions
          • Equivalent fractions with visuals
          • Equivalent fraction models
        • Equivalent fractions on the number line
          • Equivalent fraction visually
          • Creating equivalent fractions
        • Comparing fractions
          • Comparing fractions with > and < symbols
          • Comparing fractions visually
          • Comparing fractions with the same denominator
          • Comparing fractions with the same numerator
        • Comparing fractions of different wholes
          • Comparing fractions of different wholes 1
          • Comparing fractions of different wholes 2
      • Measurement and data
        • Time
          • Telling time to the nearest minute (labeled clock)
          • Telling time to the nearest minute (unlabeled clock)
          • Time differences example
          • Telling time with number line
          • Time word problem: travel time
          • Time word problem: puzzle
        • Mass
          • Understanding mass (grams and kilograms)
          • Word problems with mass
        • Volume
          • Understanding volume (liters)
          • Word problems with volume
        • Bar graphs
          • Creating picture and bar graphs 1
          • Reading bar graphs: Harry Potter
          • Reading bar graphs: movies
          • Interpreting bar graphs: colors
        • Picture graphs
          • Creating picture and bar graphs 1
          • Solving problems with pictographs 2
          • Interpreting picture graphs: paint
          • Interpreting picture graphs: notebook
        • Line plots
          • Intro to line plots
          • Graphing data on line plots
      • Geometry
        • Area introduction
          • Intro to area and unit squares
        • Count unit squares to find area
          • Intro to area and unit squares
          • Measuring rectangles with different unit squares
          • Measuring area with partial unit squares
          • Creating rectangles with a given area 1
          • Creating rectangles with a given area 2
        • Area formula intuition
          • Counting unit squares to find area formula
          • Transitioning from unit squares to area formula
        • Multiply to find area
          • Counting unit squares to find area formula
          • Finding missing side when given area
          • Comparing areas of plots of land
        • Area and the distributive property
          • Area and the distributive property
        • Decompose figures to find area
          • Decomposing shapes to find area: grids
          • Decomposing shapes to find area: add
          • Decomposing shapes to find area: subtract
          • Area word problem: house size
        • Perimeter
          • Perimeter: introduction
          • Perimeter of a shape
          • Finding perimeter when a side length is missing
          • Finding missing side length when given perimeter
        • Comparing area and perimeter
          • Comparing areas and perimeters of rectangles
        • Perimeter word problems
          • Perimeter word problem: tables
          • Perimeter word problem: skating rink
        • Quadrilaterals
          • Intro to quadrilaterals
      • Place value and rounding
        • Rounding
          • Rounding to nearest 10
          • Rounding to nearest 100
          • Rounding to nearest 10 and 100
    • 4th grade
      • Addition and subtraction
        • Adding multi-digit numbers
          • Adding 3-digit numbers
          • Adding with regrouping
          • Multi-digit addition with regrouping
        • Subtracting multi-digit numbers
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
          • Multi-digit subtraction with regrouping: 7329-6278
          • Multi-digit subtraction with regrouping: 6798-3359
          • Multi-digit subtraction with regrouping twice
      • Multiplication and division
        • Multiplication by 10s, 100s, and 1000s
          • Multiplying 1-digit numbers by 10, 100, and 1000
          • Multiplying 1-digit numbers by multiples of 10, 100, and 1000
          • Multiplying 10s
        • Multi-digit multiplication: place value and area models
          • More ways to multiply
          • Multiplying with distributive property
          • Multiplying with area model: 6 x 7981
          • Multiplying with area model: 78 x 65
          • Multiplying with area model: 16 x 27
        • Multi-digit multiplication: standard algorithm
          • Multiplying 2-digit by 1-digit
          • Multiplying 3-digit by 1-digit
          • Multiplying: 2-digit by 1-digit (regrouping)
          • Multiplying 3-digit by 1-digit (regrouping)
          • Multiplying 4-digit by 1-digit (regrouping)
        • Comparing with multiplication
          • Comparing with multiplication: basics
          • Comparing with multiplication: giraffe
          • Comparing with multiplication: money
          • Comparing with multiplication: age
          • Comparing with multiplication: magic
        • Division problems that work out nicely
          • Quotients that are multiples of 10
          • Canceling zeros when dividing
        • Division: place value and area
          • Division using place value
          • Division with area models
        • Multi-digit division
          • Intro to long division (no remainders)
          • Long division: 280÷5
        • Remainders
          • Intro to remainders
          • Worked example: Long division with remainders: 3771÷8
          • Worked example: Long division with remainders: 2292÷4
        • Multiplication, division word problems
          • Division word problem: field goals
          • Multiplication word problem: pizza
      • Fractions
        • Equivalent fractions
          • Intro to equivalent fractions
          • Visualizing equivalent fractions
          • More on equivalent fractions
          • Equivalent fractions and different wholes
        • Common denominators
          • Finding common denominators
          • Common denominators: 1/4 and 5/6
          • Common denominators: 3/5 and 7/2
        • Comparing fractions with unlike denominators visually
          • Comparing fractions: tape diagram
          • Comparing fractions: number line
          • Comparing fractions: fraction models
        • Comparing fractions with unlike denominators
          • Comparing fractions 1 (unlike denominators)
          • Ordering fractions
          • Comparing fractions 2 (unlike denominators)
        • Decomposing fractions
          • Decomposing a fraction visually
          • Decomposing a mixed number
        • Adding and subtracting fractions with like denominators
          • Adding fractions with like denominators
          • Subtracting fractions with like denominators
        • Adding and subtracting fractions: word problems
          • Fraction word problem: pizza
          • Fraction word problem: spider eyes
          • Fraction word problem: piano
        • Mixed numbers
          • Rewriting mixed numbers as improper fractions
          • Rewriting improper fractions as mixed numbers
          • Comparing improper fractions and mixed numbers
        • Adding and subtracting mixed numbers
          • Intro to adding mixed numbers
          • Intro to subtracting mixed numbers
          • Fraction word problem: lizard
          • Subtracting mixed numbers with like denominators
        • Multiplying unit fractions and whole numbers
          • Multiplying unit fractions and whole numbers
        • Multiplying whole numbers and fractions
          • Multiplying fractions and whole numbers visually
          • Equivalent fraction and whole number multiplication problems
        • Multiplying whole numbers and fractions word problems
          • Multiplying fractions word problem: milk
          • Multiplying fractions word problem: movies
          • Multiplying fractions by whole numbers word problem
        • Fractions with denominators of 10 and 100
          • Visually converting tenths and hundredths
          • Decomposing hundredths
          • Decomposing hundredths on number line
          • Adding fractions (denominators 10 & 100)
          • Adding fractions: 7/10+13/100
      • Decimals
        • Intro to decimals
          • Intro to decimals
          • Decimals as words
          • Writing decimal numbers shown in grids
        • Decimals on the number line
          • Graphing tenths from 0 to 1
          • Identifying tenths on a number line
          • Graphing hundredths from 0 to 0.1
          • Identifying hundredths on a number line
          • Plotting decimal numbers on a number line
        • Decimals vs. fractions
          • Rewriting fractions as decimals
          • Writing a number as a fraction and decimal
        • Decimals greater than one
          • Place value for decimals greater than 1
        • Converting decimals to fractions
          • Rewriting decimals as fractions: 0.15
          • Rewriting decimals as fractions: 0.8
          • Rewriting decimals as fractions: 0.36
        • Common fractions and decimals
          • Common fractions and decimals
          • Common fractions (halves)
        • Comparing decimals visually
          • Comparing decimals visually
          • Comparing numbers represented different ways
        • Comparing decimals
          • Comparing decimal numbers on a number line
          • Comparing decimals 1
          • Comparing decimals 2
      • Measurement and data
        • Estimating mass
          • Metric system: units of weight
          • Choose pounds or ounces to measure weight
          • U.S. customary units: weight
        • Estimating volume
          • Metric system: units of volume
          • U.S. customary units: fluid volume
        • Estimating length
          • Metric system: units of distance
          • U.S. customary units: distance
        • Time
          • Finding reasonable unit of measurement example
          • Time differences example
          • Time word problem: puzzle
          • Time word problem: travel time
        • Converting units of mass
          • How to convert kg to mg and T to oz
          • Converting pounds to ounces
        • Converting units of volume
          • Converting US customary units of volume
        • Converting units of length
          • Converting metric units of length
          • Converting feet to inches
          • Converting yards into inches
        • Converting units of time
          • Converting units of time
          • Time word problem: travel time
          • Time word problem: Susan's break
        • Money word problems
          • Word problem: making change
          • Currency conversion word problem
        • Conversion word problems
          • Multi-step unit conversion examples (metric)
          • Multi-step unit conversion examples (U.S. Customary)
        • Area and perimeter
          • Area & perimeter word problem: dog pen
          • Area & perimeter word problem: table
          • Comparing areas word problem
        • Line plots with fractions
          • Reading a line plot with fractions
      • Geometry
        • Lines, line segments, and rays
          • Terms & labels in geometry
          • Lines, line segments, & rays
        • Parallel and perpendicular
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines
          • Drawing parallel line segments
        • Angle introduction
          • Angles: introduction
          • Naming angles
        • Measuring angles
          • Measuring angles in degrees
          • Measuring angles using a protractor
          • Measuring angles using a protractor 2
        • Constructing angles
          • Constructing angles
        • Angles in circles
          • Angle measurement & circle arcs
          • Measuring angles with a circular protractor
          • Angles in circles word problem
        • Angle types
          • Acute, right, & obtuse angles
          • Recognizing angles
          • Drawing acute, right and obtuse angles
          • Identifying an angle
        • Decomposing angles
          • Decomposing an angle
          • Decomposing angles
        • Line of symmetry
          • Intro to reflective symmetry
          • Identifying symmetrical figures
        • Classifying triangles
          • Classifying triangles
          • Classifying triangles by angles
          • Worked example: Classifying triangles
        • Classifying geometric shapes
          • Classifying shapes by line and angles types
          • Quadrilateral properties
          • Classifying shapes by lines and angles
      • Factors, multiples and patterns
        • Factors and multiples
          • Finding factors of a number
          • Finding factors and multiples
        • Prime and composite numbers
          • Prime numbers
          • Recognizing prime and composite numbers
        • Math patterns
          • Factors and multiples: days of the week
          • Math patterns: table
          • Math patterns: toothpicks
      • Place value and rounding
        • Intro to place value
          • Finding place value
          • Creating the largest number
          • Place value blocks
        • How 10 relates to place value
          • Multiplying whole numbers by 10
          • Dividing whole numbers by 10
          • Understanding place value
          • Comparing place values
          • Place value when multiplying and dividing by 10
        • Ways to write whole numbers (expanded form and written form)
          • Writing a number in expanded form
          • Writing a number in standard form
        • Regrouping whole numbers
          • Regrouping numbers into various place values
          • Regrouping whole number place values
          • Adding whole numbers by their place values
        • Comparing multi-digit numbers
          • Comparing multi-digit numbers
          • Comparing whole number place values
          • Comparing multi-digit whole numbers word problems
        • Rounding whole numbers
          • Rounding whole numbers to nearest hundred
          • Rounding whole numbers to nearest thousand
          • Rounding whole numbers to nearest ten
          • Rounding whole numbers: missing digit
          • Rounding whole numbers word problems
    • 5th grade
      • Addition and subtraction
        • Common fractions and decimals
          • Common fractions and decimals
        • Adding decimals
          • Estimating decimal addition
          • Introduction to adding decimals: tenths
          • Adding decimals with ones and tenths parts
          • Adding decimals with hundredths
          • Adding decimals with ones, tenths and hundredths
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
        • Subtracting decimals
          • Estimating decimal subtraction
          • Strategies for subtracting basic decimals
          • Strategies for subtracting more complex decimals with tenths
          • Subtraction strategies with hundredths
          • More advanced subtraction strategies with hundredths
          • Subtracting decimals: 10.1-3.93
          • Subtracting decimals: 9.57-8.09
      • Multiplication and division
        • Multi-digit multiplication
          • Relate multiplication with area models to the standard algorithm
          • Strategies for multiplying multiples of 10, 100 and 1000
          • Estimating multi-digit multiplication
          • Multiplying multi-digit numbers
        • Multiplying decimals
          • Estimating decimal multiplication
          • Intro to multiplying decimals
          • Developing strategies for multiplying decimals
          • Multiplying decimals: place value
          • Developing strategies for multiplying 2-digit decimals
        • Multi-digit division
          • Strategies for dividing multiples of 10, 100, and 1000
          • Estimating multi-digit division
          • Introduction to dividing by 2-digits
          • Dividing by 2-digits: 9815÷65
          • Dividing by 2-digits: 7182÷42
        • Dividing decimals
          • Estimating decimal division
          • Division strategies for decimal quotients
          • Dividing whole numbers to get a decimal
          • Multi-digit division strategies for decimals
          • Strategies for dividing by tenths
      • Fractions
        • Visually adding and subtracting fractions with unlike denominators
          • Visually adding fractions: 5/6+1/4
          • Visually subtracting fractions: 3/4-5/8
        • Common denominators
          • Finding common denominators
          • Common denominators: 1/4 and 5/6
          • Common denominators: 3/5 and 7/2
        • Adding and subtracting fractions with unlike denominators
          • Adding fractions with unlike denominators
          • Subtracting fractions with unlike denominators
          • Adding and subtracting 3 fractions
          • Solving for the missing fraction
        • Adding and subtracting mixed number with unlike denominators
          • Adding mixed numbers: 19 3/18 + 18 2/3
          • Subtracting mixed numbers: 7 6/9 - 3 2/5
          • Adding mixed numbers with regrouping
          • Subtracting mixed numbers with regrouping (unlike denominators)
        • Adding and subtracting fractions with unlike denominators word problems
          • Adding fractions word problem: paint
          • Subtracting fractions word problem: tomatoes
        • Multiplication as scaling
          • Multiplication as scaling with fractions
        • Multiplying fractions and whole numbers
          • Multiplying fractions by whole numbers on a number line
          • Multiplying unit fractions and whole numbers
          • Multiplying fractions and whole numbers visually
          • Multiplying fractions and whole numbers
        • Multiplying fractions and whole numbers word problems
          • Multiplying fractions word problem: movies
          • Multiplying fractions word problem: milk
        • Multiplying fractions
          • Intro to multiplying 2 fractions
          • Multiplying 2 fractions: fraction model
          • Multiplying 2 fractions: number line
          • Multiplying 2 fractions: 5/6 x 2/3
          • Finding area with fractional sides 1
          • Finding area with fractional sides 2
        • Multiplying mixed numbers
          • Multiplying mixed numbers
        • Multiplying fractions word problems
          • Multiplying fractions word problem: muffins
          • Multiplying fractions word problem: laundry
          • Multiplying fractions word problem: bike
        • Fractions as division
          • Understanding fractions as division
          • Creating a fraction through division
          • Mixed numbers on a number line
          • Creating mixed numbers with fraction division
        • Dividing unit fractions by whole numbers
          • Visually dividing unit fraction by a whole number
          • Dividing a unit fraction by a whole number
        • Dividing whole numbers by unit fractions
          • Visually dividing whole numbers by unit fractions
          • Dividing a whole number by a unit fraction
        • Dividing fractions and whole numbers word problems
          • Dividing whole numbers by fractions: word problem
          • Dividing fractions by whole numbers: studying
        • Rewriting fractions as decimals
          • Rewriting a fraction as a decimal: 3/5
          • Rewriting a fraction as a decimal: 6/12
          • Rewriting a fraction as a decimal: 21/60
      • Place value and decimals
        • Decimal place value intro
          • Decimal place value
        • Decimals in expanded form
          • Write decimal in expanded form
          • Convert decimal from expanded form to standard form
        • Decimals in written form
          • Decimals in written form (hundredths)
          • Decimals in written form (thousandths)
        • Comparing decimals
          • Comparing decimals: 9.97 and 9.798
          • Comparing decimals: 156.378 and 156.348
          • Ordering decimals 1
          • Ordering decimals 2
          • Comparing decimals word problems
        • Rounding decimals
          • Worked example: Rounding decimals to nearest tenth
          • Rounding decimals word problems
        • Multiplying and dividing whole numbers by 10, 100, and 1000
        • Multiplying and dividing decimals by 10, 100, and 1000
        • Powers of 10
          • Patterns in powers of 10
        • Multiplying and dividing with powers of 10
          • Exponents and powers of 10 patterns
          • Multiplying a decimal by a power of 10
          • Dividing a decimal by a power of 10
          • Dividing a decimal by a power of 10: pattern
          • Fractions as division by power of 10
        • Comparing decimal place values
          • Comparing decimal place values
        • Regrouping decimal numbers
          • Regrouping whole numbers: 675
          • Regrouping whole numbers: 76,830
          • Regrouping whole numbers: 430
          • Regrouping with decimals: 21.3
          • Regrouping with decimals: 86.93
          • Regrouping with decimals: 0.25
      • Measurement and data
        • Volume intro
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
        • Finding volume
          • Volume of a rectangular prism
          • Volume in unit cubes by decomposing shape
          • Volume through decomposition
          • Volume word problem: water tank
        • Unit conversion
          • Converting units: minutes to hours
          • Converting units: metric distance
          • Converting units: centimeters to meters
          • Converting units: US volume
          • Adding units of volume
          • Same length in different units
          • U.S. customary and metric units
        • Unit measurement word problems
          • Time word problem: Susan's break
          • Measurement word problem: tea party
          • Measurement word problem: distance home
          • Measurement word problem: blood drive
          • Measurement word problem: elevator
          • Measurement word problem: running laps
        • Data
          • Line plot distribution: trail mix
      • Geometry
        • Intro to the coordinate plane
          • Introduction to the coordinate plane
          • Coordinate plane: graphing points
        • Coordinate plane word problems
          • Plotting corners of a rectangle
          • Coordinate plane graphing word problem
          • Interpreting plotted points
        • Quadrilaterals
          • Intro to quadrilateral
          • Quadrilateral properties
          • Kites as a geometric shape
          • Quadrilateral types
          • Classifying quadrilaterals
      • Algebraic thinking
        • Writing expressions
          • Constructing numerical expressions
          • Evaluating expressions with & without parentheses
          • Translating expressions with parentheses
        • Number patterns
          • Graphing patterns on coordinate plane
          • Interpreting patterns on coordinate plane
          • Interpreting relationships in ordered pairs
          • Graphing sequence relationships
    • 6th grade
      • Ratios, rates, & percentages
        • Intro to ratios
          • Intro to ratios
          • Basic ratios
          • Equivalent ratios
          • Equivalent ratios: recipe
          • Equivalent ratio word problems
          • Understanding equivalent ratios
        • Visualize ratios
          • Ratios and double number lines
          • Solving ratio problems with tables
          • Ratios with tape diagrams
        • Ratio application
          • Ratios on coordinate plane
          • Ratios and measurement
          • Part:whole ratios
          • Part:whole ratio word problem
        • Intro to rates
          • Intro to rates
          • Solving unit rate problem
          • Solving unit price problem
          • Rate problems
          • Comparing rates example
        • Intro to percents
          • The meaning of percent
          • Meaning of 109%
          • Percents from fraction models
          • Fraction, decimal, and percent from visual model
        • Percent, decimal, fraction conversions
          • Converting percents to decimals & fractions example
          • Converting decimals to percents: 0.601
          • Converting decimals to percents: 1.501
          • Converting percents to decimals: 59.2%
          • Converting percents to decimals: 113.9%
        • Percent problems
          • Finding a percent
          • Percent of a whole number
          • Percent word problem: 100 is what percent of 80?
        • Percent word problems
          • Percent word problem: recycling cans
      • Arithmetic operations
        • Adding decimals
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
        • Subtracting decimals
          • Subtracting decimals: 9.005 - 3.6
          • Subtracting decimals: 39.1 - 0.794
        • Adding and subtracting decimals word problems
          • Adding decimals word problem
          • Adding & subtracting decimals word problem
        • Multiplying decimals
          • Intro to multiplying decimals
          • Multiplying decimals: place value
          • Multiplying challenging decimals
        • Dividing whole numbers
          • Dividing by 2-digits: 9815÷65
          • Dividing by 2-digits: 7182÷42
        • Dividing decimals
          • Dividing whole numbers to get a decimal
          • Dividing a decimal by a whole number
          • Dividing a whole number by a decimal
          • Dividing decimals with hundredths
          • Dividing decimals completely
          • Long division with decimals
          • Dividing by a multi-digit decimal
        • Dividing fractions by fractions
          • Understanding division of fractions
          • Dividing a fraction by a whole number
          • Dividing a whole number by a fraction
          • Dividing fractions: 2/5 ÷ 7/3
          • Dividing fractions: 3/5 ÷ 1/2
          • Dividing mixed numbers
          • Dividing whole numbers & fractions: t-shirts
        • Exponents
          • Intro to exponents
          • The zeroth power
          • Powers of fractions
        • Order of operations
          • Intro to order of operations
          • Order of operations examples: exponents
          • Order of operations example
          • Worked example: Order of operations (PEMDAS)
          • Order of operations example
      • Negative numbers
        • Intro to negative numbers
          • Intro to negative numbers
        • Negative decimals & fractions on the number line
          • Negative decimals on the number line
          • Decimals & fractions on the number line
        • Number opposites
          • Number opposites
          • Negative symbol as opposite
          • Number opposites challenge problems
        • Comparing negative numbers
          • Ordering negative numbers
          • Ordering rational numbers
          • Negative numbers, variables, number line
        • Absolute value
          • Absolute value examples
          • Comparing absolute values
          • Placing absolute values on the number line
          • Comparing absolute values on the number line
          • Testing solutions to absolute value inequalities
          • Interpreting absolute value
        • Coordinate plane
          • Points on the coordinate plane examples
          • Plotting a point (ordered pair)
          • Finding the point not graphed
          • Quadrants of the coordinate plane
          • Points and quadrants example
          • Coordinate plane word problem examples
          • Reflecting points in the coordinate plane
      • Properties of numbers
        • Properties of numbers
        • Whole numbers & integers
        • Least common multiple
          • Least common multiple
          • Least common multiple: repeating factors
          • Least common multiple of three numbers
        • Greatest common factor
          • Greatest common factor examples
          • Greatest common factor explained
          • Factor with the distributive property
          • GCF & LCM word problems
      • Variables & expressions
        • Parts of algebraic expressions
          • What is a variable?
          • Terms, factors, & coefficients
          • Why aren't we using the multiplication sign?
        • Substitution & evaluating expressions
          • Evaluating an expression with one variable
          • Evaluating exponent expressions with variables
          • Evaluating expressions with two variables
          • Evaluating expressions with two variables: fractions & decimals
        • Expression value intuition
          • Expression value intuition
        • Evaluating expressions word problems
          • Evaluating expressions with variables: temperature
          • Evaluating expressions with variables: cubes
          • Evaluating expressions with variables: exponents
        • Writing algebraic expressions introduction
          • Writing basic expressions with variables
          • Writing expressions with variables
          • Writing expressions with variables & parentheses
        • Writing basic algebraic expressions word problems
          • Writing basic expressions word problems
          • Writing expressions word problems
        • Combining like terms
          • Intro to combining like terms
          • Combining like terms
          • Combining like terms example
          • Combining like terms with distribution
        • Distributive property with variables
          • Distributive property over addition
          • Distributive property over subtraction
          • Distributive property with variables
        • Equivalent expressions
          • Equivalent expressions
      • Equations & inequalities introduction
        • Algebraic equations basics
          • Variables, expressions, & equations
          • Testing solutions to equations
        • One-step equations intuition
          • Same thing to both sides of equations
          • Representing a relationship with an equation
          • Dividing both sides of an equation
          • One-step equations intuition
        • One-step addition & subtraction equations
          • One-step addition & subtraction equations
          • One-step subtraction equations
          • One-step addition & subtraction equations: fractions & decimals
        • One-step multiplication and division equations
          • One-step division equations
          • One-step multiplication equations
          • One-step multiplication & division equations: fractions & decimals
        • Finding mistakes in one-step equations
          • Finding mistakes in one-step equations
        • One-step equation word problems
          • Modeling with one-step equations
          • One-step equations word problem: super yoga (1 of 2)
          • One-step equations word problem: super yoga (2 of 2)
        • Intro to inequalities with variables
          • Testing solutions to inequalities
          • Plotting inequalities
          • Plotting an inequality example
          • Inequalities word problems
        • Dependent and independent variables
          • Dependent & independent variables
          • Dependent & independent variables: graphing
          • Dependent & independent variables: equation
      • Geometry
        • Areas of parallelograms
          • Area of a parallelogram
          • Finding height of a parallelogram
        • Areas of triangles
          • Area of a triangle
          • Finding area of triangle
          • Triangle missing side example
        • Area of trapezoids & composite figures
          • Area of trapezoids
          • Area of kites
          • Finding area by rearranging parts
          • Area of composite shapes
        • Geometric solids (3D shapes)
          • Counting faces and edges of 3D shapes
          • Recognizing common 3D shapes
        • Volume with fractions
          • Volume with fractional cubes
          • Volume of a rectangular prism: fractional dimensions
          • How volume changes from changing dimensions
          • Volume of a rectangular prism: word problem
        • Surface area
          • Intro to nets of polyhedra
          • Surface area using a net: triangular prism
          • Surface area using a net: rectangular prism
          • Surface area of a box (cuboid)
          • Surface area of a box using nets
          • Surface area word problem example
        • Polygons on the coordinate plane
          • Drawing a quadrilateral on the coordinate plane example
          • Area of a parallelogram on the coordinate plane
          • Coordinates of a missing vertex
          • Example of shapes on a coordinate plane
          • Dimensions of a rectangle from coordinates
          • Coordinates of rectangle example
          • Parallelogram on the coordinate plane
      • Data and statistics
        • Dot plots & frequency tables
          • Representing data
          • Frequency tables & dot plots
        • Statistical questions
          • Statistical questions
        • Histograms
          • Creating a histogram
          • Interpreting a histogram
        • Mean and median
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
        • Mean and median challenge problems
          • Missing value given the mean
          • Median & range puzzlers
          • Impact on median & mean: removing an outlier
          • Impact on median & mean: increasing an outlier
        • Interquartile range (IQR)
          • Interquartile range (IQR)
        • Box plots
          • Reading box plots
          • Constructing a box plot
          • Worked example: Creating a box plot (odd number of data points)
          • Worked example: Creating a box plot (even number of data points)
          • Interpreting box plots
        • Mean absolute deviation (MAD)
          • Mean absolute deviation (MAD)
          • Mean absolute deviation example
        • Comparing data displays
          • Comparing dot plots, histograms, and box plots
        • Shape of data distributions
          • Shapes of distributions
          • Clusters, gaps, peaks & outliers
    • 7th grade
      • Negative numbers: addition and subtraction
        • Intro to adding negative numbers
          • Adding negative numbers example
          • Adding numbers with different signs
          • Missing numbers on the number line examples
        • Intro to subtracting negative numbers
          • Adding & subtracting negative numbers
          • Subtracting a negative = adding a positive
        • Adding & subtracting with negatives on the number line
          • Adding negative numbers on the number line
          • Number equations & number lines
          • Interpreting numeric expressions example
        • Adding & subtracting integers
          • Adding numbers with different signs
          • Addition & subtraction: find the missing value (part 1 of 2)
          • Addition & subtraction: find the missing value (part 2 of 2)
        • Adding & subtracting negative fractions
          • Adding fractions with different signs
          • Adding & subtracting fractions
        • Addition & subtraction word problems with negatives
          • Interpreting negative number statements
          • Negative number word problem: temperatures
          • Negative number word problem: Alaska
        • Absolute value
          • Absolute value as distance between numbers
          • Interpreting absolute value as distance
          • Justification for |a - b| = |b - a|
        • Properties of addition & subtraction
          • Equivalent expressions with negative numbers
        • Adding & subtracting negative numbers: variables
          • Substitution with negative numbers
          • Ordering expressions
      • Negative numbers: multiplication and division
        • Multiply & divide negative numbers
          • Multiplying positive & negative numbers
          • Dividing positive and negative numbers
          • Why a negative times a negative makes sense
          • Why a negative times a negative is a positive
        • Multiplication & division word problems with negatives
          • Interpreting multiplication & division of negative numbers
        • Understanding multiplying and dividing fractions
          • Negative signs in fractions
          • Simplifying complex fractions
          • Expressions with rational numbers
        • Multiply & divide negative fractions
          • Multiplying positive and negative fractions
          • Dividing negative fractions
          • Dividing mixed numbers
        • Exponents with negative bases
          • Exponents with negative bases
          • The 0 & 1st power
          • Even & odd numbers of negatives
          • 1 and -1 to different powers
          • Sign of expressions challenge problems
          • Powers of zero
        • Dividing by zero
          • Why dividing by zero is undefined
        • Order of operations
          • Intro to order of operations
          • Worked example: Order of operations (PEMDAS)
          • Order of operations example
        • Properties of multiplication & division
          • Equivalent expressions with negative numbers (multiplication and division)
      • Fractions, decimals, & percentages
        • Converting fractions to decimals
          • Decimal to simplified fraction
          • Worked example: Converting a fraction (7/8) to a decimal
          • Fraction to decimal: 11/25
          • Fraction to decimal with rounding
        • Adding & subtracting rational numbers
          • Comparing rational numbers
          • Adding & subtracting rational numbers: 79% - 79.1 - 58 1/10
          • Adding & subtracting rational numbers: 0.79 - 4/3 - 1/2 + 150%
        • Percent word problems
          • Solving percent problems
          • Percent word problem: magic club
          • Percent word problem: guavas
          • Percent word problems: tax and discount
        • Rational number word problems
          • Rational number word problem: school report
          • Rational number word problem: cosmetics
          • Rational number word problem: cab
          • Rational number word problem: ice
          • Rational number word problem: computers
          • Rational number word problem: stock
          • Rational number word problem: checking account
      • Rates & proportional relationships
        • Rate problems with fractions
          • Rates with fractions
        • Constant of proportionality
          • Identifying constant of proportionality graphically
          • Constant of proportionality from equation
          • Constant of proportionality from table
          • Comparing proportionality constants
          • Interpret proportionality constants
        • Identifying proportional relationships
          • Intro to proportional relationships
          • Proportional relationships: movie tickets
          • Proportional relationships: bananas
          • Proportional relationships: spaghetti
          • Proportional relationships: graphs
          • Proportional relationships
          • Is side length & area proportional?
          • Is side length & perimeter proportional?
        • Graphs of proportional relationships
          • Interpreting graphs of proportional relationships
        • Writing & solving proportions
          • Worked example: Solving proportions
          • Writing proportions example
          • Proportion word problem: cookies
          • Proportion word problem: hot dogs
        • Equations of proportional relationships
          • Equations for proportional relationships
          • Writing proportional equations
      • Expressions, equations, & inequalities
        • Combining like terms
          • Intro to combining like terms
          • Simplifying expressions
          • Combining like terms challenge problem
          • Simplifying expressions with rational numbers
        • The distributive property & equivalent expressions
          • The distributive property with variables
          • Factoring with the distributive property
          • Equivalent expressions: negative numbers & distribution
        • Interpreting linear expressions
          • Interpreting linear expressions: diamonds
          • Interpreting linear expressions: flowers
        • Two-step equations intro
          • Same thing to both sides of equations
          • Intro to two-step equations
          • Two-step equations intuition
          • Worked example: two-step equations
        • Two-step equations with decimals and fractions
          • Two-step equations with decimals and fractions
          • Two-step equations with decimals and fractions
          • Find the mistake: two-step equations
        • Two-step equation word problems
          • Two-step equation word problem: computers
          • Two-step equation word problem: garden
          • Two-step equation word problem: oranges
        • One-step inequalities
          • One-step inequalities examples
          • One-step inequalities: -5c ≤ 15
          • One-step inequality word problem
        • Two-step inequalities
          • Two-step inequalities
          • Two-step inequality word problem: apples
          • Two-step inequality word problem: R&B
      • Geometry
        • Area and circumference of circles
          • Radius, diameter, circumference & π
          • Labeling parts of a circle
          • Radius & diameter from circumference
          • Relating circumference and area
          • Area of a circle
          • Partial circle area and arc length
        • Area and circumference challenge problems
          • Finding circumference of a circle when given the area
          • Area of a shaded region
          • Impact of increasing the radius
        • Vertical, complementary, and supplementary angles
          • Angles: introduction
          • Complementary & supplementary angles
          • Vertical angles
        • Missing angle problems
          • Find measure of vertical angles
          • Find measure of angles word problem
          • Equation practice with complementary angles
          • Equation practice with supplementary angles
          • Equation practice with vertical angles
        • Constructing triangles
          • Construct a right isosceles triangle
          • Construct a triangle with constraints
          • Triangle inequality theorem
          • Ordering triangle sides and angles example
        • Slicing geometric shapes
          • Slicing a rectangular pyramid
        • Scale drawings
          • Exploring scale copies
          • Corresponding points and sides of scaled shapes
          • Identifying scale copies
          • Identifying scale factors
          • Identifying values in scale drawings
          • Scale drawing: centimeters to kilometers
          • Interpreting a scale drawing
          • Making a scale drawing
          • Scale factors and area
          • Solving a scale drawing word problem
        • Volume and surface area word problems
          • Volume word problem: gold ring
          • Volume of triangular prism & cube
          • Volume of rectangles inside rectangles
      • Statistics and probability
        • Basic probability
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Experimental probability
          • Intuitive sense of probabilities
        • Probability models
          • Theoretical and experimental probabilites
          • Making predictions with probability
          • Probability models example: frozen yogurt
        • Compound events and sample spaces
          • Sample spaces for compound events
          • Die rolling probability
          • Probability of a compound event
          • Counting outcomes: flower pots
          • Count outcomes using tree diagram
        • Comparing and sampling populations
          • Reasonable samples
          • Comparing distributions with dot plots (example problem)
    • 8th grade
      • Numbers and operations
        • Repeating decimals
          • Converting a fraction to a repeating decimal
          • Converting repeating decimals to fractions (part 1 of 2)
          • Converting repeating decimals to fractions (part 2 of 2)
        • Square roots & cube roots
          • Intro to square roots
          • Intro to cube roots
          • Worked example: Cube root of a negative number
          • Square root of decimal
          • Dimensions of a cube from its volume
        • Irrational numbers
          • Intro to rational & irrational numbers
          • Classifying numbers: rational & irrational
          • Classifying numbers
          • Worked example: classifying numbers
        • Approximating irrational numbers
          • Approximating square roots
          • Comparing irrational numbers with radicals
          • Approximating square roots to hundredths
          • Comparing values with calculator
        • Exponent properties intro
          • Exponent properties with products
          • Exponent properties with parentheses
          • Exponent properties with quotients
        • Negative exponents
          • Negative exponents
          • Negative exponent intuition
        • Exponent properties (integer exponents)
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
          • Powers of zero
        • Working with powers of 10
          • Multiplying multiples of powers of 10
          • Approximating with powers of 10
        • Scientific notation intro
          • Scientific notation example: 0.0000000003457
          • Scientific notation examples
        • Arithmetic with numbers in scientific notation
          • Multiplying & dividing in scientific notation
          • Multiplying three numbers in scientific notation
          • Subtracting in scientific notation
          • Simplifying in scientific notation challenge
        • Scientific notation word problems
          • Scientific notation word problem: red blood cells
          • Scientific notation word problem: U.S. national debt
          • Scientific notation word problem: speed of light
      • Solving equations with one unknown
        • Equations with variables on both sides
          • Intro to equations with variables on both sides
          • Equations with variables on both sides: 20-7x=6x-6
          • Equation with variables on both sides: fractions
          • Equation with the variable in the denominator
        • Equations with parentheses
          • Equations with parentheses
        • Number of solutions to equations
          • Number of solutions to equations
          • Worked example: number of solutions to equations
          • Creating an equation with no solutions
          • Creating an equation with infinitely many solutions
        • Equations word problems
          • Sums of consecutive integers
          • Sum of integers challenge
        • Equations & geometry
          • Equation practice with segment addition
          • Equation practice with midpoints
          • Equation practice with vertical angles
          • Equation practice with complementary angles
          • Equation practice with supplementary angles
      • Linear equations and functions
        • Graphing proportional relationships
          • Rates & proportional relationships example
          • Rates & proportional relationships: gas mileage
          • Graphing proportional relationships: unit rate
          • Graphing proportional relationships from a table
          • Graphing proportional relationships from an equation
        • Solutions to linear equations
          • Intro to the coordinate plane
          • Solutions to 2-variable equations
          • Worked example: solutions to 2-variable equations
          • Completing solutions to 2-variable equations
        • Intercepts
          • Intro to intercepts
          • x-intercept of a line
          • Intercepts from an equation
          • Intercepts from a table
          • Worked example: intercepts from an equation
        • Slope
          • Intro to slope
          • Slope & direction of a line
          • Positive & negative slope
          • Worked example: slope from graph
          • Slope of a line: negative slope
          • Worked example: slope from two points
          • Slope from equation
          • Converting to slope-intercept form
          • Slope of a horizontal line
        • Intro to slope-intercept form
          • Intro to slope-intercept form
          • Worked examples: slope-intercept intro
        • Graphing slope-intercept form
          • Graph from slope-intercept equation
        • Writing slope-intercept equations
          • Slope-intercept equation from graph
          • Slope-intercept equation from slope & point
          • Slope-intercept equation from two points
          • Slope-intercept form problems
          • Slope-intercept form from a table
        • Functions & function notation
          • What is a function?
          • Worked example: Evaluating functions from equation
          • Function notation example
          • Worked example: evaluating functions from graph
          • Equations vs. functions
          • Manipulating formulas: temperature
          • Obtaining a function from an equation
        • Linear models
          • Linear graphs word problems
          • Linear graphs word problem: cats
          • Linear equations word problems: volcano
          • Linear equations word problems: earnings
          • Modeling with linear equations: snow
          • Linear function example: spending money
          • Fitting a line to data
        • Comparing linear functions
          • Comparing linear functions: equation vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions word problem: climb
          • Comparing linear functions word problem: walk
          • Comparing linear functions word problem: work
        • Constructing linear models for real-world relationships
          • Linear functions word problem: fuel
          • Linear functions word problem: pool
          • Modeling with linear equations: gym membership & lemonade
          • Linear functions word problem: iceberg
          • Linear functions word problem: paint
        • Recognizing functions
          • Testing if a relationship is a function
          • Relations and functions
          • Recognizing functions from graph
          • Checking if a table represents a function
          • Recognizing functions from table
          • Checking if an equation represents a function
          • Does a vertical line represent a function?
          • Recognizing functions from verbal description
          • Recognizing functions from verbal description word problem
        • Linear and nonlinear functions
          • Recognizing linear functions
          • Linear & nonlinear functions: table
          • Linear & nonlinear functions: word problem
          • Linear & nonlinear functions: missing value
          • Interpreting a graph example
      • Systems of equations
        • Systems of equations overview
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: potato chips
          • Systems of equations with substitution: potato chips
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
        • Intro to systems of equations
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Testing a solution to a system of equations
        • Systems of equations with graphing
          • Systems of equations with graphing
          • Systems of equations with graphing: 5x+3y=7 & 3x-2y=8
          • Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
          • Systems of equations with graphing: chores
          • Systems of equations with graphing: exact & approximate solutions
        • Solving systems with elimination
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: 3t+4g=6 & -6t+g=6
          • Systems of equations with elimination: x+2y=6 & 4x-2y=14
          • Systems of equations with elimination: -3y+4x=11 & y+2x=13
          • Systems of equations with elimination: 2x-y=14 & -6x+3y=-42
          • Systems of equations with elimination: 4x-2y=5 & 2x-y=2.5
          • Systems of equations with elimination: x-4y=-18 & -x+3y=11
          • Systems of equations with elimination: 6x-6y=-24 & -5x-5y=-60
          • Systems of equations with elimination: potato chips
          • Systems of equations with elimination (and manipulation)
          • Why can we subtract one equation from the other in a system of equations?
          • Worked example: equivalent systems of equations
          • Worked example: non-equivalent systems of equations
        • Solving systems with substitution
          • Systems of equations with substitution: 2y=x+7 & x=y-4
          • Systems of equations with substitution: y=4x-17.5 & y+2x=6.5
          • Systems of equations with substitution: -3x-4y=-2 & y=2x-5
          • Systems of equations with substitution: 9x+3y=15 & y-x=5
          • Systems of equations with substitution: y=-5x+8 & 10x+2y=-2
          • Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120
        • Number of solutions to systems of equations
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
          • Solutions to systems of equations: consistent vs. inconsistent
          • Solutions to systems of equations: dependent vs. independent
          • Number of solutions to a system of equations
          • Number of solutions to a system of equations graphically
          • Number of solutions to a system of equations algebraically
          • How many solutions does a system of linear equations have if there are at least two?
        • Systems of equations word problems
          • Age word problem: Imran
          • Age word problem: Ben & William
          • Age word problem: Arman & Diya
          • System of equations word problem: walk & ride
          • System of equations word problem: no solution
          • System of equations word problem: infinite solutions
          • Systems of equations with elimination: TV & DVD
          • Systems of equations with elimination: apples and oranges
          • Systems of equations with substitution: coins
          • Systems of equations with elimination: coffee and croissants
      • Geometry
        • Angles between intersecting lines
          • Angles, parallel lines, & transversals
          • Parallel & perpendicular lines
          • Missing angles with a transversal
          • Measures of angles formed by a transversal
        • Triangle angles
          • Angles in a triangle sum to 180° proof
          • Isosceles & equilateral triangles problems
          • Triangle exterior angle example
          • Worked example: Triangle angles (intersecting lines)
          • Worked example: Triangle angles (diagram)
          • Triangle angle challenge problem
          • Triangle angle challenge problem 2
        • Pythagorean theorem
          • Intro to the Pythagorean theorem
          • Pythagorean theorem example
          • Pythagorean theorem with isosceles triangle
        • Pythagorean theorem application
          • Use Pythagorean theorem to find area of an isosceles triangle
          • Pythagorean theorem word problem: carpet
          • Pythagorean theorem word problem: fishing boat
          • Pythagorean theorem in 3D
        • Pythagorean theorem and distance between points
          • Distance formula
        • Pythagorean theorem proofs
          • Garfield's proof of the Pythagorean theorem
          • Bhaskara's proof of the Pythagorean theorem
          • Pythagorean theorem proof using similarity
          • Another Pythagorean theorem proof
        • Volume
          • Cylinder volume & surface area
          • Volume of a sphere
          • Volume of a cone
      • Geometric transformations
        • Transformations intro
          • Rigid transformations intro
          • Dilations intro
          • Identifying transformations
        • Translations
          • Translating points
          • Translating shapes
          • Determining translations
        • Rotations
          • Rotating points
          • Determining rotations
        • Reflections
          • Reflecting points
          • Reflecting shapes
          • Determining reflections
        • Properties & definitions of transformations
          • Finding measures using rigid transformations
          • Rigid transformations: preserved properties
          • Mapping shapes
        • Dilations
          • Dilating points
          • Dilations: scale factor
          • Dilating shapes: expanding
          • Dilating shapes: shrinking
          • Dilations and properties
        • Congruence and similarity
          • Congruent shapes & transformations
          • Non-congruent shapes & transformations
          • Similar shapes & transformations
          • Side lengths after dilation
      • Data and modeling
        • Introduction to scatter plots
          • Constructing a scatter plot
        • Interpreting scatter plots
          • Scatter plots: studying, shoe size, and test scores
          • Scatter plot: smokers
        • Estimating lines of best fit
          • Estimating the line of best fit exercise
          • Line of best fit: smoking in 1945
          • Estimating with linear regression (linear models)
          • Interpreting a trend line
        • Two-way tables
          • Two-way frequency tables and Venn diagrams
          • Two-way relative frequency tables
          • Interpreting two-way tables
    • Class 5 (India)
      • Review of shapes, operations and large numbers
        • Revision of shapes, estimation, simple operations, large numbers
          • Comparing place values
          • Regrouping numbers into various place values
          • Comparing whole number place values
          • Place value relationships example
      • Shapes and angles
        • Angles and unit of measurement for angles
          • Angle measurement & circle arcs
          • Acute, right, & obtuse angles
          • Measuring angles in degrees
          • Recognizing angles
          • Identifying an angle
          • Measuring angles using a protractor
          • Measuring angles using a protractor 2
      • Area and perimeter - Part 1
        • Areas and patterns
          • Counting unit squares to find area formula
          • Comparing areas of plots of land
          • Area word problem: house size
          • Perimeter: introduction
          • Perimeter of a shape
          • Comparing areas and perimeters of rectangles
          • Finding area by rearranging parts
      • Parts and wholes
        • Fraction comparisons and equivalent fractions
          • Intro to equivalent fractions
          • Recognizing fractions exercise
          • Representing 1 as a fraction
          • Rational number word problem: ice
          • Creating equivalent fractions
          • Equivalent fraction visually
          • Visualizing equivalent fractions
          • Equivalent fraction word problem example
          • Equivalent fraction word problem example 3
          • Equivalent fraction word problem example 4
          • Comparing fractions of different wholes 1
          • Comparing fractions with > and < symbols
          • Comparing and ordering fractions
          • Expressions with rational numbers
          • Rate problem using fractions
          • Fraction word problem: pizza
          • Fractions cut and copy 2 exercise example
      • Symmetry
        • Line of symmetry
          • Intro to reflective symmetry
        • Rotational symmetry
          • Intro to rotational symmetry
      • Factors and multiples
        • Factors and multiples
          • Finding factors of a number
          • Finding factors and multiples
          • Finding factors
      • Patterns
        • Patterns
          • Graphing patterns on coordinate plane
          • Interpreting patterns on coordinate plane
          • Interpreting relationships in ordered pairs
          • Graphing sequence relationships
      • Visualizing 3-D objects
        • Representing 3-D objects on paper in the form of nets
          • Intro to nets of polyhedra
          • Surface area using a net: rectangular prism
      • Decimals
        • Decimals
          • Writing decimal numbers shown in grids
          • Rewriting a fraction as a decimal: 3/5
          • Rewriting a fraction as a decimal: 6/12
          • Rewriting a fraction as a decimal: 21/60
          • Rewriting decimals as fractions: 0.15
          • Rewriting decimals as fractions: 0.8
          • Rewriting decimals as fractions: 0.36
          • Comparing decimals 1
          • Converting units: centimeters to meters
          • Subtracting decimals: 9.57-8.09
          • Subtracting decimals: 10.1-3.93
          • Adding & subtracting decimals word problem
          • Rational number word problem: checking account
      • Area and perimeter - Part 2
        • Area and perimeter
          • Intro to area and unit squares
          • Counting unit squares to find area formula
          • Comparing areas word problem
          • Perimeter: introduction
          • Perimeter of a shape
          • Area & perimeter word problem: dog pen
          • Comparing areas and perimeters of rectangles
      • Ways to multiply and divide
        • Strategies for multiplication and division
          • Multiplying multi-digit numbers
          • Multiplication 7: Old video giving more examples
          • Constructing numerical expressions
          • Partial quotient method of division
          • Partial quotient method of division 2
          • Multiple units word problem: road trip
      • Volume and weight
        • Volume
          • Conversion between metric units
        • Weight
          • Metric system: units of weight
          • Multiple units word problem: drug dosage (advanced)
    • Class 6 (India)
      • Knowing our numbers
        • Comparing 3-digit numbers
          • Comparing whole numbers
        • Comparing multi-digit numbers
          • Comparing multi-digit numbers
          • Comparing whole number place values
          • Comparing multi-digit whole numbers word problems
        • Lakhs, crores, millions, and billions
          • Intro to lakhs and crores
          • Reading and writing large numbers
          • Intro to millions and billions
        • Word problems involving lakhs and crores
        • System of measurement
          • Metric system: units of distance
          • Metric system: units of volume
          • Metric system: units of weight
          • Converting metric units of length
          • How to convert kg to mg and T to oz
          • Multi-step unit conversion examples (metric)
        • Rounding
          • Rounding to nearest 10
          • Rounding to nearest 100
          • Rounding to nearest 10 and 100
        • Rounding whole numbers
          • Rounding whole numbers to nearest hundred
          • Rounding whole numbers to nearest thousand
          • Rounding whole numbers to nearest ten
          • Rounding whole numbers: missing digit
          • Rounding whole numbers word problems
          • 2-step estimation problem: marbles
          • Multiplication estimation example
        • Roman numerals
          • Intro to Roman numerals
          • Reading and writing Roman numerals
          • Roman numerals worked example
      • Whole numbers
        • Number line of whole numbers
          • Number line 1
        • Arithmetic properties of whole numbers
          • Properties of multiplication
          • Properties and patterns for multiplication
          • Commutative law of addition
          • Commutative law of multiplication
          • Associative law of addition
          • Associative law of multiplication
          • Identity property of 1
          • Identity property of 0
          • Inverse property of addition
          • Inverse property of multiplication
        • Distributive property
          • Distributive property over addition
          • Distributive property over subtraction
          • Distributive property exercise examples
      • Playing with numbers
        • Factors and multiples
          • Finding factors of a number
          • Finding factors and multiples
        • Prime and composite numbers
          • Prime numbers
          • Recognizing prime and composite numbers
        • Divisibility tests
          • Divisibility tests for 2, 3, 4, 5, 6, 9, 10
          • Recognizing divisibility
          • The why of the 3 divisibility rule
          • The why of the 9 divisibility rule
        • Prime factorisation
          • Prime factorization
          • Prime factorization exercise
          • Common divisibility examples
        • Common factors and multiples
        • Lowest common multiple
          • Least common multiple
          • Least common multiple: repeating factors
          • Least common multiple of three numbers
        • Highest common factor
          • Greatest common factor examples
          • Greatest common factor explained
          • GCF & LCM word problems
      • Basic geometrical ideas
        • Lines, line segments, and rays
          • Terms & labels in geometry
          • Lines, line segments, & rays
        • Angle introduction
          • Angles: introduction
          • Naming angles
        • Introduction to polygons
          • Cousin Fal's shape collection
          • Recognizing shapes
          • Vertices, sides, and diagonals of a polygon
        • Curves and polygons
          • Intro to curves
          • Polygons as special curves
          • Circles glossary
      • Understanding elementary shapes
        • Angle types
          • Acute, right, & obtuse angles
          • Recognizing angles
          • Drawing acute, right and obtuse angles
          • Identifying an angle
        • Measuring angles
          • Measuring angles in degrees
          • Measuring angles using a protractor
          • Measuring angles using a protractor 2
        • Constructing angles
          • Constructing angles
        • Parallel and perpendicular
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines
          • Drawing parallel line segments
        • Classifying triangles
          • Classifying triangles
          • Classifying triangles by angles
          • Worked example: Classifying triangles
        • Quadrilaterals
          • Intro to quadrilaterals
        • Geometric solids (3D shapes)
          • Counting faces and edges of 3D shapes
          • Recognizing common 3D shapes
      • Integers
        • Intro to negative numbers
          • Intro to negative numbers
        • Number opposites
          • Number opposites
          • Negative symbol as opposite
          • Number opposites challenge problems
        • Absolute value
          • Absolute value examples
          • Identify and order absolute values
          • Comparing absolute values
          • Placing absolute values on the number line
          • Absolute value as distance between numbers
          • Absolute value word problems
        • Order integers
          • Ordering negative numbers
        • Introduction to adding integers
          • Adding negative numbers example
          • Adding numbers with different signs
        • Introduction to subtracting integers
          • Adding & subtracting negative numbers
          • Subtracting a negative = adding a positive
        • Adding & subtracting integers
          • Adding numbers with different signs
          • Addition & subtraction: find the missing value (part 1 of 2)
          • Addition & subtraction: find the missing value (part 2 of 2)
        • Adding & subtracting with negatives on the number line
          • Adding negative numbers on the number line
          • Number equations & number lines
          • Interpreting numeric expressions example
      • Fractions
        • Fractions intro
          • Intro to fractions
          • Cutting shapes into equal parts
          • Identifying unit fractions word problem
          • More than one equal section
          • Recognizing fractions greater than 1
          • Identifying numerators and denominators
        • Fractions on the number line
          • Fractions on a number line
          • Fractions on number line widget
        • Fractions and whole numbers
          • Representing 1 as a fraction
          • Whole numbers as fractions
        • Equivalent fractions
          • Intro to equivalent fractions
          • Visualizing equivalent fractions
          • More on equivalent fractions
          • Equivalent fractions
          • Equivalent fractions and different wholes
        • Equivalent fractions on the number line
          • Equivalent fraction visually
          • Creating equivalent fractions
        • Common denominators
          • Finding common denominators
          • Common denominators: 3/5 and 7/2
          • Common denominators: 1/4 and 5/6
        • Comparing fractions
          • Comparing fractions with > and < symbols
          • Comparing fractions with like numerators and denominators
          • Comparing fractions
          • Comparing fractions 2 (unlike denominators)
          • Comparing and ordering fractions
          • Ordering fractions
        • Mixed numbers
          • Rewriting mixed numbers as improper fractions
          • Rewriting improper fractions as mixed numbers
          • Comparing improper fractions and mixed numbers
          • Mixed number or improper fraction on a number line
        • Decomposing fractions
          • Decomposing a fraction visually
          • Decomposing a mixed number
        • Adding and subtracting fractions with like denominators
          • Adding fractions with like denominators
          • Subtracting fractions with like denominators
        • Visually adding and subtracting fractions with unlike denominators
          • Visually adding fractions: 5/6+1/4
          • Visually subtracting fractions: 3/4-5/8
        • Adding and subtracting fractions with unlike denominators
          • Adding fractions with unlike denominators
          • Subtracting fractions with unlike denominators
          • Adding and subtracting 3 fractions
          • Solving for the missing fraction
        • Adding and subtracting mixed numbers
          • Intro to adding mixed numbers
          • Intro to subtracting mixed numbers
        • Adding and subtracting mixed number with unlike denominators
          • Adding mixed numbers: 19 3/18 + 18 2/3
          • Subtracting mixed numbers: 7 6/9 - 3 2/5
          • Adding mixed numbers with regrouping
          • Subtracting mixed numbers with regrouping (unlike denominators)
        • Adding and subtracting fractions with unlike denominators word problems
          • Adding fractions word problem: paint
          • Subtracting fractions word problem: tomatoes
      • Decimals
        • Intro to decimals
          • Intro to decimals
          • Decimals as words
          • Writing decimal numbers shown in grids
        • Decimal place value intro
          • Decimal place value
        • Decimals in expanded form
          • Write decimal in expanded form
          • Convert decimal from expanded form to standard form
        • Decimals in written form
          • Decimals in written form (hundredths)
          • Decimals in written form (thousandths)
        • Comparing decimal place values
          • Comparing decimal place values
        • Comparing decimals
          • Comparing decimals: 9.97 and 9.798
          • Comparing decimals: 156.378 and 156.348
          • Ordering decimals 1
          • Ordering decimals 2
          • Comparing decimals word problems
        • Decimals on the number line
          • Comparing decimal numbers on a number line
          • Plotting decimal numbers on a number line
        • Rewriting decimals as fractions
          • Writing a number as a fraction and decimal
          • Rewriting decimals as fractions: 0.15
          • Rewriting decimals as fractions: 0.8
          • Rewriting decimals as fractions: 0.36
          • Rewriting tricky fractions to decimals
        • Common fractions and decimals
          • Common fractions and decimals
        • Adding decimals
        • Adding decimals: standard algorithm
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
        • Subtracting decimals
          • Subtracting decimals: 9.57-8.09
          • Subtracting decimals: 10.1-3.93
        • Subtracting decimals: standard algorithm
          • Subtracting decimals: 9.005 - 3.6
          • Subtracting decimals: 39.1 - 0.794
        • Adding and subtracting decimals word problems
          • Adding decimals word problem
          • Adding & subtracting decimals word problem
      • Data Handling
        • Pictographs (Picture graphs)
          • Creating picture and bar graphs 1
          • Solving problems with pictographs 2
          • Interpreting picture graphs: paint
          • Interpreting picture graphs: notebook
        • Bar graphs
          • Reading bar graphs: bikes
          • Reading bar graphs: dog bones
          • Creating picture and bar graphs 1
      • Mensuration
        • Count unit squares to find area
          • Intro to area and unit squares
          • Measuring rectangles with different unit squares
          • Measuring area with partial unit squares
          • Creating rectangles with a given area 1
          • Creating rectangles with a given area 2
        • Multiply to find area
          • Counting unit squares to find area formula
          • Finding missing side when given area
          • Comparing areas of plots of land
        • Area and the distributive property
          • Area and the distributive property
        • Decompose figures to find area
          • Decomposing shapes to find area: grids
          • Decomposing shapes to find area: add
          • Decomposing shapes to find area: subtract
          • Area word problem: house size
        • Perimeter
          • Perimeter: introduction
          • Perimeter of a shape
          • Finding perimeter when a side length is missing
          • Finding missing side length when given perimeter
        • Comparing area and perimeter
          • Comparing areas and perimeters of rectangles
        • Area word problems
        • Perimeter word problems
          • Perimeter word problem: tables
          • Perimeter word problem: skating rink
      • Algebra
        • Overview and history of algebra
          • Origins of algebra
          • Abstract-ness
          • The beauty of algebra
          • Why all the letters in algebra?
        • Intro to variables
          • What is a variable?
          • Why aren't we using the multiplication sign?
        • Substitution & evaluating expressions
          • Evaluating an expression with one variable
          • Evaluating expressions with two variables
          • Evaluating expressions with two variables: fractions & decimals
        • Writing algebraic expressions introduction
          • Writing basic expressions with variables
          • Writing expressions with variables
          • Writing expressions with variables & parentheses
        • Writing basic algebraic expressions word problems
          • Writing basic expressions word problems
          • Writing expressions word problems
        • Algebraic equations basics
          • Variables, expressions, & equations
          • Testing solutions to equations
      • Ratio and Proportion
        • Introduction to ratios
          • Intro to ratios
          • Part:whole ratios
          • Solving ratio problems with tables
          • Solving ratio problems with graph
        • Ratio word problems
          • Scale drawing: centimeters to kilometers
          • Part:whole ratio word problem
        • Intro to rates
          • Intro to rates
          • Solving unit rate problem
          • Solving unit price problem
          • Multiple rates word problem
          • Comparing rates example
          • Finding average speed or rate
          • Speed translation
        • Identifying proportional relationships
          • Intro to proportional relationships
          • Proportional relationships: spaghetti
          • Proportional relationships: bananas
          • Proportional relationships: movie tickets
        • Writing & solving proportions
          • Worked example: Solving proportions
          • Writing proportions example
          • Proportion word problem: hot dogs
          • Proportion word problem: cookies
          • Multiple units word problem: road trip
      • Symmetry
        • Line of symmetry
          • Intro to reflective symmetry
          • Identifying symmetrical figures
      • Practical geometry
        • Constructing angles
          • Constructing angles
          • Drawing acute, right and obtuse angles
          • Geometric constructions: angle bisector
        • Constructing perpendiculars
          • Geometric constructions: perpendicular line
          • Geometric constructions: perpendicular bisector
    • Class 7 (India)
      • Integers
        • Integer addition and subtraction review
        • Arithmetic properties
          • Properties of multiplication
          • Properties and patterns for multiplication
          • Commutative law of addition
          • Commutative law of multiplication
          • Associative law of addition
          • Associative law of multiplication
          • Identity property of 1
          • Identity property of 0
          • Inverse property of addition
          • Inverse property of multiplication
        • Multiplying & dividing negative numbers
          • Multiplying positive & negative numbers
          • Multiplying numbers with different signs
          • Dividing positive and negative numbers
          • Multiplying and dividing negative numbers
          • Why a negative times a negative makes sense
          • Why a negative times a negative is a positive
      • Fractions and decimals
        • Multiplying unit fractions and whole numbers
          • Multiplying unit fractions and whole numbers
        • Multiplying whole numbers and fractions
          • Multiplying fractions and whole numbers visually
          • Equivalent fraction and whole number multiplication problems
        • Multiplication as scaling
          • Multiplication as scaling with fractions
        • Multiplying fractions
          • Intro to multiplying 2 fractions
          • Multiplying 2 fractions: fraction model
          • Multiplying 2 fractions: number line
          • Multiplying 2 fractions: 5/6 x 2/3
          • Finding area with fractional sides 1
          • Finding area with fractional sides 2
        • Multiplying mixed numbers
          • Multiplying mixed numbers
        • Multiplying fractions word problems
          • Multiplying fractions word problem: muffins
          • Multiplying fractions word problem: laundry
          • Multiplying fractions word problem: bike
          • Multiplying fractions word problem: pumpkin pie
        • Fractions as division
          • Understanding fractions as division
          • Creating a fraction through division
          • Creating mixed numbers with fraction division
        • Dividing unit fractions by whole numbers
        • Dividing whole numbers by unit fractions
        • Dividing fractions by fractions
          • Understanding division of fractions
          • Dividing fractions: 2/5 ÷ 7/3
          • Dividing fractions: 3/5 ÷ 1/2
        • Dividing fractions word problems
          • Dividing whole numbers & fractions: t-shirts
        • Multiplying decimals
          • Intro to multiplying decimals
          • Multiplying decimals: place value
        • Multiplying decimals: standard algorithm
          • Multiplying challenging decimals
        • Dividing decimals
          • Dividing whole numbers to get a decimal
        • Dividing decimals: standard algorithm
          • Dividing decimals with hundredths
          • Dividing decimals completely
          • Long division with decimals
          • Dividing by a multi-digit decimal
      • Data Handling
        • Mean and median: The basics
          • Statistics: The average
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
        • Interpreting data from bar graphs
          • Reading pictographs
          • Reading bar graphs
          • Reading bar graphs: Harry Potter
          • Creating a bar graph
          • Reading bar charts: comparing two sets of data
          • Reading bar charts: putting it together with central tendency
          • Reading pie graphs (circle graphs)
        • Basic theoretical probability
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Intuitive sense of probabilities
          • The Monty Hall problem
      • Simple equations
        • Algebraic equations basics
          • Variables, expressions, & equations
          • Testing solutions to equations
        • One-step equations intuition
          • Same thing to both sides of equations
          • Representing a relationship with an equation
          • Dividing both sides of an equation
          • One-step equations intuition
        • One-step addition & subtraction equations
          • One-step addition & subtraction equations
          • One-step subtraction equations
          • One-step addition & subtraction equations: fractions & decimals
        • One-step multiplication and division equations
          • One-step division equations
          • One-step multiplication equations
          • One-step multiplication & division equations: fractions & decimals
        • One-step equation word problems
          • One-step equations word problem: super yoga (1 of 2)
          • One-step equations word problem: super yoga (2 of 2)
          • Modeling with one-step equations
      • Lines and angles
        • Vertical, complementary, and supplementary angles
          • Complementary & supplementary angles
          • Vertical angles
        • Angles between intersecting lines
          • Angles, parallel lines, & transversals
          • Parallel & perpendicular lines
          • Missing angles with a transversal
      • The triangle and its properties
        • Triangle angles
          • Triangle exterior angle example
          • Angles in a triangle sum to 180° proof
          • Worked example: Triangle angles (intersecting lines)
          • Worked example: Triangle angles (diagram)
          • Triangle angle challenge problem
          • Triangle angle challenge problem 2
        • Equilateral and isosceles triangles
          • Proofs concerning equilateral triangles
          • Isosceles & equilateral triangles problems
        • Triangle inequality theorem
          • Triangle inequality theorem
        • Pythagoras theorem
          • Intro to the Pythagorean theorem
          • Pythagorean theorem example
      • Congruence of triangles
        • Transformations and congruence
          • Congruent shapes & transformations
          • Non-congruent shapes & transformations
          • Performing sequences of transformations
          • Determining mappings (incl. dilations)
        • Triangle congruence
          • Congruent triangles & the SSS postulate/criterion
          • Triangle congruence postulates/criteria
          • Why SSA isn't a congruence postulate/criterion
          • Determining congruent triangles
      • Comparing quantities
        • Intro to percents
          • The meaning of percent
          • Meaning of 109%
        • Percent, fraction, decimal conversions
          • Converting decimals to percents: 0.601
          • Converting decimals to percents: 1.501
          • Converting percents to decimals: 59.2%
          • Converting percents to decimals: 113.9%
          • Converting percents to decimals & fractions example
          • Ordering numeric expressions
        • Percent problems
          • Finding a percent
          • Percent of a whole number
          • Identifying percent amount and base
        • Percent word problems
          • Percent word problem: 100 is what percent of 80?
          • Percent word problem: 78 is 15% of what number?
          • Percent word problem: guavas
          • Percent word problem: recycling cans
          • Percent word problem: penguins
          • Growing by a percentage
          • Solving percent problems
          • Rational number word problem: cosmetics
          • Rational number word problem: cab
          • Rational number word problem: school report
      • Rational numbers
        • Intro to rational numbers
          • Intro to rational & irrational numbers
          • Identifying Rational Numbers
        • Comparing rational numbers
          • Comparing rational numbers
          • Ordering rational numbers
        • Adding & subtracting rational numbers
          • Addition of Rational Numbers
          • Adding fractions with different signs
          • Subtraction of Rational Numbers
          • Adding & subtracting rational numbers: 79% - 79.1 - 58 1/10
          • Adding & subtracting rational numbers: 0.79 - 4/3 - 1/2 + 150%
        • Multiplication and division of rational numbers
          • Multiplication of Rational Numbers
          • Multiplying positive and negative fractions
          • Division of Rational Numbers
          • Dividing negative fractions
        • Rational number word problems
          • Rational number word problem: school report
          • Rational number word problem: cosmetics
          • Rational number word problem: cab
          • Rational number word problem: ice
          • Rational number word problem: computers
          • Rational number word problem: stock
          • Rational number word problem: checking account
      • Practical geometry
        • Construction of triangles when the lengths of its three sides are known (SSS Criterion)
          • Construct a triangle with constraints
      • Perimeter and area
        • Area and perimeter
          • Area & perimeter word problem: dog pen
          • Area & perimeter word problem: table
          • Comparing areas word problem
        • Area of triangles
          • Area of a triangle
          • Area of triangle proof
        • Area of shapes on grids
          • Area of a triangle on a grid
          • Area of a quadrilateral on a grid
        • Area and circumference of circles
          • Radius, diameter, circumference & π
          • Labeling parts of a circle
          • Radius & diameter from circumference
          • Area of a circle
          • Area of a circle intuition
        • Area of composite figures
          • Area of composite shapes
          • Perimeter & area of composite shapes
          • Challenge problems: perimeter & area
      • Algebraic Expressions
        • The constituents of an expression
          • Terms, factors, & coefficients
        • Combining like terms- part 1/1
          • Intro to combining like terms
          • Combining like terms
          • Combining like terms example
          • Combining like terms with distribution
        • Combining like terms- part 2/1
          • Simplifying expressions
          • Combining like terms challenge problem
          • Simplifying expressions with rational numbers
        • Adding & subtracting algebraic expressions
          • Adding polynomials
          • Subtracting polynomials
          • Polynomial subtraction & set closure
          • Adding & subtracting multiple polynomials
        • Adding and subtracting algebraic expressions : two variables
          • Adding polynomials: two variables
          • Subtracting polynomials: two variables
          • Subtracting polynomials: two variables challenge
          • Finding an error in polynomial subtraction
          • Polynomials review
        • Finding the value of an expression
          • Evaluating expressions with variables: exponents
          • Evaluating exponential expressions 2
          • Evaluating expressions with variables: temperature
          • Evaluating expressions with two variables
          • Example: Evaluating expressions with 2 variables
          • Examples of evaluating variable expressions
        • Number patterns
          • Factors and multiples: days of the week
          • Math patterns: toothpicks
          • Math patterns: table
      • Powers and exponents
        • Exponents
          • Exponentiation warmup
          • Intro to exponents
          • Exponent example 1
          • Exponent example 2
          • The 0 & 1st power
          • Powers of zero
          • 1 and -1 to different powers
        • Exponent properties- part 1/1
          • Exponent properties 1
          • Exponent properties 2
          • Exponent properties 3
          • Worked example: Exponent properties
        • Exponent properties- part 2/1
          • Exponent properties with products
          • Exponent properties with parentheses
          • Exponent properties with quotients
        • Scientific notation
          • Introduction to scientific notation
          • Scientific notation examples
          • Scientific notation examples
          • Scientific notation example: 0.0000000003457
        • Orders of magnitude
          • Multiplying multiples of powers of 10
          • Exponents and powers of 10 patterns
          • Orders of magnitude exercise example 1
          • Orders of magnitude exercise example 2
      • Symmetry
        • Lines of symmetry for regular polygons
          • Intro to reflective symmetry
        • Rotational symmetry
          • Intro to rotational symmetry
      • Visualising solid shapes
        • Nets for building 3D shapes
          • Surface area using a net: rectangular prism
        • Viewing different sections of a solid
          • Slicing a rectangular pyramid
    • Class 8 (India)
      • Linear equations in one variable
        • Two-steps equations intro
          • Intro to two-step equations
          • Two-step equations intuition
          • Worked example: two-step equations
          • Describing steps when solving equations
        • Old school equations with Sal
          • Linear equations 1
          • Linear equations 2
          • Linear equations 3
          • Linear equations 4
          • Averages
          • Taking percentages
          • Growing by a percentage
          • More percent problems
          • Solving an equation for a variable
        • Linear equations with variables on both sides
          • Why we do the same thing to both sides: Variable on both sides
          • Intro to equations with variables on both sides
          • Equations with variables on both sides: 20-7x=6x-6
          • Equation with variables on both sides: fractions
          • Equation with the variable in the denominator
          • Figuring out missing algebraic step
        • Linear equations with parentheses
          • Equations with parentheses
        • Analyzing the number of solutions to linear equations
          • Number of solutions to equations
          • Worked example: number of solutions to equations
          • Creating an equation with no solutions
          • Creating an equation with infinitely many solutions
        • Linear equations word problems
          • Sums of consecutive integers
          • Sum of integers challenge
          • Linear equation word problem: saline
          • Linear equation word problem: sugary drinks
        • Two-step equation word problems
          • Two-step equation word problem: computers
          • Two-step equation word problem: garden
          • Two-step equation word problem: oranges
      • Understanding Quadrilaterals
        • Angles with polygons
          • Sum of interior angles of a polygon
          • Sum of the exterior angles of a polygon
        • Quadrilateral types
          • Intro to quadrilateral
          • Quadrilateral properties
          • Quadrilateral types
          • Classifying quadrilaterals
          • Kites as a geometric shape
        • Quadrilateral proofs and angles
          • Proof: Opposite sides of a parallelogram
          • Proof: Diagonals of a parallelogram
          • Proof: Opposite angles of a parallelogram
          • Proof: Rhombus diagonals are perpendicular bisectors
          • Whether a special quadrilateral can exist
          • Rhombus diagonals
      • Data Handling
        • Categorical data displays
          • Reading pictographs
          • Reading bar graphs
          • Reading bar graphs: Harry Potter
          • Creating a bar graph
          • Reading bar charts: comparing two sets of data
          • Reading bar charts: putting it together with central tendency
          • Reading pie graphs (circle graphs)
        • Histograms
          • Creating a histogram
          • Interpreting a histogram
        • Pie Charts
          • Reading pie graphs (circle graphs)
        • Basic theoretical probability
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Intuitive sense of probabilities
          • The Monty Hall problem
        • Probability using sample spaces
          • Probability with counting outcomes
          • Example: All the ways you can flip a coin
          • Die rolling probability
          • Subsets of sample spaces
      • Square and square roots
        • Square roots
          • Intro to square roots
          • Understanding square roots
          • Approximating square roots to hundredths
        • Simplifying square roots
          • Simplifying square roots
      • Cubes and cube roots
        • Cube roots
          • Intro to cube roots
          • Cube root of a non-perfect cube
          • Simplifying a cube root
          • Worked example: Cube root of a negative number
      • Algebraic expressions and Identities
        • Adding & subtracting polynomials (Part1/2)
          • Adding polynomials
          • Subtracting polynomials
        • Adding and subtracting polynomials (Part2/2)
          • Adding & subtracting multiple polynomials
          • Subtracting polynomials: two variables
        • Multiplying monomials
          • Multiplying monomials
          • Multiplying monomials to find area: two variables
          • Multiplying monomials to find area
          • Multiplying monomials challenge
        • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials: area model
          • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials challenge
        • Multiplying binomials
          • Multiplying binomials: area model
          • Multiplying binomials intro
          • Multiplying binomials
        • Special products of binomials
          • Special products of the form (x+a)(x-a)
          • Squaring binomials of the form (x+a)²
          • Special products of the form (ax+b)(ax-b)
          • Squaring binomials of the form (ax+b)²
          • Special products of binomials: two variables
          • More examples of special products
          • Squaring a binomial (old)
        • Multiplying binomials by polynomials
          • Multiplying binomials by polynomials
          • Multiplying binomials by polynomials: area model
          • Multiplying binomials by polynomials challenge
          • Multiplying binomials by polynomials (old)
          • Multiplying binomials with radicals (old)
        • Polynomials word problems
          • Polynomial multiplication word problem
          • Polynomial word problem: rectangle and circle area
          • Polynomial word problem: total value of bills
          • Polynomial word problem: area of a window
      • Visualising solid shapes
        • Geometric solids (3D shapes)
          • Counting faces and edges of 3D shapes
          • Recognizing common 3D shapes
        • 2D vs. 3D objects
          • Slicing a rectangular pyramid
          • Ways to cross-section a cube
          • Rotating 2D shapes in 3D
        • Scale drawing
          • Interpreting a scale drawing
          • Solving a scale drawing word problem
          • Making a scale drawing
      • Mensuration
        • Area of trapezoids & composite figures
          • Area of trapezoids
          • Area of kites
          • Finding area by rearranging parts
          • Area of composite shapes
          • Perimeter & area of composite shapes
          • Challenge problems: perimeter & area
        • Surface area
          • Intro to nets of polyhedra
          • Surface area using a net: triangular prism
          • Surface area of a box (cuboid)
          • Surface area of a box using nets
          • Surface area using a net: rectangular prism
        • Volume of rectangular prisms
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
          • Volume of a rectangular prism
          • Volume of triangular prism & cube
        • Volume and surface area of cylinder
          • Cylinder volume & surface area
      • Exponents and powers
        • Negative exponents
          • Negative exponents
          • Negative exponent intuition
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
        • Scientific notation
          • Introduction to scientific notation
          • Scientific notation examples
          • Scientific notation examples
          • Scientific notation example: 0.0000000003457
        • Orders of magnitude
          • Multiplying multiples of powers of 10
          • Exponents and powers of 10 patterns
          • Orders of magnitude exercise example 1
          • Orders of magnitude exercise example 2
        • Arithmetic with numbers in scientific notation
          • Multiplying & dividing in scientific notation
          • Multiplying three numbers in scientific notation
          • Subtracting in scientific notation
          • Simplifying in scientific notation challenge
        • Scientific notation word problems
          • Scientific notation word problem: red blood cells
          • Scientific notation word problem: U.S. national debt
          • Scientific notation word problem: speed of light
      • Direct and inverse proportion
        • Direct and inverse proportion
          • Intro to direct & inverse variation
          • Recognizing direct & inverse variation
          • Recognizing direct & inverse variation: table
          • Inverse variation word problem: string vibration
      • Factorization
        • Introduction to factorization
          • Intro to factors & divisibility
        • Factoring monomials
          • Monomial factorization
          • Worked example: finding the missing monomial factor
          • Worked example: finding missing monomial side in area model
          • Greatest common factor of monomials
        • Factoring polynomials by taking common factors
          • Factoring with the distributive property
          • Factoring binomials: common factor
          • Factoring polynomials: common factor
          • Factoring polynomials: common factor area model
          • Factoring polynomials: common binomial factor
          • Factoring polynomials: common factor (old)
          • Factoring polynomials: common factor (old)
        • Evaluating expressions with unknown variables
          • Worked example: evaluating expressions using structure
          • Worked example: evaluating expressions using structure (more examples)
        • Factoring quadratics intro
          • Factoring quadratics as (x+a)(x+b)
          • Factoring quadratics as (x+a)(x+b) (example 2)
          • More examples of factoring quadratics as (x+a)(x+b)
        • Factoring quadratics by grouping
          • Intro to grouping
          • Factoring quadratics by grouping
        • Factoring polynomials with quadratic forms
          • Factoring quadratics: common factor + grouping
          • Factoring quadratics: negative common factor + grouping
          • Factoring two-variable quadratics
          • Factoring two-variable quadratics: rearranging
          • Factoring two-variable quadratics: grouping
          • Factoring quadratics with common factor (old)
        • Factoring quadratics: Difference of squares
          • Difference of squares intro
          • Factoring difference of squares: leading coefficient ≠ 1
          • Factoring difference of squares: analyzing factorization
          • Factoring difference of squares: missing values
          • Factoring difference of squares: shared factors
        • Factoring quadratics: Perfect squares
          • Perfect square factorization intro
          • Factoring perfect squares
          • Identifying perfect square form
          • Factoring perfect squares: common factor
          • Factoring perfect squares: negative common factor
          • Factoring perfect squares: missing values
          • Factoring perfect squares: shared factors
        • Strategy in factoring quadratics
          • Strategy in factoring quadratics (part 1 of 2)
          • Strategy in factoring quadratics (part 2 of 2)
        • Factoring polynomials with special product forms
          • Factoring perfect squares: 4th degree polynomial
          • Factoring difference of squares: two variables
          • Factoring difference of squares: two variables (example 2)
      • Introduction to graphs
        • Intro to the coordinate plane
          • Introduction to the coordinate plane
          • Coordinate plane: graphing points
        • Coordinate plane word problems
          • Plotting corners of a rectangle
          • Coordinate plane graphing word problem
          • Interpreting plotted points
      • Playing with numbers
        • Divisibility tests
          • Divisibility tests for 2, 3, 4, 5, 6, 9, 10
          • Recognizing divisibility
          • The why of the 3 divisibility rule
          • The why of the 9 divisibility rule
    • Class 9 (India)
      • Number systems
        • Irrational numbers
          • Intro to rational & irrational numbers
          • Classifying numbers: rational & irrational
          • Classifying numbers
          • Worked example: classifying numbers
        • Rational numbers and repeating decimals
          • Converting a fraction to a repeating decimal
          • Converting repeating decimals to fractions (part 1 of 2)
          • Converting repeating decimals to fractions (part 2 of 2)
        • Sums and products of rational and irrational numbers
          • Proof: sum & product of two rationals is rational
          • Proof: product of rational & irrational is irrational
          • Proof: sum of rational & irrational is irrational
          • Sums and products of irrational numbers
          • Worked example: rational vs. irrational expressions
          • Worked example: rational vs. irrational expressions (unknowns)
        • Exponent properties review
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
        • Radicals
          • Intro to square roots
          • Understanding square roots
          • Intro to cube roots
          • Square root of decimal
          • 5th roots
        • Rational exponents intro
          • Intro to rational exponents
          • Rewriting roots as rational exponents
          • Exponential equation with rational answer
        • Properties of exponents (rational exponents)
          • Simplifying quotient of powers (rational exponents)
          • Simplifying mixed radical and exponential expressions
        • Advanced: exponent & radical evaluation
          • Evaluating fractional exponents
          • Evaluating fractional exponents: negative unit-fraction
          • Evaluating fractional exponents: fractional base
          • Evaluating quotient of fractional exponents
          • Evaluating mixed radicals and exponents
        • Simplifying square roots
          • Simplifying square roots
          • Simplifying square roots (variables)
          • Simplifying square-root expressions
        • Simplifying radicals (higher-index roots)
          • Simplifying higher-index roots
          • Simplifying cube root expressions
          • Simplifying cube root expressions (two variables)
          • Simplifying higher-index root expressions
        • Radicals (miscellaneous videos)
          • Simplifying square-root expressions: no variables
          • Simplifying square roots of fractions
          • Simplifying rational exponent expressions: mixed exponents and radicals
          • Simplifying square-root expressions: no variables (advanced)
          • Intro to rationalizing the denominator
          • Worked example: rationalizing the denominator
          • Simplifying radical expressions (addition)
          • Simplifying radical expressions (subtraction)
          • Simplifying radical expressions: two variables
          • Simplifying radical expressions: three variables
          • Simplifying hairy expression with fractional exponents
      • Polynomials
        • Introduction to polynomials
          • Polynomials intro
          • The parts of polynomial expressions
          • Evaluating polynomials
          • Simplifying polynomials
        • Zeroes of a polynomial
          • Evaluating polynomials
        • Long division of polynomials
          • Intro to long division of polynomials
          • Dividing polynomials: long division
        • Polynomial Remainder Theorem
          • Intro to the Polynomial Remainder Theorem
          • Using the Polynomial Remainder Theorem: finding remainders
          • Using the Polynomial Remainder Theorem: checking factors
          • Using the Polynomial Remainder Theorem: finding coefficients
          • Proof of the Polynomial Remainder Theorem
        • Advanced polynomial factorization methods
          • Factoring higher-degree polynomials
          • Factoring sum of cubes
          • Factoring difference of cubes
      • Coordinate geometry
        • Intro to the coordinate plane
          • Introduction to the coordinate plane
          • Coordinate plane: graphing points
        • Coordinate plane: quadrant 1
          • Coordinate plane: graphing points
          • Coordinate plane graphing word problem
        • Quadrants on the coordinate plane
          • Quadrants of the coordinate plane
          • Points and quadrants example
        • Coordinate plane: 4 quadrants
          • Points on the coordinate plane examples
          • Plotting a point (ordered pair)
          • Finding the point not graphed
          • Coordinate plane word problem examples
        • Coordinate plane word problems
          • Plotting corners of a rectangle
          • Coordinate plane graphing word problem
          • Interpreting plotted points
      • Linear equations in two variables
        • Introduction to linear equations in 2 variables
          • Two-variable linear equations intro
          • Solutions to 2-variable equations
          • Worked example: solutions to 2-variable equations
          • Completing solutions to 2-variable equations
          • Testing solutions to linear equations
        • Graph of a Linear Equation in Two Variables
          • Graphing a linear equation: 5x+2y=20
          • Graphing a linear equation: y=2x+7
        • Linear models word problems
          • Linear models word problem: book
          • Linear models word problem: marbles
      • Introduction to Euclid's geometry
        • Introduction to Euclidean geometry
          • Euclid as the father of geometry
      • Lines and angles
        • Geometric definitions
          • Geometric definitions example
        • Angle types
          • Acute, right, & obtuse angles
          • Recognizing angles
          • Drawing acute, right and obtuse angles
          • Identifying an angle
        • Vertical, complementary, and supplementary angles
          • Complementary & supplementary angles
          • Angle relationships example
          • Vertical angles are congruent proof
        • Angles between intersecting lines
          • Angles, parallel lines, & transversals
          • Parallel & perpendicular lines
          • Missing angles with a transversal
          • Parallel lines & corresponding angles proof
          • Missing angles (CA geometry)
          • Proving angles are congruent
          • Proofs with transformations
        • Triangle angles
          • Triangle exterior angle example
          • Angles in a triangle sum to 180° proof
          • Worked example: Triangle angles (intersecting lines)
          • Worked example: Triangle angles (diagram)
          • Triangle angle challenge problem
          • Triangle angle challenge problem 2
      • Triangles
        • Congruence of triangles and the criteria for congruence
          • Congruent triangles & the SSS postulate/criterion
          • Triangle congruence postulates/criteria
          • Why SSA isn't a congruence postulate/criterion
          • Determining congruent triangles
        • Proofs of general theorems that use triangle congruence
          • Geometry proof problem: midpoint
          • Geometry proof problem: congruent segments
          • Geometry proof problem: squared circle
        • Theorems concerning triangle properties
          • Angles in a triangle sum to 180° proof
          • Proofs concerning isosceles triangles
          • Proofs concerning equilateral triangles
        • Inequalities in a Triangle
          • Triangle inequality theorem
      • Quadrilaterals
        • Quadrilaterals
          • Intro to quadrilateral
          • Quadrilateral properties
          • Quadrilateral types
          • Classifying quadrilaterals
          • Kites as a geometric shape
        • Quadrilateral proof and angles
          • Sum of interior angles of a polygon
          • Proof: Opposite sides of a parallelogram
          • Proof: Diagonals of a parallelogram
          • Proof: Opposite angles of a parallelogram
          • Proof: Rhombus diagonals are perpendicular bisectors
          • Whether a special quadrilateral can exist
          • Rhombus diagonals
      • Areas of parallelograms and triangles
        • Area of Parallelogram
          • Area of a parallelogram
          • Area of parallelogram proof
        • Area of a triangle
          • Area of a triangle
          • Area of triangle proof
      • Circles
        • Circle basics
          • Circles glossary
          • Radius, diameter, circumference & π
          • Proof: all circles are similar
        • Chord and Circles
          • Proof: radius is perpendicular to a chord it bisects
          • Proof: perpendicular radius bisects chord
        • Points and Circles
          • Distance between a point & line
        • Inscribed angles
          • Inscribed angles
          • Inscribed angle theorem proof
        • Inscribed shapes problem solving
          • Proof: Right triangles inscribed in circles
          • Inscribed shapes: find diameter
          • Inscribed shapes: angle subtended by diameter
          • Inscribed shapes: find inscribed angle
          • Inscribed quadrilaterals proof
          • Solving inscribed quadrilaterals
          • Area of inscribed equilateral triangle
      • Constructions
        • Basic constructions
          • Geometric constructions: angle bisector
          • Geometric constructions: perpendicular bisector
      • Heron's Formula
        • Finding the area of a triangle using Heron's formula
          • Heron's formula
      • Surface areas and volumes
        • Surface area
          • Intro to nets of polyhedra
          • Surface area using a net: triangular prism
          • Surface area of a box (cuboid)
          • Surface area of a box using nets
          • Surface area using a net: rectangular prism
        • Volume of cubes and cuboids
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
          • Volume of a rectangular prism
          • Volume of triangular prism & cube
        • Volume of cones, cylinders, and spheres
          • Cylinder volume & surface area
          • Volume of a cone
          • Volume of a sphere
      • Statistics
        • Presentation of Data
          • Representing data
          • Frequency tables & dot plots
        • Graphical Representation of Data using bar graphs
          • Creating a bar graph
          • Reading bar graphs
          • Reading bar graphs: Harry Potter
          • Reading bar graphs: dog bones
          • Reading bar charts: comparing two sets of data
        • Graphical representation of data using histograms
          • Creating a histogram
          • Interpreting a histogram
        • Measures of Central Tendency
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
          • Means and medians of different distributions
          • Missing value given the mean
          • Impact on median & mean: removing an outlier
          • Impact on median & mean: increasing an outlier
          • Comparing means of distributions
          • Reading bar charts: putting it together with central tendency
      • Probability
        • Probability basics
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Experimental probability
          • Theoretical and experimental probabilites
          • Making predictions with probability
          • Intuitive sense of probabilities
          • Probability models example: frozen yogurt
    • Class 10 (India)
      • Real numbers
        • The Fundamental Theorem of Arithmetic
          • The fundamental theorem of arithmetic
          • Prime factorization exercise
        • GCF and LCM
          • Greatest common factor examples
          • Least common multiple: repeating factors
          • Least common multiple of three numbers
          • GCF & LCM word problems
        • Proofs related to irrational numbers
          • Proof: √2 is irrational
          • Proof: square roots of prime numbers are irrational
          • Proof: there's an irrational number between any two rational numbers
        • Converting fractions to repeating decimals
          • Converting a fraction to a repeating decimal
          • Converting repeating decimals to fractions (part 1 of 2)
          • Converting repeating decimals to fractions (part 2 of 2)
      • Polynomials
        • Finding zeros of polynomials
          • Finding zeros of polynomials (1 of 2)
          • Finding zeros of polynomials (2 of 2)
          • Finding zeros of polynomials (example 2)
          • Number of possible real roots of a polynomial
        • Long division of polynomials
          • Intro to long division of polynomials
          • Dividing polynomials: long division
          • Intro to the Polynomial Remainder Theorem
          • Finding zeros of polynomials (1 of 2)
          • Finding zeros of polynomials (2 of 2)
      • Pair of linear equations in two variables
        • Geometrical interpretation of pair of linear equations in two variables:
          • Number of solutions to a system of equations
          • How many solutions does a system of linear equations have if there are at least two?
        • Algebraic methods to solve pair of linear equations in two variables
          • Number of solutions to a system of equations algebraically
          • Systems of equations with substitution: 2y=x+7 & x=y-4
          • Systems of equations with substitution: y=4x-17.5 & y+2x=6.5
          • Systems of equations with substitution: -3x-4y=-2 & y=2x-5
          • Systems of equations with substitution: 9x+3y=15 & y-x=5
          • Systems of equations with substitution: y=-5x+8 & 10x+2y=-2
          • Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120
        • System of equations : word problems
          • System of equations word problem: walk & ride
          • System of equations word problem: no solution
          • System of equations word problem: infinite solutions
          • Systems of equations with elimination: TV & DVD
          • Systems of equations with elimination: apples and oranges
          • Systems of equations with substitution: coins
          • Systems of equations with elimination: coffee and croissants
      • Quadratic equations
        • Solving quadratics by using factorisation
          • Solving quadratics by factoring
          • Solving quadratics by factoring: leading coefficient ≠ 1
          • Solving quadratics using structure
          • Quadratic equations word problem: triangle dimensions
          • Quadratic equations word problem: box dimensions
        • Solution of a Quadratic Equation by Completing the Square
          • Completing the square
          • Worked example: Completing the square (intro)
          • Worked example: Rewriting expressions by completing the square
          • Worked example: Solving equations by completing the square
          • Worked example: completing the square (leading coefficient ≠ 1)
          • Solving quadratics by completing the square: no solution
        • Solving quadratics using the quadratic formula
          • The quadratic formula
          • Using the quadratic formula
          • Worked example: quadratic formula
          • Worked example: quadratic formula (example 2)
          • Worked example: quadratic formula (negative coefficients)
          • Proof of the quadratic formula
          • Using the quadratic formula: number of solutions
      • Arithmetic progressions
        • Arithmetic Progression
          • Sequences intro
          • Intro to arithmetic sequences
          • Extending arithmetic sequences
        • nth term of an AP
          • Using arithmetic sequences formulas
        • Sum of First n Terms of an AP
          • Arithmetic series intro
          • Arithmetic series formula
          • Proof of finite arithmetic series formula
      • Triangles
        • Introduction to triangle similarity
          • Intro to triangle similarity
          • Triangle similarity postulates/criteria
          • Determining similar triangles
          • Proving slope is constant using similarity
        • Solving similar triangles
          • Solving similar triangles
          • Solving similar triangles: same side plays different roles
          • Using similar & congruent triangles
        • Pythagoras Theorem
          • Pythagorean theorem proof using similarity
          • Pythagorean theorem word problem: fishing boat
          • Another Pythagorean theorem proof
      • Coordinate geometry
        • Distance Formula
          • Distance formula
          • Midpoint formula
        • Section Formula
          • Dividing line segments: graphical
          • Dividing line segments
        • Area of Triangle
          • Area of a triangle on a grid
      • Introduction to trigonometry
        • Introduction to trigonometry
          • Intro to the trigonometric ratios
          • Trigonometric ratios in right triangles
        • Trigonometric Ratios of Complementary Angles
          • Sine & cosine of complementary angles
          • Using complementary angles
          • Trig word problem: complementary angles
        • Trigonometric Identities
          • Intro to the Pythagorean trig identity
          • Using trigonometric identities
      • Some applications of trigonometry
        • Word problems: Applications of trigonometry
          • Right triangle word problem
      • Circles
        • Tangent to a Circle
          • Proof: Radius is perpendicular to tangent line
          • Proof: Segments tangent to circle from outside point are congruent
          • Tangents of circles problem (example 1)
          • Tangents of circles problem (example 2)
          • Tangents of circles problem (example 3)
      • Constructions
        • Construction of Tangents to a Circle
          • Geometric constructions: circle tangent
          • Geometric constructions: circle tangent (example 2)
      • Area Related to Circles
        • Areas of parts of a circle
          • Area of a sector
        • Areas of Combinations of Plane Figures
          • Area of a shaded region
      • Surface Areas and Volumes
        • Conversion of Solid from One Shape to Another
          • Volume word problem: gold ring
      • Probability
        • Probability basics
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Experimental probability
          • Theoretical and experimental probabilites
          • Making predictions with probability
          • Intuitive sense of probabilities
          • Probability models example: frozen yogurt
    • Class 11 (India)
      • Sets
        • Intersection and union of sets
          • Intersection and union of sets
        • Types of sets
          • Subset, strict subset, and superset
        • Operations on sets
          • Relative complement or difference between sets
          • Universal set and absolute complement
          • Bringing the set operations together
      • Relations and functions
        • Evaluating functions
          • What is a function?
          • Worked example: Evaluating functions from equation
          • Worked example: evaluating functions from graph
          • Worked example: evaluating expressions with function notation
        • Inputs and outputs of a function
          • Worked example: matching an input to a function's output (equation)
          • Worked example: matching an input to a function's output (graph)
          • Worked example: two inputs with the same output (graph)
        • Interpreting function notation
          • Function notation word problem: bank
          • Function notation word problem: beach
        • Functions and equations
          • Equations vs. functions
          • Obtaining a function from an equation
        • Introduction to the domain and range of a function
          • Intervals and interval notation
          • What is the domain of a function?
          • What is the range of a function?
          • Worked example: domain and range from graph
        • Determining the domain of a function
          • Domain of a radical function
          • Worked example: domain of algebraic functions
          • Worked example: determining domain word problem (real numbers)
          • Worked example: determining domain word problem (positive integers)
          • Worked example: determining domain word problem (all integers)
        • Recognizing functions
          • Recognizing functions from graph
          • Does a vertical line represent a function?
          • Recognizing functions from table
          • Recognizing functions from verbal description
          • Recognizing functions from verbal description word problem
        • Maximum and minimum points
          • Introduction to minimum and maximum points
          • Worked example: relative maxima and minima
        • Intervals where a function is positive, negative, increasing, or decreasing
          • Increasing, decreasing, positive or negative intervals
          • Worked example: positive & negative intervals
        • Interpreting features of graphs
          • Graph interpretation word problem: temperature
          • Graph interpretation word problem: basketball
        • Combining functions
          • Adding functions
          • Subtracting functions
          • Multiplying functions
          • Dividing functions
        • Piecewise functions
          • Introduction to piecewise functions
          • Worked example: evaluating piecewise functions
          • Worked example: graphing piecewise functions
          • Worked example: domain & range of step function
          • Worked example: domain & range of piecewise linear functions
        • Determining the domain of advanced functions
          • Domain of advanced functions
        • Determining the range of advanced functions
          • Domain and range of quadratic functions
      • Exponential and Logarithmic functions
        • Introduction to logarithms
          • Intro to logarithms
          • Evaluating logarithms (advanced)
          • Relationship between exponentials & logarithms
          • Relationship between exponentials & logarithms: graphs
          • Relationship between exponentials & logarithms: tables
        • The constant e and the natural logarithm
          • 𝑒 and compound interest
          • 𝑒 as a limit
          • Evaluating natural logarithm with calculator
        • Properties of logarithms
          • Intro to logarithm properties (1 of 2)
          • Intro to logarithm properties (2 of 2)
          • Using the logarithmic product rule
          • Using the logarithmic power rule
          • Using the properties of logarithms: multiple steps
          • Proof of the logarithm product rule
          • Proof of the logarithm quotient and power rules
        • Change of base formula for logarithms
          • Evaluating logarithms: change of base rule
          • Using the logarithm change of base rule
          • Proof of the logarithm change of base rule
        • Exponential equations: logarithms
          • Solving exponential equations using logarithms: base-10
          • Solving exponential equations using logarithms: base-2
        • Solving exponential models
          • Exponential model word problem: medication dissolve
          • Exponential model word problem: bacteria growth
        • Logarithmic scale (optional)
          • Logarithmic scale (with Vi Hart)
          • Logarithmic scale
          • Richter scale
          • Benford's law (with Vi Hart, 1 of 2)
          • Benford's law (with Vi Hart, 2 of 2)
      • Advanced: Playing with graphs
        • Shifting functions
          • Shifting functions
          • Graphing shifted functions
        • Stretching functions
          • Shifting & reflecting functions
          • Compressing functions
          • Reflecting & compressing functions
        • Symmetry of functions
          • Intro to function symmetry
          • Graphical symmetry of functions
          • Even/odd functions & numbers
        • Symmetry of polynomial functions
          • Even & odd polynomials
        • Graphs of absolute value functions
          • Shifting absolute value graphs
          • Scaling & reflecting absolute value functions: equation
          • Scaling & reflecting absolute value functions: graph
          • Graphing absolute value functions
        • Graphs of radical functions
          • Transforming the square-root function
          • Graphs of square-root functions
          • Square-root functions & their graphs
        • Graphs of exponential functions
          • Transforming exponential graphs
          • Transforming exponential graphs (example 2)
          • Graphing exponential functions
          • Exponential function graph
          • Graphs of exponential functions (old example)
        • Graphs of logarithmic functions
          • Graphs of logarithms & exponentials
          • Graphing basic logarithmic functions
          • Graphs of logarithmic functions
        • Graphs of polynomials using their zeroes
          • Zeros of polynomials & their graphs
        • End behavior of polynomial functions
          • Intro to end behavior of polynomials
          • End behavior of functions & their graphs
        • Graphs of polynomials
      • Trigonometric functions
        • Intro to radians
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
          • Radian angles & quadrants
        • Trigonometric values of special angles
          • Trig values of special angles
        • Trigonometric functions
          • Unit circle
          • The trig functions & right triangle trig ratios
        • The graphs of sine, cosine, and tangent
          • Graph of y=sin(x)
          • Graph of y=tan(x)
          • Intersection points of y=sin(x) and y=cos(x)
        • Trigonometric identities
          • Sine & cosine identities: symmetry
          • Tangent identities: symmetry
          • Sine & cosine identities: periodicity
          • Tangent identities: periodicity
          • Proof of the Pythagorean trig identity
          • Using the Pythagorean trig identity
          • Trig challenge problem: cosine of angle-sum
        • Trigonometric equations
          • Solving sinusoidal equations of the form sin(x)=d
          • Solving cos(θ)=1 and cos(θ)=-1
      • Principle of mathematical induction
        • Deductive and inductive reasoning
          • Inductive & deductive reasoning
          • Deductive reasoning
          • Using deductive reasoning
          • Inductive reasoning
          • Inductive reasoning (example 2)
          • Using inductive reasoning
          • Using inductive reasoning (example 2)
        • Principle of mathematical induction
          • Proof of finite arithmetic series formula by induction
      • Complex numbers
        • What are the imaginary numbers?
          • Intro to the imaginary numbers
          • Powers of the imaginary unit
          • Simplifying roots of negative numbers
          • i as the principal root of -1
        • What are the complex numbers?
          • Intro to complex numbers
          • How to classify complex numbers (example)
        • The complex plane
          • Plotting numbers on the complex plane
        • Adding & subtracting complex numbers
          • Adding complex numbers
          • Subtracting complex numbers
        • Multiplying complex numbers
          • Multiplying complex numbers
        • Complex conjugates & dividing complex numbers
          • Intro to complex number conjugates
          • Complex number conjugates
          • Dividing complex numbers
        • Absolute value & angle of complex numbers
          • Absolute value of complex numbers
          • Absolute value & angle of complex numbers
        • Distance & midpoint of complex numbers
          • Distance & midpoint of complex numbers
        • Polar form of complex numbers
          • Polar & rectangular forms of complex numbers
        • Multiplying & dividing complex numbers in polar form
          • Dividing complex numbers: polar & exponential form
          • Powers of complex numbers
          • Complex number equations: x³=1
      • Linear inequalities
        • Introduction to inequalities with variables
          • Testing solutions to inequalities
          • Plotting inequalities
          • Plotting an inequality example
          • Inequalities word problems
        • One-step inequalities
          • One-step inequalities examples
          • One-step inequalities: -5c ≤ 15
          • One-step inequality involving addition
          • One-step inequality word problem
          • Inequalities using addition and subtraction
          • Solving and graphing linear inequalities
        • Two-step inequalities
          • Two-step inequalities
          • Two-step inequality word problem: apples
          • Two-step inequality word problem: R&B
        • Multi-step inequalities
          • Inequalities with variables on both sides
          • Inequalities with variables on both sides (with parentheses)
          • Multi-step inequalities
        • Compound inequalities
          • Compound inequalities: OR
          • Compound inequalities: AND
          • A compound inequality with no solution
          • Double inequalities
          • Compound inequalities examples
        • Checking solutions of two-variable inequalities
          • Testing solutions to inequalities
          • Testing solutions to systems of inequalities
        • Constraining solutions of two-variable inequalities
          • Constraining solutions of two-variable inequalities
          • Constraining solutions of systems of inequalities
        • Graphing two-variable inequalities
          • Intro to graphing two-variable inequalities
          • Graphing two-variable inequalities
          • Two-variable inequalities from their graphs
          • Intro to graphing systems of inequalities
          • Graphing systems of inequalities
          • Graphing two-variable inequalities (old)
      • Permutations and combinations
        • Fundamental Principle of Counting
          • Count outcomes using tree diagram
          • Counting outcomes: flower pots
        • Permutations
          • Factorial and counting seat arrangements
          • Permutation formula
          • Possible three letter words
          • Zero factorial or 0!
          • Ways to arrange colors
          • Ways to pick officers
        • Combinations
          • Intro to combinations
          • Combination formula
          • Handshaking combinations
          • Combination example: 9 card hands
      • Binomial theorem
        • Binomial Theorem for Positive Integral Indices
          • Intro to the Binomial Theorem
          • Pascal's triangle and binomial expansion
          • Expanding binomials
          • Expanding binomials w/o Pascal's triangle
          • Binomial expansion & combinatorics
          • Pascal's triangle & combinatorics
          • Binomial expansion & combinatorics (old)
      • Sequence and series
        • Sequences
          • Sequences intro
          • Intro to arithmetic sequences
          • Sequences word problem: growth pattern
          • Explicit formulas for arithmetic sequences
        • Basic sigma notation
          • Summation notation
        • Finite arithmetic series
          • Arithmetic series intro
          • Worked example: arithmetic series (sigma notation)
          • Worked example: arithmetic series (sum expression)
          • Worked example: arithmetic series (recursive formula)
          • Proof of finite arithmetic series formula
        • Finite geometric series
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
          • Finite geometric series formula justification
        • Finite geometric series applications
          • Finite geometric series word problem: social media
          • Finite geometric series word problem: mortgage
        • Sum to n terms of special series
          • Sum of n squares (part 1)
          • Sum of n squares (part 2)
          • Sum of n squares (part 3)
      • Straight lines
        • x-intercepts and y-intercepts
          • Intro to intercepts
          • x-intercept of a line
          • Intercepts from an equation
          • Intercepts from a table
          • Graphing using intercepts (old)
        • Slope
          • Intro to slope
          • Positive & negative slope
          • Worked example: slope from graph
          • Worked example: slope from two points
          • Slope (more examples)
        • Horizontal & vertical lines
          • Slope of a horizontal line
          • Horizontal & vertical lines
        • Intro to slope-intercept form
          • Intro to slope-intercept form
          • Worked examples: slope-intercept intro
        • Graphing slope-intercept equations
          • Graph from slope-intercept equation
        • Parallel and perpendicular lines on the coordinate plane
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines from graph
        • Equations of parallel and perpendicular lines
          • Parallel lines from equation
          • Parallel lines from equation (example 2)
          • Parallel lines from equation (example 3)
          • Perpendicular lines from equation
          • Writing equations of perpendicular lines
          • Writing equations of perpendicular lines (example 2)
          • Proof: parallel lines have the same slope
          • Proof: perpendicular lines have negative reciprocal slope
        • Writing slope-intercept equations
          • Slope-intercept equation from graph
          • Slope-intercept equation from slope & point
          • Slope-intercept equation from two points
          • Slope-intercept form problems
          • Slope-intercept equation from slope & point (old)
          • Slope-intercept equation from slope & point: fractions (old)
          • Finding y intercept given slope & point (old)
        • Point-slope form
          • Intro to point-slope form
          • Point-slope & slope-intercept equations
        • Standard form
          • Intro to linear equation standard form
          • Graphing a linear equation: 5x+2y=20
          • Converting from slope-intercept to standard form
        • Summary: Forms of two-variable linear equations
          • Slope from equation
          • Writing linear equations in all forms
        • Distance between a point and a line
          • Distance between point & line
      • Conic sections
        • Introduction to conic sections
          • Intro to conic sections
        • Features of a circle
          • Graphing circles from features
          • Features of a circle from its graph
        • Standard equation of a circle
          • Features of a circle from its standard equation
          • Graphing a circle from its standard equation
          • Writing standard equation of a circle
        • Expanded equation of a circle
          • Features of a circle from its expanded equation
        • Features of an ellipse
          • Intro to ellipses
        • Equation of an ellipse
          • Ellipse standard equation from graph
          • Ellipse graph from standard equation
        • Foci of an ellipse
          • Foci of an ellipse from equation
        • Features of a parabola
          • Intro to focus & directrix
          • Equation of a parabola from focus & directrix
          • Focus & directrix of a parabola from equation
        • Features of a hyperbola
          • Intro to hyperbolas
          • Vertices & direction of a hyperbola
          • Vertices & direction of a hyperbola (example 2)
          • Graphing hyperbolas (old example)
        • Foci of a hyperbola
          • Foci of a hyperbola from equation
          • Proof of the hyperbola foci formula
        • Identifying conic sections from their expanded equations
          • Conic section from expanded equation: circle & parabola
          • Conic section from expanded equation: ellipse
          • Conic section from expanded equation: hyperbola
      • Limits
        • Limits introduction
          • Newton, Leibniz, and Usain Bolt
          • Limits intro
        • Limits from tables
          • Limits from tables
        • Limits from graphs
          • Limits from graphs: function undefined
          • Limits from graphs: limit isn't equal to the function's value
          • Limits from graphs: asymptote
        • One-sided limits
          • One-sided limits from graphs
          • One-sided limits from graphs: asymptote
          • One-sided limits from tables
        • One-sided vs. Two-sided limits
          • 1-sided vs. 2-sided limits (graphical)
          • One-sided limit from equation
          • Connecting limits and graphical behavior (more examples)
        • Formal definition of limits (optional)
          • Formal definition of limits Part 1: intuition review
          • Formal definition of limits Part 2: building the idea
          • Formal definition of limits Part 3: the definition
          • Formal definition of limits Part 4: using the definition
        • Review: Limits basics
        • Limits from equations (direct substitution)
          • Limits by direct substitution
          • Undefined limits by direct substitution
        • Limits from equations (factoring & rationalizing)
          • Limits by factoring
          • Limits by rationalizing
        • Squeeze theorem
          • Squeeze theorem intro
          • Squeeze theorem example
          • Limit of sin(x)/x as x approaches 0
        • Limits of trigonometric functions
          • Trig limit using pythagorean identity
          • Trig limit using double angle identity
        • Limits of piecewise functions
          • Analyzing functions for discontinuities (continuous example)
          • Analyzing functions for discontinuities (discontinuity example)
        • Review: Limits from equations
      • Derivatives
        • Introduction to differential calculus
          • Newton, Leibniz, and Usain Bolt
          • Derivative as slope of curve
          • Derivative & the direction of a function
        • Derivative as slope of tangent line
          • The derivative & tangent line equations
          • Interpreting derivative challenge
        • Derivative as instantaneous rate of change
          • Tangent slope as instantaneous rate of change
          • Estimating derivatives with two consecutive secant lines
          • Approximating instantaneous rate of change with average rate of change
        • Secant lines
          • Slope of a line secant to a curve
          • Secant line with arbitrary difference
          • Secant line with arbitrary point
          • Secant line with arbitrary difference (with simplification)
          • Secant line with arbitrary point (with simplification)
          • Secant lines: challenging problem 1
          • Secant lines: challenging problem 2
        • Derivative as a limit
          • Formal definition of the derivative as a limit
          • Derivative as a limit: numerical
        • Formal definition of derivative
          • Formal and alternate form of the derivative
          • Worked example: Derivative as a limit
          • Worked example: Derivative from limit expression
        • Using the formal definition of derivative
          • The derivative of x² at x=3 using the formal definition
          • The derivative of x² at any point using the formal definition
          • Limit expression for the derivative of a linear function
          • Limit expression for the derivative of cos(x) at a minimum point
          • Limit expression for the derivative of function (graphical)
        • Derivative as a function
          • The graphical relationship between a function & its derivative (part 1)
          • The graphical relationship between a function & its derivative (part 2)
          • Matching functions & their derivatives graphically
          • Connecting f and f' graphically
        • Review: Derivative basics
      • Differentiation
        • Basic differentiation rules
          • Basic derivative rules (Part 1)
          • Basic derivative rules (Part 2)
          • Basic derivative rules: find the error
          • Basic derivative rules: table
        • Power rule
          • Power rule
          • Justifying the power rule
          • Proof of power rule for positive integer powers
          • Proof of power rule for square root function
        • Polynomial functions differentiation
          • Differentiating polynomials
          • Differentiating polynomials example
          • Tangents of polynomials
        • Rational functions differentiation (intro)
          • Negative powers differentiation
          • Worked example: Tangent to the graph of 1/x
        • Radical functions differentiation (intro)
          • Fractional powers differentiation
          • Radical functions differentiation intro
        • Sine & cosine, eˣ and ln(x) derivatives
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Proof: The derivative of 𝑒ˣ is 𝑒ˣ
          • Proof: the derivative of ln(x) is 1/x
          • Worked example: Derivatives of sin(x) and cos(x)
        • Review: Basic differentiation
        • Product rule
          • Product rule
          • Differentiating products
          • Worked example: Product rule with table
          • Worked example: Product rule with mixed implicit & explicit
          • Product rule to find derivative of product of three functions
          • Product rule proof
      • Statistics
        • Measures of dispersion
          • Measures of spread: range, variance & standard deviation
        • Range
          • Range and mid-range
          • Median & range puzzlers
        • Mean absolute deviation (grouped data)
          • Mean absolute deviation (MAD)
          • Mean absolute deviation example
        • Variance and standard deviation (ungrouped data):
          • Measures of spread: range, variance & standard deviation
          • Variance of a population
          • Population standard deviation
          • Statistics: Variance of a population
          • Statistics: Alternate variance formulas
        • Comparing distributions
          • Comparing distributions with dot plots (example problem)
      • Probability
        • Basic probability
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
        • Addition rule for probability
          • Probability with Venn diagrams
          • Addition rule for probability
        • Compound probability of independent events using diagrams
          • Die rolling probability
          • Probability with counting outcomes
          • Compound events example with tree diagram
        • Compound probability of independent events using the multiplication rule
          • Compound probability of independent events
          • Probability without equally likely events
          • Independent events example: test taking
          • Die rolling probability with independent events
          • Coin flipping probability
          • Free-throw probability
          • Three-pointer vs free-throw probability
        • Probability using combinatorics
          • Probability using combinations
          • Probability & combinations (2 of 2)
          • Probability with counting outcomes
          • Getting exactly two heads (combinatorics)
          • Exactly three heads in five flips
          • Generalizing with binomial coefficients (bit advanced)
          • Example: Different ways to pick officers
          • Example: Combinatorics and probability
          • Example: Lottery probability
          • Mega millions jackpot probability
          • Birthday probability problem
    • Class 12 (India)
      • Relations and functions
        • Introduction to inverses of functions
          • Intro to inverse functions
          • Inputs & outputs of inverse functions
          • Graphing the inverse of a linear function
        • Finding inverse functions
          • Finding inverse functions: linear
          • Finding inverse functions: quadratic
          • Finding inverse functions: quadratic (example 2)
          • Finding inverse functions: radical
          • Finding inverse functions: rational
        • Verifying that functions are inverses
          • Verifying inverse functions by composition
          • Verifying inverse functions by composition: not inverse
        • Determining whether a function is invertible
          • Determining if a function is invertible
          • Restricting domains of functions to make them invertible
        • Formal definition of inverse functions (optional)
          • Introduction to the inverse of a function
          • Proof: Invertibility implies a unique solution to f(x)=y
          • Surjective (onto) and injective (one-to-one) functions
          • Relating invertibility to being onto and one-to-one
        • Binary operations
          • New operator definitions 1
          • New operator definitions 2
      • Inverse trigonometric functions
        • Basic concepts of inverse trigonometry
          • Intro to arcsine
          • Intro to arctangent
          • Intro to arccosine
          • Using inverse trig functions with a calculator
        • Domain and range of inverse trigonometric functions
          • Domain & range of inverse tangent function
          • Restricting domains of functions to make them invertible
      • Matrices
        • Introduction to matrices
          • Intro to matrices
        • Adding & subtracting matrices
          • Adding & subtracting matrices
        • Multiplying matrices by scalars
          • Multiplying matrices by scalars
        • Properties of matrix addition & scalar multiplication
        • Multiplying matrices by matrices
          • Intro to matrix multiplication
          • Multiplying matrices
        • Properties of matrix multiplication
          • Defined matrix operations
          • Intro to identity matrix
          • Dimensions of identity matrix
          • Is matrix multiplication commutative?
          • Associative property of matrix multiplication
          • Zero matrix & matrix multiplication
          • Using properties of matrix operations
          • Using identity & zero matrices
        • Transpose of a matrix
          • Transpose of a matrix
          • Transpose of a matrix product
          • Transposes of sums and inverses
        • Elementary matrix row operations
        • Invertible matrices
          • Intro to matrix inverses
          • Determining invertible matrices
        • Inverse of a matrix by elementary operations
          • Inverting a 3x3 matrix using Gaussian elimination
      • Determinants
        • Determinants
          • Determinant of a 2x2 matrix
          • Determinant of a 3x3 matrix: standard method (1 of 2)
          • Determinant of a 3x3 matrix: shortcut method (2 of 2)
        • Properties of determinants
          • Determinant when row multiplied by scalar
          • (correction) scalar multiplication of row
          • Determinant when row is added
          • Duplicate row determinant
          • Determinant after row operations
          • Upper triangular determinant
        • Adjoint and inverse of a matrix using minors and cofactors
          • Inverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrix
          • Inverting a 3x3 matrix using determinants Part 2: Adjugate matrix
        • Solving equations with inverse matrices
          • Representing linear systems with matrix equations
          • Solving linear systems with matrix equations
          • Matrix word problem: vector combination
        • Determinants in more depth (Optional)
          • Deriving a method for determining inverses
          • Example of finding matrix inverse
          • Formula for 2x2 inverse
          • 3 x 3 determinant
          • n x n determinant
          • Determinants along other rows/cols
          • Rule of Sarrus of determinants
          • Simpler 4x4 determinant
          • Determinant and area of a parallelogram
          • Determinant as scaling factor
      • Continuity & differentiability
        • Continuity at a point
          • Continuity introduction
          • Worked example: Continuity at a point
        • Review: Limits of combined & composite functions
          • Limit properties
          • Limits of combined functions
          • Limits of composite functions
        • Continuous functions
          • Functions continuous on all real numbers
          • Functions continuous at specific x-values
        • Review: Intermediate value theorem
          • Intermediate value theorem
          • Intermediate value theorem example
        • Differentiability
          • Differentiability at a point: graphical
          • Differentiability at a point: algebraic (function is differentiable)
          • Differentiability at a point: algebraic (function isn't differentiable)
        • Review: Differentiation basics
        • Product rule
          • Product rule
          • Differentiating products
          • Worked example: Product rule with table
          • Worked example: Product rule with mixed implicit & explicit
          • Product rule to find derivative of product of three functions
          • Product rule proof
        • Quotient rule
          • Quotient rule from product & chain rules
          • Worked example: Quotient rule with table
          • Tangent to y=𝑒ˣ/(2+x³)
          • Normal to y=𝑒ˣ/x²
        • Chain rule
          • Chain rule
          • Worked example: Derivative of cos³(x) using the chain rule
          • Worked example: Derivative of ln(√x) using the chain rule
          • Worked example: Derivative of √(3x²-x) using the chain rule
          • Worked example: Chain rule with table
        • Review: Product, quotient, & chain rule
        • Rational functions differentiation
          • Differentiating rational functions
        • Radical functions differentiation
          • Radical functions differentiation
        • Trigonometric functions differentiation
          • Derivatives of tan(x) and cot(x)
          • Derivatives of sec(x) and csc(x)
          • Trig functions differentiation
        • Exponential functions differentiation
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Exponential functions differentiation intro
          • Exponential functions differentiation
        • Logarithmic functions differentiation
          • Derivative of logₐx (for any positive base a≠1)
          • Logarithmic functions differentiation
          • Differentiating logarithmic functions using log properties
        • Derivatives capstone
          • Derivative of eᶜᵒˢˣ⋅cos(eˣ)
          • Derivative of sin(ln(x²))
      • Advanced differentiation
        • Implicit differentiation introduction
          • Implicit differentiation
          • Worked example: Implicit differentiation
          • Worked example: Evaluating derivative with implicit differentiation
          • Showing explicit and implicit differentiation give same result
        • Implicit differentiation (advanced examples)
          • Implicit differentiation (advanced example)
          • Implicit differentiation (advanced example)
          • Implicit differentiation (advanced example)
          • Derivative of ln(x) from derivative of 𝑒ˣ and implicit differentiation
        • Inverse trig functions differentiation
          • Derivative of inverse sine
          • Derivative of inverse cosine
          • Derivative of inverse tangent
        • Derivatives of inverse functions
          • Derivatives of inverse functions: from equation
          • Derivatives of inverse functions: from table
        • Disguised derivatives
          • Derivatives expressed as limits
        • Proofs for the derivatives of eˣ and ln(x)
          • Proof: d/dx(ln x) = 1/x
          • Proof: d/dx(eˣ) = eˣ
          • Proofs of derivatives of ln(x) and eˣ
        • Logarithmic differentiation
          • Composite exponential function differentiation
          • Worked example: Composite exponential function differentiation
        • Review: Advanced differentiation
        • Higher-order derivatives
          • Second derivatives
        • Mean value theorem
          • Mean value theorem
          • Mean value theorem example: polynomial
          • Mean value theorem example: square root function
          • Mean value theorem application
      • Playing with graphs (using differentiation)
        • Critical points
          • Introduction to minimum and maximum points
          • Identifying relative minimum and maximum values
          • Critical points introduction
          • Finding critical points
        • Increasing & decreasing intervals
          • Finding decreasing interval given the function
          • Finding increasing interval given the derivative
        • Relative minima & maxima
          • Finding relative extrema (first derivative test)
          • Analyzing a function with its derivative
          • Worked example: finding relative extrema
        • Absolute minima & maxima
          • Extreme value theorem
          • Finding absolute extrema on a closed interval
          • Absolute minima & maxima (entire domain)
        • Review: Increasing/decreasing intervals & extrema
        • Concavity
          • Concavity introduction
          • Analyzing concavity (graphical)
          • Analyzing concavity (algebraic)
          • Second derivative test
        • Points of inflection
          • Inflection points introduction
          • Inflection points (graphical)
          • Worked example: Inflection points from first derivative
          • Inflection points (algebraic)
        • Sketching graphs using calculus
          • Curve sketching with calculus: polynomial
          • Curve sketching with calculus: logarithm
        • Review: Concavity & points of inflection
      • Applications of derivatives
        • Approximations
          • Local linearity
          • Linear approximation example (old)
        • Rectilinear motion (optional)
          • Motion problems: when a particle is speeding up
          • Motion problems: finding the maximum acceleration
          • Analyzing rectilinear motion graphically
          • Total distance traveled by a particle
        • Related rates
          • Related rates intro
          • Related rates: Approaching cars
          • Related rates: Falling ladder
          • Related rates: water pouring into a cone
          • Related rates: shadow
          • Related rates: balloon
        • Tangents & Normals
          • Tangent to y=𝑒ˣ/(2+x³)
          • Worked example: Tangent to the graph of 1/x
          • Normal to y=𝑒ˣ/x²
        • Optimization
          • Optimization: sum of squares
          • Optimization: box volume (Part 1)
          • Optimization: box volume (Part 2)
          • Optimization: profit
          • Optimization: cost of materials
          • Optimization: area of triangle & square (Part 1)
          • Optimization: area of triangle & square (Part 2)
          • Optimization problem: extreme normaline to y=x²
        • Applied rates of change
          • Applied rate of change: forgetfulness
          • Applied rates of change: marginal costs
          • Marginal cost & differential calculus
        • L'Hôpital's rule
          • L'Hôpital's rule introduction
          • L'Hôpital's rule: limit at 0 example
          • L'Hôpital's rule: challenging problem
          • L'Hôpital's rule: limit at infinity example
          • L'Hôpital's rule: solve for a variable
          • L’Hôpital’s rule (composite exponential functions)
          • Proof of special case of l'Hôpital's rule
        • Review: Derivative applications
      • Indefinite Integrals
        • Antiderivatives
          • Antiderivatives and indefinite integrals
        • Indefinite integrals intro
          • Reverse power rule
          • Rewriting before integrating: challenge problem
          • Visually determining antiderivative
          • Graphs of indefinite integrals
        • Indefinite integrals of common functions
          • Indefinite integrals of sin(x), cos(x), and eˣ
          • Indefinite integral of 1/x
          • Finding specific antiderivatives: rational function
          • Finding specific antiderivatives: exponential function
        • Review: Indefinite integrals & antiderivatives
        • Integration by parts
          • Integration by parts intro
          • Integration by parts: ∫x⋅cos(x)dx
          • Integration by parts: ∫ln(x)dx
          • Integration by parts: ∫x²⋅𝑒ˣdx
          • Integration by parts: ∫𝑒ˣ⋅cos(x)dx
        • u-substitution
          • 𝘶-substitution intro
          • 𝘶-substitution: rational function
          • 𝘶-substitution: multiplying by a constant
          • 𝘶-substitution: logarithmic function
          • 𝘶-substitution: composite function
          • 𝘶-substitution: special application
          • 𝘶-substitution: definite integral of exponential function
          • 𝘶-substitution: double substitution
        • Reverse chain rule
          • Reverse chain rule introduction
          • Reverse chain rule example
          • Integral of tan x
        • Partial fraction expansion
          • Partial fraction expansion to evaluate integral
          • Dividing expressions to evaluate integral
        • Integration using trigonometric identities
          • Integral of cos^3(x)
          • Integral of sin^2(x) cos^3(x)
          • Integral of sin^4(x)
        • Trigonometric substitution
          • Introduction to trigonometric substitution
          • Substitution with x=sin(theta)
          • More trig sub practice
          • Trig and u substitution together (part 1)
          • Trig and u substitution together (part 2)
          • Trig substitution with tangent
          • More trig substitution with tangent
          • Long trig sub problem
      • Definite Integrals
        • Functions defined by integrals
          • Functions defined by integrals
          • Functions defined by integrals: switched interval
          • Functions defined by integrals: challenge problem
        • Fundamental theorem of calculus
          • Fundamental theorem of calculus (Part 1)
          • Worked example: Finding derivative with fundamental theorem of calculus
          • Finding derivative with fundamental theorem of calculus: x is on both bounds
          • Proof of fundamental theorem of calculus
        • Fundamental theorem of calculus: chain rule
          • Worked example: Finding derivative with fundamental theorem of calculus
          • Finding derivative with fundamental theorem of calculus: x is on lower bound
        • Definite integral as area
          • Worked example: Definite integral by thinking about the function's graph
        • Definite integral properties
          • Integrating scaled version of function
          • Integrating sums of functions
          • Definite integral over a single point
          • Breaking up integral interval
          • Definite integral of shifted function
          • Switching bounds of definite integral
          • Worked examples: Definite integral properties 1
          • Definite integral properties (no graph): function combination
          • Worked examples: Definite integral properties 2
          • Definite integral properties (no graph): breaking interval
          • Examples leveraging integration properties
        • Review: Definite integral basics
        • Definite integral evaluation
          • Fundamental theorem of calculus (Part 2)
          • Intuition for second part of fundamental theorem of calculus
          • Area between a curve and the x-axis
          • Area between a curve and the x-axis: negative area
          • Definite integral of rational function
          • Definite integral of radical function
          • Definite integral of trig function
          • Definite integral involving natural log
        • Definite integrals of piecewise functions
          • Definite integral of piecewise function
          • Definite integral of absolute value function
        • Challenging definite integrals
          • Challenging definite integration
        • Improper integrals
          • Introduction to improper integrals
          • Improper integral with two infinite bounds
          • Divergent improper integral
        • Riemann sums
          • Riemann approximation introduction
          • Riemann sums in summation notation
          • Motion problem with Riemann sum approximation
        • Riemann sums with sigma notation
          • Worked example: Riemann sums in summation notation
          • Riemann sums in summation notation: challenge problem
        • Trapezoidal rule
          • Midpoint and trapezoidal sums in summation notation
          • Trapezoidal sums
        • Definite integral as the limit of a Riemann sum
          • Definite integral as the limit of a Riemann sum
          • Ranking area estimates
        • Review: Riemann sums
      • Application of integrals
        • Area under simple curves
          • Area between a curve and the x-axis
        • Area between two curves
          • Area between curves
          • Area between curves with multiple boundaries
        • Area defined by polar graphs
          • Area bounded by polar curves
          • Worked example: Area enclosed by cardioid
          • Worked example: Area between two polar graphs
          • Evaluating definite integral with calculator
        • Arc length (optional)
          • Arc length intro
          • Worked example: arc length
          • Worked example: arc length (advanced)
      • Differential equations
        • Intro to differential equations
          • Differential equations introduction
          • Worked example: linear solution to differential equation
          • Writing a differential equation
        • Slope fields (optional)
          • Slope fields introduction
          • Worked example: equation from slope field
          • Worked example: slope field from equation
          • Worked example: forming a slope field
          • Solutions from slope field
          • Worked example: range of solution curve from slope field
        • Separable equations
          • Separable equations introduction
          • Addressing treating differentials algebraically
          • Worked example: identifying separable equations
          • Worked example: finding a specific solution to a separable equation
          • Worked example: separable equation with an implicit solution
          • Separable equations (old)
          • Separable equations example (old)
        • Homogeneous equations
          • First order homogenous equations
          • First order homogeneous equations 2
        • Exact equations and integrating factors
          • Exact equations intuition 1 (proofy)
          • Exact equations intuition 2 (proofy)
          • Exact equations example 1
          • Exact equations example 2
          • Exact equations example 3
          • Integrating factors 1
          • Integrating factors 2
      • Vector algebra
        • Vector basics
          • Intro to vectors & scalars
          • Recognizing vectors
          • Equivalent vectors
          • Components of vectors
          • Components of vectors (example 2)
        • Scalar multiplication
          • Scalar multiplication of vectors
          • Analyzing scalar multiplication
        • Magnitude of vectors
          • Vector magnitude from graph
          • Vector magnitude from components
          • Vector magnitude from initial & terminal points
        • Vector addition & subtraction
          • Adding & subtracting vectors
          • Graphically adding & subtracting vectors
          • Graphically subtracting vectors
          • Adding vectors algebraically & graphically
        • Combined vector operations
          • Combined vector operations
        • Unit vectors
          • Worked example: finding unit vector with given direction
          • Unit vectors intro
          • Worked example: Scaling unit vectors
        • Magnitude & direction form of vectors
          • Direction of vectors from components: 1st & 2nd quadrants
          • Direction of vectors from components: 3rd & 4th quadrants
        • Adding vectors in magnitude & direction form
          • Adding vectors in magnitude & direction form (1 of 2)
          • Adding vectors in magnitude & direction form (2 of 2)
          • Vector addition & magnitude
        • Component form of vectors
          • Vector components from magnitude & direction
        • Applications of vectors
          • Vectors word problem: pushing a box
          • Vectors word problem: tug of war
          • Vectors word problem: hiking
        • Multiplication of a vector by a scalar
          • Scalar multiplication of vectors
          • Analyzing scalar multiplication
        • Dot product: Scaler product of two vectors
          • Vector dot product and vector length
          • Proving vector dot product properties
          • Proof of the Cauchy-Schwarz inequality
          • Vector triangle inequality
        • Cross product: Vector product of two vectors
          • Cross product introduction
          • Proof: Relationship between cross product and sin of angle
          • Dot and cross product comparison/intuition
          • Vector triple product expansion (very optional)
          • Combined vector operations
      • Linear Programming
        • Modeling with linear inequalities
          • Writing two-variable inequalities word problem
          • Solving two-variable inequalities word problem
          • Graphs of two-variable inequalities word problem
          • Interpreting two-variable inequalities word problem
          • Modeling with systems of inequalities
          • Writing systems of inequalities word problem
          • Solving systems of inequalities word problem
          • Graphs of systems of inequalities word problem
          • Analyzing structure with linear inequalities: fruits
          • Analyzing structure with linear inequalities: balls
      • Probability of events
        • Review: Multiplication rule for independent events
          • Compound probability of independent events
          • Independent events example: test taking
          • Free-throw probability
        • Review: Combinatorics and probability
          • Probability using combinations
          • Probability & combinations (2 of 2)
          • Example: Different ways to pick officers
          • Example: Combinatorics and probability
          • Exactly three heads in five flips
          • Generalizing with binomial coefficients (bit advanced)
          • Example: Lottery probability
          • Conditional probability and combinations
          • Mega millions jackpot probability
          • Birthday probability problem
        • Multiplication rule for dependent events
          • Dependent probability introduction
          • Independent & dependent probability
          • Dependent probability: coins
        • Conditional probability and independence
          • Calculating conditional probability
          • Analyzing event probability for independence
          • Conditional probability explained visually
        • Baye's Theorem
          • Conditional probability with Bayes' Theorem
        • Discrete and continuous random variables and probability models
          • Random variables
          • Discrete and continuous random variables
          • Constructing a probability distribution for random variable
          • Probability models example: frozen yogurt
          • Probability density functions
        • Expected value
          • Term life insurance and death probability
          • Getting data from expected value
          • Expected profit from lottery ticket
          • Expected value while fishing
          • Comparing insurance with expected value
          • Law of large numbers
        • Variance of random variables
          • Deriving the variance of the difference of random variables
        • Binomial random variables
          • Binomial distribution
          • Visualizing a binomial distribution
          • Binomial probability example
          • Generalizing k scores in n attempts
          • Free throw binomial probability distribution
          • Graphing basketball binomial distribution
          • Expected value of binomial distribution
        • Poisson distribution (optional)
          • Poisson process 1
          • Poisson process 2
    • Arithmetic
      • Addition and subtraction
        • Basic addition and subtraction
          • Basic addition
          • Basic subtraction
        • Addition and subtraction within 20
          • Adding 8 + 7
          • Subtracting 14 - 6
        • Strategies for adding and subtracting within 100
          • Adding 53+17 by making a group of 10
          • Adding by making a group of 10
          • Strategies for adding 2-digit numbers
          • Addition and subtraction with number lines
        • Addition within 100
          • Adding 2-digit numbers without regrouping 2
          • Adding with regrouping
        • Subtraction within 100
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting with regrouping (borrowing)
        • Word problems within 100
          • Subtraction word problem: tennis balls
          • Addition word problem: horses
          • Subtraction word problem: snow
          • Subtraction word problem: crayons
        • Multi-digit addition with regrouping
          • Adding with regrouping
          • Adding 3-digit numbers with regrouping
        • Multi-digit subtraction with regrouping
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Worked example: Subtracting 3-digit numbers (regrouping twice)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
          • Subtracting in your head (no regrouping)
      • Multiplication and division
        • Multiplication intro
          • Intro to multiplication
          • Multiplication as groups of objects
          • More ways to multiply
        • Multiplication by 10s, 100s, and 1000s
          • Multiplying 1-digit numbers by multiples of 10, 100, and 1000
          • Multiplying by multiples of 10
          • Multiplying 10s
          • Multiplying 1-digit numbers by 10, 100, and 1000
        • Multiplication: place value and area models
          • More ways to multiply
          • Multiplying with distributive property
          • Multiplying with area model: 6 x 7981
          • Multiplying with area model: 78 x 65
        • Multi-digit multiplication
          • Multiplying 2-digit by 1-digit
          • Multiplying 3-digit by 1-digit
          • Multiplying 3-digit by 1-digit (regrouping)
          • Multiplying 4-digit by 1-digit (regrouping)
          • Multiplying 2-digit numbers
          • Multiplying multi-digit numbers
        • Division intro
          • The idea of division
        • Division: place value and area
          • Division using place value
          • Division with area models
        • Multi-digit division (no remainders)
          • Intro to long division (no remainders)
          • Long division: 280÷5
          • Introduction to dividing by 2-digits
        • Remainders
          • Intro to remainders
          • Worked example: Long division with remainders: 3771÷8
          • Worked example: Long division with remainders: 2292÷4
        • Multi-digit division (remainders)
          • Dividing by 2-digits: 9815÷65
          • Dividing by 2-digits: 7182÷42
        • Multiplication and division word problems
          • Division word problem: field goals
          • Multiplication word problem: pizza
      • Negative numbers
        • Intro to negative numbers
          • Intro to negative numbers
        • Order negative numbers
          • Ordering negative numbers
        • Number opposites
          • Number opposites
          • Negative symbol as opposite
        • Absolute value
          • Absolute value examples
          • Comparing absolute values
          • Placing absolute values on the number line
          • Absolute value as distance between numbers
        • Intro to adding negative numbers
          • Adding negative numbers example
          • Adding numbers with different signs
        • Intro to subtracting negative numbers
          • Adding & subtracting negative numbers
          • Subtracting a negative = adding a positive
        • Adding & subtracting negative numbers
          • Adding numbers with different signs
        • Multiplying & dividing negative numbers
          • Why a negative times a negative makes sense
          • Why a negative times a negative is a positive
          • Multiplying positive & negative numbers
          • Dividing positive and negative numbers
      • Fractions
        • Fractions intro
          • Intro to fractions
          • Cutting shapes into equal parts
          • Identifying unit fractions word problem
          • More than one equal section
          • Recognizing fractions greater than 1
          • Identifying numerators and denominators
        • Fractions on the number line
          • Fractions on a number line
          • Fractions on number line widget
        • Equivalent fractions
          • Intro to equivalent fractions
          • Visualizing equivalent fractions
          • Equivalent fraction models
          • More on equivalent fractions
        • Comparing fractions
          • Comparing fractions with > and < symbols
          • Comparing fractions with like numerators and denominators
          • Comparing fractions
          • Comparing fractions 2 (unlike denominators)
          • Comparing and ordering fractions
          • Ordering fractions
        • Common denominators
          • Finding common denominators
          • Common denominators: 1/4 and 5/6
          • Common denominators: 3/5 and 7/2
        • Decomposing fractions
          • Decomposing a fraction visually
          • Decomposing a mixed number
        • Adding and subtracting fractions with like denominators
          • Adding fractions with like denominators
          • Subtracting fractions with like denominators
          • Adding mixed numbers with like denominators
          • Subtracting mixed numbers with like denominators
        • Mixed numbers
          • Rewriting mixed numbers as improper fractions
          • Rewriting improper fractions as mixed numbers
          • Comparing improper fractions and mixed numbers
        • Adding and subtracting fractions with unlike denominators
          • Visually adding fractions: 5/6+1/4
          • Visually subtracting fractions: 3/4-5/8
          • Adding fractions with unlike denominators
          • Subtracting fractions with unlike denominators
        • Adding and subtracting mixed number with unlike denominators
          • Adding mixed numbers: 19 3/18 + 18 2/3
          • Subtracting mixed numbers: 7 6/9 - 3 2/5
          • Adding mixed numbers with regrouping
          • Subtracting mixed numbers with regrouping (unlike denominators)
        • Adding and subtracting fractions word problems
          • Fraction word problem: piano
          • Fraction word problem: lizard
          • Adding fractions word problem: paint
          • Subtracting fractions word problem: tomatoes
        • Multiplying whole numbers and fractions
          • Multiplying fractions and whole numbers visually
          • Equivalent fraction and whole number multiplication problems
        • Multiplication as scaling
          • Multiplication as scaling with fractions
        • Multiplying fractions
          • Intro to multiplying 2 fractions
          • Multiplying 2 fractions: fraction model
          • Multiplying 2 fractions: number line
          • Multiplying 2 fractions: 5/6 x 2/3
        • Multiplying mixed numbers
          • Multiplying mixed numbers
        • Multiplying fractions word problems
          • Multiplying fractions word problem: muffins
          • Multiplying fractions word problem: laundry
          • Multiplying fractions word problem: bike
        • Fractions as division
          • Understanding fractions as division
          • Creating a fraction through division
        • Dividing unit fractions and whole numbers
          • Dividing a unit fraction by a whole number
          • Dividing a whole number by a unit fraction
        • Dividing fractions by fractions
          • Understanding division of fractions
          • Dividing fractions: 2/5 ÷ 7/3
          • Dividing fractions: 3/5 ÷ 1/2
        • Dividing fractions word problems
          • Dividing whole numbers & fractions: t-shirts
      • Decimals
        • Intro to decimals
          • Intro to decimals
          • Decimals as words
          • Write decimal in expanded form
          • Writing decimal numbers shown in grids
        • Decimals on the number line
          • Graphing tenths from 0 to 1
          • Identifying tenths on a number line
          • Plotting decimal numbers on a number line
          • Graphing hundredths from 0 to 0.1
          • Identifying hundredths on a number line
        • Rounding decimals
          • Worked example: Rounding decimals to nearest tenth
        • Comparing decimals
          • Comparing decimals 2
          • Comparing decimals 3
          • Comparing decimals: 9.97 and 9.798
          • Comparing decimals: 156.378 and 156.348
          • Ordering decimals 1
          • Ordering decimals 2
        • Rewriting decimals as fractions
          • Writing a number as a fraction and decimal
          • Rewriting decimals as fractions: 0.15
          • Rewriting decimals as fractions: 0.8
          • Rewriting decimals as fractions: 0.36
          • Fraction to decimal: 11/25
          • Worked example: Converting a fraction (7/8) to a decimal
          • Converting decimals to fractions 2 (ex 1)
          • Converting decimals to fractions 2 (ex 2)
        • Adding decimals
          • Introduction to adding decimals: tenths
          • Adding decimals with hundredths
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
        • Subtracting decimals
          • Strategies for subtracting more complex decimals with tenths
          • More advanced subtraction strategies with hundredths
          • Subtracting decimals: 9.005 - 3.6
          • Subtracting decimals: 39.1 - 0.794
        • Adding and subtracting decimals word problems
          • Adding decimals word problem
          • Adding & subtracting decimals word problem
        • Multiplying decimals
          • Intro to multiplying decimals
          • Developing strategies for multiplying decimals
          • Multiplying decimals: place value
          • Multiplying challenging decimals
        • Dividing decimals
          • Dividing whole numbers to get a decimal
          • Division strategies for decimal quotients
          • Multi-digit division strategies for decimals
          • Strategies for dividing by tenths
          • Dividing decimals completely
          • Dividing decimals with hundredths
          • Long division with decimals
          • Dividing by a multi-digit decimal
    • Pre-algebra
      • Arithmetic properties
        • Place value
          • Finding place value
          • Writing a number in standard form
          • Writing a number in expanded form
          • Comparing place values
          • Understanding place value
          • Place value when multiplying and dividing by 10
          • Regrouping numbers into various place values
          • Comparing whole number place values
          • Regrouping whole number place values
          • Adding whole numbers by their place values
        • Rounding whole numbers
          • Rounding whole numbers to nearest hundred
          • Rounding whole numbers to nearest thousand
          • Rounding whole numbers to nearest ten
        • Regrouping whole numbers
          • Regrouping whole numbers: 675
          • Regrouping whole numbers: 76,830
          • Regrouping whole numbers: 430
        • Order of operations
          • Intro to order of operations
          • Order of operations example
          • Worked example: Order of operations (PEMDAS)
        • Arithmetic properties
          • Properties of multiplication
          • Properties and patterns for multiplication
          • Commutative law of addition
          • Commutative law of multiplication
          • Associative law of addition
          • Associative law of multiplication
          • Identity property of 1
          • Identity property of 0
          • Inverse property of addition
          • Inverse property of multiplication
        • Distributive property
          • Distributive property over addition
          • Distributive property over subtraction
          • Distributive property exercise examples
        • Rational and irrational numbers
          • Intro to rational & irrational numbers
          • Classifying numbers: rational & irrational
      • Factors and multiples
        • Divisibility tests
          • Divisibility tests for 2, 3, 4, 5, 6, 9, 10
          • Recognizing divisibility
          • The why of the 3 divisibility rule
          • The why of the 9 divisibility rule
        • Factors and multiples
          • Finding factors of a number
          • Finding factors and multiples
        • Prime numbers
          • Prime numbers
          • Recognizing prime and composite numbers
        • Prime factorization
          • Prime factorization
          • Prime factorization exercise
          • Common divisibility examples
        • Least common multiple
          • Least common multiple
          • Least common multiple: repeating factors
          • Least common multiple of three numbers
        • Greatest common factor
          • Greatest common factor examples
          • Greatest common factor explained
          • GCF & LCM word problems
      • Reading and interpreting data
        • Representing data
          • Representing data
        • Stem and leaf plots
          • Stem-and-leaf plots
          • Reading stem and leaf plots
        • Picture graphs, bar graphs, and histograms
          • Reading pictographs
          • Creating a bar graph
          • Reading bar graphs
          • Interpreting bar graphs: alligators
          • Histograms
          • Reading pie graphs (circle graphs)
        • Frequency tables and dot plots
          • Frequency tables & dot plots
          • Comparing dot plots, histograms, and box plots
          • Misleading line graphs
        • Number patterns
          • Factors and multiples: days of the week
          • Math patterns: toothpicks
          • Math patterns: table
      • Measurement
        • Area of rectangles
          • Intro to area and unit squares
          • Measuring rectangles with different unit squares
          • Measuring area with partial unit squares
          • Counting unit squares to find area formula
          • Creating rectangles with a given area 2
          • Creating rectangles with a given area 1
          • Transitioning from unit squares to area formula
        • Perimeter
          • Perimeter: introduction
          • Perimeter of a shape
          • Finding perimeter when a side length is missing
          • Finding missing side length when given perimeter
          • Area & perimeter word problem: table
          • Area & perimeter word problem: dog pen
        • Volume of a rectangular prism
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
          • Volume of a rectangular prism
          • Volume word problem: water tank
      • Fractions
        • Fractions intro
          • Intro to fractions
          • Cutting shapes into equal parts
          • Identifying unit fractions word problem
          • More than one equal section
          • Recognizing fractions greater than 1
          • Identifying numerators and denominators
        • Fractions on the number line
          • Fractions on a number line
          • Fractions on number line widget
        • Equivalent fractions
          • Intro to equivalent fractions
          • Visualizing equivalent fractions
          • Equivalent fraction models
          • More on equivalent fractions
        • Comparing fractions
          • Comparing fractions with > and < symbols
          • Comparing fractions with like numerators and denominators
          • Comparing fractions
          • Comparing fractions 2 (unlike denominators)
          • Comparing and ordering fractions
          • Ordering fractions
        • Common denominators
          • Finding common denominators
          • Common denominators: 3/5 and 7/2
          • Common denominators: 1/4 and 5/6
        • Decomposing fractions
          • Decomposing a fraction visually
          • Decomposing a mixed number
        • Adding and subtracting fractions with like denominators
          • Adding fractions with like denominators
          • Subtracting fractions with like denominators
          • Adding mixed numbers with like denominators
          • Subtracting mixed numbers with like denominators
        • Mixed numbers
          • Rewriting mixed numbers as improper fractions
          • Rewriting improper fractions as mixed numbers
          • Comparing improper fractions and mixed numbers
        • Adding and subtracting fractions with unlike denominators
          • Visually adding fractions: 5/6+1/4
          • Visually subtracting fractions: 3/4-5/8
          • Adding fractions with unlike denominators
          • Subtracting fractions with unlike denominators
        • Adding and subtracting mixed number with unlike denominators
          • Adding mixed numbers: 19 3/18 + 18 2/3
          • Subtracting mixed numbers: 7 6/9 - 3 2/5
          • Adding mixed numbers with regrouping
          • Subtracting mixed numbers with regrouping (unlike denominators)
        • Adding and subtracting fractions word problems
          • Fraction word problem: piano
          • Fraction word problem: lizard
          • Adding fractions word problem: paint
          • Subtracting fractions word problem: tomatoes
        • Multiplying whole numbers and fractions
          • Multiplying fractions and whole numbers visually
          • Equivalent fraction and whole number multiplication problems
        • Multiplication as scaling
          • Multiplication as scaling with fractions
        • Multiplying fractions
          • Intro to multiplying 2 fractions
          • Multiplying 2 fractions: fraction model
          • Multiplying 2 fractions: number line
          • Multiplying 2 fractions: 5/6 x 2/3
        • Multiplying mixed numbers
          • Multiplying mixed numbers
        • Multiplying fractions word problems
          • Multiplying fractions word problem: muffins
          • Multiplying fractions word problem: laundry
          • Multiplying fractions word problem: bike
        • Fractions as division
          • Understanding fractions as division
          • Creating a fraction through division
        • Dividing unit fractions and whole numbers
        • Dividing fractions by fractions
          • Understanding division of fractions
          • Dividing fractions: 2/5 ÷ 7/3
          • Dividing fractions: 3/5 ÷ 1/2
        • Dividing fractions word problems
          • Dividing whole numbers & fractions: t-shirts
          • Dividing fractions by whole numbers: studying
      • Decimals
        • Intro to decimals
          • Intro to decimals
          • Decimals as words
          • Write decimal in expanded form
          • Writing decimal numbers shown in grids
        • Decimals on the number line
          • Graphing tenths from 0 to 1
          • Identifying tenths on a number line
          • Plotting decimal numbers on a number line
          • Graphing hundredths from 0 to 0.1
          • Identifying hundredths on a number line
        • Rounding decimals
          • Worked example: Rounding decimals to nearest tenth
        • Comparing decimals
          • Comparing decimals 2
          • Comparing decimals 3
          • Comparing decimals: 9.97 and 9.798
          • Comparing decimals: 156.378 and 156.348
          • Ordering decimals 1
          • Ordering decimals 2
        • Rewriting decimals as fractions
          • Writing a number as a fraction and decimal
          • Rewriting decimals as fractions: 0.15
          • Rewriting decimals as fractions: 0.8
          • Rewriting decimals as fractions: 0.36
          • Fraction to decimal: 11/25
          • Worked example: Converting a fraction (7/8) to a decimal
          • Converting decimals to fractions 2 (ex 1)
          • Converting decimals to fractions 2 (ex 2)
        • Adding decimals
          • Introduction to adding decimals: tenths
          • Adding decimals with ones, tenths and hundredths
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
        • Subtracting decimal
          • Strategies for subtracting more complex decimals with tenths
          • More advanced subtraction strategies with hundredths
          • Subtracting decimals: 9.005 - 3.6
          • Subtracting decimals: 39.1 - 0.794
        • Adding and subtracting decimals word problems
          • Adding decimals word problem
          • Adding & subtracting decimals word problem
        • Multiplying decimals
          • Intro to multiplying decimals
          • Developing strategies for multiplying decimals
          • Multiplying decimals: place value
          • Multiplying challenging decimals
        • Dividing decimals
          • Dividing whole numbers to get a decimal
          • Division strategies for decimal quotients
          • Multi-digit division strategies for decimals
          • Strategies for dividing by tenths
          • Dividing decimals completely
          • Dividing decimals with hundredths
          • Long division with decimals
          • Dividing by a multi-digit decimal
      • Negative numbers and coordinate plane
        • Intro to negative numbers
          • Intro to negative numbers
        • Order negative numbers
          • Ordering negative numbers
        • Number opposites
          • Number opposites
          • Negative symbol as opposite
        • Absolute value
          • Absolute value examples
          • Comparing absolute values
          • Placing absolute values on the number line
          • Absolute value as distance between numbers
        • Intro to adding negative numbers
          • Adding negative numbers example
          • Adding numbers with different signs
        • Intro to subtracting negative numbers
          • Adding & subtracting negative numbers
          • Subtracting a negative = adding a positive
        • Adding & subtracting negative numbers
          • Adding numbers with different signs
        • Multiplying & dividing negative numbers
          • Why a negative times a negative makes sense
          • Why a negative times a negative is a positive
          • Multiplying positive & negative numbers
          • Dividing positive and negative numbers
        • Coordinate plane
          • Points on the coordinate plane examples
          • Plotting a point (ordered pair)
          • Finding the point not graphed
          • Quadrants of the coordinate plane
          • Points and quadrants example
          • Coordinate plane word problem examples
          • Reflecting points in the coordinate plane
          • Graphing patterns on coordinate plane
          • Interpreting patterns on coordinate plane
          • Interpreting relationships in ordered pairs
      • Ratios, rates, proportions
        • Intro to ratios
          • Intro to ratios
          • Basic ratios
          • Equivalent ratios
          • Equivalent ratios: recipe
          • Equivalent ratio word problems
          • Understanding equivalent ratios
          • Solving ratio problems with graph
        • Visualize ratios
          • Ratios and double number lines
          • Solving ratio problems with tables
          • Ratios with tape diagrams
        • Ratio application
          • Ratios on coordinate plane
          • Ratios and measurement
          • Part:whole ratio word problem
          • Part:whole ratios
        • Intro to rates
          • Intro to rates
          • Solving unit rate problem
          • Solving unit price problem
          • Multiple rates word problem
          • Comparing rates example
          • Finding average speed or rate
          • Speed translation
        • Intro to percents
          • The meaning of percent
          • Meaning of 109%
          • Percents from fraction models
          • Fraction, decimal, and percent from visual model
        • Percent, fraction, decimal conversions
          • Converting decimals to percents: 0.601
          • Converting decimals to percents: 1.501
          • Converting percents to decimals: 59.2%
          • Converting percents to decimals: 113.9%
          • Converting percents to decimals & fractions example
          • Ordering numeric expressions
        • Percent problems
          • Finding a percent
          • Percent of a whole number
          • Identifying percent amount and base
        • Percent word problems
          • Percent word problem: 100 is what percent of 80?
          • Percent word problem: 78 is 15% of what number?
          • Percent word problem: guavas
          • Percent word problem: recycling cans
          • Percent word problem: penguins
          • Growing by a percentage
          • Solving percent problems
          • Percent word problem: magic club
          • Percent word problems: tax and discount
          • Rational number word problem: cosmetics
          • Rational number word problem: cab
          • Rational number word problem: school report
        • Identifying proportional relationships
          • Intro to proportional relationships
          • Proportional relationships: spaghetti
          • Proportional relationships: bananas
          • Proportional relationships: movie tickets
          • Proportional relationships
        • Writing & solving proportions
          • Worked example: Solving proportions
          • Writing proportions example
          • Proportion word problem: hot dogs
          • Proportion word problem: cookies
          • Multiple units word problem: road trip
      • Equations, expressions, and inequalities
        • Intro to variables
          • What is a variable?
          • Why aren't we using the multiplication sign?
        • Substitution & evaluating expressions
          • Evaluating an expression with one variable
          • Evaluating expressions with two variables
          • Evaluating expressions with two variables: fractions & decimals
        • Expression value intuition
          • Expression value intuition
        • Constructing numeric expressions
          • Constructing numerical expressions
          • Evaluating expressions with & without parentheses
        • Evaluating expressions word problems
          • Evaluating expressions with variables: temperature
          • Evaluating expressions with variables: cubes
          • Evaluating expressions with variables: exponents
        • Writing algebraic expressions introduction
          • Writing basic expressions with variables
          • Writing expressions with variables
          • Writing expressions with variables & parentheses
        • Writing basic algebraic expressions word problems
          • Writing basic expressions word problems
          • Writing expressions word problems
        • Algebraic equations basics
          • Variables, expressions, & equations
          • Testing solutions to equations
        • One-step equations intuition
          • Same thing to both sides of equations
          • Representing a relationship with an equation
          • Dividing both sides of an equation
          • One-step equations intuition
        • One-step addition & subtraction equations
          • One-step addition & subtraction equations
          • One-step subtraction equations
          • One-step addition & subtraction equations: fractions & decimals
        • One-step multiplication and division equations
          • One-step division equations
          • One-step multiplication equations
          • One-step multiplication & division equations: fractions & decimals
        • One-step equation word problems
          • Modeling with one-step equations
          • One-step equations word problem: super yoga (1 of 2)
          • One-step equations word problem: super yoga (2 of 2)
        • Inequalities: Greater than and less than basics
          • Greater than and less than symbols
          • Plotting inequalities
          • Inequality word problems
          • Inequality word problem: one variable
          • Plotting an inequality example
        • Two-step equations intro
          • Same thing to both sides of equations
          • Intro to two-step equations
          • Two-step equations intuition
          • Worked example: two-step equations
        • Dependent and independent variables
          • Dependent & independent variables
          • Dependent & independent variables: graphing
          • Dependent & independent variables: equation
      • Exponents, radicals, and scientific notation
        • Exponents
          • Exponentiation warmup
          • Intro to exponents
          • The 0 & 1st power
          • 1 and -1 to different powers
          • Powers of zero
        • Square roots
          • Intro to square roots
          • Understanding square roots
          • Approximating square roots to hundredths
          • Simplifying square roots
        • Cube roots
          • Intro to cube roots
          • Cube root of a non-perfect cube
          • Simplifying a cube root
          • Worked example: Cube root of a negative number
        • Exponent properties
          • Exponent properties with products
          • Exponent properties with parentheses
          • Exponent properties with quotients
        • Negative exponents
          • Negative exponents
          • Negative exponent intuition
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
        • Scientific notation
          • Introduction to scientific notation
          • Scientific notation examples
          • Scientific notation examples
          • Scientific notation example: 0.0000000003457
        • Orders of magnitude
          • Multiplying multiples of powers of 10
          • Exponents and powers of 10 patterns
          • Orders of magnitude exercise example 1
          • Orders of magnitude exercise example 2
        • Computing with scientific notation
          • Multiplying in scientific notation example
          • Multiplying three numbers in scientific notation
          • Multiplying & dividing in scientific notation
    • Algebra basics
      • Foundations
        • Negative numbers
          • Number opposites
          • Adding numbers with different signs
          • Adding & subtracting negative numbers
          • Subtracting a negative = adding a positive
          • Multiplying positive & negative numbers
          • Why a negative times a negative is a positive
          • Why a negative times a negative makes sense
          • Dividing positive and negative numbers
        • Absolute value
          • Absolute value examples
          • Absolute value and number lines
          • Interpreting absolute value
        • Exponents
          • Intro to exponents
          • Exponent example 1
          • Exponent example 2
          • The 0 & 1st power
          • Powers of zero
          • 1 and -1 to different powers
          • Comparing exponent expressions
          • Exponents of decimals
          • Evaluating exponent expressions with variables
        • Square roots
          • Intro to square roots
          • Simplifying square roots
          • Simplifying square roots of fractions
          • Simplifying square-root expressions: no variables
        • Order of operations
          • Intro to order of operations
          • Order of operations examples: exponents
          • Worked example: Order of operations (PEMDAS)
          • Order of operations example
        • Fractions
          • Adding fractions word problem: paint
          • Adding fractions with different signs
          • Multiplying positive and negative fractions
          • Dividing negative fractions
        • Decimals, fractions and percentages
          • Worked example: Converting a fraction (7/8) to a decimal
          • Fraction to decimal with rounding
          • Rewriting decimals as fractions: 0.36
          • Converting decimals to fractions 2 (ex 1)
          • Finding a percent
          • Percent of a whole number
          • Percent word problem: guavas
          • Percent word problem: recycling cans
        • Operations with decimals
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Subtracting decimals: 9.57-8.09
          • Subtracting decimals: 39.1 - 0.794
          • Multiplying decimals example
          • Multiplying challenging decimals
          • Dividing decimals with hundredths
          • Dividing by a multi-digit decimal
        • Area of triangles
          • Area of a triangle
        • Circumference and area of circles
          • Radius, diameter, circumference & π
          • Labeling parts of a circle
          • Area of a circle
      • Algebraic expressions
        • Introduction to variables
          • What is a variable?
          • Why aren't we using the multiplication sign?
          • Evaluating an expression with one variable
        • Substitution & evaluating expressions
          • Evaluating expressions with two variables
        • Writing algebraic expressions
          • Writing basic expressions with variables
          • Writing expressions with variables
          • Writing expressions with variables & parentheses
        • Combining like terms
          • Intro to combining like terms
          • Simplifying expressions
          • Combining like terms challenge problem
          • Simplifying expressions with rational numbers
        • Distributive property
          • Distributive property with variables
        • Equivalent algebraic expressions
          • Equivalent expressions
        • Nested fractions
          • Nested fractions
      • Linear equations and inequalities
        • One-step equations
          • One-step addition & subtraction equations
          • One-step subtraction equations
          • One-step division equations
          • One-step multiplication equations
        • Two-steps equations
          • Intro to two-step equations
          • Two-step equations intuition
          • Worked example: two-step equations
          • Two-step equation word problem: computers
          • Two-step equation word problem: garden
          • Two-step equation word problem: oranges
        • Multi-step equations
          • Why we do the same thing to both sides: Variable on both sides
          • Intro to equations with variables on both sides
          • Equations with variables on both sides: 20-7x=6x-6
          • Equations with parentheses
        • One-step inequalities
          • One-step inequalities examples
          • One-step inequalities: -5c ≤ 15
          • One-step inequality word problem
        • Two-step inequalities
          • Two-step inequalities
          • Two-step inequality word problem: apples
          • Two-step inequality word problem: R&B
        • Multi-step inequalities
          • Inequalities with variables on both sides
          • Inequalities with variables on both sides (with parentheses)
          • Multi-step inequalities
        • Writing & solving proportions
          • Worked example: Solving proportions
          • Solving hairy proportions
          • Writing proportions example
          • Proportion word problem: cookies
          • Proportion word problem: hot dogs
      • Graphing lines and slope
        • Coordinate plane
          • Quadrants of the coordinate plane
          • Points and quadrants example
        • Solutions to two-variable linear equations
          • Two-variable linear equations intro
          • Solutions to 2-variable equations
          • Worked example: solutions to 2-variable equations
          • Completing solutions to 2-variable equations
        • x-intercepts and y-intercepts
          • Intro to intercepts
          • x-intercept of a line
          • Intercepts from an equation
        • Slope
          • Intro to slope
          • Positive & negative slope
          • Worked example: slope from graph
          • Worked example: slope from two points
          • Slope (more examples)
        • Horizontal & vertical lines
          • Slope of a horizontal line
        • Slope-intercept form intro
          • Intro to slope-intercept form
          • Graph from slope-intercept equation
        • Writing slope-intercept equations
          • Slope-intercept equation from graph
          • Slope-intercept equation from slope & point
          • Slope-intercept equation from two points
          • Slope-intercept form problems
        • Graphing two-variable inequalities
          • Intro to graphing two-variable inequalities
          • Graphing two-variable inequalities
      • Systems of equations
        • Systems of equations overview
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: potato chips
          • Systems of equations with substitution: potato chips
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
        • Systems of equations intro
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Testing a solution to a system of equations
          • Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
          • Systems of equations with graphing: exact & approximate solutions
        • Elimination method for systems of equations
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: x-4y=-18 & -x+3y=11
          • Systems of equations with elimination: potato chips
          • Systems of equations with elimination (and manipulation)
          • Why can we subtract one equation from the other in a system of equations?
        • Substitution method for systems of equations
          • Systems of equations with substitution: potato chips
          • Systems of equations with substitution: -3x-4y=-2 & y=2x-5
        • Number of solutions to systems of equations
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
          • Solutions to systems of equations: consistent vs. inconsistent
          • Solutions to systems of equations: dependent vs. independent
          • Number of solutions to a system of equations
          • Number of solutions to a system of equations graphically
          • Number of solutions to a system of equations algebraically
          • How many solutions does a system of linear equations have if there are at least two?
        • Solving any system of linear equations
      • Expressions with exponents
        • Exponent properties intro
          • Exponent properties with products
          • Exponent properties with parentheses
          • Exponent properties with quotients
        • Negative exponents
          • Negative exponents
          • Negative exponent intuition
        • Exponent properties (integer exponents)
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
          • Powers of zero
        • Scientific notation intro
          • Scientific notation examples
          • Scientific notation example: 0.0000000003457
          • Multiplying in scientific notation example
          • Multiplying & dividing in scientific notation
          • Multiplying three numbers in scientific notation
        • Scientific notation word problems
          • Scientific notation word problem: red blood cells
          • Scientific notation word problem: U.S. national debt
          • Scientific notation word problem: speed of light
      • Quadratics and polynomials
        • Adding & subtracting polynomials
          • Adding polynomials
          • Subtracting polynomials
        • Multiplying binomials
          • Multiplying monomials by polynomials
          • Multiplying binomials: area model
          • Multiplying binomials intro
          • Multiplying binomials
        • Special products of binomials
          • Special products of the form (x+a)(x-a)
          • Squaring binomials of the form (x+a)²
        • Factoring polynomials by taking common factors
          • Factoring with the distributive property
          • Factoring binomials: common factor
          • Factoring polynomials: common factor
          • Factoring polynomials: common factor area model
          • Factoring polynomials: common binomial factor
        • Factoring quadratics 1
          • Factoring quadratics as (x+a)(x+b)
          • Factoring quadratics as (x+a)(x+b) (example 2)
          • More examples of factoring quadratics as (x+a)(x+b)
        • Factoring quadratics 2
          • Intro to grouping
          • Factoring quadratics by grouping
        • Factoring quadratics: Difference of squares
          • Factoring difference of squares: leading coefficient ≠ 1
          • Factoring difference of squares: analyzing factorization
          • Factoring difference of squares: missing values
          • Factoring difference of squares: shared factors
        • Factoring quadratics: Perfect squares
          • Factoring perfect squares
          • Identifying perfect square form
          • Factoring perfect squares: common factor
          • Factoring perfect squares: negative common factor
          • Factoring perfect squares: missing values
          • Factoring perfect squares: shared factors
        • Solving quadratic equations by factoring
          • Solving quadratics by factoring
          • Solving quadratics by factoring: leading coefficient ≠ 1
      • Equations and geometry
        • Triangle angles
          • Angles in a triangle sum to 180° proof
        • Pythagorean theorem
          • Intro to the Pythagorean theorem
          • Intro to the Pythagorean theorem 2
          • Pythagorean theorem example
          • Thiago asks: How much time does a goalkeeper have to react to a penalty kick?
        • Triangle similarity intro
          • Intro to triangle similarity
          • Triangle similarity postulates/criteria
          • Determining similar triangles
        • Solving similar triangles
          • Solving similar triangles
          • Solving similar triangles: same side plays different roles
    • Algebra I
      • Algebra foundations
        • Overview and history of algebra
          • Origins of algebra
          • Abstract-ness
          • The beauty of algebra
          • Intro to the coordinate plane
          • Why all the letters in algebra?
        • Introduction to variables
          • What is a variable?
          • Why aren't we using the multiplication sign?
          • Evaluating an expression with one variable
        • Substitution and evaluating expressions
          • Evaluating expressions with two variables
          • Evaluating expressions with two variables: fractions & decimals
        • Evaluating expressions word problems
          • Evaluating expressions with variables: temperature
          • Evaluating expressions with variables: cubes
          • Evaluating expressions with variables: exponents
        • Writing algebraic expressions introduction
          • Writing basic expressions with variables
          • Writing expressions with variables
          • Writing expressions with variables & parentheses
        • Dependent & independent variables
          • Dependent & independent variables
          • Dependent & independent variables: graphing
          • Dependent & independent variables: equation
        • Combining like terms
          • Intro to combining like terms
          • Simplifying expressions
          • Combining like terms challenge problem
          • Simplifying expressions with rational numbers
        • Introduction to equivalent algebraic expressions
          • Equivalent expressions
        • Interpreting linear expressions
          • Interpreting linear expressions: diamonds
          • Interpreting linear expressions: flowers
        • Division by zero
          • Why dividing by zero is undefined
          • The problem with dividing zero by zero
          • Undefined & indeterminate expressions
      • Solving equations
        • Algebraic equations basics
          • Variables, expressions, & equations
          • Testing solutions to equations
        • Why we do the same thing to both sides of an equation
          • Same thing to both sides of equations
          • Representing a relationship with an equation
          • Dividing both sides of an equation
          • One-step equations intuition
        • One-step addition & subtraction equations
          • One-step addition & subtraction equations
          • One-step subtraction equations
          • One-step addition & subtraction equations: fractions & decimals
        • One-step multiplication & division equations
          • One-step division equations
          • One-step multiplication equations
          • One-step multiplication & division equations: fractions & decimals
        • Two-steps equations intro
          • Intro to two-step equations
          • Two-step equations intuition
          • Worked example: two-step equations
        • Two-step equation word problems
          • Two-step equation word problem: computers
          • Two-step equation word problem: garden
          • Two-step equation word problem: oranges
        • Linear equations with variables on both sides
          • Why we do the same thing to both sides: Variable on both sides
          • Intro to equations with variables on both sides
          • Equations with variables on both sides: 20-7x=6x-6
          • Equation with variables on both sides: fractions
          • Equation with the variable in the denominator
        • Linear equations with parentheses
          • Equations with parentheses
        • Analyzing the number of solutions to linear equations
          • Number of solutions to equations
          • Worked example: number of solutions to equations
          • Creating an equation with no solutions
          • Creating an equation with infinitely many solutions
        • Linear equations word problems
          • Sums of consecutive integers
          • Sum of integers challenge
          • Linear equation word problem: saline
          • Linear equation word problem: sugary drinks
        • Linear equations with unknown coefficients
          • Linear equations with unknown coefficients
      • Solving inequalities
        • Introduction to inequalities with variables
          • Testing solutions to inequalities
          • Plotting inequalities
          • Plotting an inequality example
          • Inequalities word problems
        • One-step inequalities
          • One-step inequalities examples
          • One-step inequalities: -5c ≤ 15
          • One-step inequality word problem
        • Two-step inequalities
          • Two-step inequalities
          • Two-step inequality word problem: apples
          • Two-step inequality word problem: R&B
        • Multi-step inequalities
          • Inequalities with variables on both sides
          • Inequalities with variables on both sides (with parentheses)
          • Multi-step inequalities
        • Compound inequalities
          • Compound inequalities: OR
          • Compound inequalities: AND
          • A compound inequality with no solution
          • Double inequalities
          • Compound inequalities examples
      • Working with units
        • Rate conversion
          • Intro to dimensional analysis
          • Same rate with different units
        • Appropriate units
          • Appropriate units
          • Interpreting units in formulas
          • Interpreting units in formulas: novel units
        • Word problems with multiple units
          • Worked example: Rate problem
          • Multiple units word problem: road trip
          • Measurement word problem: running laps
          • Multiple units word problem: drug dosage (advanced)
      • Linear equations & graphs
        • Two-variable linear equations intro
          • Two-variable linear equations intro
          • Solutions to 2-variable equations
          • Worked example: solutions to 2-variable equations
          • Completing solutions to 2-variable equations
        • x-intercepts and y-intercepts
          • Intro to intercepts
          • x-intercept of a line
          • Intercepts from an equation
          • Intercepts from a table
        • Slope
          • Intro to slope
          • Positive & negative slope
          • Worked example: slope from graph
          • Worked example: slope from two points
          • Slope (more examples)
        • Horizontal & vertical lines
          • Slope of a horizontal line
          • Horizontal & vertical lines
        • Intro to slope-intercept form
          • Intro to slope-intercept form
          • Worked examples: slope-intercept intro
        • Graphing slope-intercept equations
          • Graph from slope-intercept equation
        • Writing slope-intercept equations
          • Slope-intercept equation from graph
          • Slope-intercept equation from slope & point
          • Slope-intercept equation from two points
          • Slope-intercept form problems
        • Point-slope form
          • Intro to point-slope form
          • Point-slope & slope-intercept equations
        • Standard form
          • Intro to linear equation standard form
          • Graphing a linear equation: 5x+2y=20
          • Converting from slope-intercept to standard form
        • Summary: Forms of two-variable linear equations
          • Slope from equation
          • Writing linear equations in all forms
      • Functions
        • Evaluating functions
          • What is a function?
          • Worked example: Evaluating functions from equation
          • Worked example: evaluating functions from graph
          • Worked example: evaluating expressions with function notation
        • Inputs and outputs of a function
          • Worked example: matching an input to a function's output (equation)
          • Worked example: matching an input to a function's output (graph)
          • Worked example: two inputs with the same output (graph)
        • Functions and equations
          • Equations vs. functions
          • Obtaining a function from an equation
        • Interpreting function notation
          • Function notation word problem: bank
          • Function notation word problem: beach
        • Introduction to the domain and range of a function
          • Intervals and interval notation
          • What is the domain of a function?
          • What is the range of a function?
          • Worked example: domain and range from graph
        • Determining the domain of a function
          • Domain of a radical function
          • Worked example: domain of algebraic functions
          • Worked example: determining domain word problem (real numbers)
          • Worked example: determining domain word problem (positive integers)
          • Worked example: determining domain word problem (all integers)
        • Recognizing functions
          • Recognizing functions from graph
          • Does a vertical line represent a function?
          • Recognizing functions from table
          • Recognizing functions from verbal description
          • Recognizing functions from verbal description word problem
        • Maximum and minimum points
          • Introduction to minimum and maximum points
          • Worked example: relative maxima and minima
        • Intervals where a function is positive, negative, increasing, or decreasing
          • Increasing, decreasing, positive or negative intervals
          • Worked example: positive & negative intervals
        • Interpreting features of graphs
          • Graph interpretation word problem: temperature
          • Graph interpretation word problem: basketball
        • Average rate of change
          • Introduction to average rate of change
          • Worked example: average rate of change from graph
          • Worked example: average rate of change from table
          • Worked example: average rate of change from equation
        • Average rate of change word problems
          • Average rate of change word problem: table
          • Average rate of change word problem: graph
          • Average rate of change word problem: equation
        • Unit review
      • Linear word problems
        • Interpreting linear functions and equations
          • Linear equations word problems: earnings
          • Linear equations word problems: volcano
          • Linear graphs word problems
          • Linear graphs word problem: cats
          • Linear equations word problem: marbles
          • Linear equations word problem: file transfer
        • Comparing linear functions
          • Comparing linear functions: equation vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions word problem: climb
          • Comparing linear functions word problem: walk
          • Comparing linear functions word problem: work
        • Constructing linear models for real-world relationships
          • Linear functions word problem: fuel
          • Linear functions word problem: pool
          • Linear functions word problem: iceberg
          • Linear functions word problem: paint
        • Linear models word problems
          • Linear models word problem: book
          • Linear models word problem: marbles
      • Sequences
        • Introduction to arithmetic sequences
          • Sequences intro
          • Intro to arithmetic sequences
          • Extending arithmetic sequences
          • Using arithmetic sequences formulas
          • Worked example: using recursive formula for arithmetic sequence
        • Constructing arithmetic sequences
          • Recursive formulas for arithmetic sequences
          • Explicit formulas for arithmetic sequences
          • Arithmetic sequence problem
          • Converting recursive & explicit forms of arithmetic sequences
        • Introduction to geometric sequences
          • Intro to geometric sequences
          • Extending geometric sequences
          • Using explicit formulas of geometric sequences
          • Using recursive formulas of geometric sequences
        • Constructing geometric sequences
          • Explicit & recursive formulas for geometric sequences
          • Converting recursive & explicit forms of geometric sequences
        • Modeling with sequences
          • Sequences word problems
        • General sequences
          • Evaluating sequences in recursive form
      • Systems of equations
        • Systems of equations overview
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: potato chips
          • Systems of equations with substitution: potato chips
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
        • Introduction to systems of equations
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Testing a solution to a system of equations
          • Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
          • Systems of equations with graphing: exact & approximate solutions
        • Equivalent systems of equations and the elimination method
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: x-4y=-18 & -x+3y=11
          • Systems of equations with elimination: potato chips
          • Systems of equations with elimination (and manipulation)
          • Why can we subtract one equation from the other in a system of equations?
          • Worked example: equivalent systems of equations
          • Worked example: non-equivalent systems of equations
        • Solving systems of equations with substitution
          • Systems of equations with substitution: potato chips
          • Systems of equations with substitution: -3x-4y=-2 & y=2x-5
        • Manipulating expressions with unknown variables
          • Manipulating expressions using structure
          • Manipulating expressions using structure (example 2)
        • Number of solutions to systems of equations
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
          • Solutions to systems of equations: consistent vs. inconsistent
          • Solutions to systems of equations: dependent vs. independent
          • Number of solutions to a system of equations
          • Number of solutions to a system of equations graphically
          • Number of solutions to a system of equations algebraically
          • How many solutions does a system of linear equations have if there are at least two?
        • Solving any system of linear equations
        • Systems of equations word problems
          • Age word problem: Imran
          • Age word problem: Ben & William
          • Age word problem: Arman & Diya
          • System of equations word problem: walk & ride
          • System of equations word problem: no solution
          • System of equations word problem: infinite solutions
          • Systems of equations with elimination: TV & DVD
          • Systems of equations with elimination: apples and oranges
          • Systems of equations with substitution: coins
          • Systems of equations with elimination: coffee and croissants
      • Inequalities (systems & graphs)
        • Checking solutions of two-variable inequalities
          • Testing solutions to inequalities
          • Testing solutions to systems of inequalities
        • Graphing two-variable inequalities
          • Intro to graphing two-variable inequalities
          • Graphing two-variable inequalities
          • Two-variable inequalities from their graphs
          • Intro to graphing systems of inequalities
          • Graphing systems of inequalities
        • Modeling with linear inequalities
          • Writing two-variable inequalities word problem
          • Solving two-variable inequalities word problem
          • Graphs of two-variable inequalities word problem
          • Interpreting two-variable inequalities word problem
          • Modeling with systems of inequalities
          • Writing systems of inequalities word problem
          • Solving systems of inequalities word problem
          • Graphs of systems of inequalities word problem
          • Analyzing structure with linear inequalities: fruits
          • Analyzing structure with linear inequalities: balls
      • Absolute value & piecewise functions
        • Graphs of absolute value functions
          • Shifting absolute value graphs
          • Scaling & reflecting absolute value functions: equation
          • Scaling & reflecting absolute value functions: graph
          • Graphing absolute value functions
        • Piecewise functions
          • Introduction to piecewise functions
          • Worked example: evaluating piecewise functions
          • Worked example: graphing piecewise functions
          • Worked example: domain & range of step function
          • Worked example: domain & range of piecewise linear functions
      • Rational exponents & radicals
        • Exponent properties review
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
        • Radicals
          • Intro to square roots
          • Understanding square roots
          • Intro to cube roots
          • Square root of decimal
          • 5th roots
        • Rational exponents intro
          • Intro to rational exponents
          • Rewriting roots as rational exponents
          • Exponential equation with rational answer
        • Properties of exponents (rational exponents)
          • Simplifying quotient of powers (rational exponents)
          • Simplifying mixed radical and exponential expressions
        • Advanced exponent & radical evaluation
          • Evaluating fractional exponents
          • Evaluating fractional exponents: negative unit-fraction
          • Evaluating fractional exponents: fractional base
          • Evaluating quotient of fractional exponents
          • Evaluating mixed radicals and exponents
        • Simplifying square roots
          • Simplifying square roots
          • Simplifying square roots (variables)
          • Simplifying square-root expressions
        • Simplifying radicals (higher-index roots)
          • Simplifying higher-index roots
          • Simplifying cube root expressions
          • Simplifying cube root expressions (two variables)
          • Simplifying higher-index root expressions
      • Exponential growth & decay
        • Exponential vs. linear growth
          • Intro to exponential functions
          • Exponential vs. linear growth
          • Exponential vs. linear models: verbal
          • Exponential vs. linear models: table
        • Exponential expressions
          • Exponential expressions word problems (numerical)
          • Initial value & common ratio of exponential functions
          • Exponential expressions word problems (algebraic)
          • Interpreting exponential expression word problem
        • Graphs of exponential growth
          • Exponential function graph
          • Graphs of exponential growth
        • Exponential vs. linear growth over time
          • Exponential vs. linear growth over time
        • Exponential decay
          • Exponential decay intro
          • Graphing exponential growth & decay
          • Exponential growth & decay word problems
        • Exponential functions from tables & graphs
          • Writing exponential functions
          • Writing exponential functions from tables
          • Writing exponential functions from graphs
          • Analyzing tables of exponential functions
          • Analyzing graphs of exponential functions
          • Analyzing graphs of exponential functions: negative initial value
          • Modeling with basic exponential functions word problem
      • Polynomials
        • Intro to polynomials
          • Polynomials intro
          • The parts of polynomial expressions
          • Evaluating polynomials
          • Simplifying polynomials
        • Adding & subtracting polynomials
          • Adding polynomials
          • Subtracting polynomials
          • Polynomial subtraction & set closure
          • Adding & subtracting multiple polynomials
        • Adding and subtracting polynomials: two variables
          • Adding polynomials: two variables
          • Subtracting polynomials: two variables
          • Subtracting polynomials: two variables challenge
          • Finding an error in polynomial subtraction
          • Polynomials review
        • Multiplying monomials
          • Multiplying monomials
          • Multiplying monomials to find area: two variables
          • Multiplying monomials to find area
          • Multiplying monomials challenge
        • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials: area model
          • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials challenge
        • Multiplying binomials
          • Multiplying binomials: area model
          • Multiplying binomials intro
          • Multiplying binomials
        • Multiplying binomials by polynomials
          • Multiplying binomials by polynomials
          • Multiplying binomials by polynomials: area model
          • Multiplying binomials by polynomials challenge
        • Special products of binomials
          • Special products of the form (x+a)(x-a)
          • Squaring binomials of the form (x+a)²
          • Special products of the form (ax+b)(ax-b)
          • Squaring binomials of the form (ax+b)²
          • Special products of binomials: two variables
          • More examples of special products
        • Polynomials word problems
          • Polynomial multiplication word problem
          • Polynomial word problem: rectangle and circle area
          • Polynomial word problem: total value of bills
          • Polynomial word problem: area of a window
      • Factorization
        • Introduction to factorization
          • Intro to factors & divisibility
        • Factoring monomials
          • Monomial factorization
          • Worked example: finding the missing monomial factor
          • Worked example: finding missing monomial side in area model
          • Greatest common factor of monomials
        • Factoring polynomials by taking common factors
          • Factoring with the distributive property
          • Factoring binomials: common factor
          • Factoring polynomials: common factor
          • Factoring polynomials: common factor area model
          • Factoring polynomials: common binomial factor
        • Evaluating expressions with unknown variables
          • Worked example: evaluating expressions using structure
          • Worked example: evaluating expressions using structure (more examples)
        • Factoring quadratics intro
          • Factoring quadratics as (x+a)(x+b)
          • Factoring quadratics as (x+a)(x+b) (example 2)
          • More examples of factoring quadratics as (x+a)(x+b)
        • Factoring quadratics by grouping
          • Intro to grouping
          • Factoring quadratics by grouping
        • Factoring polynomials with quadratic forms
          • Factoring quadratics: common factor + grouping
          • Factoring quadratics: negative common factor + grouping
          • Factoring two-variable quadratics
          • Factoring two-variable quadratics: rearranging
          • Factoring two-variable quadratics: grouping
        • Factoring quadratics: Difference of squares
          • Difference of squares intro
          • Factoring difference of squares: leading coefficient ≠ 1
          • Factoring difference of squares: analyzing factorization
          • Factoring difference of squares: missing values
          • Factoring difference of squares: shared factors
        • Factoring quadratics: Perfect squares
          • Perfect square factorization intro
          • Factoring perfect squares
          • Identifying perfect square form
          • Factoring perfect squares: common factor
          • Factoring perfect squares: negative common factor
          • Factoring perfect squares: missing values
          • Factoring perfect squares: shared factors
        • Strategy in factoring quadratics
          • Strategy in factoring quadratics (part 1 of 2)
          • Strategy in factoring quadratics (part 2 of 2)
        • Factoring polynomials with special product forms
          • Factoring perfect squares: 4th degree polynomial
          • Factoring difference of squares: two variables
          • Factoring difference of squares: two variables (example 2)
      • Quadratics
        • Parabolas intro
          • Parabolas intro
        • Quadratic factored form
          • Zero product property
          • Graphing quadratics in factored form
          • Quadratic word problems (factored form)
        • Solving quadratics by taking square roots
          • Solving quadratics by taking square roots
          • Solving quadratics by taking square roots examples
          • Solving quadratics by taking square roots: strategy
          • Quadratics by taking square roots: with steps
        • Quadratic vertex form
          • Vertex form introduction
          • Graphing quadratics: vertex form
          • Quadratic word problems (vertex form)
        • Solving quadratics by factoring
          • Solving quadratics by factoring
          • Solving quadratics by factoring: leading coefficient ≠ 1
          • Solving quadratics using structure
          • Quadratic equations word problem: triangle dimensions
          • Quadratic equations word problem: box dimensions
        • Completing the square
          • Completing the square
          • Worked example: Completing the square (intro)
          • Worked example: Rewriting expressions by completing the square
          • Worked example: Solving equations by completing the square
          • Worked example: completing the square (leading coefficient ≠ 1)
          • Solving quadratics by completing the square: no solution
        • The quadratic formula
          • The quadratic formula
          • Using the quadratic formula
          • Worked example: quadratic formula
          • Worked example: quadratic formula (example 2)
          • Worked example: quadratic formula (negative coefficients)
          • Using the quadratic formula: number of solutions
          • Proof of the quadratic formula
        • Quadratic standard form
          • Finding the vertex of a parabola in standard form
          • Graphing quadratics: standard form
          • Quadratic word problem: ball
        • Features & forms of quadratic functions
          • Forms & features of quadratic functions
          • Worked examples: Forms & features of quadratic functions
          • Vertex & axis of symmetry of a parabola
          • Finding features of quadratic functions
          • Comparing features of quadratic functions
          • Comparing maximum points of quadratic functions
        • Transforming parabolas
          • Intro to parabola transformations
          • Shifting parabolas
          • Scaling & reflecting parabolas
        • Unit review
      • Irrational numbers
        • Irrational numbers
          • Intro to rational & irrational numbers
          • Classifying numbers: rational & irrational
        • Sums and products of rational and irrational numbers
          • Proof: sum & product of two rationals is rational
          • Proof: product of rational & irrational is irrational
          • Proof: sum of rational & irrational is irrational
          • Sums and products of irrational numbers
          • Worked example: rational vs. irrational expressions
          • Worked example: rational vs. irrational expressions (unknowns)
        • Proofs concerning irrational numbers
          • Proof: √2 is irrational
          • Proof: square roots of prime numbers are irrational
          • Proof: there's an irrational number between any two rational numbers
    • Algebra II
      • Functions
        • Combining functions
          • Adding functions
          • Subtracting functions
          • Multiplying functions
          • Dividing functions
        • Composing functions
          • Intro to composing functions
          • Evaluating composite functions
          • Evaluating composite functions: using tables
          • Evaluating composite functions: using graphs
          • Finding composite functions
          • Evaluating composite functions (advanced)
        • Modeling situations by combining and composing functions
          • Modeling with function combination
          • Modeling with composite functions
          • Modeling with composite functions (example 2)
        • Shifting functions
          • Shifting functions
          • Graphing shifted functions
        • Stretching functions
          • Shifting & reflecting functions
          • Compressing functions
          • Reflecting & compressing functions
        • Introduction to inverses of functions
          • Intro to inverse functions
          • Inputs & outputs of inverse functions
          • Graphing the inverse of a linear function
        • Finding inverse functions
          • Finding inverse functions: linear
          • Finding inverse functions: quadratic
          • Finding inverse functions: quadratic (example 2)
          • Finding inverse functions: radical
          • Finding inverse functions: rational
        • Verifying that functions are inverses
          • Verifying inverse functions by composition
          • Verifying inverse functions by composition: not inverse
        • Determining whether a function is invertible
          • Determining if a function is invertible
          • Restricting domains of functions to make them invertible
      • Complex numbers
        • What are the imaginary numbers?
          • Intro to the imaginary numbers
          • Simplifying roots of negative numbers
          • Powers of the imaginary unit
          • i as the principal root of -1
        • What are the complex numbers?
          • Intro to complex numbers
        • The complex plane
          • Plotting numbers on the complex plane
        • Adding and subtracting complex numbers
          • Adding complex numbers
          • Subtracting complex numbers
        • Multiplying complex numbers
          • Multiplying complex numbers
      • Arithmetic with polynomials
        • Adding and subtracting polynomials
          • Adding & subtracting multiple polynomials
          • Subtracting polynomials: two variables
        • Multiplying polynomials
          • Multiplying binomials by polynomials
          • More examples of special products
        • Long division of polynomials
          • Intro to long division of polynomials
          • Dividing polynomials: long division
        • Synthetic division of polynomials
          • Intro to polynomial synthetic division
          • Dividing polynomials: synthetic division
          • Why synthetic division works
        • Practice dividing polynomials with remainders
          • Dividing polynomials with remainders: monomial divisors
          • Dividing polynomials with remainders: monomial divisors (example 2)
          • Dividing polynomials with remainders
        • Polynomial Remainder Theorem
          • Intro to the Polynomial Remainder Theorem
          • Using the Polynomial Remainder Theorem: finding remainders
          • Using the Polynomial Remainder Theorem: checking factors
          • Using the Polynomial Remainder Theorem: finding coefficients
          • Proof of the Polynomial Remainder Theorem
      • Polynomials
        • The binomial theorem
          • Intro to the Binomial Theorem
          • Pascal's triangle and binomial expansion
          • Expanding binomials
          • Expanding binomials w/o Pascal's triangle
        • Understanding the binomial theorem
          • Binomial expansion & combinatorics
          • Pascal's triangle & combinatorics
          • Binomial expansion & combinatorics (old)
        • Factoring polynomials - Quadratic forms
          • Factoring quadratics as (x+a)(x+b) (example 2)
          • More examples of factoring quadratics as (x+a)(x+b)
          • Intro to grouping
          • Factoring quadratics by grouping
          • Factoring quadratics: common factor + grouping
          • Factoring quadratics: negative common factor + grouping
          • Factoring two-variable quadratics
          • Factoring two-variable quadratics: rearranging
          • Factoring two-variable quadratics: grouping
        • Factoring polynomials - Special product forms
          • Factoring difference of squares: two variables
          • Factoring difference of squares: leading coefficient ≠ 1
          • Factoring perfect squares: negative common factor
          • Factoring perfect squares
          • Factoring perfect squares: 4th degree polynomial
          • Factoring difference of squares: two variables (example 2)
        • Advanced polynomial factorization methods
          • Factoring higher-degree polynomials
          • Factoring sum of cubes
          • Factoring difference of cubes
        • Proving polynomial identities
          • Analyzing polynomial identities
        • Polynomial identities with complex numbers
          • Complex numbers & sum of squares factorization
          • Factoring sum of squares
        • Quadratic equations with complex numbers
          • Solving quadratic equations: complex roots
        • The Fundamental Theorem of Algebra
          • The Fundamental theorem of Algebra
          • Quadratics & the Fundamental Theorem of Algebra
          • Number of possible real roots of a polynomial
        • Finding zeros of polynomials
          • Finding zeros of polynomials (1 of 2)
          • Finding zeros of polynomials (2 of 2)
          • Finding zeros of polynomials (example 2)
        • Zeros of polynomials and their graphs
          • Zeros of polynomials & their graphs
        • End behavior of polynomial functions
          • Intro to end behavior of polynomials
          • End behavior of functions & their graphs
        • Graphs of polynomials
        • Introduction to symmetry of functions
          • Intro to function symmetry
          • Graphical symmetry of functions
          • Even/odd functions & numbers
        • Symmetry of polynomial functions
          • Even & odd polynomials
      • Radical relationships
        • Solving square-root equations
          • Intro to square-root equations & extraneous solutions
          • Solving square-root equations (basic)
          • Solving square-root equations: one solution
          • Solving square-root equations: two solutions
          • Solving square-root equations: no solution
        • Extraneous solutions of radical equations
          • Extraneous solutions of radical equations
          • Extraneous solutions of radical equations (example 2)
        • Solving cube-root equations
          • Solving cube-root equations
        • Domain of radical functions
          • Domain of a radical function
        • Graphs of radical functions
          • Transforming the square-root function
          • Graphs of square-root functions
          • Square-root functions & their graphs
      • Rational relationships
        • Simplifying rational expressions
          • Intro to rational expression simplification
          • Simplifying rational expressions: common monomial factors
          • Simplifying rational expressions: common binomial factors
          • Simplifying rational expressions: opposite common binomial factors
          • Simplifying rational expressions: grouping
          • Simplifying rational expressions: higher degree terms
          • Simplifying rational expressions: two variables
          • Simplifying rational expressions (old video)
        • Multiplying & dividing rational expressions
          • Multiplying & dividing rational expressions: monomials
          • Multiplying rational expressions
          • Dividing rational expressions
          • Multiplying rational expressions: multiple variables
          • Dividing rational expressions: unknown expression
        • Adding & subtracting rational expressions
          • Adding & subtracting rational expressions: like denominators
          • Intro to adding rational expressions with unlike denominators
          • Adding rational expression: unlike denominators
          • Subtracting rational expressions: unlike denominators
          • Least common multiple
          • Least common multiple: repeating factors
          • Subtracting rational expressions: factored denominators
          • Least common multiple of polynomials
          • Subtracting rational expressions
        • Nested fractions
          • Nested fractions
        • Solving rational equations
          • Equations with one rational expression
          • Equations with one rational expression (advanced)
          • Equations with rational expressions
          • Equations with rational expressions (example 2)
        • Direct and inverse variation
          • Intro to direct & inverse variation
          • Recognizing direct & inverse variation
          • Recognizing direct & inverse variation: table
          • Inverse variation word problem: string vibration
        • End behavior of rational functions
          • End behavior of rational functions
        • Discontinuities of rational functions
          • Discontinuities of rational functions
          • Analyzing vertical asymptotes of rational functions
        • Graphs of rational functions
          • Graphing rational functions according to asymptotes
          • Graphs of rational functions: y-intercept
          • Graphs of rational functions: horizontal asymptote
          • Graphs of rational functions: vertical asymptotes
          • Graphs of rational functions: zeros
          • Graphs of rational functions (old example)
          • Graphing rational functions 1
          • Graphing rational functions 2
          • Graphing rational functions 3
          • Graphing rational functions 4
        • Modeling with rational functions
          • Analyzing structure word problem: pet store (1 of 2)
          • Analyzing structure word problem: pet store (2 of 2)
          • Rational equations word problem: combined rates
          • Rational equations word problem: combined rates (example 2)
          • Rational equations word problem: eliminating solutions
          • Reasoning about unknown variables
          • Reasoning about unknown variables: divisibility
          • Structure in rational expression
      • Exponential growth & decay
        • Equivalent forms of exponential expressions
          • Simplifying exponential expressions
          • Rewriting exponential expressions
        • Solving exponential equations using properties of exponents
          • Solving exponential equations using exponent properties
          • Solving exponential equations using exponent properties (advanced)
        • Interpreting the rate of change of exponential models
          • Interpreting change in exponential models
          • Interpreting time in exponential models
        • Constructing exponential models according to rate of change
          • Constructing exponential models
          • Constructing exponential models: half life
          • Constructing exponential models: percent change
          • Constructing exponential models (old example)
        • Advanced interpretation of exponential models
          • Interpreting change in exponential models: with manipulation
          • Interpreting change in exponential models: changing units
          • Structure in exponential expression
        • Distinguishing between linear and exponential growth
          • Linear vs. exponential growth: from data
          • Linear vs. exponential growth: from data (example 2)
          • Comparing growth of exponential & quadratic models
      • Exponentials & logarithms
        • Introduction to logarithms
          • Intro to logarithms
          • Evaluating logarithms (advanced)
          • Relationship between exponentials & logarithms
          • Relationship between exponentials & logarithms: graphs
          • Relationship between exponentials & logarithms: tables
        • The constant e and the natural logarithm
          • 𝑒 and compound interest
          • 𝑒 as a limit
          • Evaluating natural logarithm with calculator
        • Properties of logarithms
          • Intro to logarithm properties (1 of 2)
          • Intro to logarithm properties (2 of 2)
          • Using the logarithmic product rule
          • Using the logarithmic power rule
          • Using the properties of logarithms: multiple steps
          • Proof of the logarithm product rule
          • Proof of the logarithm quotient and power rules
        • The change of base formula for logarithms
          • Evaluating logarithms: change of base rule
          • Using the logarithm change of base rule
          • Proof of the logarithm change of base rule
        • Solving exponential equations with logarithms
          • Solving exponential equations using logarithms: base-10
          • Solving exponential equations using logarithms: base-2
        • Solving exponential models
          • Exponential model word problem: medication dissolve
          • Exponential model word problem: bacteria growth
        • Graphs of exponential functions
          • Transforming exponential graphs
          • Transforming exponential graphs (example 2)
          • Graphing exponential functions
          • Exponential function graph
          • Graphs of exponential functions (old example)
        • Graphs of logarithmic functions
          • Graphs of logarithms & exponentials
          • Graphing basic logarithmic functions
          • Graphs of logarithmic functions
        • Logarithmic scale (optional)
          • Logarithmic scale (with Vi Hart)
          • Logarithmic scale
          • Richter scale
          • Benford's law (with Vi Hart, 1 of 2)
          • Benford's law (with Vi Hart, 2 of 2)
      • Trigonometry
        • Introduction to radians
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
          • Radian angles & quadrants
        • The unit circle definition of sine, cosine, and tangent
          • Unit circle
          • The trig functions & right triangle trig ratios
        • The graphs of sine, cosine, and tangent
          • Graph of y=sin(x)
          • Intersection points of y=sin(x) and y=cos(x)
          • Graph of y=tan(x)
        • Basic trigonometric identities
          • Sine & cosine identities: symmetry
          • Tangent identities: symmetry
          • Sine & cosine identities: periodicity
          • Tangent identities: periodicity
        • Trigonometric values of special angles
          • Trig values of special angles
        • The Pythagorean identity
          • Proof of the Pythagorean trig identity
          • Using the Pythagorean trig identity
        • Introduction to amplitude, midline, and extrema of sinusoidal functions
          • Features of sinusoidal functions
        • Finding amplitude and midline of sinusoidal functions from their formulas
          • Amplitude & period of sinusoidal functions from equation
        • Period of sinusoidal functions
          • Amplitude & period of sinusoidal functions from equation
        • Graphing sinusoidal functions
          • Transforming sinusoidal graphs: vertical stretch & horizontal reflection
          • Transforming sinusoidal graphs: vertical & horizontal stretches
        • Constructing sinusoidal functions
          • Sinusoidal function from graph
          • Trig word problem: modeling daily temperature
          • Trig word problem: modeling annual temperature
      • Advanced equations & functions
        • Solving equations by graphing
          • Interpreting equations graphically
          • Interpreting equations graphically (example 2)
          • Solving equations graphically (1 of 2)
          • Solving equations graphically (2 of 2)
        • Systems of quadratic equations
          • Quadratic systems: graphical solution
          • Quadratic systems: algebraic solution
          • Quadratic systems: no solution
          • Quadratic systems: both variables are squared
          • Quadratic systems: a line and a circle
        • Interpreting the symmetry of algebraic models
          • Symmetry of algebraic models
        • Interpreting the end behavior of algebraic models
          • End behavior of algebraic models
        • Interpreting the periodicity of algebraic models
          • Periodicity of algebraic models
        • Comparing features of functions
          • Comparing functions: x-intercepts
          • Comparing functions: shared features
          • Comparing maximum points of quadratic functions
      • Series
        • Arithmetic sequences
          • Sequences intro
          • Intro to arithmetic sequences
          • Explicit formulas for arithmetic sequences
          • Sequences word problem: growth pattern
        • Basic sigma notation
          • Summation notation
        • Finite arithmetic series
          • Arithmetic series intro
          • Arithmetic series formula
          • Worked example: arithmetic series (sigma notation)
          • Worked example: arithmetic series (sum expression)
          • Worked example: arithmetic series (recursive formula)
          • Proof of finite arithmetic series formula
        • Geometric sequences
          • Intro to geometric sequences
          • Explicit & recursive formulas for geometric sequences
          • Converting recursive & explicit forms of geometric sequences
        • Finite geometric series
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
          • Finite geometric series formula justification
        • Finite geometric series applications
          • Finite geometric series word problem: social media
          • Finite geometric series word problem: mortgage
      • Modeling
        • Modeling with one-variable equations and inequalities
          • Rational equation word problem
          • Quadratic inequality word problem
          • Exponential equation word problem
        • Manipulating formulas
          • Manipulating formulas: perimeter
          • Manipulating formulas: area
          • Manipulating formulas: temperature
      • Conic sections
        • Introduction to conic sections
          • Intro to conic sections
        • The features of a circle
          • Graphing circles from features
          • Features of a circle from its graph
        • Standard equation of a circle
          • Features of a circle from its standard equation
          • Graphing a circle from its standard equation
          • Writing standard equation of a circle
        • Expanded equation of a circle
          • Features of a circle from its expanded equation
        • Focus and directrix of a parabola
          • Intro to focus & directrix
          • Equation of a parabola from focus & directrix
          • Focus & directrix of a parabola from equation
    • Basic geometry
      • Lines
        • Lines, line segments, and rays
          • Terms & labels in geometry
          • Lines, line segments, & rays
        • Parallel and perpendicular
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines
          • Drawing parallel line segments
      • Angles
        • Angle introduction
          • Angles: introduction
          • Naming angles
        • Measuring and drawing angles
          • Measuring angles in degrees
          • Measuring angles using a protractor
          • Measuring angles using a protractor 2
          • Constructing angles
        • Angles in circles
          • Angle measurement & circle arcs
          • Measuring angles with a circular protractor
          • Angles in circles word problem
        • Angle types
          • Acute, right, & obtuse angles
          • Recognizing angles
          • Drawing acute, right and obtuse angles
          • Identifying an angle
        • Vertical, complementary, and supplementary angles
          • Complementary & supplementary angles
          • Vertical angles
        • Angles between intersecting lines
          • Angles, parallel lines, & transversals
          • Parallel & perpendicular lines
          • Missing angles with a transversal
      • Shapes
        • Properties of shapes
          • Cousin Fal's shape collection
          • Recognizing shapes
        • Quadrilaterals
          • Intro to quadrilateral
          • Quadrilateral properties
          • Quadrilateral types
          • Classifying quadrilaterals
          • Kites as a geometric shape
        • Triangles
          • Classifying triangles
          • Classifying triangles by angles
          • Worked example: Classifying triangles
        • Classifying geometric shapes
          • Classifying shapes by line and angles types
          • Classifying shapes by lines and angles
        • Triangle angles
          • Angles in a triangle sum to 180° proof
          • Triangle exterior angle example
          • Worked example: Triangle angles (intersecting lines)
          • Worked example: Triangle angles (diagram)
          • Triangle angle challenge problem
          • Triangle angle challenge problem 2
        • Geometric solids (3D shapes)
          • Counting faces and edges of 3D shapes
          • Recognizing common 3D shapes
      • Coordinate plane
        • Coordinate plane: quadrant 1
          • Introduction to the coordinate plane
          • Coordinate plane: graphing points
          • Coordinate plane graphing word problem
        • Coordinate plane: 4 quadrants
          • Points on the coordinate plane examples
          • Plotting a point (ordered pair)
          • Finding the point not graphed
          • Coordinate plane word problem examples
        • Quadrants on the coordinate plane
          • Quadrants of the coordinate plane
          • Points and quadrants example
        • Reflecting points on coordinate plane
          • Reflecting points in the coordinate plane
        • Quadrilaterals on the coordinate plane
          • Coordinates of a missing vertex
          • Area of a parallelogram on the coordinate plane
          • Dimensions of a rectangle from coordinates
          • Example of shapes on a coordinate plane
          • Coordinates of rectangle example
          • Plotting corners of a rectangle
        • Drawing polygons in the coordinate plane
          • Drawing a quadrilateral on the coordinate plane example
      • Area and perimeter
        • Count unit squares to find area
          • Intro to area and unit squares
          • Measuring rectangles with different unit squares
          • Measuring area with partial unit squares
          • Creating rectangles with a given area 1
          • Creating rectangles with a given area 2
        • Area of rectangles
          • Transitioning from unit squares to area formula
          • Finding missing side when given area
          • Counting unit squares to find area formula
        • Perimeter
          • Perimeter: introduction
          • Perimeter of a shape
          • Finding perimeter when a side length is missing
          • Finding missing side length when given perimeter
        • Area of parallelograms
          • Area of a parallelogram
          • Finding height of a parallelogram
        • Area of triangles
          • Area of a triangle
          • Finding area of triangle
          • Triangle missing side example
        • Area of trapezoids & composite figures
          • Area of trapezoids
          • Area of kites
          • Finding area by rearranging parts
          • Area of composite shapes
        • Area and circumference of circles
          • Radius, diameter, circumference & π
          • Labeling parts of a circle
          • Relating circumference and area
          • Area of a circle
          • Finding circumference of a circle when given the area
          • Partial circle area and arc length
      • Volume and surface area
        • Volume of rectangular prisms
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
          • Volume of a rectangular prism
          • How volume changes from changing dimensions
        • Volume with fractions
          • Volume with fractional cubes
          • Volume of a rectangular prism: fractional dimensions
          • Volume of a rectangular prism: word problem
        • Surface area
          • Intro to nets of polyhedra
          • Surface area using a net: triangular prism
          • Surface area of a box (cuboid)
          • Surface area of a box using nets
          • Surface area using a net: rectangular prism
          • Surface area word problem example
        • Volume of cones, cylinders, and spheres
          • Cylinder volume & surface area
          • Volume of a cone
          • Volume of a sphere
      • Pythagorean theorem
        • Pythagorean theorem
          • Intro to the Pythagorean theorem
          • Pythagorean theorem example
          • Use Pythagorean theorem to find area of an isosceles triangle
        • Pythagorean theorem application
          • Use Pythagorean theorem to find area of an isosceles triangle
          • Pythagorean theorem word problem: carpet
          • Pythagorean theorem word problem: fishing boat
          • Pythagorean theorem in 3D
        • Pythagorean theorem and distance between points
          • Finding distance with Pythagorean theorem
          • Distance formula
        • Pythagorean theorem proofs
          • Garfield's proof of the Pythagorean theorem
          • Bhaskara's proof of the Pythagorean theorem
          • Pythagorean theorem proof using similarity
          • Another Pythagorean theorem proof
      • Transformations, congruence, and similarity
        • Line of symmetry
          • Intro to reflective symmetry
          • Identifying symmetrical figures
        • Transformations intro
          • Rigid transformations intro
          • Identifying type of transformation
        • Translations
          • Performing translations
          • Translating shapes
          • Determining translations
        • Rotations
          • Performing rotations
          • Determining rotations
        • Reflections
          • Performing reflections
          • Reflecting shapes: diagonal line of reflection
          • Determining reflections (advanced)
        • Properties & definitions of transformations
          • Congruent shapes & transformations
        • Dilations
          • Performing dilations
          • Dilating shapes: expanding
          • Dilating shapes: shrinking
          • Side lengths after dilation
        • Congruence and similarity
          • Congruent shapes & transformations
          • Non-congruent shapes & transformations
          • Dilating shapes: shrinking
          • Similar shapes & transformations
          • Side lengths after dilation
    • High school geometry
      • Geometry foundations
        • Intro to Euclidean geometry
          • Euclid as the father of geometry
          • Terms & labels in geometry
          • Lines, line segments, & rays
          • Specifying planes in three dimensions
          • Geometric definitions example
        • Angles
          • Measuring angles in degrees
          • Measuring angles using a protractor
          • Measuring angles using a protractor 2
          • Acute, right, & obtuse angles
          • Vertical angles
          • Equation practice with vertical angles
          • Complementary & supplementary angles
          • Equation practice with complementary angles
          • Equation practice with supplementary angles
          • Angles, parallel lines, & transversals
          • Parallel & perpendicular lines
          • Missing angles with a transversal
          • Measures of angles formed by a transversal
          • Proving angles are congruent
        • Polygons
          • Angles in a triangle sum to 180° proof
          • Triangle exterior angle example
          • Worked example: Triangle angles (intersecting lines)
          • Worked example: Triangle angles (diagram)
          • Triangle angle challenge problem 2
          • Sum of interior angles of a polygon
          • Sum of the exterior angles of a polygon
          • Triangle inequality theorem
          • Intro to quadrilateral
          • Quadrilateral properties
          • Kites as a geometric shape
          • Quadrilateral types
          • Proof: Opposite sides of a parallelogram
          • Proof: Opposite angles of a parallelogram
        • Area
          • Perimeter & area
          • Area of a triangle
          • Area of a parallelogram
          • Area of trapezoids
          • Area of kites
          • Area of a triangle on a grid
          • Area of a quadrilateral on a grid
          • Area of composite shapes
          • Perimeter & area of composite shapes
          • Challenge problems: perimeter & area
          • Radius, diameter, circumference & π
          • Area of a circle
          • Radius & diameter from circumference
          • Area of a circle intuition
          • Intro to nets of polyhedra
          • Surface area using a net: rectangular prism
      • Transformations
        • Introduction to rigid transformations
          • Rigid transformations intro
          • Dilations intro
          • Identifying transformations
        • Translations
          • Translating points
          • Determining translations
          • Translating shapes
          • Translation challenge problem
        • Rotations
          • Rotating points
          • Determining rotations
          • Rotating shapes
        • Reflections
          • Reflecting points
          • Determining reflections
          • Determining reflections (advanced)
          • Reflecting shapes
          • Reflecting shapes: diagonal line of reflection
        • Rigid transformations overview
          • Finding measures using rigid transformations
          • Rigid transformations: preserved properties
          • Mapping shapes
        • Dilations
          • Dilating points
          • Dilations: scale factor
          • Dilations: center
          • Dilating shapes: expanding
          • Dilating shapes: shrinking
          • Dilations and properties
        • Properties & definitions of transformations
          • Sequences of transformations
          • Defining transformations
          • Proofs with transformations
          • Identifying type of transformation
        • Symmetry
          • Intro to reflective symmetry
          • Intro to rotational symmetry
          • Finding a quadrilateral from its symmetries
          • Finding a quadrilateral from its symmetries (example 2)
      • Congruence
        • Transformations & congruence
          • Congruent shapes & transformations
          • Non-congruent shapes & transformations
        • Triangle congruence
          • Congruent triangles & the SSS postulate/criterion
          • Triangle congruence postulates/criteria
          • Why SSA isn't a congruence postulate/criterion
          • Determining congruent triangles
        • Theorems concerning triangle properties
          • Angles in a triangle sum to 180° proof
          • Proofs concerning isosceles triangles
          • Proofs concerning equilateral triangles
        • Working with triangles
          • Corresponding angles in congruent triangles
          • Isosceles & equilateral triangles problems
          • Finding angles in isosceles triangles
          • Finding angles in isosceles triangles (example 2)
        • Theorems concerning quadrilateral properties
          • Proof: Opposite sides of a parallelogram
          • Proof: Diagonals of a parallelogram
          • Proof: Opposite angles of a parallelogram
          • Proof: The diagonals of a kite are perpendicular
          • Proof: Rhombus diagonals are perpendicular bisectors
          • Proof: Rhombus area
        • Proofs of general theorems that use triangle congruence
          • Geometry proof problem: midpoint
          • Geometry proof problem: congruent segments
          • Geometry proof problem: squared circle
        • Constructing bisectors of lines & angles
          • Geometric constructions: perpendicular bisector
          • Geometric constructions: perpendicular line
          • Geometric constructions: angle bisector
      • Similarity
        • Definitions of similarity
          • Similar shapes & transformations
        • Introduction to triangle similarity
          • Intro to triangle similarity
          • Triangle similarity postulates/criteria
          • Determining similar triangles
          • Proving slope is constant using similarity
        • Solving similar triangles
          • Solving similar triangles
          • Solving similar triangles: same side plays different roles
        • Angle bisector theorem
          • Intro to angle bisector theorem
          • Using the angle bisector theorem
        • Solving problems with similar & congruent triangles
          • Using similar & congruent triangles
          • Challenging similarity problem
        • Solving modeling problems with similar & congruent triangles
          • Geometry word problem: the golden ratio
          • Geometry word problem: Earth & Moon radii
          • Geometry word problem: a perfect pool shot
      • Right triangles & trigonometry
        • Pythagorean theorem
          • Intro to the Pythagorean theorem
          • Intro to the Pythagorean theorem 2
          • Pythagorean theorem word problem: fishing boat
          • Pythagorean theorem example
          • Pythagorean theorem word problem: carpet
          • Pythagorean theorem with isosceles triangle
          • Introduction to the Pythagorean theorem
          • Pythagorean theorem II
        • Pythagorean theorem proofs
          • Garfield's proof of the Pythagorean theorem
          • Bhaskara's proof of the Pythagorean theorem
          • Pythagorean theorem proof using similarity
          • Another Pythagorean theorem proof
        • Special right triangles
          • Special right triangles intro (part 1)
          • Special right triangles intro (part 2)
          • 30-60-90 triangle example problem
          • Special right triangles proof (part 1)
          • Special right triangles proof (part 2)
          • Area of a regular hexagon
        • Introduction to the trigonometric ratios
          • Intro to the trigonometric ratios
          • Trigonometric ratios in right triangles
        • Solving for a side in a right triangle using the trigonometric ratios
          • Solving for a side in right triangles with trigonometry
        • Solving for an angle in a right triangle using the trigonometric ratios
        • Modeling with right triangles
          • Right triangle word problem
        • Trigonometric ratios & similarity
          • Triangle similarity & the trigonometric ratios
          • Trig challenge problem: verify identities
          • Trig challenge problem: trig values & side ratios
        • Sine & cosine of complementary angles
          • Sine & cosine of complementary angles
          • Using complementary angles
          • Trig word problem: complementary angles
        • Law of sines
          • Solving for a side with the law of sines
          • Solving for an angle with the law of sines
          • Proof of the law of sines
        • Law of cosines
          • Solving for a side with the law of cosines
          • Solving for an angle with the law of cosines
          • Proof of the law of cosines
        • Solving general triangles
          • Trig word problem: stars
      • Solid geometry
        • Solid geometry intro
          • Volume of triangular prism & cube
          • Volume of a cone
          • Cylinder volume & surface area
          • Volume of a sphere
        • Density
          • Density word problem: blimp
        • 2D vs. 3D objects
          • Slicing a rectangular pyramid
          • Ways to cross-section a cube
          • Rotating 2D shapes in 3D
      • Analytic geometry
        • Distance and midpoints
          • Distance formula
          • Midpoint formula
        • Dividing line segments
          • Dividing line segments: graphical
          • Dividing line segments
        • Problem solving with distance on the coordinate plane
          • Area of trapezoid on the coordinate plane
          • Points inside/outside/on a circle
          • Challenge problem: Points on two circles
          • Coordinate plane word problem
          • Classifying quadrilaterals on the coordinate plane
        • Parallel & perpendicular lines on the coordinate plane
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines from graph
        • Equations of parallel & perpendicular lines
          • Parallel lines from equation
          • Parallel lines from equation (example 2)
          • Parallel lines from equation (example 3)
          • Perpendicular lines from equation
          • Writing equations of perpendicular lines
          • Writing equations of perpendicular lines (example 2)
          • Proof: parallel lines have the same slope
          • Proof: perpendicular lines have negative reciprocal slope
      • Circles
        • Circle basics
          • Circles glossary
          • Radius, diameter, circumference & π
          • Proof: all circles are similar
        • Arc measure
          • Intro to arc measure
          • Finding arc measures
          • Finding arc measures with equations
        • Arc length (degrees)
          • Arc length from subtended angle
          • Subtended angle from arc length
        • Introduction to radians
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
        • Arc length (radians)
          • Arc length as fraction of circumference
          • Arc length from subtended angle: radians
        • Sectors
          • Area of a sector
        • Inscribed angles
          • Inscribed angles
          • Inscribed angle theorem proof
        • Inscribed shapes problem solving
          • Proof: Right triangles inscribed in circles
          • Inscribed shapes: find diameter
          • Inscribed shapes: angle subtended by diameter
          • Inscribed shapes: find inscribed angle
          • Inscribed quadrilaterals proof
          • Solving inscribed quadrilaterals
          • Proof: radius is perpendicular to a chord it bisects
          • Proof: perpendicular radius bisects chord
          • Area of inscribed equilateral triangle
        • Properties of tangents
          • Proof: Radius is perpendicular to tangent line
          • Proof: Segments tangent to circle from outside point are congruent
          • Tangents of circles problem (example 1)
          • Tangents of circles problem (example 2)
          • Tangents of circles problem (example 3)
        • Standard equation of a circle
          • Features of a circle from its standard equation
          • Graphing a circle from its standard equation
          • Writing standard equation of a circle
        • Expanded equation of a circle
          • Features of a circle from its expanded equation
        • Constructing regular polygons inscribed in circles
          • Geometric constructions: circle-inscribed square
          • Geometric constructions: circle-inscribed equilateral triangle
          • Geometric constructions: circle-inscribed regular hexagon
        • Constructing circumcircles & incircles
          • Geometric constructions: triangle-inscribing circle
          • Geometric constructions: triangle-circumscribing circle
        • Constructing a line tangent to a circle
          • Geometric constructions: circle tangent
          • Geometric constructions: circle tangent (example 2)
    • Trigonometry
      • Trigonometry with right triangles
        • Introduction to the trigonometric ratios
          • Intro to the trigonometric ratios
          • Trigonometric ratios in right triangles
        • Solving for a side in a right triangle using the trigonometric ratios
          • Solving for a side in right triangles with trigonometry
        • Solving for an angle in a right triangle using the trigonometric ratios
        • Modeling with right triangles
          • Right triangle word problem
        • Trigonometric ratios and similarity
          • Triangle similarity & the trigonometric ratios
          • Trig challenge problem: verify identities
          • Trig challenge problem: trig values & side ratios
        • Sine and cosine of complementary angles
          • Sine & cosine of complementary angles
          • Using complementary angles
          • Trig word problem: complementary angles
        • Trigonometric ratios of special triangles
        • Introduction to the Pythagorean trigonometric identity
          • Intro to the Pythagorean trig identity
        • The reciprocal trigonometric ratios
          • Finding reciprocal trig ratios
          • Using reciprocal trig ratios
      • Trigonometry with general triangles
        • The law of sines
          • Solving for a side with the law of sines
          • Solving for an angle with the law of sines
          • Proof of the law of sines
        • The law of cosines
          • Solving for a side with the law of cosines
          • Solving for an angle with the law of cosines
          • Proof of the law of cosines
        • Solving general triangles
          • Trig word problem: stars
      • The unit circle definition of sine, cosine, and tangent
        • Introduction to radians
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
          • Radian angles & quadrants
        • The unit circle definition of sine, cosine, and tangent
          • Unit circle
          • The trig functions & right triangle trig ratios
        • The graphs of sine, cosine, and tangent
          • Graph of y=sin(x)
          • Graph of y=tan(x)
          • Intersection points of y=sin(x) and y=cos(x)
        • Basic trigonometric identities
          • Sine & cosine identities: symmetry
          • Tangent identities: symmetry
          • Sine & cosine identities: periodicity
          • Tangent identities: periodicity
        • Trigonometric values of special angles
          • Trig values of special angles
        • The Pythagorean identity
          • Proof of the Pythagorean trig identity
          • Using the Pythagorean trig identity
        • Long live Tau
          • Tau versus pi
          • Pi is (still) wrong
      • Graphs of trigonometric functions
        • The graphs of sine, cosine, and tangent
          • Graph of y=sin(x)
          • Intersection points of y=sin(x) and y=cos(x)
          • Graph of y=tan(x)
        • Introduction to amplitude, midline, and extrema of sinusoidal functions
          • Features of sinusoidal functions
        • Finding amplitude and midline of sinusoidal functions from their formulas
          • Amplitude & period of sinusoidal functions from equation
        • Period of sinusoidal functions
          • Amplitude & period of sinusoidal functions from equation
        • Graphing sinusoidal functions
          • Transforming sinusoidal graphs: vertical stretch & horizontal reflection
          • Transforming sinusoidal graphs: vertical & horizontal stretches
        • Constructing sinusoidal functions
          • Sinusoidal function from graph
          • Trig word problem: modeling daily temperature
          • Trig word problem: modeling annual temperature
          • Trig word problem: length of day (phase shift)
      • Trigonometric equations and identities
        • The inverse trigonometric functions
          • Intro to arcsine
          • Intro to arctangent
          • Intro to arccosine
          • Restricting domains of functions to make them invertible
          • Domain & range of inverse tangent function
          • Using inverse trig functions with a calculator
        • Solving basic sinusoidal equations
          • Solving sinusoidal equations of the form sin(x)=d
          • Solving cos(θ)=1 and cos(θ)=-1
        • Solving advanced sinusoidal equations
        • Solving sinusoidal models
          • Trig word problem: solving for temperature
        • Introduction to the trigonometric angle addition identities
          • Review of trig angle addition identities
          • Using the cosine angle addition identity
          • Using the cosine double-angle identity
          • Proof of the sine angle addition identity
          • Proof of the cosine angle addition identity
        • Using trigonometric identities to solve problems
          • Finding trig values using angle addition identities
          • Using trig angle addition identities: finding side lengths
          • Using trig angle addition identities: manipulating expressions
          • Using trigonometric identities
        • Challenging trigonometry problems
          • Trig challenge problem: area of a triangle
          • Trig challenge problem: area of a hexagon
          • Trig challenge problem: cosine of angle-sum
          • Trig challenge problem: arithmetic progression
          • Trig challenge problem: maximum value
          • Trig challenge problem: multiple constraints
          • Trig challenge problem: system of equations
    • Statistics and probability
      • Analyzing categorical data
        • Analyzing one categorical variable
          • Identifying individuals, variables and categorical variables in a data set
          • Reading pictographs
          • Reading bar graphs
          • Reading bar graphs: Harry Potter
          • Creating a bar graph
          • Reading bar charts: comparing two sets of data
          • Reading bar charts: putting it together with central tendency
          • Reading pie graphs (circle graphs)
        • Two-way tables
          • Two-way frequency tables and Venn diagrams
          • Two-way relative frequency tables
          • Interpreting two-way tables
          • Categorical data example
          • Analyzing trends in categorical data
        • Distributions in two-way tables
          • Marginal distribution and conditional distribution
      • Displaying and comparing quantitative data
        • Displaying quantitative data with graphs
          • Representing data
          • Frequency tables & dot plots
          • Creating a histogram
          • Interpreting a histogram
          • Stem-and-leaf plots
          • Reading stem and leaf plots
        • Describing and comparing distributions
          • Shapes of distributions
          • Clusters, gaps, peaks & outliers
          • Comparing distributions with dot plots (example problem)
          • Comparing dot plots, histograms, and box plots
          • Example: Comparing distributions
        • More on data displays
          • Reading line graphs
          • Misleading line graphs
      • Summarizing quantitative data
        • Measuring center in quantitative data
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
        • More on mean and median
          • Comparing means of distributions
          • Means and medians of different distributions
          • Impact on median & mean: removing an outlier
          • Impact on median & mean: increasing an outlier
          • Missing value given the mean
          • Median & range puzzlers
        • Interquartile range (IQR)
          • Interquartile range (IQR)
        • Variance and standard deviation of a population
          • Measures of spread: range, variance & standard deviation
          • Variance of a population
          • Population standard deviation
          • Mean and standard deviation versus median and IQR
          • Statistics: Alternate variance formulas
        • Variance and standard deviation of a sample
          • Sample variance
          • Sample standard deviation and bias
        • More on standard deviation
          • Why we divide by n - 1 in variance
          • Simulation showing bias in sample variance
          • Simulation providing evidence that (n-1) gives us unbiased estimate
          • Review and intuition why we divide by n-1 for the unbiased sample variance
        • Box and whisker plots
          • Worked example: Creating a box plot (odd number of data points)
          • Worked example: Creating a box plot (even number of data points)
          • Constructing a box plot
          • Reading box plots
          • Interpreting box plots
          • Judging outliers in a dataset
        • Other measures of spread
          • Range and mid-range
          • Mean absolute deviation (MAD)
          • Mean absolute deviation example
      • Modeling data distributions
        • Percentiles
          • Calculating percentile
          • Analyzing a cumulative relative frequency graph
        • Z-scores
          • Normal distribution problem: z-scores (from ck12.org)
          • Comparing with z-scores
        • Effects of linear transformations
          • How parameters change as data is shifted and scaled
        • Density curves
          • Density Curves
          • Median, mean and skew from density curves
          • Density curve worked example
          • Worked example finding area under density curves
        • Normal distributions and the empirical rule
          • Qualitative sense of normal distributions (from ck12.org)
          • Normal distribution problems: Empirical rule (from ck12.org)
          • Standard normal distribution and the empirical rule (from ck12.org)
          • More empirical rule and z-score practice (from ck12.org)
          • Introduction to the normal distribution
          • Normal distribution excel exercise
        • Normal distribution calculations
          • Standard normal table for proportion below
          • Standard normal table for proportion above
          • Standard normal table for proportion between values
          • Finding z-score for a percentile
          • Threshold for low percentile
      • Exploring bivariate numerical data
        • Introduction to scatterplots
          • Constructing a scatter plot
          • Scatter plots: studying, shoe size, and test scores
          • Scatter plot: smokers
          • Bivariate relationship linearity, strength and direction
        • Correlation coefficients
          • Example: Correlation coefficient intuition
          • Calculating correlation coefficient r
        • Introduction to trend lines
          • Fitting a line to data
          • Estimating the line of best fit exercise
          • Estimating with linear regression (linear models)
          • Line of best fit: smoking in 1945
        • Least-squares regression equations
          • Introduction to residuals and least squares regression
          • Calculating residual example
          • Calculating the equation of a regression line
          • Interpreting slope of regression line
          • Interpreting y-intercept in regression model
          • Interpreting a trend line
        • Assessing the fit in least-squares regression
          • Residual plots
          • R-squared or coefficient of determination
          • Standard deviation of residuals or Root-mean-square error (RMSD)
          • Interpreting computer regression data
          • Impact of removing outliers on regression lines
        • More on regression
          • Squared error of regression line
          • Proof (part 1) minimizing squared error to regression line
          • Proof (part 2) minimizing squared error to regression line
          • Proof (part 3) minimizing squared error to regression line
          • Proof (part 4) minimizing squared error to regression line
          • Regression line example
          • Second regression example
          • R-squared or coefficient of determination
          • Calculating R-squared
          • Covariance and the regression line
      • Study design
        • Statistical questions
          • Statistical questions
          • Statistical and non statistical questions
        • Sampling and observational studies
          • Reasonable samples
          • Identifying a sample and population
          • Examples of bias in surveys
          • Example of under coverage introducing bias
          • Correlation and causality
        • Sampling methods
          • Picking fairly
          • Techniques for generating a simple random sample
          • Techniques for random sampling and avoiding bias
        • Types of studies (experimental vs. observational)
          • Types of statistical studies
          • Worked example identifying experiment
          • Worked example identifying observational study
          • Worked example identifying sample study
          • Appropriate statistical study example
        • Experiments
          • Introduction to experiment design
          • Matched pairs experiment design
      • Probability
        • Basic theoretical probability
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Intuitive sense of probabilities
          • The Monty Hall problem
        • Probability using sample spaces
          • Probability with counting outcomes
          • Example: All the ways you can flip a coin
          • Die rolling probability
          • Subsets of sample spaces
        • Basic set operations
          • Intersection and union of sets
          • Relative complement or difference between sets
          • Universal set and absolute complement
          • Subset, strict subset, and superset
          • Bringing the set operations together
        • Experimental probability
          • Experimental probability
          • Theoretical and experimental probabilites
          • Making predictions with probability
        • Randomness, probability, and simulation
          • Experimental versus theoretical probability simulation
          • Random number list to run experiment
          • Random numbers for experimental probability
          • Statistical significance of experiment
        • Addition rule
          • Probability with Venn diagrams
          • Addition rule for probability
        • Multiplication rule for independent events
          • Sample spaces for compound events
          • Compound probability of independent events
          • Probability of a compound event
          • Coin flipping probability
          • Free-throw probability
          • Three-pointer vs free-throw probability
          • Probability without equally likely events
          • Independent events example: test taking
          • Die rolling probability with independent events
        • Multiplication rule for dependent events
          • Dependent probability introduction
          • Dependent probability: coins
          • Dependent probability example
          • Independent & dependent probability
          • Dependent probability
        • Conditional probability and independence
          • Calculating conditional probability
          • Conditional probability explained visually
          • Conditional probability tree diagram example
          • Conditional probability and independence
          • Analyzing event probability for independence
      • Counting, permutations, and combinations
        • Counting principle and factorial
          • Count outcomes using tree diagram
          • Counting outcomes: flower pots
        • Permutations
          • Permutation formula
          • Zero factorial or 0!
          • Factorial and counting seat arrangements
          • Possible three letter words
          • Ways to arrange colors
          • Ways to pick officers
        • Combinations
          • Intro to combinations
          • Combination formula
          • Handshaking combinations
          • Combination example: 9 card hands
        • Combinatorics and probability
          • Probability using combinations
          • Probability & combinations (2 of 2)
          • Example: Different ways to pick officers
          • Example: Combinatorics and probability
          • Getting exactly two heads (combinatorics)
          • Exactly three heads in five flips
          • Generalizing with binomial coefficients (bit advanced)
          • Example: Lottery probability
          • Conditional probability and combinations
          • Mega millions jackpot probability
          • Birthday probability problem
      • Random variables
        • Discrete random variables
          • Random variables
          • Discrete and continuous random variables
          • Constructing a probability distribution for random variable
          • Probability models example: frozen yogurt
          • Valid discrete probability distribution examples
          • Probability with discrete random variable example
          • Mean (expected value) of a discrete random variable
          • Variance and standard deviation of a discrete random variable
        • Continuous random variables
          • Probability density functions
          • Probabilities from density curves
        • Transforming random variables
          • Impact of transforming (scaling and shifting) random variables
          • Example: Transforming a discrete random variable
        • Combining random variables
          • Mean of sum and difference of random variables
          • Variance of sum and difference of random variables
          • Intuition for why independence matters for variance of sum
          • Deriving the variance of the difference of random variables
          • Example: Analyzing distribution of sum of two normally distributed random variables
          • Example: Analyzing the difference in distributions
        • Binomial random variables
          • Binomial variables
          • Recognizing binomial variables
          • 10% Rule of assuming "independence" between trials
          • Binomial distribution
          • Visualizing a binomial distribution
          • Binomial probability example
          • Generalizing k scores in n attempts
          • Free throw binomial probability distribution
          • Graphing basketball binomial distribution
          • Binompdf and binomcdf functions
        • Binomial mean and standard deviation formulas
          • Mean and variance of Bernoulli distribution example
          • Bernoulli distribution mean and variance formulas
          • Expected value of a binomial variable
          • Variance of a binomial variable
          • Finding the mean and standard deviation of a binomial random variable
        • Geometric random variables
          • Geometric random variables introduction
          • Probability for a geometric random variable
          • Cumulative geometric probability (greater than a value)
          • Cumulative geometric probability (less than a value)
          • TI-84 geometpdf and geometcdf functions
          • Proof of expected value of geometric random variable
        • More on expected value
          • Term life insurance and death probability
          • Getting data from expected value
          • Expected profit from lottery ticket
          • Expected value while fishing
          • Comparing insurance with expected value
          • Law of large numbers
        • Poisson distribution
          • Poisson process 1
          • Poisson process 2
      • Sampling distributions
        • What is a sampling distribution?
          • Introduction to sampling distributions
          • Sample statistic bias worked example
        • Sampling distribution of a sample proportion
          • Sampling distribution of sample proportion part 1
          • Sampling distribution of sample proportion part 2
          • Normal conditions for sampling distributions of sample proportions
          • Probability of sample proportions example
        • Sampling distribution of a sample mean
          • Inferring population mean from sample mean
          • Central limit theorem
          • Sampling distribution of the sample mean
          • Sampling distribution of the sample mean 2
          • Standard error of the mean
          • Example: Probability of sample mean exceeding a value
          • Example: Probability of sample mean being within some distance of true mean
      • One-sample confidence intervals
        • Introduction to confidence intervals
          • Confidence intervals and margin of error
          • Confidence interval simulation
          • Interpreting confidence level example
        • Estimating a population proportion
          • Confidence interval example
          • Margin of error 1
          • Margin of error 2
          • Conditions for valid confidence intervals
          • Conditions for confidence intervals worked examples
          • Critical value (z*) for a given confidence level
          • Example constructing and interpreting a confidence interval for p
          • Determining sample size based on confidence and margin of error
        • Estimating a population mean
          • Example: Probability of sample mean being within some distance of true mean
          • Small sample size confidence intervals
          • T-statistic confidence interval
      • One-sample z and t significance tests
        • The idea of significance tests
          • Simple hypothesis testing
          • Idea behind hypothesis testing
          • Type 1 errors
        • Tests about a population proportion
          • Large sample proportion hypothesis testing
        • Tests about a population mean
          • Hypothesis testing and p-values
          • One-tailed and two-tailed tests
          • Z-statistics vs. T-statistics
          • Small sample hypothesis test
      • Two-sample inference for the difference between groups
        • Comparing two proportions
          • Comparing population proportions 1
          • Comparing population proportions 2
          • Hypothesis test comparing population proportions
        • Comparing two means
          • Statistical significance of experiment
          • Statistical significance on bus speeds
          • Difference of sample means distribution
          • Confidence interval of difference of means
          • Clarification of confidence interval of difference of means
          • Hypothesis test for difference of means
      • Inference for categorical data (chi-square tests)
        • Chi-square goodness-of-fit tests
          • Chi-square distribution introduction
          • Pearson's chi square test (goodness of fit)
        • Chi-square tests for homogeneity and association/independence
          • Filling out frequency table for independent events
          • Contingency table chi-square test
      • Advanced regression (inference and transforming)
        • Nonlinear regression
          • Comparing models to fit data example
      • Analysis of variance (ANOVA)
        • Analysis of variance (ANOVA)
          • ANOVA 1: Calculating SST (total sum of squares)
          • ANOVA 2: Calculating SSW and SSB (total sum of squares within and between)
          • ANOVA 3: Hypothesis test with F-statistic
    • High school statistics
      • Scatterplots
        • Creating and interpreting scatterplots
          • Constructing a scatter plot
          • Scatter plots: studying, shoe size, and test scores
          • Example: Correlation coefficient intuition
          • Correlation and causality
        • Estimating with trend lines
          • Estimating the line of best fit exercise
          • Line of best fit: smoking in 1945
          • Interpreting a trend line
      • Data distributions
        • Displays of distributions
          • Frequency tables & dot plots
          • Creating a histogram
          • Interpreting a histogram
          • Shapes of distributions
        • Summarizing center of distributions (central tendency)
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
          • Missing value given the mean
          • Impact on median & mean: removing an outlier
          • Impact on median & mean: increasing an outlier
        • Summarizing spread of distributions
          • Interquartile range (IQR)
          • Measures of spread: range, variance & standard deviation
        • Box and whisker plots
          • Constructing a box plot
          • Reading box plots
          • Interpreting box plots
        • Comparing distributions
          • Comparing distributions with dot plots (example problem)
      • Two-way tables
        • Two-way frequency tables
        • Two-way relative frequency tables
          • Two-way relative frequency tables
          • Interpreting two-way tables
      • Study design
        • Samples and surveys
          • Reasonable samples
        • Observational studies and experiments
          • Types of statistical studies
          • Correlation and causality
          • Appropriate statistical study example
      • Probability
        • Probability basics
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Experimental probability
          • Theoretical and experimental probabilites
          • Making predictions with probability
          • Intuitive sense of probabilities
          • Probability models example: frozen yogurt
        • Counting with permutations
          • Counting outcomes: flower pots
          • Count outcomes using tree diagram
          • Permutation formula
          • Possible three letter words
          • Ways to arrange colors
          • Ways to pick officers
        • Counting with combinations
          • Intro to combinations
          • Combination formula
          • Handshaking combinations
          • Combination example: 9 card hands
        • Probability with counting, permutations, combinations
          • Probability with counting outcomes
          • Example: All the ways you can flip a coin
          • Example: Different ways to pick officers
          • Example: Combinatorics and probability
          • Mega millions jackpot probability
        • Multiplication rule for independent events
          • Compound probability of independent events
          • Independent events example: test taking
          • Free-throw probability
        • Multiplication rule for dependent events
          • Dependent probability introduction
          • Independent & dependent probability
          • Dependent probability: coins
        • Addition rule for probability
          • Probability with Venn diagrams
          • Addition rule for probability
        • Conditional probability
        • Simulation and randomness
        • Expected value
      • Binomial probability
        • Binomial probability
          • Binomial probability example
          • Generalizing k scores in n attempts
          • Free throw binomial probability distribution
          • Graphing basketball binomial distribution
      • Normal distributions
        • Normal distributions
          • Normal distribution problems: Empirical rule (from ck12.org)
          • Qualitative sense of normal distributions (from ck12.org)
    • AP Statistics
      • Analyzing categorical data
        • Welcome to AP Statistics
          • Meet Jeff, a creator of AP Statistics on Khan Academy
        • Analyzing one categorical variable
          • Identifying individuals, variables and categorical variables in a data set
          • Creating a bar graph
          • Reading bar charts: comparing two sets of data
        • Two-way tables
          • Two-way frequency tables and Venn diagrams
          • Two-way relative frequency tables
          • Interpreting two-way tables
        • Distributions in two-way tables
          • Marginal distribution and conditional distribution
      • Displaying and comparing quantitative data
        • Frequency tables and dot plots
          • Frequency tables & dot plots
        • Histograms and stem-and-leaf plots
          • Creating a histogram
          • Interpreting a histogram
          • Stem-and-leaf plots
          • Reading stem and leaf plots
        • Describing and comparing distributions
          • Shapes of distributions
          • Example: Describing a distribution
          • Example: Comparing distributions
      • Summarizing quantitative data
        • Measuring center in quantitative data
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
        • More on mean and median
          • Missing value given the mean
          • Impact on median & mean: increasing an outlier
          • Impact on median & mean: removing an outlier
        • Measuring spread in quantitative data
          • Interquartile range (IQR)
          • Variance of a population
          • Population standard deviation
          • Sample variance
          • Sample standard deviation and bias
          • Mean and standard deviation versus median and IQR
        • More on standard deviation
          • Why we divide by n - 1 in variance
          • Simulation showing bias in sample variance
          • Simulation providing evidence that (n-1) gives us unbiased estimate
          • Review and intuition why we divide by n-1 for the unbiased sample variance
        • Box and whisker plots
          • Worked example: Creating a box plot (odd number of data points)
          • Worked example: Creating a box plot (even number of data points)
          • Constructing a box plot
          • Reading box plots
          • Interpreting box plots
          • Judging outliers in a dataset
      • Modeling data distributions
        • Percentiles (cumulative relative frequency)
          • Calculating percentile
          • Analyzing a cumulative relative frequency graph
        • Z-scores
          • Normal distribution problem: z-scores (from ck12.org)
          • Comparing with z-scores
        • Effects of linear transformations
          • How parameters change as data is shifted and scaled
        • Density curves
          • Density Curves
          • Median, mean and skew from density curves
          • Density curve worked example
          • Worked example finding area under density curves
        • Normal distributions and the empirical rule
          • Qualitative sense of normal distributions (from ck12.org)
          • Normal distribution problems: Empirical rule (from ck12.org)
        • Normal distribution calculations
          • Standard normal table for proportion below
          • Standard normal table for proportion above
          • Standard normal table for proportion between values
          • Finding z-score for a percentile
          • Threshold for low percentile
      • Exploring bivariate numerical data
        • Introduction to scatterplots
          • Constructing a scatter plot
          • Scatter plots: studying, shoe size, and test scores
          • Bivariate relationship linearity, strength and direction
        • Correlation coefficients
          • Calculating correlation coefficient r
          • Example: Correlation coefficient intuition
        • Least-squares regression equations
          • Estimating with linear regression (linear models)
          • Introduction to residuals and least squares regression
          • Calculating residual example
          • Calculating the equation of a regression line
          • Interpreting slope of regression line
          • Interpreting y-intercept in regression model
        • Assessing the fit in least-squares regression
          • Residual plots
          • R-squared or coefficient of determination
          • Standard deviation of residuals or Root-mean-square error (RMSD)
          • Interpreting computer regression data
          • Impact of removing outliers on regression lines
      • Study design
        • Sampling and observational studies
          • Identifying a sample and population
          • Examples of bias in surveys
          • Example of under coverage introducing bias
          • Correlation and causality
        • Sampling methods
          • Techniques for generating a simple random sample
          • Techniques for random sampling and avoiding bias
        • Types of studies (experimental vs. observational)
          • Types of statistical studies
          • Worked example identifying experiment
          • Worked example identifying observational study
        • Experiments
          • Introduction to experiment design
          • Matched pairs experiment design
      • Probability
        • Randomness, probability, and simulation
          • Intro to theoretical probability
          • Experimental versus theoretical probability simulation
          • Random number list to run experiment
          • Random numbers for experimental probability
          • Statistical significance of experiment
        • Addition rule
          • Probability with Venn diagrams
          • Addition rule for probability
        • Multiplication rule
          • Compound probability of independent events
          • Independent events example: test taking
          • Free-throw probability
          • Three-pointer vs free-throw probability
          • Dependent probability introduction
          • Coin flipping probability
        • Conditional probability
          • Conditional probability and independence
          • Conditional probability with Bayes' Theorem
          • Conditional probability tree diagram example
      • Random variables
        • Discrete random variables
          • Constructing a probability distribution for random variable
          • Valid discrete probability distribution examples
          • Probability with discrete random variable example
          • Mean (expected value) of a discrete random variable
          • Variance and standard deviation of a discrete random variable
        • Continuous random variables
          • Probabilities from density curves
        • Transforming random variables
          • Impact of transforming (scaling and shifting) random variables
          • Example: Transforming a discrete random variable
        • Combining random variables
          • Mean of sum and difference of random variables
          • Variance of sum and difference of random variables
          • Intuition for why independence matters for variance of sum
          • Deriving the variance of the difference of random variables
          • Example: Analyzing distribution of sum of two normally distributed random variables
          • Example: Analyzing the difference in distributions
        • Binomial random variables
          • Binomial variables
          • Recognizing binomial variables
          • 10% Rule of assuming "independence" between trials
          • Binomial probability example
          • Generalizing k scores in n attempts
          • Free throw binomial probability distribution
          • Graphing basketball binomial distribution
          • Binompdf and binomcdf functions
        • Binomial mean and standard deviation formulas
          • Expected value of a binomial variable
          • Variance of a binomial variable
          • Finding the mean and standard deviation of a binomial random variable
        • Geometric random variables
          • Geometric random variables introduction
          • Probability for a geometric random variable
          • Cumulative geometric probability (greater than a value)
          • Cumulative geometric probability (less than a value)
          • TI-84 geometpdf and geometcdf functions
          • Proof of expected value of geometric random variable
      • Sampling distributions
        • What is a sampling distribution?
          • Introduction to sampling distributions
          • Sample statistic bias worked example
        • Sampling distribution of a sample proportion
          • Sampling distribution of sample proportion part 1
          • Sampling distribution of sample proportion part 2
          • Normal conditions for sampling distributions of sample proportions
          • Probability of sample proportions example
        • Sampling distribution of a sample mean
          • Central limit theorem
          • Sampling distribution of the sample mean
          • Sampling distribution of the sample mean 2
          • Standard error of the mean
          • Example: Probability of sample mean exceeding a value
      • One-sample confidence intervals
        • Introduction to confidence intervals
          • Confidence intervals and margin of error
          • Confidence interval simulation
          • Interpreting confidence level example
        • One-sample z interval for a proportion
          • Conditions for valid confidence intervals
          • Conditions for confidence intervals worked examples
          • Critical value (z*) for a given confidence level
          • Example constructing and interpreting a confidence interval for p
          • Determining sample size based on confidence and margin of error
    • Precalculus
      • Trigonometry
        • The inverse trigonometric functions
          • Intro to arcsine
          • Intro to arctangent
          • Intro to arccosine
          • Restricting domains of functions to make them invertible
          • Domain & range of inverse tangent function
          • Using inverse trig functions with a calculator
        • Solving basic sinusoidal equations
          • Solving sinusoidal equations of the form sin(x)=d
          • Solving cos(θ)=1 and cos(θ)=-1
        • Solving advanced sinusoidal equations
        • Solving sinusoidal models
          • Trig word problem: solving for temperature
        • Introduction to the trigonometric angle addition identities
          • Review of trig angle addition identities
          • Using the cosine angle addition identity
          • Using the cosine double-angle identity
          • Proof of the sine angle addition identity
          • Proof of the cosine angle addition identity
        • Using trigonometric identities to solve problems
          • Finding trig values using angle addition identities
          • Using trig angle addition identities: finding side lengths
          • Using trig angle addition identities: manipulating expressions
          • Using trigonometric identities
      • Conic sections
        • Introduction to conic sections
          • Intro to conic sections
        • The features of a circle
          • Graphing circles from features
          • Features of a circle from its graph
        • Standard equation of a circle
          • Features of a circle from its standard equation
          • Graphing a circle from its standard equation
          • Writing standard equation of a circle
        • Expanded equation of a circle
          • Features of a circle from its expanded equation
        • Center and radii of an ellipse
          • Intro to ellipses
          • Ellipse standard equation from graph
          • Ellipse graph from standard equation
        • Foci of an ellipse
          • Foci of an ellipse from equation
        • Focus and directrix of a parabola
          • Intro to focus & directrix
          • Equation of a parabola from focus & directrix
          • Focus & directrix of a parabola from equation
        • Introduction to hyperbolas
          • Intro to hyperbolas
          • Vertices & direction of a hyperbola
          • Vertices & direction of a hyperbola (example 2)
          • Graphing hyperbolas (old example)
        • Foci of a hyperbola
          • Foci of a hyperbola from equation
          • Proof of the hyperbola foci formula
        • Hyperbolas not centered at the origin
          • Equation of a hyperbola not centered at the origin
        • Identifying conic sections from their expanded equations
          • Conic section from expanded equation: circle & parabola
          • Conic section from expanded equation: ellipse
          • Conic section from expanded equation: hyperbola
        • Challenging conic section problems (IIT JEE)
          • Representing a line tangent to a hyperbola
          • Common tangent of circle & hyperbola (1 of 5)
          • Common tangent of circle & hyperbola (2 of 5)
          • Common tangent of circle & hyperbola (3 of 5)
          • Common tangent of circle & hyperbola (4 of 5)
          • Common tangent of circle & hyperbola (5 of 5)
          • Intersection of circle & hyperbola
      • Vectors
        • Vector basics
          • Intro to vectors & scalars
          • Recognizing vectors
          • Equivalent vectors
          • Components of vectors
          • Components of vectors (example 2)
        • Magnitude of vectors
          • Vector magnitude from graph
          • Vector magnitude from components
          • Vector magnitude from initial & terminal points
        • Scalar multiplication
          • Scalar multiplication of vectors
          • Analyzing scalar multiplication
        • Vector addition and subtraction
          • Adding & subtracting vectors
          • Graphically adding & subtracting vectors
          • Graphically subtracting vectors
          • Adding vectors algebraically & graphically
        • Combined vector operations
          • Combined vector operations
        • Unit vectors
          • Unit vectors intro
          • Worked example: finding unit vector with given direction
          • Worked example: Scaling unit vectors
        • Magnitude and direction form of vectors
          • Direction of vectors from components: 1st & 2nd quadrants
          • Direction of vectors from components: 3rd & 4th quadrants
        • Component form of vectors
          • Vector components from magnitude & direction
        • Adding vectors in magnitude and direction form
          • Adding vectors in magnitude & direction form (1 of 2)
          • Adding vectors in magnitude & direction form (2 of 2)
          • Vector addition & magnitude
        • Applications of vectors
          • Vectors word problem: pushing a box
          • Vectors word problem: tug of war
          • Vectors word problem: hiking
      • Matrices
        • Introduction to matrices
          • Intro to matrices
        • Representing linear systems of equations with augmented matrices
        • Elementary matrix row operations
        • Row-echelon form and Gaussian elimination
          • Solving linear systems with matrices
        • Adding and subtracting matrices
          • Adding & subtracting matrices
        • Multiplying matrices by scalars
          • Multiplying matrices by scalars
        • Properties of matrix addition & scalar multiplication
        • Multiplying matrices by matrices
          • Intro to matrix multiplication
          • Multiplying matrices
        • Properties of matrix multiplication
          • Defined matrix operations
          • Intro to identity matrix
          • Dimensions of identity matrix
          • Is matrix multiplication commutative?
          • Associative property of matrix multiplication
          • Zero matrix & matrix multiplication
          • Using properties of matrix operations
          • Using identity & zero matrices
        • Matrices as transformations
          • Transforming vectors using matrices
          • Transforming polygons using matrices
          • Visual representation of transformation from matrix
        • The determinant of a 2x2 matrix
          • Determinant of a 2x2 matrix
        • Introduction to matrix inverses
          • Intro to matrix inverses
          • Determining invertible matrices
        • Finding the inverse of a matrix using its determinant
          • Finding inverse of a 2x2 matrix using determinant & adjugate
        • Practice finding the inverses of 2x2 matrices
        • Solving equations with inverse matrices
          • Representing linear systems with matrix equations
          • Solving linear systems with matrix equations
          • Matrix word problem: vector combination
        • Model real-world situations with matrices
          • Matrix word problem: prices
      • Complex numbers
        • What are the imaginary numbers?
          • Intro to the imaginary numbers
          • Powers of the imaginary unit
          • Simplifying roots of negative numbers
          • i as the principal root of -1
        • What are the complex numbers?
          • Intro to complex numbers
          • How to classify complex numbers (example)
        • The complex plane
          • Plotting numbers on the complex plane
        • Adding and subtracting complex numbers
          • Adding complex numbers
          • Subtracting complex numbers
        • Distance and midpoint of complex numbers
          • Distance & midpoint of complex numbers
        • Multiplying complex numbers
          • Multiplying complex numbers
        • Complex conjugates and dividing complex numbers
          • Intro to complex number conjugates
          • Complex number conjugates
          • Dividing complex numbers
        • Absolute value and angle of complex numbers
          • Absolute value of complex numbers
          • Absolute value & angle of complex numbers
        • Polar form of complex numbers
          • Polar & rectangular forms of complex numbers
        • Multiplying and dividing complex numbers in polar form
          • Dividing complex numbers: polar & exponential form
          • Powers of complex numbers
          • Complex number equations: x³=1
        • Challenging complex number problems
          • Challenging complex numbers problem (1 of 3)
          • Challenging complex numbers problem (2 of 3)
          • Challenging complex numbers problem (3 of 3)
          • Challenging complex numbers problem: complex determinant
      • Probability and combinatorics
        • Basic probability
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
        • Venn diagrams and the addition rule
          • Probability with Venn diagrams
          • Addition rule for probability
        • Compound probability of independent events using diagrams
          • Die rolling probability
          • Probability with counting outcomes
          • Compound events example with tree diagram
        • Compound probability of independent events using the multiplication rule
          • Compound probability of independent events
          • Probability without equally likely events
          • Independent events example: test taking
          • Die rolling probability with independent events
          • Coin flipping probability
          • Free-throw probability
          • Three-pointer vs free-throw probability
        • Dependent events
          • Dependent probability introduction
          • Dependent probability: coins
          • Dependent probability example
          • Independent & dependent probability
          • The Monty Hall problem
        • Permutations
          • Factorial and counting seat arrangements
          • Permutation formula
          • Possible three letter words
          • Zero factorial or 0!
          • Ways to arrange colors
          • Ways to pick officers
        • Combinations
          • Intro to combinations
          • Combination formula
          • Handshaking combinations
          • Combination example: 9 card hands
        • Probability using combinatorics
          • Probability using combinations
          • Probability & combinations (2 of 2)
          • Probability with counting outcomes
          • Getting exactly two heads (combinatorics)
          • Exactly three heads in five flips
          • Generalizing with binomial coefficients (bit advanced)
          • Example: Different ways to pick officers
          • Example: Combinatorics and probability
          • Example: Lottery probability
          • Mega millions jackpot probability
          • Birthday probability problem
      • Series
        • Arithmetic sequences
          • Sequences intro
          • Intro to arithmetic sequences
          • Sequences word problem: growth pattern
          • Explicit formulas for arithmetic sequences
        • Basic sigma notation
          • Summation notation
        • Finite arithmetic series
          • Arithmetic series intro
          • Worked example: arithmetic series (sigma notation)
          • Worked example: arithmetic series (sum expression)
          • Worked example: arithmetic series (recursive formula)
          • Proof of finite arithmetic series formula
        • Geometric sequences
          • Intro to geometric sequences
          • Intro to geometric sequences (advanced)
          • Explicit & recursive formulas for geometric sequences
          • Converting recursive & explicit forms of geometric sequences
        • Finite geometric series
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
          • Finite geometric series formula justification
        • Finite geometric series applications
          • Finite geometric series word problem: social media
          • Finite geometric series word problem: mortgage
    • Differential calculus
      • Limits basics
        • Limits introduction
          • Newton, Leibniz, and Usain Bolt
          • Limits intro
          • Introduction to limits (old)
          • Limits from tables
          • Limits from graphs: function undefined
          • Limits from graphs: limit isn't equal to the function's value
          • Limits from graphs: asymptote
        • One-sided limits
          • One-sided limits from graphs
          • One-sided limits from graphs: asymptote
          • One-sided limits from tables
          • 1-sided vs. 2-sided limits (graphical)
          • One-sided limit from equation
          • Connecting limits and graphical behavior (more examples)
        • Formal definition of limits (epsilon-delta)
          • Formal definition of limits Part 1: intuition review
          • Formal definition of limits Part 2: building the idea
          • Formal definition of limits Part 3: the definition
          • Formal definition of limits Part 4: using the definition
        • Review: Limits basics
      • Continuity
        • Continuity at a point
          • Continuity introduction
          • Worked example: Continuity at a point
        • Limits of combined & composite functions
          • Limit properties
          • Limits of combined functions
          • Limits of composite functions
        • Continuous functions
          • Functions continuous on all real numbers
          • Functions continuous at specific x-values
        • Intermediate value theorem
          • Intermediate value theorem
          • Intermediate value theorem example
        • Review: Continuity
      • Limits from equations
        • Limits from equations (direct substitution)
          • Limits by direct substitution
          • Undefined limits by direct substitution
        • Limits from equations (factoring & rationalizing)
          • Limits by factoring
          • Limits by rationalizing
        • Squeeze theorem
          • Squeeze theorem intro
          • Squeeze theorem example
          • Limit of sin(x)/x as x approaches 0
        • Limits of trigonometric functions
          • Trig limit using pythagorean identity
          • Trig limit using double angle identity
        • Limits of piecewise functions
          • Analyzing functions for discontinuities (continuous example)
          • Analyzing functions for discontinuities (discontinuity example)
        • Removable discontinuities
          • Removing discontinuities (factoring)
          • Removing discontinuities (rationalization)
        • Review: Limits from equations
      • Infinite limits
        • Unbounded limits (vertical asymptotes)
          • Infinite limits intro
          • Unbounded limits: graphical
          • Analyzing unbounded limits: rational function
          • Analyzing unbounded limits: mixed function
        • Limits at infinity (horizontal asymptotes)
          • Infinite limits intro
          • Limits at infinity of quotients (Part 1)
          • Limits at infinity of quotients (Part 2)
          • Limits at infinity of quotients with square roots (odd power)
          • Limits at infinity of quotients with square roots (even power)
          • Limits at infinity of quotients with trig
          • Limits at infinity of quotients with trig (limit undefined)
          • Limit at infinity of a difference of functions
        • Review: Infinite limits
      • Derivative introduction
        • Introduction to differential calculus
          • Newton, Leibniz, and Usain Bolt
          • Derivative as slope of curve
          • Derivative & the direction of a function
        • Derivative as slope of tangent line
          • The derivative & tangent line equations
          • Interpreting derivative challenge
        • Derivative as instantaneous rate of change
          • Tangent slope as instantaneous rate of change
          • Estimating derivatives with two consecutive secant lines
          • Approximating instantaneous rate of change with average rate of change
        • Secant lines
          • Slope of a line secant to a curve
          • Secant line with arbitrary difference
          • Secant line with arbitrary point
          • Secant line with arbitrary difference (with simplification)
          • Secant line with arbitrary point (with simplification)
          • Secant lines: challenging problem 1
          • Secant lines: challenging problem 2
        • Derivative as a limit
          • Formal definition of the derivative as a limit
          • Derivative as a limit: numerical
        • Formal definition of derivative
          • Formal and alternate form of the derivative
          • Worked example: Derivative as a limit
          • Worked example: Derivative from limit expression
        • Using the formal definition of derivative
          • The derivative of x² at x=3 using the formal definition
          • The derivative of x² at any point using the formal definition
          • Limit expression for the derivative of a linear function
          • Limit expression for the derivative of cos(x) at a minimum point
          • Limit expression for the derivative of function (graphical)
        • Differentiability
          • Differentiability at a point: graphical
          • Differentiability at a point: algebraic (function is differentiable)
          • Differentiability at a point: algebraic (function isn't differentiable)
        • Derivative as a function
          • The graphical relationship between a function & its derivative (part 1)
          • The graphical relationship between a function & its derivative (part 2)
          • Matching functions & their derivatives graphically
          • Connecting f and f' graphically
        • Review: Derivative basics
      • Basic differentiation
        • Basic differentiation rules
          • Basic derivative rules (Part 1)
          • Basic derivative rules (Part 2)
          • Basic derivative rules: find the error
          • Basic derivative rules: table
        • Power rule
          • Power rule
          • Justifying the power rule
          • Proof of power rule for positive integer powers
          • Proof of power rule for square root function
        • Polynomial functions differentiation
          • Differentiating polynomials
          • Differentiating polynomials example
          • Tangents of polynomials
        • Rational functions differentiation (intro)
          • Negative powers differentiation
          • Worked example: Tangent to the graph of 1/x
        • Radical functions differentiation (intro)
          • Fractional powers differentiation
          • Radical functions differentiation intro
        • Sine & cosine derivatives
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Worked example: Derivatives of sin(x) and cos(x)
        • eˣ and ln(x) derivatives
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
        • Review: Basic differentiation
      • Product, quotient, & chain rules
        • Product rule
          • Product rule
          • Differentiating products
          • Worked example: Product rule with table
          • Worked example: Product rule with mixed implicit & explicit
          • Product rule to find derivative of product of three functions
          • Product rule proof
        • Chain rule
          • Chain rule
          • Worked example: Derivative of cos³(x) using the chain rule
          • Worked example: Derivative of ln(√x) using the chain rule
          • Worked example: Derivative of √(3x²-x) using the chain rule
          • Worked example: Chain rule with table
        • Chain rule proof
          • Proof: Differentiability implies continuity
          • If function u is continuous at x, then Δu→0 As Δx→0
          • Chain rule proof
        • Quotient rule
          • Quotient rule from product & chain rules
          • Worked example: Quotient rule with table
          • Tangent to y=𝑒ˣ/(2+x³)
          • Normal to y=𝑒ˣ/x²
        • Review: Product, quotient, & chain rule
      • Differentiating common functions
        • Rational functions differentiation
          • Differentiating rational functions
        • Radical functions differentiation
          • Radical functions differentiation
        • Trigonometric functions differentiation
          • Derivatives of tan(x) and cot(x)
          • Derivatives of sec(x) and csc(x)
          • Trig functions differentiation
        • Exponential functions differentiation
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Exponential functions differentiation intro
          • Exponential functions differentiation
        • Logarithmic functions differentiation
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Derivative of logₐx (for any positive base a≠1)
          • Logarithmic functions differentiation
          • Differentiating logarithmic functions using log properties
        • Derivatives capstone
          • Derivative of eᶜᵒˢˣ⋅cos(eˣ)
          • Derivative of sin(ln(x²))
      • Advanced differentiation
        • Implicit differentiation introduction
          • Implicit differentiation
          • Worked example: Implicit differentiation
          • Worked example: Evaluating derivative with implicit differentiation
          • Showing explicit and implicit differentiation give same result
        • Implicit differentiation (advanced examples)
          • Implicit differentiation (advanced example)
          • Implicit differentiation (advanced example)
          • Implicit differentiation (advanced example)
          • Derivative of ln(x) from derivative of 𝑒ˣ and implicit differentiation
        • Inverse trig functions differentiation
          • Derivative of inverse sine
          • Derivative of inverse cosine
          • Derivative of inverse tangent
        • Derivatives of inverse functions
          • Derivatives of inverse functions: from equation
          • Derivatives of inverse functions: from table
        • Disguised derivatives
          • Derivatives expressed as limits
        • Proofs for the derivatives of eˣ and ln(x)
          • Proof: The derivative of 𝑒ˣ is 𝑒ˣ
          • Proof: the derivative of ln(x) is 1/x
        • Logarithmic differentiation
          • Composite exponential function differentiation
          • Worked example: Composite exponential function differentiation
        • Parametric & vector-valued function differentiation
          • Vector-valued functions intro
          • Vector-valued functions differentiation
          • Parametric functions differentiation
        • Review: Advanced differentiation
        • Higher-order derivatives
          • Second derivatives
        • Higher-order derivatives (parametric & vector-valued functions)
          • Second derivatives (vector-valued functions)
          • Second derivatives (parametric functions)
      • Analyzing functions with calculus
        • Critical points
          • Introduction to minimum and maximum points
          • Identifying relative minimum and maximum values
          • Critical points introduction
          • Finding critical points
        • Increasing & decreasing intervals
          • Finding decreasing interval given the function
          • Finding increasing interval given the derivative
        • Relative minima & maxima
          • Finding relative extrema (first derivative test)
          • Analyzing a function with its derivative
          • Worked example: finding relative extrema
        • Absolute minima & maxima
          • Extreme value theorem
          • Finding absolute extrema on a closed interval
          • Absolute minima & maxima (entire domain)
        • Review: Increasing/decreasing intervals & extrema
        • Concavity
          • Concavity introduction
          • Analyzing concavity (graphical)
          • Analyzing concavity (algebraic)
          • Second derivative test
        • Points of inflection
          • Inflection points introduction
          • Inflection points (graphical)
          • Worked example: Inflection points from first derivative
          • Inflection points (algebraic)
        • Sketching graphs using calculus
          • Curve sketching with calculus: polynomial
          • Curve sketching with calculus: logarithm
        • Review: Concavity & points of inflection
      • Derivative applications
        • Linear approximation
          • Local linearity
          • Linear approximation of a rational function
        • Rectilinear motion
          • Motion problems: when a particle is speeding up
          • Motion problems: finding the maximum acceleration
          • Analyzing rectilinear motion graphically
          • Total distance traveled by a particle
        • Planar motion
          • Planar motion example: acceleration vector
          • Motion along a curve: finding rate of change
          • Motion along a curve: finding velocity magnitude
        • Related rates
          • Related rates intro
          • Related rates: Approaching cars
          • Related rates: Falling ladder
          • Related rates: water pouring into a cone
          • Related rates: shadow
          • Related rates: balloon
        • Optimization
          • Optimization: sum of squares
          • Optimization: box volume (Part 1)
          • Optimization: box volume (Part 2)
          • Optimization: profit
          • Optimization: cost of materials
          • Optimization: area of triangle & square (Part 1)
          • Optimization: area of triangle & square (Part 2)
          • Optimization problem: extreme normaline to y=x²
        • Applied rates of change
          • Applied rate of change: forgetfulness
          • Applied rates of change: marginal costs
          • Marginal cost & differential calculus
        • Mean value theorem
          • Mean value theorem
          • Mean value theorem example: polynomial
          • Mean value theorem example: square root function
          • Mean value theorem application
        • L'Hôpital's rule
          • L'Hôpital's rule introduction
          • L'Hôpital's rule: limit at 0 example
          • L'Hôpital's rule: challenging problem
          • L'Hôpital's rule: limit at infinity example
          • L'Hôpital's rule: solve for a variable
          • L’Hôpital’s rule (composite exponential functions)
          • Proof of special case of l'Hôpital's rule
        • Review: Derivative applications
    • Integral calculus
      • Indefinite integrals
        • Antiderivatives
          • Antiderivatives and indefinite integrals
        • Indefinite integrals intro
          • Reverse power rule
          • Rewriting before integrating: challenge problem
          • Visually determining antiderivative
          • Graphs of indefinite integrals
        • Indefinite integrals of common functions
          • Indefinite integrals of sin(x), cos(x), and eˣ
          • Indefinite integral of 1/x
          • Finding specific antiderivatives: rational function
          • Finding specific antiderivatives: exponential function
        • Review: Indefinite integrals & antiderivatives
      • Definite integrals introduction
        • Definite integral as area
          • Worked example: Definite integral by thinking about the function's graph
        • Definite integral properties
          • Integrating scaled version of function
          • Integrating sums of functions
          • Definite integral over a single point
          • Breaking up integral interval
          • Definite integral of shifted function
          • Switching bounds of definite integral
          • Worked examples: Definite integral properties 2
          • Worked examples: Definite integral properties 1
          • Definite integral properties (no graph): function combination
          • Definite integral properties (no graph): breaking interval
          • Examples leveraging integration properties
        • Review: Definite integral basics
      • Riemann sums
        • Riemann sums
          • Riemann approximation introduction
          • Riemann sums in summation notation
          • Motion problem with Riemann sum approximation
        • Riemann sums with sigma notation
          • Worked example: Riemann sums in summation notation
          • Riemann sums in summation notation: challenge problem
        • Trapezoidal rule
          • Midpoint and trapezoidal sums in summation notation
          • Trapezoidal sums
        • Definite integral as the limit of a Riemann sum
          • Definite integral as the limit of a Riemann sum
          • Ranking area estimates
        • Review: Riemann sums
      • Fundamental theorem of calculus
        • Functions defined by integrals
          • Functions defined by integrals
          • Functions defined by integrals: switched interval
          • Functions defined by integrals: challenge problem
        • Fundamental theorem of calculus
          • Fundamental theorem of calculus (Part 1)
          • Worked example: Finding derivative with fundamental theorem of calculus
          • Finding derivative with fundamental theorem of calculus: x is on both bounds
          • Proof of fundamental theorem of calculus
        • Fundamental theorem of calculus: chain rule
          • Worked example: Finding derivative with fundamental theorem of calculus
          • Finding derivative with fundamental theorem of calculus: x is on lower bound
      • Definite integral evaluation
        • Definite integral evaluation
          • Fundamental theorem of calculus (Part 2)
          • Intuition for second part of fundamental theorem of calculus
          • Area between a curve and the x-axis
          • Area between a curve and the x-axis: negative area
          • Definite integral of rational function
          • Definite integral of radical function
          • Definite integral of trig function
          • Definite integral involving natural log
        • Definite integrals of piecewise functions
          • Definite integral of piecewise function
          • Definite integral of absolute value function
        • Improper integrals
          • Introduction to improper integrals
          • Improper integral with two infinite bounds
          • Divergent improper integral
      • Integration techniques
        • Integration by parts
          • Integration by parts intro
          • Integration by parts: ∫x⋅cos(x)dx
          • Integration by parts: ∫ln(x)dx
          • Integration by parts: ∫x²⋅𝑒ˣdx
          • Integration by parts: ∫𝑒ˣ⋅cos(x)dx
          • Challenging definite integration
        • u-substitution
          • 𝘶-substitution intro
          • 𝘶-substitution: rational function
          • 𝘶-substitution: multiplying by a constant
          • 𝘶-substitution: logarithmic function
          • 𝘶-substitution: composite function
          • 𝘶-substitution: special application
          • 𝘶-substitution: definite integral of exponential function
          • 𝘶-substitution: double substitution
        • Reverse chain rule
          • Reverse chain rule introduction
          • Reverse chain rule example
          • Integral of tan x
        • Partial fraction expansion
          • Partial fraction expansion to evaluate integral
          • Dividing expressions to evaluate integral
        • Integration using trigonometric identities
          • Integral of cos^3(x)
          • Integral of sin^2(x) cos^3(x)
          • Integral of sin^4(x)
        • Trigonometric substitution
          • Introduction to trigonometric substitution
          • Substitution with x=sin(theta)
          • More trig sub practice
          • Trig and u substitution together (part 1)
          • Trig and u substitution together (part 2)
          • Trig substitution with tangent
          • More trig substitution with tangent
          • Long trig sub problem
      • Area & arc length using calculus
        • Area between curves
          • Area between curves
          • Area between curves with multiple boundaries
        • Arc length
          • Arc length intro
          • Worked example: arc length
          • Worked example: arc length (advanced)
        • Area defined by polar graphs
          • Area bounded by polar curves
          • Worked example: Area enclosed by cardioid
          • Worked example: Area between two polar graphs
          • Evaluating definite integral with calculator
        • Arc length of polar graphs
          • Arc length of polar curves
          • Worked example: Arc length of polar curves
      • Integration applications
        • Area & net change
          • Worked example: problem involving definite integral (graphical)
          • Area under rate function gives the net change
        • Rectilinear motion
          • Worked example: motion problems (with definite integrals)
        • Average value of a function
          • Average value over a closed interval
          • Calculating average value of function over interval
          • Average acceleration over interval
          • Mean value theorem for integrals
      • Volume using calculus
        • Solids with known cross sections
          • Volume with cross sections: semicircle
          • Volume with cross sections: triangle
        • Disc method
          • Disc method around x-axis
          • Generalizing disc method around x-axis
          • Disc method around y-axis
          • Disc method rotation around horizontal line
          • Disc method rotating around vertical line
          • Calculating integral disc around vertical line
        • Washer method
          • Solid of revolution between two functions (leading up to the washer method)
          • Generalizing the washer method
          • Washer method rotating around horizontal line (not x-axis), part 1
          • Washer method rotating around horizontal line (not x-axis), part 2
          • Washer method rotating around vertical line (not y-axis), part 1
          • Washer method rotating around vertical line (not y-axis), part 2
        • Shell method
          • Shell method for rotating around vertical line
          • Evaluating integral for shell method example
          • Shell method for rotating around horizontal line
          • Shell method with two functions of x
          • Calculating integral with shell method
          • Shell method with two functions of y
          • Part 2 of shell method with 2 functions of y
      • Sequences & series intro
        • Sequences review
          • Sequences intro
          • Worked example: sequence explicit formula
          • Worked example: sequence recursive formula
          • Geometric sequence review
          • Extending geometric sequences
          • Using explicit formulas of geometric sequences
          • Using recursive formulas of geometric sequences
        • Infinite sequences
          • Convergent and divergent sequences
          • Worked example: sequence convergence/divergence
          • Formal definition for limit of a sequence
          • Proving a sequence converges using the formal definition
        • Series review
          • Summation notation
          • Converting explicit series terms to summation notation
          • Converting explicit series terms to summation notation (n ≥ 2)
          • Arithmetic series formula
          • Worked example: arithmetic series (sigma notation)
          • Worked example: arithmetic series (sum expression)
          • Worked example: arithmetic series (recursive formula)
        • Finite geometric series
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
          • Finite geometric series formula justification
        • Partial sums
          • Partial sums intro
          • Partial sums: formula for nth term from partial sum
          • Partial sums: term value from partial sum
          • Infinite series as limit of partial sums
        • Infinite geometric series
          • Worked example: convergent geometric series
          • Worked example: divergent geometric series
          • Infinite geometric series formula intuition
          • Proof of infinite geometric series as a limit
          • Infinite geometric series word problem: bouncing ball
          • Infinite geometric series word problem: repeating decimal
          • Convergent & divergent geometric series (with manipulation)
        • Series basics challenge
          • Telescoping series
          • Divergent telescoping series
          • Sum of n squares (part 1)
          • Sum of n squares (part 2)
          • Sum of n squares (part 3)
          • Worked example: evaluating series (advanced)
      • Series convergence & estimation
        • Basic convergence tests
          • nth term divergence test
          • Integral test
          • Worked example: Integral test
          • Worked example: p-series
          • Proof of p-series convergence criteria
        • Comparison tests
          • Direct comparison test
          • Worked example: direct comparison test
          • Limit comparison test
          • Worked example: limit comparison test
          • Proof: harmonic series diverges
        • Ratio & alternating series tests
          • Ratio test
          • Alternating series test
          • Conditional & absolute convergence
          • Worked example: alternating series
        • Estimating infinite series
          • Series estimation with integrals
          • Worked example: Series estimation with integrals
          • Alternating series remainder
          • Worked example: alternating series remainder
      • Power series
        • Power series intro
          • Power series intro
          • Worked example: interval of convergence
          • Integrating power series
          • Differentiating power series
          • Interval of convergence for derivative and integral
        • Taylor & Maclaurin polynomials intro
          • Taylor & Maclaurin polynomials intro (part 1)
          • Taylor & Maclaurin polynomials intro (part 2)
          • Worked example: Maclaurin polynomial
          • Worked example: coefficient in Maclaurin polynomial
          • Worked example: coefficient in Taylor polynomial
          • Taylor polynomial remainder (part 1)
          • Taylor polynomial remainder (part 2)
          • Worked example: estimating sin(0.4) using Lagrange error bound
          • Worked example: estimating eˣ using Lagrange error bound
          • Visualizing Taylor polynomial approximations
          • Worked example: Taylor polynomial of derivative function
        • Maclaurin series of sin(x), cos(x), and eˣ
          • Maclaurin series of cos(x)
          • Maclaurin series of sin(x)
          • Maclaurin series of eˣ
          • Euler's formula & Euler's identity
          • Worked example: power series from cos(x)
          • Worked example: cosine function from power series
          • Worked example: recognizing function from Taylor series
          • Visualizing Taylor series approximations
        • Power series function representation
          • Function as a geometric series
          • Geometric series as a function
          • Geometric series interval of convergence
          • Power series of arctan(2x)
          • Power series of ln(1+x³)
          • Finding function from power series by integrating
        • Challenge series exercises
    • AP Calculus AB
      • Limits and continuity
        • About the course
          • Khan Academy in the classroom
          • Sal interviews the AP Calculus Lead at College Board
        • Analyzing limits graphically
          • Limits intro
          • Approximating limits
          • Limits from graphs
        • Analyzing limits numerically
          • Approximating limits using tables
          • Estimating limits from tables
        • One-sided limits
          • One-sided limits from graphs
          • One-sided limits from graphs: asymptote
          • One-sided limits from tables
          • Connecting limits and graphical behavior
        • Continuity
          • Continuity at a point
          • Worked example: Continuity at a point
          • Continuity over an interval
          • Functions continuous on all real numbers
          • Functions continuous at specific x-values
        • Limit rules for sums, differences, products, quotients, & composites
          • Limit properties
          • Limits of combined functions
          • Limits of composite functions
        • Determining limits using direct substitution
          • Limits by direct substitution
          • Undefined limits by direct substitution
        • Determining limits by factoring & rationalizing
          • Limits by factoring
          • Limits by rationalizing
        • Trigonometric limits & squeeze theorem
          • Limits of trigonometric functions
          • Trig limit using pythagorean identity
          • Trig limit using double angle identity
          • Squeeze theorem intro
          • Squeeze theorem example
          • Limit of sin(x)/x as x approaches 0
        • Infinite limits intro
          • Infinite limits intro
          • Infinite limits and asymptotes
          • Functions with same limit at infinity
          • Analyzing unbounded limits: rational function
          • Analyzing unbounded limits: mixed function
        • Analyzing limits at infinity
          • Limits at infinity of quotients (Part 1)
          • Limits at infinity of quotients (Part 2)
          • Limits at infinity of quotients with square roots (odd power)
          • Limits at infinity of quotients with square roots (even power)
          • Limits at infinity of quotients with trig
          • Limits at infinity of quotients with trig (limit undefined)
        • Analyzing functions for discontinuities
          • Types of discontinuities
          • Analyzing functions for discontinuities (continuous example)
          • Analyzing functions for discontinuities (discontinuity example)
          • Removing discontinuities (factoring)
          • Removing discontinuities (rationalization)
        • Strategy in finding limits
          • Strategy in finding limits
        • Optional videos
          • Formal definition of limits Part 1: intuition review
          • Formal definition of limits Part 2: building the idea
          • Formal definition of limits Part 3: the definition
          • Formal definition of limits Part 4: using the definition
      • Derivatives introduction
        • The derivative: An intuitive introduction
          • Newton, Leibniz, and Usain Bolt
          • Derivative as a concept
          • Derivative as slope of curve
          • Derivative & the direction of a function
        • Estimating derivatives with secant lines and average rate of change
          • Secant lines & average rate of change
          • Estimating derivatives
        • Secant lines & average rate of change with arbitrary points
          • Secant line with arbitrary difference
          • Secant line with arbitrary point
          • Secant line with arbitrary difference (with simplification)
          • Secant line with arbitrary point (with simplification)
        • Formal definition of the derivative as a limit
          • Formal definition of the derivative as a limit
          • Formal and alternate form of the derivative
          • Worked example: Derivative as a limit
          • Worked example: Derivative from limit expression
          • The derivative of x² at x=3 using the formal definition
          • The derivative of x² at any point using the formal definition
        • Differentiability
          • Differentiability and continuity
          • Differentiability at a point: graphical
          • Differentiability at a point: algebraic (function is differentiable)
          • Differentiability at a point: algebraic (function isn't differentiable)
        • Tangent lines and local linearity
          • The derivative & tangent line equations
          • Local linearity
          • Local linearity and differentiability
        • Optional videos
          • Proof: Differentiability implies continuity
      • Derivative rules
        • Basic differentiation rules
          • Basic derivative rules (Part 1)
          • Basic derivative rules (Part 2)
          • Basic derivative rules: find the error
          • Basic derivative rules: table
        • Power rule
          • Power rule
        • Polynomial functions differentiation
          • Differentiating polynomials
          • Differentiating polynomials example
          • Tangents of polynomials
        • Derivatives of negative and fractional powers with power rule
          • Negative powers differentiation
          • Fractional powers differentiation
          • Radical functions differentiation intro
          • Worked example: Tangent to the graph of 1/x
        • Derivatives of sin(x) and cos(x)
          • Derivatives of sin(x) and cos(x)
          • Worked example: Derivatives of sin(x) and cos(x)
        • Product rule
          • Product rule
          • Worked example: Product rule with table
          • Worked example: Product rule with mixed implicit & explicit
        • Quotient rule
          • Quotient rule
          • Worked example: Quotient rule with table
        • Rational functions differentiation
          • Differentiating rational functions
          • Linear approximation of a rational function
        • Chain rule
          • Chain rule
          • Identifying composite functions
          • Worked example: Derivative of cos³(x) using the chain rule
          • Worked example: Derivative of √(3x²-x) using the chain rule
          • Common chain rule misunderstandings
          • Worked example: Chain rule with table
        • Radical functions differentiation
          • Radical functions differentiation
        • Trigonometric functions differentiation
          • Derivatives of tan(x) and cot(x)
          • Derivatives of sec(x) and csc(x)
          • Trig functions differentiation
        • Strategy in differentiating functions
          • Differentiating functions: Find the error
          • Manipulating functions before differentiation
        • Optional videos
          • Justifying the power rule
          • Proof of power rule for positive integer powers
          • Proof of power rule for square root function
          • Limit of sin(x)/x as x approaches 0
          • Limit of (1-cos(x))/x as x approaches 0
          • Proof of the derivative of sin(x)
          • Proof of the derivative of cos(x)
          • Product rule proof
          • Proof: Differentiability implies continuity
          • If function u is continuous at x, then Δu→0 As Δx→0
          • Chain rule proof
          • Quotient rule from product & chain rules
      • Advanced derivatives
        • Differentiating using multiple rules
          • Differentiating using multiple rules: strategy
          • Applying the chain rule and product rule
          • Applying the chain rule twice
        • Second derivatives
          • Second derivatives
        • Implicit differentiation
          • Implicit differentiation
          • Worked example: Implicit differentiation
          • Worked example: Evaluating derivative with implicit differentiation
          • Differentiating related functions intro
          • Worked example: Differentiating related functions
          • Showing explicit and implicit differentiation give same result
        • Derivatives of inverse functions
          • Derivatives of inverse functions
          • Derivatives of inverse functions: from equation
          • Derivatives of inverse functions: from table
        • Derivatives of inverse trigonometric functions
          • Derivative of inverse sine
          • Derivative of inverse cosine
          • Derivative of inverse tangent
        • Derivatives of exponential functions
          • 𝑒 and compound interest
          • 𝑒 as a limit
          • Derivative of 𝑒ˣ
          • Exponential functions differentiation intro
          • Exponential functions differentiation
          • Derivative of eᶜᵒˢˣ⋅cos(eˣ)
        • Derivatives of logarithmic functions
          • Derivative of ln(x)
          • Derivative of logₐx (for any positive base a≠1)
          • Logarithmic functions differentiation
          • Worked example: Derivative of ln(√x) using the chain rule
          • Derivative of sin(ln(x²))
          • Differentiating logarithmic functions using log properties
        • Optional videos
          • Proof: The derivative of 𝑒ˣ is 𝑒ˣ
          • Proof: the derivative of ln(x) is 1/x
          • Derivative of ln(x) from derivative of 𝑒ˣ and implicit differentiation
      • Existence theorems
        • Intermediate and extreme value theorems
          • Existence theorems intro
          • Intermediate value theorem
          • Extreme value theorem
          • Conditions for IVT and EVT: graph
          • Conditions for IVT and EVT: table
          • Intermediate value theorem example
        • Mean value theorem
          • Mean value theorem
          • Conditions for MVT: graph
          • Conditions for MVT: table
          • Mean value theorem example: polynomial
          • Mean value theorem example: square root function
          • Mean value theorem application
      • Using derivatives to analyze functions
        • L'Hôpital's rule
          • L'Hôpital's rule introduction
          • L'Hôpital's rule: limit at 0 example
          • L'Hôpital's rule: limit at infinity example
        • Justifying properties of functions using the first derivative
          • Introduction to minimum and maximum points
          • Justification using first derivative
          • Calculus based justification for function increasing
          • Analyzing a function with its derivative
        • Critical points & intervals of increase or decrease
          • Critical points introduction
          • Finding critical points
          • Finding decreasing interval given the function
          • Finding increasing interval given the derivative
        • Finding relative extrema (first derivative test)
          • Finding relative extrema (first derivative test)
          • Worked example: finding relative extrema
          • Analyzing mistakes when finding extrema (example 1)
          • Analyzing mistakes when finding extrema (example 2)
        • Finding absolute extrema
          • Finding absolute extrema on a closed interval
          • Absolute minima & maxima (entire domain)
        • Concavity & inflection points intro
          • Concavity introduction
          • Analyzing concavity (graphical)
          • Inflection points introduction
          • Inflection points (graphical)
        • Justifying properties of functions using the second derivative
          • Inflection points from graphs of function & derivatives
          • Justification using second derivative: maximum point
          • Justification using second derivative: inflection point
          • Worked example: Inflection points from first derivative
          • Worked example: Inflection points from second derivative
          • Second derivative test
        • Finding inflection points & analyzing concavity
          • Analyzing concavity (algebraic)
          • Inflection points (algebraic)
          • Mistakes when finding inflection points: second derivative undefined
          • Mistakes when finding inflection points: not checking candidates
        • Connecting ƒ, ƒ’, and ƒ’’
          • The graphical relationship between a function & its derivative (part 1)
          • The graphical relationship between a function & its derivative (part 2)
          • Matching functions & their derivatives graphically
          • Connecting f, f', and f'' graphically
          • Connecting f, f', and f'' graphically (another example)
          • Curve sketching with calculus: polynomial
          • Curve sketching with calculus: logarithm
        • Optional videos
          • Proof of special case of l'Hôpital's rule
      • Applications of derivatives
        • Rates of change in applied contexts
          • Applied rate of change: forgetfulness
          • Marginal cost & differential calculus
        • Related rates intro
          • Related rates intro
          • Analyzing related rates problems: expressions
          • Analyzing related rates problems: equations (Pythagoras)
          • Analyzing related rates problems: equations (trig)
        • Related rates problems
          • Related rates: Approaching cars
          • Related rates: Falling ladder
          • Related rates: water pouring into a cone
          • Related rates: shadow
          • Related rates: balloon
        • Optimization
          • Optimization: sum of squares
          • Optimization: box volume (Part 1)
          • Optimization: box volume (Part 2)
          • Optimization: profit
          • Optimization: cost of materials
          • Optimization: area of triangle & square (Part 1)
          • Optimization: area of triangle & square (Part 2)
          • Optimization problem: extreme normaline to y=x²
        • Motion problems
          • Motion problems: when a particle is speeding up
          • Motion problems: finding the maximum acceleration
      • Accumulation and Riemann sums
        • Definite integrals intro
          • Introduction to integral calculus
          • Definite integrals intro
          • Worked example: Definite integral by thinking about the function's graph
        • Left & right Riemann sums
          • Riemann approximation introduction
          • Over- and under-estimation of Riemann sums
          • Worked example: finding a Riemann sum using a table
          • Worked example: over- and under-estimation of Riemann sums
        • Midpoint & trapezoidal sums
          • Midpoint sums
          • Trapezoidal sums
        • Summation notation
          • Summation notation
          • Worked examples: Summation notation
        • Riemann sums in summation notation
          • Riemann sums in summation notation
          • Worked example: Riemann sums in summation notation
        • Definite integral as the limit of a Riemann sum
          • Definite integral as the limit of a Riemann sum
          • Worked example: Rewriting definite integral as limit of Riemann sum
          • Worked example: Rewriting limit of Riemann sum as definite integral
        • Definite integrals properties
          • Breaking up integral interval
          • Definite integral over a single point
          • Switching bounds of definite integral
          • Worked examples: Definite integral properties 1
          • Integrating sums of functions
          • Integrating scaled version of function
          • Worked examples: Definite integral properties 2
          • Definite integral properties (no graph): function combination
          • Definite integral properties (no graph): breaking interval
        • Functions defined by integrals
          • Functions defined by integrals
          • Functions defined by integrals: switched interval
      • Antiderivatives and the fundamental theorem of calculus
        • Antiderivatives and indefinite integrals intro
          • Antiderivatives and indefinite integrals
        • Fundamental theorem of calculus
          • Fundamental theorem of calculus (Part 1)
          • Fundamental theorem of calculus (Part 2)
          • Worked example: Finding derivative with fundamental theorem of calculus
          • Finding derivative with fundamental theorem of calculus: x is on lower bound
          • Finding derivative with fundamental theorem of calculus: x is on both bounds
        • Indefinite integrals: reverse power rule
          • Reverse power rule
          • Indefinite integrals: sums & multiples
          • Rewriting before integrating
        • Indefinite integrals of sin(x), cos(x), eˣ, and 1/x
          • Indefinite integral of 1/x
          • Indefinite integrals of sin(x), cos(x), and eˣ
        • Finding definite integrals
          • Definite integrals: reverse power rule
          • Definite integral of rational function
          • Definite integral of radical function
          • Definite integral of trig function
          • Definite integral involving natural log
          • Definite integral of piecewise function
          • Definite integral of absolute value function
        • 𝘶-substitution
          • 𝘶-substitution intro
          • 𝘶-substitution: multiplying by a constant
          • 𝘶-substitution: defining 𝘶
          • 𝘶-substitution: defining 𝘶 (more examples)
          • 𝘶-substitution: rational function
          • 𝘶-substitution: logarithmic function
          • 𝘶-substitution: definite integrals
          • 𝘶-substitution: definite integral of exponential function
        • Average value of a function
          • Average value over a closed interval
          • Calculating average value of function over interval
          • Mean value theorem for integrals
        • Interpreting behavior of 𝑔 from graph of 𝑔'=ƒ
          • Interpreting behavior of 𝑔 from graph of 𝑔'=ƒ
          • Visually determining antiderivative
        • Optional videos
          • Proof of fundamental theorem of calculus
          • Intuition for second part of fundamental theorem of calculus
      • Differential equations
        • Differential equations intro
          • Differential equations introduction
          • Writing a differential equation
        • Separable differential equations
          • Separable equations introduction
          • Addressing treating differentials algebraically
          • Worked example: separable differential equations
          • Worked example: identifying separable equations
        • Differential equations: specific solutions
          • Finding specific antiderivatives: rational function
          • Finding specific antiderivatives: exponential function
          • Worked example: finding a specific solution to a separable equation
          • Worked example: separable equation with an implicit solution
        • Slope fields
          • Slope fields introduction
          • Worked example: equation from slope field
          • Worked example: slope field from equation
          • Worked example: forming a slope field
          • Solutions from slope field
          • Worked example: range of solution curve from slope field
        • Exponential models
          • Exponential models & differential equations (Part 1)
          • Exponential models & differential equations (Part 2)
          • Worked example: exponential solution to differential equation
      • Applications of definite integrals
        • Word problems involving definite integrals
          • Area under rate function gives the net change
          • Interpreting definite integral as net change
          • Worked examples: interpreting definite integrals in context
          • Analyzing problems involving definite integrals
          • Worked example: problem involving definite integral (graphical)
          • Worked example: problem involving definite integral (algebraic)
        • Motion problems (with integrals)
          • Motion problems with integrals: displacement vs. distance
          • Analyzing motion problems: position
          • Analyzing motion problems: total distance traveled
          • Worked example: motion problems (with definite integrals)
          • Average acceleration over interval
        • Area: vertical areas between curves
          • Area between a curve and the x-axis
          • Area between a curve and the x-axis: negative area
          • Area between curves
          • Worked example: area between curves
        • Area: horizontal areas between curves
          • Area between a curve and and the 𝘺-axis
          • Horizontal area between curves
        • Volume: known cross sections
          • Volume with cross sections: semicircle
          • Volume with cross sections: triangle
        • Volume: disc method
          • Disc method around x-axis
          • Generalizing disc method around x-axis
          • Disc method around y-axis
          • Disc method rotation around horizontal line
          • Disc method rotating around vertical line
          • Calculating integral disc around vertical line
        • Volume: washer method
          • Solid of revolution between two functions (leading up to the washer method)
          • Generalizing the washer method
          • Washer method rotating around horizontal line (not x-axis), part 1
          • Washer method rotating around horizontal line (not x-axis), part 2
          • Washer method rotating around vertical line (not y-axis), part 1
          • Washer method rotating around vertical line (not y-axis), part 2
        • Volume: shell method (optional)
          • Shell method for rotating around vertical line
          • Evaluating integral for shell method example
          • Shell method for rotating around horizontal line
          • Shell method with two functions of x
          • Calculating integral with shell method
          • Shell method with two functions of y
          • Part 2 of shell method with 2 functions of y
      • AP Calculus AB solved exams
        • AP Calculus AB 2017
          • 2017 AP Calculus AB/BC 4a
          • 2017 AP Calculus AB/BC 4b
          • 2017 AP Calculus AB/BC 4c
        • AP Calculus AB 2015
          • 2015 AP Calculus AB/BC 1ab
          • 2015 AP Calculus AB/BC 1c
          • 2015 AP Calculus AB/BC 1d
          • 2015 AP Calculus AB 2a
          • 2015 AP Calculus AP 2b
          • 2015 AP Calculus 2c
          • 2015 AP Calculus AB/BC 3a
          • 2015 AP Calculus AB/BC 3b
          • 2015 AP Calculus AB/BC 3cd
          • 2015 AP Calculus AB/BC 4ab
          • 2015 AP Calculus AB/BC 4cd
          • 2015 AP Calculus AB 5a
          • 2015 AP Calculus AB 5b
          • 2015 AP Calculus AB 5c
          • 2015 AP Calculus AB 5d
          • 2015 AP Calculus AB 6a
          • 2015 AP Calculus AB 6b
          • 2015 AP Calculus AB 6c
        • AP Calculus AB 2011
          • 2011 Calculus AB free response #1a
          • 2011 Calculus AB Free Response #1 parts b c d
          • 2011 Calculus AB free response #2 (a & b)
          • 2011 Calculus AB free response #2 (c & d)
          • 2011 Calculus AB free response #3 (a & b)
          • 2011 Calculus AB free response #3 (c)
          • 2011 Calculus AB free response #4a
          • 2011 Calculus AB free response #4b
          • 2011 Calculus AB free response #4c
          • 2011 Calculus AB free response #4d
          • 2011 Calculus AB free response #5a
          • 2011 Calculus AB free response #5b
          • 2011 Calculus AB free response #5c.
          • 2011 Calculus AB free response #6a
          • 2011 Calculus AB free response #6b
          • 2011 Calculus AB free response #6c
    • AP Calculus BC
      • Limits and continuity
        • About the course
          • Khan Academy in the classroom
          • Sal interviews the AP Calculus Lead at College Board
        • Analyzing limits graphically
          • Limits intro
          • Approximating limits
          • Limits from graphs
        • Analyzing limits numerically
          • Approximating limits using tables
          • Estimating limits from tables
        • One-sided limits
          • One-sided limits from graphs
          • One-sided limits from graphs: asymptote
          • One-sided limits from tables
          • Connecting limits and graphical behavior
        • Continuity
          • Continuity at a point
          • Worked example: Continuity at a point
          • Continuity over an interval
          • Functions continuous on all real numbers
          • Functions continuous at specific x-values
        • Limit rules for sums, differences, products, quotients, & composites
          • Limit properties
          • Limits of combined functions
          • Limits of composite functions
        • Determining limits using direct substitution
          • Limits by direct substitution
          • Undefined limits by direct substitution
        • Determining limits by factoring & rationalizing
          • Limits by factoring
          • Limits by rationalizing
        • Trigonometric limits & squeeze theorem
          • Limits of trigonometric functions
          • Trig limit using pythagorean identity
          • Trig limit using double angle identity
          • Squeeze theorem intro
          • Squeeze theorem example
          • Limit of sin(x)/x as x approaches 0
        • Infinite limits intro
          • Infinite limits intro
          • Infinite limits and asymptotes
          • Functions with same limit at infinity
          • Analyzing unbounded limits: rational function
          • Analyzing unbounded limits: mixed function
        • Analyzing limits at infinity
          • Limits at infinity of quotients (Part 1)
          • Limits at infinity of quotients (Part 2)
          • Limits at infinity of quotients with square roots (odd power)
          • Limits at infinity of quotients with square roots (even power)
          • Limits at infinity of quotients with trig
          • Limits at infinity of quotients with trig (limit undefined)
        • Analyzing functions for discontinuities
          • Types of discontinuities
          • Analyzing functions for discontinuities (continuous example)
          • Analyzing functions for discontinuities (discontinuity example)
          • Removing discontinuities (factoring)
          • Removing discontinuities (rationalization)
        • Strategy in finding limits
          • Strategy in finding limits
        • Optional videos
          • Formal definition of limits Part 1: intuition review
          • Formal definition of limits Part 2: building the idea
          • Formal definition of limits Part 3: the definition
          • Formal definition of limits Part 4: using the definition
      • Derivatives introduction
        • The derivative: An intuitive introduction
          • Newton, Leibniz, and Usain Bolt
          • Derivative as a concept
          • Derivative as slope of curve
          • Derivative & the direction of a function
        • Estimating derivatives with secant lines and average rate of change
          • Secant lines & average rate of change
          • Estimating derivatives
        • Secant lines & average rate of change with arbitrary points
          • Secant line with arbitrary difference
          • Secant line with arbitrary point
          • Secant line with arbitrary difference (with simplification)
          • Secant line with arbitrary point (with simplification)
        • Formal definition of the derivative as a limit
          • Formal definition of the derivative as a limit
          • Formal and alternate form of the derivative
          • Worked example: Derivative as a limit
          • Worked example: Derivative from limit expression
          • The derivative of x² at x=3 using the formal definition
          • The derivative of x² at any point using the formal definition
        • Differentiability
          • Differentiability and continuity
          • Differentiability at a point: graphical
          • Differentiability at a point: algebraic (function is differentiable)
          • Differentiability at a point: algebraic (function isn't differentiable)
        • Tangent lines and local linearity
          • The derivative & tangent line equations
          • Local linearity
          • Local linearity and differentiability
        • Optional videos
          • Proof: Differentiability implies continuity
      • Derivative rules
        • Basic differentiation rules
          • Basic derivative rules (Part 1)
          • Basic derivative rules (Part 2)
          • Basic derivative rules: find the error
          • Basic derivative rules: table
        • Power rule
          • Power rule
        • Polynomial functions differentiation
          • Differentiating polynomials
          • Differentiating polynomials example
          • Tangents of polynomials
        • Derivatives of negative and fractional powers with power rule
          • Negative powers differentiation
          • Fractional powers differentiation
          • Radical functions differentiation intro
          • Worked example: Tangent to the graph of 1/x
        • Derivatives of sin(x) and cos(x)
          • Derivatives of sin(x) and cos(x)
          • Worked example: Derivatives of sin(x) and cos(x)
        • Product rule
          • Product rule
          • Differentiating products
          • Worked example: Product rule with table
          • Worked example: Product rule with mixed implicit & explicit
        • Quotient rule
          • Quotient rule
          • Worked example: Quotient rule with table
        • Rational functions differentiation
          • Differentiating rational functions
          • Linear approximation of a rational function
        • Chain rule
          • Chain rule
          • Identifying composite functions
          • Worked example: Derivative of cos³(x) using the chain rule
          • Worked example: Derivative of √(3x²-x) using the chain rule
          • Common chain rule misunderstandings
          • Worked example: Chain rule with table
        • Radical functions differentiation
          • Radical functions differentiation
        • Trigonometric functions differentiation
          • Derivatives of tan(x) and cot(x)
          • Derivatives of sec(x) and csc(x)
          • Trig functions differentiation
        • Strategy in differentiating functions
          • Differentiating functions: Find the error
          • Manipulating functions before differentiation
        • Optional videos
          • Justifying the power rule
          • Proof of power rule for positive integer powers
          • Proof of power rule for square root function
          • Limit of sin(x)/x as x approaches 0
          • Limit of (1-cos(x))/x as x approaches 0
          • Proof of the derivative of sin(x)
          • Proof of the derivative of cos(x)
          • Product rule proof
          • Proof: Differentiability implies continuity
          • If function u is continuous at x, then Δu→0 As Δx→0
          • Chain rule proof
          • Quotient rule from product & chain rules
      • Advanced derivatives
        • Differentiating using multiple rules
          • Differentiating using multiple rules: strategy
          • Applying the chain rule and product rule
          • Applying the chain rule twice
        • Second derivatives
          • Second derivatives
        • Implicit differentiation
          • Implicit differentiation
          • Worked example: Implicit differentiation
          • Worked example: Evaluating derivative with implicit differentiation
          • Differentiating related functions intro
          • Worked example: Differentiating related functions
          • Showing explicit and implicit differentiation give same result
        • Derivatives of inverse functions
          • Derivatives of inverse functions
          • Derivatives of inverse functions: from equation
          • Derivatives of inverse functions: from table
        • Derivatives of inverse trigonometric functions
          • Derivative of inverse sine
          • Derivative of inverse cosine
          • Derivative of inverse tangent
        • Derivatives of exponential functions
          • 𝑒 and compound interest
          • 𝑒 as a limit
          • Derivative of 𝑒ˣ
          • Exponential functions differentiation intro
          • Exponential functions differentiation
          • Derivative of eᶜᵒˢˣ⋅cos(eˣ)
        • Derivatives of logarithmic functions
          • Derivative of ln(x)
          • Derivative of logₐx (for any positive base a≠1)
          • Logarithmic functions differentiation
          • Worked example: Derivative of ln(√x) using the chain rule
          • Derivative of sin(ln(x²))
          • Differentiating logarithmic functions using log properties
        • Parametric & vector-valued function differentiation
          • Parametric functions intro
          • Parametric functions differentiation
          • Vector-valued functions intro
          • Vector-valued functions differentiation
          • Second derivatives (parametric functions)
          • Second derivatives (vector-valued functions)
        • Polar functions differentiation
          • Polar functions derivatives
          • Worked example: differentiating polar functions
        • Optional videos
          • Proof: The derivative of 𝑒ˣ is 𝑒ˣ
          • Proof: the derivative of ln(x) is 1/x
          • Derivative of ln(x) from derivative of 𝑒ˣ and implicit differentiation
          • Derivatives expressed as limits
      • Existence theorems
        • Intermediate and extreme value theorems
          • Existence theorems intro
          • Intermediate value theorem
          • Extreme value theorem
          • Conditions for IVT and EVT: graph
          • Conditions for IVT and EVT: table
          • Intermediate value theorem example
        • Mean value theorem
          • Mean value theorem
          • Conditions for MVT: graph
          • Conditions for MVT: table
          • Mean value theorem example: polynomial
          • Mean value theorem example: square root function
          • Mean value theorem application
      • Using derivatives to analyze functions
        • L'Hôpital's rule
          • L'Hôpital's rule introduction
          • L'Hôpital's rule: limit at 0 example
          • L'Hôpital's rule: limit at infinity example
        • Justifying properties of functions using the first derivative
          • Introduction to minimum and maximum points
          • Justification using first derivative
          • Calculus based justification for function increasing
          • Analyzing a function with its derivative
        • Critical points & intervals of increase or decrease
          • Critical points introduction
          • Finding critical points
          • Finding decreasing interval given the function
          • Finding increasing interval given the derivative
        • Finding relative extrema (first derivative test)
          • Finding relative extrema (first derivative test)
          • Worked example: finding relative extrema
          • Analyzing mistakes when finding extrema (example 1)
          • Analyzing mistakes when finding extrema (example 2)
        • Finding absolute extrema
          • Finding absolute extrema on a closed interval
          • Absolute minima & maxima (entire domain)
        • Concavity & inflection points intro
          • Concavity introduction
          • Analyzing concavity (graphical)
          • Inflection points introduction
          • Inflection points (graphical)
        • Justifying properties of functions using the second derivative
          • Inflection points from graphs of function & derivatives
          • Justification using second derivative: maximum point
          • Justification using second derivative: inflection point
          • Worked example: Inflection points from first derivative
          • Worked example: Inflection points from second derivative
          • Second derivative test
        • Finding inflection points & analyzing concavity
          • Analyzing concavity (algebraic)
          • Inflection points (algebraic)
          • Mistakes when finding inflection points: second derivative undefined
          • Mistakes when finding inflection points: not checking candidates
        • Connecting ƒ, ƒ’, and ƒ’’
          • The graphical relationship between a function & its derivative (part 1)
          • The graphical relationship between a function & its derivative (part 2)
          • Matching functions & their derivatives graphically
          • Connecting f, f', and f'' graphically
          • Connecting f, f', and f'' graphically (another example)
          • Curve sketching with calculus: polynomial
          • Curve sketching with calculus: logarithm
        • Optional videos
          • Proof of special case of l'Hôpital's rule
      • Applications of derivatives
        • Rates of change in applied contexts
          • Applied rate of change: forgetfulness
          • Marginal cost & differential calculus
        • Related rates intro
          • Related rates intro
          • Analyzing related rates problems: expressions
          • Analyzing related rates problems: equations (Pythagoras)
          • Analyzing related rates problems: equations (trig)
        • Related rates problems
          • Related rates: Approaching cars
          • Related rates: Falling ladder
          • Related rates: water pouring into a cone
          • Related rates: shadow
          • Related rates: balloon
        • Optimization
          • Optimization: sum of squares
          • Optimization: box volume (Part 1)
          • Optimization: box volume (Part 2)
          • Optimization: profit
          • Optimization: cost of materials
          • Optimization: area of triangle & square (Part 1)
          • Optimization: area of triangle & square (Part 2)
          • Optimization problem: extreme normaline to y=x²
        • Planar motion
          • Planar motion example: acceleration vector
          • Motion along a curve: finding rate of change
          • Motion along a curve: finding velocity magnitude
      • Accumulation and Riemann sums
        • Definite integrals intro
          • Introduction to integral calculus
          • Definite integrals intro
          • Worked example: Definite integral by thinking about the function's graph
        • Left & right Riemann sums
          • Riemann approximation introduction
          • Over- and under-estimation of Riemann sums
          • Worked example: finding a Riemann sum using a table
          • Worked example: over- and under-estimation of Riemann sums
        • Midpoint & trapezoidal sums
          • Midpoint sums
          • Trapezoidal sums
        • Summation notation
          • Summation notation
          • Worked examples: Summation notation
        • Riemann sums in summation notation
          • Riemann sums in summation notation
          • Worked example: Riemann sums in summation notation
        • Definite integral as the limit of a Riemann sum
          • Definite integral as the limit of a Riemann sum
          • Worked example: Rewriting definite integral as limit of Riemann sum
          • Worked example: Rewriting limit of Riemann sum as definite integral
        • Definite integrals properties
          • Breaking up integral interval
          • Definite integral over a single point
          • Switching bounds of definite integral
          • Worked examples: Definite integral properties 1
          • Integrating sums of functions
          • Integrating scaled version of function
          • Worked examples: Definite integral properties 2
          • Definite integral properties (no graph): function combination
          • Definite integral properties (no graph): breaking interval
        • Functions defined by integrals
          • Functions defined by integrals
          • Functions defined by integrals: switched interval
      • Antiderivatives and the fundamental theorem of calculus
        • Antiderivatives and indefinite integrals intro
          • Antiderivatives and indefinite integrals
        • Fundamental theorem of calculus
          • Fundamental theorem of calculus (Part 1)
          • Fundamental theorem of calculus (Part 2)
          • Worked example: Finding derivative with fundamental theorem of calculus
          • Finding derivative with fundamental theorem of calculus: x is on lower bound
          • Finding derivative with fundamental theorem of calculus: x is on both bounds
        • Indefinite integrals: reverse power rule
          • Reverse power rule
          • Indefinite integrals: sums & multiples
          • Rewriting before integrating
        • Indefinite integrals of sin(x), cos(x), eˣ, and 1/x
          • Indefinite integral of 1/x
          • Indefinite integrals of sin(x), cos(x), and eˣ
        • Finding definite integrals
          • Definite integrals: reverse power rule
          • Definite integral of rational function
          • Definite integral of radical function
          • Definite integral of trig function
          • Definite integral involving natural log
          • Definite integral of piecewise function
          • Definite integral of absolute value function
        • Improper integrals
          • Introduction to improper integrals
          • Improper integral with two infinite bounds
          • Divergent improper integral
        • 𝘶-substitution
          • 𝘶-substitution intro
          • 𝘶-substitution: multiplying by a constant
          • 𝘶-substitution: defining 𝘶
          • 𝘶-substitution: defining 𝘶 (more examples)
          • 𝘶-substitution: rational function
          • 𝘶-substitution: logarithmic function
          • 𝘶-substitution: definite integrals
          • 𝘶-substitution: definite integral of exponential function
        • Integration by parts
          • Integration by parts intro
          • Integration by parts: ∫x⋅cos(x)dx
          • Integration by parts: ∫ln(x)dx
          • Integration by parts: ∫x²⋅𝑒ˣdx
          • Integration by parts: ∫𝑒ˣ⋅cos(x)dx
          • Integration by parts: definite integrals
        • Partial fraction expansion
          • Partial fraction expansion to evaluate integral
          • Dividing expressions to evaluate integral
        • Average value of a function
          • Average value over a closed interval
          • Calculating average value of function over interval
          • Mean value theorem for integrals
        • Interpreting behavior of 𝑔 from graph of 𝑔'=ƒ
          • Interpreting behavior of 𝑔 from graph of 𝑔'=ƒ
          • Visually determining antiderivative
        • Optional videos
          • Proof of fundamental theorem of calculus
          • Intuition for second part of fundamental theorem of calculus
      • Differential equations
        • Differential equations intro
          • Differential equations introduction
          • Writing a differential equation
        • Separable differential equations
          • Separable equations introduction
          • Addressing treating differentials algebraically
          • Worked example: separable differential equations
          • Worked example: separable differential equation (with taking log of both sides)
          • Worked example: separable differential equation (with taking exp of both sides)
          • Worked example: identifying separable equations
        • Differential equations: specific solutions
          • Finding specific antiderivatives: rational function
          • Finding specific antiderivatives: exponential function
          • Worked example: finding a specific solution to a separable equation
          • Worked example: separable equation with an implicit solution
        • Slope fields
          • Slope fields introduction
          • Worked example: equation from slope field
          • Worked example: slope field from equation
          • Worked example: forming a slope field
          • Solutions from slope field
          • Worked example: range of solution curve from slope field
        • Euler's Method
          • Euler's method
          • Worked example: Euler's method
        • Exponential models
          • Exponential models & differential equations (Part 1)
          • Exponential models & differential equations (Part 2)
          • Worked example: exponential solution to differential equation
        • Logistic models
          • Logistic models & differential equations (Part 1)
          • Logistic models & differential equations (Part 2)
          • Worked example: Logistic model word problem
          • Logistic equations (Part 1)
          • Logistic equations (Part 2)
          • Worked example: logistic model equations
      • Applications of definite integrals
        • Word problems involving definite integrals
          • Area under rate function gives the net change
          • Interpreting definite integral as net change
          • Worked examples: interpreting definite integrals in context
          • Analyzing problems involving definite integrals
          • Worked example: problem involving definite integral (graphical)
          • Worked example: problem involving definite integral (algebraic)
        • Motion problems (with integrals)
          • Motion problems with integrals: displacement vs. distance
          • Analyzing motion problems: position
          • Analyzing motion problems: total distance traveled
          • Worked example: motion problems (with definite integrals)
          • Planar motion (with integrals)
          • Average acceleration over interval
        • Area: vertical areas between curves
          • Area between a curve and the x-axis
          • Area between a curve and the x-axis: negative area
          • Area between curves
          • Worked example: area between curves
        • Area: horizontal areas between curves
          • Area between a curve and and the 𝘺-axis
          • Horizontal area between curves
        • Area: polar curves
          • Area bounded by polar curves
          • Worked example: Area enclosed by cardioid
          • Worked example: Area between two polar graphs
          • Evaluating definite integral with calculator
        • Arc length
          • Arc length intro
          • Worked example: arc length
          • Parametric curve arc length
          • Worked example: Parametric arc length
        • Volume: known cross sections
          • Volume with cross sections: semicircle
          • Volume with cross sections: triangle
        • Volume: disc method
          • Disc method around x-axis
          • Generalizing disc method around x-axis
          • Disc method around y-axis
          • Disc method rotation around horizontal line
          • Disc method rotating around vertical line
          • Calculating integral disc around vertical line
        • Volume: washer method
          • Solid of revolution between two functions (leading up to the washer method)
          • Generalizing the washer method
          • Washer method rotating around horizontal line (not x-axis), part 1
          • Washer method rotating around horizontal line (not x-axis), part 2
          • Washer method rotating around vertical line (not y-axis), part 1
          • Washer method rotating around vertical line (not y-axis), part 2
        • Volume: shell method (optional)
          • Shell method for rotating around vertical line
          • Evaluating integral for shell method example
          • Shell method for rotating around horizontal line
          • Shell method with two functions of x
          • Calculating integral with shell method
          • Shell method with two functions of y
          • Part 2 of shell method with 2 functions of y
      • Series
        • Infinite sequences
          • Sequences intro
          • Worked example: sequence explicit formula
          • Worked example: sequence recursive formula
          • Convergent and divergent sequences
          • Worked example: sequence convergence/divergence
        • Finite geometric series
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
        • Partial sums
          • Partial sums intro
          • Partial sums: formula for nth term from partial sum
          • Partial sums: term value from partial sum
          • Infinite series as limit of partial sums
        • Infinite geometric series
          • Worked example: convergent geometric series
          • Worked example: divergent geometric series
          • Infinite geometric series word problem: bouncing ball
          • Infinite geometric series word problem: repeating decimal
        • Basic convergence tests
          • nth term divergence test
          • Integral test
          • Worked example: Integral test
          • Worked example: p-series
        • Comparison tests
          • Direct comparison test
          • Worked example: direct comparison test
          • Limit comparison test
          • Worked example: limit comparison test
          • Proof: harmonic series diverges
        • Ratio & alternating series tests
          • Ratio test
          • Alternating series test
          • Conditional & absolute convergence
          • Worked example: alternating series
        • Estimating infinite series
          • Series estimation with integrals
          • Worked example: Series estimation with integrals
          • Alternating series remainder
          • Worked example: alternating series remainder
        • Power series intro
          • Power series intro
          • Worked example: interval of convergence
          • Integrating power series
          • Differentiating power series
          • Interval of convergence for derivative and integral
        • Taylor & Maclaurin polynomials intro
          • Taylor & Maclaurin polynomials intro (part 1)
          • Taylor & Maclaurin polynomials intro (part 2)
          • Worked example: Maclaurin polynomial
          • Worked example: coefficient in Maclaurin polynomial
          • Worked example: coefficient in Taylor polynomial
          • Taylor polynomial remainder (part 1)
          • Taylor polynomial remainder (part 2)
          • Worked example: estimating sin(0.4) using Lagrange error bound
          • Worked example: estimating eˣ using Lagrange error bound
          • Visualizing Taylor polynomial approximations
          • Worked example: Taylor polynomial of derivative function
        • Maclaurin series of sin(x), cos(x), and eˣ
          • Maclaurin series of cos(x)
          • Maclaurin series of sin(x)
          • Maclaurin series of eˣ
          • Euler's formula & Euler's identity
          • Worked example: power series from cos(x)
          • Worked example: cosine function from power series
          • Worked example: recognizing function from Taylor series
          • Visualizing Taylor series approximations
        • Power series function representation
          • Function as a geometric series
          • Geometric series as a function
          • Geometric series interval of convergence
          • Power series of arctan(2x)
          • Power series of ln(1+x³)
          • Finding function from power series by integrating
        • Optional videos
          • Formal definition for limit of a sequence
          • Proving a sequence converges using the formal definition
          • Finite geometric series formula justification
          • Infinite geometric series formula intuition
          • Proof of infinite geometric series as a limit
          • Proof of p-series convergence criteria
      • AP Calculus BC solved exams
        • AP Calculus BC 2017
          • 2017 AP Calculus AB/BC 4a
          • 2017 AP Calculus AB/BC 4b
          • 2017 AP Calculus AB/BC 4c
        • AP Calculus BC 2015
          • 2015 AP Calculus BC 2a
          • 2015 AP Calculus BC 2b
          • 2015 AP Calculus BC 2c
          • 2015 AP Calculus BC 2d
          • 2015 AP Calculus BC 5a
          • 2015 AP Calculus BC 5b
          • 2015 AP Calculus BC 5c
          • 2015 AP Calculus BC 5d
          • 2015 AP Calculus BC 6a
          • 2015 AP Calculus BC 6b
          • 2015 AP Calculus BC 6c
        • AP Calculus BC 2011
          • 2011 Calculus BC free response #1a
          • 2011 Calculus BC free response #1 (b & c)
          • 2011 Calculus BC free response #1d
          • 2011 Calculus BC free response #3a
          • 2011 Calculus BC free response #3 (b & c)
          • 2011 Calculus BC free response #6a
          • 2011 Calculus BC free response #6b
          • 2011 Calculus BC free response #6c
          • 2011 Calculus BC free response #6d
        • AP Calculus BC 2008
          • AP Calculus BC exams: 2008 1 a
          • AP Calculus BC exams: 2008 1 b&c
          • AP Calculus BC exams: 2008 1 c&d
          • AP Calculus BC exams: 2008 1 d
          • Calculus BC 2008 2 a
          • Calculus BC 2008 2 b &c
          • Calculus BC 2008 2d
    • Multivariable calculus
      • Thinking about multivariable functions
        • Introduction to multivariable calculus
          • Multivariable functions
        • Skill check
        • Visualizing scalar-valued functions
          • Representing points in 3d
          • Introduction to 3d graphs
          • Interpreting graphs with slices
          • Contour plots
        • Visualizing vector-valued functions
          • Parametric curves
          • Parametric surfaces
          • Vector fields, introduction
          • Fluid flow and vector fields
          • 3d vector fields, introduction
          • 3d vector field example
        • Transformations
          • Transformations, part 1
          • Transformations, part 2
          • Transformations, part 3
        • Visualizing multivariable functions (articles)
      • Derivatives of multivariable functions
        • Skill checks
        • Partial derivatives
          • Partial derivatives, introduction
          • Graphical understanding of partial derivatives
          • Formal definition of partial derivatives
          • Symmetry of second partial derivatives
        • Gradient and directional derivatives
          • Gradient
          • Gradient and graphs
          • Gradient and contour maps
          • Directional derivative
          • Directional derivative, formal definition
          • Directional derivatives and slope
          • Why the gradient is the direction of steepest ascent
        • Partial derivative and gradient (articles)
        • Differentiating parametric curves
          • Vector-valued functions intro
          • Vector-valued functions differentiation
          • Differential of a vector valued function
          • Vector valued function derivative example
        • Multivariable chain rule
          • Multivariable chain rule
          • Multivariable chain rule intuition
          • Vector form of the multivariable chain rule
          • Multivariable chain rule and directional derivatives
          • More formal treatment of multivariable chain rule
        • Curvature
          • Curvature intuition
          • Curvature formula, part 1
          • Curvature formula, part 2
          • Curvature formula, part 3
          • Curvature formula, part 4
          • Curvature formula, part 5
          • Curvature of a helix, part 1
          • Curvature of a helix, part 2
          • Curvature of a cycloid
        • Partial derivatives of vector-valued functions
          • Computing the partial derivative of a vector-valued function
          • Partial derivative of a parametric surface, part 1
          • Partial derivative of a parametric surface, part 2
          • Partial derivatives of vector fields
          • Partial derivatives of vector fields, component by component
        • Differentiating vector-valued functions (articles)
        • Divergence
          • Divergence intuition, part 1
          • Divergence intuition, part 2
          • Divergence formula, part 1
          • Divergence formula, part 2
          • Divergence example
          • Divergence notation
        • Curl
          • 2d curl intuition
          • 2d curl formula
          • 2d curl example
          • 2d curl nuance
          • Describing rotation in 3d with a vector
          • 3d curl intuition, part 1
          • 3d curl intuition, part 2
          • 3d curl formula, part 1
          • 3d curl formula, part 2
          • 3d curl computation example
        • Divergence and curl (articles)
        • Laplacian
          • Laplacian intuition
          • Laplacian computation example
          • Explicit Laplacian formula
          • Harmonic Functions
        • Jacobian
          • Jacobian prerequisite knowledge
          • Local linearity for a multivariable function
          • The Jacobian matrix
          • Computing a Jacobian matrix
          • The Jacobian Determinant
      • Applications of multivariable derivatives
        • Tangent planes and local linearization
          • What is a tangent plane
          • Controlling a plane in space
          • Computing a tangent plane
          • Local linearization
        • Quadratic approximations
          • What do quadratic approximations look like
          • Quadratic approximation formula, part 1
          • Quadratic approximation formula, part 2
          • Quadratic approximation example
          • The Hessian matrix
          • Expressing a quadratic form with a matrix
          • Vector form of multivariable quadratic approximation
        • Optimizing multivariable functions
          • Multivariable maxima and minima
          • Saddle points
          • Warm up to the second partial derivative test
          • Second partial derivative test
          • Second partial derivative test intuition
          • Second partial derivative test example, part 1
          • Second partial derivative test example, part 2
        • Optimizing multivariable functions (articles)
        • Lagrange multipliers and constrained optimization
          • Constrained optimization introduction
          • Lagrange multipliers, using tangency to solve constrained optimization
          • Finishing the intro lagrange multiplier example
          • Lagrange multiplier example, part 1
          • Lagrange multiplier example, part 2
          • The Lagrangian
          • Meaning of the Lagrange multiplier
          • Proof for the meaning of Lagrange multipliers
        • Constrained optimization (articles)
      • Integrating multivariable functions
        • Line integrals for scalar functions (videos)
          • Introduction to the line integral
          • Line integral example 1
          • Line integral example 2 (part 1)
          • Line integral example 2 (part 2)
        • Line integrals for scalar functions (articles)
        • Line integrals in vector fields (videos)
          • Line integrals and vector fields
          • Using a line integral to find work
          • Parametrization of a reverse path
          • Scalar field line integral independent of path direction
          • Vector field line integrals dependent on path direction
          • Path independence for line integrals
          • Closed curve line integrals of conservative vector fields
          • Example of closed line integral of conservative field
          • Second example of line integral of conservative vector field
        • Line integrals in vector fields (articles)
        • Double integrals (videos)
          • Double integral 1
          • Double integrals 2
          • Double integrals 3
          • Double integrals 4
          • Double integrals 5
          • Double integrals 6
        • Double integrals (articles)
          • Polar coordinates
        • Triple integrals (videos)
          • Triple integrals 1
          • Triple integrals 2
          • Triple integrals 3
        • Triple integrals (articles)
        • Surface integral preliminaries (videos)
          • Parametrizing a surface, part 1
          • Determining a position vector-valued function for a parametrization of two parameters
          • Partial derivatives of vector-valued functions
        • Surface integrals (videos)
          • Introduction to the surface integral
          • Example of calculating a surface integral part 1
          • Example of calculating a surface integral part 2
          • Example of calculating a surface integral part 3
          • Surface integral example, part 1
          • Surface integral example part 2
          • Surface integral example part 3: The home stretch
          • Surface integral ex2 part 1
          • Surface integral ex2 part 2
          • Surface integral ex3 part 1
          • Surface integral ex3 part 2
          • Surface integral ex3 part 3
          • Surface integral ex3 part 4
        • Surface integrals (articles)
        • Flux in 3D (videos)
          • Conceptual understanding of flux
          • Constructing a unit normal vector
          • Vector representation of a surface integral
        • Flux in 3D (articles)
      • Green's, Stokes', and the divergence theorems
        • Formal definitions of div and curl (optional reading)
        • Green's theorem (videos)
          • Green's theorem proof (part 1)
          • Green's theorem proof (part 2)
          • Green's theorem example 1
          • Green's theorem example 2
        • Green's theorem (articles)
        • 2D divergence theorem
          • Constructing a unit normal vector to a curve
          • 2D divergence theorem
          • Conceptual clarification for 2D divergence theorem
        • Stokes' theorem (videos)
          • Stokes' theorem intuition
          • Green's and Stokes' theorem relationship
          • Orienting boundary with surface
          • Orientation and stokes
          • Conditions for stokes theorem
          • Stokes example part 1
          • Stokes example part 2
          • Stokes example part 3
          • Stokes example part 4
          • Evaluating line integral directly - part 1
          • Evaluating line integral directly - part 2
        • Stokes' theorem (articles)
        • 3D divergence theorem (videos)
          • 3D divergence theorem intuition
          • Divergence theorem example 1
          • Explanation of example 1
        • Divergence theorem (articles)
        • Proof of Stokes' theorem
          • Stokes' theorem proof part 1
          • Stokes' theorem proof part 2
          • Stokes' theorem proof part 3
          • Stokes' theorem proof part 4
          • Stokes' theorem proof part 5
          • Stokes' theorem proof part 6
          • Stokes' theorem proof part 7
        • Types of regions in three dimensions
          • Type I regions in three dimensions
          • Type II regions in three dimensions
          • Type III regions in three dimensions
        • Divergence theorem proof
          • Divergence theorem proof (part 1)
          • Divergence theorem proof (part 2)
          • Divergence theorem proof (part 3)
          • Divergence theorem proof (part 4)
          • Divergence theorem proof (part 5)
    • Differential equations
      • First order differential equations
        • Intro to differential equations
          • Differential equations introduction
          • Writing a differential equation
          • Worked example: linear solution to differential equation
        • Slope fields
          • Slope fields introduction
          • Worked example: equation from slope field
          • Worked example: slope field from equation
          • Worked example: forming a slope field
          • Solutions from slope field
          • Worked example: range of solution curve from slope field
        • Euler's Method
          • Euler's method
          • Worked example: Euler's method
        • Separable equations
          • Separable equations introduction
          • Addressing treating differentials algebraically
          • Worked example: identifying separable equations
          • Worked example: finding a specific solution to a separable equation
          • Worked example: separable equation with an implicit solution
          • Separable equations (old)
          • Separable equations example (old)
        • Exponential models
          • Exponential models & differential equations (Part 1)
          • Exponential models & differential equations (Part 2)
          • Worked example: exponential solution to differential equation
          • Newton's Law of Cooling
          • Worked example: Newton's law of cooling
        • Logistic models
          • Logistic models & differential equations (Part 1)
          • Logistic models & differential equations (Part 2)
          • Logistic equations (Part 1)
          • Logistic equations (Part 2)
          • Worked example: logistic model equations
        • Exact equations and integrating factors
          • Exact equations intuition 1 (proofy)
          • Exact equations intuition 2 (proofy)
          • Exact equations example 1
          • Exact equations example 2
          • Exact equations example 3
          • Integrating factors 1
          • Integrating factors 2
        • Homogeneous equations
          • First order homogenous equations
          • First order homogeneous equations 2
      • Second order linear equations
        • Linear homogeneous equations
          • 2nd order linear homogeneous differential equations 1
          • 2nd order linear homogeneous differential equations 2
          • 2nd order linear homogeneous differential equations 3
          • 2nd order linear homogeneous differential equations 4
        • Complex and repeated roots of characteristic equation
          • Complex roots of the characteristic equations 1
          • Complex roots of the characteristic equations 2
          • Complex roots of the characteristic equations 3
          • Repeated roots of the characteristic equation
          • Repeated roots of the characteristic equations part 2
        • Method of undetermined coefficients
          • Undetermined coefficients 1
          • Undetermined coefficients 2
          • Undetermined coefficients 3
          • Undetermined coefficients 4
      • Laplace transform
        • Laplace transform
          • Laplace transform 1
          • Laplace transform 2
          • L{sin(at)} - transform of sin(at)
          • Part 2 of the transform of the sin(at)
        • Properties of the Laplace transform
          • Laplace as linear operator and Laplace of derivatives
          • Laplace transform of cos t and polynomials
          • "Shifting" transform by multiplying function by exponential
          • Laplace transform of t: L{t}
          • Laplace transform of t^n: L{t^n}
          • Laplace transform of the unit step function
          • Inverse Laplace examples
          • Dirac delta function
          • Laplace transform of the dirac delta function
        • Laplace transform to solve a differential equation
          • Laplace transform to solve an equation
          • Laplace transform solves an equation 2
          • Using the Laplace transform to solve a nonhomogeneous eq
          • Laplace/step function differential equation
        • The convolution integral
          • Introduction to the convolution
          • The convolution and the laplace transform
          • Using the convolution theorem to solve an initial value prob
    • Linear algebra
      • Vectors and spaces
        • Vectors
          • Vector intro for linear algebra
          • Real coordinate spaces
          • Adding vectors algebraically & graphically
          • Multiplying a vector by a scalar
          • Vector examples
          • Unit vectors intro
          • Parametric representations of lines
        • Linear combinations and spans
          • Linear combinations and span
        • Linear dependence and independence
          • Introduction to linear independence
          • More on linear independence
          • Span and linear independence example
        • Subspaces and the basis for a subspace
          • Linear subspaces
          • Basis of a subspace
        • Vector dot and cross products
          • Vector dot product and vector length
          • Proving vector dot product properties
          • Proof of the Cauchy-Schwarz inequality
          • Vector triangle inequality
          • Defining the angle between vectors
          • Defining a plane in R3 with a point and normal vector
          • Cross product introduction
          • Proof: Relationship between cross product and sin of angle
          • Dot and cross product comparison/intuition
          • Vector triple product expansion (very optional)
          • Normal vector from plane equation
          • Point distance to plane
          • Distance between planes
        • Matrices for solving systems by elimination
          • Solving a system of 3 equations and 4 variables using matrix row-echelon form
          • Solving linear systems with matrices
          • Using matrix row-echelon form in order to show a linear system has no solutions
        • Null space and column space
          • Matrix vector products
          • Introduction to the null space of a matrix
          • Null space 2: Calculating the null space of a matrix
          • Null space 3: Relation to linear independence
          • Column space of a matrix
          • Null space and column space basis
          • Visualizing a column space as a plane in R3
          • Proof: Any subspace basis has same number of elements
          • Dimension of the null space or nullity
          • Dimension of the column space or rank
          • Showing relation between basis cols and pivot cols
          • Showing that the candidate basis does span C(A)
      • Matrix transformations
        • Functions and linear transformations
          • A more formal understanding of functions
          • Vector transformations
          • Linear transformations
          • Matrix vector products as linear transformations
          • Linear transformations as matrix vector products
          • Image of a subset under a transformation
          • im(T): Image of a transformation
          • Preimage of a set
          • Preimage and kernel example
          • Sums and scalar multiples of linear transformations
          • More on matrix addition and scalar multiplication
        • Linear transformation examples
          • Linear transformation examples: Scaling and reflections
          • Linear transformation examples: Rotations in R2
          • Rotation in R3 around the x-axis
          • Unit vectors
          • Introduction to projections
          • Expressing a projection on to a line as a matrix vector prod
        • Transformations and matrix multiplication
          • Compositions of linear transformations 1
          • Compositions of linear transformations 2
          • Matrix product examples
          • Matrix product associativity
          • Distributive property of matrix products
        • Inverse functions and transformations
          • Introduction to the inverse of a function
          • Proof: Invertibility implies a unique solution to f(x)=y
          • Surjective (onto) and injective (one-to-one) functions
          • Relating invertibility to being onto and one-to-one
          • Determining whether a transformation is onto
          • Exploring the solution set of Ax = b
          • Matrix condition for one-to-one transformation
          • Simplifying conditions for invertibility
          • Showing that inverses are linear
        • Finding inverses and determinants
          • Deriving a method for determining inverses
          • Example of finding matrix inverse
          • Formula for 2x2 inverse
          • 3 x 3 determinant
          • n x n determinant
          • Determinants along other rows/cols
          • Rule of Sarrus of determinants
        • More determinant depth
          • Determinant when row multiplied by scalar
          • (correction) scalar multiplication of row
          • Determinant when row is added
          • Duplicate row determinant
          • Determinant after row operations
          • Upper triangular determinant
          • Simpler 4x4 determinant
          • Determinant and area of a parallelogram
          • Determinant as scaling factor
        • Transpose of a matrix
          • Transpose of a matrix
          • Determinant of transpose
          • Transpose of a matrix product
          • Transposes of sums and inverses
          • Transpose of a vector
          • Rowspace and left nullspace
          • Visualizations of left nullspace and rowspace
          • rank(a) = rank(transpose of a)
          • Showing that A-transpose x A is invertible
      • Alternate coordinate systems (bases)
        • Orthogonal complements
          • Orthogonal complements
          • dim(v) + dim(orthogonal complement of v) = n
          • Representing vectors in rn using subspace members
          • Orthogonal complement of the orthogonal complement
          • Orthogonal complement of the nullspace
          • Unique rowspace solution to Ax = b
          • Rowspace solution to Ax = b example
        • Orthogonal projections
          • Projections onto subspaces
          • Visualizing a projection onto a plane
          • A projection onto a subspace is a linear transformation
          • Subspace projection matrix example
          • Another example of a projection matrix
          • Projection is closest vector in subspace
          • Least squares approximation
          • Least squares examples
          • Another least squares example
        • Change of basis
          • Coordinates with respect to a basis
          • Change of basis matrix
          • Invertible change of basis matrix
          • Transformation matrix with respect to a basis
          • Alternate basis transformation matrix example
          • Alternate basis transformation matrix example part 2
          • Changing coordinate systems to help find a transformation matrix
        • Orthonormal bases and the Gram-Schmidt process
          • Introduction to orthonormal bases
          • Coordinates with respect to orthonormal bases
          • Projections onto subspaces with orthonormal bases
          • Finding projection onto subspace with orthonormal basis example
          • Example using orthogonal change-of-basis matrix to find transformation matrix
          • Orthogonal matrices preserve angles and lengths
          • The Gram-Schmidt process
          • Gram-Schmidt process example
          • Gram-Schmidt example with 3 basis vectors
        • Eigen-everything
          • Introduction to eigenvalues and eigenvectors
          • Proof of formula for determining eigenvalues
          • Example solving for the eigenvalues of a 2x2 matrix
          • Finding eigenvectors and eigenspaces example
          • Eigenvalues of a 3x3 matrix
          • Eigenvectors and eigenspaces for a 3x3 matrix
          • Showing that an eigenbasis makes for good coordinate systems
    • 6th grade (Ontario)
      • Number Sense and Numeration
        • Ratios, rates and percentages
        • Fractions
          • Fractions on number line widget
        • Multiplication and division
        • Place value and decimal operations
        • Factors and multiples
        • Properties of numbers
          • Commutative law of addition
          • Commutative law of multiplication
      • Measurement
        • Unit conversion
          • Converting units: centimeters to meters
          • Converting units: US volume
        • Area
          • Multiplying with area model: 78 x 65
          • Area of composite shapes
        • Volume
          • Measuring volume as area times length
          • Volume of triangular prism & cube
      • Geometry and Spatial Sense
        • Coordinate plane
          • Intro to the coordinate plane
          • Points on the coordinate plane examples
          • Plotting a point (ordered pair)
        • Angles and polygons
        • Transformations
      • Patterning and Algebra
        • Number patterns
          • Graphing patterns on coordinate plane
        • Variables and expressions
          • Expressions with two variables
          • Combining like terms
          • Combining like terms challenge problem
      • Data Management and Probability
        • Data management
        • Statistics
          • Mean, median, & mode example
          • Missing value given the mean
        • Probability
          • Intro to theoretical probability
    • 6th grade (WNCP)
      • Number
        • Place value and decimals
          • Regrouping numbers into various place values
          • Comparing whole number place values
        • Prime numbers, factors and multiples
        • Mixed numbers and improper fractions
        • Ratios and percentages
        • Integers negative numbers
        • Multiplication and division
          • Multiplying a decimal by a power of 10
          • Intro to multiplying decimals
          • Dividing a decimal by a power of 10: pattern
        • Properties of numbers and order of operations
          • Commutative law of addition
          • Commutative law of multiplication
      • Patterns and Relations
        • Number patterns
          • Factors and multiples: days of the week
          • Math patterns: toothpicks
          • Interpreting relationships in ordered pairs
          • Graphing sequence relationships
        • Variables and equations
          • Expressions with two variables
          • Combining like terms
          • Combining like terms challenge problem
      • Shape and Space
        • Angles
          • Angles in a triangle sum to 180° proof
        • Perimeter and Area
          • Perimeter & area
          • Perimeter & area of composite shapes
        • Volume
        • Triangles and Polygons
          • Classifying triangles
          • Classifying triangles by angles
          • Sum of interior angles of a polygon
        • Coordinate plane
          • Coordinate plane: graphing points
          • Plotting a point (ordered pair)
        • Transformations
          • Intro to reflective symmetry
          • Rigid transformations intro
          • Performing translations
          • Performing rotations
          • Performing reflections
          • Finding a quadrilateral from its symmetries
          • Finding a quadrilateral from its symmetries (example 2)
      • Statistics and Probability
        • Data
          • Reading line graphs
          • Misleading line graphs
        • Probability
          • Intro to theoretical probability
          • Coin flipping probability
          • Example: All the ways you can flip a coin
          • Die rolling probability
    • 3rd grade (Eureka Math/EngageNY)
      • Module 1: Properties of multiplication and division and solving problems with units of 2–5 and 10
        • Topic A: Multiplication and the meaning of the factors
          • Intro to multiplication
          • Multiplication as groups of objects
        • Topic B: Division as an unknown factor problem
          • The idea of division
        • Topic C: Multiplication using units of 2 and 3
          • Properties of multiplication
          • Properties and patterns for multiplication
        • Topic D: Division using units of 2 and 3
        • Topic E: Multiplication and division using units of 4
          • Relating division to multiplication
          • Multiplication word problem: parking lot
          • Division word problem: school building
          • Multiplication word problem: soda party
          • Division word problem: blueberries
        • Topic F: Distributive property and problem solving using units of 2–5 and 10
          • Properties of multiplication
          • 2-step word problem: theater
          • 2-step word problem: truffles
          • 2-step word problem: running
          • 2-step estimation problem: marbles
      • Module 2: Place value and problem solving with units of measure
        • Topic A: Time measurement and problem solving
          • Telling time (labeled clock)
          • Telling time (unlabeled clock)
          • Telling time to the nearest minute (labeled clock)
          • Telling time to the nearest minute (unlabeled clock)
          • Time differences example
          • Telling time with number line
          • Time word problem: puzzle
          • Time word problem: travel time
        • Topic B: Measuring weight and liquid volume in metric units
          • Understanding mass (grams and kilograms)
          • Word problems with mass
          • Understanding volume (liters)
          • Word problems with volume
        • Topic C: Rounding to the nearest ten and hundred
          • Rounding to nearest 10
          • Rounding to nearest 100
          • Rounding to nearest 10 and 100
        • Topic D: Two- and three-digit measurement addition using the standard algorithm
          • Intro to place value
          • Adding 3-digit numbers
          • Adding with regrouping
          • Breaking apart 3-digit addition problems
          • Addition using groups of 10 and 100
          • Adding 3-digit numbers with regrouping
        • Topic E: Two- and three-digit measurement subtraction using the standard algorithm
          • Adding and subtracting on number line 2
          • Subtracting in your head (no regrouping)
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Subtracting 3-digit numbers (regrouping) 2
          • Worked example: Subtracting 3-digit numbers (regrouping twice)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
          • Missing number for addition and subtraction within 1000
      • Module 3: Multiplication and division with units of 0, 1, 6–9, and multiples of 10
        • Topic A: The properties of multiplication and division
          • Unknowns with multiplication and division
        • Topic B: Multiplication and division using units of 6 and 7
        • Topic C: Multiplication and division using units up to 8
          • More ways to multiply
        • Topic D: Multiplication and division using units of 9
          • Intro to even and odd numbers
          • Finding patterns in numbers
        • Topic E: Analysis of patterns and problem solving including units of 0 and 1
          • Identity property of 1
          • Patterns in multiplication tables
          • 2-step estimation problem: marbles
          • 2-step word problem: running
      • Module 4: Multiplication and area
        • Topic A: Foundations for understanding area
          • Intro to area and unit squares
          • Measuring rectangles with different unit squares
        • Topic B: Concepts of area measurement
          • Measuring area with partial unit squares
          • Counting unit squares to find area formula
          • Transitioning from unit squares to area formula
          • Finding missing side when given area
        • Topic C: Arithmetic properties using area models
          • Area and the distributive property
          • Creating rectangles with a given area 1
          • Creating rectangles with a given area 2
        • Topic D: Applications of area using side lengths of figures
          • Decomposing shapes to find area: grids
          • Decomposing shapes to find area: subtract
          • Decomposing shapes to find area: add
          • Area word problem: house size
          • Comparing areas of plots of land
      • Module 5: Fractions as numbers on the number line
        • Topic A: Partitioning a whole into equal parts
          • Intro to fractions
          • Cutting shapes into equal parts
          • Identifying unit fractions word problem
        • Topic B: Unit fractions and their relation to the whole
          • More than one equal section
          • Recognizing fractions greater than 1
          • Identifying numerators and denominators
        • Topic C: Comparing unit fractions and specifying the whole
          • Comparing fractions of different wholes 1
          • Comparing fractions of different wholes 2
        • Topic D: Fractions on the number line
          • Fractions on a number line
          • Fractions on number line widget
          • Representing 1 as a fraction
        • Topic E: Equivalent fractions
          • Equivalent fractions with visuals
          • Equivalent fraction models
          • Equivalent fraction visually
          • Creating equivalent fractions
          • Whole numbers as fractions
        • Topic F: Comparison, order, and size of fractions
          • Comparing fractions with > and < symbols
          • Comparing fractions visually
          • Comparing fractions with the same denominator
          • Comparing fractions with the same numerator
      • Module 6: Collecting and displaying data
        • Topic A: Generate and analyze categorical data
          • Solving problems with pictographs 2
          • Interpreting picture graphs: paint
          • Interpreting picture graphs: notebook
          • Creating a bar graph
          • Reading bar graphs: Harry Potter
          • Reading bar graphs: movies
          • Interpreting bar graphs: colors
          • Interpreting bar graphs: alligators
        • Topic B: Generate and analyze measurement data
          • Intro to line plots
          • Graphing data on line plots
      • Module 7: Geometry and measurement word problems
        • Topic A: Solving word problems
        • Topic B: Attributes of two-dimensional figures
          • Intro to quadrilaterals
        • Topic C: Problem solving with perimeter
          • Perimeter: introduction
          • Perimeter of a shape
          • Finding perimeter when a side length is missing
          • Finding missing side length when given perimeter
          • Perimeter word problem: skating rink
          • Perimeter word problem: tables
        • Topic D: Recording perimeter and area data on line plots
        • Topic E: Problem solving with perimeter and area
          • Comparing areas and perimeters of rectangles
          • Area & perimeter word problem: table
          • Area & perimeter word problem: dog pen
    • 4th grade (Eureka Math/EngageNY)
      • Module 1: Place value, rounding, and algorithms for addition and subtraction
        • Topic A: Place value of multi-digit whole numbers
          • Comparing with multiplication: basics
          • Comparing with multiplication: giraffe
          • Comparing with multiplication: age
          • Finding place value
          • Creating the largest number
          • Place value blocks
          • Multiplying whole numbers by 10
          • Dividing whole numbers by 10
          • Understanding place value
          • Comparing place values
          • Place value when multiplying and dividing by 10
          • Writing a number in expanded form
          • Writing a number in standard form
          • Regrouping numbers into various place values
          • Regrouping whole number place values
          • Adding whole numbers by their place values
        • Topic B: Comparing multi-digit whole numbers
          • Comparing multi-digit numbers
          • Comparing whole number place values
          • Comparing multi-digit whole numbers word problems
        • Topic C: Rounding multi-digit whole numbers
          • Rounding whole numbers to nearest hundred
          • Rounding whole numbers to nearest thousand
          • Rounding whole numbers to nearest ten
          • Rounding whole numbers word problems
          • Rounding whole numbers: missing digit
        • Topic D: Multi-digit whole number addition
          • Adding 3-digit numbers
          • Adding with regrouping
          • Multi-digit addition with regrouping
        • Topic E: Multi-digit whole number subtraction
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
          • Multi-digit subtraction with regrouping: 6798-3359
          • Multi-digit subtraction with regrouping: 7329-6278
          • Multi-digit subtraction with regrouping twice
        • Topic F: Addition and subtraction word problems
      • Module 2: Unit conversions and problem solving with metric measurement
        • Topic A: Metric unit conversions
          • Metric system: units of weight
          • Metric system: units of distance
          • Metric system: units of volume
        • Topic B: Application of metric unit conversions
          • How to convert kg to mg and T to oz
          • Converting metric units of length
          • Multi-step unit conversion examples (metric)
      • Module 3: Multi-digit multiplication and division
        • Topic A: Multiplicative comparison word problems
          • Comparing with multiplication: magic
          • Comparing with multiplication: money
          • Area & perimeter word problem: table
          • Area & perimeter word problem: dog pen
          • Comparing areas word problem
        • Topic B: Multiplication by 10, 100, and 1,000
          • Multiplying 1-digit numbers by 10, 100, and 1000
          • Multiplying 1-digit numbers by multiples of 10, 100, and 1000
          • Multiplying 10s
        • Topic C: Multiplication of up to four digits by single-digit numbers
          • More ways to multiply
          • Multiplying with distributive property
          • Multiplying with area model: 6 x 7981
          • Multiplying with area model: 78 x 65
          • Multiplying with area model: 16 x 27
          • Multiplying 2-digit by 1-digit
          • Multiplying 3-digit by 1-digit
          • Multiplying: 2-digit by 1-digit (regrouping)
          • Multiplying 3-digit by 1-digit (regrouping)
          • Multiplying 4-digit by 1-digit (regrouping)
        • Topic D: Multiplication word problems
          • 2-step estimation problem: marbles
        • Topic E: Division of tens and ones with successive remainders
          • Quotients that are multiples of 10
          • Division using place value
          • Division with area models
          • Intro to remainders
          • Division word problem: field goals
          • Multiplication word problem: pizza
        • Topic F: Reasoning with divisibility
          • Finding factors of a number
          • Finding factors and multiples
          • Prime numbers
          • Recognizing prime and composite numbers
          • Factors and multiples: days of the week
          • Math patterns: table
          • Math patterns: toothpicks
        • Topic G: Division of thousands, hundreds, tens, and ones
          • Canceling zeros when dividing
          • Worked example: Long division with remainders: 2292÷4
          • Worked example: Long division with remainders: 3771÷8
          • Long division: 280÷5
          • Intro to long division (no remainders)
        • Topic H: Multiplication of two-digit by two-digit numbers
          • Multiplying 2-digit numbers
      • Module 4: Angle measure and plane figures
        • Topic A: Line and angles
          • Terms & labels in geometry
          • Lines, line segments, & rays
          • Angles: introduction
          • Naming angles
          • Recognizing angles
          • Acute, right, & obtuse angles
          • Drawing acute, right and obtuse angles
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines
          • Drawing parallel line segments
        • Topic B: Angle measurement
          • Angle measurement & circle arcs
          • Measuring angles with a circular protractor
          • Angles in circles word problem
          • Identifying an angle
          • Measuring angles in degrees
          • Measuring angles using a protractor
          • Measuring angles using a protractor 2
          • Constructing angles
        • Topic C: Problem solving with the addition of angle measures
          • Decomposing an angle
          • Decomposing angles
        • Topic D: Two-dimensional figures and symmetry
          • Intro to reflective symmetry
          • Identifying symmetrical figures
          • Classifying triangles by angles
          • Classifying triangles
          • Worked example: Classifying triangles
          • Classifying shapes by line and angles types
          • Quadrilateral properties
          • Classifying shapes by lines and angles
      • Module 5: Fraction equivalence, ordering, and operations
        • Topic A: Decomposition and fraction equivalence
          • Decomposing a fraction visually
          • Decomposing a mixed number
        • Topic B: Fraction equivalence using multiplication and division
          • Intro to equivalent fractions
          • Visualizing equivalent fractions
          • More on equivalent fractions
          • Equivalent fractions and different wholes
          • Finding common denominators
          • Common denominators: 3/5 and 7/2
        • Topic C: Fraction comparison
          • Comparing fractions: tape diagram
          • Comparing fractions: number line
          • Comparing fractions: fraction models
          • Comparing fractions 1 (unlike denominators)
          • Comparing fractions 2 (unlike denominators)
          • Ordering fractions
        • Topic D: Fraction addition and subtraction
          • Adding fractions with like denominators
          • Subtracting fractions with like denominators
          • Fraction word problem: pizza
          • Fraction word problem: piano
          • Fraction word problem: lizard
          • Fraction word problem: spider eyes
        • Topic E: Extending fraction equivalence to fractions greater than 1
          • Multiplying unit fractions and whole numbers
          • Rewriting mixed numbers as improper fractions
          • Rewriting improper fractions as mixed numbers
          • Comparing improper fractions and mixed numbers
          • Reading a line plot with fractions
        • Topic F: Addition and subtraction of fractions by decomposition
          • Intro to adding mixed numbers
          • Intro to subtracting mixed numbers
          • Fraction word problem: lizard
          • Subtracting mixed numbers with like denominators
        • Topic G: Repeated addition of fractions as multiplication
          • Multiplying fractions and whole numbers visually
          • Equivalent fraction and whole number multiplication problems
          • Multiplying fractions word problem: movies
          • Multiplying fractions word problem: milk
          • Multiplying fractions by whole numbers word problem
        • Topic H: Exploring a fraction pattern
      • Module 6: Decimal fractions
        • Topic A: Exploration of tenths
          • Intro to decimals
          • Decimals as words
          • Graphing tenths from 0 to 1
          • Identifying tenths on a number line
          • Visually converting tenths and hundredths
          • Decomposing hundredths on number line
          • Decomposing hundredths
        • Topic B: Tenths and hundredths
          • Writing decimal numbers shown in grids
          • Graphing hundredths from 0 to 0.1
          • Identifying hundredths on a number line
          • Plotting decimal numbers on a number line
          • Rewriting fractions as decimals
          • Writing a number as a fraction and decimal
          • Place value for decimals greater than 1
        • Topic C: Decimal comparison
          • Comparing decimals visually
          • Comparing numbers represented different ways
          • Comparing decimal numbers on a number line
          • Comparing decimals 1
          • Comparing decimals 2
          • Comparing decimals 3
        • Topic D: Addition with tenths and hundredths
          • Rewriting decimals as fractions: 0.15
          • Rewriting decimals as fractions: 0.8
          • Rewriting decimals as fractions: 0.36
          • Adding fractions (denominators 10 & 100)
          • Adding fractions: 7/10+13/100
        • Topic E: Money amounts as decimal numbers
      • Module 7: Exploring measurement with multiplication
        • Topic A: Measurement conversion tables
          • Converting US customary units of volume
          • Converting feet to inches
          • Converting units of time
          • Word problem: making change
          • Currency conversion word problem
        • Topic B: Problem solving with measurement
          • Choose pounds or ounces to measure weight
          • U.S. customary units: weight
          • U.S. customary units: distance
          • U.S. customary units: fluid volume
          • Multi-step unit conversion examples (U.S. Customary)
          • Time differences example
          • Time word problem: travel time
          • Time word problem: Susan's break
        • Topic C: Investigation of measurements expressed as mixed numbers
    • 5th grade (Eureka Math/EngageNY)
      • Module 1: Place value and decimal fractions
        • Topic A: Multiplicative patterns on the place value chart
          • Decimal place value
          • Patterns in powers of 10
          • Exponents and powers of 10 patterns
          • Multiplying a decimal by a power of 10
          • Dividing a decimal by a power of 10
          • Dividing a decimal by a power of 10: pattern
          • Fractions as division by power of 10
          • Comparing decimal place values
        • Topic B: Decimal fractions and place value patterns
          • Write decimal in expanded form
          • Convert decimal from expanded form to standard form
          • Decimals in written form (hundredths)
          • Decimals in written form (thousandths)
          • Comparing decimals: 9.97 and 9.798
          • Comparing decimals: 156.378 and 156.348
          • Ordering decimals 1
          • Ordering decimals 2
          • Comparing decimals word problems
        • Topic C: Place value and rounding decimal fractions
          • Worked example: Rounding decimals to nearest tenth
          • Rounding decimals word problems
        • Topic D: Adding and subtracting decimals
          • Introduction to adding decimals: tenths
          • Adding decimals with ones and tenths parts
          • Adding decimals with hundredths
          • Adding decimals with ones, tenths and hundredths
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
          • Strategies for subtracting basic decimals
          • Strategies for subtracting more complex decimals with tenths
          • More advanced subtraction strategies with hundredths
          • Subtraction strategies with hundredths
          • Subtracting decimals: 10.1-3.93
          • Subtracting decimals: 9.57-8.09
        • Topic E: Multiplying decimals
          • Intro to multiplying decimals
          • Developing strategies for multiplying decimals
        • Topic F: Dividing decimals
          • Dividing whole numbers to get a decimal
          • Division strategies for decimal quotients
          • Multi-digit division strategies for decimals
          • Strategies for dividing by tenths
      • Module 2: Multi-digit whole number and decimal fraction operations
        • Topic A: Mental strategies for multi-digit whole number multiplication
          • Multiplying by multiples of 10
          • Strategies for multiplying multiples of 10, 100 and 1000
        • Topic B: The standard algorithm for multi-digit whole number multiplication
          • More ways to multiply
          • Multiplying with distributive property
          • Multiplying with area model: 6 x 7981
          • Multiplying with area model: 78 x 65
          • Multiplying with area model: 16 x 27
          • Multiplying multi-digit numbers
        • Topic C: Decimal multi-digit multiplication
          • Multiplying decimals: place value
          • Developing strategies for multiplying 2-digit decimals
        • Topic D: Measurement word problems with whole number and decimal multiplication
          • Converting units: minutes to hours
          • Converting units: metric distance
          • Converting units: centimeters to meters
          • Converting units: US volume
          • Adding units of volume
          • Same length in different units
          • Time word problem: Susan's break
          • Measurement word problem: tea party
          • Measurement word problem: blood drive
          • Measurement word problem: distance home
          • Measurement word problem: elevator
          • Measurement word problem: running laps
        • Topic E: Mental strategies for multi-digit whole number division
        • Topic F: Partial quotients and multi-digit whole number division
          • Strategies for dividing multiples of 10, 100, and 1000
          • Introduction to dividing by 2-digits
          • Dividing by 2-digits: 9815÷65
          • Dividing by 2-digits: 7182÷42
        • Topic G: Partial quotients and multi-digit decimal division
          • Division strategies for decimal quotients
          • Multi-digit division strategies for decimals
          • Strategies for dividing by tenths
        • Topic H: Measurement word problems with multi-digit division
      • Module 3: Addition and subtractions of fractions
        • Topic A: Equivalent fractions
          • Intro to equivalent fractions
          • Visualizing equivalent fractions
          • More on equivalent fractions
          • Decomposing a fraction visually
          • Decomposing a mixed number
          • Rewriting mixed numbers as improper fractions
          • Rewriting improper fractions as mixed numbers
        • Topic B: Making like units pictorially
          • Visually adding fractions: 5/6+1/4
          • Visually subtracting fractions: 3/4-5/8
          • Finding common denominators
          • Adding fractions with unlike denominators
          • Subtracting fractions with unlike denominators
          • Adding fractions word problem: paint
          • Subtracting fractions word problem: tomatoes
        • Topic C: Making like units numerically
          • Adding mixed numbers: 19 3/18 + 18 2/3
          • Subtracting mixed numbers: 7 6/9 - 3 2/5
          • Adding mixed numbers with regrouping
          • Subtracting mixed numbers with regrouping (unlike denominators)
        • Topic D: Further applications
          • Adding and subtracting 3 fractions
          • Solving for the missing fraction
      • Module 4: Multiplication and division of fractions and decimal fractions
        • Topic A: Line plots of fraction measurements
          • Line plot distribution: trail mix
        • Topic B: Fractions as division
          • Understanding fractions as division
          • Creating mixed numbers with fraction division
          • Mixed numbers on a number line
          • Creating a fraction through division
        • Topic C: Multiplication of a whole number by a fraction
          • Multiplying fractions by whole numbers on a number line
          • Multiplying fractions and whole numbers visually
          • Multiplying unit fractions and whole numbers
          • Multiplying fractions and whole numbers
          • Multiplying fractions word problem: movies
          • Multiplying fractions word problem: milk
        • Topic D: Fraction expressions and word problems
          • Constructing numerical expressions
          • Evaluating expressions with & without parentheses
        • Topic E: Multiplication of a fraction by a fraction
          • Intro to multiplying 2 fractions
          • Multiplying 2 fractions: fraction model
          • Multiplying 2 fractions: number line
          • Multiplying 2 fractions: 5/6 x 2/3
          • Multiplying mixed numbers
          • Multiplying fractions word problem: laundry
          • Multiplying fractions word problem: muffins
          • Multiplying fractions word problem: bike
        • Topic F: Multiplication with fractions and decimals as scaling and word problems
          • Multiplication as scaling with fractions
        • Topic G: Division of fractions and decimal fractions
          • Visually dividing whole numbers by unit fractions
          • Dividing a whole number by a unit fraction
          • Visually dividing unit fraction by a whole number
          • Dividing a unit fraction by a whole number
          • Dividing whole numbers by fractions: word problem
          • Dividing fractions by whole numbers: studying
          • Rewriting a fraction as a decimal: 3/5
          • Rewriting a fraction as a decimal: 21/60
        • Topic H: Interpretation of numerical expressions
          • Translating expressions with parentheses
      • Module 5: Addition and multiplication with volume and area
        • Topic A: Concepts of volume
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
        • Topic B: Volume and the operations of multiplication and addition
          • Volume of a rectangular prism
          • Volume in unit cubes by decomposing shape
          • Volume through decomposition
          • Volume word problem: water tank
        • Topic C: Area of rectangular figures with fractional side lengths
          • Finding area with fractional sides 1
          • Finding area with fractional sides 2
        • Topic D: Drawing, analysis, and classification of two-dimensional shapes
          • Intro to quadrilateral
          • Quadrilateral properties
          • Kites as a geometric shape
          • Quadrilateral types
          • Classifying quadrilaterals
      • Module 6: Problem solving with the coordinate plane
        • Topic A: Coordinate systems
          • Introduction to the coordinate plane
          • Coordinate plane: graphing points
        • Topic B: Patterns in the coordinate plane and graphing number patterns from rules
          • Graphing patterns on coordinate plane
          • Interpreting patterns on coordinate plane
          • Interpreting relationships in ordered pairs
          • Graphing sequence relationships
        • Topic C: Drawing figures in the coordinate plane
        • Topic D: Problem solving in the coordinate plane
          • Plotting corners of a rectangle
          • Interpreting plotted points
          • Coordinate plane graphing word problem
    • 6th grade (Eureka Math/EngageNY)
      • Module 1: Ratios and unit rates
        • Topic A: Representing and reasoning about ratios
          • Intro to ratios
          • Basic ratios
          • Equivalent ratios
          • Equivalent ratios: recipe
          • Equivalent ratio word problems
          • Understanding equivalent ratios
        • Topic B: Collections of equivalent ratios
          • Part:whole ratios
          • Part:whole ratio word problem
        • Topic C: Unit rates
          • Intro to rates
          • Solving unit rate problem
          • Solving unit price problem
          • Comparing rates example
        • Topic D: Percents
          • The meaning of percent
          • Meaning of 109%
          • Converting percents to decimals & fractions example
          • Converting decimals to percents: 0.601
          • Converting decimals to percents: 1.501
          • Converting percents to decimals: 59.2%
          • Converting percents to decimals: 113.9%
          • Finding a percent
          • Percent of a whole number
          • Percent word problem: 100 is what percent of 80?
          • Percent word problem: 78 is 15% of what number?
          • Percent word problem: guavas
          • Percent word problem: penguins
          • Percent word problem: recycling cans
      • Module 2: Arithmetic operations including dividing by a fraction
        • Topic A: Dividing fractions by fractions
          • Understanding division of fractions
          • Dividing a fraction by a whole number
          • Dividing a whole number by a fraction
          • Dividing fractions: 2/5 ÷ 7/3
          • Dividing fractions: 3/5 ÷ 1/2
          • Dividing mixed numbers
          • Dividing whole numbers & fractions: t-shirts
        • Topic B: Multi‐digit decimal operations—adding, subtracting, and multiplying
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
          • Subtracting decimals: 39.1 - 0.794
          • Subtracting decimals: 9.005 - 3.6
          • Adding decimals word problem
          • Adding & subtracting decimals word problem
          • Intro to multiplying decimals
          • Multiplying decimals: place value
          • Multiplying challenging decimals
        • Topic C: Dividing whole numbers and decimals
          • Dividing by 2-digits: 9815÷65
          • Dividing by 2-digits: 7182÷42
          • Dividing whole numbers to get a decimal
          • Dividing a whole number by a decimal
          • Dividing a decimal by a whole number
          • Dividing decimals with hundredths
          • Dividing decimals completely
          • Long division with decimals
          • Dividing by a multi-digit decimal
        • Topic D: Number theory—thinking logically about multiplicative arithmetic
          • Divisibility tests for 2, 3, 4, 5, 6, 9, 10
          • Recognizing divisibility
          • The why of the 3 divisibility rule
          • The why of the 9 divisibility rule
          • Intro to even and odd numbers
          • Greatest common factor examples
          • Greatest common factor explained
          • Least common multiple
          • Least common multiple: repeating factors
          • Least common multiple of three numbers
          • GCF & LCM word problems
      • Module 3: Rational numbers
        • Topic A: Understanding positive and negative numbers on the number line
          • Intro to negative numbers
          • Negative decimals on the number line
          • Decimals & fractions on the number line
          • Missing numbers on the number line examples
          • Number opposites
          • Negative symbol as opposite
          • Number opposites challenge problems
        • Topic B: Order and absolute value
          • Ordering negative numbers
          • Ordering rational numbers
          • Negative numbers, variables, number line
          • Absolute value examples
          • Comparing absolute values
          • Placing absolute values on the number line
          • Comparing absolute values on the number line
          • Testing solutions to absolute value inequalities
          • Interpreting absolute value
        • Topic C: Rational numbers and the coordinate plane
          • Points on the coordinate plane examples
          • Finding the point not graphed
          • Plotting a point (ordered pair)
          • Quadrants of the coordinate plane
          • Points and quadrants example
          • Coordinate plane word problem examples
          • Reflecting points in the coordinate plane
      • Module 4: Expressions and equations
        • Topic A: Relationships of the operations
        • Topic B: Special notations of operations
          • Intro to exponents
          • Comparing exponent expressions
          • Exponents of decimals
          • The zeroth power
          • Powers of fractions
          • Intro to order of operations
          • Order of operations examples: exponents
          • Order of operations example
          • Worked example: Order of operations (PEMDAS)
          • Order of operations example
        • Topic C: Replacing letters and numbers
          • What is a variable?
        • Topic D: Expanding, factoring, and distributing expressions
          • Expression value intuition
          • Terms, factors, & coefficients
          • Intro to combining like terms
          • Combining like terms
          • Combining like terms example
          • Combining like terms with distribution
          • Distributive property over addition
          • Distributive property over subtraction
          • Factor with the distributive property
          • Distributive property with variables
          • Equivalent expressions
        • Topic E: Expressing operations in algebraic form
          • Writing basic expressions with variables
          • Why aren't we using the multiplication sign?
          • Writing expressions with variables
          • Writing expressions with variables & parentheses
          • Writing basic expressions word problems
          • Writing expressions word problems
        • Topic F: Writing and evaluating expressions and formulas
          • Evaluating an expression with one variable
          • Evaluating expressions with variables: temperature
          • Evaluating expressions with variables: cubes
          • Evaluating expressions with variables: exponents
          • Evaluating expressions with two variables
          • Evaluating expressions with two variables: fractions & decimals
          • Evaluating exponent expressions with variables
        • Topic G: Solving equations
          • Variables, expressions, & equations
          • Testing solutions to equations
          • Same thing to both sides of equations
          • Representing a relationship with an equation
          • One-step equations intuition
          • Dividing both sides of an equation
          • One-step addition & subtraction equations
          • One-step subtraction equations
          • One-step addition & subtraction equations: fractions & decimals
          • One-step multiplication equations
          • One-step division equations
          • One-step multiplication & division equations: fractions & decimals
          • Finding mistakes in one-step equations
        • Topic H: Applications of equations
          • One-step equations word problem: super yoga (1 of 2)
          • One-step equations word problem: super yoga (2 of 2)
          • Modeling with one-step equations
          • Testing solutions to inequalities
          • Plotting inequalities
          • Plotting an inequality example
          • Inequalities word problems
          • Dependent & independent variables
          • Dependent & independent variables: graphing
          • Dependent & independent variables: equation
      • Module 5: Area, surface area, and volume problems
        • Topic A: Area of triangles, quadrilaterals, and polygons
          • Area of a parallelogram
          • Finding height of a parallelogram
          • Area of a triangle
          • Finding area of triangle
          • Triangle missing side example
          • Area of a triangle on a grid
          • Area of a quadrilateral on a grid
          • Area of trapezoids
          • Area of kites
          • Finding area by rearranging parts
          • Area of composite shapes
        • Topic B: Polygons on the coordinate plane
          • Drawing a quadrilateral on the coordinate plane example
          • Coordinates of a missing vertex
          • Area of a parallelogram on the coordinate plane
          • Dimensions of a rectangle from coordinates
          • Coordinates of rectangle example
          • Example of shapes on a coordinate plane
          • Parallelogram on the coordinate plane
        • Topic C: Volume of right rectangular prisms
          • Volume with fractional cubes
          • Volume of a rectangular prism: fractional dimensions
          • How volume changes from changing dimensions
          • Volume of a rectangular prism: word problem
        • Topic D: Nets and surface area
          • Counting faces and edges of 3D shapes
          • Recognizing common 3D shapes
          • Intro to nets of polyhedra
          • Surface area using a net: rectangular prism
          • Surface area using a net: triangular prism
          • Surface area of a box using nets
          • Surface area of a box (cuboid)
          • Surface area word problem example
      • Module 6: Statistics
        • Topic A: Understanding distributions
          • Representing data
          • Frequency tables & dot plots
          • Statistical questions
          • Creating a histogram
          • Interpreting a histogram
        • Topic B: Summarizing a distribution that is approximately symmetric using the mean and mean absolute deviation
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
          • Missing value given the mean
          • Median & range puzzlers
          • Impact on median & mean: removing an outlier
          • Impact on median & mean: increasing an outlier
          • Mean absolute deviation (MAD)
          • Mean absolute deviation example
        • Topic C: Summarizing a distribution that is skewed using the median and the interquartile range
          • Interquartile range (IQR)
          • Constructing a box plot
          • Reading box plots
          • Worked example: Creating a box plot (odd number of data points)
          • Worked example: Creating a box plot (even number of data points)
          • Interpreting box plots
        • Topic D: Summarizing and describing distributions
          • Comparing dot plots, histograms, and box plots
          • Shapes of distributions
          • Clusters, gaps, peaks & outliers
    • 7th grade (Eureka Math/EngageNY)
      • Module 1: Ratios and proportional relationships
        • Topic A: Proportional relationships
          • Intro to proportional relationships
          • Proportional relationships: spaghetti
          • Proportional relationships: movie tickets
          • Proportional relationships: bananas
          • Proportional relationships: graphs
          • Proportional relationships
          • Is side length & perimeter proportional?
          • Is side length & area proportional?
        • Topic B: Unit rate and constant of proportionality
          • Intro to rates
          • Solving unit rate problem
          • Solving unit price problem
          • Constant of proportionality from equation
          • Identifying constant of proportionality graphically
          • Constant of proportionality from table
          • Comparing proportionality constants
          • Interpret proportionality constants
          • Worked example: Solving proportions
          • Writing proportions example
          • Proportion word problem: cookies
          • Proportion word problem: hot dogs
          • Equations for proportional relationships
          • Writing proportional equations
          • Interpreting graphs of proportional relationships
        • Topic C: Ratios and rates involving fractions
          • Rates with fractions
        • Topic D: Ratios of scale drawings
          • Exploring scale copies
          • Corresponding points and sides of scaled shapes
          • Identifying scale copies
          • Identifying scale factors
          • Identifying values in scale drawings
          • Scale drawing: centimeters to kilometers
          • Making a scale drawing
          • Interpreting a scale drawing
          • Solving a scale drawing word problem
      • Module 2: Rational numbers
        • Topic A: Addition and subtraction of integers and rational numbers
          • Adding negative numbers example
          • Adding & subtracting negative numbers
          • Subtracting a negative = adding a positive
          • Adding negative numbers on the number line
          • Number equations & number lines
          • Interpreting numeric expressions example
          • Absolute value as distance between numbers
          • Equivalent expressions with negative numbers
          • Interpreting absolute value as distance
          • Adding fractions with different signs
          • Adding & subtracting fractions
          • Comparing rational numbers
          • Adding & subtracting rational numbers: 79% - 79.1 - 58 1/10
          • Adding & subtracting rational numbers: 0.79 - 4/3 - 1/2 + 150%
        • Topic B: Multiplication and division of integers and rational numbers
          • Why a negative times a negative is a positive
          • Why a negative times a negative makes sense
          • Multiplying positive & negative numbers
          • Dividing positive and negative numbers
          • Decimal to simplified fraction
          • Fraction to decimal: 11/25
          • Worked example: Converting a fraction (7/8) to a decimal
          • Fraction to decimal with rounding
          • Multiplying positive and negative fractions
          • Dividing negative fractions
          • Negative signs in fractions
          • Dividing mixed numbers
          • Simplifying complex fractions
          • Expressions with rational numbers
          • Equivalent expressions with negative numbers (multiplication and division)
          • Why dividing by zero is undefined
        • Topic C: Applying operations with rational numbers to expressions and equations
          • Order of operations example
          • Negative number word problem: temperatures
          • Negative number word problem: Alaska
          • Interpreting negative number statements
          • Interpreting multiplication & division of negative numbers
          • Addition & subtraction: find the missing value (part 1 of 2)
          • Addition & subtraction: find the missing value (part 2 of 2)
          • Substitution with negative numbers
          • Ordering expressions
      • Module 3: Expressions and equations
        • Topic A: Use properties of operations to generate equivalent expressions
          • Intro to combining like terms
          • Simplifying expressions
          • Combining like terms challenge problem
          • Simplifying expressions with rational numbers
          • The distributive property with variables
          • Factoring with the distributive property
          • Equivalent expressions: negative numbers & distribution
          • Interpreting linear expressions: flowers
          • Interpreting linear expressions: diamonds
        • Topic B: Solve problems using expressions, equations, and inequalities
          • Intro to two-step equations
          • Same thing to both sides of equations
          • Two-step equations intuition
          • Worked example: two-step equations
          • Two-step equations with decimals and fractions
          • Two-step equations with decimals and fractions
          • Find the mistake: two-step equations
          • Two-step equation word problem: computers
          • Two-step equation word problem: garden
          • Two-step equation word problem: oranges
          • One-step inequalities examples
          • One-step inequalities: -5c ≤ 15
          • One-step inequality word problem
          • Two-step inequalities
          • Two-step inequality word problem: apples
          • Two-step inequality word problem: R&B
        • Topic C: Use equations and inequalities to solve geometry problems
          • Radius, diameter, circumference & π
          • Labeling parts of a circle
          • Area of a circle
          • Relating circumference and area
          • Partial circle area and arc length
          • Area of composite shapes
          • Perimeter & area of composite shapes
          • Surface area of a box (cuboid)
          • Surface area of a box using nets
          • Volume of triangular prism & cube
          • Volume of a rectangular prism: fractional dimensions
          • Volume with fractional cubes
      • Module 4: Percent and proportional relationships
        • Topic A: Finding the whole
          • Finding a percent
          • Percent word problem: 100 is what percent of 80?
          • Percent word problem: 78 is 15% of what number?
          • Percent word problem: penguins
          • Percent word problem: recycling cans
          • Percent word problem: guavas
          • Growing by a percentage
        • Topic B: Percent problems including more than one whole
          • Solving percent problems
          • Percent word problem: magic club
          • Percent word problems: tax and discount
        • Topic C: Scale drawings
          • Exploring scale copies
          • Corresponding points and sides of scaled shapes
          • Identifying scale copies
          • Identifying scale factors
          • Identifying values in scale drawings
          • Solving a scale drawing word problem
          • Interpreting a scale drawing
          • Making a scale drawing
          • Scale factors and area
        • Topic D: Population, mixture, and counting problems involving percents
      • Module 5: Statistics and probability
        • Topic A: Calculating and interpreting probabilities
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Experimental probability
          • Sample spaces for compound events
          • Die rolling probability
          • Probability of a compound event
          • Count outcomes using tree diagram
          • Counting outcomes: flower pots
        • Topic B: Estimating probabilities
          • Theoretical and experimental probabilites
          • Intuitive sense of probabilities
          • Making predictions with probability
          • Probability models example: frozen yogurt
        • Topic C: Random sampling and estimating population characteristics
          • Reasonable samples
        • Topic D: Comparing populations
          • Comparing distributions with dot plots (example problem)
      • Module 6: Geometry
        • Topic A: Unknown angles
          • Angles: introduction
          • Complementary & supplementary angles
          • Vertical angles
          • Find measure of vertical angles
          • Find measure of angles word problem
          • Equation practice with supplementary angles
          • Equation practice with complementary angles
        • Topic B: Constructing triangles
          • Construct a right isosceles triangle
          • Construct a triangle with constraints
          • Triangle inequality theorem
          • Ordering triangle sides and angles example
        • Topic C: Slicing solids
          • Slicing a rectangular pyramid
          • Ways to cross-section a cube
        • Topic D: Problems involving area and surface area
          • Finding circumference of a circle when given the area
          • Area of a shaded region
          • Impact of increasing the radius
        • Topic E: Problems involving volume
          • Volume word problem: gold ring
          • Volume of triangular prism & cube
          • Volume of rectangles inside rectangles
          • Volume of a rectangular prism: word problem
    • 8th grade (Eureka Math/EngageNY)
      • Module 1: Integer exponents and scientific notation
        • Topic A: Exponential notation and properties of integer exponents
          • Exponent properties with products
          • Exponent properties with parentheses
          • Exponent properties with quotients
          • Negative exponents
          • Negative exponent intuition
          • Powers of zero
          • Multiplying multiples of powers of 10
          • Approximating with powers of 10
        • Topic B: Magnitude and scientific notation
          • Scientific notation examples
          • Scientific notation example: 0.0000000003457
          • Multiplying in scientific notation example
          • Multiplying & dividing in scientific notation
          • Multiplying three numbers in scientific notation
          • Subtracting in scientific notation
          • Simplifying in scientific notation challenge
          • Scientific notation word problem: red blood cells
          • Scientific notation word problem: U.S. national debt
          • Scientific notation word problem: speed of light
      • Module 2: The concept of congruence
        • Topic A: Definitions and properties of the basic rigid motions
          • Introduction to geometric transformations
          • Identifying type of transformation
          • Performing translations
          • Translating shapes
          • Determining translations
          • Performing rotations
          • Determining rotations
          • Performing reflections
          • Reflecting shapes: diagonal line of reflection
          • Determining reflections (advanced)
        • Topic B: Sequencing the basic rigid motions
          • Congruent shapes & transformations
          • Non-congruent shapes & transformations
        • Topic C: Congruence and angle relationships
          • Congruence and similarity — Basic example
          • Congruence and similarity — Harder example
          • Angles, parallel lines, & transversals
          • Parallel & perpendicular lines
          • Missing angles with a transversal
          • Measures of angles formed by a transversal
          • Angles in a triangle sum to 180° proof
          • Triangle exterior angle example
          • Worked example: Triangle angles (intersecting lines)
          • Worked example: Triangle angles (diagram)
          • Triangle angle challenge problem
          • Triangle angle challenge problem 2
        • Topic D: The Pythagorean theorem
      • Module 3: Similarity
        • Topic A: Dilation
          • Dilating shapes: expanding
          • Dilating shapes: shrinking
          • Performing dilations
          • Side lengths after dilation
        • Topic B: Similar figures
          • Intro to triangle similarity
          • Triangle similarity postulates/criteria
          • Determining similar triangles
          • Similar shapes & transformations
        • Topic C: The Pythagorean theorem
          • Intro to the Pythagorean theorem
          • Pythagorean theorem example
          • Pythagorean theorem word problem: carpet
          • Pythagorean theorem with isosceles triangle
          • Thiago asks: How much time does a goalkeeper have to react to a penalty kick?
          • Garfield's proof of the Pythagorean theorem
          • Bhaskara's proof of the Pythagorean theorem
          • Pythagorean theorem proof using similarity
          • Another Pythagorean theorem proof
      • Module 4: Linear equations
        • Topic A: Writing and solving linear equations
          • Intro to equations with variables on both sides
          • Equations with variables on both sides: 20-7x=6x-6
          • Equation with variables on both sides: fractions
          • Equation with the variable in the denominator
          • Equations with parentheses
          • Number of solutions to equations
          • Worked example: number of solutions to equations
          • Creating an equation with no solutions
          • Creating an equation with infinitely many solutions
        • Topic B: Linear equations in two variables and their graphs
          • Rates & proportional relationships example
          • Rates & proportional relationships: gas mileage
          • Graphing proportional relationships: unit rate
          • Graphing proportional relationships from a table
          • Graphing proportional relationships from an equation
          • Points on the coordinate plane examples
          • Testing solutions to linear equations
          • Graphing a linear equation: 5x+2y=20
          • Graphing a linear equation: y=2x+7
        • Topic C: Slope and equations of lines
          • x-intercept of a line
          • Intercepts from a table
          • Intercepts from an equation
          • Worked example: intercepts from an equation
          • Slope & direction of a line
          • Worked example: slope from graph
          • Slope of a line: negative slope
          • Worked example: slope from two points
          • Slope of a horizontal line
          • Converting to slope-intercept form
          • Proving slope is constant using similarity
          • Intro to point-slope form
          • Graph from slope-intercept equation
          • Slope-intercept equation from graph
          • Slope-intercept equation from slope & point
          • Slope-intercept equation from two points
          • Slope-intercept form from a table
          • Slope-intercept form problems
        • Topic D: Systems of linear equations and their solutions
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Testing a solution to a system of equations
          • Systems of equations with graphing
          • Systems of equations with graphing: 5x+3y=7 & 3x-2y=8
          • Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
          • Systems of equations with graphing: chores
          • Systems of equations with elimination: 3t+4g=6 & -6t+g=6
          • Systems of equations with elimination: x+2y=6 & 4x-2y=14
          • Systems of equations with elimination: -3y+4x=11 & y+2x=13
          • Systems of equations with elimination: 2x-y=14 & -6x+3y=-42
          • Systems of equations with elimination: 4x-2y=5 & 2x-y=2.5
          • Systems of equations with elimination: x-4y=-18 & -x+3y=11
          • Systems of equations with elimination: 6x-6y=-24 & -5x-5y=-60
          • Systems of equations with substitution: 2y=x+7 & x=y-4
          • Systems of equations with substitution: y=4x-17.5 & y+2x=6.5
          • Systems of equations with substitution: -3x-4y=-2 & y=2x-5
          • Systems of equations with substitution: 9x+3y=15 & y-x=5
          • Systems of equations with substitution: y=-5x+8 & 10x+2y=-2
          • Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120
          • Systems of equations with elimination: apples and oranges
          • Systems of equations with elimination: TV & DVD
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: Sum/difference of numbers
          • Systems of equations with elimination: potato chips
          • Systems of equations with elimination: coffee and croissants
          • Systems of equations with substitution: coins
          • Systems of equations with substitution: potato chips
          • Systems of equations with substitution: shelves
          • Age word problem: Imran
          • Age word problem: Ben & William
          • Age word problem: Arman & Diya
          • Solutions to systems of equations: consistent vs. inconsistent
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
          • Systems of equations number of solutions: y=3x+1 & 2y+4=6x
          • Solutions to systems of equations: dependent vs. independent
          • Number of solutions to a system of equations graphically
          • Forming systems of equations with different numbers of solutions
          • Comparing Celsius and Fahrenheit temperature scales
          • Converting Fahrenheit to Celsius
      • Module 5: Examples of functions from geometry
        • Topic A: Functions
          • What is a function?
          • Worked example: Evaluating functions from equation
          • Worked example: evaluating functions from graph
          • Function notation example
          • Equations vs. functions
          • Manipulating formulas: temperature
          • Testing if a relationship is a function
          • Relations and functions
          • Recognizing functions from graph
          • Checking if a table represents a function
          • Recognizing functions from verbal description
          • Recognizing functions from table
          • Recognizing functions from verbal description word problem
          • Checking if an equation represents a function
          • Does a vertical line represent a function?
        • Topic B: Volume
          • Cylinder volume & surface area
          • Volume of a sphere
          • Volume of a cone
      • Module 6: Linear functions
        • Topic A: Linear functions
          • Recognizing linear functions
          • Linear & nonlinear functions: table
          • Linear & nonlinear functions: word problem
          • Linear & nonlinear functions: missing value
          • Interpreting a graph example
          • Comparing linear functions: equation vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions: table vs. graph
        • Topic B: Bivariate numerical data
          • Constructing a scatter plot
          • Scatter plots: studying, shoe size, and test scores
          • Scatter plot: smokers
          • Estimating the line of best fit exercise
          • Line of best fit: smoking in 1945
          • Interpreting a trend line
        • Topic C: Linear and nonlinear models
          • Linear graphs word problems
          • Linear graphs word problem: cats
          • Linear equations word problems: volcano
          • Linear equations word problems: earnings
          • Modeling with linear equations: snow
          • Comparing linear functions word problem: climb
          • Comparing linear functions word problem: walk
          • Comparing linear functions word problem: work
          • Linear function example: spending money
          • Modeling with linear equations: gym membership & lemonade
          • Fitting a line to data
        • Topic D: Bivariate categorical data
          • Two-way frequency tables and Venn diagrams
          • Two-way relative frequency tables
          • Interpreting two-way tables
      • Module 7: Introduction to irrational numbers using geometry
        • Topic A: Square and cube roots
          • Intro to square roots
          • Intro to cube roots
          • Worked example: Cube root of a negative number
          • Square root of decimal
          • Dimensions of a cube from its volume
          • Intro to rational & irrational numbers
          • Classifying numbers: rational & irrational
          • Classifying numbers
          • Approximating square roots
          • Approximating square roots to hundredths
        • Topic B: Decimal expansions of numbers
          • Converting a fraction to a repeating decimal
          • Converting repeating decimals to fractions (part 1 of 2)
          • Converting repeating decimals to fractions (part 2 of 2)
          • Comparing irrational numbers with radicals
          • Comparing values with calculator
        • Topic C: The Pythagorean theorem
          • Finding distance with Pythagorean theorem
          • Distance formula
          • Use Pythagorean theorem to find area of an isosceles triangle
          • Pythagorean theorem word problem: carpet
          • Pythagorean theorem word problem: fishing boat
          • Pythagorean theorem in 3D
        • Topic D: Applications of radicals and roots
    • Algebra I (Eureka Math/EngageNY)
      • Module 1: Relationships between quantities and reasoning with equations and their graphs
        • Topic A: Lessons 1-3: Piecewise, quadratic, and exponential functions
          • Introduction to piecewise functions
          • Worked example: evaluating piecewise functions
          • Parabolas intro
          • Intro to exponential functions
          • Exponential vs. linear growth
          • Exponential vs. linear models: verbal
          • Exponential vs. linear models: table
        • Topic A: Lessons 4-5: Analyzing graphs
          • Graph interpretation word problem: temperature
          • Graph interpretation word problem: basketball
          • Linear graphs word problems
          • Linear graphs word problem: cats
          • System of equations word problem: walk & ride
        • Topic B: Lessons 6-7: Distributive, commutative, and associative properties
        • Topic B: Lesson 8: Adding and subtracting polynomials
          • Polynomials intro
          • The parts of polynomial expressions
          • Evaluating polynomials
          • Simplifying polynomials
          • Adding polynomials
          • Subtracting polynomials
          • Polynomial subtraction & set closure
          • Adding & subtracting multiple polynomials
        • Topic B: Lesson 8: Adding and subtracting polynomials with 2 variables
          • Adding polynomials: two variables
          • Subtracting polynomials: two variables
          • Subtracting polynomials: two variables challenge
          • Finding an error in polynomial subtraction
        • Topic B: Lesson 9: Multiplying monomials
          • Multiplying monomials
          • Multiplying monomials to find area: two variables
          • Multiplying monomials to find area
          • Multiplying monomials challenge
        • Topic B: Lesson 9: Multiplying monomials by polynomials
          • Multiplying monomials by polynomials: area model
          • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials challenge
        • Topic B: Lesson 9: Multiplying binomials
          • Multiplying binomials: area model
          • Multiplying binomials intro
          • Multiplying binomials
        • Topic C: Lessons 10-13: Solving Equations
          • Testing solutions to equations
          • Number of solutions to equations
          • Worked example: number of solutions to equations
          • Creating an equation with no solutions
          • Creating an equation with infinitely many solutions
          • Same thing to both sides of equations
          • Representing a relationship with an equation
          • Dividing both sides of an equation
          • Why we do the same thing to both sides: Variable on both sides
        • Topic C: Lesson 14: Solving inequalities
          • Testing solutions to inequalities
          • Plotting inequalities
          • Plotting an inequality example
          • One-step inequalities examples
          • One-step inequalities: -5c ≤ 15
          • Two-step inequalities
          • Inequalities with variables on both sides
          • Inequalities with variables on both sides (with parentheses)
          • Multi-step inequalities
        • Topic C: Lessons 15-16 Compound inequalities
          • Compound inequalities: OR
          • Compound inequalities: AND
          • A compound inequality with no solution
          • Double inequalities
          • Compound inequalities examples
        • Topic C: Lessons 17-19: Advanced equations
          • Zero product property
          • Solving quadratics by factoring
          • Linear equations with unknown coefficients
        • Topic C: Lesson 20: Solution sets to equations with two variables
          • Two-variable linear equations intro
          • Solutions to 2-variable equations
          • Worked example: solutions to 2-variable equations
          • Completing solutions to 2-variable equations
        • Topic C: Lesson 21: Solution sets to inequalities with two variables
          • Testing solutions to inequalities
          • Intro to graphing two-variable inequalities
          • Graphing two-variable inequalities
          • Two-variable inequalities from their graphs
        • Topic C: Lesson 22: Solution sets to simultaneous equations
          • Testing a solution to a system of equations
          • Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
          • Systems of equations with graphing: exact & approximate solutions
          • Testing solutions to systems of inequalities
          • Intro to graphing systems of inequalities
          • Graphing systems of inequalities
        • Topic C: Lesson 23: Solution sets to simultaneous equations
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: x-4y=-18 & -x+3y=11
          • Systems of equations with elimination: potato chips
          • Systems of equations with elimination (and manipulation)
          • Why can we subtract one equation from the other in a system of equations?
          • Worked example: equivalent systems of equations
          • Worked example: non-equivalent systems of equations
        • Topic C: Lesson 24: Applications of systems of equations and inequalities
          • Age word problem: Imran
          • Age word problem: Ben & William
          • Age word problem: Arman & Diya
          • Modeling with systems of inequalities
          • Writing systems of inequalities word problem
          • Solving systems of inequalities word problem
          • Graphs of systems of inequalities word problem
        • Topic D: Creating equations to solve problems
          • Worked example: Rate problem
          • Sequences intro
          • Evaluating sequences in recursive form
      • Module 2: Descriptive statistics
        • Topic A: Lesson 1: Dot plots and histograms
          • Representing data
          • Frequency tables & dot plots
          • Creating a histogram
          • Interpreting a histogram
        • Topic A: Lesson 1: Box plots and shape
          • Worked example: Creating a box plot (odd number of data points)
          • Worked example: Creating a box plot (even number of data points)
          • Constructing a box plot
          • Reading box plots
          • Shapes of distributions
        • Topic A: Lesson 2: Describing the center of a distribution
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
        • Topic A: Lesson 3: Estimating centers and interpreting the mean as a balance point
          • Comparing means of distributions
          • Means and medians of different distributions
          • Impact on median & mean: removing an outlier
          • Impact on median & mean: increasing an outlier
          • Missing value given the mean
        • Topic B: Lesson 4: Summarizing deviations from the mean
          • Example: Comparing distributions
          • Mean absolute deviation example
        • Topic B: Lessons 5-6: Standard deviation and variability
          • Population standard deviation
          • Sample standard deviation and bias
        • Topic B: Lesson 7: Measuring variability for skewed distributions (interquartile range)
          • Interquartile range (IQR)
          • Interpreting box plots
          • Judging outliers in a dataset
        • Topic B: Lesson 8: Comparing distributions
          • Comparing distributions with dot plots (example problem)
          • Comparing dot plots, histograms, and box plots
        • Topic C: Lessons 9-10: Bivariate categorical data
          • Two-way frequency tables and Venn diagrams
          • Two-way relative frequency tables
        • Topic C: Lesson 11: Conditional relative frequencies and association
          • Interpreting two-way tables
          • Categorical data example
          • Analyzing trends in categorical data
        • Topic D: Lessons 12-13: Relationships between two numerical variables
          • Constructing a scatter plot
          • Scatter plots: studying, shoe size, and test scores
        • Topic D: Lesson 14: Modeling relationships with a line
          • Fitting a line to data
          • Estimating the line of best fit exercise
          • Estimating with linear regression (linear models)
          • Interpreting slope of regression line
          • Interpreting y-intercept in regression model
        • Topic D: Lessons 15-18: Residuals
          • Introduction to residuals and least squares regression
          • Calculating residual example
          • Residual plots
        • Topic D: Lesson 19: Interpreting correlation
          • Bivariate relationship linearity, strength and direction
          • Example: Correlation coefficient intuition
          • Correlation and causality
          • Calculating correlation coefficient r
      • Module 3: Linear and exponential functions
        • Topic A: Lessons 1-3: Arithmetic sequence intro
          • Sequences intro
          • Intro to arithmetic sequences
          • Extending arithmetic sequences
          • Using arithmetic sequences formulas
          • Worked example: using recursive formula for arithmetic sequence
        • Topic A: Lessons 1-3: Geometric sequence intro
          • Intro to geometric sequences
          • Extending geometric sequences
          • Using explicit formulas of geometric sequences
          • Using recursive formulas of geometric sequences
        • Topic A: Lessons 1-3: Arithmetic sequence formulas
          • Explicit formulas for arithmetic sequences
          • Arithmetic sequence problem
          • Recursive formulas for arithmetic sequences
          • Converting recursive & explicit forms of arithmetic sequences
        • Topic A: Lessons 1-3: Geometric sequence formulas
          • Explicit & recursive formulas for geometric sequences
          • Converting recursive & explicit forms of geometric sequences
          • Sequences word problems
        • Topic A: Lessons 4-6: Exponential growth
          • Intro to exponential functions
          • Exponential vs. linear growth
          • Exponential vs. linear models: verbal
          • Exponential vs. linear models: table
          • Exponential expressions word problems (numerical)
          • Initial value & common ratio of exponential functions
          • Exponential expressions word problems (algebraic)
          • Interpreting exponential expression word problem
        • Topic A: Lesson 7: Exponential decay
          • Exponential function graph
          • Graphs of exponential growth
          • Exponential decay intro
          • Graphing exponential growth & decay
          • Exponential growth & decay word problems
        • Topic B: Lessons 8-12: Functions intro
          • What is a function?
          • Worked example: Evaluating functions from equation
          • Worked example: evaluating functions from graph
          • Worked example: evaluating expressions with function notation
        • Topic B: Lessons 8-12: Function domain and range
          • What is the domain of a function?
          • What is the range of a function?
          • Intervals and interval notation
          • Worked example: domain and range from graph
          • Domain of a radical function
          • Worked example: domain of algebraic functions
        • Topic B: Lessons 8-12: Recognizing functions
          • Recognizing functions from graph
          • Does a vertical line represent a function?
          • Recognizing functions from table
          • Recognizing functions from verbal description
          • Recognizing functions from verbal description word problem
        • Topic B: Lesson 13: Interpreting the graph of a function
          • Increasing, decreasing, positive or negative intervals
          • Worked example: positive & negative intervals
          • Introduction to minimum and maximum points
          • Worked example: relative maxima and minima
          • Graph interpretation word problem: temperature
          • Graph interpretation word problem: basketball
        • Topic B: Lesson 14: Linear and exponential Models—comparing growth rates
          • Exponential vs. linear growth over time
        • Topic C: Lesson 15: Piecewise functions
          • Introduction to piecewise functions
          • Worked example: evaluating piecewise functions
          • Worked example: graphing piecewise functions
          • Worked example: domain & range of step function
          • Worked example: domain & range of piecewise linear functions
        • Topic C: Lessons 16-20: Graphing absolute value functions
          • Shifting absolute value graphs
          • Scaling & reflecting absolute value functions: equation
          • Scaling & reflecting absolute value functions: graph
          • Graphing absolute value functions
        • Topic D: Using functions and graphs to solve problems
      • Module 4: Polynomial and quadratic expressions, equations, and functions
        • Topic A: Lessons 1-2: Factoring monomials
          • Intro to factors & divisibility
          • Monomial factorization
          • Worked example: finding the missing monomial factor
          • Worked example: finding missing monomial side in area model
        • Topic A: Lessons 1-2: Common factor
          • Greatest common factor of monomials
          • Factoring with the distributive property
          • Factoring binomials: common factor
          • Factoring polynomials: common factor
          • Factoring polynomials: common factor area model
          • Factoring polynomials: common binomial factor
        • Topic A: Lessons 1-2: Factoring binomials intro
          • Difference of squares intro
          • Perfect square factorization intro
          • Factoring quadratics as (x+a)(x+b)
          • Factoring quadratics as (x+a)(x+b) (example 2)
          • More examples of factoring quadratics as (x+a)(x+b)
        • Topic A: Lessons 3-4: Special forms
          • Factoring difference of squares: leading coefficient ≠ 1
          • Factoring difference of squares: analyzing factorization
          • Factoring difference of squares: missing values
          • Factoring difference of squares: shared factors
          • Factoring perfect squares
          • Identifying perfect square form
          • Factoring perfect squares: common factor
          • Factoring perfect squares: negative common factor
          • Factoring perfect squares: missing values
          • Factoring perfect squares: shared factors
        • Topic A: Lessons 3-4: Factoring by grouping
          • Intro to grouping
          • Factoring quadratics by grouping
          • Strategy in factoring quadratics (part 1 of 2)
          • Strategy in factoring quadratics (part 2 of 2)
        • Topic A: Lesson 5: The zero product property
          • Zero product property
          • Solving quadratics by factoring
          • Solving quadratics by factoring: leading coefficient ≠ 1
          • Solving quadratics using structure
        • Topic A: Lessons 6-7: Solving basic one-variable quadratic equations
          • Solving quadratics by taking square roots
          • Solving quadratics by taking square roots examples
          • Solving quadratics by taking square roots: strategy
          • Quadratics by taking square roots: with steps
          • Quadratic equations word problem: triangle dimensions
          • Quadratic equations word problem: box dimensions
        • Topic A: Lessons 8-10: Parabolas intro
          • Parabolas intro
          • Graphing quadratics in factored form
          • Quadratic word problems (factored form)
        • Topic B: Lessons 11-13: Completing the square
          • Completing the square
          • Worked example: Completing the square (intro)
          • Worked example: Rewriting expressions by completing the square
          • Worked example: Solving equations by completing the square
          • Worked example: completing the square (leading coefficient ≠ 1)
          • Solving quadratics by completing the square: no solution
        • Topic B: Lessons 14-15: The quadratic formula
          • The quadratic formula
          • Using the quadratic formula
          • Worked example: quadratic formula
          • Worked example: quadratic formula (example 2)
          • Worked example: quadratic formula (negative coefficients)
          • Using the quadratic formula: number of solutions
          • Proof of the quadratic formula
        • Topic B: Lesson 16: Graphing quadratic equations from the vertex form
          • Vertex form introduction
          • Vertex & axis of symmetry of a parabola
          • Graphing quadratics: vertex form
          • Quadratic word problems (vertex form)
        • Topic B: Lesson 17: Graphing quadratic functions from the standard form
          • Finding the vertex of a parabola in standard form
          • Graphing quadratics: standard form
          • Quadratic word problem: ball
          • Forms & features of quadratic functions
          • Worked examples: Forms & features of quadratic functions
          • Finding features of quadratic functions
        • Topic C: Lessons 18-19: Translating graphs of functions
          • Intro to parabola transformations
          • Shifting parabolas
          • Shifting functions
          • Graphing shifted functions
        • Topic C: Lessons 20-22: Scaling and transforming graphs
          • Scaling & reflecting parabolas
          • Shifting & reflecting functions
          • Compressing functions
          • Reflecting & compressing functions
          • Transforming the square-root function
          • Graphs of square-root functions
          • Square-root functions & their graphs
        • Topic C: Lessons 23–24: Modeling with quadratic functions
      • Module 5: A synthesis of modeling with equations and functions
        • Module 5: A synthesis of modeling with equations and functions
    • Geometry (Eureka Math/EngageNY)
      • Module 1: Congruence, proof, and constructions
        • Topic A: Basic constructions
          • Geometric constructions: angle bisector
          • Geometric constructions: perpendicular line
          • Geometric constructions: perpendicular bisector
        • Topic B: Lesson 6: Solve for unknown angles—angles and lines at a point
          • Equation practice with vertical angles
          • Complementary & supplementary angles
          • Equation practice with complementary angles
          • Equation practice with supplementary angles
        • Topic B: Lesson 7: Solve for unknown angles—transversals
          • Angles, parallel lines, & transversals
          • Parallel & perpendicular lines
          • Missing angles with a transversal
          • Measures of angles formed by a transversal
        • Topic B: Lesson 8: Solve for unknown angles—angles in a triangle
          • Angles in a triangle sum to 180° proof
          • Triangle exterior angle example
          • Worked example: Triangle angles (intersecting lines)
          • Worked example: Triangle angles (diagram)
          • Triangle angle challenge problem 2
          • Proving angles are congruent
        • Topic B: Lessons 9-11: Unknown angle proofs
        • Topic C: Lesson 12: Transformations—The next level
          • Rigid transformations intro
        • Topic C: Lesson 13: Rotations
          • Rotating shapes
          • Determining rotations
        • Topic C: Lesson 14: Reflections
          • Reflecting shapes: diagonal line of reflection
          • Determining reflections (advanced)
        • Topic C: Lesson 15: Rotations, reflections, and symmetry
          • Intro to reflective symmetry
          • Intro to rotational symmetry
          • Finding a quadrilateral from its symmetries
          • Finding a quadrilateral from its symmetries (example 2)
        • Topic C: Lesson 16: Translations
          • Translating shapes
          • Determining translations
        • Topic C: Lessons 17-21: Applications of transformations
          • Proofs with transformations
        • Topic D: Lessons 22-25: Congruence criteria
          • Congruent triangles & the SSS postulate/criterion
          • Triangle congruence postulates/criteria
          • Why SSA isn't a congruence postulate/criterion
          • Determining congruent triangles
          • Proofs concerning isosceles triangles
          • Proofs concerning equilateral triangles
          • Isosceles & equilateral triangles problems
          • Corresponding angles in congruent triangles
        • Topic D: Lessons 26-27: Triangle congruency proofs
          • Angles in a triangle sum to 180° proof
          • Geometry proof problem: midpoint
          • Geometry proof problem: congruent segments
          • Geometry proof problem: squared circle
        • Topic E: Proving properties of geometric figures
          • Proof: Opposite sides of a parallelogram
          • Proof: Diagonals of a parallelogram
          • Proof: Opposite angles of a parallelogram
          • Proof: The diagonals of a kite are perpendicular
          • Proof: Rhombus diagonals are perpendicular bisectors
          • Proof: Rhombus area
          • Triangle medians & centroids
          • Triangle medians and centroids (2D proof)
          • Dividing triangles with medians
          • Exploring medial triangles
          • Centroid & median proof
          • Median, centroid example
        • Topic F: Advanced constructions
        • Topic G: Axiomatic systems
      • Module 2: Similarity, proof, and trigonometry
        • Topic A: Scale drawings
        • Topic B: Dilations
          • Performing dilations
          • Dilating shapes: expanding
          • Dilating shapes: shrinking
        • Topic C: Lessons 12-14: Similarity intro
          • Similar shapes & transformations
          • Proof: all circles are similar
        • Topic C: Lessons 15-17: Triangle similarity
          • Intro to triangle similarity
          • Triangle similarity postulates/criteria
          • Determining similar triangles
          • Solving similar triangles
          • Solving similar triangles: same side plays different roles
        • Topic C: Lessons 18-20: Similarity applications
          • Intro to angle bisector theorem
          • Using the angle bisector theorem
          • Using similar & congruent triangles
          • Challenging similarity problem
          • Geometry word problem: the golden ratio
          • Geometry word problem: Earth & Moon radii
          • Geometry word problem: a perfect pool shot
        • Topic D: Applying similarity to right triangles
          • Simplifying square roots
          • Simplifying square roots (variables)
          • Simplifying square-root expressions
          • Pythagorean theorem proof using similarity
        • Topic E: Lessons 25-26: Trigonometry intro
          • Intro to the trigonometric ratios
          • Trigonometric ratios in right triangles
          • Triangle similarity & the trigonometric ratios
          • Solving for a side in right triangles with trigonometry
        • Topic E: Lesson 27: Sine and cosine of complementary angles and special angles
          • Sine & cosine of complementary angles
          • Using complementary angles
          • Trig word problem: complementary angles
          • Special right triangles intro (part 1)
          • Special right triangles intro (part 2)
          • 30-60-90 triangle example problem
          • Special right triangles proof (part 1)
          • Special right triangles proof (part 2)
        • Topic E: Lesson 34: Unknown angles
        • Topic E: Lessons 28-31: Trigonometry applications
          • Right triangle word problem
        • Topic E: Lessons 32-33: Laws of sines and cosines
          • Solving for a side with the law of sines
          • Solving for an angle with the law of sines
          • Proof of the law of sines
          • Solving for a side with the law of cosines
          • Solving for an angle with the law of cosines
          • Proof of the law of cosines
          • Trig word problem: stars
      • Module 3: Extending to three dimensions
        • Topic A: Area
          • Intro to area and unit squares
          • Area of composite shapes
          • Perimeter & area of composite shapes
        • Topic B: Lessons 5-9: Volume properties
          • Specifying planes in three dimensions
          • Slicing a rectangular pyramid
          • Ways to cross-section a cube
        • Topic B: Lessons 10-13: Volume formulas
          • Volume of triangular prism & cube
          • Cylinder volume & surface area
          • Volume of a cone
          • Volume of a sphere
      • Module 4: Connecting algebra and geometry through coordinates
        • Topic A: Rectangular and triangular regions defined by inequalities
          • Intro to graphing two-variable inequalities
          • Graphing two-variable inequalities
          • Two-variable inequalities from their graphs
          • Intro to graphing systems of inequalities
          • Graphing systems of inequalities
        • Topic B: Perpendicular and parallel lines in the Cartesian plane
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines from graph
          • Parallel lines from equation
          • Parallel lines from equation (example 2)
          • Parallel lines from equation (example 3)
          • Perpendicular lines from equation
          • Writing equations of perpendicular lines
          • Writing equations of perpendicular lines (example 2)
          • Proof: parallel lines have the same slope
          • Proof: perpendicular lines have negative reciprocal slope
        • Topic C: Perimeters and areas of polygonal regions in the Cartesian plane
          • Area of trapezoid on the coordinate plane
          • Coordinate plane word problem
        • Topic D: Partitioning and extending segments and parameterization of lines
          • Midpoint formula
          • Dividing line segments: graphical
          • Dividing line segments
          • Distance between point & line
      • Module 5: Circles with and without coordinates
        • Topic A: Central and inscribed angles
          • Inscribed angles
          • Inscribed angle theorem proof
          • Proof: Right triangles inscribed in circles
          • Inscribed shapes: find diameter
          • Inscribed shapes: angle subtended by diameter
          • Inscribed shapes: find inscribed angle
          • Inscribed quadrilaterals proof
          • Solving inscribed quadrilaterals
          • Geometric constructions: circle-inscribed square
          • Geometric constructions: circle-inscribed equilateral triangle
          • Geometric constructions: circle-inscribed regular hexagon
          • Proof: radius is perpendicular to a chord it bisects
          • Proof: perpendicular radius bisects chord
        • Topic B: Lessons 7-8: Angles and arcs
          • Intro to arc measure
          • Finding arc measures
          • Finding arc measures with equations
        • Topic B: Lessons 9-10: Arc length and sector area
          • Arc length from subtended angle
          • Subtended angle from arc length
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
          • Arc length as fraction of circumference
          • Arc length from subtended angle: radians
          • Area of a sector
        • Topic C: Secants and tangents
          • Proof: Radius is perpendicular to tangent line
          • Proof: Segments tangent to circle from outside point are congruent
          • Tangents of circles problem (example 1)
          • Tangents of circles problem (example 2)
          • Tangents of circles problem (example 3)
          • Geometric constructions: circle tangent
          • Geometric constructions: circle tangent (example 2)
        • Topic D: Equations for circles and their tangents
          • Features of a circle from its standard equation
          • Graphing a circle from its standard equation
          • Writing standard equation of a circle
          • Features of a circle from its expanded equation
        • Topic E: Cyclic quadrilaterals and Ptolemy’s theorem
    • Algebra II (Eureka Math/EngageNY)
      • Module 1: Polynomial, rational, and radical relationships
        • Topic A: Polynomials—from base ten to base X
          • Intro to long division of polynomials
          • Dividing polynomials: long division
          • Dividing polynomials with remainders: monomial divisors
          • Dividing polynomials with remainders: monomial divisors (example 2)
          • Dividing polynomials with remainders
        • Topic B: Lessons 12-13: Factorization
          • Factoring quadratics: common factor + grouping
          • Factoring quadratics: negative common factor + grouping
          • Factoring two-variable quadratics
          • Factoring two-variable quadratics: rearranging
          • Factoring two-variable quadratics: grouping
          • Factoring perfect squares: 4th degree polynomial
          • Factoring difference of squares: two variables
          • Factoring difference of squares: two variables (example 2)
          • Factoring higher-degree polynomials
          • Factoring sum of cubes
          • Factoring difference of cubes
          • Finding zeros of polynomials (1 of 2)
          • Finding zeros of polynomials (2 of 2)
          • Finding zeros of polynomials (example 2)
        • Topic B: Lesson 14: Graphing factored polynomials
          • Zeros of polynomials & their graphs
          • Intro to end behavior of polynomials
          • End behavior of functions & their graphs
        • Topic B: Lesson 15: Structure in graphs of polynomial functions
          • Intro to function symmetry
          • Graphical symmetry of functions
          • Even/odd functions & numbers
          • Even & odd polynomials
        • Topic B: Lessons 16-21: Remainder theorem and modeling
          • Intro to the Polynomial Remainder Theorem
          • Using the Polynomial Remainder Theorem: finding remainders
          • Using the Polynomial Remainder Theorem: checking factors
          • Using the Polynomial Remainder Theorem: finding coefficients
          • Proof of the Polynomial Remainder Theorem
        • Topic C: Lessons 22-23: Simplifying rational expressions
          • Intro to rational expression simplification
          • Simplifying rational expressions: common monomial factors
          • Simplifying rational expressions: common binomial factors
          • Simplifying rational expressions: opposite common binomial factors
          • Simplifying rational expressions: grouping
          • Simplifying rational expressions: higher degree terms
          • Simplifying rational expressions: two variables
        • Topic C: Lesson 24: Multiplying and dividing rational expressions
          • Multiplying & dividing rational expressions: monomials
          • Multiplying rational expressions
          • Dividing rational expressions
          • Multiplying rational expressions: multiple variables
          • Dividing rational expressions: unknown expression
        • Topic C: Lesson 25: Adding and subtracting rational expressions
          • Adding & subtracting rational expressions: like denominators
          • Intro to adding rational expressions with unlike denominators
          • Adding rational expression: unlike denominators
          • Subtracting rational expressions: unlike denominators
          • Least common multiple
          • Least common multiple: repeating factors
          • Subtracting rational expressions: factored denominators
          • Least common multiple of polynomials
          • Subtracting rational expressions
          • Nested fractions
        • Topic C: Lessons 26-27: Rational equations
          • Equations with one rational expression
          • Equations with one rational expression (advanced)
          • Equations with rational expressions
          • Equations with rational expressions (example 2)
          • Rational equations word problem: combined rates
          • Rational equations word problem: combined rates (example 2)
          • Rational equations word problem: eliminating solutions
        • Topic C: Lessons 28-29: Radical equations
          • Intro to square-root equations & extraneous solutions
          • Solving square-root equations (basic)
          • Solving square-root equations: one solution
          • Solving square-root equations: two solutions
          • Solving square-root equations: no solution
          • Extraneous solutions of radical equations
          • Extraneous solutions of radical equations (example 2)
          • Solving cube-root equations
        • Topic C: Lesson 30: Linear systems in three variables
          • Intro to linear systems with 3 variables
          • Solving linear systems with 3 variables
          • Solving linear systems with 3 variables: no solution
          • 3-variable linear system word problem
        • Topic C: Lesson 31: Systems of equations
          • Quadratic systems: graphical solution
          • Quadratic systems: algebraic solution
          • Quadratic systems: no solution
          • Quadratic systems: both variables are squared
          • Quadratic systems: a line and a circle
        • Topic C: Lesson 32: Graphing systems of equations
          • Interpreting equations graphically
          • Interpreting equations graphically (example 2)
          • Solving equations graphically (1 of 2)
          • Solving equations graphically (2 of 2)
        • Topic C: Lessons 33-35: Parabolas
          • Intro to focus & directrix
          • Equation of a parabola from focus & directrix
          • Focus & directrix of a parabola from equation
        • Topic D: Lesson 37: Complex numbers intro
          • Intro to the imaginary numbers
          • Simplifying roots of negative numbers
          • i as the principal root of -1
          • Intro to complex numbers
          • Plotting numbers on the complex plane
        • Topic D: Lesson 37: Complex number operations
          • Adding complex numbers
          • Subtracting complex numbers
          • Multiplying complex numbers
          • Powers of the imaginary unit
        • Topic D: Lessons 38-40: Complex roots and the fundamental theorem of algebra
          • Solving quadratic equations: complex roots
          • Complex numbers & sum of squares factorization
          • Factoring sum of squares
          • The Fundamental theorem of Algebra
          • Quadratics & the Fundamental Theorem of Algebra
          • Number of possible real roots of a polynomial
      • Module 2: Trigonometric functions
        • Topic A: Lessons 1-7: Unit circle definition of trig functions
          • Unit circle
          • The trig functions & right triangle trig ratios
          • Trig values of special angles
          • Sine & cosine identities: symmetry
          • Sine & cosine identities: periodicity
          • Tangent identities: symmetry
          • Tangent identities: periodicity
        • Topic A: Lessons 8-10: Trigonometric graphs and radians
          • Graph of y=sin(x)
          • Intersection points of y=sin(x) and y=cos(x)
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
          • Radian angles & quadrants
        • Topic B: Lesson 11: Graphing sinusoidal functions
          • Transforming sinusoidal graphs: vertical stretch & horizontal reflection
          • Transforming sinusoidal graphs: vertical & horizontal stretches
        • Topic B: Lesson 11: Features of sinusoidal graphs
          • Features of sinusoidal functions
          • Amplitude & period of sinusoidal functions from equation
        • Topic B: Lessons 12-13: Modeling with sinusoidal functions
          • Sinusoidal function from graph
          • Trig word problem: modeling daily temperature
          • Trig word problem: modeling annual temperature
          • Trig word problem: length of day (phase shift)
        • Topic B: Lesson 14: Graphing the tangent function
          • Graph of y=tan(x)
          • Tangent identities: symmetry
          • Tangent identities: periodicity
        • Topic B: Lessons 15-17: Trigonometric identities
          • Proof of the Pythagorean trig identity
          • Using the Pythagorean trig identity
          • Review of trig angle addition identities
          • Using the cosine angle addition identity
          • Using the cosine double-angle identity
          • Proof of the sine angle addition identity
          • Proof of the cosine angle addition identity
      • Module 3: Exponential and logarithmic functions
        • Topic A: Lesson 1: Integer exponents
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
        • Topic A: Lesson 2: Scientific notation
          • Scientific notation example: 0.0000000003457
          • Scientific notation examples
          • Multiplying & dividing in scientific notation
          • Multiplying three numbers in scientific notation
          • Subtracting in scientific notation
          • Scientific notation word problem: red blood cells
          • Scientific notation word problem: U.S. national debt
          • Scientific notation word problem: speed of light
        • Topic A: Lessons 3-6: Rational exponents
          • Intro to rational exponents
          • Rewriting roots as rational exponents
          • Simplifying quotient of powers (rational exponents)
          • Simplifying mixed radical and exponential expressions
          • 𝑒 and compound interest
          • 𝑒 as a limit
        • Topic B: Lessons 7-9: Logarithms intro
          • Solving exponential equations using exponent properties
          • Intro to logarithms
          • Evaluating logarithms (advanced)
        • Topic B: Lessons 10-12: Logarithm properties
          • Intro to logarithm properties (1 of 2)
          • Intro to logarithm properties (2 of 2)
          • Using the logarithmic product rule
          • Using the logarithmic power rule
          • Using the properties of logarithms: multiple steps
          • Proof of the logarithm product rule
          • Proof of the logarithm quotient and power rules
        • Topic B: Lesson 13: Changing the base
          • Evaluating logarithms: change of base rule
          • Using the logarithm change of base rule
          • Proof of the logarithm change of base rule
        • Topic B: Lessons 13-15: Logarithms and equations
          • Solving exponential equations using logarithms: base-10
          • Solving exponential equations using logarithms: base-2
          • Logarithmic equations: variable in the argument
          • Logarithmic equations: variable in the base
        • Topic C: Lesson 16: Rational and Irrational Numbers
          • Approximating square roots
          • Comparing irrational numbers with radicals
          • Approximating square roots to hundredths
          • Comparing values with calculator
          • Proof: sum & product of two rationals is rational
          • Proof: product of rational & irrational is irrational
          • Proof: sum of rational & irrational is irrational
          • Sums and products of irrational numbers
          • Worked example: rational vs. irrational expressions
          • Worked example: rational vs. irrational expressions (unknowns)
        • Topic C: Lessons 17-22: Graphs of exponential and logarithmic functions
          • Relationship between exponentials & logarithms
          • Relationship between exponentials & logarithms: graphs
          • Relationship between exponentials & logarithms: tables
          • Transforming exponential graphs
          • Transforming exponential graphs (example 2)
          • Graphing exponential functions
          • Graphs of logarithms & exponentials
          • Graphing basic logarithmic functions
          • Graphs of logarithmic functions
        • Topic D: Lessons 23-24: Solving exponential models
          • Solving exponential equations using exponent properties (advanced)
          • Exponential model word problem: medication dissolve
          • Exponential model word problem: bacteria growth
        • Topic D: Lessons 25-28: Modeling with exponential functions
          • Interpreting change in exponential models
          • Interpreting time in exponential models
          • Constructing exponential models
          • Constructing exponential models: half life
          • Constructing exponential models: percent change
          • Interpreting change in exponential models: with manipulation
          • Interpreting change in exponential models: changing units
        • Topic E: Lesson 29: Geometric series
          • Summation notation
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
          • Finite geometric series formula justification
          • Finite geometric series word problem: social media
          • Finite geometric series word problem: mortgage
        • Topic E: Lesson 29: Arithmetic series
          • Arithmetic series intro
          • Arithmetic series formula
          • Worked example: arithmetic series (sigma notation)
          • Worked example: arithmetic series (sum expression)
          • Worked example: arithmetic series (recursive formula)
          • Proof of finite arithmetic series formula
        • Topic E: Lessons 30-33: Series applications
      • Module 4: Inferences and conclusions from data
        • Topic A: Lesson 1: Chance experiments, sample spaces, and events
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Intuitive sense of probabilities
        • Topic A: Lessons 2-4: Two-way tables, conditional probability, and independence
          • Calculating conditional probability
          • Conditional probability explained visually
          • Conditional probability and independence
          • Analyzing event probability for independence
        • Topic A: Lessons 5-7: Venn diagrams and probability rules
          • Intersection and union of sets
          • Compound probability of independent events
          • Independent events example: test taking
          • Coin flipping probability
          • Probability with Venn diagrams
          • Addition rule for probability
        • Topic B: Lesson 8: Distributions—Center, shape, and spread
          • Shapes of distributions
          • Comparing means of distributions
          • Means and medians of different distributions
          • Mean and standard deviation versus median and IQR
          • Population standard deviation
          • Sample standard deviation and bias
        • Topic B: Lesson 9: Using a curve to model a data distribution
          • Density Curves
          • Median, mean and skew from density curves
          • Density curve worked example
          • Worked example finding area under density curves
        • Topic B: Lessons 10-11: Normal distributions
          • Normal distribution problem: z-scores (from ck12.org)
          • Comparing with z-scores
          • Qualitative sense of normal distributions (from ck12.org)
          • Normal distribution problems: Empirical rule (from ck12.org)
          • Standard normal table for proportion below
          • Standard normal table for proportion above
          • Standard normal table for proportion between values
        • Topic C: Lesson 12: Types of statistical studies
          • Types of statistical studies
          • Worked example identifying experiment
          • Worked example identifying observational study
          • Worked example identifying sample study
          • Correlation and causality
        • Topic C: Lesson 13: Using sample data to estimate a population characteristic
          • Identifying a sample and population
          • Reasonable samples
          • Techniques for generating a simple random sample
        • Topic C: Lessons 14-17: Sample proportions, sampling variability, margin of error
        • Topic C: Lessons 18-19: Sampling variability in the sample mean
          • Inferring population mean from sample mean
          • Central limit theorem
          • Sampling distribution of the sample mean
          • Sampling distribution of the sample mean 2
          • Standard error of the mean
        • Topic C: Lessons 20-22: Margin of error with sample means; evaluating reports with sample data
        • Topic D: Lesson 23: Experiments and the role of random assignment
          • Types of statistical studies
          • Worked example identifying experiment
          • Worked example identifying observational study
          • Introduction to experiment design
          • Matched pairs experiment design
        • Topic D: Lessons 24-27: Ruling out differences due to random assignment
        • Topic D: Lessons 28-29: Drawing a conclusion from an experiment
        • Topic D: Lesson 30: Evaluating reports based on data from an experiment
    • Precalculus (Eureka Math/EngageNY)
      • Module 1: Complex numbers and transformations
        • Topic A: Lessons 1-6: Complex numbers review
          • Powers of the imaginary unit
          • Plotting numbers on the complex plane
          • Adding complex numbers
          • Subtracting complex numbers
          • Multiplying complex numbers
        • Topic A: Lessons 7-8: Complex number division
          • Intro to complex number conjugates
          • Complex number conjugates
          • Dividing complex numbers
        • Topic B: Lessons 9-12: Distance and midpoint of complex numbers
          • Distance & midpoint of complex numbers
        • Topic B: Lesson 13: Trigonometry and complex numbers
          • Absolute value of complex numbers
          • Absolute value & angle of complex numbers
          • Polar & rectangular forms of complex numbers
        • Topic B: Lessons 14-17: Multiplying and dividing complex numbers in polar form
          • Dividing complex numbers: polar & exponential form
        • Topic C: Lessons 18-19: Exploiting the connection to trigonometry
          • Powers of complex numbers
          • Complex number equations: x³=1
        • Topic C: Lessons 21-23: Transforming vectors with matrices
          • Intro to matrices
          • Intro to matrix multiplication
          • Multiplying matrices
          • Transforming vectors using matrices
        • Topic C: Lesson 24: Matrix notation encompasses new transformations!
          • Transforming polygons using matrices
          • Visual representation of transformation from matrix
        • Topic C: Lesson 25: Properties of matrix addition
          • Adding & subtracting matrices
          • Multiplying matrices by scalars
        • Topic C: Lesson 25: Properties of matrix multiplication
          • Intro to identity matrix
          • Dimensions of identity matrix
          • Is matrix multiplication commutative?
          • Associative property of matrix multiplication
          • Zero matrix & matrix multiplication
        • Topic C: Lessons 26-30: Matrix inverses
          • Determinant of a 2x2 matrix
          • Intro to matrix inverses
          • Determining invertible matrices
          • Finding inverse of a 2x2 matrix using determinant & adjugate
      • Module 2: Vectors and matrices
        • Topic A: Networks and matrices
        • Topic B: Linear transformations of planes and space
        • Topic C: Systems of linear equations
          • Representing linear systems with matrix equations
          • Solving linear systems with matrix equations
          • Matrix word problem: vector combination
          • Matrix word problem: prices
        • Topic D: Lessons 17-18: Vectors in the coordinate plane
          • Intro to vectors & scalars
          • Recognizing vectors
          • Equivalent vectors
          • Vector magnitude from graph
          • Vector magnitude from components
          • Vector magnitude from initial & terminal points
          • Scalar multiplication of vectors
          • Adding & subtracting vectors
        • Topic D: Lesson 19: Directed line segments and vectors
          • Components of vectors
          • Components of vectors (example 2)
          • Analyzing scalar multiplication
          • Graphically adding & subtracting vectors
          • Graphically subtracting vectors
          • Adding vectors algebraically & graphically
          • Combined vector operations
        • Topic D: Lessons 20: Vectors and stone bridges
          • Direction of vectors from components: 1st & 2nd quadrants
          • Direction of vectors from components: 3rd & 4th quadrants
          • Vector components from magnitude & direction
          • Adding vectors in magnitude & direction form (1 of 2)
          • Adding vectors in magnitude & direction form (2 of 2)
          • Vector addition & magnitude
        • Topic D: Lessons 21-22: Vectors and linear transformations
        • Topic D: Lessons 23-24: Why are vectors useful?
          • Vectors word problem: pushing a box
          • Vectors word problem: tug of war
          • Vectors word problem: hiking
        • Topic E: First-person video games—projection matrices
      • Module 3: Rational and exponential functions
        • Topic A: Lesson 1: Solutions to polynomial equations
          • Factoring higher-degree polynomials
          • Factoring sum of cubes
          • Factoring difference of cubes
          • Complex numbers & sum of squares factorization
          • Factoring sum of squares
          • Finding zeros of polynomials (1 of 2)
          • Finding zeros of polynomials (2 of 2)
          • Finding zeros of polynomials (example 2)
        • Topic A: Lessons 2-3: Roots of complex numbers
          • Powers of complex numbers
          • Complex number equations: x³=1
        • Topic A: Lessons 4-5: The binomial theorem
          • Intro to the Binomial Theorem
          • Pascal's triangle and binomial expansion
          • Expanding binomials
          • Expanding binomials w/o Pascal's triangle
          • Binomial expansion & combinatorics
          • Pascal's triangle & combinatorics
        • Topic A: Lessons 6-7: Ellipses
          • Intro to ellipses
          • Ellipse standard equation from graph
          • Ellipse graph from standard equation
          • Foci of an ellipse from equation
        • Topic A: Lesson 8: Hyperbolas
          • Intro to hyperbolas
          • Vertices & direction of a hyperbola
          • Vertices & direction of a hyperbola (example 2)
          • Foci of a hyperbola from equation
          • Proof of the hyperbola foci formula
          • Equation of a hyperbola not centered at the origin
        • Topic A: Lesson 9: Volume and Cavalieri’s principle
        • Topic B: Lessons 10-11: Simplifying rational expressions
          • Intro to rational expression simplification
          • Simplifying rational expressions: common monomial factors
          • Simplifying rational expressions: common binomial factors
          • Simplifying rational expressions: opposite common binomial factors
          • Simplifying rational expressions: grouping
          • Simplifying rational expressions: higher degree terms
          • Simplifying rational expressions: two variables
        • Topic B: Lessons 10-11: Multiplying & dividing rational expressions
          • Multiplying & dividing rational expressions: monomials
          • Multiplying rational expressions
          • Dividing rational expressions
          • Multiplying rational expressions: multiple variables
          • Dividing rational expressions: unknown expression
        • Topic B: Lessons 10-11: Adding & subtracting rational expressions
          • Adding & subtracting rational expressions: like denominators
          • Intro to adding rational expressions with unlike denominators
          • Adding rational expression: unlike denominators
          • Subtracting rational expressions: unlike denominators
          • Least common multiple
          • Least common multiple: repeating factors
          • Subtracting rational expressions: factored denominators
          • Least common multiple of polynomials
          • Subtracting rational expressions
          • Nested fractions
        • Topic B: Lessons 12-13: End behavior and asymptotes of rational functions
          • End behavior of rational functions
          • Discontinuities of rational functions
          • Analyzing vertical asymptotes of rational functions
        • Topic B: Lessons 14-15: Graphing rational functions
          • Graphing rational functions according to asymptotes
          • Graphs of rational functions: y-intercept
          • Graphs of rational functions: horizontal asymptote
          • Graphs of rational functions: vertical asymptotes
          • Graphs of rational functions: zeros
          • Graphing rational functions 1
          • Graphing rational functions 2
          • Graphing rational functions 3
          • Graphing rational functions 4
        • Topic B: Lessons 16-17: Function composition
          • Intro to composing functions
          • Evaluating composite functions
          • Evaluating composite functions: using tables
          • Evaluating composite functions: using graphs
          • Finding composite functions
          • Evaluating composite functions (advanced)
          • Modeling with composite functions
          • Modeling with composite functions (example 2)
        • Topic C: Lesson 18: Inverse functions
          • Intro to inverse functions
          • Inputs & outputs of inverse functions
          • Graphing the inverse of a linear function
          • Finding inverse functions: linear
          • Finding inverse functions: quadratic
          • Finding inverse functions: quadratic (example 2)
          • Finding inverse functions: radical
          • Finding inverse functions: rational
        • Topic C: Lesson 19: Restricting the domain
          • Verifying inverse functions by composition
          • Verifying inverse functions by composition: not inverse
          • Determining if a function is invertible
          • Restricting domains of functions to make them invertible
        • Topic C: Lessons 20-21: Inverse relationship of exponentials and logarithms
          • Relationship between exponentials & logarithms
          • Relationship between exponentials & logarithms: graphs
          • Relationship between exponentials & logarithms: tables
      • Module 4: Trigonometry
        • Topic A: Lessons 1-2: Special values and properties of trigonometric functions
          • Trig values of special angles
          • Graph of y=sin(x)
          • Graph of y=tan(x)
          • Intersection points of y=sin(x) and y=cos(x)
          • Sine & cosine identities: symmetry
          • Tangent identities: symmetry
          • Sine & cosine identities: periodicity
          • Tangent identities: periodicity
        • Topic A: Lessons 3-4: Addition and subtraction formulas
          • Review of trig angle addition identities
          • Using the cosine angle addition identity
          • Using the cosine double-angle identity
          • Proof of the sine angle addition identity
          • Proof of the cosine angle addition identity
          • Finding trig values using angle addition identities
        • Topic A: Lessons 5-6: Tangent lines and wave functions
        • Topic B: Trigonometry and triangles
          • Solving for a side with the law of sines
          • Solving for an angle with the law of sines
          • Proof of the law of sines
          • Solving for a side with the law of cosines
          • Solving for an angle with the law of cosines
          • Proof of the law of cosines
          • Trig word problem: stars
        • Topic C: Inverse trigonometric functions
          • Intro to arcsine
          • Intro to arctangent
          • Intro to arccosine
          • Restricting domains of functions to make them invertible
          • Domain & range of inverse tangent function
          • Using inverse trig functions with a calculator
          • Solving sinusoidal equations of the form sin(x)=d
          • Solving cos(θ)=1 and cos(θ)=-1
          • Trig word problem: solving for temperature
      • Module 5: Probability and statistics
        • Topic A: Lesson 1: The general multiplication rule
          • Compound probability of independent events
          • Independent events example: test taking
          • Coin flipping probability
          • Dependent probability introduction
        • Topic A: Lesson 2: Counting rules—The fundamental counting principle and permutations
          • Count outcomes using tree diagram
          • Counting outcomes: flower pots
          • Permutation formula
          • Zero factorial or 0!
          • Possible three letter words
          • Ways to arrange colors
          • Ways to pick officers
        • Topic A: Lesson 3: Counting rules—Combinations
          • Intro to combinations
          • Combination formula
          • Handshaking combinations
          • Combination example: 9 card hands
        • Topic A: Lesson 4: Using permutations and combinations to compute probabilities
          • Getting exactly two heads (combinatorics)
          • Exactly three heads in five flips
          • Example: Different ways to pick officers
          • Example: Combinatorics and probability
          • Example: Lottery probability
          • Mega millions jackpot probability
        • Topic B: Lessons 5-6: Discrete random variables
          • Random variables
          • Discrete and continuous random variables
          • Valid discrete probability distribution examples
          • Probability with discrete random variable example
        • Topic B: Lessons 7-8: Expected value
          • Mean (expected value) of a discrete random variable
        • Topic B: Lessons 9-12: Creating probability distributions
          • Constructing a probability distribution for random variable
          • Probability models example: frozen yogurt
        • Topic C: Lessons 13-19: Applications of expected value and probability
          • Term life insurance and death probability
          • Expected profit from lottery ticket
          • Expected value while fishing
          • Comparing insurance with expected value
    • 3rd grade foundations (Eureka Math/EngageNY)
      • Module 1: Foundations
        • Topic A, B, C, D, E, & F: Foundations
          • Repeated addition: haircuts
          • Addition word problem: horses
          • Subtraction word problem: crayons
      • Module 2: Foundations
        • Topic A: Foundations
          • Measuring lengths 2
          • Measuring lengths with different units
          • Telling time (labeled clock)
        • Topic B: Foundations
          • Estimating lengths
          • Comparing lengths
        • Topic C: Foundations
          • Comparing whole numbers
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting two-digit numbers without regrouping (example 2)
          • Subtracting a 1-digit number with regrouping
        • Topic D & E: Foundations
          • Adding 1s, 10s, and 100s
          • Breaking apart 3-digit addition problems
          • Adding and subtracting on number line 2
          • Adding and subtracting on number line word problems
      • Module 3: Foundations
        • Topic A, C, D, & E: Foundations
          • Addition word problem: horses
          • Subtraction word problem: tennis balls
          • Repeated addition: haircuts
        • Topic B: Foundations
          • Adding 53+17 by making a group of 10
          • Adding 2-digit numbers without regrouping 2
          • Adding with regrouping
          • Subtracting a 1-digit number with regrouping
        • Topic F: Foundations
          • Intro to place value
      • Module 4: Foundations
        • Topic A: Foundations
          • Measuring lengths 2
          • Measuring lengths with different units
        • Topic B, C, & D: Foundations
          • Repeated addition: haircuts
          • Length word problems
          • Measuring length: golden statue
      • Module 5: Foundations
        • Topic A, B, C, D, E, & F: Foundations
          • Equal parts of circles and rectangles
          • Adding and subtracting on number line word problems
          • Addition word problem: horses
          • Subtraction word problem: tennis balls
      • Module 6: Foundations
        • Topic A: Foundations
          • Length word problems
          • Measuring length: golden statue
          • Adding and subtracting on number line word problems
        • Topic B: Foundations
          • Intro to line plots
          • Making picture graphs and line plots
          • Reading bar graphs: bikes
          • Creating picture and bar graphs 1
          • Picture graphs
      • Module 7: Foundations
        • Topic A, B, C, D, & E: Foundations
          • Measuring lengths 2
          • Measuring lengths with different units
          • Adding and subtracting on number line word problems
          • Recognizing shapes
    • 4th grade foundations (Eureka Math/EngageNY)
      • Module 1: Foundations
        • Topic A & B: Foundations
          • Comparing whole numbers
        • Topic C: Foundations
          • Rounding to nearest 10 and 100
          • Rounding to nearest 10
          • Rounding to nearest 100
        • Topic D, E, & F: Foundations
          • 2-step word problem: truffles
          • 2-step word problem: theater
          • Breaking apart 3-digit addition problems
      • Module 2: Foundations
        • Topic A & B: Foundations
          • Measuring lengths with different units
          • Comparing lengths
          • Length word problems
          • Understanding mass (grams and kilograms)
          • Word problems with mass
      • Module 3: Foundations
        • Topic A: Foundations
          • 2-step estimation problem: marbles
          • Unknowns with multiplication and division
          • Comparing areas and perimeters of rectangles
        • Topic B & C: Foundations
          • Properties and patterns for multiplication
          • Multiplying 1-digit numbers by multiples of 10, 100, and 1000
          • Multiplying by tens word problem
        • Topic D: Foundations
          • 2-step word problem: theater
          • 2-step word problem: running
          • 2-step word problem: truffles
        • Topic E: Foundations
          • Division word problem: school building
          • The idea of division
        • Topic F: Foundations
        • Topic G: Foundations
          • 2-step word problem: theater
          • 2-step word problem: running
          • 2-step word problem: truffles
          • Properties and patterns for multiplication
          • The idea of division
        • Topic H: Foundations
      • Module 4: Foundations
        • Topic A: Foundations
          • Intro to quadrilateral
          • Quadrilateral types
          • Classifying quadrilaterals
        • Topic C: Foundations
          • 2-step word problem: theater
          • 2-step word problem: running
          • 2-step word problem: truffles
      • Module 5: Foundations
        • Topic A, D, & F: Foundations
          • Intro to fractions
          • Identifying unit fractions word problem
          • Identifying numerators and denominators
          • Representing 1 as a fraction
          • Fractions on a number line
        • Topic B: Foundations
          • Equivalent fractions with visuals
          • Creating equivalent fractions
          • Whole numbers as fractions
        • Topic C: Foundations
          • Comparing fractions of different wholes 1
          • Comparing fractions with > and < symbols
          • Comparing fractions with the same numerator
          • Comparing fractions with the same denominator
        • Topic E: Foundations
          • Intro to line plots
          • Graphing data on line plots
        • Topic H: Foundations
          • Finding patterns in numbers
          • Intro to even and odd numbers
          • Patterns in multiplication tables
      • Module 6: Foundations
        • Topic A: Foundations
          • Understanding mass (grams and kilograms)
          • Word problems with mass
          • Multiplying 1-digit numbers by multiples of 10, 100, and 1000
          • Multiplying by tens word problem
        • Topic B, C, & D: Foundations
          • Intro to fractions
          • Identifying unit fractions word problem
          • Identifying numerators and denominators
          • Representing 1 as a fraction
          • Fractions on a number line
          • Equivalent fractions with visuals
          • Creating equivalent fractions
      • Module 7: Foundations
        • Topic A: Foundations
          • Intro to multiplication
        • Topic B: Foundations
          • 2-step word problem: theater
          • 2-step word problem: running
          • 2-step word problem: truffles
        • Topic C: Foundations
          • Understanding mass (grams and kilograms)
          • Word problems with mass
    • 5th grade foundations (Eureka Math/EngageNY)
      • Module 1: Foundations
        • Topic A: Foundations
          • Place value when multiplying and dividing by 10
          • Understanding place value
          • Intro to decimals
          • Decimals as words
          • Rewriting decimals as fractions: 0.15
        • Topic B: Foundations
          • Writing a number in expanded form
          • Comparing whole number place values
          • Comparing decimals 1
          • Comparing numbers represented different ways
        • Topic C: Foundations
          • Rounding whole numbers to nearest hundred
          • Rounding whole numbers to nearest thousand
          • Rounding whole numbers to nearest ten
        • Topic D & F: Foundations
          • Adding 3-digit numbers
          • Adding with regrouping
          • Adding 3-digit numbers with regrouping
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Worked example: Subtracting 3-digit numbers (regrouping twice)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
        • Topic E: Foundations
          • Multiplication tables for 2-9
          • Multiplying 1-digit numbers by 10, 100, and 1000
          • Multiplying by multiples of 10
      • Module 2: Foundations
        • Topic A: Foundations
          • Multiplying whole numbers by 10
          • Creating the largest number
        • Topic B: Foundations
          • Comparing with multiplication: giraffe
          • Multiplying with area model: 6 x 7981
          • Multiplying with area model: 78 x 65
        • Topic C: Foundations
          • Rewriting fractions as decimals
          • Graphing tenths from 0 to 1
          • Identifying tenths on a number line
          • Graphing hundredths from 0 to 0.1
          • Identifying hundredths on a number line
        • Topic D: Foundations
          • Multiplying 2-digit numbers
          • Multiplying multi-digit numbers
        • Topic E: Foundations
          • Dividing whole numbers by 10
          • Quotients that are multiples of 10
          • Division using place value
        • Topic F: Foundations
          • Division with area models
          • Intro to long division (no remainders)
          • Long division: 280÷5
        • Topic G: Foundations
          • Adding 3-digit numbers
          • Adding with regrouping
          • Adding 3-digit numbers with regrouping
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Worked example: Subtracting 3-digit numbers (regrouping twice)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
        • Topic H: Foundations
          • Worked example: Long division with remainders: 3771÷8
          • Worked example: Long division with remainders: 2292÷4
          • Division word problem: field goals
      • Module 3: Foundations
        • Topic A: Foundations
          • Equivalent fractions with visuals
          • Equivalent fraction models
          • Equivalent fraction visually
          • Creating equivalent fractions
        • Topic B: Foundations
          • Adding fractions with like denominators
          • Subtracting fractions with like denominators
          • Adding mixed numbers with like denominators
          • Subtracting mixed numbers with like denominators
        • Topic C: Foundations
          • Intro to adding mixed numbers
          • Intro to subtracting mixed numbers
          • Comparing fractions: tape diagram
          • Comparing fractions: number line
        • Topic D: Foundations
          • Fraction word problem: pizza
          • Fraction word problem: piano
          • Fraction word problem: lizard
      • Module 4: Foundations
        • Topic A: Foundations
          • Reading a line plot with fractions
        • Topic B: Foundations
          • Intro to multiplication
          • The idea of division
          • Comparing with multiplication: money
        • Topic C: Foundations
          • Multiplying unit fractions and whole numbers
          • Equivalent fraction and whole number multiplication problems
        • Topic D: Foundations
          • More ways to multiply
          • Multiplication word problem: pizza
          • Multiplying fractions word problem: milk
        • Topic E: Foundations
          • Division word problem: field goals
        • Topic F: Foundations
          • Comparing with multiplication: age
          • Comparing with multiplication: magic
        • Topic G: Foundations
          • Relating division to multiplication
          • Division word problem: blueberries
          • Recognizing fractions greater than 1
      • Module 5: Foundations
        • Topic A: Foundations
          • Intro to area and unit squares
          • Creating rectangles with a given area 1
          • Creating rectangles with a given area 2
        • Topic B: Foundations
          • Area & perimeter word problem: table
        • Topic C: Foundations
          • Multiplying fractions and whole numbers visually
          • Multiplication word problem: carrots
          • Division word problem: row boat
          • Multi-step unit conversion examples (U.S. Customary)
        • Topic D: Foundations
          • Intro to quadrilaterals
          • Quadrilateral properties
          • Quadrilateral types
      • Module 6: Foundations
        • Topic A: Foundations
          • Fractions on a number line
          • Representing 1 as a fraction
          • Whole numbers as fractions
        • Topic B & D: Foundations
          • Factors and multiples: days of the week
          • Math patterns: toothpicks
          • Math patterns: table
        • Topic C: Foundations
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines
          • Drawing parallel line segments
        • Topic E: Foundations
          • Fraction word problem: pizza
          • Multiplying fractions word problem: movies
          • Multiplying fractions word problem: milk
    • 6th grade foundations (Eureka Math/EngageNY)
      • Module 1: Foundations
        • Topic A: Foundations
          • Comparing with multiplication: basics
          • Comparing with multiplication: giraffe
          • Understanding fractions as division
          • Creating a fraction through division
        • Topic B: Foundations
          • Coordinate plane: graphing points
          • Coordinate plane graphing word problem
          • Drawing a quadrilateral on the coordinate plane example
        • Topic C: Foundations
          • Converting US customary units of volume
          • Converting units: metric distance
          • Creating a fraction through division
      • Module 2: Foundations
        • Topic A: Foundations
          • Multiplying 2 fractions: fraction model
          • Multiplying mixed numbers
          • Dividing fractions by whole numbers: studying
        • Topic B: Foundations
          • Multiplying 3-digit by 1-digit
          • Multiplying 4-digit by 1-digit (regrouping)
          • Multiplying multi-digit numbers
          • Adding decimals: 0.822+5.65
          • Subtracting decimals: 9.005 - 3.6
          • Intro to multiplying decimals
          • Multiplying decimals: place value
          • Dividing whole numbers to get a decimal
        • Topic C: Foundations
          • Dividing by 2-digits: 7182÷42
          • Intro to remainders
        • Topic D: Foundations
          • Finding factors of a number
          • Finding factors and multiples
          • Recognizing prime and composite numbers
      • Module 3: Foundations
        • Topic A: Foundations
          • Coordinate plane: graphing points
          • Intro to reflective symmetry
          • Fractions on a number line
        • Topic B: Foundations
          • Representing 1 as a fraction
          • Whole numbers as fractions
        • Topic C: Foundations
          • Coordinate plane graphing word problem
          • Fractions on number line widget
      • Module 4: Foundations
        • Topic A & D: Foundations
          • Properties of multiplication
          • Associative law of multiplication
          • Distributive property over addition
          • Distributive property over subtraction
        • Topic B: Foundations
          • Finding factors and multiples
          • Exponents and powers of 10 patterns
          • Multiplying a decimal by a power of 10
          • Dividing a decimal by a power of 10
        • Topic C: Foundations
          • Constructing numerical expressions
          • Evaluating expressions with & without parentheses
        • Topic E & F: Foundations
          • Graphing patterns on coordinate plane
          • Interpreting patterns on coordinate plane
          • Interpreting relationships in ordered pairs
          • Graphing sequence relationships
        • Topic G: Foundations
          • Visually adding fractions: 5/6+1/4
          • Visually subtracting fractions: 3/4-5/8
          • Solving for the missing fraction
          • Multiplying fractions and whole numbers visually
        • Topic H: Foundations
          • Identifying an angle
          • Constructing angles
          • Measuring angles using a protractor 2
          • Decomposing an angle
          • Coordinate plane: graphing points
          • Coordinate plane graphing word problem
          • Drawing a quadrilateral on the coordinate plane example
      • Module 5: Foundations
        • Topic A: Foundations
          • Area & perimeter word problem: table
          • Area & perimeter word problem: dog pen
          • Worked example: Classifying triangles
        • Topic B: Foundations
          • Coordinate plane graphing word problem
        • Topic C: Foundations
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
          • Multiplying 2 fractions: fraction model
        • Topic D: Foundations
          • Volume through decomposition
          • Volume word problem: water tank
      • Module 6: Foundations
        • Topics A, B, C, & D: Foundations
          • Line plot distribution: trail mix
          • Multiplying 3-digit by 1-digit
          • Multiplying 4-digit by 1-digit (regrouping)
          • Multiplying multi-digit numbers
          • Dividing by 2-digits: 7182÷42
          • Intro to remainders
          • Dividing whole numbers to get a decimal
    • 7th grade foundations (Eureka Math/EngageNY)
      • Module 1: Foundations
        • Topic A: Foundations
          • Intro to ratios
          • Part:whole ratios
        • Topic B: Foundations
          • Intro to rates
          • Solving unit rate problem
          • Modeling with one-step equations
        • Topic C: Foundations
          • Dividing fractions: 3/5 ÷ 1/2
          • Comparing rates example
          • Finding average speed or rate
          • One-step multiplication & division equations: fractions & decimals
        • Topic D: Foundations
          • Area of a parallelogram
          • Area of a triangle
      • Module 2: Foundations
        • Topic A: Foundations
          • Ordering negative numbers
          • Negative symbol as opposite
        • Topic B: Foundations
          • Multiplying 2 fractions: 5/6 x 2/3
          • Multiplying mixed numbers
          • Dividing fractions: 2/5 ÷ 7/3
        • Topic C: Foundations
          • Adding decimals: 9.087+15.31
          • Subtracting decimals: 39.1 - 0.794
          • Multiplying challenging decimals
          • Distributive property over addition
      • Module 3: Foundations
        • Topic A: Foundations
          • Combining like terms
          • Combining like terms with distribution
          • Distributive property with variables
          • Equivalent expressions
        • Topic B: Foundations
          • One-step equations word problem: super yoga (1 of 2)
          • One-step equations word problem: super yoga (2 of 2)
          • Inequality word problem: one variable
          • Plotting an inequality example
        • Topic C: Foundations
          • Area of composite shapes
          • Area of a quadrilateral on a grid
          • Surface area of a box using nets
      • Module 4: Foundations
        • Topic A: Foundations
          • The meaning of percent
          • Converting percents to decimals & fractions example
          • Finding a percent
          • Percent of a whole number
          • Identifying percent amount and base
        • Topic B: Foundations
          • Meaning of 109%
          • Percent word problem: 100 is what percent of 80?
          • Percent word problem: recycling cans
          • Percent word problem: penguins
          • Solving percent problems
        • Topic C & D: Foundations
          • Intro to rates
          • Solving unit rate problem
          • Comparing rates example
          • Finding average speed or rate
          • Area of composite shapes
          • Area of a quadrilateral on a grid
      • Module 5: Foundations
        • Topic A, B, C, & D: Foundations
          • Statistical questions
          • Frequency tables & dot plots
          • Creating a histogram
          • Interpreting a histogram
          • Shapes of distributions
          • Clusters, gaps, peaks & outliers
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
          • Impact on median & mean: removing an outlier
          • Impact on median & mean: increasing an outlier
          • Median & range puzzlers
      • Module 6: Foundations
        • Topic A: Foundations
          • Angles in circles word problem
          • Measuring angles using a protractor
          • Decomposing an angle
          • Decomposing angles
        • Topic B: Foundations
        • Topic D: Foundations
          • Intro to nets of polyhedra
          • Surface area of a box using nets
          • Surface area of a box (cuboid)
          • Area of a parallelogram
          • Area of a triangle
          • Area of composite shapes
          • Area of a quadrilateral on a grid
        • Topic E: Foundations
          • Volume through decomposition
          • Volume word problem: water tank
          • Volume of a rectangular prism: fractional dimensions
          • Volume of a rectangular prism: word problem
    • 8th grade foundations (Eureka Math/EngageNY)
      • Module 1: Foundations
        • Topic A: Foundations
          • Intro to exponents
          • The zeroth power
          • Powers of fractions
        • Topic B: Foundations
          • Patterns in powers of 10
          • Multiplying a decimal by a power of 10
          • Dividing a decimal by a power of 10
      • Module 2: Foundations
        • Topic A: Foundations
          • Classifying shapes by line and angles types
          • Angle measurement & circle arcs
          • Naming angles
        • Topic B: Foundations
          • Lines, line segments, & rays
          • Acute, right, & obtuse angles
          • Classifying triangles
          • Parallel & perpendicular lines intro
          • Coordinate plane: graphing points
        • Topic C: Foundations
          • Complementary & supplementary angles
          • Vertical angles
        • Topic D: Foundations
          • Perimeter & area
          • Area of a parallelogram
          • Area of trapezoids
          • Area of composite shapes
      • Module 3: Foundations
        • Topic A: Foundations
          • Intro to proportional relationships
          • Proportional relationships
          • Interpreting a scale drawing
          • Making a scale drawing
        • Topic B: Foundations
          • Writing proportions example
          • Worked example: Solving proportions
          • Complementary & supplementary angles
          • Vertical angles
        • Topic C: Foundations
          • Construct a right isosceles triangle
          • Construct a triangle with constraints
          • Triangle inequality theorem
      • Module 4: Foundations
        • Topic A: Foundations
          • Writing basic expressions with variables
          • Evaluating an expression with one variable
          • Intro to two-step equations
          • Worked example: two-step equations
        • Topic B: Foundations
          • Solving unit rate problem
          • Solving unit price problem
          • Equations for proportional relationships
          • Writing proportional equations
        • Topic C: Foundations
          • Comparing rates example
          • Proportional relationships: graphs
          • Interpreting graphs of proportional relationships
        • Topic D: Foundations
          • Variables, expressions, & equations
          • Testing solutions to equations
      • Module 5: Foundations
        • Topic A: Foundations
          • Intro to proportional relationships
          • Proportional relationships
          • Equations for proportional relationships
          • Writing proportional equations
          • Proportional relationships: graphs
          • Interpreting graphs of proportional relationships
        • Topic B: Foundations
          • Radius, diameter, circumference & π
          • Area of a circle
          • Measuring volume as area times length
      • Module 6: Foundations
        • Topic A: Foundations
          • Intro to two-step equations
          • Worked example: two-step equations
          • Two-step equations with decimals and fractions
          • Two-step equation word problem: computers
          • Two-step equation word problem: garden
          • Two-step equation word problem: oranges
        • Topic B & C: Foundations
          • Points on the coordinate plane examples
          • Plotting a point (ordered pair)
          • Finding the point not graphed
          • Coordinate plane word problem examples
      • Module 7: Foundations
        • Topic A & B: Foundations
          • Dividing by 2-digits: 7182÷42
          • Decimals & fractions on the number line
          • Fraction to decimal: 11/25
          • Variables, expressions, & equations
          • Testing solutions to equations
        • Topic C: Foundations
          • Perimeter & area
          • Area of a parallelogram
          • Area of trapezoids
          • Area of composite shapes
          • Dimensions of a rectangle from coordinates
        • Topic D: Foundations
          • Radius, diameter, circumference & π
          • Area of a circle
          • Measuring volume as area times length
    • Mathematics I
      • Solving equations & inequalities
        • Linear equations with variables on both sides
          • Why we do the same thing to both sides: Variable on both sides
          • Intro to equations with variables on both sides
          • Equations with variables on both sides: 20-7x=6x-6
          • Equation with variables on both sides: fractions
          • Equation with the variable in the denominator
        • Linear equations with parentheses
          • Equations with parentheses
        • Analyzing the number of solutions to linear equations
          • Number of solutions to equations
          • Worked example: number of solutions to equations
          • Creating an equation with no solutions
          • Creating an equation with infinitely many solutions
        • Linear equations with unknown coefficients
          • Linear equations with unknown coefficients
        • Multi-step inequalities
          • Inequalities with variables on both sides
          • Inequalities with variables on both sides (with parentheses)
          • Multi-step inequalities
        • Compound inequalities
          • Compound inequalities: OR
          • Compound inequalities: AND
          • A compound inequality with no solution
          • Double inequalities
          • Compound inequalities examples
      • Working with units
        • Rate conversion
          • Intro to dimensional analysis
          • Same rate with different units
        • Appropriate units
          • Appropriate units
          • Interpreting units in formulas
          • Interpreting units in formulas: novel units
        • Word problems with multiple units
          • Worked example: Rate problem
          • Multiple units word problem: road trip
          • Measurement word problem: running laps
          • Multiple units word problem: drug dosage (advanced)
      • Linear equations & graphs
        • Two-variable linear equations intro
          • Two-variable linear equations intro
          • Solutions to 2-variable equations
          • Worked example: solutions to 2-variable equations
          • Completing solutions to 2-variable equations
        • x-intercepts and y-intercepts
          • Intro to intercepts
          • x-intercept of a line
          • Intercepts from an equation
          • Intercepts from a table
        • Slope
          • Intro to slope
          • Positive & negative slope
          • Worked example: slope from graph
          • Worked example: slope from two points
          • Slope (more examples)
        • Horizontal & vertical lines
          • Slope of a horizontal line
          • Horizontal & vertical lines
        • Intro to slope-intercept form
          • Intro to slope-intercept form
          • Worked examples: slope-intercept intro
        • Graphing slope-intercept equations
          • Graph from slope-intercept equation
        • Writing slope-intercept equations
          • Slope-intercept equation from graph
          • Slope-intercept equation from slope & point
          • Slope-intercept equation from two points
          • Slope-intercept form problems
        • Point-slope form
          • Intro to point-slope form
          • Point-slope & slope-intercept equations
        • Standard form
          • Intro to linear equation standard form
          • Graphing a linear equation: 5x+2y=20
          • Converting from slope-intercept to standard form
        • Summary: Forms of two-variable linear equations
          • Slope from equation
          • Writing linear equations in all forms
      • Functions
        • Evaluating functions
          • What is a function?
          • Worked example: Evaluating functions from equation
          • Worked example: evaluating functions from graph
          • Worked example: evaluating expressions with function notation
        • Inputs and outputs of a function
          • Worked example: matching an input to a function's output (equation)
          • Worked example: matching an input to a function's output (graph)
          • Worked example: two inputs with the same output (graph)
        • Functions and equations
          • Equations vs. functions
          • Obtaining a function from an equation
        • Interpreting function notation
          • Function notation word problem: bank
          • Function notation word problem: beach
        • Introduction to the domain and range of a function
          • Intervals and interval notation
          • What is the domain of a function?
          • What is the range of a function?
          • Worked example: domain and range from graph
        • Determining the domain of a function
          • Domain of a radical function
          • Worked example: domain of algebraic functions
          • Worked example: determining domain word problem (real numbers)
          • Worked example: determining domain word problem (positive integers)
          • Worked example: determining domain word problem (all integers)
        • Recognizing functions
          • Recognizing functions from graph
          • Does a vertical line represent a function?
          • Recognizing functions from table
          • Recognizing functions from verbal description
          • Recognizing functions from verbal description word problem
        • Maximum and minimum points
          • Introduction to minimum and maximum points
          • Worked example: relative maxima and minima
        • Intervals where a function is positive, negative, increasing, or decreasing
          • Increasing, decreasing, positive or negative intervals
          • Worked example: positive & negative intervals
        • Interpreting features of graphs
          • Graph interpretation word problem: temperature
          • Graph interpretation word problem: basketball
        • Average rate of change
          • Introduction to average rate of change
          • Worked example: average rate of change from graph
          • Worked example: average rate of change from table
          • Worked example: average rate of change from equation
        • Average rate of change word problems
          • Average rate of change word problem: table
          • Average rate of change word problem: graph
          • Average rate of change word problem: equation
      • Linear word problems
        • Interpreting linear functions and equations
          • Linear equations word problems: earnings
          • Linear equations word problems: volcano
          • Linear graphs word problems
          • Linear graphs word problem: cats
          • Linear equations word problem: marbles
          • Linear equations word problem: file transfer
        • Comparing linear functions
          • Comparing linear functions: equation vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions word problem: climb
          • Comparing linear functions word problem: walk
          • Comparing linear functions word problem: work
        • Constructing linear models for real-world relationships
          • Linear functions word problem: fuel
          • Linear functions word problem: pool
          • Linear functions word problem: iceberg
          • Linear functions word problem: paint
        • Linear models word problems
          • Linear models word problem: book
          • Linear models word problem: marbles
      • Systems of equations
        • Systems of equations overview
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: potato chips
          • Systems of equations with substitution: potato chips
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
        • Introduction to systems of equations
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Testing a solution to a system of equations
          • Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
          • Systems of equations with graphing: exact & approximate solutions
        • Equivalent systems of equations and the elimination method
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: x-4y=-18 & -x+3y=11
          • Systems of equations with elimination: potato chips
          • Systems of equations with elimination (and manipulation)
          • Why can we subtract one equation from the other in a system of equations?
          • Worked example: equivalent systems of equations
          • Worked example: non-equivalent systems of equations
        • Solving systems of equations with substitution
          • Systems of equations with substitution: potato chips
          • Systems of equations with substitution: -3x-4y=-2 & y=2x-5
        • Manipulating expressions with unknown variables
          • Manipulating expressions using structure
          • Manipulating expressions using structure (example 2)
        • Number of solutions to systems of equations
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
          • Solutions to systems of equations: consistent vs. inconsistent
          • Solutions to systems of equations: dependent vs. independent
          • Number of solutions to a system of equations graphically
          • Number of solutions to a system of equations algebraically
          • How many solutions does a system of linear equations have if there are at least two?
        • Solving any system of linear equations
        • Systems of equations word problems
          • Age word problem: Imran
          • Age word problem: Ben & William
          • Age word problem: Arman & Diya
          • System of equations word problem: walk & ride
          • System of equations word problem: no solution
          • System of equations word problem: infinite solutions
          • Systems of equations with elimination: TV & DVD
          • Systems of equations with elimination: apples and oranges
          • Systems of equations with substitution: coins
          • Systems of equations with elimination: coffee and croissants
      • Inequalities (systems & graphs)
        • Checking solutions of two-variable inequalities
          • Testing solutions to inequalities
          • Testing solutions to systems of inequalities
        • Graphing two-variable inequalities
          • Intro to graphing two-variable inequalities
          • Graphing two-variable inequalities
          • Two-variable inequalities from their graphs
          • Intro to graphing systems of inequalities
          • Graphing systems of inequalities
        • Modeling with linear inequalities
          • Writing two-variable inequalities word problem
          • Interpreting two-variable inequalities word problem
          • Solving two-variable inequalities word problem
          • Graphs of two-variable inequalities word problem
          • Modeling with systems of inequalities
          • Writing systems of inequalities word problem
          • Solving systems of inequalities word problem
          • Graphs of systems of inequalities word problem
          • Analyzing structure with linear inequalities: fruits
          • Analyzing structure with linear inequalities: balls
      • Scatterplots
        • Creating and interpreting scatterplots
          • Constructing a scatter plot
          • Scatter plots: studying, shoe size, and test scores
          • Example: Correlation coefficient intuition
          • Correlation and causality
        • Estimating with trend lines
          • Estimating the line of best fit exercise
          • Line of best fit: smoking in 1945
          • Interpreting a trend line
      • Data distributions
        • Displays of distributions
          • Frequency tables & dot plots
          • Creating a histogram
          • Interpreting a histogram
          • Shapes of distributions
        • Summarizing center of distributions (central tendency)
          • Statistics intro: Mean, median, & mode
          • Mean, median, & mode example
          • Missing value given the mean
          • Impact on median & mean: removing an outlier
          • Impact on median & mean: increasing an outlier
        • Box and whisker plots
          • Constructing a box plot
          • Reading box plots
          • Interpreting box plots
        • Comparing distributions
          • Comparing distributions with dot plots (example problem)
      • Two-way tables
        • Two-way frequency tables
        • Two-way relative frequency tables
          • Two-way relative frequency tables
          • Interpreting two-way tables
      • Sequences
        • Introduction to arithmetic sequences
          • Sequences intro
          • Intro to arithmetic sequences
          • Extending arithmetic sequences
          • Using arithmetic sequences formulas
          • Worked example: using recursive formula for arithmetic sequence
        • Constructing arithmetic sequences
          • Recursive formulas for arithmetic sequences
          • Explicit formulas for arithmetic sequences
          • Arithmetic sequence problem
          • Converting recursive & explicit forms of arithmetic sequences
        • Introduction to geometric sequences
          • Intro to geometric sequences
          • Extending geometric sequences
          • Using explicit formulas of geometric sequences
          • Using recursive formulas of geometric sequences
        • Constructing geometric sequences
          • Explicit & recursive formulas for geometric sequences
          • Converting recursive & explicit forms of geometric sequences
        • Modeling with sequences
          • Sequences word problems
        • General sequences
          • Evaluating sequences in recursive form
      • Rational exponents & radicals
        • Exponent properties review
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
        • Radicals
          • Intro to square roots
          • Understanding square roots
          • Intro to cube roots
          • Square root of decimal
          • 5th roots
        • Rational exponents intro
          • Intro to rational exponents
          • Rewriting roots as rational exponents
          • Exponential equation with rational answer
        • Properties of exponents (rational exponents)
          • Simplifying quotient of powers (rational exponents)
          • Simplifying mixed radical and exponential expressions
        • Advanced exponent & radical evaluation
          • Evaluating fractional exponents
          • Evaluating fractional exponents: negative unit-fraction
          • Evaluating fractional exponents: fractional base
          • Evaluating quotient of fractional exponents
          • Evaluating mixed radicals and exponents
        • Simplifying square roots
          • Simplifying square roots
          • Simplifying square roots (variables)
          • Simplifying square-root expressions
        • Simplifying radicals (higher-index roots)
          • Simplifying higher-index roots
          • Simplifying cube root expressions
          • Simplifying cube root expressions (two variables)
          • Simplifying higher-index root expressions
      • Exponential growth & decay
        • Exponential vs. linear growth
          • Intro to exponential functions
          • Exponential vs. linear growth
          • Exponential vs. linear models: verbal
          • Exponential vs. linear models: table
        • Exponential expressions
          • Exponential expressions word problems (numerical)
          • Initial value & common ratio of exponential functions
          • Exponential expressions word problems (algebraic)
          • Interpreting exponential expression word problem
        • Graphs of exponential growth
          • Exponential function graph
          • Graphs of exponential growth
        • Exponential vs. linear growth over time
          • Exponential vs. linear growth over time
        • Exponential decay
          • Exponential decay intro
          • Graphing exponential growth & decay
          • Exponential growth & decay word problems
        • Exponential functions from tables & graphs
          • Writing exponential functions
          • Writing exponential functions from tables
          • Writing exponential functions from graphs
          • Analyzing tables of exponential functions
          • Analyzing graphs of exponential functions
          • Analyzing graphs of exponential functions: negative initial value
          • Modeling with basic exponential functions word problem
      • Geometry transformations
        • Introduction to rigid transformations
          • Rigid transformations intro
        • Translations
          • Translating shapes
          • Determining translations
          • Translation challenge problem
        • Rotations
          • Rotating shapes
          • Determining rotations
        • Reflections
          • Reflecting shapes: diagonal line of reflection
          • Determining reflections (advanced)
        • Rigid transformations overview
        • Dilations
          • Performing dilations
          • Dilating shapes: expanding
          • Dilating shapes: shrinking
        • Properties & definitions of transformations
          • Identifying type of transformation
        • Symmetry
          • Intro to reflective symmetry
          • Intro to rotational symmetry
          • Finding a quadrilateral from its symmetries
          • Finding a quadrilateral from its symmetries (example 2)
      • Congruence
        • Transformations & congruence
          • Congruent shapes & transformations
          • Non-congruent shapes & transformations
        • Triangle congruence
          • Congruent triangles & the SSS postulate/criterion
          • Triangle congruence postulates/criteria
          • Why SSA isn't a congruence postulate/criterion
          • Determining congruent triangles
        • Theorems concerning triangle properties
          • Angles in a triangle sum to 180° proof
          • Proofs concerning isosceles triangles
          • Proofs concerning equilateral triangles
        • Working with triangles
          • Corresponding angles in congruent triangles
          • Isosceles & equilateral triangles problems
          • Finding angles in isosceles triangles
          • Finding angles in isosceles triangles (example 2)
        • Theorems concerning quadrilateral properties
          • Proof: Opposite sides of a parallelogram
          • Proof: Diagonals of a parallelogram
          • Proof: Opposite angles of a parallelogram
          • Proof: The diagonals of a kite are perpendicular
          • Proof: Rhombus diagonals are perpendicular bisectors
          • Proof: Rhombus area
        • Proofs of general theorems that use triangle congruence
          • Geometry proof problem: midpoint
          • Geometry proof problem: congruent segments
          • Geometry proof problem: squared circle
        • Constructing bisectors of lines & angles
          • Geometric constructions: perpendicular bisector
          • Geometric constructions: perpendicular line
          • Geometric constructions: angle bisector
      • Analytic geometry
        • Distance and midpoints
          • Distance formula
          • Midpoint formula
        • Dividing line segments
          • Dividing line segments: graphical
          • Dividing line segments
        • Problem solving with distance on the coordinate plane
          • Area of trapezoid on the coordinate plane
          • Points inside/outside/on a circle
          • Challenge problem: Points on two circles
          • Coordinate plane word problem
        • Parallel & perpendicular lines on the coordinate plane
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines from graph
        • Equations of parallel & perpendicular lines
          • Parallel lines from equation
          • Parallel lines from equation (example 2)
          • Parallel lines from equation (example 3)
          • Perpendicular lines from equation
          • Writing equations of perpendicular lines
          • Writing equations of perpendicular lines (example 2)
          • Proof: parallel lines have the same slope
          • Proof: perpendicular lines have negative reciprocal slope
        • Coordinate plane proofs
          • Classifying quadrilaterals on the coordinate plane
    • Mathematics II
      • Absolute value & piecewise functions
        • Graphs of absolute value functions
          • Shifting absolute value graphs
          • Scaling & reflecting absolute value functions: equation
          • Scaling & reflecting absolute value functions: graph
          • Graphing absolute value functions
        • Piecewise functions
          • Introduction to piecewise functions
          • Worked example: evaluating piecewise functions
          • Worked example: graphing piecewise functions
          • Worked example: domain & range of step function
          • Worked example: domain & range of piecewise linear functions
      • Polynomials
        • Intro to polynomials
          • Polynomials intro
          • The parts of polynomial expressions
          • Evaluating polynomials
          • Simplifying polynomials
        • Adding & subtracting polynomials
          • Adding polynomials
          • Subtracting polynomials
          • Polynomial subtraction & set closure
          • Adding & subtracting multiple polynomials
        • Adding & subtracting polynomials: two variables
          • Adding polynomials: two variables
          • Subtracting polynomials: two variables
          • Subtracting polynomials: two variables challenge
          • Finding an error in polynomial subtraction
          • Polynomials review
        • Multiplying monomials
          • Multiplying monomials
          • Multiplying monomials to find area: two variables
          • Multiplying monomials to find area
          • Multiplying monomials challenge
        • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials: area model
          • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials challenge
        • Multiplying binomials
          • Multiplying binomials: area model
          • Multiplying binomials intro
          • Multiplying binomials
        • Multiplying binomials by polynomials
          • Multiplying binomials by polynomials
          • Multiplying binomials by polynomials: area model
          • Multiplying binomials by polynomials challenge
        • Special products of binomials
          • Special products of the form (x+a)(x-a)
          • Squaring binomials of the form (x+a)²
          • Special products of the form (ax+b)(ax-b)
          • Squaring binomials of the form (ax+b)²
          • Special products of binomials: two variables
          • More examples of special products
        • Polynomials word problems
          • Polynomial multiplication word problem
          • Polynomial word problem: rectangle and circle area
          • Polynomial word problem: total value of bills
          • Polynomial word problem: area of a window
        • Long division of polynomials
          • Intro to long division of polynomials
          • Dividing polynomials: long division
        • Synthetic division of polynomials
          • Intro to polynomial synthetic division
          • Dividing polynomials: synthetic division
          • Why synthetic division works
        • Practice dividing polynomials with remainders
          • Dividing polynomials with remainders: monomial divisors
          • Dividing polynomials with remainders: monomial divisors (example 2)
          • Dividing polynomials with remainders
      • Factorization
        • Introduction to factorization
          • Intro to factors & divisibility
        • Factoring monomials
          • Monomial factorization
          • Worked example: finding the missing monomial factor
          • Worked example: finding missing monomial side in area model
          • Greatest common factor of monomials
        • Factoring polynomials by taking common factors
          • Factoring with the distributive property
          • Factoring binomials: common factor
          • Factoring polynomials: common factor
          • Factoring polynomials: common factor area model
          • Factoring polynomials: common binomial factor
        • Evaluating expressions with unknown variables
          • Worked example: evaluating expressions using structure
          • Worked example: evaluating expressions using structure (more examples)
        • Factoring quadratics intro
          • Factoring quadratics as (x+a)(x+b)
          • Factoring quadratics as (x+a)(x+b) (example 2)
          • More examples of factoring quadratics as (x+a)(x+b)
        • Factoring quadratics by grouping
          • Intro to grouping
          • Factoring quadratics by grouping
        • Factoring polynomials with quadratic forms
          • Factoring quadratics: common factor + grouping
          • Factoring quadratics: negative common factor + grouping
          • Factoring two-variable quadratics
          • Factoring two-variable quadratics: rearranging
          • Factoring two-variable quadratics: grouping
        • Factoring quadratics: Difference of squares
          • Difference of squares intro
          • Factoring difference of squares: leading coefficient ≠ 1
          • Factoring difference of squares: analyzing factorization
          • Factoring difference of squares: missing values
          • Factoring difference of squares: shared factors
        • Factoring quadratics: Perfect squares
          • Perfect square factorization intro
          • Factoring perfect squares
          • Identifying perfect square form
          • Factoring perfect squares: common factor
          • Factoring perfect squares: negative common factor
          • Factoring perfect squares: missing values
          • Factoring perfect squares: shared factors
        • Strategy in factoring quadratics
          • Strategy in factoring quadratics (part 1 of 2)
          • Strategy in factoring quadratics (part 2 of 2)
        • Factoring polynomials with special product forms
          • Factoring perfect squares: 4th degree polynomial
          • Factoring difference of squares: two variables
          • Factoring difference of squares: two variables (example 2)
        • Polynomial Remainder Theorem
          • Intro to the Polynomial Remainder Theorem
          • Using the Polynomial Remainder Theorem: finding remainders
          • Using the Polynomial Remainder Theorem: checking factors
          • Using the Polynomial Remainder Theorem: finding coefficients
          • Proof of the Polynomial Remainder Theorem
      • Quadratics
        • Parabolas intro
          • Parabolas intro
        • Quadratic factored form
          • Zero product property
          • Graphing quadratics in factored form
          • Quadratic word problems (factored form)
        • Solving quadratics by taking square roots
          • Solving quadratics by taking square roots
          • Solving quadratics by taking square roots examples
          • Solving quadratics by taking square roots: strategy
          • Quadratics by taking square roots: with steps
        • Quadratic vertex form
          • Vertex form introduction
          • Graphing quadratics: vertex form
          • Quadratic word problems (vertex form)
        • Solving quadratics by factoring
          • Solving quadratics by factoring
          • Solving quadratics by factoring: leading coefficient ≠ 1
          • Solving quadratics using structure
          • Quadratic equations word problem: triangle dimensions
          • Quadratic equations word problem: box dimensions
        • Completing the square
          • Completing the square
          • Worked example: Completing the square (intro)
          • Worked example: Rewriting expressions by completing the square
          • Worked example: Solving equations by completing the square
          • Worked example: completing the square (leading coefficient ≠ 1)
          • Solving quadratics by completing the square: no solution
        • The quadratic formula
          • The quadratic formula
          • Using the quadratic formula
          • Worked example: quadratic formula
          • Worked example: quadratic formula (example 2)
          • Worked example: quadratic formula (negative coefficients)
          • Using the quadratic formula: number of solutions
          • Proof of the quadratic formula
        • Quadratic standard form
          • Finding the vertex of a parabola in standard form
          • Graphing quadratics: standard form
          • Quadratic word problem: ball
        • Features & forms of quadratic functions
          • Forms & features of quadratic functions
          • Worked examples: Forms & features of quadratic functions
          • Vertex & axis of symmetry of a parabola
          • Finding features of quadratic functions
          • Comparing features of quadratic functions
          • Comparing maximum points of quadratic functions
        • Transforming parabolas
          • Intro to parabola transformations
          • Shifting parabolas
          • Scaling & reflecting parabolas
      • Irrational numbers
        • Irrational numbers
          • Intro to rational & irrational numbers
          • Classifying numbers: rational & irrational
        • Sums & products of rational/irrational numbers
          • Proof: sum & product of two rationals is rational
          • Proof: product of rational & irrational is irrational
          • Proof: sum of rational & irrational is irrational
          • Sums and products of irrational numbers
          • Worked example: rational vs. irrational expressions
          • Worked example: rational vs. irrational expressions (unknowns)
        • Proofs concerning irrational numbers
          • Proof: √2 is irrational
          • Proof: square roots of prime numbers are irrational
          • Proof: there's an irrational number between any two rational numbers
      • Complex numbers
        • What are the imaginary numbers?
          • Intro to the imaginary numbers
          • Simplifying roots of negative numbers
          • Powers of the imaginary unit
          • i as the principal root of -1
        • What are the complex numbers?
          • Intro to complex numbers
        • The complex plane
          • Plotting numbers on the complex plane
        • Adding & subtracting complex numbers
          • Adding complex numbers
          • Subtracting complex numbers
        • Multiplying complex numbers
          • Multiplying complex numbers
      • Exponential growth & decay
        • Equivalent forms of exponential expressions
          • Simplifying exponential expressions
          • Rewriting exponential expressions
        • Exponential equations: properties of exponents
          • Solving exponential equations using exponent properties
          • Solving exponential equations using exponent properties (advanced)
        • Interpreting the rate of change of exponential models
          • Interpreting change in exponential models
          • Interpreting time in exponential models
        • Constructing exponential models according to rate of change
          • Constructing exponential models
          • Constructing exponential models: half life
          • Constructing exponential models: percent change
          • Constructing exponential models (old example)
        • Advanced interpretation of exponential models
          • Interpreting change in exponential models: with manipulation
          • Interpreting change in exponential models: changing units
          • Structure in exponential expression
        • Distinguishing between linear and exponential growth
          • Linear vs. exponential growth: from data
          • Linear vs. exponential growth: from data (example 2)
          • Comparing growth of exponential & quadratic models
      • Modeling with algebra
        • Modeling with one-variable equations & inequalities
          • Rational equation word problem
          • Quadratic inequality word problem
          • Exponential equation word problem
        • Manipulating formulas
          • Manipulating formulas: perimeter
          • Manipulating formulas: area
          • Manipulating formulas: temperature
      • Similarity
        • Definitions of similarity
          • Similar shapes & transformations
        • Introduction to triangle similarity
          • Intro to triangle similarity
          • Triangle similarity postulates/criteria
          • Determining similar triangles
          • Proving slope is constant using similarity
        • Solving similar triangles
          • Solving similar triangles
          • Solving similar triangles: same side plays different roles
        • Angle bisector theorem
          • Intro to angle bisector theorem
          • Using the angle bisector theorem
        • Solving problems with similar & congruent triangles
          • Using similar & congruent triangles
          • Challenging similarity problem
        • Solving modeling problems with similar & congruent triangles
          • Geometry word problem: the golden ratio
          • Geometry word problem: Earth & Moon radii
          • Geometry word problem: a perfect pool shot
      • Right triangles & trigonometry
        • Pythagorean theorem
          • Intro to the Pythagorean theorem
          • Intro to the Pythagorean theorem 2
          • Pythagorean theorem word problem: fishing boat
          • Pythagorean theorem example
          • Pythagorean theorem word problem: carpet
          • Introduction to the Pythagorean theorem
          • Pythagorean theorem II
        • Pythagorean theorem proofs
          • Garfield's proof of the Pythagorean theorem
          • Bhaskara's proof of the Pythagorean theorem
          • Pythagorean theorem proof using similarity
          • Another Pythagorean theorem proof
        • Pursuit of a Pythagorean proof (optional)
        • Special right triangles
          • Special right triangles intro (part 1)
          • Special right triangles intro (part 2)
          • 30-60-90 triangle example problem
          • Special right triangles proof (part 1)
          • Special right triangles proof (part 2)
          • Area of a regular hexagon
        • Introduction to the trigonometric ratios
          • Intro to the trigonometric ratios
          • Trigonometric ratios in right triangles
        • Solving for a side in a right triangle using the trigonometric ratios
          • Solving for a side in right triangles with trigonometry
        • Solving for an angle in a right triangle using the trigonometric ratios
        • Modeling with right triangles
          • Right triangle word problem
        • Trigonometric ratios & similarity
          • Triangle similarity & the trigonometric ratios
          • Trig challenge problem: verify identities
          • Trig challenge problem: trig values & side ratios
        • Sine & cosine of complementary angles
          • Sine & cosine of complementary angles
          • Using complementary angles
          • Trig word problem: complementary angles
      • Solid geometry
        • Solid geometry intro
          • Volume of triangular prism & cube
          • Volume of a cone
          • Cylinder volume & surface area
          • Volume of a sphere
        • Density
          • Density word problem: blimp
        • 2D vs. 3D objects
          • Slicing a rectangular pyramid
          • Ways to cross-section a cube
          • Rotating 2D shapes in 3D
      • Circles
        • Circle basics
          • Circles glossary
          • Radius, diameter, circumference & π
          • Proof: all circles are similar
        • Arc measure
          • Intro to arc measure
          • Finding arc measures
          • Finding arc measures with equations
        • Arc length (degrees)
          • Arc length from subtended angle
          • Subtended angle from arc length
        • Introduction to radians
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
        • Arc length (radians)
          • Arc length as fraction of circumference
          • Arc length from subtended angle: radians
        • Sectors
          • Area of a sector
        • Inscribed angles
          • Inscribed angles
          • Inscribed angle theorem proof
        • Inscribed shapes problem solving
          • Proof: Right triangles inscribed in circles
          • Inscribed shapes: find diameter
          • Inscribed shapes: angle subtended by diameter
          • Inscribed shapes: find inscribed angle
          • Inscribed quadrilaterals proof
          • Solving inscribed quadrilaterals
        • Properties of tangents
          • Proof: Radius is perpendicular to tangent line
          • Proof: Segments tangent to circle from outside point are congruent
          • Tangents of circles problem (example 1)
          • Tangents of circles problem (example 2)
          • Tangents of circles problem (example 3)
        • Area of inscribed triangle
          • Proof: radius is perpendicular to a chord it bisects
          • Proof: perpendicular radius bisects chord
          • Area of inscribed equilateral triangle
        • Constructing regular polygons inscribed in circles
          • Geometric constructions: circle-inscribed square
          • Geometric constructions: circle-inscribed equilateral triangle
          • Geometric constructions: circle-inscribed regular hexagon
        • Constructing circumcircles & incircles
          • Geometric constructions: triangle-inscribing circle
          • Geometric constructions: triangle-circumscribing circle
        • Constructing a line tangent to a circle
          • Geometric constructions: circle tangent
          • Geometric constructions: circle tangent (example 2)
      • Conic sections
        • Introduction to conic sections
          • Intro to conic sections
        • Features of a circle
          • Graphing circles from features
          • Features of a circle from its graph
        • Standard equation of a circle
          • Features of a circle from its standard equation
          • Graphing a circle from its standard equation
          • Writing standard equation of a circle
        • Expanded equation of a circle
          • Features of a circle from its expanded equation
        • Focus & directrix of a parabola
          • Intro to focus & directrix
          • Equation of a parabola from focus & directrix
          • Focus & directrix of a parabola from equation
      • Probability
        • Probability basics
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Experimental probability
          • Theoretical and experimental probabilites
          • Making predictions with probability
          • Intuitive sense of probabilities
          • Probability models example: frozen yogurt
        • Counting with combinations
          • Intro to combinations
          • Combination formula
          • Handshaking combinations
          • Combination example: 9 card hands
        • Probability with counting, permutations, combinations
          • Probability with counting outcomes
          • Example: All the ways you can flip a coin
          • Example: Different ways to pick officers
          • Example: Combinatorics and probability
          • Mega millions jackpot probability
        • Multiplication rule for independent events
          • Compound probability of independent events
          • Independent events example: test taking
          • Free-throw probability
        • Multiplication rule for dependent events
          • Dependent probability introduction
          • Independent & dependent probability
          • Dependent probability: coins
        • Addition rule for probability
          • Probability with Venn diagrams
          • Addition rule for probability
        • Conditional probability
        • Simulation and randomness
        • Expected value
    • Mathematics III
      • Functions
        • Combining functions
          • Adding functions
          • Subtracting functions
          • Multiplying functions
          • Dividing functions
        • Composing functions
          • Intro to composing functions
          • Evaluating composite functions
          • Evaluating composite functions: using tables
          • Evaluating composite functions: using graphs
          • Finding composite functions
          • Evaluating composite functions (advanced)
        • Modeling situations by combining & composing functions
          • Modeling with function combination
          • Modeling with composite functions
          • Modeling with composite functions (example 2)
        • Shifting functions
          • Shifting functions
          • Graphing shifted functions
        • Stretching functions
          • Shifting & reflecting functions
          • Compressing functions
          • Reflecting & compressing functions
        • Introduction to inverses of functions
          • Intro to inverse functions
          • Inputs & outputs of inverse functions
          • Graphing the inverse of a linear function
        • Finding inverse functions
          • Finding inverse functions: linear
          • Finding inverse functions: quadratic
          • Finding inverse functions: quadratic (example 2)
          • Finding inverse functions: radical
          • Finding inverse functions: rational
        • Verifying that functions are inverses
          • Verifying inverse functions by composition
          • Verifying inverse functions by composition: not inverse
        • Determining whether a function is invertible
          • Determining if a function is invertible
          • Restricting domains of functions to make them invertible
      • Polynomials
        • Binomial theorem
          • Intro to the Binomial Theorem
          • Pascal's triangle and binomial expansion
          • Expanding binomials
          • Expanding binomials w/o Pascal's triangle
        • Understanding the binomial theorem
          • Binomial expansion & combinatorics
          • Pascal's triangle & combinatorics
          • Binomial expansion & combinatorics (old)
        • Advanced polynomial factorization methods
          • Factoring higher-degree polynomials
          • Factoring sum of cubes
          • Factoring difference of cubes
        • Proving polynomial identities
          • Analyzing polynomial identities
        • Polynomial identities with complex numbers
          • Complex numbers & sum of squares factorization
          • Factoring sum of squares
        • Quadratic equations with complex numbers
          • Solving quadratic equations: complex roots
        • Fundamental theorem of Algebra
          • The Fundamental theorem of Algebra
          • Quadratics & the Fundamental Theorem of Algebra
          • Number of possible real roots of a polynomial
        • Finding zeros of polynomials
          • Finding zeros of polynomials (1 of 2)
          • Finding zeros of polynomials (2 of 2)
          • Finding zeros of polynomials (example 2)
        • Zeros of polynomials and their graphs
          • Zeros of polynomials & their graphs
        • End behavior of polynomial functions
          • Intro to end behavior of polynomials
          • End behavior of functions & their graphs
        • Graphs of polynomials
        • Introduction to symmetry of functions
          • Intro to function symmetry
          • Graphical symmetry of functions
          • Even/odd functions & numbers
        • Symmetry of polynomial functions
          • Even & odd polynomials
      • Radical relationships
        • Square-root equations
          • Intro to square-root equations & extraneous solutions
          • Solving square-root equations (basic)
          • Solving square-root equations: one solution
          • Solving square-root equations: two solutions
          • Solving square-root equations: no solution
        • Extraneous solutions of radical equations
          • Extraneous solutions of radical equations
          • Extraneous solutions of radical equations (example 2)
        • Cube-root equations
          • Solving cube-root equations
        • Domain of radical functions
          • Domain of a radical function
        • Graphs of radical functions
          • Transforming the square-root function
          • Graphs of square-root functions
          • Square-root functions & their graphs
      • Rational relationships
        • Simplifying rational expressions
          • Intro to rational expression simplification
          • Simplifying rational expressions: common monomial factors
          • Simplifying rational expressions: common binomial factors
          • Simplifying rational expressions: opposite common binomial factors
          • Simplifying rational expressions: grouping
          • Simplifying rational expressions: higher degree terms
          • Simplifying rational expressions: two variables
          • Simplifying rational expressions (old video)
        • Multiplying & dividing rational expressions
          • Multiplying & dividing rational expressions: monomials
          • Multiplying rational expressions
          • Dividing rational expressions
          • Multiplying rational expressions: multiple variables
          • Dividing rational expressions: unknown expression
        • Adding & subtracting rational expressions
          • Adding & subtracting rational expressions: like denominators
          • Intro to adding rational expressions with unlike denominators
          • Adding rational expression: unlike denominators
          • Subtracting rational expressions: unlike denominators
          • Least common multiple
          • Least common multiple: repeating factors
          • Subtracting rational expressions: factored denominators
          • Least common multiple of polynomials
          • Subtracting rational expressions
        • Nested fractions
          • Nested fractions
        • Rational equations
          • Equations with one rational expression
          • Equations with one rational expression (advanced)
          • Equations with rational expressions
          • Equations with rational expressions (example 2)
          • Equation with two rational expressions (old example)
          • Equation with two rational expressions (old example 2)
          • Equation with two rational expressions (old example 3)
        • Direct and inverse variation
          • Intro to direct & inverse variation
          • Recognizing direct & inverse variation
          • Recognizing direct & inverse variation: table
          • Inverse variation word problem: string vibration
        • End behavior of rational functions
          • End behavior of rational functions
        • Discontinuities of rational functions
          • Discontinuities of rational functions
          • Analyzing vertical asymptotes of rational functions
        • Graphs of rational functions
          • Graphing rational functions according to asymptotes
          • Graphs of rational functions: y-intercept
          • Graphs of rational functions: horizontal asymptote
          • Graphs of rational functions: vertical asymptotes
          • Graphs of rational functions: zeros
          • Graphs of rational functions (old example)
          • Graphing rational functions 1
          • Graphing rational functions 2
          • Graphing rational functions 3
          • Graphing rational functions 4
        • Modeling with rational functions
          • Analyzing structure word problem: pet store (1 of 2)
          • Analyzing structure word problem: pet store (2 of 2)
          • Rational equations word problem: combined rates
          • Rational equations word problem: combined rates (example 2)
          • Rational equations word problem: eliminating solutions
          • Reasoning about unknown variables
          • Reasoning about unknown variables: divisibility
          • Structure in rational expression
      • Exponentials & logarithms
        • Introduction to logarithms
          • Intro to logarithms
          • Evaluating logarithms (advanced)
          • Relationship between exponentials & logarithms
          • Relationship between exponentials & logarithms: graphs
          • Relationship between exponentials & logarithms: tables
        • The constant e and the natural logarithm
          • 𝑒 and compound interest
          • 𝑒 as a limit
          • Evaluating natural logarithm with calculator
        • Properties of logarithms
          • Intro to logarithm properties (1 of 2)
          • Intro to logarithm properties (2 of 2)
          • Using the logarithmic product rule
          • Using the logarithmic power rule
          • Using the properties of logarithms: multiple steps
          • Proof of the logarithm product rule
          • Proof of the logarithm quotient and power rules
        • Change of base formula for logarithms
          • Evaluating logarithms: change of base rule
          • Using the logarithm change of base rule
          • Proof of the logarithm change of base rule
        • Exponential equations: logarithms
          • Solving exponential equations using logarithms: base-10
          • Solving exponential equations using logarithms: base-2
        • Solving exponential models
          • Exponential model word problem: medication dissolve
          • Exponential model word problem: bacteria growth
        • Graphs of exponential functions
          • Transforming exponential graphs
          • Transforming exponential graphs (example 2)
          • Graphing exponential functions
          • Exponential function graph
          • Graphs of exponential functions (old example)
        • Graphs of logarithmic functions
          • Graphs of logarithms & exponentials
          • Graphing basic logarithmic functions
          • Graphs of logarithmic functions
        • Logarithmic scale (optional)
          • Logarithmic scale (with Vi Hart)
          • Logarithmic scale
          • Richter scale
          • Benford's law (with Vi Hart, 1 of 2)
          • Benford's law (with Vi Hart, 2 of 2)
      • Trigonometry
        • Law of sines
          • Solving for a side with the law of sines
          • Solving for an angle with the law of sines
          • Proof of the law of sines
        • Law of cosines
          • Solving for a side with the law of cosines
          • Solving for an angle with the law of cosines
          • Proof of the law of cosines
        • Solving general triangles
          • Trig word problem: stars
        • Introduction to radians
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
          • Radian angles & quadrants
        • Unit circle definition of sine, cosine, & tangent
          • Unit circle
          • The trig functions & right triangle trig ratios
        • Graphs of sine, cosine, & tangent
          • Graph of y=sin(x)
          • Graph of y=tan(x)
          • Intersection points of y=sin(x) and y=cos(x)
        • Basic trigonometric identities
          • Sine & cosine identities: symmetry
          • Tangent identities: symmetry
          • Sine & cosine identities: periodicity
          • Tangent identities: periodicity
        • Trigonometric values of special angles
          • Trig values of special angles
        • Pythagorean identity
          • Proof of the Pythagorean trig identity
          • Using the Pythagorean trig identity
        • Introduction to amplitude, midline, & extrema of sinusoidal functions
          • Features of sinusoidal functions
        • Finding amplitude & midline from formula
          • Amplitude & period of sinusoidal functions from equation
        • Period of sinusoidal functions
          • Amplitude & period of sinusoidal functions from equation
        • Graphing sinusoidal functions
          • Transforming sinusoidal graphs: vertical stretch & horizontal reflection
          • Transforming sinusoidal graphs: vertical & horizontal stretches
        • Constructing sinusoidal functions
          • Sinusoidal function from graph
          • Trig word problem: modeling daily temperature
          • Trig word problem: modeling annual temperature
      • Advanced equations & functions
        • Solving equations by graphing
          • Interpreting equations graphically
          • Interpreting equations graphically (example 2)
          • Solving equations graphically (1 of 2)
          • Solving equations graphically (2 of 2)
        • Systems of quadratic equations
          • Quadratic systems: graphical solution
          • Quadratic systems: algebraic solution
          • Quadratic systems: no solution
          • Quadratic systems: both variables are squared
          • Quadratic systems: a line and a circle
        • Interpreting the symmetry of algebraic models
          • Symmetry of algebraic models
        • Interpreting the end behavior of algebraic models
          • End behavior of algebraic models
        • Interpreting the periodicity of algebraic models
          • Periodicity of algebraic models
        • Comparing features of functions
          • Comparing functions: x-intercepts
          • Comparing functions: shared features
          • Comparing maximum points of quadratic functions
      • Series
        • Basic sigma notation
          • Summation notation
        • Finite arithmetic series
          • Arithmetic series intro
          • Arithmetic series formula
          • Worked example: arithmetic series (sigma notation)
          • Worked example: arithmetic series (sum expression)
          • Worked example: arithmetic series (recursive formula)
          • Proof of finite arithmetic series formula
        • Finite geometric series
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
          • Finite geometric series formula justification
        • Finite geometric series applications
          • Finite geometric series word problem: social media
          • Finite geometric series word problem: mortgage
      • Study Design
        • Samples and surveys
          • Reasonable samples
        • Observational studies and experiments
          • Types of statistical studies
          • Correlation and causality
          • Appropriate statistical study example
      • Binomial probability
        • Binomial probability
          • Binomial probability example
          • Generalizing k scores in n attempts
          • Free throw binomial probability distribution
          • Graphing basketball binomial distribution
      • Normal distributions
        • Normal distributions
          • Normal distribution problems: Empirical rule (from ck12.org)
          • Qualitative sense of normal distributions (from ck12.org)
    • Arithmetic (all content)
      • Addition and subtraction
        • Basic addition and subtraction
          • Basic addition
          • Basic subtraction
          • Add and subtract: pieces of fruit
          • Relating addition and subtraction
        • Making 10
          • Getting to 10 by filling boxes
          • Adding to 10
        • Addition and subtraction word problems within 10
          • Addition word problems within 10
          • Subtraction word problems within 10
        • Addition and subtraction within 20
          • Adding 8 + 7
          • Subtracting different ways
          • Subtracting 14 - 6
        • Word problems within 20
          • Addition and subtraction word problems: superheroes
          • Addition and subtraction word problems: gorillas
        • Word problems with "more" and "fewer"
          • Comparison word problems: marbles
          • Comparison word problems: roly-polies
        • Intro to addition with 2-digit numbers
          • Adding 2-digit numbers without regrouping 1
          • Example: Adding 2-digit numbers (no carrying)
          • Breaking apart 2-digit addition problems
          • Regrouping to add 1-digit number
        • Intro to subtraction with 2-digit numbers
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting a 1-digit number with regrouping
        • Strategies for adding and subtracting within 100
          • Adding 53+17 by making a group of 10
          • Adding by making a group of 10
          • Strategies for adding 2-digit numbers
          • Addition and subtraction with number lines
        • Addition within 100
          • Adding 2-digit numbers without regrouping 2
          • Adding with regrouping
        • Subtraction within 100
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting with regrouping (borrowing)
        • Word problems within 100
          • Adding and subtracting on number line word problems
          • Subtraction word problem: tennis balls
          • Addition word problem: horses
          • Subtraction word problem: snow
          • Subtraction word problem: crayons
        • Adding 1s, 10s, and 100s
          • Adding 10 or 100
          • Adding 1s, 10s, and 100s
          • Adding 3-digit numbers (no regrouping)
        • Subtracting 1s, 10s, and 100s
          • Subtracting 1, 10, or 100
          • Subtracting 1s, 10s, and 100s
          • Subtracting 3-digit numbers (no regrouping)
        • Strategies for adding 2- and 3-digit numbers
          • Breaking apart 3-digit addition problems
          • Solving 3-digit addition in your head
          • Addition using groups of 10 and 100
          • Adding and subtracting on number line 2
        • Addition with regrouping within 1000
          • Adding with regrouping
          • Adding 3-digit numbers with regrouping
        • Subtraction with regrouping within 1000
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Worked example: Subtracting 3-digit numbers (regrouping twice)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
          • Subtracting in your head (no regrouping)
        • Addition and subtraction missing value problems
          • Missing number for addition and subtraction within 1000
        • Addition and subtraction greater than 1000
          • Multi-digit addition with regrouping
          • Multi-digit subtraction with regrouping: 6798-3359
          • Multi-digit subtraction with regrouping: 7329-6278
          • Multi-digit subtraction with regrouping twice
          • Alternate mental subtraction method
      • Multiplication and division
        • Multiplication intro
          • Intro to multiplication
          • Basic multiplication
          • Multiplication as groups of objects
          • More ways to multiply
          • Ways to represent multiplication
        • Multiplication facts
          • Multiplication tables for 2-9
          • Multiplication tables for 10, 11, 12
        • Properties of multiplication
          • Properties of multiplication
          • Properties and patterns for multiplication
        • Multiplying by tens
          • Multiplying by multiples of 10
          • Multiplying by tens word problem
        • Multiplication by 10s, 100s, and 1000s
          • Multiplying 10s
          • Multiplying 1-digit numbers by multiples of 10, 100, and 1000
          • Multiplying 1-digit numbers by 10, 100, and 1000
        • Multiplication: place value and area models
          • More ways to multiply
          • Multiplying with distributive property
          • Multiplying with area model: 6 x 7981
          • Multiplying with area model: 78 x 65
          • Lattice multiplication
          • Why lattice multiplication works
        • Multi-digit multiplication
          • Multiplying 2-digit by 1-digit
          • Multiplying 3-digit by 1-digit
          • Multiplying 3-digit by 1-digit (regrouping)
          • Multiplying 4-digit by 1-digit (regrouping)
          • Multiplying 2-digit numbers
          • Multiplying 2-digit by 2-digit: 36x23
          • Multiplying 2-digit by 2-digit: 23x44
          • Multiplying multi-digit numbers
        • Relating multiplication and division
          • Relating division to multiplication
          • Multiplication word problem: parking lot
          • Division word problem: school building
        • Division intro
          • The idea of division
          • Intro to division
          • Ways to represent division
        • Division facts
        • Division problems that work out nicely
          • Quotients that are multiples of 10
          • Canceling zeros when dividing
        • Division: place value and area
          • Division using place value
          • Division with area models
        • Multi-digit division (no remainders)
          • Intro to long division (no remainders)
          • Long division: 280÷5
        • Remainders
          • Intro to remainders
          • Worked example: Long division with remainders: 3771÷8
          • Worked example: Long division with remainders: 2292÷4
        • Multi-digit division (remainders)
          • Intro to long division (remainders)
          • Dividing by 2-digits: 6250÷25
          • Dividing by 2-digits: 9815÷65
          • Dividing by 2-digits: 7182÷42
        • Multiplication and division word problems
          • Multiplication word problem: carrots
          • Division word problem: row boat
          • Multiplication word problem: pizza
          • Division word problem: field goals
          • Multiplication, division word problem: pedaling
      • Negative numbers
        • Intro to negative numbers
          • Intro to negative numbers
        • Number opposites
          • Number opposites
          • Negative symbol as opposite
          • Number opposites challenge problems
        • Absolute value
          • Absolute value examples
          • Identify and order absolute values
          • Comparing absolute values
          • Placing absolute values on the number line
          • Absolute value as distance between numbers
          • Absolute value word problems
        • Intro to adding negative numbers
          • Adding negative numbers example
          • Adding numbers with different signs
        • Intro to subtracting negative numbers
          • Adding & subtracting negative numbers
          • Subtracting a negative = adding a positive
        • Adding & subtracting negative numbers
          • Adding numbers with different signs
          • Addition & subtraction: find the missing value (part 1 of 2)
          • Addition & subtraction: find the missing value (part 2 of 2)
        • Adding & subtracting with negatives on the number line
          • Adding negative numbers on the number line
          • Number equations & number lines
          • Interpreting numeric expressions example
        • Multiplying & dividing negative numbers
          • Multiplying positive & negative numbers
          • Multiplying numbers with different signs
          • Why a negative times a negative makes sense
          • Dividing positive and negative numbers
          • Multiplying and dividing negative numbers
          • Why a negative times a negative is a positive
      • Fractions
        • Fractions intro
          • Intro to fractions
          • More about fractions
          • Cutting shapes into equal parts
          • Identifying unit fractions word problem
        • What fractions mean
          • More than one equal section
          • Recognizing fractions greater than 1
          • Identifying numerators and denominators
        • Fractions on the number line
          • Fractions on a number line
          • Fractions on number line widget
        • Fractions and whole numbers
          • Representing 1 as a fraction
          • Whole numbers as fractions
        • Comparing fractions
          • Comparing fractions with > and < symbols
          • Comparing fractions with like numerators and denominators
          • Comparing fractions
          • Comparing fractions 2 (unlike denominators)
          • Comparing and ordering fractions
          • Ordering fractions
        • Equivalent fractions 1
          • Equivalent fractions with visuals
          • Equivalent fraction models
        • Equivalent fractions on the number line
          • Equivalent fraction visually
          • Creating equivalent fractions
        • Equivalent fractions 2
          • Intro to equivalent fractions
          • Visualizing equivalent fractions
          • More on equivalent fractions
          • Equivalent fractions
          • Equivalent fractions and different wholes
        • Common denominators
          • Finding common denominators
          • Common denominators: 3/5 and 7/2
          • Common denominators: 1/4 and 5/6
        • Decomposing fractions
          • Decomposing a fraction visually
          • Decomposing a mixed number
        • Adding and subtracting fractions with like denominators
          • Adding fractions with like denominators
          • Subtracting fractions with like denominators
        • Mixed numbers
          • Rewriting mixed numbers as improper fractions
          • Rewriting improper fractions as mixed numbers
          • Comparing improper fractions and mixed numbers
          • Mixed number or improper fraction on a number line
        • Adding and subtracting mixed numbers
          • Intro to adding mixed numbers
          • Intro to subtracting mixed numbers
        • Visually adding and subtracting fractions with unlike denominators
          • Visually adding fractions: 5/6+1/4
          • Visually subtracting fractions: 3/4-5/8
        • Adding and subtracting fractions with unlike denominators
          • Adding fractions with unlike denominators
          • Subtracting fractions with unlike denominators
          • Adding and subtracting 3 fractions
          • Solving for the missing fraction
        • Adding and subtracting mixed number with unlike denominators
          • Adding mixed numbers: 19 3/18 + 18 2/3
          • Subtracting mixed numbers: 7 6/9 - 3 2/5
          • Adding mixed numbers with regrouping
          • Subtracting mixed numbers with regrouping (unlike denominators)
        • Adding and subtracting fractions with unlike denominators word problems
          • Adding fractions word problem: paint
          • Subtracting fractions word problem: tomatoes
        • Multiplying unit fractions and whole numbers
          • Multiplying unit fractions and whole numbers
        • Multiplying whole numbers and fractions
          • Multiplying fractions and whole numbers visually
          • Equivalent fraction and whole number multiplication problems
        • Multiplication as scaling
          • Multiplication as scaling with fractions
        • Multiplying fractions
          • Intro to multiplying 2 fractions
          • Multiplying 2 fractions: fraction model
          • Multiplying 2 fractions: number line
          • Multiplying 2 fractions: 5/6 x 2/3
          • Finding area with fractional sides 1
          • Finding area with fractional sides 2
        • Multiplying mixed numbers
          • Multiplying mixed numbers
        • Multiplying fractions word problems
          • Multiplying fractions word problem: muffins
          • Multiplying fractions word problem: laundry
          • Multiplying fractions word problem: bike
          • Multiplying fractions word problem: pumpkin pie
        • Fractions as division
          • Understanding fractions as division
          • Creating a fraction through division
          • Creating mixed numbers with fraction division
        • Dividing unit fractions by whole numbers
        • Dividing whole numbers by unit fractions
        • Dividing fractions by fractions
          • Understanding division of fractions
          • Dividing fractions: 2/5 ÷ 7/3
          • Dividing fractions: 3/5 ÷ 1/2
        • Dividing fractions word problems
          • Dividing whole numbers & fractions: t-shirts
      • Decimals
        • Intro to decimals
          • Intro to decimals
          • Decimals as words
          • Writing decimal numbers shown in grids
        • Decimal place value intro
          • Decimal place value
        • Decimals on the number line
          • Comparing decimal numbers on a number line
          • Plotting decimal numbers on a number line
        • Rewriting decimals as fractions
          • Writing a number as a fraction and decimal
          • Rewriting decimals as fractions: 0.15
          • Rewriting decimals as fractions: 0.8
          • Rewriting decimals as fractions: 0.36
          • Rewriting tricky fractions to decimals
        • Adding decimals
        • Adding decimals: standard algorithm
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
        • Subtracting decimals
          • Subtracting decimals: 9.57-8.09
          • Subtracting decimals: 10.1-3.93
        • Subtracting decimals: standard algorithm
          • Subtracting decimals: 9.005 - 3.6
          • Subtracting decimals: 39.1 - 0.794
        • Adding and subtracting decimals word problems
          • Adding decimals word problem
          • Adding & subtracting decimals word problem
        • Multiplying decimals
          • Intro to multiplying decimals
          • Multiplying decimals: place value
        • Multiplying decimals: standard algorithm
          • Multiplying challenging decimals
        • Dividing decimals
          • Dividing whole numbers to get a decimal
        • Dividing decimals: standard algorithm
          • Dividing decimals with hundredths
          • Dividing decimals completely
          • Long division with decimals
          • Dividing by a multi-digit decimal
        • Significant figures
          • Intro to significant figures
          • Rules of significant figures
          • Addition and subtraction with significant figures
          • Multiplying and dividing with significant figures
    • Algebra (all content)
      • Introduction to algebra
        • Overview and history of algebra
          • Origins of algebra
          • Abstract-ness
          • The beauty of algebra
          • Intro to the coordinate plane
          • Why all the letters in algebra?
        • Introduction to variables
          • What is a variable?
          • Why aren't we using the multiplication sign?
          • Evaluating an expression with one variable
        • Substitution and evaluating expressions
          • Evaluating expressions with two variables
          • Evaluating expressions with two variables: fractions & decimals
        • Evaluating expressions word problems
          • Evaluating expressions with variables: temperature
          • Evaluating expressions with variables: cubes
          • Evaluating expressions with variables: exponents
        • Writing algebraic expressions introduction
          • Writing basic expressions with variables
          • Writing expressions with variables
          • Writing expressions with variables & parentheses
        • Introduction to equivalent algebraic expressions
          • Equivalent expressions
        • Dependent & independent variables
          • Dependent & independent variables
          • Dependent & independent variables: graphing
          • Dependent & independent variables: equation
        • Combining like terms
          • Intro to combining like terms
          • Simplifying expressions
          • Combining like terms challenge problem
          • Simplifying expressions with rational numbers
        • Interpreting linear expressions
          • Interpreting linear expressions: diamonds
          • Interpreting linear expressions: flowers
        • Irrational numbers
          • Intro to rational & irrational numbers
          • Classifying numbers: rational & irrational
          • Square roots and real numbers (old)
        • Sums and products of rational and irrational numbers
          • Proof: sum & product of two rationals is rational
          • Proof: product of rational & irrational is irrational
          • Proof: sum of rational & irrational is irrational
          • Sums and products of irrational numbers
          • Worked example: rational vs. irrational expressions
          • Worked example: rational vs. irrational expressions (unknowns)
        • Proofs concerning irrational numbers
          • Proof: √2 is irrational
          • Proof: square roots of prime numbers are irrational
          • Proof: there's an irrational number between any two rational numbers
        • Division by zero
          • Why dividing by zero is undefined
          • The problem with dividing zero by zero
          • Undefined & indeterminate expressions
        • Binary and hexadecimal number systems
          • Introduction to number systems and binary
          • Hexadecimal number system
          • Converting from decimal to binary
          • Converting larger number from decimal to binary
          • Converting from decimal to hexadecimal representation
          • Adding in binary
          • Multiplying in binary
          • Converting directly from binary to hexadecimal
          • How to count to 1000 on two hands
      • Solving basic equations & inequalities (one variable, linear)
        • Algebraic equations basics
          • Variables, expressions, & equations
          • Testing solutions to equations
        • Why we do the same thing to both sides of an equation
          • Same thing to both sides of equations
          • Representing a relationship with an equation
          • Dividing both sides of an equation
          • One-step equations intuition
        • One-step addition & subtraction equations
          • One-step addition & subtraction equations
          • One-step subtraction equations
          • One-step addition & subtraction equations: fractions & decimals
        • One-step multiplication & division equations
          • One-step division equations
          • One-step multiplication equations
          • One-step multiplication & division equations: fractions & decimals
        • Two-steps equations intro
          • Intro to two-step equations
          • Two-step equations intuition
          • Worked example: two-step equations
          • Describing steps when solving equations
        • Two-step equation word problems
          • Two-step equation word problem: computers
          • Two-step equation word problem: garden
          • Two-step equation word problem: oranges
        • Linear equations with variables on both sides
          • Why we do the same thing to both sides: Variable on both sides
          • Intro to equations with variables on both sides
          • Equations with variables on both sides: 20-7x=6x-6
          • Equation with variables on both sides: fractions
          • Equation with the variable in the denominator
          • Figuring out missing algebraic step
        • Linear equations with parentheses
          • Equations with parentheses
        • Analyzing the number of solutions to linear equations
          • Number of solutions to equations
          • Worked example: number of solutions to equations
          • Creating an equation with no solutions
          • Creating an equation with infinitely many solutions
        • Linear equations word problems
          • Sums of consecutive integers
          • Sum of integers challenge
          • Linear equation word problem: saline
          • Linear equation word problem: sugary drinks
        • Linear equations with unknown coefficients
          • Linear equations with unknown coefficients
        • Old school equations with Sal
          • Linear equations 1
          • Linear equations 2
          • Linear equations 3
          • Linear equations 4
          • Averages
          • Taking percentages
          • Growing by a percentage
          • More percent problems
          • Solving an equation for a variable
        • Introduction to inequalities with variables
          • Testing solutions to inequalities
          • Plotting inequalities
          • Plotting an inequality example
          • Inequalities word problems
        • One-step inequalities
          • One-step inequalities examples
          • One-step inequalities: -5c ≤ 15
          • One-step inequality involving addition
          • One-step inequality word problem
          • Inequalities using addition and subtraction
          • Solving and graphing linear inequalities
        • Two-step inequalities
          • Two-step inequalities
          • Two-step inequality word problem: apples
          • Two-step inequality word problem: R&B
        • Multi-step inequalities
          • Inequalities with variables on both sides
          • Inequalities with variables on both sides (with parentheses)
          • Multi-step inequalities
        • Compound inequalities
          • Compound inequalities: OR
          • Compound inequalities: AND
          • A compound inequality with no solution
          • Double inequalities
          • Compound inequalities examples
      • Linear equations, functions, & graphs
        • Two-variable linear equations intro
          • Two-variable linear equations intro
          • Solutions to 2-variable equations
          • Worked example: solutions to 2-variable equations
          • Completing solutions to 2-variable equations
          • Solutions to 2-variable equations: substitution (old)
          • Solutions to 2-variable equations: graphical (old)
        • x-intercepts and y-intercepts
          • Intro to intercepts
          • x-intercept of a line
          • Intercepts from an equation
          • Intercepts from a table
          • Graphing using intercepts (old)
        • Slope
          • Intro to slope
          • Positive & negative slope
          • Worked example: slope from graph
          • Worked example: slope from two points
          • Slope (more examples)
        • Horizontal & vertical lines
          • Slope of a horizontal line
          • Horizontal & vertical lines
        • Intro to slope-intercept form
          • Intro to slope-intercept form
          • Worked examples: slope-intercept intro
        • Graphing slope-intercept equations
          • Graph from slope-intercept equation
        • Writing slope-intercept equations
          • Slope-intercept equation from graph
          • Slope-intercept equation from slope & point
          • Slope-intercept equation from two points
          • Slope-intercept form problems
          • Slope-intercept equation from slope & point (old)
          • Slope-intercept equation from slope & point: fractions (old)
          • Finding y intercept given slope & point (old)
        • Point-slope form
          • Intro to point-slope form
          • Point-slope & slope-intercept equations
        • Standard form
          • Intro to linear equation standard form
          • Graphing a linear equation: 5x+2y=20
          • Converting from slope-intercept to standard form
        • Summary: Forms of two-variable linear equations
          • Slope from equation
          • Writing linear equations in all forms
        • Interpreting linear functions and equations
          • Linear equations word problems: earnings
          • Linear equations word problems: volcano
          • Linear graphs word problems
          • Linear graphs word problem: cats
          • Linear equations word problem: marbles
          • Linear equations word problem: file transfer
        • Comparing linear functions
          • Comparing linear functions: equation vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions: table vs. graph
          • Comparing linear functions word problem: climb
          • Comparing linear functions word problem: walk
          • Comparing linear functions word problem: work
        • Constructing linear models for real-world relationships
          • Linear functions word problem: fuel
          • Linear functions word problem: pool
          • Linear functions word problem: iceberg
          • Linear functions word problem: paint
        • Linear models word problems
          • Linear models word problem: book
          • Linear models word problem: marbles
      • Sequences
        • Introduction to arithmetic sequences
          • Sequences intro
          • Intro to arithmetic sequences
          • Extending arithmetic sequences
          • Using arithmetic sequences formulas
          • Worked example: using recursive formula for arithmetic sequence
        • Constructing arithmetic sequences
          • Recursive formulas for arithmetic sequences
          • Explicit formulas for arithmetic sequences
          • Arithmetic sequence problem
          • Converting recursive & explicit forms of arithmetic sequences
        • Introduction to geometric sequences
          • Intro to geometric sequences
          • Extending geometric sequences
          • Using explicit formulas of geometric sequences
          • Using recursive formulas of geometric sequences
        • Constructing geometric sequences
          • Explicit & recursive formulas for geometric sequences
          • Converting recursive & explicit forms of geometric sequences
        • Modeling with sequences
          • Sequences word problems
        • General sequences
          • Evaluating sequences in recursive form
      • System of equations
        • Systems of equations overview
          • Systems of equations: trolls, tolls (1 of 2)
          • Systems of equations: trolls, tolls (2 of 2)
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: potato chips
          • Systems of equations with substitution: potato chips
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
        • Introduction to systems of equations
          • Systems of equations: trolls, tolls (1 of 2)
          • Testing a solution to a system of equations
          • Systems of equations: trolls, tolls (2 of 2)
          • Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
          • Systems of equations with graphing: exact & approximate solutions
        • Equivalent systems of equations and the elimination method
          • Systems of equations with elimination: King's cupcakes
          • Systems of equations with elimination: x-4y=-18 & -x+3y=11
          • Systems of equations with elimination: potato chips
          • Systems of equations with elimination (and manipulation)
          • Why can we subtract one equation from the other in a system of equations?
          • Worked example: equivalent systems of equations
          • Worked example: non-equivalent systems of equations
          • Solving systems of equations by elimination (old)
        • Solving systems of equations with substitution
          • Systems of equations with substitution: potato chips
          • Systems of equations with substitution: -3x-4y=-2 & y=2x-5
          • Solving linear systems by substitution (old)
        • Manipulating expressions with unknown variables
          • Manipulating expressions using structure
          • Manipulating expressions using structure (example 2)
        • Number of solutions to systems of equations
          • Systems of equations number of solutions: fruit prices (1 of 2)
          • Systems of equations number of solutions: fruit prices (2 of 2)
          • Solutions to systems of equations: consistent vs. inconsistent
          • Solutions to systems of equations: dependent vs. independent
          • Number of solutions to a system of equations
          • Number of solutions to a system of equations graphically
          • Number of solutions to a system of equations algebraically
          • How many solutions does a system of linear equations have if there are at least two?
        • Solving any system of linear equations
        • Systems of equations word problems
          • Age word problem: Imran
          • Age word problem: Ben & William
          • Age word problem: Arman & Diya
          • System of equations word problem: walk & ride
          • System of equations word problem: no solution
          • System of equations word problem: infinite solutions
          • Systems of equations with elimination: TV & DVD
          • Systems of equations with elimination: apples and oranges
          • Systems of equations with substitution: coins
          • Systems of equations with elimination: coffee and croissants
        • Solving equations by graphing (Algebra 2 level)
          • Interpreting equations graphically
          • Interpreting equations graphically (example 2)
          • Solving equations graphically (1 of 2)
          • Solving equations graphically (2 of 2)
        • Systems with three variables
          • Intro to linear systems with 3 variables
          • Solving linear systems with 3 variables
          • Solving linear systems with 3 variables: no solution
          • 3-variable linear system word problem
      • Two-variable inequalities
        • Checking solutions of two-variable inequalities
          • Testing solutions to inequalities
          • Testing solutions to systems of inequalities
        • Constraining solutions of two-variable inequalities
          • Constraining solutions of two-variable inequalities
          • Constraining solutions of systems of inequalities
        • Graphing two-variable inequalities
          • Intro to graphing two-variable inequalities
          • Graphing two-variable inequalities
          • Two-variable inequalities from their graphs
          • Intro to graphing systems of inequalities
          • Graphing systems of inequalities
          • Graphing two-variable inequalities (old)
        • Modeling with linear inequalities
          • Writing two-variable inequalities word problem
          • Interpreting two-variable inequalities word problem
          • Solving two-variable inequalities word problem
          • Graphs of two-variable inequalities word problem
          • Modeling with systems of inequalities
          • Writing systems of inequalities word problem
          • Solving systems of inequalities word problem
          • Graphs of systems of inequalities word problem
          • Analyzing structure with linear inequalities: fruits
          • Analyzing structure with linear inequalities: balls
      • Functions
        • Evaluating functions
          • What is a function?
          • Worked example: Evaluating functions from equation
          • Worked example: evaluating functions from graph
          • Worked example: evaluating expressions with function notation
        • Inputs and outputs of a function
          • Worked example: matching an input to a function's output (equation)
          • Worked example: matching an input to a function's output (graph)
          • Worked example: two inputs with the same output (graph)
        • Functions and equations
          • Equations vs. functions
          • Obtaining a function from an equation
        • Interpreting function notation
          • Function notation word problem: bank
          • Function notation word problem: beach
        • Introduction to the domain and range of a function
          • Intervals and interval notation
          • What is the domain of a function?
          • What is the range of a function?
          • Worked example: domain and range from graph
        • Determining the domain of a function
          • Domain of a radical function
          • Worked example: domain of algebraic functions
          • Worked example: determining domain word problem (real numbers)
          • Worked example: determining domain word problem (positive integers)
          • Worked example: determining domain word problem (all integers)
        • Recognizing functions
          • Recognizing functions from graph
          • Does a vertical line represent a function?
          • Recognizing functions from table
          • Recognizing functions from verbal description
          • Recognizing functions from verbal description word problem
        • Piecewise functions
          • Introduction to piecewise functions
          • Worked example: evaluating piecewise functions
          • Worked example: graphing piecewise functions
          • Worked example: domain & range of step function
          • Worked example: domain & range of piecewise linear functions
        • Maximum and minimum points
          • Introduction to minimum and maximum points
          • Worked example: relative maxima and minima
        • Intervals where a function is positive, negative, increasing, or decreasing
          • Increasing, decreasing, positive or negative intervals
          • Worked example: positive & negative intervals
        • Interpreting features of graphs
          • Graph interpretation word problem: temperature
          • Graph interpretation word problem: basketball
        • Average rate of change
          • Introduction to average rate of change
          • Worked example: average rate of change from graph
          • Worked example: average rate of change from table
          • Worked example: average rate of change from equation
        • Average rate of change word problems
          • Average rate of change word problem: table
          • Average rate of change word problem: graph
          • Average rate of change word problem: equation
        • Combining functions
          • Adding functions
          • Subtracting functions
          • Multiplying functions
          • Dividing functions
        • Composing functions (Algebra 2 level)
          • Intro to composing functions
          • Evaluating composite functions
          • Evaluating composite functions: using tables
          • Evaluating composite functions: using graphs
          • Finding composite functions
          • Evaluating composite functions (advanced)
        • Shifting functions
          • Shifting functions
          • Graphing shifted functions
        • Stretching functions
          • Shifting & reflecting functions
          • Compressing functions
          • Reflecting & compressing functions
        • Modeling situations by combining and composing functions (Algebra 2 level)
          • Modeling with function combination
          • Modeling with composite functions
          • Modeling with composite functions (example 2)
        • Introduction to inverses of functions (Algebra 2 level)
          • Intro to inverse functions
          • Inputs & outputs of inverse functions
          • Graphing the inverse of a linear function
        • Finding inverse functions (Algebra 2 level)
          • Finding inverse functions: linear
          • Finding inverse functions: quadratic
          • Finding inverse functions: quadratic (example 2)
          • Finding inverse functions: radical
          • Finding inverse functions: rational
        • Verifying that functions are inverses (Algebra 2 level)
          • Verifying inverse functions by composition
          • Verifying inverse functions by composition: not inverse
        • Determining whether a function is invertible (Algebra 2 level)
          • Determining if a function is invertible
          • Restricting domains of functions to make them invertible
        • Determining the domain of advanced functions (Algebra 2 level)
          • Domain of advanced functions
        • Determining the range of a function (Algebra 2 level)
          • Domain and range of quadratic functions
        • Graphing nonlinear piecewise functions (Algebra 2 level)
          • Graphs of nonlinear piecewise functions
        • Interpreting the symmetry of algebraic models (Algebra 2 level)
          • Symmetry of algebraic models
        • Interpreting the end behavior of algebraic models (Algebra 2 level)
          • End behavior of algebraic models
        • Interpreting the periodicity of algebraic models (Algebra 2 level)
          • Periodicity of algebraic models
        • Comparing features of functions (Algebra 2 level)
          • Comparing functions: x-intercepts
          • Comparing functions: shared features
          • Comparing maximum points of quadratic functions
        • Two-variable functions
          • New operator definitions 1
          • New operator definitions 2
      • Absolute value equations, functions, & inequalities
        • Graphs of absolute value functions
          • Shifting absolute value graphs
          • Scaling & reflecting absolute value functions: equation
          • Scaling & reflecting absolute value functions: graph
          • Graphing absolute value functions
        • Solving absolute value equations
          • Intro to absolute value equations and graphs
          • Worked example: absolute value equation with two solutions
          • Worked example: absolute value equations with one solution
          • Worked example: absolute value equations with no solution
        • Solving absolute value inequalities
          • Intro to absolute value inequalities
          • Solving absolute value inequalities 1
          • Solving absolute value inequalities 2
          • Solving absolute value inequalities: fractions
          • Solving absolute value inequalities: no solution
          • Absolute value inequalities word problem
      • Quadratic equations & functions
        • Parabolas intro
          • Parabolas intro
        • Quadratic factored form
          • Zero product property
          • Graphing quadratics in factored form
          • Quadratic word problems (factored form)
        • Solving quadratics by taking square roots
          • Solving quadratics by taking square roots
          • Solving quadratics by taking square roots examples
          • Solving quadratics by taking square roots: strategy
          • Quadratics by taking square roots: with steps
          • Solving quadratics by taking square roots: challenge
        • Quadratic vertex form
          • Vertex form introduction
          • Graphing quadratics: vertex form
          • Quadratic word problems (vertex form)
        • Solving quadratics by factoring
          • Solving quadratics by factoring
          • Solving quadratics by factoring: leading coefficient ≠ 1
          • Solving quadratics using structure
          • Quadratic equations word problem: triangle dimensions
          • Quadratic equations word problem: box dimensions
          • Solving quadratic equations by factoring (old)
        • Completing the square
          • Completing the square
          • Worked example: Completing the square (intro)
          • Worked example: Rewriting expressions by completing the square
          • Worked example: Solving equations by completing the square
          • Worked example: completing the square (leading coefficient ≠ 1)
          • Solving quadratics by completing the square: no solution
        • The quadratic formula
          • The quadratic formula
          • Using the quadratic formula
          • Worked example: quadratic formula
          • Worked example: quadratic formula (example 2)
          • Worked example: quadratic formula (negative coefficients)
          • Using the quadratic formula: number of solutions
          • Proof of the quadratic formula
        • Quadratic standard form
          • Finding the vertex of a parabola in standard form
          • Graphing quadratics: standard form
          • Quadratic word problem: ball
        • Features & forms of quadratic functions
          • Forms & features of quadratic functions
          • Worked examples: Forms & features of quadratic functions
          • Vertex & axis of symmetry of a parabola
          • Finding features of quadratic functions
          • Comparing features of quadratic functions
          • Comparing maximum points of quadratic functions
        • Transforming parabolas
          • Intro to parabola transformations
          • Shifting parabolas
          • Scaling & reflecting parabolas
        • Systems of quadratic equations
          • Quadratic systems: graphical solution
          • Quadratic systems: algebraic solution
          • Quadratic systems: no solution
          • Quadratic systems: both variables are squared
          • Quadratic systems: a line and a circle
        • Quadratic inequalities
          • Quadratic inequalities
          • Quadratic inequalities (example 2)
          • Quadratic inequalities: graphical approach
      • Polynomial expressions, equations, & functions
        • Intro to polynomials
          • Polynomials intro
          • The parts of polynomial expressions
          • Evaluating polynomials
          • Simplifying polynomials
        • Adding & subtracting polynomials
          • Adding polynomials
          • Subtracting polynomials
          • Polynomial subtraction & set closure
          • Adding & subtracting multiple polynomials
          • Adding polynomials (old)
        • Adding & subtracting polynomials: two variables
          • Adding polynomials: two variables
          • Subtracting polynomials: two variables
          • Subtracting polynomials: two variables challenge
          • Finding an error in polynomial subtraction
          • Polynomials review
        • Multiplying monomials
          • Multiplying monomials
          • Multiplying monomials to find area: two variables
          • Multiplying monomials to find area
          • Multiplying monomials challenge
        • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials: area model
          • Multiplying monomials by polynomials
          • Multiplying monomials by polynomials challenge
        • Multiplying binomials
          • Multiplying binomials: area model
          • Multiplying binomials intro
          • Multiplying binomials
        • Special products of binomials
          • Special products of the form (x+a)(x-a)
          • Squaring binomials of the form (x+a)²
          • Special products of the form (ax+b)(ax-b)
          • Squaring binomials of the form (ax+b)²
          • Special products of binomials: two variables
          • More examples of special products
          • Squaring a binomial (old)
        • Multiplying binomials by polynomials
          • Multiplying binomials by polynomials
          • Multiplying binomials by polynomials: area model
          • Multiplying binomials by polynomials challenge
          • Multiplying binomials by polynomials (old)
          • Multiplying binomials with radicals (old)
        • Polynomials word problems
          • Polynomial multiplication word problem
          • Polynomial word problem: rectangle and circle area
          • Polynomial word problem: total value of bills
          • Polynomial word problem: area of a window
        • Introduction to factorization
          • Intro to factors & divisibility
        • Factoring monomials
          • Monomial factorization
          • Worked example: finding the missing monomial factor
          • Worked example: finding missing monomial side in area model
          • Greatest common factor of monomials
        • Factoring polynomials by taking common factors
          • Factoring with the distributive property
          • Factoring binomials: common factor
          • Factoring polynomials: common factor
          • Factoring polynomials: common factor area model
          • Factoring polynomials: common binomial factor
          • Factoring polynomials: common factor (old)
          • Factoring polynomials: common factor (old)
        • Evaluating expressions with unknown variables
          • Worked example: evaluating expressions using structure
          • Worked example: evaluating expressions using structure (more examples)
        • Factoring quadratics intro
          • Factoring quadratics as (x+a)(x+b)
          • Factoring quadratics as (x+a)(x+b) (example 2)
          • More examples of factoring quadratics as (x+a)(x+b)
        • Factoring quadratics by grouping
          • Intro to grouping
          • Factoring quadratics by grouping
        • Factoring polynomials with quadratic forms
          • Factoring quadratics: common factor + grouping
          • Factoring quadratics: negative common factor + grouping
          • Factoring two-variable quadratics
          • Factoring two-variable quadratics: rearranging
          • Factoring two-variable quadratics: grouping
          • Factoring quadratics with common factor (old)
        • Factoring quadratics: Difference of squares
          • Difference of squares intro
          • Factoring difference of squares: leading coefficient ≠ 1
          • Factoring difference of squares: analyzing factorization
          • Factoring difference of squares: missing values
          • Factoring difference of squares: shared factors
        • Factoring quadratics: Perfect squares
          • Perfect square factorization intro
          • Factoring perfect squares
          • Identifying perfect square form
          • Factoring perfect squares: common factor
          • Factoring perfect squares: negative common factor
          • Factoring perfect squares: missing values
          • Factoring perfect squares: shared factors
        • Strategy in factoring quadratics
          • Strategy in factoring quadratics (part 1 of 2)
          • Strategy in factoring quadratics (part 2 of 2)
        • Factoring polynomials with special product forms
          • Factoring perfect squares: 4th degree polynomial
          • Factoring difference of squares: two variables
          • Factoring difference of squares: two variables (example 2)
        • Long division of polynomials
          • Intro to long division of polynomials
          • Dividing polynomials: long division
        • Synthetic division of polynomials
          • Intro to polynomial synthetic division
          • Dividing polynomials: synthetic division
          • Why synthetic division works
        • Practice dividing polynomials with remainders
          • Dividing polynomials with remainders: monomial divisors
          • Dividing polynomials with remainders: monomial divisors (example 2)
          • Dividing polynomials with remainders
        • Polynomial Remainder Theorem
          • Intro to the Polynomial Remainder Theorem
          • Using the Polynomial Remainder Theorem: finding remainders
          • Using the Polynomial Remainder Theorem: checking factors
          • Using the Polynomial Remainder Theorem: finding coefficients
          • Proof of the Polynomial Remainder Theorem
        • Binomial theorem
          • Intro to the Binomial Theorem
          • Pascal's triangle and binomial expansion
          • Expanding binomials
          • Expanding binomials w/o Pascal's triangle
        • Understanding the binomial theorem
          • Binomial expansion & combinatorics
          • Pascal's triangle & combinatorics
          • Binomial expansion & combinatorics (old)
        • Advanced polynomial factorization methods
          • Factoring higher-degree polynomials
          • Factoring sum of cubes
          • Factoring difference of cubes
        • Proving polynomial identities
          • Analyzing polynomial identities
        • Polynomial identities with complex numbers
          • Complex numbers & sum of squares factorization
          • Factoring sum of squares
        • Quadratic equations with complex numbers
          • Solving quadratic equations: complex roots
        • Fundamental Theorem of Algebra
          • The Fundamental theorem of Algebra
          • Quadratics & the Fundamental Theorem of Algebra
          • Number of possible real roots of a polynomial
        • Finding zeros of polynomials
          • Finding zeros of polynomials (1 of 2)
          • Finding zeros of polynomials (2 of 2)
          • Finding zeros of polynomials (example 2)
        • Zeros of polynomials and their graphs
          • Zeros of polynomials & their graphs
        • End behavior of polynomial functions
          • Intro to end behavior of polynomials
          • End behavior of functions & their graphs
        • Graphs of polynomials
        • Introduction to symmetry of functions
          • Intro to function symmetry
          • Graphical symmetry of functions
          • Even/odd functions & numbers
        • Symmetry of polynomial functions
          • Even & odd polynomials
      • Exponential & logarithmic functions
        • Exponent properties review
          • Multiplying & dividing powers (integer exponents)
          • Powers of products & quotients (integer exponents)
        • Radicals
          • Intro to square roots
          • Understanding square roots
          • Intro to cube roots
          • Square root of decimal
          • 5th roots
        • Rational exponents intro
          • Intro to rational exponents
          • Rewriting roots as rational exponents
          • Exponential equation with rational answer
        • Properties of exponents (rational exponents)
          • Simplifying quotient of powers (rational exponents)
          • Simplifying mixed radical and exponential expressions
        • Advanced exponent & radical evaluation
          • Evaluating fractional exponents
          • Evaluating fractional exponents: negative unit-fraction
          • Evaluating fractional exponents: fractional base
          • Evaluating quotient of fractional exponents
          • Evaluating mixed radicals and exponents
        • Simplifying square roots
          • Simplifying square roots
          • Simplifying square roots (variables)
          • Simplifying square-root expressions
        • Simplifying radicals (higher-index roots)
          • Simplifying higher-index roots
          • Simplifying cube root expressions
          • Simplifying cube root expressions (two variables)
          • Simplifying higher-index root expressions
        • Radicals (miscellaneous videos)
          • Simplifying square-root expressions: no variables
          • Simplifying square roots of fractions
          • Simplifying rational exponent expressions: mixed exponents and radicals
          • Simplifying square-root expressions: no variables (advanced)
          • Intro to rationalizing the denominator
          • Worked example: rationalizing the denominator
          • Simplifying radical expressions (addition)
          • Simplifying radical expressions (subtraction)
          • Simplifying radical expressions: two variables
          • Simplifying radical expressions: three variables
          • Simplifying hairy expression with fractional exponents
        • Exponential vs. linear growth
          • Intro to exponential functions
          • Exponential vs. linear growth
          • Exponential vs. linear models: verbal
          • Exponential vs. linear models: table
        • Exponential expressions
          • Exponential expressions word problems (numerical)
          • Initial value & common ratio of exponential functions
          • Exponential expressions word problems (algebraic)
          • Interpreting exponential expression word problem
        • Graphs of exponential growth
          • Exponential function graph
          • Graphs of exponential growth
        • Exponential vs. linear growth over time
          • Exponential vs. linear growth over time
        • Exponential decay
          • Exponential decay intro
          • Graphing exponential growth & decay
          • Exponential growth & decay word problems
        • Exponential functions from tables & graphs
          • Writing exponential functions
          • Writing exponential functions from tables
          • Writing exponential functions from graphs
          • Analyzing tables of exponential functions
          • Analyzing graphs of exponential functions
          • Analyzing graphs of exponential functions: negative initial value
          • Modeling with basic exponential functions word problem
        • Equivalent forms of exponential expressions
          • Simplifying exponential expressions
          • Rewriting exponential expressions
        • Solving exponential equations using properties of exponents
          • Solving exponential equations using exponent properties
          • Solving exponential equations using exponent properties (advanced)
        • Introduction to rate of exponential growth and decay
          • Exponential growth & decay word problems
        • Interpreting the rate of change of exponential models (Algebra 2 level)
          • Interpreting change in exponential models
          • Interpreting time in exponential models
        • Constructing exponential models according to rate of change (Algebra 2 level)
          • Constructing exponential models
          • Constructing exponential models: half life
          • Constructing exponential models: percent change
          • Constructing exponential models (old example)
        • Advanced interpretation of exponential models (Algebra 2 level)
          • Interpreting change in exponential models: with manipulation
          • Interpreting change in exponential models: changing units
          • Structure in exponential expression
        • Distinguishing between linear and exponential growth (Algebra 2 level)
          • Linear vs. exponential growth: from data
          • Linear vs. exponential growth: from data (example 2)
          • Comparing growth of exponential & quadratic models
        • Introduction to logarithms (Algebra 2 level)
          • Intro to logarithms
          • Evaluating logarithms (advanced)
          • Relationship between exponentials & logarithms
          • Relationship between exponentials & logarithms: graphs
          • Relationship between exponentials & logarithms: tables
        • The constant e and the natural logarithm (Algebra 2 level)
          • 𝑒 and compound interest
          • 𝑒 as a limit
          • Evaluating natural logarithm with calculator
        • Properties of logarithms (Algebra 2 level)
          • Intro to logarithm properties (1 of 2)
          • Intro to logarithm properties (2 of 2)
          • Using the logarithmic product rule
          • Using the logarithmic power rule
          • Using the properties of logarithms: multiple steps
          • Proof of the logarithm product rule
          • Proof of the logarithm quotient and power rules
        • The change of base formula for logarithms (Algebra 2 level)
          • Evaluating logarithms: change of base rule
          • Using the logarithm change of base rule
          • Proof of the logarithm change of base rule
        • Logarithmic equations (Algebra 2 level)
          • Logarithmic equations: variable in the argument
          • Logarithmic equations: variable in the base
        • Solving exponential equations with logarithms (Algebra 2 level)
          • Solving exponential equations using logarithms: base-10
          • Solving exponential equations using logarithms: base-2
        • Solving exponential models (Algebra 2 level)
          • Exponential model word problem: medication dissolve
          • Exponential model word problem: bacteria growth
        • Graphs of exponential functions (Algebra 2 level)
          • Transforming exponential graphs
          • Transforming exponential graphs (example 2)
          • Graphing exponential functions
          • Exponential function graph
          • Graphs of exponential functions (old example)
        • Graphs of logarithmic functions (Algebra 2 level)
          • Graphs of logarithms & exponentials
          • Graphing basic logarithmic functions
          • Graphs of logarithmic functions
          • Vertical asymptote of natural log
        • Logarithmic scale (Algebra 2 level)
          • Logarithmic scale (with Vi Hart)
          • Logarithmic scale
          • Richter scale
          • Benford's law (with Vi Hart, 1 of 2)
          • Benford's law (with Vi Hart, 2 of 2)
      • Radical equations & functions
        • Solving square-root equations
          • Intro to square-root equations & extraneous solutions
          • Solving square-root equations (basic)
          • Solving square-root equations: one solution
          • Solving square-root equations: two solutions
          • Solving square-root equations: no solution
        • Extraneous solutions of radical equations
          • Extraneous solutions of radical equations
          • Extraneous solutions of radical equations (example 2)
        • Solving cube-root equations
          • Solving cube-root equations
        • Domain of radical functions
          • Domain of a radical function
        • Graphs of radical functions
          • Transforming the square-root function
          • Graphs of square-root functions
          • Square-root functions & their graphs
      • Rational expressions, equations, & functions
        • Intro to rational expressions
        • Simplifying rational expressions
          • Intro to rational expression simplification
          • Simplifying rational expressions: common monomial factors
          • Simplifying rational expressions: common binomial factors
          • Simplifying rational expressions: opposite common binomial factors
          • Simplifying rational expressions: grouping
          • Simplifying rational expressions: higher degree terms
          • Simplifying rational expressions: two variables
          • Simplifying rational expressions (old video)
        • Multiplying & dividing rational expressions
          • Multiplying & dividing rational expressions: monomials
          • Multiplying rational expressions
          • Dividing rational expressions
          • Multiplying rational expressions: multiple variables
          • Dividing rational expressions: unknown expression
        • Adding & subtracting rational expressions
          • Adding & subtracting rational expressions: like denominators
          • Intro to adding rational expressions with unlike denominators
          • Adding rational expression: unlike denominators
          • Subtracting rational expressions: unlike denominators
          • Least common multiple
          • Least common multiple: repeating factors
          • Subtracting rational expressions: factored denominators
          • Least common multiple of polynomials
          • Subtracting rational expressions
        • Nested fractions
          • Nested fractions
        • Solving rational equations
          • Equations with one rational expression
          • Equations with one rational expression (advanced)
          • Equations with rational expressions
          • Equations with rational expressions (example 2)
          • Equation with two rational expressions (old example)
          • Equation with two rational expressions (old example 2)
          • Equation with two rational expressions (old example 3)
        • Direct and inverse variation
          • Intro to direct & inverse variation
          • Recognizing direct & inverse variation
          • Recognizing direct & inverse variation: table
          • Direct variation word problem: filling gas
          • Direct variation word problem: space travel
          • Inverse variation word problem: string vibration
          • Proportionality constant for direct variation
        • End behavior of rational functions
          • End behavior of rational functions
        • Discontinuities of rational functions
          • Discontinuities of rational functions
          • Analyzing vertical asymptotes of rational functions
        • Graphs of rational functions
          • Graphing rational functions according to asymptotes
          • Graphs of rational functions: y-intercept
          • Graphs of rational functions: horizontal asymptote
          • Graphs of rational functions: vertical asymptotes
          • Graphs of rational functions: zeros
          • Graphs of rational functions (old example)
          • Graphing rational functions 1
          • Graphing rational functions 2
          • Graphing rational functions 3
          • Graphing rational functions 4
        • Modeling with rational functions
          • Analyzing structure word problem: pet store (1 of 2)
          • Analyzing structure word problem: pet store (2 of 2)
          • Rational equations word problem: combined rates
          • Rational equations word problem: combined rates (example 2)
          • Rational equations word problem: eliminating solutions
          • Reasoning about unknown variables
          • Reasoning about unknown variables: divisibility
          • Structure in rational expression
        • Rational inequalities
          • Rational inequalities: one side is zero
          • Rational inequalities: both sides are not zero
        • Partial fraction expansion
          • Intro to partial fraction expansion
          • Partial fraction expansion
          • Partial fraction expansion: repeated factors
      • Trigonometric functions
        • Introduction to radians
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
          • Radian angles & quadrants
        • The unit circle definition of sine, cosine, & tangent
          • Unit circle
          • The trig functions & right triangle trig ratios
        • The graphs of sine, cosine, & tangent
          • Graph of y=sin(x)
          • Graph of y=tan(x)
          • Intersection points of y=sin(x) and y=cos(x)
        • Basic trigonometric identities
          • Sine & cosine identities: symmetry
          • Tangent identities: symmetry
          • Sine & cosine identities: periodicity
          • Tangent identities: periodicity
        • Trigonometric values of special angles
          • Trig values of special angles
        • Pythagorean identity
          • Proof of the Pythagorean trig identity
          • Using the Pythagorean trig identity
        • Introduction to amplitude, midline, & extrema of sinusoidal functions
          • Features of sinusoidal functions
        • Finding amplitude & midline of sinusoidal functions from their formulas
          • Amplitude & period of sinusoidal functions from equation
        • Period of sinusoidal functions
          • Amplitude & period of sinusoidal functions from equation
        • Graphing sinusoidal functions
          • Transforming sinusoidal graphs: vertical stretch & horizontal reflection
          • Transforming sinusoidal graphs: vertical & horizontal stretches
        • Constructing sinusoidal functions
          • Sinusoidal function from graph
          • Trig word problem: modeling daily temperature
          • Trig word problem: modeling annual temperature
        • The inverse trigonometric functions
          • Intro to arcsine
          • Intro to arctangent
          • Intro to arccosine
          • Restricting domains of functions to make them invertible
          • Domain & range of inverse tangent function
          • Using inverse trig functions with a calculator
        • Solving basic sinusoidal equations
          • Solving sinusoidal equations of the form sin(x)=d
          • Solving cos(θ)=1 and cos(θ)=-1
        • Solving advanced sinusoidal equations
        • Solving sinusoidal models
          • Trig word problem: solving for temperature
        • Introduction to the trigonometric angle addition identities
          • Review of trig angle addition identities
          • Using the cosine angle addition identity
          • Using the cosine double-angle identity
          • Proof of the sine angle addition identity
          • Proof of the cosine angle addition identity
        • Using trigonometric identities to solve problems
          • Finding trig values using angle addition identities
          • Using trig angle addition identities: finding side lengths
          • Using trig angle addition identities: manipulating expressions
          • Using trigonometric identities
        • Parametric equations
          • Parametric functions intro
          • Removing the parameter in parametric equations
          • Removing the parameter in parametric equations (example 2)
          • Parametric equations with the same graph
      • Algebraic modeling
        • Rate conversion
          • Intro to dimensional analysis
          • Same rate with different units
        • Appropriate units
          • Appropriate units
          • Interpreting units in formulas
          • Interpreting units in formulas: novel units
        • Word problems with multiple units
          • Worked example: Rate problem
          • Multiple units word problem: road trip
          • Measurement word problem: running laps
          • Multiple units word problem: drug dosage (advanced)
        • Modeling with one-variable equations & inequalities
          • Rational equation word problem
          • Quadratic inequality word problem
          • Exponential equation word problem
        • Manipulating formulas
          • Manipulating formulas: perimeter
          • Manipulating formulas: area
          • Manipulating formulas: temperature
      • Complex numbers
        • What are the imaginary numbers?
          • Intro to the imaginary numbers
          • Powers of the imaginary unit
          • Simplifying roots of negative numbers
          • i as the principal root of -1
        • What are the complex numbers?
          • Intro to complex numbers
          • How to classify complex numbers (example)
        • The complex plane
          • Plotting numbers on the complex plane
        • Adding & subtracting complex numbers
          • Adding complex numbers
          • Subtracting complex numbers
        • Multiplying complex numbers
          • Multiplying complex numbers
        • Complex conjugates & dividing complex numbers
          • Intro to complex number conjugates
          • Complex number conjugates
          • Dividing complex numbers
        • Absolute value & angle of complex numbers
          • Absolute value of complex numbers
          • Absolute value & angle of complex numbers
        • Distance & midpoint of complex numbers
          • Distance & midpoint of complex numbers
        • Polar form of complex numbers
          • Polar & rectangular forms of complex numbers
        • Multiplying & dividing complex numbers in polar form
          • Dividing complex numbers: polar & exponential form
          • Powers of complex numbers
          • Complex number equations: x³=1
        • Challenging complex number problems
          • Challenging complex numbers problem (1 of 3)
          • Challenging complex numbers problem (2 of 3)
          • Challenging complex numbers problem (3 of 3)
          • Challenging complex numbers problem: complex determinant
      • Conic sections
        • Introduction to conic sections
          • Intro to conic sections
        • The features of a circle
          • Graphing circles from features
          • Features of a circle from its graph
        • Standard equation of a circle
          • Features of a circle from its standard equation
          • Graphing a circle from its standard equation
          • Writing standard equation of a circle
        • Expanded equation of a circle
          • Features of a circle from its expanded equation
        • Center and radii of an ellipse
          • Intro to ellipses
          • Ellipse standard equation from graph
          • Ellipse graph from standard equation
        • Foci of an ellipse
          • Foci of an ellipse from equation
        • Focus and directrix of a parabola
          • Intro to focus & directrix
          • Equation of a parabola from focus & directrix
          • Focus & directrix of a parabola from equation
        • Introduction to hyperbolas
          • Intro to hyperbolas
          • Vertices & direction of a hyperbola
          • Vertices & direction of a hyperbola (example 2)
          • Graphing hyperbolas (old example)
        • Foci of a hyperbola
          • Foci of a hyperbola from equation
          • Proof of the hyperbola foci formula
        • Hyperbolas not centered at the origin
          • Equation of a hyperbola not centered at the origin
        • Identifying conic sections from their expanded equations
          • Conic section from expanded equation: circle & parabola
          • Conic section from expanded equation: ellipse
          • Conic section from expanded equation: hyperbola
        • Challenging conic section problems (IIT JEE)
          • Representing a line tangent to a hyperbola
          • Common tangent of circle & hyperbola (1 of 5)
          • Common tangent of circle & hyperbola (2 of 5)
          • Common tangent of circle & hyperbola (3 of 5)
          • Common tangent of circle & hyperbola (4 of 5)
          • Common tangent of circle & hyperbola (5 of 5)
          • Intersection of circle & hyperbola
      • Series & induction
        • Basic sigma notation
          • Summation notation
        • Arithmetic series
          • Arithmetic series intro
          • Worked example: arithmetic series (sigma notation)
          • Worked example: arithmetic series (sum expression)
          • Worked example: arithmetic series (recursive formula)
          • Proof of finite arithmetic series formula
        • Geometric sequences
          • Intro to geometric sequences
          • Intro to geometric sequences (advanced)
          • Explicit & recursive formulas for geometric sequences
          • Converting recursive & explicit forms of geometric sequences
        • Finite geometric series
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
          • Finite geometric series formula justification
        • Finite geometric series applications
          • Finite geometric series word problem: social media
          • Finite geometric series word problem: mortgage
        • Advanced sigma notation
          • Arithmetic series in sigma notation
          • Finite geometric series in sigma notation
          • Worked example: evaluating series (advanced)
          • Partial sums intro
          • Partial sums: formula for nth term from partial sum
          • Partial sums: term value from partial sum
        • Infinite geometric series
          • Infinite geometric series formula intuition
          • Proof of infinite geometric series as a limit
          • Worked example: convergent geometric series
          • Worked example: divergent geometric series
        • Infinite geometric series applications
          • Infinite geometric series word problem: bouncing ball
          • Infinite geometric series word problem: repeating decimal
        • Deductive and inductive reasoning
          • Inductive & deductive reasoning
          • Deductive reasoning
          • Using deductive reasoning
          • Inductive reasoning
          • Inductive reasoning (example 2)
          • Using inductive reasoning
          • Using inductive reasoning (example 2)
        • Induction
          • Proof of finite arithmetic series formula by induction
      • Vectors
        • Vector basics
          • Intro to vectors & scalars
          • Recognizing vectors
          • Equivalent vectors
          • Components of vectors
          • Components of vectors (example 2)
        • Magnitude of vectors
          • Vector magnitude from graph
          • Vector magnitude from components
          • Vector magnitude from initial & terminal points
        • Scalar multiplication
          • Scalar multiplication of vectors
          • Analyzing scalar multiplication
        • Vector addition & subtraction
          • Adding & subtracting vectors
          • Graphically adding & subtracting vectors
          • Graphically subtracting vectors
          • Adding vectors algebraically & graphically
        • Combined vector operations
          • Combined vector operations
        • Unit vectors
          • Worked example: finding unit vector with given direction
          • Unit vectors intro
          • Worked example: Scaling unit vectors
        • Magnitude & direction form of vectors
          • Direction of vectors from components: 1st & 2nd quadrants
          • Direction of vectors from components: 3rd & 4th quadrants
        • Component form of vectors
          • Vector components from magnitude & direction
        • Adding vectors in magnitude & direction form
          • Adding vectors in magnitude & direction form (1 of 2)
          • Adding vectors in magnitude & direction form (2 of 2)
          • Vector addition & magnitude
        • Applications of vectors
          • Vectors word problem: pushing a box
          • Vectors word problem: tug of war
          • Vectors word problem: hiking
      • Matrices
        • Introduction to matrices
          • Intro to matrices
        • Representing linear systems of equations with augmented matrices
        • Elementary matrix row operations
        • Row-echelon form & Gaussian elimination
          • Solving linear systems with matrices
        • Adding & subtracting matrices
          • Adding & subtracting matrices
        • Multiplying matrices by scalars
          • Multiplying matrices by scalars
        • Properties of matrix addition & scalar multiplication
        • Multiplying matrices by matrices
          • Intro to matrix multiplication
          • Multiplying matrices
        • Properties of matrix multiplication
          • Defined matrix operations
          • Intro to identity matrix
          • Dimensions of identity matrix
          • Is matrix multiplication commutative?
          • Associative property of matrix multiplication
          • Zero matrix & matrix multiplication
          • Using properties of matrix operations
          • Using identity & zero matrices
        • Matrices as transformations
          • Transforming vectors using matrices
          • Transforming polygons using matrices
          • Visual representation of transformation from matrix
        • Determinant of a 2x2 matrix
          • Determinant of a 2x2 matrix
        • Introduction to matrix inverses
          • Intro to matrix inverses
          • Determining invertible matrices
        • Finding the inverse of a matrix using its determinant
          • Finding inverse of a 2x2 matrix using determinant & adjugate
        • Practice finding the inverses of 2x2 matrices
        • Determinants & inverses of large matrices
          • Determinant of a 3x3 matrix: standard method (1 of 2)
          • Determinant of a 3x3 matrix: shortcut method (2 of 2)
          • Inverting a 3x3 matrix using Gaussian elimination
          • Inverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrix
          • Inverting a 3x3 matrix using determinants Part 2: Adjugate matrix
        • Solving equations with inverse matrices
          • Representing linear systems with matrix equations
          • Solving linear systems with matrix equations
          • Matrix word problem: vector combination
        • Model real-world situations with matrices
          • Matrix word problem: prices
    • Geometry (all content)
      • Lines
        • Lines, line segments, and rays
          • Euclid as the father of geometry
          • Terms & labels in geometry
          • Lines, line segments, & rays
          • Identifying rays
        • Measuring segments
          • Measuring Line Segments
          • Congruent line segments
        • Parallel and perpendicular
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines
          • Drawing parallel line segments
        • Points, lines, & planes
          • Specifying planes in three dimensions
        • Geometric definitions
          • Geometric definitions example
        • The golden ratio
          • The golden ratio
      • Angles
        • Angle introduction
          • Angles: introduction
          • Naming angles
        • Measuring angles
          • Measuring angles in degrees
          • Measuring angles using a protractor
          • Measuring angles using a protractor 2
        • Constructing angles
          • Constructing angles
        • Angles in circles
          • Angle measurement & circle arcs
        • Angle types
          • Recognizing angles
          • Drawing acute, right and obtuse angles
          • Identifying an angle
        • Vertical, complementary, and supplementary angles
          • Complementary & supplementary angles
          • Angle relationships example
          • Vertical angles are congruent proof
        • Angles between intersecting lines
          • Angles, parallel lines, & transversals
          • Parallel & perpendicular lines
          • Missing angles with a transversal
          • Parallel lines & corresponding angles proof
          • Missing angles (CA geometry)
          • Proving angles are congruent
          • Proofs with transformations
        • Sal's old angle videos
          • Intro to angles (old)
          • Angles (part 2)
          • Angles (part 3)
          • Angles formed between transversals and parallel lines
          • Angles of parallel lines 2
          • The angle game
          • The angle game (part 2)
          • Acute, right, & obtuse angles
      • Shapes
        • Properties of shapes
          • Cousin Fal's shape collection
          • Recognizing shapes
        • Classifying geometric shapes
          • Classifying shapes by line and angles types
        • Angles with polygons
          • Sum of interior angles of a polygon
          • Sum of the exterior angles of a polygon
        • Geometric solids (3D shapes)
      • Triangles
        • Triangle types
          • Classifying triangles
          • Classifying triangles by angles
          • Worked example: Classifying triangles
        • Triangle angles
          • Angles in a triangle sum to 180° proof
          • Triangle exterior angle example
          • Worked example: Triangle angles (intersecting lines)
          • Worked example: Triangle angles (diagram)
          • Triangle angle challenge problem
          • Triangle angle challenge problem 2
        • Triangle inequality theorem
          • Triangle inequality theorem
        • Perpendicular bisectors
          • Circumcenter of a triangle
          • Circumcenter of a right triangle
          • Three points defining a circle
          • Area circumradius formula proof
          • 2003 AIME II problem 7
        • Angle bisectors
          • Distance between a point & line
          • Incenter and incircles of a triangle
          • Inradius, perimeter, & area
        • Medians & centroids
          • Triangle medians & centroids
          • Triangle medians and centroids (2D proof)
          • Dividing triangles with medians
          • Exploring medial triangles
          • Centroid & median proof
          • Median, centroid example
        • Altitudes
          • Proof: Triangle altitudes are concurrent (orthocenter)
          • Common orthocenter and centroid
        • Bringing it all together
          • Review of triangle properties
          • Euler line
          • Euler's line proof
      • Quadrilaterals
        • Quadrilateral types
          • Intro to quadrilateral
          • Quadrilateral properties
          • Quadrilateral types
          • Classifying quadrilaterals
          • Kites as a geometric shape
        • Quadrilateral proofs & angles
          • Proof: Opposite sides of a parallelogram
          • Proof: Diagonals of a parallelogram
          • Proof: Opposite angles of a parallelogram
          • Proof: Rhombus diagonals are perpendicular bisectors
          • Whether a special quadrilateral can exist
          • Rhombus diagonals
      • Coordinate plane
        • Coordinate plane: quadrant 1
          • Coordinate plane: graphing points
          • Coordinate plane graphing word problem
        • Coordinate plane: 4 quadrants
          • Points on the coordinate plane examples
          • Plotting a point (ordered pair)
          • Finding the point not graphed
          • Coordinate plane word problem examples
        • Quadrants on the coordinate plane
          • Quadrants of the coordinate plane
          • Points and quadrants example
        • Reflecting points on coordinate plane
          • Reflecting points in the coordinate plane
        • Quadrilaterals on the coordinate plane
          • Coordinates of a missing vertex
          • Area of a parallelogram on the coordinate plane
          • Dimensions of a rectangle from coordinates
          • Coordinates of rectangle example
        • Drawing polygons in the coordinate plane
          • Drawing a quadrilateral on the coordinate plane example
          • Drawing a trapezoid on the coordinate plane example
      • Area and perimeter
        • Count unit squares to find area
          • Intro to area and unit squares
          • Measuring rectangles with different unit squares
          • Measuring area with partial unit squares
          • Creating rectangles with a given area 1
          • Creating rectangles with a given area 2
        • Area of rectangles
          • Transitioning from unit squares to area formula
          • Finding missing side when given area
          • Counting unit squares to find area formula
        • Perimeter
          • Perimeter: introduction
          • Perimeter of a shape
          • Finding perimeter when a side length is missing
          • Finding missing side length when given perimeter
          • Perimeter & area
          • Perimeter and unit conversion
          • Applying the metric system to perimeter
        • Area of parallelograms
          • Area of a parallelogram
        • Area of triangles
          • Area of a triangle
          • Area of triangle proof
        • Area of shapes on grids
          • Area of a triangle on a grid
          • Area of a quadrilateral on a grid
        • Area of trapezoids & composite figures
          • Area of trapezoids
          • Area of kites
          • Finding area by rearranging parts
          • Area of composite shapes
          • Perimeter & area of composite shapes
          • Challenge problems: perimeter & area
        • Area and circumference of circles
          • Radius, diameter, circumference & π
          • Labeling parts of a circle
          • Radius & diameter from circumference
          • Area of a circle
          • Area of a circle intuition
        • Advanced area with triangles
          • Area of equilateral triangle
          • Area of equilateral triangle (advanced)
          • Area of diagonal-generated triangles
      • Volume and surface area
        • Volume of rectangular prisms
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
          • Volume of a rectangular prism
          • Volume of triangular prism & cube
        • Volume with fractions
          • Volume with fractional cubes
          • Volume of a rectangular prism: fractional dimensions
          • Volume of a rectangular prism: word problem
        • Surface area
          • Intro to nets of polyhedra
          • Surface area using a net: triangular prism
          • Surface area of a box (cuboid)
          • Surface area of a box using nets
          • Surface area using a net: rectangular prism
        • Volume of cones, cylinders, and spheres
          • Cylinder volume & surface area
          • Volume of a cone
          • Volume of a sphere
        • Solid geometry
        • Surface and volume density
          • Density word problem: blimp
        • Cross sections of 3D objects
          • Slicing a rectangular pyramid
          • Ways to cross-section a cube
          • Rotating 2D shapes in 3D
        • Koch snowflake fractal
          • Koch snowflake fractal
          • Area of Koch snowflake (1 of 2)
          • Area of Koch snowflake (2 of 2)
        • Heron's formula
          • Heron's formula
          • Proof of Heron's formula (1 of 2)
          • Proof of Heron's formula (2 of 2)
      • Pythagorean theorem
        • Pythagorean theorem
          • Intro to the Pythagorean theorem
          • Pythagorean theorem example
        • Pythagorean theorem application
          • Pythagorean theorem word problem: carpet
          • Pythagorean theorem word problem: fishing boat
          • Pythagorean theorem in 3D
        • Pythagorean theorem and distance between points
          • Distance formula
        • Pythagorean theorem proofs
          • Garfield's proof of the Pythagorean theorem
          • Bhaskara's proof of the Pythagorean theorem
          • Pythagorean theorem proof using similarity
          • Another Pythagorean theorem proof
      • Transformations
        • Introduction to rigid transformations
          • Rigid transformations intro
        • Translations
          • Translating shapes
          • Determining translations
          • Translation challenge problem
        • Rotations
          • Rotating shapes
          • Determining rotations
          • Rotating shapes: center ≠ (0,0)
        • Reflections
          • Reflecting shapes: diagonal line of reflection
          • Determining reflections (advanced)
        • Rigid transformations overview
        • Dilations
          • Performing dilations
          • Dilating shapes: shrinking
          • Dilating shapes: expanding
        • Properties and definitions of transformations
          • Identifying type of transformation
        • Symmetry
          • Intro to reflective symmetry
          • Intro to rotational symmetry
          • Finding a quadrilateral from its symmetries
          • Finding a quadrilateral from its symmetries (example 2)
        • Old transformations videos
          • Performing translations (old)
          • Performing rotations (old)
          • Performing reflections: rectangle (old)
          • Performing reflections: line (old)
          • Determining translations (old)
          • Rotation examples (old)
          • Determining rotations (old)
          • Dilating lines
      • Congruence
        • Transformations and congruence
          • Congruent shapes & transformations
          • Non-congruent shapes & transformations
        • Triangle congruence
          • Congruent triangles & the SSS postulate/criterion
          • Triangle congruence postulates/criteria
          • Why SSA isn't a congruence postulate/criterion
          • Determining congruent triangles
        • Theorems concerning triangle properties
          • Angles in a triangle sum to 180° proof
          • Proofs concerning isosceles triangles
          • Proofs concerning equilateral triangles
        • Working with triangles
          • Corresponding angles in congruent triangles
          • Isosceles & equilateral triangles problems
          • Finding angles in isosceles triangles
          • Finding angles in isosceles triangles (example 2)
        • Theorems concerning quadrilateral properties
          • Proof: Opposite sides of a parallelogram
          • Proof: Diagonals of a parallelogram
          • Proof: Opposite angles of a parallelogram
          • Proof: The diagonals of a kite are perpendicular
          • Proof: Rhombus diagonals are perpendicular bisectors
          • Proof: Rhombus area
        • Proofs of general theorems that use triangle congruence
          • Geometry proof problem: midpoint
          • Geometry proof problem: congruent segments
          • Geometry proof problem: squared circle
      • Similarity
        • Definitions of similarity
          • Similar shapes & transformations
        • Introduction to triangle similarity
          • Intro to triangle similarity
          • Triangle similarity postulates/criteria
          • Determining similar triangles
          • Proving slope is constant using similarity
        • Solving similar triangles
          • Solving similar triangles
          • Solving similar triangles: same side plays different roles
        • Angle bisector theorem
          • Intro to angle bisector theorem
          • Using the angle bisector theorem
        • Solving problems with similar and congruent triangles
          • Using similar & congruent triangles
          • Challenging similarity problem
        • Solving modeling problems with similar and congruent triangles
          • Geometry word problem: the golden ratio
          • Geometry word problem: Earth & Moon radii
          • Geometry word problem: a perfect pool shot
      • Trigonometry
        • Special right triangles
          • Special right triangles intro (part 1)
          • Special right triangles intro (part 2)
          • 30-60-90 triangle example problem
          • Special right triangles proof (part 1)
          • Special right triangles proof (part 2)
          • Area of a regular hexagon
        • Introduction to the trigonometric ratios
          • Intro to the trigonometric ratios
          • Trigonometric ratios in right triangles
        • Solving for a side in a right triangle using the trigonometric ratios
          • Solving for a side in right triangles with trigonometry
        • Solving for an angle in a right triangle using the trigonometric ratios
        • Modeling with right triangles
          • Right triangle word problem
        • Trigonometric ratios and similarity
          • Triangle similarity & the trigonometric ratios
          • Trig challenge problem: verify identities
          • Trig challenge problem: trig values & side ratios
        • Sine and cosine of complementary angles
          • Sine & cosine of complementary angles
          • Using complementary angles
          • Trig word problem: complementary angles
        • The reciprocal trigonometric ratios
          • Finding reciprocal trig ratios
          • Using reciprocal trig ratios
        • The law of sines
          • Solving for a side with the law of sines
          • Solving for an angle with the law of sines
          • Proof of the law of sines
        • The law of cosines
          • Solving for a side with the law of cosines
          • Solving for an angle with the law of cosines
          • Proof of the law of cosines
        • Solving general triangles
          • Trig word problem: stars
      • Circles
        • Circle basics
          • Circles glossary
          • Radius, diameter, circumference & π
          • Proof: all circles are similar
        • Arc measure
          • Intro to arc measure
          • Finding arc measures
          • Finding arc measures with equations
        • Arc length (degrees)
          • Arc length from subtended angle
          • Subtended angle from arc length
        • Introduction to radians
          • Intro to radians
          • Radians & degrees
          • Degrees to radians
          • Radians to degrees
        • Arc length (radians)
          • Arc length as fraction of circumference
          • Arc length from subtended angle: radians
        • Sectors
          • Area of a sector
        • Inscribed angles
          • Inscribed angles
          • Inscribed angle theorem proof
        • Inscribed shapes problem solving
          • Proof: Right triangles inscribed in circles
          • Inscribed shapes: find diameter
          • Inscribed shapes: angle subtended by diameter
          • Inscribed shapes: find inscribed angle
          • Inscribed quadrilaterals proof
          • Solving inscribed quadrilaterals
        • Properties of tangents
          • Proof: Radius is perpendicular to tangent line
          • Proof: Segments tangent to circle from outside point are congruent
          • Tangents of circles problem (example 1)
          • Tangents of circles problem (example 2)
          • Tangents of circles problem (example 3)
        • Area of inscribed triangle
          • Proof: radius is perpendicular to a chord it bisects
          • Proof: perpendicular radius bisects chord
          • Area of inscribed equilateral triangle
        • Standard equation of a circle
          • Features of a circle from its standard equation
          • Graphing a circle from its standard equation
          • Writing standard equation of a circle
        • Expanded equation of a circle
          • Features of a circle from its expanded equation
      • Analytic geometry
        • Distance and midpoints
          • Distance formula
          • Midpoint formula
        • Dividing line segments
          • Dividing line segments: graphical
          • Dividing line segments
        • Problem solving with distance on the coordinate plane
          • Area of trapezoid on the coordinate plane
          • Points inside/outside/on a circle
          • Challenge problem: Points on two circles
          • Coordinate plane word problem
          • Classifying quadrilaterals on the coordinate plane
        • Parallel and perpendicular lines on the coordinate plane
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines from graph
        • Equations of parallel and perpendicular lines
          • Parallel lines from equation
          • Parallel lines from equation (example 2)
          • Parallel lines from equation (example 3)
          • Perpendicular lines from equation
          • Writing equations of perpendicular lines
          • Writing equations of perpendicular lines (example 2)
          • Proof: parallel lines have the same slope
          • Proof: perpendicular lines have negative reciprocal slope
        • Challenge: Distance between a point and a line
          • Distance between point & line
      • Geometric constructions
        • Constructing bisectors of lines and angles
          • Geometric constructions: perpendicular bisector
          • Geometric constructions: perpendicular line
          • Geometric constructions: angle bisector
        • Constructing regular polygons inscribed in circles
          • Geometric constructions: circle-inscribed square
          • Geometric constructions: circle-inscribed equilateral triangle
          • Geometric constructions: circle-inscribed regular hexagon
        • Constructing circumcircles and incircles
          • Geometric constructions: triangle-inscribing circle
          • Geometric constructions: triangle-circumscribing circle
        • Constructing a line tangent to a circle
          • Geometric constructions: circle tangent
          • Geometric constructions: circle tangent (example 2)
      • Miscellaneous
        • Worked examples
          • Challenge problems: perimeter & area
          • Challenging perimeter problem
          • CA Geometry: Deductive reasoning
          • CA Geometry: Proof by contradiction
          • CA Geometry: More proofs
          • CA Geometry: Similar triangles 1
          • CA Geometry: More on congruent and similar triangles
          • CA Geometry: Triangles and parallelograms
          • CA Geometry: Area, pythagorean theorem
          • CA Geometry: Area, circumference, volume
          • CA Geometry: Pythagorean theorem, area
          • CA Geometry: Exterior angles
          • CA Geometry: Pythagorean theorem, compass constructions
          • CA Geometry: Compass construction
          • CA Geometry: Basic trigonometry
          • CA Geometry: More trig
          • CA Geometry: Circle area chords tangent
    • Calculus (all content, first year)
      • Limits and continuity
        • Limits introduction
          • Newton, Leibniz, and Usain Bolt
          • Limits intro
          • Introduction to limits (old)
        • Limits from tables
          • Limits from tables
        • Limits from graphs
          • Limits from graphs: function undefined
          • Limits from graphs: limit isn't equal to the function's value
          • Limits from graphs: asymptote
        • One-sided limits
          • One-sided limits from graphs
          • One-sided limits from graphs: asymptote
          • One-sided limits from tables
          • 1-sided vs. 2-sided limits (graphical)
          • One-sided limit from equation
          • Connecting limits and graphical behavior (more examples)
        • Formal definition of limits (epsilon-delta)
          • Formal definition of limits Part 1: intuition review
          • Formal definition of limits Part 2: building the idea
          • Formal definition of limits Part 3: the definition
          • Formal definition of limits Part 4: using the definition
        • Review: Limits basics
        • Continuity at a point
          • Continuity introduction
          • Worked example: Continuity at a point
          • Discontinuity points challenge example
        • Limits of combined and composite functions
          • Limit properties
          • Limits of combined functions
          • Limits of composite functions
        • Continuous functions
          • Functions continuous on all real numbers
          • Functions continuous at specific x-values
        • Intermediate value theorem
          • Intermediate value theorem
          • Intermediate value theorem example
        • Review: Continuity
        • Limits from equations (direct substitution)
          • Limits by direct substitution
          • Undefined limits by direct substitution
        • Limits from equations (factoring & rationalizing)
          • Limits by factoring
          • Limits by rationalizing
          • Finding limits by factoring (cubic)
        • Squeeze theorem
          • Squeeze theorem intro
          • Squeeze theorem example
          • Limit of sin(x)/x as x approaches 0
          • Squeeze theorem intro (old)
        • Limits of trigonometric functions
          • Trig limit using pythagorean identity
          • Trig limit using double angle identity
        • Limits of piecewise functions
          • Analyzing functions for discontinuities (continuous example)
          • Analyzing functions for discontinuities (discontinuity example)
        • Removable discontinuities
          • Removing discontinuities (factoring)
          • Removing discontinuities (rationalization)
        • Review: Limits from equations
        • Unbounded limits (vertical asymptotes)
          • Infinite limits intro
          • Unbounded limits: graphical
          • Analyzing unbounded limits: rational function
          • Analyzing unbounded limits: mixed function
          • Visually determining vertical asymptotes (old)
        • Limits at infinity (horizontal asymptotes)
          • Infinite limits intro
          • Limits at infinity of quotients (Part 1)
          • Limits at infinity of quotients (Part 2)
          • Limits at infinity of quotients with square roots (odd power)
          • Limits at infinity of quotients with square roots (even power)
          • Limits at infinity of quotients with trig
          • Limits at infinity of quotients with trig (limit undefined)
          • Limit at infinity of a difference of functions
        • Review: Infinite limits
        • Limits skill checks
      • Taking derivatives
        • Introduction to differential calculus
          • Newton, Leibniz, and Usain Bolt
          • Derivative as slope of curve
          • Derivative & the direction of a function
        • Derivative as slope of tangent line
          • The derivative & tangent line equations
          • Interpreting derivative challenge
        • Derivative as instantaneous rate of change
          • Tangent slope as instantaneous rate of change
          • Estimating derivatives with two consecutive secant lines
          • Approximating instantaneous rate of change with average rate of change
        • Secant lines
          • Slope of a line secant to a curve
          • Secant line with arbitrary difference
          • Secant line with arbitrary point
          • Secant line with arbitrary difference (with simplification)
          • Secant line with arbitrary point (with simplification)
          • Secant lines: challenging problem 1
          • Secant lines: challenging problem 2
        • Derivative as a limit
          • Formal definition of the derivative as a limit
          • Derivative as a limit: numerical
        • Formal definition of derivative
          • Formal and alternate form of the derivative
          • Worked example: Derivative as a limit
          • Worked example: Derivative from limit expression
        • Using the formal definition of derivative
          • The derivative of x² at x=3 using the formal definition
          • The derivative of x² at any point using the formal definition
          • Limit expression for the derivative of a linear function
          • Limit expression for the derivative of cos(x) at a minimum point
          • Limit expression for the derivative of function (graphical)
        • Differentiability
          • Differentiability at a point: graphical
          • Differentiability at a point: algebraic (function is differentiable)
          • Differentiability at a point: algebraic (function isn't differentiable)
          • Differentiability at a point (old)
        • Derivative as a function
          • The graphical relationship between a function & its derivative (part 1)
          • The graphical relationship between a function & its derivative (part 2)
          • Matching functions & their derivatives graphically
          • Connecting f and f' graphically
          • Matching functions & their derivatives graphically (old)
        • Review: Derivative basics
        • Basic differentiation rules
          • Basic derivative rules (Part 1)
          • Basic derivative rules (Part 2)
          • Basic derivative rules: find the error
          • Basic derivative rules: table
        • Power rule
          • Power rule
          • Justifying the power rule
          • Proof of power rule for positive integer powers
          • Proof of power rule for square root function
        • Polynomial functions differentiation
          • Differentiating polynomials
          • Differentiating polynomials example
          • Tangents of polynomials
        • Rational functions differentiation (intro)
          • Negative powers differentiation
          • Worked example: Tangent to the graph of 1/x
        • Radical functions differentiation (intro)
          • Fractional powers differentiation
          • Radical functions differentiation intro
        • Sine & cosine derivatives
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Worked example: Derivatives of sin(x) and cos(x)
        • eˣ and ln(x) derivatives
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
        • Review: Basic differentiation
        • Product rule
          • Product rule
          • Differentiating products
          • Worked example: Product rule with table
          • Worked example: Product rule with mixed implicit & explicit
          • Product rule to find derivative of product of three functions
          • Product rule proof
        • Chain rule
          • Chain rule
          • Worked example: Derivative of cos³(x) using the chain rule
          • Worked example: Derivative of ln(√x) using the chain rule
          • Worked example: Derivative of √(3x²-x) using the chain rule
          • Worked example: Chain rule with table
          • Quotient rule from product & chain rules
          • Chain rule with the power rule
          • Applying the chain rule graphically 1 (old)
          • Applying the chain rule graphically 2 (old)
          • Applying the chain rule graphically 3 (old)
        • Chain rule proof
          • Proof: Differentiability implies continuity
          • If function u is continuous at x, then Δu→0 As Δx→0
          • Chain rule proof
        • Quotient rule
          • Quotient rule from product & chain rules
          • Worked example: Quotient rule with table
          • Tangent to y=𝑒ˣ/(2+x³)
          • Normal to y=𝑒ˣ/x²
        • Review: Product, quotient, & chain rule
        • Rational functions differentiation
          • Differentiating rational functions
        • Radical functions differentiation
          • Radical functions differentiation
        • Trigonometric functions differentiation
          • Derivatives of tan(x) and cot(x)
          • Derivatives of sec(x) and csc(x)
          • Trig functions differentiation
          • Derivative of tan(x) (old)
        • Exponential functions differentiation
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Exponential functions differentiation intro
          • Exponential functions differentiation
          • Derivative of 2ˣ (old)
        • Logarithmic functions differentiation
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Derivative of logₐx (for any positive base a≠1)
          • Logarithmic functions differentiation
          • Differentiating logarithmic functions using log properties
          • Derivative of logarithm for any base (old)
        • Derivatives capstone
          • Derivative of eᶜᵒˢˣ⋅cos(eˣ)
          • Derivative of sin(ln(x²))
        • Implicit differentiation introduction
          • Implicit differentiation
          • Worked example: Implicit differentiation
          • Worked example: Evaluating derivative with implicit differentiation
          • Showing explicit and implicit differentiation give same result
        • Implicit differentiation (advanced examples)
          • Implicit differentiation (advanced example)
          • Implicit differentiation (advanced example)
          • Implicit differentiation (advanced example)
          • Derivative of ln(x) from derivative of 𝑒ˣ and implicit differentiation
        • Inverse trig functions differentiation
          • Derivative of inverse sine
          • Derivative of inverse cosine
          • Derivative of inverse tangent
        • Derivatives of inverse functions
          • Derivatives of inverse functions: from equation
          • Derivatives of inverse functions: from table
        • Disguised derivatives
          • Derivatives expressed as limits
        • Proofs for the derivatives of eˣ and ln(x)
          • Proof: The derivative of 𝑒ˣ is 𝑒ˣ
          • Proof: the derivative of ln(x) is 1/x
        • Logarithmic differentiation
          • Composite exponential function differentiation
          • Worked example: Composite exponential function differentiation
        • Parametric & vector-valued function differentiation
          • Vector-valued functions intro
          • Vector-valued functions differentiation
          • Parametric functions differentiation
        • Review: Advanced differentiation
        • Higher-order derivatives
          • Second derivatives
        • Higher-order derivatives (parametric & vector-valued functions)
          • Second derivatives (vector-valued functions)
          • Second derivatives (parametric functions)
        • Taking derivatives skill checks
        • Polar curve differentiation
      • Derivative applications
        • Critical points
          • Introduction to minimum and maximum points
          • Identifying relative minimum and maximum values
          • Critical points introduction
          • Finding critical points
        • Increasing & decreasing intervals
          • Finding decreasing interval given the function
          • Finding increasing interval given the derivative
        • Relative minima & maxima
          • Finding relative extrema (first derivative test)
          • Analyzing a function with its derivative
          • Worked example: finding relative extrema
        • Absolute minima & maxima
          • Extreme value theorem
          • Finding absolute extrema on a closed interval
          • Absolute minima & maxima (entire domain)
        • Review: Increasing/decreasing intervals & extrema
        • Concavity
          • Concavity introduction
          • Analyzing concavity (graphical)
          • Analyzing concavity (algebraic)
          • Second derivative test
        • Points of inflection
          • Inflection points introduction
          • Inflection points (graphical)
          • Worked example: Inflection points from first derivative
          • Inflection points (algebraic)
        • Sketching graphs using calculus
          • Curve sketching with calculus: polynomial
          • Curve sketching with calculus: logarithm
        • Review: Concavity & points of inflection
        • Linear approximation
          • Local linearity
          • Linear approximation of a rational function
          • Linear approximation example (old)
        • Rectilinear motion
          • Motion problems: when a particle is speeding up
          • Motion problems: finding the maximum acceleration
          • Analyzing rectilinear motion graphically
          • Total distance traveled by a particle
        • Planar motion
          • Planar motion example: acceleration vector
          • Motion along a curve: finding rate of change
          • Motion along a curve: finding velocity magnitude
        • Related rates
          • Related rates intro
          • Related rates: Approaching cars
          • Related rates: Falling ladder
          • Related rates: water pouring into a cone
          • Related rates: shadow
          • Related rates: balloon
        • Optimization
          • Optimization: sum of squares
          • Optimization: box volume (Part 1)
          • Optimization: box volume (Part 2)
          • Optimization: profit
          • Optimization: cost of materials
          • Optimization: area of triangle & square (Part 1)
          • Optimization: area of triangle & square (Part 2)
          • Optimization problem: extreme normaline to y=x²
        • Applied rates of change
          • Applied rate of change: forgetfulness
          • Applied rates of change: marginal costs
          • Marginal cost & differential calculus
        • Mean value theorem
          • Mean value theorem
          • Mean value theorem example: polynomial
          • Mean value theorem example: square root function
          • Mean value theorem application
          • Mean value theorem (old)
        • L'Hôpital's rule
          • L'Hôpital's rule introduction
          • L'Hôpital's rule: limit at 0 example
          • L'Hôpital's rule: challenging problem
          • L'Hôpital's rule: limit at infinity example
          • L'Hôpital's rule: solve for a variable
          • L’Hôpital’s rule (composite exponential functions)
          • Proof of special case of l'Hôpital's rule
        • Review: Derivative applications
        • Derivative applications skill check
      • Integration
        • Antiderivatives
          • Antiderivatives and indefinite integrals
        • Indefinite integrals intro
          • Reverse power rule
          • Rewriting before integrating: challenge problem
          • Visually determining antiderivative
          • Graphs of indefinite integrals
        • Indefinite integrals of common functions
          • Indefinite integrals of sin(x), cos(x), and eˣ
          • Indefinite integral of 1/x
          • Finding specific antiderivatives: rational function
          • Finding specific antiderivatives: exponential function
        • Review: Indefinite integrals & antiderivatives
        • Definite integral as area
          • Worked example: Definite integral by thinking about the function's graph
        • Definite integral properties
          • Integrating scaled version of function
          • Integrating sums of functions
          • Definite integral over a single point
          • Breaking up integral interval
          • Definite integral of shifted function
          • Switching bounds of definite integral
          • Worked examples: Definite integral properties 2
          • Worked examples: Definite integral properties 1
          • Definite integral properties (no graph): function combination
          • Definite integral properties (no graph): breaking interval
          • Examples leveraging integration properties
        • Review: Definite integral basics
        • Riemann sums
          • Riemann approximation introduction
          • Riemann sums in summation notation
          • Motion problem with Riemann sum approximation
        • Riemann sums with sigma notation
          • Worked example: Riemann sums in summation notation
          • Riemann sums in summation notation: challenge problem
        • Trapezoidal rule
          • Midpoint and trapezoidal sums in summation notation
          • Trapezoidal sums
        • Definite integral as the limit of a Riemann sum
          • Definite integral as the limit of a Riemann sum
          • Ranking area estimates
        • Review: Riemann sums
        • Functions defined by integrals
          • Functions defined by integrals
          • Functions defined by integrals: switched interval
          • Functions defined by integrals: challenge problem
        • Fundamental theorem of calculus
          • Fundamental theorem of calculus (Part 1)
          • Worked example: Finding derivative with fundamental theorem of calculus
          • Finding derivative with fundamental theorem of calculus: x is on both bounds
          • Proof of fundamental theorem of calculus
        • Fundamental theorem of calculus: chain rule
          • Worked example: Finding derivative with fundamental theorem of calculus
          • Finding derivative with fundamental theorem of calculus: x is on lower bound
        • Definite integral evaluation
          • Fundamental theorem of calculus (Part 2)
          • Intuition for second part of fundamental theorem of calculus
          • Area between a curve and the x-axis
          • Area between a curve and the x-axis: negative area
          • Definite integral of rational function
          • Definite integral of radical function
          • Definite integral of trig function
          • Definite integral involving natural log
        • Definite integrals of piecewise functions
          • Definite integral of piecewise function
          • Definite integral of absolute value function
        • Improper integrals
          • Introduction to improper integrals
          • Improper integral with two infinite bounds
          • Divergent improper integral
      • Integration techniques
        • Integration by parts
          • Integration by parts intro
          • Integration by parts: ∫x⋅cos(x)dx
          • Integration by parts: ∫ln(x)dx
          • Integration by parts: ∫x²⋅𝑒ˣdx
          • Integration by parts: ∫𝑒ˣ⋅cos(x)dx
          • Challenging definite integration
        • u-substitution
          • 𝘶-substitution intro
          • 𝘶-substitution: rational function
          • 𝘶-substitution: multiplying by a constant
          • 𝘶-substitution: logarithmic function
          • 𝘶-substitution: composite function
          • 𝘶-substitution: special application
          • 𝘶-substitution: definite integral of exponential function
          • 𝘶-substitution: double substitution
        • Reverse chain rule
          • Reverse chain rule introduction
          • Reverse chain rule example
          • Integral of tan x
        • Partial fraction expansion
          • Partial fraction expansion to evaluate integral
          • Dividing expressions to evaluate integral
        • Integration using trigonometric identities
          • Integral of cos^3(x)
          • Integral of sin^2(x) cos^3(x)
          • Integral of sin^4(x)
        • Trigonometric substitution
          • Introduction to trigonometric substitution
          • Substitution with x=sin(theta)
          • More trig sub practice
          • Trig and u substitution together (part 1)
          • Trig and u substitution together (part 2)
          • Trig substitution with tangent
          • More trig substitution with tangent
          • Long trig sub problem
      • Integration applications
        • Area between curves
          • Area between curves
          • Area between curves with multiple boundaries
        • Area defined by polar graphs
          • Area bounded by polar curves
          • Worked example: Area enclosed by cardioid
          • Worked example: Area between two polar graphs
          • Evaluating definite integral with calculator
        • Arc length
          • Arc length intro
          • Worked example: arc length
          • Worked example: arc length (advanced)
        • Arc length of polar graphs
          • Arc length of polar curves
          • Worked example: Arc length of polar curves
        • Average value of a function
          • Average value over a closed interval
          • Calculating average value of function over interval
          • Average acceleration over interval
          • Mean value theorem for integrals
        • Area & net change
          • Worked example: problem involving definite integral (graphical)
          • Area under rate function gives the net change
        • Rectilinear motion (integral calc)
          • Worked example: motion problems (with definite integrals)
        • Solids with known cross sections
          • Volume with cross sections: semicircle
          • Volume with cross sections: triangle
        • Disc method
          • Disc method around x-axis
          • Generalizing disc method around x-axis
          • Disc method around y-axis
          • Disc method rotation around horizontal line
          • Disc method rotating around vertical line
          • Calculating integral disc around vertical line
        • Washer method
          • Solid of revolution between two functions (leading up to the washer method)
          • Generalizing the washer method
          • Washer method rotating around horizontal line (not x-axis), part 1
          • Washer method rotating around horizontal line (not x-axis), part 2
          • Washer method rotating around vertical line (not y-axis), part 1
          • Washer method rotating around vertical line (not y-axis), part 2
        • Shell method
          • Shell method for rotating around vertical line
          • Evaluating integral for shell method example
          • Shell method for rotating around horizontal line
          • Shell method with two functions of x
          • Calculating integral with shell method
          • Shell method with two functions of y
          • Part 2 of shell method with 2 functions of y
      • Series
        • Sequences review
          • Sequences intro
          • Worked example: sequence explicit formula
          • Worked example: sequence recursive formula
          • Geometric sequence review
          • Extending geometric sequences
          • Using explicit formulas of geometric sequences
          • Using recursive formulas of geometric sequences
        • Infinite sequences
          • Convergent and divergent sequences
          • Worked example: sequence convergence/divergence
          • Formal definition for limit of a sequence
          • Proving a sequence converges using the formal definition
        • Series review
          • Summation notation
          • Converting explicit series terms to summation notation
          • Converting explicit series terms to summation notation (n ≥ 2)
          • Arithmetic series formula
          • Worked example: arithmetic series (sigma notation)
          • Worked example: arithmetic series (sum expression)
          • Worked example: arithmetic series (recursive formula)
        • Finite geometric series
          • Geometric series intro
          • Geometric series with sigma notation
          • Worked example: finite geometric series (sigma notation)
          • Worked examples: finite geometric series
          • Finite geometric series formula justification
        • Partial sums
          • Partial sums intro
          • Partial sums: formula for nth term from partial sum
          • Infinite series as limit of partial sums
          • Partial sums: term value from partial sum
        • Infinite geometric series
          • Worked example: convergent geometric series
          • Worked example: divergent geometric series
          • Infinite geometric series formula intuition
          • Proof of infinite geometric series as a limit
          • Infinite geometric series word problem: bouncing ball
          • Infinite geometric series word problem: repeating decimal
          • Convergent & divergent geometric series (with manipulation)
        • Series basics challenge
          • Telescoping series
          • Divergent telescoping series
          • Sum of n squares (part 1)
          • Sum of n squares (part 2)
          • Sum of n squares (part 3)
          • Worked example: evaluating series (advanced)
        • Basic convergence tests
          • nth term divergence test
          • Integral test
          • Worked example: Integral test
          • Worked example: p-series
          • Proof of p-series convergence criteria
        • Comparison tests
          • Direct comparison test
          • Worked example: direct comparison test
          • Limit comparison test
          • Worked example: limit comparison test
          • Proof: harmonic series diverges
        • Ratio & alternating series tests
          • Ratio test
          • Alternating series test
          • Conditional & absolute convergence
          • Worked example: alternating series
        • Estimating infinite series
          • Series estimation with integrals
          • Worked example: Series estimation with integrals
          • Alternating series remainder
          • Worked example: alternating series remainder
        • Power series intro
          • Power series intro
          • Worked example: interval of convergence
          • Integrating power series
          • Differentiating power series
          • Interval of convergence for derivative and integral
        • Taylor & Maclaurin polynomials intro
          • Taylor & Maclaurin polynomials intro (part 1)
          • Taylor & Maclaurin polynomials intro (part 2)
          • Worked example: Maclaurin polynomial
          • Worked example: coefficient in Maclaurin polynomial
          • Worked example: coefficient in Taylor polynomial
          • Taylor polynomial remainder (part 1)
          • Taylor polynomial remainder (part 2)
          • Worked example: estimating sin(0.4) using Lagrange error bound
          • Worked example: estimating eˣ using Lagrange error bound
          • Visualizing Taylor polynomial approximations
          • Worked example: Taylor polynomial of derivative function
        • Maclaurin series of sin(x), cos(x), and eˣ
          • Maclaurin series of cos(x)
          • Maclaurin series of sin(x)
          • Maclaurin series of eˣ
          • Euler's formula & Euler's identity
          • Worked example: power series from cos(x)
          • Worked example: cosine function from power series
          • Worked example: recognizing function from Taylor series
          • Visualizing Taylor series approximations
        • Power series function representation
          • Function as a geometric series
          • Geometric series as a function
          • Geometric series interval of convergence
          • Power series of arctan(2x)
          • Power series of ln(1+x³)
          • Finding function from power series by integrating
        • Challenge series exercises
      • AP Calculus practice questions
        • AP Calculus AB questions
          • Sal interviews the AP Calculus Lead at College Board
          • 2015 AP Calculus AB/BC 1ab
          • 2015 AP Calculus AB/BC 1c
          • 2015 AP Calculus AB/BC 1d
          • 2015 AP Calculus AB 2a
          • 2015 AP Calculus AP 2b
          • 2015 AP Calculus 2c
          • 2015 AP Calculus AB/BC 3a
          • 2015 AP Calculus AB/BC 3b
          • 2015 AP Calculus AB/BC 3cd
          • 2015 AP Calculus AB/BC 4ab
          • 2015 AP Calculus AB/BC 4cd
          • 2015 AP Calculus AB 5a
          • 2015 AP Calculus AB 5b
          • 2015 AP Calculus AB 5c
          • 2015 AP Calculus AB 5d
          • 2015 AP Calculus AB 6a
          • 2015 AP Calculus AB 6b
          • 2015 AP Calculus AB 6c
          • 2011 Calculus AB free response #1a
          • 2011 Calculus AB Free Response #1 parts b c d
          • 2011 Calculus AB free response #2 (a & b)
          • 2011 Calculus AB free response #2 (c & d)
          • 2011 Calculus AB free response #3 (a & b)
          • 2011 Calculus AB free response #3 (c)
          • 2011 Calculus AB free response #4a
          • 2011 Calculus AB free response #4b
          • 2011 Calculus AB free response #4c
          • 2011 Calculus AB free response #4d
          • 2011 Calculus AB free response #5a
          • 2011 Calculus AB free response #5b
          • 2011 Calculus AB free response #5c.
          • 2011 Calculus AB free response #6a
          • 2011 Calculus AB free response #6b
          • 2011 Calculus AB free response #6c
        • AP Calculus BC questions
          • 2015 AP Calculus BC 2a
          • 2015 AP Calculus BC 2b
          • 2015 AP Calculus BC 2c
          • 2015 AP Calculus BC 2d
          • 2015 AP Calculus BC 5a
          • 2015 AP Calculus BC 5b
          • 2015 AP Calculus BC 5c
          • 2015 AP Calculus BC 5d
          • 2015 AP Calculus BC 6a
          • 2015 AP Calculus BC 6b
          • 2015 AP Calculus BC 6c
          • AP Calculus BC exams: 2008 1 a
          • AP Calculus BC exams: 2008 1 b&c
          • AP Calculus BC exams: 2008 1 c&d
          • AP Calculus BC exams: 2008 1 d
          • Calculus BC 2008 2 a
          • Calculus BC 2008 2 b &c
          • Calculus BC 2008 2d
          • 2011 Calculus BC free response #1a
          • 2011 Calculus BC free response #1 (b & c)
          • 2011 Calculus BC free response #1d
          • 2011 Calculus BC free response #3a
          • 2011 Calculus BC free response #3 (b & c)
          • 2011 Calculus BC free response #6a
          • 2011 Calculus BC free response #6b
          • 2011 Calculus BC free response #6c
          • 2011 Calculus BC free response #6d
    • Math for fun and glory
      • Doodling in Math and more
        • Spirals, Fibonacci and being a plant
          • Doodling in math: Spirals, Fibonacci, and being a plant [1 of 3]
          • Doodling in math: Spirals, Fibonacci, and being a plant [2 of 3]
          • Doodling in math: Spirals, Fibonacci, and being a plant [3 of 3]
          • Open letter to Nickelodeon, re: SpongeBob's pineapple under the sea
          • Angle-a-trons
        • Doodling in math
          • Doodling in math: Infinity elephants
          • Doodling in math: Stars
          • Doodling in math: Binary trees
          • Doodling in math: Sick number games
          • Doodling in math: Squiggle inception
          • Doodling in math: Connecting dots
          • Doodling in math: Triangle party
          • Doodling in math: Snakes and graphs
          • Doodling in math: Dragons
          • Doodling in math: Dragon dungeons
          • Doodling in math: Dragon scales
        • Hexaflexagons
          • Hexaflexagons
          • Hexaflexagons 2
          • Hexaflexagon safety guide
          • Flex Mex
        • About pi and tau
          • A song about a circle constant
          • Pi is (still) wrong
          • Rhapsody on the proof of pi = 4
          • Are Shakespeare's plays encoded within pi?
          • Anti-Pi Rant, 3/14/14
        • Singing (and noises)
          • Doodle music
          • Binary hand dance
          • What is up with noises? (The science and mathematics of sound, frequency, and pitch)
          • Folding space-time
        • Mobius strips
          • Math improv: Fruit by the foot
          • Mobius story: Wind and Mr. Ug
        • Thanksgiving math
          • Green bean matherole
          • Borromean onion rings
          • Optimal potatoes
          • Thanksgiving turduckenen-duckenen
        • Infinity . . .
          • 9.999... reasons that .999... = 1
          • How many kinds of infinity are there?
          • Proof some infinities are bigger than other infinities
        • Other cool stuff
          • What was up with Pythagoras?
          • Origami proof of the Pythagorean theorem
          • Wau: The most amazing, ancient, and singular number
          • Dialogue for 2
          • Fractal fractions
          • How to snakes
          • Re: Visual multiplication and 48/2(9+3)
          • The Gauss Christmath Special
          • Snowflakes, starflakes, and swirlflakes
          • Sphereflakes
          • Reel
          • How I Feel About Logarithms
      • Puzzles
        • Brain teasers
          • Cheryl's birthday
          • Heavier ball
          • Liar truth-teller brain teaser
          • Toggler brain teaser
          • Alien abduction brain teaser
          • Blue forehead room brain teaser
          • Blue forehead room solution
          • Forehead numbers brain teaser
          • Light bulb switching brain teaser
          • Path counting brain teaser
          • 3D path counting brain teaser
        • Transformation Puzzles
        • Lights Puzzles
      • AMC 10
        • 2013 AMC 10 A
          • 2013 AMC 10 A #21 / AMC 12 A #17
          • 2013 AMC 10 A #22 / AMC 12 A #18
          • 2013 AMC 10 A #23 / AMC 12 A #19
          • 2013 AMC 10 A #24
          • 2013 AMC 10 A #25
      • AIME
        • 2003 AIME
          • 2003 AIME II problem 1
          • 2003 AIME II problem 3
          • Sum of factors of 27000
          • Sum of factors 2
          • 2003 AIME II problem 4 (part 1)
          • 2003 AIME II problem 4 (part 2)
          • 2003 AIME II problem 5
          • 2003 AIME II problem 5 minor correction
          • Area circumradius formula proof
          • 2003 AIME II problem 6
          • 2003 AIME II problem 7
          • 2003 AIME II problem 8
          • Sum of polynomial roots (proof)
          • Sum of squares of polynomial roots
          • 2003 AIME II problem 9
          • 2003 AIME II problem 10
          • Trig challenge problem: area of a triangle
          • 2003 AIME II problem 12
          • 2003 AIME II problem 13
          • Trig challenge problem: area of a hexagon
          • 2003 AIME II problem 15 (part 1)
          • 2003 AIME II problem 15 (part 2)
          • 2003 AIME II problem 15 (part 3)
      • Math warmups
        • Expectation warmup
          • Problem of Points
          • Problem of Points 2
        • Random sampling warmup
          • Introduction to random sampling
          • Random sampling intuition
        • Distribution warmup
        • Arithmetic warmups
          • Exponentiation warmup
          • Intro to place value
          • Addition with carrying warmup
    • Class 8 (India)
      • Rational numbers
        • Introduction to rational numbers
          • Intro to rational & irrational numbers
      • Linear equations in one variable
        • Solving equations with linear expressions on one side and numbers on the other
        • Some applications
        • Solving linear equations with variables on both sides
          • Why we do the same thing to both sides: Variable on both sides
          • Intro to equations with variables on both sides
          • Equations with variables on both sides: 20-7x=6x-6
          • Equation with variables on both sides: fractions
        • Reducing equations to simpler form
          • Equations with parentheses
          • Ex 2: Distributive property to simplify
          • Ex 3: Distributive property to simplify
        • Some more applications
          • Equation with the variable in the denominator
          • Age word problem: Arman & Diya
          • Age word problem: Imran
          • Age word problem: Ben & William
        • Algebraic equations review
          • Variables, expressions, & equations
          • Testing solutions to equations
      • Understanding quadrilaterals
        • Polygons
          • Sum of interior angles of a polygon
        • Sum of the measures of the exterior angles of a polygon
          • Sum of the exterior angles of a polygon
        • Kinds of quadrilaterals
          • Intro to quadrilateral
          • Quadrilateral properties
          • Quadrilateral types
          • Classifying quadrilaterals
          • Kites as a geometric shape
        • Quadrilateral proofs & angles
          • Proof: Opposite sides of a parallelogram
          • Proof: Diagonals of a parallelogram
          • Proof: Opposite angles of a parallelogram
          • Proof: Rhombus diagonals are perpendicular bisectors
          • Proof: Rhombus area
          • Area of parallelogram proof
          • Whether a special quadrilateral can exist
          • Rhombus diagonals
      • Data handling
        • Organizing data
          • Frequency tables & dot plots
        • Grouping Data
          • Creating a histogram
          • Interpreting a histogram
          • Histograms
        • Circle Graphs or Pie Charts
          • Reading pie graphs (circle graphs)
        • Chance and Probability
          • Intro to theoretical probability
          • Simple probability: yellow marble
          • Simple probability: non-blue marble
          • Die rolling probability
          • Die rolling probability
          • Count outcomes using tree diagram
          • Probability with counting outcomes
          • Coin flipping probability
          • Free-throw probability
          • Three-pointer vs free-throw probability
      • Squares and square roots
        • Squares
        • Square roots
          • Intro to square roots
          • Understanding square roots
          • Simplifying square roots
          • Simplifying radicals
          • Square root of decimal
        • Estimating square roots
          • Approximating square roots to hundredths
      • Cubes and cubes roots
        • Cube roots
          • Intro to cube roots
          • Worked example: Cube root of a negative number
          • Simplifying a cube root
          • Cube root of a non-perfect cube
      • Algebraic expressions and identities
        • What are expressions?
          • Variables, expressions, & equations
        • Addition and subtraction of algebraic expressions
        • Multiplying a monomial by a monomial
          • Multiplying monomials to find area
          • Multiplying monomials to find area: two variables
        • Multiplying a monomial by a polynomial
          • Multiplying monomials by polynomials
        • Multiplying a polynomial by a polynomial
          • Multiplying binomials
          • Multiplying binomials by polynomials
        • Standard identities
          • More examples of special products
          • Special products of the form (ax+b)(ax-b)
          • Squaring binomials of the form (ax+b)²
          • Special products of binomials: two variables
      • Visualising solid shapes
        • Mapping space around us
          • Interpreting a scale drawing
          • Solving a scale drawing word problem
          • Making a scale drawing
      • Mensuration
        • Let us recall
        • Area of a trapezium
          • Area of trapezoids
        • Area of general quadrilaterals
          • Proof: Rhombus area
        • Area of polygons
          • Area of composite shapes
        • Surface area
          • Intro to nets of polyhedra
          • Surface area using a net: triangular prism
          • Surface area of a box (cuboid)
          • Surface area of a box using nets
          • Surface area using a net: rectangular prism
        • Volume of cube, cuboid, and cylinder
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
          • Volume of a rectangular prism: fractional dimensions
          • Volume of a rectangular prism
          • Volume word problem: water tank
          • Volume word problem: gold ring
          • Volume of triangular prism & cube
          • Volume of rectangles inside rectangles
          • Volume of a rectangular prism: word problem
          • Cylinder volume & surface area
      • Exponents and powers
        • Powers with negative exponents
          • Negative exponents
          • Negative exponent intuition
        • Laws of exponents
          • Exponent rules intro
          • Exponent rules part (composite problems)
          • Exponent properties with quotients
          • Worked example: Exponent properties
        • Use of Exponents to Express Small Numbers in Standard Form
          • Introduction to scientific notation
          • Scientific notation examples
          • Scientific notation examples
          • Scientific notation example: 0.0000000003457
      • Direct and inverse proportions
        • Direct and inverse proportions
          • Intro to direct & inverse variation
          • Recognizing direct & inverse variation
          • Proportionality constant for direct variation
          • Direct variation word problem: filling gas
          • Direct variation word problem: space travel
      • Factorization
        • Introduction to factorization
          • Intro to factors & divisibility
        • Factoring polynomials by taking common factors
          • Factoring with the distributive property
          • Factoring binomials: common factor
          • Factoring polynomials: common factor
          • Factoring polynomials: common factor area model
          • Factoring polynomials: common binomial factor
        • Factoring quadratics 1
          • Factoring quadratics as (x+a)(x+b)
          • Factoring quadratics as (x+a)(x+b) (example 2)
          • More examples of factoring quadratics as (x+a)(x+b)
        • Factoring quadratics using identities: Difference of squares
          • Factoring difference of squares: leading coefficient ≠ 1
          • Factoring difference of squares: analyzing factorization
          • Factoring difference of squares: missing values
          • Factoring difference of squares: shared factors
        • Factoring quadratics using identities: Perfect squares
          • Factoring perfect squares
          • Identifying perfect square form
          • Factoring perfect squares: common factor
          • Factoring perfect squares: negative common factor
          • Factoring perfect squares: missing values
          • Factoring perfect squares: shared factors
        • Factoring quadratics by grouping
          • Intro to grouping
          • Factoring quadratics by grouping
      • Introduction to graphs
        • Bar graphs and histograms review
          • Creating a bar graph
          • Reading bar graphs
          • Interpreting bar graphs: alligators
          • Histograms
          • Reading pie graphs (circle graphs)
        • Linear graphs
          • Reading line graphs
        • Intro to the coordinate plane
          • Coordinate plane: graphing points
        • Coordinate plane word problems
          • Coordinate plane graphing word problem
        • Dependent & independent variables
          • Dependent & independent variables
          • Dependent & independent variables: graphing
          • Dependent & independent variables: equation
        • Some applications
          • Linear graphs word problems
          • Linear graphs word problem: cats
          • Linear equations word problems: volcano
          • Linear equations word problems: earnings
          • Modeling with linear equations: snow
      • Playing with numbers
        • Numbers in general form
          • Writing a number in expanded form
          • Writing a number in standard form
          • Regrouping numbers into various place values
        • Tests of divisibility
          • Divisibility tests for 2, 3, 4, 5, 6, 9, 10
          • Recognizing divisibility
          • The why of the 3 divisibility rule
          • The why of the 9 divisibility rule
          • Common divisibility examples
    • Math - Indian curriculum
    • 1ère primaire
      • Nombres
        • Plus grand ….plus petit… comparaison de quantités
          • Comparing numbers of objects
          • Comparing numbers on the number line
        • Les symboles égal ou pas égal
        • L'ordinalité
          • Counting in order
        • Dénombrer la quantité 1 à 5
          • Counting with small numbers
          • Counting in order
          • Counting in pictures
          • Counting objects 1
          • Counting by category
        • Additions et soustractions de 1 à 5
          • Intro to addition
          • Relating addition and subtraction
          • Intro to subtraction
        • Décomposer et recomposer les nombres de 1à 5 (ex : 5 c'est 1+1+3 ou 5 c'est 2 +2+1)
          • Making 5
        • Reconnaître et ordonner des quantités jusque 10
          • Counting with small numbers
          • Counting objects 1
          • Counting objects 2
        • Additionner et soustraire jusqu'à 10
          • Getting to 10 by filling boxes
          • Add and subtract: pieces of fruit
          • Addition and subtraction within 10
          • Adding to 10
        • L'arbre de 10
          • Getting to 10 by filling boxes
        • Découverte de la multiplication
          • More ways to multiply
          • Ways to represent multiplication
        • Décomposition additive de nombre plus grands que 10
          • Adding 7 + 6
          • Adding 8 + 7
          • Teen numbers: monkeys
        • La découverte de la division
          • Intro to division
        • Ordonner des nombres jusque 20
          • Teens as sums with 10
        • Additionner soustraire jusque 20
          • Subtracting 14 - 6
          • Adding 5 + 3 + 6
          • Comparison word problems: roly-polies
      • Solides et figures
        • Situer ma position
        • Reconnaitre des figures
          • Cousin Fal's shape collection
          • Recognizing shapes
      • Grandeurs
        • Comparaison : plus petit, plus grand
          • Ordering by length
      • Traitement de données
        • Le tableau à double entrée
          • Picture graphs
        • Opérations habillées sur l'addition et la soustraction
          • Addition word problems within 10
          • Subtraction word problems within 10
          • Addition and subtraction word problems: superheroes
          • Addition and subtraction word problems: gorillas
          • Comparison word problems: marbles
          • Repeated addition: haircuts
    • 2ème primaire
      • Nombres
        • Révisions additions
          • Adding 1 vs. adding 10
        • Additions de dizaines (D+D)
          • Understanding place value when adding tens
        • Comptage par 10
        • Les tables de multiplication de 2 à 9
          • Multiplication tables for 2-9
        • Valeur et position des chiffres dans un nombre à 2 chiffres
          • Intro to place value
          • Number grid
        • Les nombres 25 et 42
          • Place value example: 25
          • Place value example: 42
        • D + U = DU U +D = DU (sans passage de la dizaine)
          • Adding 2-digit numbers without regrouping 2
        • D-U (sans passage de la dizaine)
          • Subtracting 1s using place value
        • Du +Du =D
        • DU-DU ou U (avec passage)
          • Subtracting a 1-digit number with regrouping
        • DU+U/U+DU/DU+U
        • DU + DU(avec passage)
          • Breaking apart 2-digit addition problems
          • Regrouping to add 1-digit number
          • Adding 53+17 by making a group of 10
          • Strategies for adding 2-digit numbers
          • Addition and subtraction with number lines
          • Adding 2-digit numbers without regrouping 2
          • Adding with regrouping
        • DU-DU = D, DU-DU=DU(avec passage)
          • Subtracting 1 vs. subtracting 10
          • Subtracting 10s using place value
          • Subtracting 2-digit numbers without regrouping 1
          • Subtracting with regrouping (borrowing)
        • Soustraire avec 2 nombres ou plus
          • Subtracting two-digit numbers without regrouping (example 2)
        • Introduction à la multiplication
          • Intro to multiplication
          • Multiplication as groups of objects
          • Basic multiplication
        • Table de 2 de division
        • Table de division
        • Initiation aux fractions
        • Divisions avec reste (ex : 37 : 5 = 7 reste 2)
          • Intro to division
        • Classement des nombres jusque 100
          • Missing numbers between 0 and 120
      • Solides et figures
        • Reconnaître et comparer des solides
          • Cousin Fal's shape collection
          • Recognizing shapes
      • Traitement de données
        • Opérations habillées pour l'addition
          • Adding and subtracting on number line word problems
          • Addition word problem: horses
          • Addition word problem: spots
        • Opérations habillées pour la soustraction
          • Adding and subtracting on number line word problems
          • Subtraction word problem: snow
          • Subtraction word problem: tennis balls
          • Subtraction word problem: crayons
          • Subtraction word problem: basketball
        • Additions ET soustractions habillées
          • Addition word problem: starfish
    • 3ème primaire
      • Nombres
        • Révisions signe =
          • Equal sign
        • Ecrire et lire les nombres de 0 à 1000
          • Comparing whole numbers
          • Regrouping whole numbers: 675
          • Regrouping whole numbers: 76,830
          • Regrouping whole numbers: 430
        • Les classer dans un abaque (CDU et MCDU)
          • Comparing whole number place values
        • Situer des nombres de 0 à 100 sur une droite graduée
        • Décomposition des nombres 100 et 1000
        • Multiplier et diviser par 10, 100 et 1000
          • Multiplying 1-digit numbers by 10, 100, and 1000
          • Place value when multiplying and dividing by 10
        • Addition écrite sans et avec report (nombres à deux et trois chiffres)
          • Adding 10 or 100
          • Adding 1s, 10s, and 100s
          • Adding 3-digit numbers (no regrouping)
          • Subtracting 3-digit numbers (no regrouping)
          • Adding 3-digit numbers
          • Adding with regrouping
          • Adding 3-digit numbers with regrouping
          • Adding whole numbers by their place values
        • Soustraction écrite (avec emprunt/compensation, nombres à deux et trois chiffres)
          • Subtracting 1, 10, or 100
          • Subtracting 1s, 10s, and 100s
          • Worked example: Subtracting 3-digit numbers (regrouping)
          • Worked example: Subtracting 3-digit numbers (regrouping twice)
          • Worked example: Subtracting 3-digit numbers (regrouping from 0)
          • Missing number for addition and subtraction within 1000
        • Multiplication écrite sans/avec une retenue (multiplicateur à un chiffre)
          • Multiplying 2-digit by 1-digit
          • Multiplying 3-digit by 1-digit
          • Multiplying: 2-digit by 1-digit (regrouping)
          • Multiplying 3-digit by 1-digit (regrouping)
          • Multiplying 4-digit by 1-digit (regrouping)
          • Multiplication word problem: parking lot
        • Division écrite (diviseur à un chiffre)
          • Intro to division
        • Calcul mental : addition et soustraction avec compensation
          • Skip-counting by 5s
          • Missing numbers in addition and subtraction
          • Counting by 10s
          • Skip-counting by 100s
          • Breaking apart 3-digit addition problems
          • Solving 3-digit addition in your head
          • Addition using groups of 10 and 100
        • Calcul mental : addition et soustraction avec décomposition
          • Breaking apart 3-digit addition problems
          • Solving 3-digit addition in your head
          • Adding and subtracting on number line 2
          • Subtracting in your head (no regrouping)
          • Alternate mental subtraction method
      • Solides et figures
        • Droite, ½ droite, segment
          • Terms & labels in geometry
          • Lines, line segments, & rays
        • Droites parallèles, sécantes, perpendiculaires
          • Parallel & perpendicular lines intro
          • Parallel & perpendicular lines
        • Tracer et mesurer des segments
          • Drawing parallel line segments
          • Measuring Line Segments
          • Congruent line segments
        • Notion d’angle ; angles droits, aigus, obtus
          • Angles: introduction
          • Naming angles
          • Measuring angles in degrees
          • Acute, right, & obtuse angles
          • Recognizing angles
          • Identifying an angle
        • Tracer des angles
          • Drawing acute, right and obtuse angles
        • Connaître le vocabulaire (faces, arêtes, sommets) des solides
        • Classement des triangles selon la longueur des côtés et les amplitudes d’angles
          • Classifying triangles
          • Classifying triangles by angles
          • Worked example: Classifying triangles
        • Distinguer les polygones des non polygones
          • Cousin Fal's shape collection
          • Classifying shapes by line and angles types
          • Classifying shapes by lines and angles
          • Recognizing shapes
      • Grandeurs
        • Lire les heures et les ½ heures
          • Telling time (labeled clock)
          • Telling time (unlabeled clock)
        • Mesurer des durées (heures et minutes)
        • Mesurer des longueurs et conversion via abaque (cm, dm, m, dam, hm, km)
          • Ordering by length
          • Measuring lengths 2
          • Measuring lengths with different units
          • Estimating lengths
          • Length word problems
        • Mesurer des capacités (litre et fractions simples - ½, ¼-du litre, dl, cl)
        • Mesurer des masses (kg, ½ kg, ¼ kg, g, t)
        • Proportionnalité directe simple (multiplier les ingrédients dans une recette ou calculer l’échelle d’un plan ; x 4, : 4, p ex)
          • Intro to ratios
          • Part:whole ratios
          • Solving ratio problems with tables
          • Scale drawing: centimeters to kilometers
          • Part:whole ratio word problem
      • Traitement de données
        • Dans un problème : sélectionner les informations utiles et organiser ces informations
          • Multiplication word problem: parking lot
          • Division word problem: school building
        • Manipuler les € dans des achats
          • Counting dollars
        • Lecture et construction de graphiques (températures, par exemple)
          • Making picture graphs and line plots
        • Lecture d’un tableau à double entrée (programmes TV, par exemple)
          • Picture graphs
    • 4ème primaire
      • Nombres
        • Ecrire et lire les nombres jusqu’à 100000
          • Regrouping numbers into various place values
          • Regrouping whole number place values
          • Adding whole numbers by their place values
          • Regrouping whole numbers: 675
          • Regrouping whole numbers: 76,830
          • Finding place value
          • Writing a number in standard form
          • Writing a number in expanded form
          • Comparing place values
          • Understanding place value
          • Creating the largest number
          • Rounding whole numbers to nearest ten
          • Rounding whole numbers to nearest hundred
          • Rounding whole numbers to nearest thousand
        • Nombres décimaux (jusqu’à deux chiffres après la virgule)
          • Intro to decimals
          • Write decimal in expanded form
          • Writing decimal numbers shown in grids
        • Ecriture décimale des nombres fractionnaires dont le dénominateur est 10 ou 100
          • Writing a number as a fraction and decimal
          • Rewriting decimals as fractions: 0.15
          • Rewriting decimals as fractions: 0.8
          • Rewriting decimals as fractions: 0.36
          • Converting decimals to fractions 2 (ex 2)
        • Placer des fractions ou des nombres décimaux sur une droite graduée
          • Graphing tenths from 0 to 1
          • Identifying tenths on a number line
          • Plotting decimal numbers on a number line
          • Graphing hundredths from 0 to 0.1
          • Identifying hundredths on a number line
          • Fractions on a number line
          • Fractions on number line widget
        • Diviseurs et multiples
        • Critères de divisibilité par 2, 4, 5, 10, 25 et 100
          • Divisibility tests for 2, 3, 4, 5, 6, 9, 10
          • Recognizing divisibility
          • The why of the 3 divisibility rule
          • The why of the 9 divisibility rule
          • Finding factors of a number
          • Finding factors and multiples
        • Addition, soustraction écrites avec des nombres jusqu’à 100000
          • Multi-digit addition with regrouping
          • Multi-digit subtraction with regrouping: 6798-3359
          • Multi-digit subtraction with regrouping: 7329-6278
          • Multi-digit subtraction with regrouping twice
        • Multiplication, division écrites avec deuxième facteur à 2 ou 3 chiffres
          • Multiplying 2-digit numbers
          • Multiplying 2-digit by 2-digit: 36x23
          • Multiplying 2-digit by 2-digit: 23x44
          • Multiplying multi-digit numbers
          • Comparing with multiplication: basics
        • Calcul mental : multiplication avec décomposition
          • Multiplying 10s
          • Multiplying by multiples of 10
          • Multiplying by tens word problem
          • Multiplying 1-digit numbers by multiples of 10, 100, and 1000
          • Multiplying with distributive property
          • Multiplying with area model: 6 x 7981
          • Multiplying with area model: 78 x 65
          • Multiplying with area model: 16 x 27
        • Commutativité, associativité, distributivité
          • Properties of multiplication
          • Properties and patterns for multiplication
          • Commutative law of addition
          • Commutative law of multiplication
          • Associative law of addition
          • Associative law of multiplication
          • Distributive property over addition
          • Distributive property over subtraction
      • Solides et figures
        • Caractéristiques des quadrilatères (carré, rectangle, losange, parallélogramme et trapèze)
          • Quadrilateral types
          • Classifying quadrilaterals
          • Kites as a geometric shape
        • Se repérer, se déplacer sur un quadrillage : introduction des coordonnées
          • Coordinate plane: graphing points
          • Coordinate plane graphing word problem
        • Symétrie orthogonale : l’axe de symétrie d’une figure
          • Intro to reflective symmetry
          • Identifying symmetrical figures
      • Grandeurs
        • Mesurer des durées (heures, minutes et introduction de la seconde)
        • Utilisation du rapporteur
          • Measuring angles in degrees
          • Measuring angles using a protractor
          • Measuring angles using a protractor 2
          • Constructing angles
        • Aire d'un triangle
        • Proportionnalité directe simple : la règle de 3
          • Intro to proportional relationships
          • Proportional relationships: bananas
          • Proportional relationships: movie tickets
        • Fractions inférieures/supérieures à 1
          • Identifying numerators and denominators
          • Whole numbers as fractions
        • Fraction d’un nombre (Ex : ½ de 8) – multiplier une fraction par un nombre
          • Understanding fractions as division
          • Creating a fraction through division
        • Fractions équivalentes (suite) : passer d’une fraction réductible à son équivalent irréductible
          • Intro to equivalent fractions
          • Visualizing equivalent fractions
          • More on equivalent fractions
          • Equivalent fractions
          • Equivalent fractions and different wholes
        • Représentations semi-abstraites des fractions (bandelettes, disque,…)
          • Intro to fractions
          • More about fractions
          • Cutting shapes into equal parts
          • Identifying unit fractions word problem
          • More than one equal section
          • Recognizing fractions greater than 1
          • Representing 1 as a fraction
          • Whole numbers as fractions
          • Equivalent fractions with visuals
          • Equivalent fraction models
          • Equivalent fraction visually
          • Creating equivalent fractions
          • Intro to equivalent fractions
          • Halves and fourths
          • Equal parts of circles and rectangles
        • Addition et soustraction de deux fractions ayant même dénominateur
          • Adding fractions (denominators 10 & 100)
        • Introduction aux pourcentages (représentation sur des quadrillages 10 x 10, relation avec fraction ayant un dénominateur égal à 100, calcul de ristournes)
          • The meaning of percent
          • Meaning of 109%
          • Finding a percent
          • Percent of a whole number
          • Identifying percent amount and base
      • Traitement de données
        • Introduction aux histogrammes et tableaux à double entrée et graphes
          • Reading bar graphs: dog bones
          • Reading bar graphs: bikes
          • Creating picture and bar graphs 1
          • Intro to line plots
          • Reading pictographs
          • Creating a bar graph
          • Interpreting bar graphs: alligators
          • Reading bar graphs: Harry Potter
          • Making picture graphs and line plots
          • Reading bar graphs
    • 5ème primaire
      • Nombres
        • Ecriture décimale des nombres fractionnaires
          • Convert decimal from expanded form to standard form
          • Decimal place value
        • Décomposition d’un nombre en produits de facteurs premiers
          • Finding factors of a number
          • Finding factors and multiples
        • Critères de divisibilité par 3, 9, 8, 125, 1000
          • Divisibility tests for 2, 3, 4, 5, 6, 9, 10
          • Recognizing divisibility
          • The why of the 3 divisibility rule
          • The why of the 9 divisibility rule
          • Finding factors of a number
          • Finding factors and multiples
        • Comparaison de nombres décimaux
          • Comparing decimal numbers on a number line
          • Worked example: Rounding decimals to nearest tenth
          • Comparing decimals 2
          • Comparing decimals 3
          • Comparing decimals: 9.97 and 9.798
          • Comparing decimals: 156.378 and 156.348
        • Addition, soustraction, multiplication et division écrites avec des nombres décimaux
          • Fraction to decimal: 11/25
          • Worked example: Converting a fraction (7/8) to a decimal
      • Solides et figures
        • Caractéristiques de l’octogone
      • Grandeurs
        • Comparaison de fractions (même numérateur/même dénominateur)
          • Comparing fractions with > and < symbols
          • Comparing fractions with like numerators and denominators
      • Traitement de données
        • Introduction aux graphiques circulaires
          • Reading pie graphs (circle graphs)
    • 6ème primaire
      • Nombres
        • Priorité des opérations
          • Intro to order of operations
          • Order of operations example
          • Worked example: Order of operations (PEMDAS)
        • Le PGCD et le PPCM
          • Distributive property exercise examples
        • Introduction à la notion de division
          • The idea of division
          • Intro to division
          • Ways to represent division
        • Quotients multiples de 10
          • Quotients that are multiples of 10
          • Canceling zeros when dividing
          • Division using place value
          • Division with area models
          • Intro to long division (no remainders)
          • Long division: 280÷5
          • Intro to long division (remainders)
        • Introduction à la division avec reste non nul
          • Intro to remainders
          • Worked example: Long division with remainders: 3771÷8
          • Worked example: Long division with remainders: 2292÷4
          • Dividing by 2-digits: 6250÷25
          • Dividing by 2-digits: 9815÷65
          • Dividing by 2-digits: 7182÷42
        • Introduction aux longues divisions
        • Problèmes avec des multiplications ou des divisions
          • Multiplication word problem: carrots
          • Division word problem: row boat
          • Multiplication word problem: pizza
          • Division word problem: field goals
          • Multiplication, division word problem: pedaling
        • Addition des nombres décimaux
          • Adding decimals: 9.087+15.31
          • Adding decimals: 0.822+5.65
          • Adding three decimals
        • Soustraire des nombres décimaux
          • Subtracting decimals: 9.57-8.09
          • Subtracting decimals: 10.1-3.93
        • Soustraire des nombres décimaux-Exemple 2
          • Subtracting decimals: 9.005 - 3.6
          • Subtracting decimals: 39.1 - 0.794
        • Additionner ou soustraire des nombres décimaux dans un exercice concret
          • Adding decimals word problem
          • Adding & subtracting decimals word problem
        • Multiplier un nombre entier par un nombre décimal
          • Intro to multiplying decimals
          • Multiplying decimals: place value
        • Poser une multiplication de nombres décimaux
          • Multiplying challenging decimals
        • Pousser la division après la virgule
          • Dividing whole numbers to get a decimal
        • Exercices : Diviser des nombres décimaux
      • Grandeurs
        • Première approche de la notion d'aire
          • Intro to area and unit squares
          • More ways to multiply
          • Measuring rectangles with different unit squares
          • Measuring area with partial unit squares
          • Creating rectangles with a given area 1
          • Creating rectangles with a given area 2
        • Le périmètre
          • Perimeter: introduction
          • Perimeter of a shape
          • Finding perimeter when a side length is missing
          • Finding missing side length when given perimeter
        • L'aire du rectangle
          • Counting unit squares to find area formula
          • Transitioning from unit squares to area formula
          • Finding missing side when given area
          • Comparing areas of plots of land
        • Calculs d'aires et distributivité de la multiplication sur l'addition
          • Area and the distributive property
          • Area word problem: house size
        • Aire et périmètre
          • Comparing areas and perimeters of rectangles
          • Area & perimeter word problem: table
          • Perimeter & area
          • Perimeter and unit conversion
          • Applying the metric system to perimeter
          • Area & perimeter word problem: dog pen
        • Aire d'un triangle
          • Area of a triangle
          • Area of triangle proof
          • Area of equilateral triangle
        • Rayon, diamètre et circonférence d'un cercle
          • Radius, diameter, circumference & π
          • Labeling parts of a circle
          • Area of a circle
        • L'aire d'un étrange quadrilatère
          • Area of a quadrilateral on a grid
        • Volume, aire, longueur: comment mesure-t-on ?
          • Volume intro
          • Measuring volume with unit cubes
          • Measuring volume as area times length
          • Volume of a rectangular prism
          • Volume word problem: water tank
        • Des volumes et des fractions
          • Volume with fractional cubes
          • Volume of a rectangular prism: fractional dimensions
          • Volume of a rectangular prism: word problem
      • Traitement de données
        • Intérêt, taux, capital
          • Solving unit rate problem
          • Intro to rates
          • Solving unit price problem
        • Les différents types de diagrammes - révision
  • Science
    • Physics
      • One-dimensional motion
        • Skill check for one-dimensional motion
        • Introduction to physics
          • Introduction to physics
        • Displacement, velocity, and time
          • Intro to vectors & scalars
          • Introduction to reference frames
          • Calculating average velocity or speed
          • Solving for time
          • Displacement from time and velocity example
          • Instantaneous speed and velocity
          • Position vs. time graphs
        • Acceleration
          • Acceleration
          • Airbus A380 take-off time
          • Airbus A380 take-off distance
          • Why distance is area under velocity-time line
          • Acceleration vs. time graphs
        • Kinematic formulas and projectile motion
          • Average velocity for constant acceleration
          • Acceleration of aircraft carrier take-off
          • Airbus A380 take-off distance
          • Deriving displacement as a function of time, acceleration, and initial velocity
          • Plotting projectile displacement, acceleration, and velocity
          • Projectile height given time
          • Deriving max projectile displacement given time
          • Impact velocity from given height
          • Viewing g as the value of Earth's gravitational field near the surface
          • Choosing kinematic equations
        • Old videos on projectile motion
          • Projectile motion (part 1)
          • Projectile motion (part 2)
          • Projectile motion (part 3)
          • Projectile motion (part 4)
          • Projectile motion (part 5)
      • Two-dimensional motion
        • Skill check for two-dimensional motion
        • Two-dimensional projectile motion
          • Horizontally launched projectile
          • Visualizing vectors in 2 dimensions
          • Projectile at an angle
          • Launching and landing on different elevations
          • Total final velocity for projectile
          • Correction to total final velocity for projectile
          • Projectile on an incline
          • Unit vectors and engineering notation
          • Unit vector notation
          • Unit vector notation (part 2)
          • Projectile motion with ordered set notation
        • Optimal angle for a projectile
          • Optimal angle for a projectile part 1: Components of initial velocity
          • Optimal angle for a projectile part 2: Hangtime
          • Optimal angle for a projectile part 3: Horizontal distance as a function of angle (and speed)
          • Optimal angle for a projectile part 4: Finding the optimal angle and distance with a bit of calculus
      • Forces and Newton's laws of motion
        • Skill check for Forces and Newton's Laws of motion
        • Newton's laws of motion
          • Newton's first law of motion
          • Newton's first law of motion concepts
          • More on Newton's first law of motion
          • Newton's second law of motion
          • More on Newton's second law
          • Newton's third law of motion
          • More on Newton's third law
        • Normal force and contact force
          • Normal force and contact force
          • Normal force in an elevator
          • More on Normal force (shoe on floor)
          • More on Normal force (shoe on wall)
        • Balanced and unbalanced forces
          • Balanced and unbalanced forces
          • Unbalanced forces and motion
        • Slow sock on Lubricon VI
          • Slow sock on Lubricon VI
          • Normal forces on Lubricon VI
        • Inclined planes and friction
          • Inclined plane force components
          • Ice accelerating down an incline
          • Force of friction keeping the block stationary
          • Correction to force of friction keeping the block stationary
          • Force of friction keeping velocity constant
          • Intuition on static and kinetic friction comparisons
          • Static and kinetic friction example
        • Tension
          • The force of tension
          • Mild and medium tension
          • Super hot tension
          • Introduction to tension
          • Introduction to tension (part 2)
          • Tension in an accelerating system and pie in the face
        • Treating systems
          • Treating systems (the hard way)
          • Treating systems (the easy way)
          • Two masses hanging from a pulley
          • Three box system problem
          • Masses on incline system problem
      • Centripetal force and gravitation
        • Skill check for centripetal force and gravitation
        • Circular motion and centripetal acceleration
          • Race cars with constant speed around curve
          • Centripetal force and acceleration intuition
          • Visual understanding of centripetal acceleration formula
          • Optimal turns at Indianapolis Motor Speedway with JR Hildebrand
          • Calculus proof of centripetal acceleration formula
          • Loop de loop question
          • Loop de loop answer part 1
          • Loop de loop answer part 2
        • Centripetal forces
          • Centripetal force problem solving
          • Yo-yo in vertical circle example
          • Bowling ball in vertical loop
          • Mass swinging in a horizontal circle
        • Newton's law of gravitation
          • Introduction to gravity
          • Mass and weight clarification
          • Gravity for astronauts in orbit
          • Would a brick or feather fall faster?
          • Acceleration due to gravity at the space station
          • Space station speed in orbit
          • Introduction to Newton's law of gravitation
          • Gravitation (part 2)
      • Work and energy
        • Skill check for work and energy
        • Work and energy
          • Introduction to work and energy
          • Work and energy (part 2)
          • Conservation of energy
          • Work and the work-energy principle
          • Work as the transfer of energy
          • Work example problems
          • Work as area under curve
          • Thermal energy from friction
          • Work/energy problem with friction
          • Conservative forces
          • Power
        • Springs and Hooke's law
          • Intro to springs and Hooke's law
          • Potential energy stored in a spring
          • Spring potential energy example (mistake in math)
          • LOL diagrams
          • Vertical springs and energy conservation
        • Mechanical advantage
          • Introduction to mechanical advantage
          • Mechanical advantage (part 2)
          • Mechanical advantage (part 3)
      • Impacts and linear momentum
        • Skill check for Impacts and linear momentum
        • Momentum and Impulse
          • Introduction to momentum
          • Impulse and momentum dodgeball example
          • Bouncing fruit collision example
          • Momentum: Ice skater throws a ball
          • 2-dimensional momentum problem
          • 2-dimensional momentum problem (part 2)
          • Force vs. time graphs
        • Elastic and inelastic collisions
          • Elastic and inelastic collisions
          • Solving elastic collision problems the hard way
          • Deriving the shortcut to solve elastic collision problems
          • How to use the shortcut for solving elastic collisions
        • Center of mass
          • Center of mass
          • Equation for center of mass
      • Torque and angular momentum
        • Skill check: Moments, torque, and angular momentum
        • Rotational kinematics
          • Angular motion variables
          • Relating angular and regular motion variables
          • Relationship between angular velocity and speed
          • Rotational kinematic formulas
        • Torque, moments, and angular momentum
          • Introduction to torque
          • Moments
          • Moments (part 2)
          • Finding torque for angled forces
          • Rotational version of Newton's second law
          • More on moment of inertia
          • Rotational kinetic energy
          • Rolling without slipping problems
          • Angular momentum
          • Constant angular momentum when no net torque
          • Angular momentum of an extended object
          • Ball hits rod angular momentum example
          • Cross product and torque
      • Oscillations and mechanical waves
        • Skill check: Oscillations and mechanical waves
        • Simple harmonic motion
          • Intuition about simple harmonic oscillators
          • Definition of Amplitude and Period
          • Equation for simple harmonic oscillators
          • Period dependence for mass on spring
          • Phase constant
          • Pendulums
        • Simple harmonic motion (with calculus)
          • Introduction to harmonic motion
          • Harmonic motion part 2 (calculus)
          • Harmonic motion part 3 (no calculus)
        • Introduction to mechanical waves
          • Introduction to waves
          • Properties of periodic waves
          • The equation of a wave
        • Sound
          • Production of sound
          • Sound Properties (Amplitude, Period, Frequency, Wavelength)
          • Speed of Sound
          • Relative speed of sound in solids, liquids, and gases
          • Mach numbers
          • Decibel Scale
          • Why do sounds get softer?
          • Ultrasound medical imaging
        • The Doppler effect
          • Doppler effect introduction
          • Doppler effect formula for observed frequency
          • Doppler effect formula when source is moving away
          • When the source and the wave move at the same velocity
          • Doppler effect for a moving observer
          • Doppler effect: reflection off a moving object
        • Wave interference
          • Wave interference
          • Constructive and Destructive interference
          • Standing waves on strings
          • Standing waves in open tubes
          • Standing waves in closed tubes
          • Beat frequency
          • Derivation of beat frequency formula
      • Fluids
        • Skill check for fluids
        • Density and Pressure
          • Specific gravity
          • Pressure and Pascal's principle (part 1)
          • Pressure and Pascal's principle (part 2)
          • Pressure at a depth in a fluid
          • Finding height of fluid in a barometer
        • Buoyant Force and Archimedes' Principle
          • Archimedes principle and buoyant force
          • Buoyant force example problems
        • Fluid Dynamics
          • Volume flow rate and equation of continuity
          • Bernoulli's equation (part 1)
          • Bernoulli's equation (part 2)
          • Bernoulli's equation (part 3)
          • Bernoulli's equation (part 4)
          • Bernoulli's example problem
          • Viscosity and Poiseuille flow
          • Turbulence at high velocities and Reynold's number
          • Venturi effect and Pitot tubes
          • Surface Tension and Adhesion
      • Thermodynamics
        • Skill check for thermodynamics
        • Temperature, kinetic theory, and the ideal gas law
          • Thermodynamics part 1: Molecular theory of gases
          • Thermodynamics part 2: Ideal gas law
          • Thermodynamics part 3: Kelvin scale and Ideal gas law example
          • Thermodynamics part 4: Moles and the ideal gas law
          • Thermodynamics part 5: Molar ideal gas law problem
          • Maxwell Boltzmann distribution
        • Specific heat and heat transfer
          • Specific heat and latent heat of fusion and vaporization
          • Thermal conduction, convection, and radiation
          • Thermal conduction
          • Thermal conductivity of metal and wood
          • Intuition behind formula for thermal conductivity
        • Laws of thermodynamics
          • Macrostates and microstates
          • Quasistatic and reversible processes
          • First law of thermodynamics / internal energy
          • More on internal energy
          • Work from expansion
          • PV-diagrams and expansion work
          • Proof: U = (3/2)PV or U = (3/2)nRT
          • Work done by isothermic process
          • Carnot cycle and Carnot engine
          • Proof: Volume ratios in a Carnot cycle
          • Proof: S (or entropy) is a valid state variable
          • Thermodynamic entropy definition clarification
          • Reconciling thermodynamic and state definitions of entropy
          • Entropy intuition
          • Maxwell's demon
          • More on entropy
          • Efficiency of a Carnot engine
          • Carnot efficiency 2: Reversing the cycle
          • Carnot efficiency 3: Proving that it is the most efficient
      • Electric charge, field, and potential
        • Skill check: Electric charge, field, and electric potential
        • Charge and electric force (Coulomb's law)
          • Triboelectric effect and charge
          • Coulomb's Law
          • Conductors and Insulators
          • Conservation of Charge
        • Electric field
          • Electric field definition
          • Electric field direction
          • Magnitude of electric field created by a charge
          • Net electric field from multiple charges in 1D
          • Net electric field from multiple charges in 2D
          • Electric field
          • Proof: Field from infinite plate (part 1)
          • Proof: Field from infinite plate (part 2)
        • Electric potential energy, electric potential, and voltage
          • Electric potential energy
          • Electric potential energy (part 2-- involves calculus)
          • Voltage
          • Electric potential energy of charges
          • Electric potential at a point in space
          • Electric potential charge configuration
      • Circuits
        • Skill Check: Circuits
        • Ohm's law and circuits with resistors
          • Introduction to circuits and Ohm's law
          • Resistors in series
          • Resistors in parallel
          • Example: Analyzing a more complex resistor circuit
          • Analyzing a resistor circuit with two batteries
          • Resistivity and conductivity
          • Kirchhoff's current law
          • Kirchhoff's voltage law
          • Electric power
          • Voltmeters and Ammeters
          • Electrolytic conductivity
        • Circuits with capacitors
          • Capacitors and capacitance
          • Capacitance
          • Energy of a capacitor
          • Capacitors in series
          • Capacitors in parallel
          • Dielectrics in capacitors
      • Magnetic forces, magnetic fields, and Faraday's law
        • Skill check for Magnetic fields, forces, and Faraday's law
        • Magnets and Magnetic Force
          • Introduction to magnetism
          • Magnetic force on a charge
          • Cross product 1
          • Cross product 2
          • Magnetic force on a proton example (part 1)
          • Magnetic force on a proton example (part 2)
          • Magnetic force on a current carrying wire
        • Magnetic field created by a current
          • Magnetic field created by a current carrying wire
          • Magnetic force between two currents going in the same direction
          • Magnetic force between two currents going in opposite directions
          • Induced current in a wire
        • Electric motors
          • Electric motors (part 1)
          • Electric motors (part 2)
          • Electric motors (part 3)
          • The dot product
          • Dot vs. cross product
          • Calculating dot and cross products with unit vector notation
        • Magnetic flux and Faraday's law
          • Flux and magnetic flux
          • Faraday's Law Introduction
          • Lenz's Law
          • Faraday's Law example
          • Emf induced in rod traveling through magnetic field
          • Faraday's Law for generating electricity
      • Electromagnetic waves and interference
        • Skill Check: Electromagnetic waves and interference
        • Introduction to electromagnetic waves
          • Electromagnetic waves and the electromagnetic spectrum
          • Polarization of light, linear and circular
        • Interference of electromagnetic waves
          • Constructive and Destructive interference
          • Young's double slit introduction
          • Young's double slit equation
          • Young's double slit problem solving
          • Diffraction grating
          • Single slit interference
          • More on single slit interference
          • Thin Film Interference part 1
          • Thin Film Interference part 2
      • Geometric optics
        • Skill check: Geometric optics
        • Reflection and refraction
          • Specular and diffuse reflection
          • Specular and diffuse reflection 2
          • Refraction and Snell's law
          • Refraction in water
          • Snell's law example 1
          • Snell's law example 2
          • Total internal reflection
          • Dispersion
        • Mirrors
          • Virtual image
          • Parabolic mirrors and real images
          • Parabolic mirrors 2
          • Convex parabolic mirrors
          • Derivation of the mirror equation
          • Mirror equation example problems
        • Lenses
          • Convex lenses
          • Convex lens examples
          • Concave lenses
          • Object image and focal distance relationship (proof of formula)
          • Object image height and distance relationship
          • Thin lens equation and problem solving
          • Multiple lens systems
          • Diopters, Aberration, and the Human Eye
      • Special relativity
        • Michelson and Morley's luminiferous ether experiment
          • Light and the luminiferous ether
          • Potential ways to detect an ether wind
          • Michelson–Morley Experiment introduction
        • Minkowski spacetime
          • Starting to set up a Newtonian path–time diagram
          • Visualizing multiple Newtonian path–time diagrams
          • Galilean transformation and contradictions with light
          • Introduction to special relativity and Minkowski spacetime diagrams
          • Measuring time in meters in Minkowski spacetime
          • Angle of x' axis in Minkowski spacetime
        • Lorentz transformation
          • Introduction to the Lorentz transformation
          • Evaluating a Lorentz transformation
          • Algebraically manipulating Lorentz transformation
          • Lorentz transformation derivation part 1
          • Deriving Lorentz transformation part 2
          • Lorentz transformation derivation part 3
        • Einstein velocity addition
          • Lorentz transformation for change in coordinates
          • Einstein velocity addition formula derivation
          • Applying Einstein velocity addition
          • Finding an in-between frame of reference
          • Calculating neutral velocity
          • Time dilation
      • Quantum Physics
        • Photons
          • Photon Energy
          • Photon Momentum
          • Photoelectric effect
        • Atoms and electrons
          • De Broglie wavelength
          • Quantum Wavefunction
          • Atomic Energy Levels
          • Bohr model radii (derivation using physics)
          • Bohr model radii
          • Bohr model energy levels (derivation using physics)
          • Bohr model energy levels
          • Absorption and emission
          • Emission spectrum of hydrogen
        • Quantum numbers and orbitals
          • Heisenberg uncertainty principle
          • Quantum numbers
          • Quantum numbers for the first four shells
        • Nuclei
          • Mass defect and binding energy
          • Nuclear stability and nuclear equations
          • Types of decay
          • Writing nuclear equations for alpha, beta, and gamma decay
          • Half-life and carbon dating
          • Half-life plot
          • Exponential decay formula proof (can skip, involves calculus)
          • Introduction to exponential decay
          • More exponential decay examples
          • Exponential decay and semi-log plots
      • Discoveries and projects
        • Discovery of magnetism
          • Discovery of magnetism
          • Compass: Which way is north?
          • Compass build (stroke direction)
          • Floating magnet
          • Compass build (magnet orientation)
          • Neutralize a compass
          • Compass interactions
        • Discovery of magnetic fields
          • Magnet near compass
          • Tracing a magnetic field
          • Discovery of magnetic fields
          • Magnetic permeability
          • Increase strength of magnet
        • Measuring magnetic fields
          • Are 2 magnets stronger than 1?
          • Earth's magnetic field (how to measure)
          • Inverse cube law (deflection method)
          • Inverse cube law (method of oscillation)
        • Discovery of batteries
          • Zinc copper cell (reduction-oxidation)
          • Battery meter (galvanometer)
          • Electrolyte test (pure water vs. vinegar)
          • Reverse electrodes (polarity)
          • Electrolyte (strong acid test)
          • Electrolyte (salt test)
          • Electrode (distance test)
          • Electrolyte (temperature test)
          • Electrode (surface area test)
          • Standard cell
          • How much electrolyte does a single cell need?
          • The battery and electromagnetism
        • Discovery of electromagnetism
          • Discovery of electromagnetism
          • Electromagnetic field (above vs. below)
          • Electromagnetic field (forward vs. reverse)
          • Electromagnetic field (loop)
          • Battery meter (galvanometer)
        • Discovery of resistors
          • Series load
          • Pencil resistor
          • Variable resistor (pencil)
          • Listen to variable resistance
        • Electric motor
          • Spinning compass
          • Build your own motor
        • Electrostatics
          • Discovery of triboelectric effect
          • Pith ball electroscope
          • Foil leaf electroscope
          • Electrostatic telegraphs (case study)
        • Projectile launcher
          • Ping pong ball launcher introduction
          • Build a ping pong ball launcher
          • Update and modify your ping pong ball launcher
        • Thermo can
          • Thermo can
        • Simple machines explorations
      • Review for AP Physics 1 exam
        • AP Physics 1 concept review
          • AP Physics 1 review of 1D motion
          • AP Physics 1 review of 2D motion and vectors
          • AP Physics 1 review of Forces and Newton's Laws
          • AP Physics 1 review of Centripetal Forces
          • AP Physics 1 review of Energy and Work
          • AP Physics 1 review of Momentum and Impulse
          • AP Physics 1 review of Torque and Angular momentum
          • AP Physics 1 review of Waves and Harmonic motion
          • AP Physics 1 Review of Charge and Circuits
        • AP Physics 1 free response questions 2015
          • Question 1a: 2015 AP Physics 1 free response
          • Question 1b: 2015 AP Physics 1 free response
          • Question 1c: 2015 AP Physics 1 free response
          • Question 2ab: 2015 AP Physics 1 free response
          • Question 2cd: 2015 AP Physics 1 free response
          • Question 3a: 2015 AP Physics 1 free response
          • Question 3b: 2015 AP Physics 1 free response
          • Question 3c: 2015 AP Physics 1 free response
          • Question 3d: 2015 AP Physics 1 free response
          • Question 4: 2015 AP Physics 1 free response
          • Question 5: 2015 AP Physics 1 free response
    • AP Physics 1
      • One-dimensional motion
        • Introduction to physics
          • Introduction to physics
          • Meet Sean, a creator of AP Physics on Khan Academy
        • Displacement, velocity, and time
          • Intro to vectors & scalars
          • Introduction to reference frames
          • Calculating average velocity or speed
          • Solving for time
          • Displacement from time and velocity example
          • Instantaneous speed and velocity
          • Position vs. time graphs
        • Acceleration
          • Acceleration
          • Airbus A380 take-off time
          • Airbus A380 take-off distance
          • Why distance is area under velocity-time line
          • Acceleration vs. time graphs
        • Kinematic formulas and projectile motion
          • Average velocity for constant acceleration
          • Acceleration of aircraft carrier take-off
          • Airbus A380 take-off distance
          • Deriving displacement as a function of time, acceleration, and initial velocity
          • Plotting projectile displacement, acceleration, and velocity
          • Projectile height given time
          • Deriving max projectile displacement given time
          • Impact velocity from given height
          • Viewing g as the value of Earth's gravitational field near the surface
          • Choosing kinematic equations
        • Old videos on projectile motion
          • Projectile motion (part 1)
          • Projectile motion (part 2)
          • Projectile motion (part 3)
          • Projectile motion (part 4)
          • Projectile motion (part 5)
      • Two-dimensional motion
        • Two-dimensional projectile motion
          • Horizontally launched projectile
          • Visualizing vectors in 2 dimensions
          • Projectile at an angle
          • Launching and landing on different elevations
          • Total final velocity for projectile
          • Correction to total final velocity for projectile
          • Projectile on an incline
          • Unit vectors and engineering notation
          • Unit vector notation
          • Unit vector notation (part 2)
          • Projectile motion with ordered set notation
        • Optimal angle for a projectile
          • Optimal angle for a projectile part 1: Components of initial velocity
          • Optimal angle for a projectile part 2: Hangtime
          • Optimal angle for a projectile part 3: Horizontal distance as a function of angle (and speed)
          • Optimal angle for a projectile part 4: Finding the optimal angle and distance with a bit of calculus
      • Forces and Newton's laws of motion
        • Newton's laws of motion
          • Newton's first law of motion
          • Newton's first law of motion concepts
          • More on Newton's first law of motion
          • Newton's second law of motion
          • More on Newton's second law
          • Newton's third law of motion
          • More on Newton's third law
        • Normal force and contact force
          • Normal force and contact force
          • Normal force in an elevator
          • More on Normal force (shoe on floor)
          • More on Normal force (shoe on wall)
        • Balanced and unbalanced forces
          • Balanced and unbalanced forces
          • Unbalanced forces and motion
        • Slow sock on Lubricon VI
          • Slow sock on Lubricon VI
          • Normal forces on Lubricon VI
        • Inclined planes and friction
          • Inclined plane force components
          • Ice accelerating down an incline
          • Force of friction keeping the block stationary
          • Correction to force of friction keeping the block stationary
          • Force of friction keeping velocity constant
          • Intuition on static and kinetic friction comparisons
          • Static and kinetic friction example
        • Tension
          • The force of tension
          • Mild and medium tension
          • Super hot tension
          • Introduction to tension
          • Introduction to tension (part 2)
          • Tension in an accelerating system and pie in the face
        • Treating systems
          • Treating systems (the hard way)
          • Treating systems (the easy way)
          • Two masses hanging from a pulley
          • Three box system problem
          • Masses on incline system problem
      • Centripetal force and gravitation
        • Circular motion and centripetal acceleration
          • Race cars with constant speed around curve
          • Centripetal force and acceleration intuition
          • Visual understanding of centripetal acceleration formula
          • Optimal turns at Indianapolis Motor Speedway with JR Hildebrand
          • Calculus proof of centripetal acceleration formula
          • Loop de loop question
          • Loop de loop answer part 1
          • Loop de loop answer part 2
        • Centripetal forces
          • Centripetal force problem solving
          • Yo-yo in vertical circle example
          • Bowling ball in vertical loop
          • Mass swinging in a horizontal circle
        • Newton's law of gravitation
          • Introduction to gravity
          • Mass and weight clarification
          • Gravity for astronauts in orbit
          • Would a brick or feather fall faster?
          • Acceleration due to gravity at the space station
          • Space station speed in orbit
          • Introduction to Newton's law of gravitation
          • Gravitation (part 2)
      • Work and energy
        • Work and energy
          • Introduction to work and energy
          • Work and energy (part 2)
          • Conservation of energy
          • Work and the work-energy principle
          • Work as the transfer of energy
          • Work example problems
          • Work as area under curve
          • Thermal energy from friction
          • Work/energy problem with friction
          • Conservative forces
          • Power
        • Springs and Hooke's law
          • Intro to springs and Hooke's law
          • Potential energy stored in a spring
          • Spring potential energy example (mistake in math)
          • LOL diagrams
          • Vertical springs and energy conservation
        • Mechanical advantage
          • Introduction to mechanical advantage
          • Mechanical advantage (part 2)
          • Mechanical advantage (part 3)
      • Impacts and linear momentum
        • Momentum and Impulse
          • Introduction to momentum
          • Impulse and momentum dodgeball example
          • Bouncing fruit collision example
          • Momentum: Ice skater throws a ball
          • 2-dimensional momentum problem
          • 2-dimensional momentum problem (part 2)
          • Force vs. time graphs
        • Elastic and inelastic collisions
          • Elastic and inelastic collisions
          • Solving elastic collision problems the hard way
          • Deriving the shortcut to solve elastic collision problems
          • How to use the shortcut for solving elastic collisions
        • Center of mass
          • Center of mass
          • Equation for center of mass
      • Torque and angular momentum
        • Rotational kinematics
          • Angular motion variables
          • Relating angular and regular motion variables
          • Relationship between angular velocity and speed
          • Rotational kinematic formulas
        • Torque, moments, and angular momentum
          • Introduction to torque
          • Moments
          • Moments (part 2)
          • Finding torque for angled forces
          • Rotational version of Newton's second law
          • More on moment of inertia
          • Rotational kinetic energy
          • Rolling without slipping problems
          • Angular momentum
          • Constant angular momentum when no net torque
          • Angular momentum of an extended object
          • Ball hits rod angular momentum example
          • Cross product and torque
      • Oscillations and mechanical waves
        • Simple harmonic motion
          • Intuition about simple harmonic oscillators
          • Definition of Amplitude and Period
          • Equation for simple harmonic oscillators
          • Period dependence for mass on spring
          • Phase constant
          • Pendulums
        • Simple harmonic motion (with calculus)
          • Introduction to harmonic motion
          • Harmonic motion part 2 (calculus)
          • Harmonic motion part 3 (no calculus)
        • Introduction to mechanical waves
          • Introduction to waves
          • Properties of periodic waves
          • The equation of a wave
        • Sound
          • Production of sound
          • Sound Properties (Amplitude, Period, Frequency, Wavelength)
          • Speed of Sound
          • Relative speed of sound in solids, liquids, and gases
          • Mach numbers
          • Decibel Scale
          • Why do sounds get softer?
          • Ultrasound medical imaging
        • The Doppler effect
          • Doppler effect introduction
          • Doppler effect formula for observed frequency
          • Doppler effect formula when source is moving away
          • When the source and the wave move at the same velocity
          • Doppler effect for a moving observer
          • Doppler effect: reflection off a moving object
        • Wave interference
          • Wave interference
          • Constructive and Destructive interference
          • Standing waves on strings
          • Standing waves in open tubes
          • Standing waves in closed tubes
          • Beat frequency
          • Derivation of beat frequency formula
      • Electric charge, field, and potential
        • Charge and electric force (Coulomb's law)
          • Triboelectric effect and charge
          • Coulomb's Law
          • Conductors and Insulators
          • Conservation of Charge
        • Electric field
          • Electric field definition
          • Electric field direction
          • Magnitude of electric field created by a charge
          • Net electric field from multiple charges in 1D
          • Net electric field from multiple charges in 2D
          • Electric field
          • Proof: Field from infinite plate (part 1)
          • Proof: Field from infinite plate (part 2)
        • Electric potential energy, electric potential, and voltage
          • Electric potential energy
          • Electric potential energy (part 2-- involves calculus)
          • Voltage
          • Electric potential energy of charges
          • Electric potential at a point in space
          • Electric potential charge configuration
      • Circuits
        • Ohm's law and circuits with resistors
          • Introduction to circuits and Ohm's law
          • Resistors in series
          • Resistors in parallel
          • Example: Analyzing a more complex resistor circuit
          • Analyzing a resistor circuit with two batteries
          • Resistivity and conductivity
          • Kirchhoff's current law
          • Kirchhoff's voltage law
          • Electric power
          • Voltmeters and Ammeters
          • Electrolytic conductivity
        • Circuits with capacitors
          • Capacitors and capacitance
          • Capacitance
          • Energy of a capacitor
          • Capacitors in series
          • Capacitors in parallel
          • Dielectrics in capacitors
      • Review for AP Physics 1 exam
        • AP Physics 1 concept review
          • AP Physics 1 review of 1D motion
          • AP Physics 1 review of 2D motion and vectors
          • AP Physics 1 review of Forces and Newton's Laws
          • AP Physics 1 review of Centripetal Forces
          • AP Physics 1 review of Energy and Work
          • AP Physics 1 review of Momentum and Impulse
          • AP Physics 1 review of Torque and Angular momentum
          • AP Physics 1 review of Waves and Harmonic motion
          • AP Physics 1 Review of Charge and Circuits
        • AP Physics 1 free response questions 2015
          • Question 1a: 2015 AP Physics 1 free response
          • Question 1b: 2015 AP Physics 1 free response
          • Question 1c: 2015 AP Physics 1 free response
          • Question 2ab: 2015 AP Physics 1 free response
          • Question 2cd: 2015 AP Physics 1 free response
          • Question 3a: 2015 AP Physics 1 free response
          • Question 3b: 2015 AP Physics 1 free response
          • Question 3c: 2015 AP Physics 1 free response
          • Question 3d: 2015 AP Physics 1 free response
          • Question 4: 2015 AP Physics 1 free response
          • Question 5: 2015 AP Physics 1 free response
    • AP Physics 2
      • Fluids
        • Density and Pressure
          • Specific gravity
          • Pressure and Pascal's principle (part 1)
          • Pressure and Pascal's principle (part 2)
          • Pressure at a depth in a fluid
          • Finding height of fluid in a barometer
        • Buoyant Force and Archimedes' Principle
          • Archimedes principle and buoyant force
          • Buoyant force example problems
        • Fluid Dynamics
          • Volume flow rate and equation of continuity
          • Bernoulli's equation (part 1)
          • Bernoulli's equation (part 2)
          • Bernoulli's equation (part 3)
          • Bernoulli's equation (part 4)
          • Bernoulli's example problem
          • Viscosity and Poiseuille flow
          • Turbulence at high velocities and Reynold's number
          • Venturi effect and Pitot tubes
          • Surface Tension and Adhesion
      • Thermodynamics
        • Temperature, kinetic theory, and the ideal gas law
          • Thermodynamics part 1: Molecular theory of gases
          • Thermodynamics part 2: Ideal gas law
          • Thermodynamics part 3: Kelvin scale and Ideal gas law example
          • Thermodynamics part 4: Moles and the ideal gas law
          • Thermodynamics part 5: Molar ideal gas law problem
          • Maxwell Boltzmann distribution
        • Specific heat and heat transfer
          • Specific heat and latent heat of fusion and vaporization
          • Thermal conduction, convection, and radiation
          • Thermal conduction
          • Thermal conductivity of metal and wood
          • Intuition behind formula for thermal conductivity
        • Laws of thermodynamics
          • Macrostates and microstates
          • Quasistatic and reversible processes
          • First law of thermodynamics / internal energy
          • More on internal energy
          • Work from expansion
          • PV-diagrams and expansion work
          • Proof: U = (3/2)PV or U = (3/2)nRT
          • Work done by isothermic process
          • Carnot cycle and Carnot engine
          • Proof: Volume ratios in a Carnot cycle
          • Proof: S (or entropy) is a valid state variable
          • Thermodynamic entropy definition clarification
          • Reconciling thermodynamic and state definitions of entropy
          • Entropy intuition
          • Maxwell's demon
          • More on entropy
          • Efficiency of a Carnot engine
          • Carnot efficiency 2: Reversing the cycle
          • Carnot efficiency 3: Proving that it is the most efficient
      • Electric charge, field, and potential
        • Charge and electric force (Coulomb's law)
          • Triboelectric effect and charge
          • Coulomb's Law
          • Conductors and Insulators
          • Conservation of Charge
        • Copy of Electric field
          • Electric field definition
          • Electric field direction
          • Magnitude of electric field created by a charge
          • Net electric field from multiple charges in 1D
          • Net electric field from multiple charges in 2D
          • Electric field
          • Proof: Field from infinite plate (part 1)
          • Proof: Field from infinite plate (part 2)
        • Electric potential energy, electric potential, and voltage
          • Electric potential energy
          • Electric potential energy (part 2-- involves calculus)
          • Voltage
          • Electric potential energy of charges
          • Electric potential at a point in space
          • Electric potential charge configuration
      • Magnetic forces, magnetic fields, and Faraday's law
        • Magnets and Magnetic Force
          • Introduction to magnetism
          • Magnetic force on a charge
          • Cross product 1
          • Cross product 2
          • Magnetic force on a proton example (part 1)
          • Magnetic force on a proton example (part 2)
          • Magnetic force on a current carrying wire
        • Magnetic field created by a current
          • Magnetic field created by a current carrying wire
          • Magnetic force between two currents going in the same direction
          • Magnetic force between two currents going in opposite directions
          • Induced current in a wire
        • Electric motors
          • Electric motors (part 1)
          • Electric motors (part 2)
          • Electric motors (part 3)
          • The dot product
          • Dot vs. cross product
          • Calculating dot and cross products with unit vector notation
        • Magnetic flux and Faraday's law
          • Flux and magnetic flux
          • Faraday's Law Introduction
          • Lenz's Law
          • Faraday's Law example
          • Emf induced in rod traveling through magnetic field
          • Faraday's Law for generating electricity
      • Electromagnetic waves and interference
        • Introduction to electromagnetic waves
          • Electromagnetic waves and the electromagnetic spectrum
          • Polarization of light, linear and circular
        • Interference of electromagnetic waves
          • Constructive and Destructive interference
          • Young's double slit introduction
          • Young's double slit equation
          • Young's double slit problem solving
          • Diffraction grating
          • Single slit interference
          • More on single slit interference
          • Thin Film Interference part 1
          • Thin Film Interference part 2
      • Geometric optics
        • Reflection and refraction
          • Specular and diffuse reflection
          • Specular and diffuse reflection 2
          • Refraction and Snell's law
          • Refraction in water
          • Snell's law example 1
          • Snell's law example 2
          • Total internal reflection
          • Dispersion
        • Mirrors
          • Virtual image
          • Parabolic mirrors and real images
          • Parabolic mirrors 2
          • Convex parabolic mirrors
          • Derivation of the mirror equation
          • Mirror equation example problems
        • Lenses
          • Convex lenses
          • Convex lens examples
          • Concave lenses
          • Object image and focal distance relationship (proof of formula)
          • Object image height and distance relationship
          • Thin lens equation and problem solving
          • Multiple lens systems
          • Diopters, Aberration, and the Human Eye
      • Quantum Physics
        • Photons
          • Photon Energy
          • Photon Momentum
          • Photoelectric effect
        • Atoms and electrons
          • De Broglie wavelength
          • Quantum Wavefunction
          • Atomic Energy Levels
          • Bohr model radii (derivation using physics)
          • Bohr model radii
          • Bohr model energy levels (derivation using physics)
          • Bohr model energy levels
          • Absorption and emission
          • Emission spectrum of hydrogen
        • Quantum numbers and orbitals
          • Heisenberg uncertainty principle
          • Quantum numbers
          • Quantum numbers for the first four shells
        • Nuclear physics
          • Mass defect and binding energy
          • Nuclear stability and nuclear equations
          • Types of decay
          • Writing nuclear equations for alpha, beta, and gamma decay
          • Half-life and carbon dating
          • Half-life plot
          • Exponential decay formula proof (can skip, involves calculus)
          • Introduction to exponential decay
          • More exponential decay examples
          • Exponential decay and semi-log plots
    • Class 11 Physics (India)
      • Physical world
        • Physics and its laws
          • Introduction to physics
          • Four fundamental forces
      • Units and measurement
        • Physical quantities and their measurement
          • Scale of the large
          • Parallax in observing stars
          • Stellar distance using parallax
          • Stellar parallax clarification
          • Scale of the small
        • Scientific notation
          • Scientific notation example: 0.0000000003457
          • Scientific notation examples
        • Significant figures
          • Intro to significant figures
          • Rules of significant figures
          • Multiplying and dividing with significant figures
          • Addition and subtraction with significant figures
      • Basic math concepts for physics (Prerequisite)
        • Introduction to logarithms
          • Intro to logarithms
          • Evaluating logarithms (advanced)
          • Relationship between exponentials & logarithms
        • The constant e and the natural logarithm
          • 𝑒 and compound interest
          • 𝑒 as a limit
          • Evaluating natural logarithm with calculator
        • Slope
          • Intro to slope
          • Positive & negative slope
          • Worked example: slope from two points
          • Slope of a horizontal line
        • Basics of trigonometry
        • The unit circle definition of sine, cosine, and tangent
          • Unit circle
          • The trig functions & right triangle trig ratios
        • The graphs of sine, cosine, and tangent
          • Graph of y=sin(x)
          • Graph of y=tan(x)
          • Intersection points of y=sin(x) and y=cos(x)
        • Basic trigonometric identities
          • Sine & cosine identities: symmetry
          • Tangent identities: symmetry
          • Sine & cosine identities: periodicity
          • Tangent identities: periodicity
        • Evaluating functions
          • What is a function?
          • Worked example: Evaluating functions from equation
          • Worked example: evaluating functions from graph
        • Inputs and outputs of a function
          • Worked example: matching an input to a function's output (equation)
          • Worked example: matching an input to a function's output (graph)
          • Worked example: two inputs with the same output (graph)
        • Maximum and minimum points
          • Introduction to minimum and maximum points
          • Worked example: relative maxima and minima
        • Intervals where a function is positive, negative, increasing, or decreasing
          • Increasing, decreasing, positive or negative intervals
          • Worked example: positive & negative intervals
        • Interpreting features of graphs
          • Graph interpretation word problem: temperature
          • Graph interpretation word problem: basketball
      • Differentiation for physics (Prerequisite)
        • Slope
          • Intro to slope
          • Positive & negative slope
          • Worked example: slope from two points
          • Slope of a horizontal line
        • Introduction to differential calculus
          • Newton, Leibniz, and Usain Bolt
          • Derivative as slope of curve
          • Derivative & the direction of a function
        • Secant lines
          • Slope of a line secant to a curve
          • Secant line with arbitrary difference
        • Limit basics
          • Introduction to limits (old)
          • Limits from tables
          • Formal definition of limits Part 1: intuition review
          • Limits by direct substitution
          • Undefined limits by direct substitution
          • Limits by factoring
        • Derivative as a limit
          • Formal definition of the derivative as a limit
          • Formal and alternate form of the derivative
          • Worked example: Derivative as a limit
          • Worked example: Derivative from limit expression
        • Using the formal definition of derivative
          • The derivative of x² at x=3 using the formal definition
          • The derivative of x² at any point using the formal definition
          • Limit expression for the derivative of a linear function
        • Derivative as a function
          • Matching functions & their derivatives graphically
          • Connecting f and f' graphically
        • Basic differentiation rules
          • Basic derivative rules (Part 1)
          • Basic derivative rules (Part 2)
          • Basic derivative rules: find the error
        • Power rule
          • Power rule
          • Proof of power rule for square root function
        • Polynomial functions differentiation
          • Differentiating polynomials
          • Differentiating polynomials example
        • Sine & cosine derivatives
          • Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
          • Worked example: Derivatives of sin(x) and cos(x)
        • Product rule
          • Product rule
          • Differentiating products
          • Product rule proof
        • Chain rule
          • Chain rule
          • Worked example: Derivative of cos³(x) using the chain rule
          • Worked example: Derivative of ln(√x) using the chain rule
          • Quotient rule from product & chain rules
      • Integration for physics (Prerequisite)
        • Antiderivatives
          • Antiderivatives and indefinite integrals
          • Reverse power rule
          • Rewriting before integrating: challenge problem
        • Indefinite integrals of common functions
          • Indefinite integrals of sin(x), cos(x), and eˣ
          • Finding specific antiderivatives: rational function
        • Definite integral as area
          • Worked example: Definite integral by thinking about the function's graph
          • Worked examples: Definite integral properties 2
          • Worked examples: Definite integral properties 1
        • Riemann sums
          • Riemann approximation introduction
          • Midpoint and trapezoidal sums in summation notation
          • Definite integral as the limit of a Riemann sum
        • Definite integral evaluation
          • Area between a curve and the x-axis
          • Area between a curve and the x-axis: negative area
        • u-substitution
          • 𝘶-substitution intro
          • 𝘶-substitution: rational function
          • 𝘶-substitution: multiplying by a constant
        • Area & net change
          • Worked example: problem involving definite integral (graphical)
          • Area under rate function gives the net change
      • Motion in a straight line
        • Skill check for one-dimensional motion
        • Displacement, velocity, and time
          • Intro to vectors & scalars
          • Introduction to reference frames
          • Calculating average velocity or speed
          • Solving for time
          • Displacement from time and velocity example
          • Instantaneous speed and velocity
          • Position vs. time graphs
        • Acceleration
          • Acceleration
          • Airbus A380 take-off time
          • Why distance is area under velocity-time line
          • Airbus A380 take-off distance
          • Acceleration vs. time graphs
        • Kinematic formulas and projectile motion
          • Average velocity for constant acceleration
          • Acceleration of aircraft carrier take-off
          • Airbus A380 take-off distance
          • Deriving displacement as a function of time, acceleration, and initial velocity
          • Plotting projectile displacement, acceleration, and velocity
          • Projectile height given time
          • Deriving max projectile displacement given time
          • Impact velocity from given height
          • Viewing g as the value of Earth's gravitational field near the surface
          • Choosing kinematic equations
        • Rectilinear motion (integral calc)
          • Worked example: motion problems (with definite integrals)
      • Vectors (Prerequisite)
        • Vector basics
          • Intro to vectors & scalars
          • Recognizing vectors
          • Equivalent vectors
          • Components of vectors
          • Components of vectors (example 2)
        • Magnitude of vectors
          • Vector magnitude from graph
          • Vector magnitude from components
          • OLD Magnitude of a vector from initial & terminal points
        • Scalar multiplication
          • Scalar multiplication of vectors
          • Analyzing scalar multiplication
        • Vector addition and subtraction
          • Adding & subtracting vectors
          • Graphically adding & subtracting vectors
          • Graphically subtracting vectors
          • Adding vectors algebraically & graphically
        • Combined vector operations
          • Combined vector operations
        • Unit vectors
          • Unit vectors intro
          • Worked example: finding unit vector with given direction
          • Worked example: Scaling unit vectors
        • Magnitude and direction form of vectors
          • Direction of vectors from components: 1st & 2nd quadrants
          • Direction of vectors from components: 3rd & 4th quadrants
        • Component form of vectors
          • Vector components from magnitude & direction
        • Adding vectors in magnitude and direction form
          • Adding vectors in magnitude & direction form (1 of 2)
          • Adding vectors in magnitude & direction form (2 of 2)
          • Vector addition & magnitude
        • Applications of vectors
          • Vectors word problem: pushing a box
          • Vectors word problem: hiking
      • Motion in a plane
        • Skill check for two-dimensional motion
        • Two-dimensional projectile motion
          • Horizontally launched projectile
          • Visualizing vectors in 2 dimensions
          • Projectile at an angle
          • Launching and landing on different elevations
          • Total displacement for projectile
          • Total final velocity for projectile
          • Correction to total final velocity for projectile
          • Projectile on an incline
        • Review: Unit vectors
          • Unit vectors and engineering notation
          • Unit vector notation
          • Unit vector notation (part 2)
        • Optimal angle for a projectile
          • Optimal angle for a projectile part 1: Components of initial velocity
          • Optimal angle for a projectile part 2: Hangtime
          • Optimal angle for a projectile part 3: Horizontal distance as a function of angle (and speed)
          • Optimal angle for a projectile part 4: Finding the optimal angle and distance with a bit of calculus
        • Circular motion and centripetal acceleration
          • Race cars with constant speed around curve
          • Centripetal force and acceleration intuition
          • Visual understanding of centripetal acceleration formula
          • Optimal turns at Indianapolis Motor Speedway with JR Hildebrand
          • Angular motion variables
          • Relating angular and regular motion variables
          • Calculus proof of centripetal acceleration formula
          • Loop de loop question
          • Loop de loop answer part 1
          • Loop de loop answer part 2
      • Laws of motion
        • Skill check for Forces and Newton's Laws of motion
        • Newton's laws of motion
          • Newton's first law of motion
          • Newton's first law of motion concepts
          • More on Newton's first law of motion
          • Newton's second law of motion
          • More on Newton's second law
          • Newton's third law of motion
          • More on Newton's third law
        • Impulse, momentum and Newton's second law
          • Introduction to momentum
          • Impulse and momentum dodgeball example
          • Bouncing fruit collision example
          • Momentum: Ice skater throws a ball
          • 2-dimensional momentum problem
          • 2-dimensional momentum problem (part 2)
          • Force vs. time graphs
        • Normal force and contact force
          • Normal force and contact force
          • Normal force in an elevator
          • More on Normal force (shoe on floor)
          • More on Normal force (shoe on wall)
        • Balanced and unbalanced forces
          • Balanced and unbalanced forces
          • Unbalanced forces and motion
        • Slow sock on Lubricon VI
          • Slow sock on Lubricon VI
          • Normal forces on Lubricon VI
        • Inclined planes and friction
          • Inclined plane force components
          • Ice accelerating down an incline
          • Force of friction keeping the block stationary
          • Correction to force of friction keeping the block stationary
          • Force of friction keeping velocity constant
          • Intuition on static and kinetic friction comparisons
          • Static and kinetic friction example
        • Tension
          • The force of tension
          • Mild and medium tension
          • Super hot tension
          • Tension in an accelerating system and pie in the face
        • Treating systems
          • Treating systems (the hard way)
          • Treating systems (the easy way)
          • Two masses hanging from a pulley
          • Three box system problem
          • Masses on incline system problem
        • Centripetal forces
          • Centripetal force problem solving
          • Yo-yo in vertical circle example
          • Bowling ball in vertical loop
          • Mass swinging in a horizontal circle
      • Work, energy and power
        • Skill check for work and energy
        • Work and energy
          • Introduction to work and energy
          • Work and energy (part 2)
          • Conservation of energy
          • Work and the work-energy principle
          • Work as the transfer of energy
          • Work example problems
          • Work as area under curve
          • The dot product
        • Conservative and non conservative forces
          • Thermal energy from friction
          • Work/energy problem with friction
          • Conservative forces
        • Springs and Hooke's law
          • Intro to springs and Hooke's law
          • Potential energy stored in a spring
          • Spring potential energy example (mistake in math)
          • LOL diagrams
          • Vertical springs and energy conservation
        • Power
          • Power
        • Elastic and inelastic collisions
          • Elastic and inelastic collisions
          • Solving elastic collision problems the hard way
          • Deriving the shortcut to solve elastic collision problems
          • How to use the shortcut for solving elastic collisions
      • System of particles and rotational motion
        • Skill check: Moments, torque, and angular momentum
        • Centre of mass
          • Center of mass
          • Equation for center of mass
        • Cross product
          • Cross product 1
          • Cross product 2
        • Rotational kinematics
          • Angular motion variables
          • Relating angular and regular motion variables
          • Relationship between angular velocity and speed
          • Rotational kinematic formulas
        • Torque, moments, and angular momentum
          • Introduction to torque
          • Moments
          • Moments (part 2)
          • Finding torque for angled forces
          • Rotational version of Newton's second law
          • More on moment of inertia
          • Rotational kinetic energy
          • Rolling without slipping problems
          • Angular momentum
          • Constant angular momentum when no net torque
          • Angular momentum of an extended object
          • Ball hits rod angular momentum example
          • Cross product and torque
      • Gravitation
        • Newton's law of gravitation
          • Introduction to gravity
          • Mass and weight clarification
          • Gravity for astronauts in orbit
          • Would a brick or feather fall faster?
          • Acceleration due to gravity at the space station
          • Space station speed in orbit
          • Introduction to Newton's law of gravitation
          • Gravitation (part 2)
      • Mechanical properties of fluid
        • Skill check for fluids
        • Density and pressure
          • Specific gravity
          • Pressure and Pascal's principle (part 1)
          • Pressure and Pascal's principle (part 2)
          • Pressure at a depth in a fluid
          • Finding height of fluid in a barometer
        • Buoyant force and Archimedes' principle
          • Archimedes principle and buoyant force
          • Buoyant force example problems
        • Fluid dynamics
          • Volume flow rate and equation of continuity
          • Bernoulli's equation (part 1)
          • Bernoulli's equation (part 2)
          • Bernoulli's equation (part 3)
          • Bernoulli's equation (part 4)
          • Bernoulli's example problem
          • Venturi effect and Pitot tubes
        • Viscosity
          • Viscosity and Poiseuille flow
          • Turbulence at high velocities and Reynold's number
        • Surface tension
          • Surface Tension and Adhesion
          • Capillary action and why we see a meniscus
          • Surface tension
      • Thermal properties of matter
        • Skill check for thermodynamics
        • Temperature, kinetic theory, and the ideal gas law
          • Thermodynamics part 1: Molecular theory of gases
          • Thermodynamics part 2: Ideal gas law
          • Thermodynamics part 3: Kelvin scale and Ideal gas law example
          • Absolute temperature and the kelvin scale
          • Thermodynamics part 4: Moles and the ideal gas law
          • Thermodynamics part 5: Molar ideal gas law problem
          • Maxwell Boltzmann distribution
        • Specific heat and heat transfer
          • Specific heat and latent heat of fusion and vaporization
          • Thermal conduction, convection, and radiation
          • Thermal conduction
          • Thermal conductivity of metal and wood
          • Intuition behind formula for thermal conductivity
          • Newton's Law of Cooling
          • Worked example: Newton's law of cooling
      • Thermodynamics
        • Zeroth and first law of thermodynamics
          • Macrostates and microstates
          • Quasistatic and reversible processes
          • First law of thermodynamics / internal energy
          • More on internal energy
          • Work done by isothermic process
        • Work from PV graph and internal energy
          • Work from expansion
          • PV-diagrams and expansion work
          • Proof: U = (3/2)PV or U = (3/2)nRT
          • Heat capacity at constant volume and pressure
        • Thermodynamic processes and Carnot cycle
          • Work done by isothermic process
          • Carnot cycle and Carnot engine
          • Proof: Volume ratios in a Carnot cycle
          • Efficiency of a Carnot engine
          • Carnot efficiency 2: Reversing the cycle
          • Carnot efficiency 3: Proving that it is the most efficient
      • Kinetic theory
        • Ideal gas equation
          • Ideal gas equation: PV = nRT
          • Deflate gate
          • Ideal gas equation example 2
          • Ideal gas equation example 4
          • Derivation of gas constants using molar volume and STP
          • Boyle's law
          • Charles's law
          • Avogadro's law
          • Introduction to partial pressure
          • Partial pressure example
          • Maxwell Boltzmann distribution
        • Kinetic molecular theory of gases
          • Boltzmann's constant
          • Heat capacity at constant volume and pressure
          • Kinetic molecular theory of gases
      • Oscillations
        • Simple harmonic motion
          • Intuition about simple harmonic oscillators
          • Definition of Amplitude and Period
          • Equation for simple harmonic oscillators
          • Period dependence for mass on spring
          • Phase constant
          • Pendulums
        • Simple harmonic motion (with calculus)
          • Introduction to harmonic motion
          • Harmonic motion part 2 (calculus)
          • Harmonic motion part 3 (no calculus)
      • Waves
        • Skill check: Oscillations and mechanical waves
        • Introduction to mechanical waves
          • Introduction to waves
          • Properties of periodic waves
          • The equation of a wave
        • Sound
          • Production of sound
          • Sound Properties (Amplitude, Period, Frequency, Wavelength)
          • Speed of Sound
          • Relative speed of sound in solids, liquids, and gases
          • Mach numbers
          • Decibel Scale
          • Why do sounds get softer?
          • Ultrasound medical imaging
        • The Doppler effect
          • Doppler effect introduction
          • Doppler effect formula for observed frequency
          • Doppler effect formula when source is moving away
          • When the source and the wave move at the same velocity
          • Doppler effect for a moving observer
          • Doppler effect: reflection off a moving object
        • Wave interference
          • Wave interference
          • Constructive and Destructive interference
          • Standing waves on strings
          • Standing waves in open tubes
          • Standing waves in closed tubes
          • Beat frequency
          • Derivation of beat frequency formula
    • Class 12 Physics (India)
      • Electric charges and field
        • Skill check: Electric charge, field, and electric potential
        • Charge and electric force (Coulomb's law)
          • Triboelectric effect and charge
          • Coulomb's Law
          • Conductors and Insulators
          • Conservation of Charge
        • Electric field
          • Electric field definition
          • Electric field direction
          • Magnitude of electric field created by a charge
          • Net electric field from multiple charges in 1D
          • Net electric field from multiple charges in 2D
          • Proof: Field from infinite plate (part 1)
          • Proof: Field from infinite plate (part 2)
          • Electric field
      • Electrostatic potential and capacitance
        • Electric potential energy, electric potential, and voltage
          • Electric potential energy
          • Electric potential energy (part 2-- involves calculus)
          • Voltage
          • Electric potential energy of charges
          • Electric potential at a point in space
          • Electric potential charge configuration
        • Circuits with capacitors
          • Capacitors and capacitance
          • Capacitance
          • Energy of a capacitor
          • Capacitors in series
          • Capacitors in parallel
          • Dielectrics in capacitors
      • Current electricity
        • Skill Check: Circuits
        • Electric current and voltage
          • Current
          • Current direction
          • Voltage
          • Conventional current
        • Ohm's law and power
          • Introduction to circuits and Ohm's law
          • Resistors in series
          • Resistors in parallel
          • Parallel resistors (part 3)
          • Simplifying resistor networks
          • Example: Analyzing a more complex resistor circuit
          • Analyzing a resistor circuit with two batteries
          • Resistivity and conductivity
          • Electric power
          • Voltmeters and Ammeters
          • Electrolytic conductivity
        • Circuit networks
          • Circuit terminology
          • Kirchhoff's current law
          • Kirchhoff's voltage law
      • Moving charges and magnetism
        • Skill check for Magnetic fields, forces, and Faraday's law
        • Cross product (Prerequisite)
          • Cross product 1
          • Cross product 2
          • The dot product
          • Dot vs. cross product
          • Calculating dot and cross products with unit vector notation
        • Magnets and magnetic force
          • Introduction to magnetism
          • Magnetic force on a charge
          • Magnetic force on a proton example (part 1)
          • Magnetic force on a proton example (part 2)
          • Mass spectrometer
          • Magnetic force on a current carrying wire
        • Magnetic field created by a current
          • Magnetic field created by a current carrying wire
          • Magnetic force between two currents going in the same direction
          • Magnetic force between two currents going in opposite directions
          • Induced current in a wire
        • Electric motors
          • Electric motors (part 1)
          • Electric motors (part 2)
          • Electric motors (part 3)
      • Electromagnetic induction
        • Magnetic flux and Faraday's law
          • Flux and magnetic flux
          • Faraday's Law Introduction
          • Lenz's Law
          • Faraday's Law example
          • Emf induced in rod traveling through magnetic field
          • Faraday's Law for generating electricity
      • Electromagnetic waves
        • Introduction to electromagnetic waves
          • Electromagnetic waves and the electromagnetic spectrum
      • Ray optics and optical instruments
        • Skill check: Geometric optics
        • Reflection and refraction
          • Specular and diffuse reflection
          • Specular and diffuse reflection 2
          • Refraction and Snell's law
          • Refraction in water
          • Snell's law example 1
          • Snell's law example 2
          • Total internal reflection
          • Dispersion
        • Mirrors
          • Virtual image
          • Parabolic mirrors and real images
          • Parabolic mirrors 2
          • Convex parabolic mirrors
          • Derivation of the mirror equation
          • Mirror equation example problems
        • Lenses
          • Convex lenses
          • Convex lens examples
          • Concave lenses
          • Object image and focal distance relationship (proof of formula)
          • Object image height and distance relationship
          • Thin lens equation and problem solving
          • Multiple lens systems
          • Diopters, Aberration, and the Human Eye
      • Wave optics
        • Skill Check: Electromagnetic waves and interference
        • Introduction to electromagnetic waves
          • Electromagnetic waves and the electromagnetic spectrum
          • Red shift
          • Polarization of light, linear and circular
        • Interference of electromagnetic waves
          • Constructive and Destructive interference
          • Young's double slit introduction
          • Young's double slit equation
          • Young's double slit problem solving
          • Diffraction grating
          • Single slit interference
          • More on single slit interference
      • Dual nature of radiation and matter
        • Photons
          • Photon Energy
          • Photon Momentum
          • Photoelectric effect
          • De Broglie wavelength
      • Atoms
        • History of atomic structure
          • The history of atomic chemistry
          • Rutherford’s gold foil experiment
        • Atoms and electrons
          • Atomic Energy Levels
          • Bohr model radii (derivation using physics)
          • Bohr model radii
          • Bohr model energy levels (derivation using physics)
          • Bohr model energy levels
          • Absorption and emission
          • Emission spectrum of hydrogen
          • Heisenberg uncertainty principle
      • Nuclei
        • Nuclear physics
          • Mass defect and binding energy
          • Nuclear stability and nuclear equations
          • Types of decay
          • Writing nuclear equations for alpha, beta, and gamma decay
        • Half life and decay rate
          • Half-life and carbon dating
          • Half-life plot
          • Exponential decay formula proof (can skip, involves calculus)
          • Introduction to exponential decay
          • More exponential decay examples
          • Exponential decay and semi-log plots
    • Chemistry
      • Atoms, compounds, and ions
        • Skill check for atoms, compounds, and ions
        • Introduction to the atom
          • Introduction to chemistry
          • Elements and atoms
          • Atomic weight and atomic mass
          • Worked example: Atomic weight calculation
          • The mole and Avogadro's number
          • Atomic number, mass number, and isotopes
          • Worked example: Identifying isotopes and ions
        • Ions and compounds
          • Introduction to ions
          • Naming ions and ionic compounds
          • Empirical, molecular, and structural formulas
          • Molecular mass and molecular weight
        • Names and formulas of ionic compounds
          • Naming ions and ionic compounds
          • Common polyatomic ions
          • Naming ionic compound with polyvalent ion
          • Finding formula for ionic compounds
        • Mass spectrometry
      • Chemical reactions and stoichiometry
        • Skill check for chemical reactions and stoichiometry
        • Balancing chemical equations
          • Chemical reactions introduction
          • Balancing chemical equations
          • Balancing more complex chemical equations
          • Visually understanding balancing chemical equations
          • Balancing another combustion reaction
          • Balancing chemical equation with substitution
        • Stoichiometry
          • Stoichiometry
          • Stoichiometry example problem 1
          • Stoichiometry example problem 2
        • Limiting reagent stoichiometry
          • Stoichiometry: Limiting reagent
          • Limiting reactant example problem 1
          • 2015 AP Chemistry free response 2a (part 1 of 2)
          • 2015 AP Chemistry free response 2a (part 2/2) and b
        • Molecular composition
          • Empirical, molecular, and structural formulas
          • Molecular mass percentage
          • Molecular weight percentages
          • Empirical formula from mass composition
          • Another mass composition problem
        • Types of chemical reactions
          • Introduction to redox reactions
          • Dissolution and precipitation
          • Precipitation reactions
          • 2015 AP Chemistry free response 3a
          • Complete ionic and net ionic equations
      • Electronic structure of atoms
        • Skill checks for electronic structure of atoms
        • History of atomic structure
          • The history of atomic chemistry
          • Rutherford’s gold foil experiment
        • Bohr's model of the hydrogen atom
          • Introduction to light
          • Photoelectric effect
          • Bohr model radii (derivation using physics)
          • Bohr model radii
          • Bohr model energy levels (derivation using physics)
          • Bohr model energy levels
          • Absorption and emission
          • Emission spectrum of hydrogen
        • Quantum numbers and orbitals
          • Heisenberg uncertainty principle
          • Quantum numbers
          • Quantum numbers for the first four shells
        • Electron configurations
          • Orbitals
          • More on orbitals and electron configuration
          • Noble gas configuration
          • Electron configurations 2
          • Electron configurations for the first period
          • Electron configurations for the second period
          • Electron configurations for the third and fourth periods
          • Electron configuration for d block element
          • Electron configurations in the 3d orbitals
          • Electron configuration for f block element Nd
          • Paramagnetism and diamagnetism
      • Periodic table
        • Skill checks for the periodic table
        • Introduction to the periodic table
          • The periodic table - classification of elements
          • Groups of the periodic table
          • Counting valence electrons for main group elements
          • The periodic table - transition metals
          • Valence electrons and bonding
        • Periodic table trends
          • Atomic radius trends on periodic table
          • Atomic and ionic radii
          • Mini-video on ion size
          • Ionization energy trends
          • Period trend for ionization energy
          • First and second ionization energy
          • Electron affinity: period trend
          • Electronegativity
          • Electronegativity and bonding
          • Metallic nature
      • Chemical bonds
        • Skill checks for chemical bonds
        • Types of chemical bonds
          • Ionic, covalent, and metallic bonds
          • Ionic bonds and Coulombs law
          • Electronegativity
          • Electronegativity and bonding
          • Metallic nature
          • Covalent networks, metallic crystals, and ionic crystals
        • Dot structures and molecular geometry
          • Drawing dot structures
          • Formal charge and dot structures
          • Resonance and dot structures
          • VSEPR for 2 electron clouds
          • VSEPR for 3 electron clouds
          • More on the dot structure for sulfur dioxide
          • VSEPR for 4 electron clouds
          • VSEPR for 5 electron clouds
          • VSEPR for 5 electron clouds (continued)
          • VSEPR for 6 electron clouds
          • Dipole moment
          • 2015 AP Chemistry free response 2d and e
        • Hybridization and hybrid orbitals
          • sp³ hybridization
          • Steric number and sp3 hybridization
          • sp² hybridization
          • Sp hybridization
          • Organic hybridization practice
      • Gases and kinetic molecular theory
        • Ideal gas equation
          • Ideal gas equation: PV = nRT
          • Deflate gate
          • Ideal gas equation example 1
          • Ideal gas equation example 2
          • Ideal gas equation example 3
          • Ideal gas equation example 4
          • Introduction to partial pressure
          • Partial pressure example
          • Vapor pressure example
          • Maxwell Boltzmann distribution
        • Non-ideal gas behavior
          • Real vs ideal gas behavior
          • Van der Waals equation
      • States of matter and intermolecular forces
        • States of matter
          • States of matter
          • States of matter follow-up
          • Specific heat and latent heat of fusion and vaporization
          • Specific heat, heat of fusion and vaporization example
          • Chilling water problem
          • Change of state example
          • Vapor pressure
          • Phase diagrams
        • Introduction to intermolecular forces
          • Van der Waals forces
          • Solubility and intermolecular forces
          • Surface tension
          • Capillary action and why we see a meniscus
          • Boiling points of organic compounds
          • Boiling point comparison: AP Chemistry multiple choice
          • Solubility of organic compounds
          • 2015 AP Chemistry free response 2f
        • Mixtures and solutions
          • Suspensions, colloids and solutions
          • Boiling point elevation and freezing point depression
      • Chemical equilibrium
        • Equilibrium constant
          • Reactions in equilibrium
          • Keq intuition
          • Keq derivation intuition
          • Heterogeneous equilibrium
          • Small x approximation for small Kc
          • Small x approximation for large Kc
        • Factors that affect chemical equilibrium
          • Le Chatelier's principle
          • Le Chatelier's principle: Worked example
          • Introduction to reaction quotient Qc
          • Comparing Q vs K example
      • Acids and bases
        • Skill checks for acids and bases
        • Acids, bases, and pH
          • Arrhenius definition of acids and bases
          • Bronsted-Lowry definition of acids and bases
          • Autoionization of water
          • Definition of pH
          • pH, pOH of strong acids and bases
          • Strong acids and strong bases
          • Acid strength, anion size, and bond energy
        • Acid-base equilibria
          • Conjugate acid-base pairs
          • pKa and pKb relationship
          • Relationship between Ka and Kb
          • Ka and acid strength
          • Weak acid equilibrium
          • Weak base equilibrium
          • Acid-base properties of salts
          • pH of salt solutions
      • Buffers, titrations, and solubility equilibria
        • Skill checks for buffers, titrations, and solubility equilibria
        • Buffer solutions
          • Introduction to buffers
          • Buffers, the acid rain slayer
          • Common ion effect and buffers
          • Buffers and Henderson-Hasselbalch
          • Buffer solution pH calculations
          • pH and pKa relationship for buffers
          • Buffer capacity
          • Ways to get a buffer solution
        • Titrations
          • Titration introduction
          • Titration calculation example
          • Titration of a strong acid with a strong base
          • Titration of a strong acid with a strong base (continued)
          • Titration of a weak acid with a strong base
          • Titration of a weak acid with a strong base (continued)
          • Titration of a weak base with a strong acid
          • Titration of a weak base with a strong acid (continued)
          • 2015 AP Chemistry free response 3b
          • 2015 AP Chemistry free response 3c
          • 2015 AP Chemistry free response 3d
          • 2015 AP Chemistry free response 3e
          • 2015 AP Chemistry free response 3f
          • Titration curves and acid-base indicators
          • Titration roundup
          • Acid base titration example
          • Redox titration
        • Solubility equilibria
          • Dissolution and precipitation
          • Common polyatomic ions
          • Introduction to solubility and solubility product constant
          • Solubility from the solubility product constant
          • 2015 AP Chemistry free response 4
          • Solubility and the common-ion effect
          • Solubility and the pH of the solution
          • Solubility and complex ion formation
      • Thermodynamics
        • Internal energy
          • First Law of Thermodynamics introduction
          • More on internal energy
          • Calculating internal energy and work example
          • Specific heat and latent heat of fusion and vaporization
          • Chilling water problem
          • Macrostates and microstates
          • Quasistatic and reversible processes
          • Work from expansion
          • PV-diagrams and expansion work
        • Enthalpy
          • Calorimetry and enthalpy introduction
          • Enthalpy
          • Heat of formation
          • Hess's law and reaction enthalpy change
          • Hess's law example
          • Bond enthalpy and enthalpy of reaction
          • 2015 AP Chemistry free response 7
        • Entropy
          • Introduction to entropy
          • Second Law of Thermodynamics
          • Entropy: Embrace the chaos
          • Work done by isothermic process
          • Carnot cycle and Carnot engine
          • Proof: Volume ratios in a Carnot cycle
          • Proof: S (or entropy) is a valid state variable
          • Thermodynamic entropy definition clarification
          • Reconciling thermodynamic and state definitions of entropy
          • Entropy intuition
          • More on entropy
          • Maxwell's demon
        • Gibbs free energy
          • Gibbs free energy and spontaneity
          • Gibbs free energy example
          • More rigorous Gibbs free energy / spontaneity relationship
          • A look at a seductive but wrong Gibbs spontaneity proof
          • Changes in free energy and the reaction quotient
          • Standard change in free energy and the equilibrium constant
          • 2015 AP Chemistry free response 2c
      • Redox reactions and electrochemistry
        • Skill checks on redox reactions and electrochemistry
        • Oxidation-reduction reactions
          • Oxidation and reduction
          • Oxidation state trends in periodic table
          • Practice determining oxidation states
          • Unusual oxygen oxidation states
          • Introduction to redox reactions
          • Redox reaction with iron
          • Oxidizing and reducing agents
          • Disproportionation
          • Balancing redox reactions in acid
          • Balancing redox reactions in base
          • Redox titration
        • Galvanic cells
          • Redox reaction from dissolving zinc in copper sulfate
          • Introduction to galvanic/voltaic cells
          • Electrodes and voltage of Galvanic cell
          • Shorthand notation for galvanic/voltaic cells
          • Lead storage battery
          • Nickel-cadmium battery
          • 2015 AP Chemistry free response 1d
        • Standard cell potentials
          • Standard reduction potentials
          • Using reduction potentials
          • Voltage as an intensive property
          • Spontaneity and redox reactions
          • 2015 AP Chemistry free response 1a
        • Electrochemistry, thermodynamics, and equilibrium
          • Free energy and cell potential
          • Standard cell potential and the equilibrium constant
          • Calculating the equilibrium constant from the standard cell potential
        • Cell potentials under nonstandard conditions
          • Galvanic cells and changes in free energy
          • Nernst equation
          • Using the Nernst equation
          • Concentration cell
          • 2015 AP Chemistry free response 1b and c
        • Electrolytic cells and electrolysis
          • Introduction to electrolysis
          • Quantitative electrolysis
          • Electrolysis of molten sodium chloride
      • Kinetics
        • Skill checks for kinetics
        • Reaction rates and rate laws
          • Introduction to kinetics
          • Rate of reaction
          • Rate law and reaction order
          • Finding units of rate constant k
          • Experimental determination of rate laws
        • Relationship between reaction concentrations and time
          • First-order reaction (with calculus)
          • Plotting data for a first-order reaction
          • Half-life of a first-order reaction
          • Half-life and carbon dating
          • First-order reaction example
          • Rate constant k from half-life example
          • Second-order reaction (with calculus)
          • Plotting data for a second-order reaction
          • Half-life of a second-order reaction
          • Second-order reaction example
          • Zero-order reaction (with calculus)
          • 2015 AP Chemistry free response 5
        • Arrhenius equation and reaction mechanisms
          • Collision theory
          • Arrhenius equation
          • Forms of the Arrhenius equation
          • Using the Arrhenius equation
          • Maxwell Boltzmann distribution
          • Elementary rate laws
          • Mechanisms and the rate-determining step
          • Catalysts
        • Spectrophotometry
          • Spectrophotometry introduction
          • Spectrophotometry example
      • Nuclear chemistry
        • Radioactive decay
          • Mass defect and binding energy
          • Nuclear stability and nuclear equations
          • Types of decay
          • Writing nuclear equations for alpha, beta, and gamma decay
          • Half-life and carbon dating
          • Half-life plot
          • Exponential decay formula proof (can skip, involves calculus)
          • Introduction to exponential decay
          • More exponential decay examples
          • Exponential decay and semi-log plots
      • Studying for the AP Chemistry exam?
        • 2015 AP Chemistry free response questions
          • 2015 AP Chemistry free response 1a
          • 2015 AP Chemistry free response 1b and c
          • 2015 AP Chemistry free response 1d
          • 2015 AP Chemistry free response 1e
          • 2015 AP Chemistry free response 2a (part 1 of 2)
          • 2015 AP Chemistry free response 2a (part 2/2) and b
          • 2015 AP Chemistry free response 2c
          • 2015 AP Chemistry free response 2d and e
          • 2015 AP Chemistry free response 2f
          • 2015 AP Chemistry free response 3a
          • 2015 AP Chemistry free response 3b
          • 2015 AP Chemistry free response 3c
          • 2015 AP Chemistry free response 3d
          • 2015 AP Chemistry free response 3e
          • 2015 AP Chemistry free response 3f
          • 2015 AP Chemistry free response 4
          • 2015 AP Chemistry free response 5
          • 2015 AP Chemistry free response 5a: Finding order of reaction
          • 2015 AP Chemistry free response 6
          • 2015 AP Chemistry free response 7
      • Meet the chemistry professional
        • Meet the chemistry professional
          • Introduction to "Meet a chemistry professional"
    • AP Chemistry
      • Atoms, compounds, and ions
        • Introduction to the atom
          • Introduction to chemistry
          • Elements and atoms
          • Atomic number, mass number, and isotopes
          • Atomic weight and atomic mass
          • The mole and Avogadro's number
        • Compounds and ions
          • Empirical, molecular, and structural formulas
          • Molecular mass and molecular weight
          • Common polyatomic ions
        • Mass Spectrometry
      • Stoichiometry and molecular composition
        • Balancing chemical equations
          • Chemical reactions introduction
          • Balancing chemical equations
          • Balancing more complex chemical equations
          • Visually understanding balancing chemical equations
          • Balancing another combustion reaction
          • Balancing chemical equation with substitution
        • Stoichiometry
          • Stoichiometry
          • Stoichiometry example problem 1
          • Stoichiometry example problem 2
        • Limiting reagent stoichiometry
          • Stoichiometry: Limiting reagent
          • Limiting reactant example problem 1
          • 2015 AP Chemistry free response 2a (part 1 of 2)
          • 2015 AP Chemistry free response 2a (part 2/2) and b
        • Molecular composition
          • Empirical, molecular, and structural formulas
          • Molecular mass percentage
          • Molecular weight percentages
          • Empirical formula from mass composition
          • Another mass composition problem
      • Chemical reactions
        • Types of chemical reactions
          • Introduction to redox reactions
          • Dissolution and precipitation
          • Precipitation reactions
          • 2015 AP Chemistry free response 3a
      • Electronic structure of atoms
        • History of atomic structure
          • The history of atomic chemistry
          • Rutherford’s gold foil experiment
        • Bohr's model of the hydrogen atom
          • Introduction to light
          • Photoelectric effect
          • Bohr model radii (derivation using physics)
          • Bohr model radii
          • Bohr model energy levels (derivation using physics)
          • Bohr model energy levels
          • Absorption and emission
          • Emission spectrum of hydrogen
        • Electron configurations
          • Orbitals
          • More on orbitals and electron configuration
          • Noble gas configuration
          • Electron configurations 2
          • Electron configurations for the first period
          • Electron configurations for the second period
          • Electron configurations for the third and fourth periods
          • Electron configuration for d block element
          • Electron configurations in the 3d orbitals
          • Electron configuration for f block element Nd
          • Paramagnetism and diamagnetism
      • Periodic table
        • Introduction to the periodic table
          • The periodic table - classification of elements
          • Groups of the periodic table
          • Counting valence electrons for main group elements
          • The periodic table - transition metals
          • Valence electrons and bonding
        • Periodic table trends
          • Atomic radius trends on periodic table
          • Atomic and ionic radii
          • Mini-video on ion size
          • Ionization energy trends
          • Group trend for ionization energy
          • Period trend for ionization energy
          • First and second ionization energy
          • Electron affinity: period trend
          • Electronegativity
          • Electronegativity and bonding
          • Metallic nature
      • Chemical bonds
        • Types of chemical bonds
          • Ionic, covalent, and metallic bonds
          • Ionic bonds and Coulombs law
          • Electronegativity
          • Electronegativity and bonding
          • Metallic nature
          • Covalent networks, metallic crystals, and ionic crystals
        • Dot structures and molecular geometry
          • Drawing dot structures
          • Formal charge and dot structures
          • Resonance and dot structures
          • VSEPR for 2 electron clouds
          • VSEPR for 3 electron clouds
          • More on the dot structure for sulfur dioxide
          • VSEPR for 4 electron clouds
          • VSEPR for 5 electron clouds
          • VSEPR for 5 electron clouds (continued)
          • VSEPR for 6 electron clouds
          • Dipole moment
          • 2015 AP Chemistry free response 2d and e
        • Hybridization and hybrid orbitals
          • sp³ hybridization
          • Steric number and sp3 hybridization
          • sp² hybridization
          • Sp hybridization
          • Organic hybridization practice
      • States of matter and intermolecular forces
        • States of matter
          • States of matter
          • States of matter follow-up
          • Specific heat and latent heat of fusion and vaporization
          • Specific heat, heat of fusion and vaporization example
          • Chilling water problem
          • Change of state example
          • Vapor pressure
        • Introduction to intermolecular forces
          • Van der Waals forces
          • Solubility and intermolecular forces
          • Surface tension
          • Capillary action and why we see a meniscus
          • Boiling points of organic compounds
          • Boiling point comparison: AP Chemistry multiple choice
          • Solubility of organic compounds
          • 2015 AP Chemistry free response 2f
        • Mixtures and solutions
          • Suspensions, colloids and solutions
      • Gases and kinetic molecular theory
        • Ideal gas equation
          • Ideal gas equation: PV = nRT
          • Deflate gate
          • Ideal gas equation example 1
          • Ideal gas equation example 2
          • Ideal gas equation example 3
          • Ideal gas equation example 4
          • Introduction to partial pressure
          • Partial pressure example
          • Vapor pressure example
          • Maxwell Boltzmann distribution
      • Kinetics
        • Reaction rates and rate laws
          • Introduction to kinetics
          • Rate of reaction
          • Rate law and reaction order
          • Finding units of rate constant k
          • Experimental determination of rate laws
        • Relationship between reaction concentrations and time
          • First-order reaction (with calculus)
          • Plotting data for a first-order reaction
          • Half-life of a first-order reaction
          • Half-life and carbon dating
          • First-order reaction example
          • Rate constant k from half-life example
          • Second-order reaction (with calculus)
          • Plotting data for a second-order reaction
          • Half-life of a second-order reaction
          • Second-order reaction example
          • Zero-order reaction (with calculus)
          • 2015 AP Chemistry free response 5
        • Arrhenius equation and reaction mechanisms
          • Collision theory
          • Arrhenius equation
          • Forms of the Arrhenius equation
          • Using the Arrhenius equation
          • Maxwell Boltzmann distribution
          • Elementary rate laws
          • Mechanisms and the rate-determining step
          • Catalysts
        • Spectrophotometry
          • Spectrophotometry introduction
          • Spectrophotometry example
      • Thermodynamics
        • Internal energy
          • First Law of Thermodynamics introduction
          • More on internal energy
          • Calculating internal energy and work example
          • Specific heat and latent heat of fusion and vaporization
          • Chilling water problem
          • Work from expansion
        • Enthalpy
          • Calorimetry and enthalpy introduction
          • Enthalpy
          • Heat of formation
          • Hess's law and reaction enthalpy change
          • Hess's law example
          • Bond enthalpy and enthalpy of reaction
          • 2015 AP Chemistry free response 7
        • Entropy
          • Introduction to entropy
          • Second Law of Thermodynamics
          • Entropy: Embrace the chaos
          • Reconciling thermodynamic and state definitions of entropy
          • Entropy intuition
        • Gibbs free energy
          • Gibbs free energy and spontaneity
          • Gibbs free energy example
          • More rigorous Gibbs free energy / spontaneity relationship
          • A look at a seductive but wrong Gibbs spontaneity proof
          • Changes in free energy and the reaction quotient
          • Standard change in free energy and the equilibrium constant
          • 2015 AP Chemistry free response 2c
      • Chemical equilibrium
        • Equilibrium constant
          • Reactions in equilibrium
          • Keq intuition
          • Keq derivation intuition
          • Heterogeneous equilibrium
          • Small x approximation for small Kc
          • Small x approximation for large Kc
        • Factors that affect chemical equilibrium
          • Le Chatelier's principle
          • Le Chatelier's principle: Worked example
          • Introduction to reaction quotient Qc
          • Comparing Q vs K example
      • Acids and bases
        • Acids, bases, and pH
          • Arrhenius definition of acids and bases
          • Bronsted-Lowry definition of acids and bases
          • Autoionization of water
          • Definition of pH
          • pH, pOH of strong acids and bases
          • Strong acids and strong bases
          • Acid strength, anion size, and bond energy
        • Acid-base equilibria
          • Conjugate acid-base pairs
          • pKa and pKb relationship
          • Relationship between Ka and Kb
          • Ka and acid strength
          • Weak acid equilibrium
          • Weak base equilibrium
          • Acid-base properties of salts
          • pH of salt solutions
      • Buffers, titrations, and solubility equilibria
        • Buffer solutions
          • Introduction to buffers
          • Buffers, the acid rain slayer
          • Common ion effect and buffers
          • Buffers and Henderson-Hasselbalch
          • Buffer solution pH calculations
          • pH and pKa relationship for buffers
          • Buffer capacity
          • Ways to get a buffer solution
        • Titrations
          • Titration introduction
          • Titration calculation example
          • Titration of a strong acid with a strong base
          • Titration of a strong acid with a strong base (continued)
          • Titration of a weak acid with a strong base
          • Titration of a weak acid with a strong base (continued)
          • Titration of a weak base with a strong acid
          • Titration of a weak base with a strong acid (continued)
          • 2015 AP Chemistry free response 3b
          • 2015 AP Chemistry free response 3c
          • 2015 AP Chemistry free response 3d
          • 2015 AP Chemistry free response 3e
          • 2015 AP Chemistry free response 3f
          • Titration curves and acid-base indicators
          • Titration roundup
          • Acid base titration example
          • Redox titration
        • Solubility equilibria
          • Dissolution and precipitation
          • Common polyatomic ions
          • Introduction to solubility and solubility product constant
          • Solubility from the solubility product constant
          • 2015 AP Chemistry free response 4
          • Solubility and the common-ion effect
          • Solubility and the pH of the solution
          • Solubility and complex ion formation
      • Redox reactions and electrochemistry
        • Oxidation-reduction reactions
          • Oxidation and reduction
          • Oxidation state trends in periodic table
          • Practice determining oxidation states
          • Unusual oxygen oxidation states
          • Introduction to redox reactions
          • Redox reaction with iron
          • Oxidizing and reducing agents
          • Disproportionation
          • Balancing redox reactions in acid
          • Balancing redox reactions in base
          • Redox titration
        • Galvanic cells
          • Redox reaction from dissolving zinc in copper sulfate
          • Introduction to galvanic/voltaic cells
          • Electrodes and voltage of Galvanic cell
          • Shorthand notation for galvanic/voltaic cells
          • Lead storage battery
          • Nickel-cadmium battery
          • 2015 AP Chemistry free response 1d
        • Standard cell potentials
          • Standard reduction potentials
          • Using reduction potentials
          • Voltage as an intensive property
          • Spontaneity and redox reactions
          • 2015 AP Chemistry free response 1a
        • Electrochemistry, thermodynamics, and equilibrium
          • Free energy and cell potential
          • Standard cell potential and the equilibrium constant
          • Calculating the equilibrium constant from the standard cell potential
        • Cell potentials under nonstandard conditions
          • Galvanic cells and changes in free energy
          • Nernst equation
          • Using the Nernst equation
          • Concentration cell
          • 2015 AP Chemistry free response 1b and c
        • Electrolytic cells and electrolysis
          • Introduction to electrolysis
          • Quantitative electrolysis
          • Electrolysis of molten sodium chloride
      • Studying for the AP Chemistry exam?
        • 2015 AP Chemistry free response questions
          • 2015 AP Chemistry free response 1a
          • 2015 AP Chemistry free response 1b and c
          • 2015 AP Chemistry free response 1d
          • 2015 AP Chemistry free response 1e
          • 2015 AP Chemistry free response 2a (part 1 of 2)
          • 2015 AP Chemistry free response 2a (part 2/2) and b
          • 2015 AP Chemistry free response 2c
          • 2015 AP Chemistry free response 2d and e
          • 2015 AP Chemistry free response 2f
          • 2015 AP Chemistry free response 3a
          • 2015 AP Chemistry free response 3b
          • 2015 AP Chemistry free response 3c
          • 2015 AP Chemistry free response 3d
          • 2015 AP Chemistry free response 3e
          • 2015 AP Chemistry free response 3f
          • 2015 AP Chemistry free response 4
          • 2015 AP Chemistry free response 5
          • 2015 AP Chemistry free response 5a: Finding order of reaction
          • 2015 AP Chemistry free response 6
          • 2015 AP Chemistry free response 7
    • Organic chemistry
      • Structure and bonding
        • Dot structures
          • Dot structures I: Single bonds
          • Dot structures II: Multiple bonds
        • Hybridization
          • sp3 hybridized orbitals and sigma bonds
          • Pi bonds and sp2 hybridized orbitals
          • sp³ hybridization
          • Steric number and sp3 hybridization
          • sp² hybridization
          • Sp hybridization
          • Organic hybridization practice
          • Tetrahedral bond angle proof
        • Bond-line structures
          • Condensed structures
          • Bond-line structures
          • Three-dimensional bond-line structures
          • Structural (constitutional) isomers
        • Electronegativity
          • Electronegativity and bonding
          • Dipole moment
          • Intermolecular forces
          • Boiling points of organic compounds
          • Solubility of organic compounds
      • Resonance and acid-base chemistry
        • Counting electrons
          • Comparing formal charges to oxidation states
          • Formal charge on carbon
          • Formal charge on nitrogen
          • Formal charge on oxygen
          • Oxidation states of carbon
          • Organic oxidation-reduction reactions
        • Resonance structures
          • Resonance structures
          • Resonance structure patterns
          • Resonance structures for benzene and the phenoxide anion
          • Common mistakes when drawing resonance structures
          • Resonance structures and hybridization
        • Organic acid-base chemistry
          • Acid-base definitions
          • Organic acid-base mechanisms
          • Ka and acid strength
          • Ka and pKa review
          • Using a pKa table
          • Using pKa values to predict the position of equilibrium
          • Stabilization of a conjugate base: electronegativity
          • Acid strength, anion size, and bond energy
          • Stabilization of a conjugate base: resonance
          • Stabilization of a conjugate base: induction
          • Stabilization of a conjugate base: hybridization
          • Stabilization of a conjugate base: solvation
      • Alkanes, cycloalkanes, and functional groups
        • Naming alkanes
          • Representing structures of organic molecules
          • Naming simple alkanes
          • Naming alkanes with alkyl groups
          • Correction - 2-propylheptane should never be the name!
          • Common and systematic naming: iso-, sec-, and tert- prefixes
          • Naming alkanes with ethyl groups
          • Alkane with isopropyl group
          • Organic chemistry naming examples 2
          • Organic chemistry naming examples 3
          • Naming a cycloalkane
          • Naming two isobutyl groups systematically
          • Organic chemistry naming examples 4
        • Naming alkanes, cycloalkanes, and bicyclic compounds
          • Alkane and cycloalkane nomenclature I
          • Alkane and cycloalkane nomenclature II
          • Alkane and cycloalkane nomenclature III
          • Bicyclic compounds
          • Naming cubane
          • Heats of combustion of alkanes
        • Conformations of alkanes
          • Newman projections
          • Newman projections 2
          • Conformations of ethane
          • Conformational analysis of ethane
          • Conformational analysis of propane
          • Conformational analysis of butane
          • Newman projection practice 1
          • Newman projection practice 2
        • Conformations of cycloalkanes
          • Chair and boat shapes for cyclohexane
          • Double Newman diagram for methylcyclohexane
          • Stability of cycloalkanes
          • Conformations of cyclohexane
          • Drawing chair conformations
          • Monosubstituted cyclohexane
          • Disubstituted cyclohexane
          • Polysubstituted cyclohexane
        • Functional groups
          • Functional groups
          • More functional groups
          • Identifying functional groups
      • Stereochemistry
        • Chirality
          • Introduction to chirality
          • Chiral examples 1
          • Chiral examples 2
          • Chiral vs achiral
          • Stereoisomers, enantiomers, and chirality centers
          • Identifying chirality centers
        • Enantiomers
          • Drawing enantiomers
          • Cahn-Ingold-Prelog system for naming enantiomers
          • R,S system
          • R,S (Cahn-Ingold-Prelog) naming system example 2
          • R,S system practice
          • More R,S practice
          • Fischer projection introduction
          • Fischer projection practice
          • Optical activity
          • Optical activity calculations
        • Stereoisomeric relationships
          • Stereoisomers, enantiomers, diastereomers, constitutional isomers and meso compounds
          • Enantiomers and diastereomers
          • Meso compounds
      • Substitution and elimination reactions
        • Free radical reaction
          • Free radical reactions
        • Nucleophilicity and basicity
          • Nucleophilicity (nucleophile strength)
          • Nucleophilicity vs. basicity
        • Elimination reactions
          • E2 reactions
          • E1 reactions
          • Zaitsev's rule
        • Sn1/Sn2/E1/E2
          • Comparing E2, E1, Sn2, Sn1 reactions
          • E2 E1 Sn2 Sn1 reactions example 2
          • E2 E1 Sn2 Sn1 reactions example 3
        • Sn1 and Sn2
          • Identifying nucleophilic and electrophilic centers
          • Curly arrow conventions in organic chemistry
          • Intro to organic mechanisms
          • Alkyl halide nomenclature and classification
          • Sn1 mechanism: kinetics and substrate
          • Sn1 mechanism: stereochemistry
          • Carbocation stability and rearrangement introduction
          • Carbocation rearrangement practice
          • Sn1 mechanism: carbocation rearrangement
          • Sn1 carbocation rearrangement (advanced)
          • Sn2 mechanism: kinetics and substrate
          • Sn2 mechanism: stereospecificity
          • Sn1 and Sn2: leaving group
          • Sn1 vs Sn2: Solvent effects
          • Sn1 vs Sn2: Summary
        • E1 and E2 reactions
          • E1 mechanism: kinetics and substrate
          • E1 elimination: regioselectivity
          • E1 mechanism: stereoselectivity
          • E1 mechanism: carbocations and rearrangements
          • E2 mechanism: kinetics and substrate
          • E2 mechanism: regioselectivity
          • E2 elimination: Stereoselectivity
          • E2 elimination: Stereospecificity
          • E2 elimination: Substituted cyclohexanes
          • Regioselectivity, stereoselectivity, and stereospecificity
        • Sn1/Sn2/E1/E2
          • Elimination vs substitution: reagent
          • Elimination vs substitution: primary substrate
          • Elimination vs substitution: secondary substrate
          • Elimination vs substitution: tertiary substrate
      • Alkenes and alkynes
        • Naming alkenes
          • Naming alkenes examples
          • cis-trans and E-Z naming scheme for alkenes
          • Entgegen-Zusammen naming scheme for alkenes examples
        • Alkene reactions
          • Introduction to reaction mechanisms
          • Markovnikov's rule and carbocations
          • Addition of water (acid-catalyzed) mechanism
          • Polymerization of alkenes with acid
        • Alkene nomenclature
          • Alkene structure and classification
          • Alkene nomenclature
          • Cis-trans isomerism
          • E-Z system
          • Alkene stability
        • Alkene reactions
          • Hydrogenation
          • Hydrohalogenation
          • Hydration
          • Hydroboration-oxidation
          • Hydroboration-oxidation: Mechanism
          • Halogenation
          • Halohydrin formation
          • Epoxide formation and anti dihydroxylation
          • Syn dihydroxylation
          • Ozonolysis
        • Naming and preparing alkynes
          • Alkyne nomenclature
          • Alkyne acidity and alkylation
          • Preparation of alkynes
        • Alkyne reactions
          • Reduction of alkynes
          • Hydrohalogenation of alkynes
          • Hydration of alkynes
          • Hydroboration-oxidation of alkynes
          • Halogenation and ozonolysis of alkynes
        • Synthesis using alkynes
          • Synthesis using alkynes
      • Alcohols, ethers, epoxides, sulfides
        • Alcohol nomenclature and properties
          • Alcohols
          • Triple bonds cause linear configurations
          • Alcohol properties
          • Alcohol nomenclature
          • Physical properties of alcohols and preparation of alkoxides
        • Synthesis of alcohols
          • Preparation of alcohols using NaBH4
          • Preparation of alcohols using LiAlH4
          • Synthesis of alcohols using Grignard reagents I
          • Synthesis of alcohols using Grignard reagents II
        • Reactions of alcohols
          • Oxidation of alcohols I: Mechanism and oxidation states
          • Oxidation of alcohols II: Examples
          • Biological redox reactions
          • Protection of alcohols
          • Preparation of mesylates and tosylates
          • SN1 and SN2 reactions of alcohols
          • Formation of nitrate esters
          • Preparation of alkyl halides from alcohols
        • Nomenclature and properties of ethers
          • Ether naming and introduction
          • Ether nomenclature
          • Properties of ethers and crown ethers
        • Synthesis and cleavage of ethers
          • Williamson ether synthesis
          • Acidic cleavage of ethers
        • Nomenclature and preparation of epoxides
          • Cyclic ethers and epoxide naming
          • Nomenclature and preparation of epoxides
          • Preparation of epoxides: Stereochemistry
        • Ring-opening reactions of epoxides
          • Ring-opening reactions of epoxides: Strong nucleophiles
          • Ring opening reactions of epoxides: Acid-catalyzed
        • Thiols and sulfides
          • Preparation of sulfides
      • Conjugated systems and pericyclic reactions
        • Diels-Alder reaction
          • Diels-Alder reaction
          • Diels-Alder: stereochemistry of dienophile
          • Diels-Alder: stereochemistry of diene
          • Diels-Alder: endo rule
          • Diels-Alder: intramolecular
          • Diels-Alder: regiochemistry
      • Aromatic compounds
        • Naming benzene derivatives
          • Naming benzene derivatives introduction
          • Naming benzene derivatives
        • Reactions of benzene
          • Electrophilic aromatic substitution
          • Bromination of benzene
          • Friedel-Crafts acylation
          • Friedel crafts acylation addendum
        • Aromatic stability
          • Aromatic stability I
          • Aromatic stability II
          • Aromatic stability III
          • Aromatic stability IV
          • Aromatic stability V
          • Aromatic heterocycles I
          • Aromatic heterocycles II
        • Electrophilic aromatic substitution
          • Electrophilic aromatic substitution mechanism
          • Halogenation
          • Nitration
          • Sulfonation
          • Friedel-Crafts alkylation
          • Friedel-Crafts acylation
        • Directing effects
          • Ortho-para directors I
          • Ortho-para directors II
          • Ortho-para directors III
          • Meta directors I
          • Meta directors II
          • Multiple substituents
        • Other reactions and synthesis
          • Birch reduction I
          • Birch reduction II
          • Reactions at the benzylic position
          • Synthesis of substituted benzene rings I
          • Synthesis of substituted benzene rings II
        • Nucleophilic aromatic substitution
          • Nucleophilic aromatic substitution I
          • Nucleophilic aromatic substitution II
      • Aldehydes and ketones
        • Introduction to aldehydes and ketones
          • Aldehyde introduction
          • Ketone naming
          • Nomenclature of aldehydes and ketones
          • Physical properties of aldehydes and ketones
          • Reactivity of aldehydes and ketones
        • Reactions of aldehydes and ketones
          • Formation of hydrates
          • Formation of hemiacetals and hemiketals
          • Acid and base catalyzed formation of hydrates and hemiacetals
          • Formation of acetals
          • Acetals as protecting groups and thioacetals
          • Formation of imines and enamines
          • Formation of oximes and hydrazones
          • Addition of carbon nucleophiles to aldehydes and ketones
          • Formation of alcohols using hydride reducing agents
          • Oxidation of aldehydes using Tollens' reagent
      • Carboxylic acids and derivatives
        • Naming carboxylic acids
          • Carboxylic acid introduction
          • Carboxylic acid naming
        • Formation of carboxylic acid derivatives
          • Fischer esterification
          • Acid chloride formation
          • Amides, anhydrides, esters, and acyl chlorides
          • Relative stability of amides, esters, anhydrides, and acyl chlorides
          • Amide formation from acyl chloride
        • Nomenclature and reactions of carboxylic acids
          • Carboxylic acid nomenclature and properties
          • Reduction of carboxylic acids
          • Preparation of acyl (acid) chlorides
          • Preparation of acid anhydrides
          • Preparation of esters via Fischer esterification
          • Preparation of amides using DCC
          • Decarboxylation
        • Nomenclature and reactions of carboxylic acid derivatives
          • Nomenclature and properties of acyl (acid) halides and acid anhydrides
          • Nomenclature and properties of esters
          • Nomenclature and properties of amides
          • Reactivity of carboxylic acid derivatives
          • Nucleophilic acyl substitution
          • Acid-catalyzed ester hydrolysis
          • Acid and base-catalyzed hydrolysis of amides
          • Beta-lactam antibiotics
      • Alpha carbon chemistry
        • Formation of enolate anions
          • Keto-enol tautomerization (by Sal)
          • Keto-enol tautomerization (by Jay)
          • Enolate formation from aldehydes
          • Enolate formation from ketones
          • Kinetic and thermodynamic enolates
        • Aldol condensations
          • Aldol reaction
          • Aldol condensation
          • Mixed (crossed) aldol condensation
          • Mixed (crossed) aldol condensation using a lithium enolate
          • Retro-aldol and retrosynthesis
          • Intramolecular aldol condensation
      • Amines
        • Naming amines
          • Amine naming introduction
          • Amine naming 2
      • Spectroscopy
        • Infrared spectroscopy
          • Introduction to infrared spectroscopy
          • Bonds as springs
          • Signal characteristics - wavenumber
          • IR spectra for hydrocarbons
          • Signal characteristics - intensity
          • Signal characteristics - shape
          • Symmetric and asymmetric stretching
          • IR signals for carbonyl compounds
          • IR spectra practice
        • UV/Vis Spectroscopy
          • UV/Vis spectroscopy
          • Absorption in the visible region
          • Conjugation and color
        • Proton NMR
          • Introduction to proton NMR
          • Nuclear shielding
          • Chemical equivalence
          • Chemical shift
          • Electronegativity and chemical shift
          • Diamagnetic anisotropy
          • Integration
          • Spin-spin splitting (coupling)
          • Multiplicity: n + 1 rule
          • Coupling constant
          • Complex splitting
          • Hydrogen deficiency index
          • Proton NMR practice 1
          • Proton NMR practice 2
          • Proton NMR practice 3
    • Biology
      • Intro to biology
        • Intro to biology skill check
        • Welcome to biology!
          • Biology overview
        • The science of biology
          • The scientific method
      • Chemistry of life
        • Chemistry of life skill checks
        • Elements and atoms
          • Elements and atoms
          • Introduction to the atom
        • Electron shells and orbitals
          • Orbitals
          • More on orbitals and electron configuration
          • Valence electrons and bonding
          • Groups of the periodic table
        • Chemical bonds and reactions
          • Ionic, covalent, and metallic bonds
          • Electronegativity
          • Electronegativity and bonding
          • Intermolecular forces
          • Chemical reactions introduction
      • Water, acids, and bases
        • Properties of water skill check
        • Hydrogen bonding in water
          • Hydrogen bonding in water
          • Water as a solvent
        • Cohesion and adhesion
          • Capillary action and why we see a meniscus
          • Surface tension
        • Temperature and state changes in water
          • LeBron Asks: Why does sweating cool you down?
          • Evaporative cooling
          • Heat of vaporization of water and ethanol
          • Specific heat of water
          • Liquid water denser than solid water (ice)
        • Acids, bases, and pH
          • Autoionization of water
          • Arrhenius definition of acids and bases
          • Bronsted-Lowry definition of acids and bases
          • Definition of pH
          • Introduction to buffers
      • Properties of carbon
        • Properties of carbon skill check
        • Carbon
          • Carbon as a building block of life
          • Silicon-based life
          • Representing structures of organic molecules
        • Hydrocarbon structures and functional groups
          • Hydrocarbon overview
          • Isomers
          • Functional groups
      • Macromolecules
        • Macromolecules skill check
        • Introduction to macromolecules
        • Carbohydrates
          • Molecular structure of glucose
          • Dehydration synthesis or a condensation reaction
          • Hydrolysis
          • Molecular structure of fructose
        • Lipids
          • Molecular structure of triglycerides (fats)
          • Saturated fats, unsaturated fats, and trans fats
          • Lipid overview
        • Nucleic acids
          • DNA
          • DNA replication and RNA transcription and translation
          • Molecular structure of DNA
          • Antiparallel structure of DNA strands
          • Molecular structure of RNA
        • Proteins
          • Introduction to amino acids
          • Peptide bond formation
          • Overview of protein structure
          • Tertiary structure of proteins
      • Energy and enzymes
        • Skill check: Energy and enzymes
        • Energy in metabolism
          • Introduction to metabolism: Anabolism and catabolism
        • Laws of thermodynamics
          • Introduction to energy
          • First Law of Thermodynamics introduction
          • Introduction to entropy
          • Second Law of Thermodynamics
          • Second Law of Thermodynamics and entropy
          • Why heat increases entropy
        • Free energy
          • Gibbs free energy and spontaneous reactions
          • Endergonic, exergonic, exothermic, and endothermic
        • ATP and reaction coupling
          • ATP: Adenosine triphosphate
          • ATP hydrolysis mechanism
          • Reaction coupling to create glucose-6-phosphate
        • Introduction to enzymes
          • Introduction to kinetics
          • Enzymes
        • Enzyme regulation
          • Enzyme cofactors and coenzymes
          • Competitive inhibition
          • Noncompetitive inhibition
      • Structure of a cell
        • Skill check: Structure of a cell
        • Introduction to cells
          • Scale of cells
          • Cell theory
        • Prokaryotic and eukaryotic cells
          • Cell size
          • Prokaryotic and eukaryotic cells
        • Tour of a eukaryotic cell
          • Endoplasmic reticulum and Golgi bodies
          • Endomembrane system
          • Mitochondria
          • The cytoskeleton
        • Extracellular structures and cell-cell junctions
          • Extracellular matrix
          • Plant cell walls
          • Overview of animal and plant cells
      • Membranes and transport
        • Membranes and transport skill checks
        • The plasma membrane
          • Fluid mosaic model of cell membranes
        • Diffusion and osmosis
          • Diffusion - Introduction
          • Concentration gradients
          • Osmosis
          • Hypotonic, isotonic, and hypertonic solutions (tonicity)
          • Diffusion and osmosis
        • Passive transport
          • Passive transport and selective permeability
          • Facilitated diffusion
        • Active transport
          • Sodium potassium pump
          • Electrochemical gradients and secondary active transport
          • Uniporters, symporters and antiporters
        • Bulk transport
          • Endocytosis, phagocytosis, and pinocytosis
          • Exocytosis
      • Cellular respiration
        • Cellular respiration skill check
        • Introduction to cellular respiration
          • ATP: Adenosine triphosphate
          • ATP hydrolysis mechanism
          • Cellular respiration introduction
          • Oxidation and reduction review from biological point-of-view
          • Oxidation and reduction in cellular respiration
        • Steps of cellular respiration
          • Overview of cellular respiration
        • Glycolysis
          • Overview of glycolysis
          • Steps of glycolysis
        • Pyruvate oxidation and the citric acid cycle
          • Krebs / citric acid cycle
        • Oxidative phosphorylation
          • Oxidative phosphorylation and the electron transport chain
        • Variations on cellular respiration
          • Lactic acid fermentation
          • Alcohol or ethanol fermentation
      • Photosynthesis
        • Photosynthesis skill check
        • Intro to photosynthesis
          • ATP: Adenosine triphosphate
          • ATP hydrolysis mechanism
          • Photosynthesis
          • Breaking down photosynthesis stages
        • The light-dependent reactions
          • Conceptual overview of light dependent reactions
          • Light dependent reactions actors
          • Photosynthesis: Overview of the light-dependent reactions
        • The Calvin cycle
          • Photosynthesis: Calvin cycle
        • Photorespiration: C3, C4, and CAM plants
          • Photorespiration
          • C4 photosynthesis
          • CAM plants
      • Cell signaling
        • Skill check: Cell signaling
        • How cells signal to each other
          • Overview of cell signaling
          • Example of a signal transduction pathway
        • Communication in single-celled organisms
          • Cell signaling in yeast reproduction
      • Cell division
        • Skill check: Cell division
        • Introduction to cell division
          • Fertilization terminology: gametes, zygotes, haploid, diploid
          • Zygote differentiating into somatic and germ cells
          • Chromosomes, chromatids and chromatin
        • The cell cycle and mitosis
          • Interphase
          • Mitosis
        • Meiosis
          • Comparing mitosis and meiosis
          • Chromosomal crossover in meiosis I
          • Phases of meiosis I
          • Phases of meiosis II
        • Cell cycle regulation, cancer, and stem cells
          • Embryonic stem cells
          • Cancer
          • Apoptosis
      • Classical and molecular genetics
        • Skill check: Classical genetics
        • Mendelian genetics
          • Introduction to heredity
          • Worked example: Punnett squares
        • Variations on Mendelian genetics
          • Worked example: Punnett squares
        • Chromosomal basis of genetics
          • Boveri-Sutton chromosome theory
          • Thomas Hunt Morgan and fruit flies
        • Sex linkage, chromosomal mutations, & non-nuclear inheritance
          • Sex-linked traits
        • Molecular basis of genetics
          • DNA
          • DNA replication and RNA transcription and translation
          • Alleles and genes
      • DNA as the genetic material
        • DNA skill checks
        • Structure of DNA
          • DNA
          • Molecular structure of DNA
          • Antiparallel structure of DNA strands
        • Discovery of DNA
          • DNA as the "transforming principle"
          • Hershey and Chase: DNA is the genetic material
          • The discovery of the double helix structure of DNA
        • DNA replication
          • DNA replication and RNA transcription and translation
          • Leading and lagging strands in DNA replication
          • Speed and precision of DNA replication
          • Molecular structure of DNA
      • Central dogma (DNA to RNA to protein)
        • Central dogma skill check
        • Central dogma and the genetic code
          • DNA replication and RNA transcription and translation
          • Alleles and genes
        • Transcription
          • DNA replication and RNA transcription and translation
          • Transcription and mRNA processing
          • Molecular structure of RNA
        • Translation
          • DNA replication and RNA transcription and translation
          • Translation (mRNA to protein)
      • Gene regulation
        • Gene regulation in bacteria
          • Operons and gene regulation in bacteria
          • Trp operon
          • Lac operon
        • Gene regulation in eukaryotes
      • Biotechnology
        • Skill check: Biotechnology
        • Intro to biotechnology
        • DNA cloning
          • DNA cloning and recombinant DNA
        • DNA analysis methods
          • Polymerase chain reaction (PCR)
          • Gel electrophoresis
        • Stem cells
          • Embryonic stem cells
      • Developmental biology
        • Development & differentiation
          • Cellular specialization (differentiation)
          • Zygote differentiating into somatic and germ cells
          • Embryonic stem cells
        • Signaling & transcription factors in development
        • Apoptosis
          • Apoptosis
      • Bacteria and archaea
        • Prokaryotes skill check
        • Prokaryote structure
          • Bacteria
          • Prokaryotic and eukaryotic cells
        • Prokaryote metabolism and ecology
      • Viruses
        • Viruses
          • Viruses
      • Evolution and the tree of life
        • Skill check: Evolution and natural selection
        • Evolution and natural selection
          • Introduction to evolution and natural selection
          • Ape clarification
          • Intelligent design and evolution
          • Evolution clarification
          • Natural selection and the owl butterfly
          • Variation in a species
          • Evidence for evolution
        • Population genetics
          • Allele frequency
          • Hardy-Weinberg equation
          • Applying the Hardy-Weinberg equation
          • Discussions of conditions for Hardy Weinberg
          • Genetic drift, bottleneck effect, and founder effect
        • Speciation and evolutionary trees
          • Taxonomy and the tree of life
          • Species
          • Speciation
          • Biodiversity and natural selection
          • Genetic variation, gene flow, and new species
          • Discovering the tree of life
          • Understanding and building phylogenetic trees
          • 2015 AP Biology free response 3 - phylogenetic trees
      • History of life on Earth
        • Formation of Earth and early life
          • Earth formation
          • Beginnings of life
          • Origins of life
        • The diversification of life
          • Ozone layer and eukaryotes show up in the Proterozoic eon
          • First living things on land clarification
          • Biodiversity flourishes in Phanerozoic eon
          • Human evolution overview
        • Radiometric dating
          • Chronometric revolution
          • Carbon 14 dating 1
          • Carbon 14 dating 2
          • Potassium-argon (K-Ar) dating
          • K-Ar dating calculation
      • Ecology
        • Intro to ecology
          • Ecology introduction
        • Intro to population ecology
          • Life history strategies and fecundity
        • Population growth & regulation
          • Exponential and logistic growth in populations
          • Population regulation
          • Predator-prey cycles
          • Thomas Malthus and population growth
        • Intro to community ecology
          • Interactions between populations
          • Predator-prey cycles
        • Community structure & diversity
          • Ecosystems and ecological networks
          • Introduced species and biodiversity
          • Ecological succession
        • Intro to ecosystems
          • Ecosystems and biomes
          • Flow of energy and matter through ecosystems
        • Biogeochemical cycles
          • Biogeochemical cycles overview
          • The water cycle
          • The carbon cycle
          • The nitrogen cycle
          • Phosphorous cycle
          • Eutrophication and dead zones
        • Biogeography
          • How biodiversity is distributed globally
          • Why biodiversity is distributed unevenly
          • New localities lead to new biodiversity
      • Biodiversity and conservation
        • Ecosystems & ecosystem services
          • Ecosystems and ecological networks
          • Ecosystem services
        • Global distribution of biodiversity
          • What is a biodiversity hotspot?
          • How biodiversity is distributed globally
          • Why biodiversity is distributed unevenly
        • Threats to biodiversity
          • Human activities that threaten biodiversity
          • Introduced species and biodiversity
          • How does climate change affect biodiversity?
          • Demystifying ocean acidification and biodiversity impacts
          • Biodiversity and extinction, then and now
        • Protecting biodiversity
          • Conservation and the race to save biodiversity
          • Protecting biodiversity: the power of the individual
          • Protecting biodiversity: local and global policies
        • Levels of biodiversity
          • Genes and biodiversity
          • Ecosystem biodiversity
          • Discovering the tree of life
      • Behavioral biology
        • Animal behavior
          • Animal communication
          • Animal behavior: foraging
      • Principles of physiology
        • Body structure & homeostasis
        • Metabolism & thermoregulation
      • Human biology
        • Circulatory and pulmonary systems
          • The lungs and pulmonary system
          • Red blood cells
          • Circulatory system and the heart
          • Hemoglobin
        • The neuron and nervous system
          • Anatomy of a neuron
          • Sodium-potassium pump
          • Correction to sodium-potassium pump video
          • Electrotonic and action potentials
          • Saltatory conduction in neurons
          • Neuronal synapses (chemical)
          • Overview of the functions of the cerebral cortex
        • The kidney and nephron
          • The kidney and nephron
          • Secondary active transport in the nephron
        • Muscles
          • Myosin and actin
          • How tropomyosin and troponin regulate muscle contraction
          • Role of the sarcoplasmic reticulum in muscle cells
          • Anatomy of a skeletal muscle fiber
        • Immunology
          • Role of phagocytes in innate or nonspecific immunity
          • Types of immune responses: Innate and adaptive, humoral vs. cell-mediated
          • B lymphocytes (B cells)
          • Professional antigen presenting cells (APC) and MHC II complexes
          • Helper T cells
          • Cytotoxic T cells and MHC I complexes
          • Review of B cells, CD4+ T cells and CD8+ T cells
          • Inflammatory response
      • Plant biology
        • Plant responses to light
          • Phototropism
          • 2015 AP Biology free response 5
          • Photoperiodism
      • Crash Course: Biology and Ecology
        • Crash Course: Biology
          • Why carbon is everywhere
          • Water - Liquid awesome
          • Biological molecules - You are what you eat
          • Eukaryopolis - The city of animal cells
          • In da club - Membranes & transport
          • Plant cells
          • ATP & respiration
          • Photosynthesis
          • Heredity
          • DNA, hot pockets, & the longest word ever
          • Mitosis: Splitting up is complicated
          • Meiosis: Where the sex starts
          • Natural Selection
          • Speciation: Of ligers & men
          • Animal development: We're just tubes
          • Evolutionary development: Chicken teeth
          • Population genetics: When Darwin met Mendel
          • Taxonomy: Life's filing system
          • Evolution: It's a Thing
          • Comparative anatomy: What makes us animals
          • Simple animals: Sponges, jellies, & octopuses
          • Complex animals: Annelids & arthropods
          • Chordates
          • Animal behavior
          • The nervous system
          • Circulatory & respiratory systems
          • The digestive system
          • The excretory system: From your heart to the toilet
          • The skeletal system: It's ALIVE!
          • Big Guns: The Muscular System
          • Your immune system: Natural born killer
          • Great glands - Your endocrine system
          • The reproductive system: How gonads go
          • Old & Odd: Archaea, Bacteria & Protists
          • The sex lives of nonvascular plants
          • Vascular plants = Winning!
          • The plants & the bees: Plant reproduction
          • Fungi: Death Becomes Them
          • Ecology - Rules for living on earth
        • Crash Course: Ecology
          • The history of life on earth
          • Population ecology: The Texas mosquito mystery
          • Human population growth
          • Community ecology: Feel the love
          • Community ecology II: Predators
          • Ecological succession: Change is good
          • Ecosystem ecology: Links in the chain
          • The hydrologic and carbon cycles: Always recycle!
          • Nitrogen and phosphorus cycles: Always recycle!
          • 5 human impacts on the environment
          • Pollution
          • Conservation and restoration ecology
      • Meet the biology professional
        • Meet the biology professional
    • AP Biology
      • Intro to biology
        • Welcome to biology!
          • Biology overview
        • The science of biology
          • The scientific method
      • Chemistry of life
        • Elements and atoms
          • Elements and atoms
          • Introduction to the atom
        • Electron shells and orbitals
          • Orbitals
          • More on orbitals and electron configuration
          • Valence electrons and bonding
          • Groups of the periodic table
        • Chemical bonds and reactions
          • Ionic, covalent, and metallic bonds
          • Electronegativity
          • Electronegativity and bonding
          • Intermolecular forces
          • Chemical reactions introduction
      • Water, acids, and bases
        • Hydrogen bonding in water
          • Hydrogen bonding in water
          • Water as a solvent
        • Cohesion and adhesion
          • Capillary action and why we see a meniscus
          • Surface tension
        • Temperature and state changes in water
          • LeBron Asks: Why does sweating cool you down?
          • Evaporative cooling
          • Heat of vaporization of water and ethanol
          • Specific heat of water
          • Liquid water denser than solid water (ice)
        • Acids, bases, and pH
          • Autoionization of water
          • Arrhenius definition of acids and bases
          • Bronsted-Lowry definition of acids and bases
          • Definition of pH
          • Introduction to buffers
      • Properties of carbon
        • Carbon
          • Carbon as a building block of life
          • Silicon-based life
          • Representing structures of organic molecules
        • Hydrocarbon structures and functional groups
          • Hydrocarbon overview
          • Isomers
          • Functional groups
      • Macromolecules
        • Introduction to macromolecules
        • Carbohydrates
          • Molecular structure of glucose
          • Dehydration synthesis or a condensation reaction
          • Hydrolysis
          • Molecular structure of fructose
        • Lipids
          • Molecular structure of triglycerides (fats)
          • Saturated fats, unsaturated fats, and trans fats
          • Lipid overview
        • Nucleic acids
          • DNA
          • DNA replication and RNA transcription and translation
          • Molecular structure of DNA
          • Antiparallel structure of DNA strands
          • Molecular structure of RNA
        • Proteins
          • Introduction to amino acids
          • Peptide bond formation
          • Overview of protein structure
          • Tertiary structure of proteins
      • Energy and enzymes
        • Energy in metabolism
          • Introduction to metabolism: Anabolism and catabolism
        • Laws of thermodynamics
          • Introduction to energy
          • First Law of Thermodynamics introduction
          • Introduction to entropy
          • Second Law of Thermodynamics
          • Second Law of Thermodynamics and entropy
          • Why heat increases entropy
        • Free energy
          • Gibbs free energy and spontaneous reactions
          • Endergonic, exergonic, exothermic, and endothermic
        • ATP and reaction coupling
          • ATP: Adenosine triphosphate
          • ATP hydrolysis mechanism
          • Reaction coupling to create glucose-6-phosphate
        • Introduction to enzymes
          • Introduction to kinetics
          • Enzymes
        • Enzyme regulation
          • Enzyme cofactors and coenzymes
          • Competitive inhibition
          • Noncompetitive inhibition
      • Structure of a cell
        • Introduction to cells
          • Scale of cells
          • Cell theory
        • Prokaryotic and eukaryotic cells
          • Cell size
          • Prokaryotic and eukaryotic cells
        • Tour of a eukaryotic cell
          • Endoplasmic reticulum and Golgi bodies
          • Endomembrane system
          • Mitochondria
          • The cytoskeleton
        • Extracellular structures and cell-cell junctions
          • Extracellular matrix
          • Plant cell walls
          • Overview of animal and plant cells
      • Membranes and transport
        • The plasma membrane
          • Fluid mosaic model of cell membranes
        • Diffusion and osmosis
          • Diffusion - Introduction
          • Concentration gradients
          • Osmosis
          • Hypotonic, isotonic, and hypertonic solutions (tonicity)
          • Diffusion and osmosis
        • Passive transport
          • Passive transport and selective permeability
          • Facilitated diffusion
        • Active transport
          • Sodium potassium pump
          • Electrochemical gradients and secondary active transport
          • Uniporters, symporters and antiporters
        • Bulk transport
          • Endocytosis, phagocytosis, and pinocytosis
          • Exocytosis
      • Cellular respiration
        • Introduction to cellular respiration
          • ATP: Adenosine triphosphate
          • ATP hydrolysis mechanism
          • Cellular respiration introduction
          • Oxidation and reduction review from biological point-of-view
          • Oxidation and reduction in cellular respiration
        • Steps of cellular respiration
          • Overview of cellular respiration
        • Glycolysis
          • Overview of glycolysis
          • Steps of glycolysis
        • Pyruvate oxidation and the citric acid cycle
          • Krebs / citric acid cycle
        • Oxidative phosphorylation
          • Oxidative phosphorylation and the electron transport chain
        • Variations on cellular respiration
          • Lactic acid fermentation
          • Alcohol or ethanol fermentation
      • Photosynthesis
        • Intro to photosynthesis
          • ATP: Adenosine triphosphate
          • ATP hydrolysis mechanism
          • Photosynthesis
          • Breaking down photosynthesis stages
        • The light-dependent reactions
          • Conceptual overview of light dependent reactions
          • Light dependent reactions actors
          • Photosynthesis: Overview of the light-dependent reactions
        • The Calvin cycle
          • Photosynthesis: Calvin cycle
        • Photorespiration: C3, C4, and CAM plants
          • Photorespiration
          • C4 photosynthesis
          • CAM plants
      • Cell signaling
        • How cells signal to each other
          • Overview of cell signaling
          • Example of a signal transduction pathway
        • Communication in single-celled organisms
          • Cell signaling in yeast reproduction
      • Cell division
        • Introduction to cell division
          • Fertilization terminology: gametes, zygotes, haploid, diploid
          • Zygote differentiating into somatic and germ cells
          • Chromosomes, chromatids and chromatin
        • The cell cycle and mitosis
          • Interphase
          • Mitosis
        • Meiosis
          • Comparing mitosis and meiosis
          • Chromosomal crossover in meiosis I
          • Phases of meiosis I
          • Phases of meiosis II
        • Cell cycle regulation, cancer, and stem cells
          • Embryonic stem cells
          • Cancer
          • Apoptosis
      • Classical and molecular genetics
        • Mendelian genetics
          • Introduction to heredity
          • Worked example: Punnett squares
        • Variations on Mendelian genetics
          • Worked example: Punnett squares
        • Chromosomal basis of genetics
          • Boveri-Sutton chromosome theory
          • Thomas Hunt Morgan and fruit flies
        • Sex linkage, chromosomal mutations, & non-nuclear inheritance
          • Sex-linked traits
        • Molecular basis of genetics
          • DNA
          • DNA replication and RNA transcription and translation
          • Alleles and genes
      • DNA as the genetic material
        • Structure of DNA
          • DNA
          • Molecular structure of DNA
          • Antiparallel structure of DNA strands
        • Discovery of DNA
          • DNA as the "transforming principle"
          • Hershey and Chase: DNA is the genetic material
          • The discovery of the double helix structure of DNA
        • DNA replication
          • DNA replication and RNA transcription and translation
          • Leading and lagging strands in DNA replication
          • Speed and precision of DNA replication
          • Molecular structure of DNA
      • Central dogma (DNA to RNA to protein)
        • Central dogma and the genetic code
          • DNA replication and RNA transcription and translation
          • Alleles and genes
        • Transcription
          • DNA replication and RNA transcription and translation
          • Transcription and mRNA processing
          • Molecular structure of RNA
        • Translation
          • DNA replication and RNA transcription and translation
          • Translation (mRNA to protein)
      • Gene regulation
        • Gene regulation in bacteria
          • Operons and gene regulation in bacteria
          • Trp operon
          • Lac operon
        • Gene regulation in eukaryotes
      • Biotechnology
        • Intro to biotechnology
        • DNA cloning
          • DNA cloning and recombinant DNA
        • DNA analysis methods
          • Polymerase chain reaction (PCR)
          • Gel electrophoresis
        • Stem cells
          • Embryonic stem cells
      • Developmental biology
        • Development & differentiation
          • Cellular specialization (differentiation)
          • Zygote differentiating into somatic and germ cells
          • Embryonic stem cells
        • Signaling & transcription factors in development
        • Apoptosis
          • Apoptosis
      • Bacteria and archaea
        • Prokaryote structure
          • Bacteria
          • Prokaryotic and eukaryotic cells
        • Prokaryote metabolism and ecology
      • Viruses
        • Viruses
          • Viruses
      • Evolution and the tree of life
        • Evolution and natural selection
          • Introduction to evolution and natural selection
          • Ape clarification
          • Intelligent design and evolution
          • Evolution clarification
          • Natural selection and the owl butterfly
          • Variation in a species
          • Evidence for evolution
        • Population genetics
          • Allele frequency
          • Hardy-Weinberg equation
          • Applying the Hardy-Weinberg equation
          • Discussions of conditions for Hardy Weinberg
          • Genetic drift, bottleneck effect, and founder effect
        • Speciation and evolutionary trees
          • Taxonomy and the tree of life
          • Species
          • Speciation
          • Biodiversity and natural selection
          • Genetic variation, gene flow, and new species
          • Discovering the tree of life
          • Understanding and building phylogenetic trees
          • 2015 AP Biology free response 3 - phylogenetic trees
      • History of life on Earth
        • Formation of Earth and early life
          • Earth formation
          • Beginnings of life
          • Origins of life
        • The diversification of life
          • Ozone layer and eukaryotes show up in the Proterozoic eon
          • First living things on land clarification
          • Biodiversity flourishes in Phanerozoic eon
          • Human evolution overview
        • Radiometric dating
          • Chronometric revolution
          • Carbon 14 dating 1
          • Carbon 14 dating 2
          • Potassium-argon (K-Ar) dating
          • K-Ar dating calculation
      • Ecology
        • Intro to ecology
          • Ecology introduction
        • Intro to population ecology
          • Life history strategies and fecundity
        • Population growth & regulation
          • Exponential and logistic growth in populations
          • Population regulation
          • Predator-prey cycles
          • Thomas Malthus and population growth
        • Intro to community ecology
          • Interactions between populations
          • Predator-prey cycles
        • Community structure & diversity
          • Ecosystems and ecological networks
          • Introduced species and biodiversity
          • Ecological succession
        • Intro to ecosystems
          • Ecosystems and biomes
          • Flow of energy and matter through ecosystems
        • Biogeochemical cycles
          • Biogeochemical cycles overview
          • The water cycle
          • The carbon cycle
          • The nitrogen cycle
          • Phosphorous cycle
          • Eutrophication and dead zones
        • Biogeography
          • How biodiversity is distributed globally
          • Why biodiversity is distributed unevenly
          • New localities lead to new biodiversity
      • Biodiversity and conservation
        • Ecosystems & ecosystem services
          • Ecosystems and ecological networks
          • Ecosystem services
        • Global distribution of biodiversity
          • What is a biodiversity hotspot?
          • How biodiversity is distributed globally
          • Why biodiversity is distributed unevenly
        • Threats to biodiversity
          • Human activities that threaten biodiversity
          • Introduced species and biodiversity
          • How does climate change affect biodiversity?
          • Demystifying ocean acidification and biodiversity impacts
          • Biodiversity and extinction, then and now
        • Protecting biodiversity
          • Conservation and the race to save biodiversity
          • Protecting biodiversity: the power of the individual
          • Protecting biodiversity: local and global policies
        • Levels of biodiversity
          • Genes and biodiversity
          • Ecosystem biodiversity
          • Discovering the tree of life
      • Behavioral biology
        • Animal behavior
          • Animal communication
          • Animal behavior: foraging
      • Principles of physiology
        • Body structure & homeostasis
        • Metabolism & thermoregulation
      • Human biology
        • Circulatory and pulmonary systems
          • The lungs and pulmonary system
          • Red blood cells
          • Circulatory system and the heart
          • Hemoglobin
        • The neuron and nervous system
          • Anatomy of a neuron
          • Sodium-potassium pump
          • Correction to sodium-potassium pump video
          • Electrotonic and action potentials
          • Saltatory conduction in neurons
          • Neuronal synapses (chemical)
          • Overview of the functions of the cerebral cortex
        • The kidney and nephron
          • The kidney and nephron
          • Secondary active transport in the nephron
        • Muscles
          • Myosin and actin
          • How tropomyosin and troponin regulate muscle contraction
          • Role of the sarcoplasmic reticulum in muscle cells
          • Anatomy of a skeletal muscle fiber
        • Immunology
          • Role of phagocytes in innate or nonspecific immunity
          • Types of immune responses: Innate and adaptive, humoral vs. cell-mediated
          • B lymphocytes (B cells)
          • Professional antigen presenting cells (APC) and MHC II complexes
          • Helper T cells
          • Cytotoxic T cells and MHC I complexes
          • Review of B cells, CD4+ T cells and CD8+ T cells
          • Inflammatory response
      • Plant biology
        • Plant responses to light
          • Phototropism
          • 2015 AP Biology free response 5
          • Photoperiodism
      • Meet the biology professional
        • Meet the biology professional
    • Cosmology and astronomy
      • Scale of the universe
        • Scale of earth, sun, galaxy and universe
          • Scale of the large
          • Scale of the small
          • Scale of earth and sun
          • Scale of solar system
          • Scale of distance to closest stars
          • Scale of the galaxy
          • Intergalactic scale
          • Hubble image of galaxies
        • Time scale of the cosmos
          • Cosmological time scale 1
          • Cosmological time scale 2
        • Light and fundamental forces
          • Introduction to light
          • Four fundamental forces
        • Big bang and expansion of the universe
          • Big bang introduction
          • Radius of observable universe
          • Radius of observable universe (correction)
          • Red shift
          • Cosmic background radiation
          • Cosmic background radiation 2
          • Hubble's law
          • A universe smaller than the observable
      • Stars, black holes and galaxies
        • Life and death of stars
          • Birth of stars
          • Accreting mass due to gravity simulation
          • Becoming a red giant
          • White and black dwarfs
          • Star field and nebula images
          • Lifecycle of massive stars
          • Supernova (supernovae)
          • Supernova clarification
          • Black holes
          • Supermassive black holes
        • Quasars and galactive collisions
          • Quasars
          • Quasar correction
          • Galactic collisions
        • Stellar parallax
          • Parallax in observing stars
          • Stellar parallax
          • Stellar distance using parallax
          • Stellar parallax clarification
          • Parsec definition
        • Cepheid variables
          • Cepheid variables 1
          • Why cepheids pulsate
          • Why gravity gets so strong near dense objects
      • Earth geological and climatic history
        • Plate tectonics
          • Plate tectonics: Difference between crust and lithosphere
          • Structure of the earth
          • Plate tectonics: Evidence of plate movement
          • Plate tectonics: Geological features of divergent plate boundaries
          • Plate tectonics: Geological features of convergent plate boundaries
          • Plates moving due to convection in mantle
          • Hawaiian islands formation
          • Pangaea
          • Compositional and mechanical layers of the earth
          • How we know about the earth's core
        • Seismic waves and how we know earth's structure
          • Seismic waves
          • Why S-waves only travel in solids
          • Refraction of seismic waves
          • The mohorovicic seismic discontinuity
        • Earth's rotation and tilt
          • Seasons aren't dictated by closeness to sun
          • How Earth's tilt causes seasons
          • Are southern hemisphere seasons more severe?
          • Milankovitch cycles precession and obliquity
          • Precession causing perihelion to happen later
          • What causes precession and other orbital changes
          • Apsidal precession (perihelion precession) and Milankovitch cycles
        • Moon phases and eclipses
          • Intro to Moon phases
          • Solar and lunar eclipses
      • Life on earth and in the universe
        • History of life on earth
          • Earth formation
          • Beginnings of life
          • Ozone layer and eukaryotes show up in the Proterozoic eon
          • Biodiversity flourishes in Phanerozoic eon
          • First living things on land clarification
        • Humanity on earth
          • Human evolution overview
          • Understanding calendar notation
          • Correction calendar notation
          • Development of agriculture and writing
          • Firestick farming in early Australia
          • Collective learning
          • Thomas Malthus and population growth
          • Agriculture and human population
          • Energy inputs for tilling a hectare of land
          • Random predictions for 2060
        • Measuring age on earth
          • Chronometric revolution
          • Carbon 14 dating 1
          • Carbon 14 dating 2
          • Potassium-argon (K-Ar) dating
          • K-Ar dating calculation
        • Life in the universe
          • Detectable civilizations in our galaxy 1
          • Detectable civilizations in our galaxy 2
          • Detectable civilizations in our galaxy 3
          • Detectable civilizations in our galaxy 4
          • Detectable civilizations in our galaxy 5
    • Electrical engineering
      • Introduction to electrical engineering
        • Getting started
          • Current
          • Current direction
          • Voltage
          • Conventional current
      • Circuit analysis
        • Circuit elements
          • Ideal circuit elements
          • Ideal sources
          • Circuit terminology
          • Sign convention for passive components
        • Resistor circuits
          • Series resistors
          • Parallel resistors (part 1)
          • Parallel resistors (part 2)
          • Parallel resistors (part 3)
          • Simplifying resistor networks
          • Voltage divider
          • Analyzing a resistor circuit with two batteries
        • DC circuit analysis
          • Kirchhoff's current law
          • Kirchhoff's voltage law
          • Labeling voltages
          • Application of the fundamental laws (setup)
          • Application of the fundamental laws (solve)
          • Node voltage method (steps 1 to 4)
          • Node voltage method (step 5)
          • Mesh current method (steps 1 to 3)
          • Mesh current method (step 4)
        • Natural and forced response
          • Capacitor i-v equations
          • A capacitor integrates current
          • Inductor equations
          • Inductor kickback (1 of 2)
          • Inductor kickback (2 of 2)
          • RC natural response - intuition
          • RC natural response - derivation
          • RC natural response - example
          • RC step response - intuition
          • RC step response setup (1 of 3)
          • RC step response solve (2 of 3)
          • RC step response example (3 of 3)
          • Sketching exponentials
          • Sketching exponentials - examples
          • LC natural response intuition 1
          • LC natural response intuition 2
          • LC natural response derivation 1
          • LC natural response derivation 2
          • LC natural response derivation 3
          • LC natural response derivation 4
          • LC natural response example
        • AC circuit analysis
          • AC analysis intro 1
          • AC analysis intro 2
          • Trigonometry review
          • Sine and cosine come from circles
          • Sine of time
          • Sine and cosine from rotating vector
          • Lead Lag
          • Complex numbers
          • Multiplying by j is rotation
          • Complex rotation
          • Euler's formula
          • Complex exponential magnitude
          • Complex exponentials spin
          • Euler's sine wave
          • Euler's cosine wave
          • Negative frequency
          • AC analysis superposition
          • Impedance
          • Impedance vs frequency
          • ELI the ICE man
          • Impedance of simple networks
          • KVL in the frequency domain
      • Amplifiers
        • Operational amplifier
          • What is an operational amplifier?
          • Non-inverting op-amp
          • Feedback
          • Inverting op-amp
          • Virtual ground
          • Virtual ground - examples
          • Summing op-amp
      • Semiconductor devices
        • Diode
          • Diode
          • Diode graphical solution
      • Electrostatics
        • Electric force and electric field
          • Coulomb's Law
        • Fields, potential, and voltage
          • Proof: Field from infinite plate (part 1)
          • Proof: Field from infinite plate (part 2)
          • Electric potential energy
      • Signals and systems
        • Fourier series
          • Fourier Series introduction
          • Integral of sin(mt) and cos(mt)
          • Integral of sine times cosine
          • Integral of product of sines
          • Integral of product of cosines
          • First term in a Fourier series
          • Fourier coefficients for cosine terms
          • Fourier coefficients for sine terms
          • Finding Fourier coefficients for square wave
          • Visualizing the Fourier expansion of a square wave
      • Home-made robots
        • All about Spout
          • 3rd graders build robots at Santa Rita Elementary School
          • SPDT switch
          • Spout's sliding SPDT switch
          • Spout lights
          • Spout motor
          • Spout battery configuration and polarization
          • Spout's reversing circuit and final assembly
        • Spout Bot with Solder
          • Parts for Spout
          • Spout tools
          • Soldering Safety
          • Spout switches
          • Spout motor mount
          • Spout motor wiring
          • Spout wiring for LED eyes
          • Spout antennae and tail
        • Spout Bot without Solder
          • Tools and parts to build a Spout
          • Connect the SPDT switches
          • Attach the LED eyes
          • Wire the SPDT switches
          • Create the motor mounts
          • Secure and wire the motors
          • Install on/off switches
          • Connect the LEDs to an on/off switch
          • Attach Spout's antennae
          • Add Spout's tail
          • Give Spout some grippy feet
        • Spider Bot
          • 6th graders learn to build a Spider robot
          • Fun with Spider Bot
          • Spider parts and tools
          • Spider's click n' stick
          • Battery and motor mounts for Spider
          • Click n' stick base & batteries
          • Spider's motor controller
          • Spider's power switch
          • Spider's bezel
          • Spider's wheels
          • Spider's Arduino Nano
          • Motor controller connections
          • Spider's LED eyes
          • Spider's stabilizer bar
          • Spider's romance
          • Ben Eater's Spider
        • Bit-zee Bot
          • Bit-zeeeeeeeeeee
          • Bit-zeeeeeeeeeee (long version)
          • Introduction
          • Planning and propulsion
          • Parts
          • Chassis/frame
          • Wheel mounts and fenders
          • Component mounting holes
          • Batteries/power
          • Battery wires
          • Power wires and on/off switch
          • Motors/propulsion
          • Motor controller functions
          • Motor controller
          • Motor controller connections
          • Arduino connections
          • Digital camera connections
          • Digital camera connections II
          • 5 volt power distribution board
          • Digital recorder/player connections
          • Power connector for the Arduino
          • Prototype board
          • Motor controller connection to Arduino
          • Camera connection to the Arduino
          • Bumper switches
          • LED eyes
          • IR sensor
          • Chassis modifications
          • Camera wiring update
      • Lego robotics
        • Introduction
          • Anything is possible!
          • DLab: Inspired by sumo robots
          • Introduction to programming using lego NXT
        • Light guitar
          • Demo: light guitar
          • Data wires
          • Math block
        • Coin detector
          • Demo: coin detector
          • Construction
          • Rotation sensor
          • Switch statements & logic
          • Design review (calibration)
          • Polling sensors
        • Ant bot
          • Demo: ant bot
          • Robot construction
          • Move block
          • Switch block (nested)
        • Programming basics
          • Move block
          • Loop block
          • Switch block
          • Switch block (nested)
          • Switch statements & logic
          • Math block
          • Variable block (counter)
          • Data wires
          • Polling sensors
      • Reverse engineering
        • Household items
          • What is inside an alarm clock radio?
          • What is inside a coffee maker?
          • What is inside a tap light?
        • Hair dryer
          • What is inside a hair dryer? (1 of 2)
          • What is inside a hair dryer? (2 of 2)
          • Compare the hair dryer motor to the one you can build
        • DVD player
          • What is inside a DVD player? (1 of 5)
          • What is inside a DVD player? (2 of 5)
          • What is inside a DVD player? (3 of 5)
          • What is inside a DVD player? (4 of 5)
          • What is inside a DVD player? (5 of 5)
          • What is inside a universal remote control?
        • Digital camera
          • What is inside a digital camera? (1 of 2)
          • What is inside a digital camera? (2 of 2)
    • Health and medicine
      • Human anatomy and physiology
        • Circulatory system introduction
          • Meet the heart!
          • Flow through the heart
          • Two circulations in the body
          • Lub dub
          • Layers of the heart
          • Thermoregulation in the circulatory system
          • Arteries vs. veins - what's the difference?
          • Arteries, arterioles, venules, and veins
          • Circulatory system and the heart
        • Respiratory system introduction
          • Meet the lungs
          • People and plants
          • The bronchial tree
          • Inhaling and exhaling
          • How does lung volume change?
          • Thermoregulation in the lungs
          • The lungs and pulmonary system
        • Urinary system introduction
          • Meet the kidneys!
          • Kidney function and anatomy
          • Glomerular filtration in the nephron
          • Changing glomerular filtration rate
          • Countercurrent multiplication in the kidney
          • Urination
          • The kidney and nephron
          • Secondary active transport in the nephron
        • Hematologic system introduction
          • What's inside of blood?
          • Red blood cells
          • Blood types
          • Blood cell lineages
          • Life and times of RBCs and platelets
          • Hemoglobin
          • Hemoglobin moves O2 and CO2
          • Fetal hemoglobin and hematocrit
          • Oxygen content
          • How do we make blood clots?
          • Coagulation cascade
          • Bohr effect vs. Haldane effect
        • Immunologic system introduction
          • Role of phagocytes in innate or nonspecific immunity
          • Types of immune responses: Innate and adaptive, humoral vs. cell-mediated
          • B lymphocytes (B cells)
          • Professional antigen presenting cells (APC) and MHC II complexes
          • Helper T cells
          • Cytotoxic T cells
          • Review of B cells, CD4+ T cells and CD8+ T cells
          • Clonal selection
          • Self vs. non-self immunity
          • How white blood cells move around
          • Inflammatory response
          • Blood cell lineages
        • Gastrointestinal system introduction
          • Meet the gastrointestinal tract!
          • Mouth
          • Teeth
          • Esophagus
          • Stomach
          • Small intestine 1: Structure
          • Small intestine 2: Digestion
          • Small intestine 3: Absorption
          • Liver
          • Hepatic lobule
          • Biliary tree
          • Exocrine pancreas
          • Endocrine pancreas
          • Colon, rectum, and anus
          • Control of the GI tract
        • Nervous system introduction
          • Introduction to neural cell types
          • Anatomy of a neuron
          • Overview of neuron structure
          • Overview of neuron function
          • Sodium-potassium pump
          • Correction to sodium-potassium pump video
          • Electrotonic and action potentials
          • Saltatory conduction in neurons
          • Synapse structure
          • Neuronal synapses (chemical)
          • Types of neurotransmitters
          • Types of neurotransmitter receptors
          • Structure of the nervous system
          • Functions of the nervous system
          • Motor unit
          • Peripheral somatosensation
          • Muscle stretch reflex
          • Autonomic nervous system
          • Upper motor neurons
          • Somatosensory tracts
          • Cerebral cortex
        • Muscular system introduction
          • Myosin and actin
          • How tropomyosin and troponin regulate muscle contraction
          • Role of the sarcoplasmic reticulum in muscle cells
          • Anatomy of a skeletal muscle cell
          • Three types of muscle
          • Motor neurons
          • Neuromuscular junction, motor end-plate
          • Type 1 and type 2 muscle fibers
          • Calcium puts myosin to work
          • Muscle innervation
          • Autonomic vs somatic nervous system
          • Thermoregulation by muscles
        • Skeletal system introduction
          • Skeletal structure and function
          • Microscopic structure of bone - the Haversian system
          • Cellular structure of bone
          • Skeletal endocrine control
          • Cartilage
          • Ligaments, tendons, and joints
        • Endocrine system introduction
          • Endocrine gland hormone review
          • The hypothalamus and pituitary gland
          • Hormone concentration metabolism and negative feedback
          • Types of hormones
          • Cellular mechanism of hormone action
        • Integumentary system introduction
          • Meet the skin! (Overview)
          • What is skin? (Epidermis)
          • What lies beneath the epidermis? (Dermis and Hypodermis)
          • Where do our nails and hair come from?
          • What's in sweat? (Holocrine, Apocrine, Merocrine Glands)
          • LeBron Asks: Why does sweating cool you down?
          • Overview of Sensation and Meissner's Corpuscle
          • Pacinian's Corpuscle and Merkel's Disk
          • Ruffini's Ending and Hair Follicle Receptor
          • Pain and temperature
          • Thermoregulation by muscles
        • Lymphatic system introduction
          • Why we need a lymphatic system
          • How lymphatic vessels move fluid
          • The lymphatic system's role in immunity
          • Lipid and protein transport in the lymphatic system
          • What is actually in lymph
        • Reproductive system introduction
          • Welcome to the reproductive system
          • Anatomy of the male reproductive system
          • Transport of sperm via erection and ejaculation
          • Spermatogenesis
          • Testosterone
          • Basics of egg development
          • The ovarian cycle
          • Meet the placenta!
          • Reproductive cycle graph - Follicular phase
          • Reproductive cycle graph - Luteal phase
          • Estrogen
          • Maternal changes in pregnancy
          • Labor (parturition)
          • Breast anatomy and lactation
        • Pregnancy and pregnancy complications
          • Diagnosis of pregnancy
          • Pregnancy physiology I
          • Labor
          • Postpartum physiology
          • Preeclampsia
          • Placenta previa
          • Placental abruption
          • Placenta accreta
          • UTIs in pregnancy
          • Blood conditions in pregnancy
          • Sheehan syndrome
          • Postpartum hemorrhage
          • Uterine inversion
          • Diabetes in pregnancy
          • Preterm labor
          • C-section
      • Advanced circulatory system physiology
        • Circulatory system introduction
          • Meet the heart!
          • Flow through the heart
          • Two circulations in the body
          • Lub dub
          • Layers of the heart
          • Thermoregulation in the circulatory system
          • Arteries vs. veins - what's the difference?
          • Arteries, arterioles, venules, and veins
          • Circulatory system and the heart
        • Blood pressure
          • What is blood pressure?
          • Learn how a stethoscope can help determine blood pressure
          • Resistance in a tube
          • Adding up resistance in series and in parallel
          • Adding up resistance problem
          • Flow and perfusion
          • Putting it all together: Pressure, flow, and resistance
          • Blood pressure changes over time
        • Blood pressure control
          • Regulation of blood pressure with baroreceptors
          • Parts of a nephron
          • General overview of the RAAS system: Cells and hormones
          • Renin production in the kidneys
          • Activating angiotensin 2
          • Angiotensin 2 raises blood pressure
          • Aldosterone raises blood pressure and lowers potassium
          • Aldosterone removes acid from the blood
          • ADH secretion
          • ADH effects on blood pressure
          • Aldosterone and ADH
        • Fetal circulation
          • Meet the placenta!
          • Umbilical vessels and the ductus venosus
          • Flow through the heart
          • Hypoxic pulmonary vasoconstriction
          • Foramen ovale and ductus arteriosus
          • Fetal hemoglobin and hematocrit
          • Double Bohr effect
          • Fetal circulation right before birth
          • Baby circulation right after birth
          • Fetal structures in an adult
        • Blood vessels
          • Layers of a blood vessel
          • Three types of capillaries
          • Pre-capillary sphincters
        • Arterial stiffness
          • Compliance and elastance
          • Bernoulli's equation of total energy
          • Stored elastic energy in large and middle sized arteries
          • Compliance - decreased blood pressure
          • Compliance - increased blood flow
        • Heart muscle contraction
          • Three types of muscle
          • Heart cells up close!
          • Myosin and actin
          • How tropomyosin and troponin regulate muscle contraction
          • Calcium puts myosin to work
          • Sympathetic nerves affect myosin activity
          • Why doesn't the heart rip?
        • Heart depolarization
          • Membrane potentials - part 1
          • Membrane potentials - part 2
          • Permeability and membrane potentials
          • Action potentials in pacemaker cells
          • Action potentials in cardiac myocytes
          • Resetting cardiac concentration gradients
          • Electrical system of the heart
          • Depolarization waves flowing through the heart
          • A race to keep pace!
          • Thinking about heartbeats
          • New perspective on the heart
        • Nerve regulation of the heart
          • Changing the AV node delay - chronotropic effect
          • Changing the heart rate - chronotropic effect
          • Increasing ventricular contractility - inotropic effect
          • Autonomic nervous system effects on the heart
          • Getting a new heart
        • Preload and afterload
          • Why doesn't the heart rip?
          • What is preload?
          • Preload and pressure
          • Preload stretches out the heart cells
          • Frank-Starling mechanism
          • Sarcomere length-tension relationship
          • Active contraction vs. passive recoil
          • What is afterload?
          • Increasing the heart's force of contraction
        • Pressure volume loops
          • Pressure in the left heart - part 1
          • Pressure in the left heart - part 2
          • Pressure in the left heart - part 3
          • Left ventricular pressure vs. time
          • Left ventricular volume vs. time
          • Drawing a pressure-volume loop
          • Understanding the pressure-volume loop
          • End diastolic pressure-volume relationship (EDPVR)
          • End systolic pressure-volume relationship (ESPVR)
          • Reimagine the pressure volume relationship
          • What is preload?
          • Why doesn't the heart rip?
        • Changing the PV loop
          • What is preload?
          • What is afterload?
          • Increasing the heart's force of contraction
          • Reimagine the pressure volume relationship
          • What is contractility?
          • Getting Ea (arterial elastance) from the PV loop
          • Arterial elastance (Ea) and afterload
          • Arterial elastance (Ea) and preload
          • Stroke work in PV loops and boxes
          • Contractility, Ea, and preload effects on PV boxes
      • Circulatory system diseases
        • Coronary artery disease
          • What is coronary artery disease?
          • Risk factors for coronary artery disease
          • Atherosclerosis
          • Heart attack (myocardial infarction) pathophysiology
          • Heart attack (myocardial infarct) diagnosis
          • Heart attack (myocardial infarct) medications
          • Heart attack (myocardial infarction) interventions and treatment
          • Healing after a heart attack (myocardial infarction)
          • Complications after a heart attack (myocardial infarction)
        • Atherosclerosis, arteriosclerosis, and arteriolosclerosis
          • Arteries vs. veins - what's the difference?
          • Arteriosclerosis, arteriolosclerosis, and atherosclerosis
          • Atherosclerosis - part 1
          • Atherosclerosis - part 2
          • Arteriolosclerosis - part 1
          • Arteriolosclerosis - part 2
        • Stroke
          • Cerebral blood supply: Part 1
          • Cerebral blood supply: Part 2
          • What is a stroke?
          • Risk factors for stroke
          • Ischemic stroke
          • Hemorrhagic strokes
          • Ischemic core and penumbra
          • The ischemic cascade in stroke
          • Blood brain barrier and vasogenic edema
          • Post stroke inflammation
          • How do you know if someone is having a stroke: Think FAST!
          • Common stroke signs and symptoms
          • Diagnosing strokes by history and physical exam
          • Diagnosing strokes with imaging CT, MRI, and Angiography
          • Diagnosing strokes with lab tests
          • Acute treatment of stroke with medications
          • Treatment of stroke with interventions
          • Preventing further strokes
        • Hypertension
          • What is hypertension?
          • Hypertension symptoms and categories
          • Hypertension effects on the blood vessels
          • Hypertension effects on the heart
          • 4 lifestyle changes to help manage hypertension
        • Heart failure
          • What is heart failure?
          • Systolic heart failure pathophysiology
          • Diastolic heart failure pathophysiology
          • Compensation and decompensation in heart failure
          • Symptoms of left sided heart failure
          • Symptoms of right sided heart failure
          • Heart failure diagnosis
          • Heart failure treatment - Early stages
          • Heart failure treatment - Late stages
          • Heart failure treatment - Devices and surgery
        • Aortic dissection and aneurysm
          • What is an aortic dissection?
          • Thoracic aortic aneurysms
          • Abdominal aortic aneurysms
          • Aortic dissection treatment
          • Aneurysm treatment
        • Myocarditis and pericarditis
          • What is myocarditis and pericarditis?
          • Causes of myocarditis
          • Causes of pericarditis
          • Diagnosis of myocarditis and pericarditis (part 1)
          • Diagnosis of myocarditis and pericarditis (part 2)
          • Myocarditis and pericarditis treatment
        • Cardiomyopathy
          • What is cardiomyopathy?
          • Cardiomyopathy signs and symptoms
          • Dilated cardiomyopathy: Pathophysiology and diagnosis
          • Hypertrophic cardiomyopathy: Pathophysiology and diagnosis
          • Restrictive cardiomyopathy: Pathophysiology and diagnosis
          • Cardiomyopathy treatment
        • Cardiac dysrhythmias and tachycardias
          • Electrical conduction in heart cells
          • Normal sinus rhythm on an EKG
          • Supraventricular tachycardia (SVT)
          • Atrial fibrillation (Afib)
          • Atrial flutter (AFL)
          • Multifocal atrial tachycardia (MAT)
          • Atrioventricular reentrant tachycardia (AVRT) & AV nodal reentrant tachycardia (AVNRT)
          • Ventricular tachycardia (Vtach)
          • Torsades de pointes
          • What is ventricle fibrillation (Vfib)?
          • Pulseless electrical activity (PEA) and asystole
          • Electrocardioversion
          • Pacemakers
          • Antiarrhythmics
          • Ablation
        • Heart valve diseases
          • What is valvular heart disease?
          • Valvular heart disease causes
          • How to identify murmurs
          • Systolic murmurs, diastolic murmurs, and extra heart sounds - Part 1
          • Systolic murmurs, diastolic murmurs, and extra heart sounds - Part 2
          • Aortic stenosis and aortic regurgitation
          • Mitral valve regurgitation and mitral valve prolapse
          • Mitral stenosis
          • Valvular heart disease diagnosis and treatment
        • Acyanotic heart diseases
          • What is acyanotic heart disease?
          • Acyanotic heart disease diagnosis
          • What is Eisenmenger syndrome?
          • Atrial septal defect
          • Ventricular septal defect
          • Patent ductus arteriosus
        • Cyanotic heart diseases
          • What is cyanotic heart disease
          • Shunting in the heart
          • Einsenmenger coarctation of aorta
          • Tetralogy of fallot
          • Truncus arteriosus
          • Total anomalous pulmonary venous return
          • Tricuspid atresia
          • Transposition of great arteries
          • Ebstein's anomaly
          • Hypoplastic left heart syndrome and norwood glenn fontan
          • Cyanotic heart diseases - Diagnosis and treatment
        • Shock
          • What is shock?
          • Shock - hemodynamics
          • Shock - oxygen delivery and metabolism
          • Shock - diagnosis and treatment
          • Cardiogenic shock
          • Sepsis: Systemic inflammatory response syndrome (SIRS) to multiple organ dysfunction syndrome (MODS)
          • Septic shock - pathophysiology and symptoms
          • Septic shock: Diagnosis and treatment
          • Hypovolemic shock
          • Neurogenic shock
          • Obstructive shock
          • Anaphylactic shock
          • Dissociative shock
          • Differentiating shock
        • Vasculitis
          • What is vasculitis?
          • Vasculitis pathophysiology
          • Temporal arteritis
          • Takayasu arteritis
          • Polyarteritis nodosa
          • What is Kawasaki disease?
          • Kawasaki disease: diagnosis and treatment
          • Buerger disease
          • Wegener’s granulomatosis and microscopic polyangiitis
          • Churg-Strauss syndrome
          • Henoch-Schonlein purpura
        • Hypertension
          • What is hypertension
          • Stages of hypertension
          • Primary hypertension
          • Hypertension and heart damage
          • Hypertension and blood vessel damage
          • Complications of hypertension
          • Secondary hypertension
          • Diagnosis of hypertension
          • Treatment of hypertension
          • Hypertensive crisis
      • Advanced respiratory system physiology
        • Respiratory system introduction
          • Meet the lungs
          • People and plants
          • The bronchial tree
          • Inhaling and exhaling
          • How does lung volume change?
          • Thermoregulation in the lungs
          • The lungs and pulmonary system
        • Gas exchange
          • Alveolar gas equation - part 1
          • Alveolar gas equation - part 2
          • Henry's law
          • O2 and CO2 solubility
          • Graham's law of diffusion
          • Fick's law of diffusion
          • Oxygen movement from alveoli to capillaries
        • Breathing control
          • Peripheral chemoreceptors
          • Central chemoreceptors
          • The respiratory center
          • Sneeze, cough, and hiccup
      • Respiratory system diseases
        • Introduction to pulmonary diseases
          • Types of pulmonary diseases
          • Respiratory distress
          • Streptococcus pneumoniae and flu vaccines
        • Asthma
          • What is asthma?
          • Asthma pathophysiology
          • Asthma diagnosis
          • Asthma severity
          • Asthma shortterm treatments
          • Asthma longterm treatment
        • Emphysema (COPD)
          • What is emphysema?
          • Emphysema pathophysiology
          • Emphysema diagnosis
          • Diffusing capacity of the lung for carbon monoxide (DLCO)
          • Centriacinar emphysema vs panacinar emphysema
          • Emphysema treatment
        • Chronic Bronchitis (COPD)
          • What is chronic bronchitis?
          • Chronic bronchitis pathophysiology
          • Chronic bronchitis treatment
        • Cystic fibrosis
          • What is cystic fibrosis?
          • Cystic fibrosis pathophysiology
          • Cystic fibrosis diagnosis
          • Cystic fibrosis treatments
          • Cystic fibrosis complications
        • Occupational lung diseases
          • Occupational asthma and mesothelioma
          • Asbestosis, silicosis, sick building syndrome
        • Pulmonary hypertension
          • What is pulmonary hypertension?
          • Pulmonary hypertension symptoms and causes
          • Hypoxic vasoconstriction
          • Pulmonary hypertension diagnosis Swan Ganz catheterization
          • Pulmonary hypertension diagnosis
          • Pulmonary hypertension treatment
        • Pneumonia
          • What is pneumonia?
          • Streptococcus pneumoniae and flu vaccines
          • Pulmonary abscess
          • Pneumonia vs. pneumonitis
        • Pleural Effusion
          • What is a pleural effusion?
          • Pleural effusion diagnosis and treatment
          • Transudate vs exudate
          • Neurovascular bundle
        • Lung cancer
          • What is lung cancer?
          • Lung cancer diagnosis
          • Lung cancer types
          • Lung cancer complications
          • Lung cancer metastasis
          • Lung cancer staging
          • Lung cancer treatment
        • Nose, sinus, and upper respiratory conditions
          • What is an upper respiratory infection (URI)?
          • What is allergic rhinitis?
          • Allergic rhinitis diagnosis and treatment
          • Nasal polyps
          • What is sinusitis?
          • Sinusitis diagnosis and treatment
        • Throat conditions
          • Tonsillitis
          • Pharyngitis
          • Tonsillopharyngitis - Complications, diagnosis, and treatment
          • What is a tonsillectomy?
        • Laryngeal conditions
          • What is epiglottitis?
          • Epiglottitis diagnosis and treatment
          • Hib vaccine
          • What is laryngitis ?
          • Laryngitis diagnosis, treatment, and prevention
          • Laryngomalacia
          • Vocal cord growths
        • Bronchiolitis
          • What is bronchiolitis?
          • Bronchiolitis pathophysiology
          • Bronchiolitis diagnosis
          • Bronchiolitis treatment
      • Advanced hematologic system physiology
        • Hematologic system introduction
          • What's inside of blood?
          • Red blood cells
          • Blood types
          • Blood cell lineages
          • Life and times of RBCs and platelets
          • Hemoglobin
          • Hemoglobin moves O2 and CO2
          • Fetal hemoglobin and hematocrit
          • Oxygen content
          • How do we make blood clots?
          • Coagulation cascade
          • Bohr effect vs. Haldane effect
        • Bleeding and impaired hemostasis
          • Hemostasis
          • Primary hemostasis
          • Secondary hemostasis
          • Anticoagulation and thrombolysis
      • Hematologic system diseases
        • Anemia
          • What is anemia?
          • Anemia pathophysiology
          • Microcytic, normocytic, and macrocytic anemias
        • Leukemia
          • Hematopoiesis
          • What is leukemia?
          • Leukemia pathophysiology
          • Leukemia diagnosis
          • Leukemia classifications
          • Acute leukemia
          • Chronic leukemia
          • Myelodysplastic syndrome
          • Leukemia treatment
        • Iron deficiency anemia and anemia of chronic disease
          • What is iron deficiency?
          • Microcytic anemia
          • Iron deficency anemia diagnosis
          • Anemia of chronic disease
          • Chronic disease vs iron deficiency anemia
          • Iron deficiency and anemia of chronic disease treatment
        • Plasma cell dyscrasias
          • What are plasma cell dyscrasias?
          • What is Monoclonal gammopathy of undetermined significance (MGUS)?
          • What is multiple myeloma?
          • What is waldenstrom macroglobulinemia?
        • Myeloproliferative disorders
          • What are myeloproliferative disorders?
          • What is polycythemia vera?
          • What is essential thrombocythemia?
          • What is chronic myelogenous leukemia?
          • What is primary myelofibrosis?
      • Advanced endocrine system physiology
        • Endocrine system introduction
          • Intro to the endocrine system
          • Endocrine gland hormone review
          • The hypothalamus and pituitary gland
          • Hormone concentration metabolism and negative feedback
          • Types of hormones
          • Cellular mechanism of hormone action
        • Biological basis of behavior: Endocrine system
          • From terpenes to steroids part 1: Terpenes
          • From terpenes to steroids part 2: Squalene, cholesterol, and steroids
          • Endocrine system and influence on behavior - Part 1
          • Endocrine system and influence on behavior - Part 2
      • Endocrine system diseases
        • Diabetes
          • What is diabetes mellitus?
          • Breaking down diabetes
          • Types of diabetes
          • Pathophysiology - Type I diabetes
          • Pathophysiology - Type II diabetes
          • Diagnosing diabetes
          • Treating type I diabetes
          • Treating type II diabetes - Pharmacology
          • Treating type II diabetes - A practical approach
          • Acute complications of diabetes - Diabetic ketoacidosis
          • Acute complications of diabetes - Hyperosmolar hyperglycemic nonketotic state
          • Diabetic nephropathy - Mechanisms
          • Diabetic nephropathy - Clinical presentation & treatment
          • Diabetic retinopathy
      • Advanced nervous system physiology
        • Nervous system introduction
          • Introduction to neural cell types
          • Anatomy of a neuron
          • Overview of neuron structure
          • Overview of neuron function
          • Sodium-potassium pump
          • Correction to sodium-potassium pump video
          • Electrotonic and action potentials
          • Saltatory conduction in neurons
          • Synapse structure
          • Neuronal synapses (chemical)
          • Types of neurotransmitters
          • Types of neurotransmitter receptors
          • Structure of the nervous system
          • Functions of the nervous system
          • Motor unit
          • Peripheral somatosensation
          • Muscle stretch reflex
          • Autonomic nervous system
          • Upper motor neurons
          • Somatosensory tracts
          • Cerebral cortex
        • Neural cells and neurotransmitters
          • Neurotransmitter release
          • Neurotransmitter removal
          • Neuroplasticity
          • Astrocytes
          • Microglia
          • Ependymal cells
          • Oligodendrocytes
          • Schwann cells
        • Neuron membrane potentials
          • Neuron resting potential description
          • Neuron resting potential mechanism
          • Neuron graded potential description
          • Neuron graded potential mechanism
          • Neuron action potential description
          • Neuron action potential mechanism
          • Effects of axon diameter and myelination
          • Action potential patterns
        • Sensory perception
          • Visual cues
          • Sensory adaptation
          • Weber's law and thresholds
          • Somatosensation
          • Signal detection theory - part 1
          • Signal detection theory - part 2
          • Bottom-up vs. top-down processing
          • Gestalt principles
        • Sight (vision)
          • The structure of the eye
          • Visual sensory information
          • The phototransduction cascade
          • Photoreceptors (rods vs cones)
          • Photoreceptor distribution in the fovea
          • Visual field processing
          • Feature detection and parallel processing
        • Sound (Audition)
          • Auditory structure - part 1
          • Auditory structure - part 2
          • Auditory processing
          • Cochlear implants
        • Somatosensation
          • Somatosensation
          • Sensory adaptation and amplification
          • Somatosensory homunculus
          • Proprioception and kinesthesia
          • Pain and temperature
        • Taste (gustation) and smell (olfaction)
          • Olfaction - structure and function
          • Pheromones
          • Gustation - structure and function
      • Nervous system diseases
        • Parkinson's disease
          • What is Parkinson's disease?
          • Movement signs and symptoms of Parkinson's disease
          • Non-movement symptoms of Parkinson's disease
          • The basal ganglia - The direct pathway
          • The basal ganglia - Concepts of the indirect pathway
          • The basal ganglia - Details of the indirect pathway
          • Putting it all together - Pathophysiology of Parkinson's disease
          • Genetics and Parkinson's disease
          • Diagnosing Parkinson's disease
          • Managing Parkinson's disease with medications
          • Managing Parkinson's disease with surgery
        • Multiple sclerosis
          • What is multiple sclerosis?
          • Multiple sclerosis signs and symptoms
          • Clinical progression of multiple sclerosis
          • Multiple sclerosis pathophysiology
          • Multiple sclerosis risk factors
          • Multiple sclerosis diagnosis
          • Multiple sclerosis treatment
      • Advanced gastrointestinal physiology
        • Gastrointestinal system introduction
          • Meet the gastrointestinal tract!
      • Gastrointestinal system diseases
        • Gastroenteritis
          • What is gastroenteritis?
          • What is staphylococcal enteritis?
          • What is viral gastroenteritis?
          • What is cholera?
          • What is campylobacteriosis?
          • What is shigellosis?
          • What is salmonella poisoning?
          • What is traveler's diarrhea?
        • Inflammatory bowel disease
          • Inflammatory bowel disease vs irritable bowel syndrome
          • Inflammatory bowel disease: Epidemiology & pathology
          • Inflammatory bowel disease: Diagnosis
          • Inflammatory bowel disease: Treatment
        • Celiac disease
          • What is gluten?
          • What is celiac disease
        • Colon diseases
          • Normal colon tissue
          • Hyperplasia in colon tissue
          • Colon dysplasia
          • Cancerous colon tissue
      • Advanced muscular-skeletal system physiology
        • Muscular system introduction
          • Myosin and actin
        • Skeletal system introduction
          • Skeletal structure and function
          • Microscopic structure of bone - the Haversian system
      • Muscular-skeletal diseases
        • Arthritis and rheumatoid arthritis
          • What is arthritis?
          • Osteoarthritis vs rheumatoid arthritis symptoms
          • Osteoarthritis vs rheumatoid arthritis pathophysiology
          • Osteoarthritis vs rheumatoid arthritis treatments
          • Ankylosing spondylitis
          • Reiter's syndrome
          • Infectious arthritis
          • Gout and pseudogout
          • Gout pathophysiology
      • Executive systems of the brain
        • Cognition
          • Piaget's stages of cognitive development
          • Problem solving
          • Decision making
          • Semantic networks and spreading activation
          • Intelligence
          • Aging and cognitive abilities
        • Attention and language
          • Divided attention, selective attention, inattentional blindness, & change blindness
          • Theories of selective attention
          • Theories of language and cognition
          • Theories of language development: Nativist, learning, interactionist
          • Language and the brain: Aphasia and split-brain patients
        • Sleep and consciousness
          • States of consciousness
          • Sleep stages and circadian rhythms
          • Dreaming
          • Sleep disorders
          • Hypnosis and meditation
        • Memory
          • Information processing model: Sensory, working, and long term memory
          • Encoding strategies
          • Retrieval cues
          • Retrieval: Free recall, cued recall, and recognition
          • Memory reconstruction, source monitoring, and emotional memories
          • Long term potentiation and synaptic plasticity
          • Decay and interference
          • Aging and cognitive abilities
          • Alzheimer's disease and Korsakoff's syndrome
        • Emotion
          • Emotions: limbic system
          • Emotions: cerebral hemispheres and prefrontal cortex
          • Autonomic nervous system (ANS) and physiologic markers of emotion
          • Three components of emotion and the universal emotions
          • Theories of Emotion
      • Mental health
        • Drug abuse and drug addictions
          • Overview of psychoactive drugs
          • Psychoactive drugs: Depressants and opiates
          • Psychoactive drugs: Stimulants
          • Psychoactive drugs: Hallucinogens
          • Routes of drug entry
          • Drug dependence and homeostasis
          • Reward pathway in the brain
          • Tolerance and withdrawal
          • Risk factors for drug use and drug abuse
          • Substance use disorders
          • The development of substance use - Why do people use legal and illegal substances?
          • Why do some people but not others develop substance use problems?
          • Treatments and triggers for drug dependence
          • How does substance use develop into substance abuse
          • Drug use prevention - school programming and protective factors
        • Depression and related disorders
          • Introduction to psychology - Depression and major depressive disorder
          • Diagnosing depression
          • Introduction to psychology - Depression and bipolar disorder
          • Diagnosing bipolar disorder
          • Types of depression and bipolar disorder in the DSM5
          • Biological basis of depression
          • Risk factors for bipolar disorder
          • Treating depression with antidepressants
          • Treatments for depression - Psychological therapies
          • Treatments for bipolar disorder
        • Anxiety
          • Generalized anxiety disorder
          • Phobias
          • Panic disorder
          • Obsessive compulsive disorder
          • Post traumatic stress disorder
        • Psychotic disorders
          • Psychosis
          • Schizophrenia symptoms
          • Schizophrenia diagnosis
          • Schizophrenia treatment
          • Schizoaffective disorder
          • Schizotypal personality disorder
          • Delusional disorder
        • Dementia, delirium, and Alzheimer's
          • What are dementia and Alzheimer's
          • Alzheimer's disease: Plaques and tangles
          • Other types of dementia
          • Risk factors for dementia
          • Stages of dementia and Alzheimer's disease
          • Diagnosis of dementia and Alzheimer's disease
          • Treatment of dementia and Alzheimer's disease
          • What is delirium?
        • Neurodevelopmental disorders
          • Intro to neurodevelopment milestones
          • Neurodevelopmental disorders: Sufficient and necessary causes
          • What is cerebral palsy and what causes it?
          • Types of cerebral palsy part 1 - Spastic
          • Types of cerebral palsy part 2 - Dyskinetic and ataxic
          • Diagnosing cerebral palsy
          • Managing cerebral palsy
          • What is autism spectrum disorder?
          • What is Tourette's?
          • Diagnosing autism spectrum disorder
          • Managing autism spectrum disorder
          • What is asperger syndrome?
      • Infectious diseases
        • Tuberculosis
          • What is TB?
          • TB epidemiology
          • TB pathogenesis
          • Primary and Secondary TB
          • Pulmonary TB
          • Extrapulmonary TB (part 1)
          • Extrapulmonary TB (Part 2)
          • Mantoux test (aka. PPD or TST)
          • Interpreting the PPD
          • Diagnosing active TB
          • Preventing TB transmission
          • Preventing TB using the "4 I's"
          • Treatment of Active TB
          • Drug-resistant TB
          • TB and HIV
        • Pertussis
          • What is pertussis?
          • Pertussis complications
          • Pathophysiology of pertussis
          • Pertussis diagnosis and treatment
          • Pertussis vaccine
        • Influenza
          • What is the flu?
          • Catching and spreading the flu
          • When flu viruses attack!
          • Three types of flu
          • Naming the flu: H-something, N-something
          • Testing for the flu
          • Antiviral drugs for the flu
          • Genetic shift in flu
          • Flu vaccine efficacy
          • Flu shift and drift
          • Two flu vaccines (TIV and LAIV)
          • Flu vaccine risks and benefits
          • Making flu vaccine each year
          • 5 common flu vaccine excuses
          • Vaccines and the autism myth - part 1
          • Vaccines and the autism myth - part 2
          • Flu surveillance
        • Polio
          • What is polio?
          • Poliomyelitis pathophysiology
          • Polio epidemiology
          • Poliomyelitis diagnosis and treatment
          • Polio vaccinations
        • Sexually transmitted diseases
          • What is gonorrhea?
          • Pathophysiology, diagnosis, treatment, and prevention of gonorrhea
          • What is chlamydia?
          • Pathophysiology of chlamydia
          • Diagnosis, treatment, and prevention of chlamydia
          • What is syphilis?
          • What is tertiary syphilis?
          • What is congenital syphilis?
          • Diagnosis, treatment, and prevention of syphilis
          • What is chancroid?
          • What is trichomoniasis?
          • What is herpes?
          • Pathophysiology of herpes
          • What is neonatal herpes?
          • Diagnosis, treatment, and prevention of herpes
          • What are warts?
          • What is bacterial vaginosis?
        • HIV and AIDS
          • What is HIV and AIDS?
          • Transmission of HIV
          • How HIV infects us: Mucous membranes, dendritic cells, and lymph nodes
          • How HIV infects us: CD4 (T-helper) lymphocyte infection
          • How HIV kills so many CD4 T cells
          • Diagnosing HIV - Concepts and tests
          • Treating HIV: Antiretroviral drugs
          • HAART treatment for HIV - Who, what, why, when, and how
          • Defining AIDS and AIDS defining illnesses
          • Immune reconstitution inflammatory syndrome (IRIS) in AIDS
          • Preventing an HIV infection
        • Malaria
          • The basics of malaria
          • Diagnosing malaria
          • Treatment of malaria
          • Preventing malaria
      • Lab values and concentrations
        • Introduction to lab values and normal ranges
        • What's inside of blood?
        • Units for common medical lab values
        • What is an equivalent?
        • The mole and Avogadro's number
        • Molarity vs. molality
        • Molarity vs. osmolarity
        • Calculate your own osmolarity
        • Molarity, molality, osmolarity, osmolality, and tonicity - what's the difference?
        • Tonicity - comparing 2 solutions
      • Current events in health and medicine
        • Antibiotics and antibiotic resistance
        • Ebola outbreak
          • Understanding the Ebola virus outbreak
          • Understanding R nought
          • R nought and vaccine coverage
      • Health care system
        • Blue Cross and Blue Shield
        • Conversation about drug pricing
        • Health care costs in US vs Europe
        • Healthcare system overview
        • Paying doctors
        • Medicare overview
      • Miscellaneous
        • Heart disease and heart attacks
        • Stenosis, ischemia and heart failure
        • Thromboemboli and thromboembolisms
        • Strokes
        • ALS - Amyotrophic lateral sclerosis
        • Low salt diet
        • Calculate your own body mass index
        • Spotting salty foods
        • Adequacy of the lateral cervical spine X-ray
        • Assessing alignment of the lateral cervical spine (neck) X-ray
        • Cervical spine protection in airway management (not a substitute for formal training)
        • Precocious puberty
        • Childhood growth
        • Bacterial meningitis
        • Ritual of the bedside exam
        • Systemic thinking about cancer
        • Inflammation
  • Economics and finance
    • Microeconomics
      • Supply, demand, and market equilibrium
        • Introduction to economics
          • Introduction to economics
        • The demand curve
          • Law of demand
          • Price of related products and demand
          • Change in expected future prices and demand
          • Changes in income, population, or preferences
          • Normal and inferior goods
          • Inferior goods clarification
        • The supply curve
          • Law of supply
          • Factors affecting supply
          • Long term supply curve
        • Market equilibrium
          • Market equilibrium
          • Changes in market equilibrium
        • What drives oil prices
          • Breakdown of gas prices
          • Short-run oil prices
      • Elasticity
        • Price elasticity
          • Price elasticity of demand
          • More on elasticity of demand
          • Perfect inelasticity and perfect elasticity of demand
          • Constant unit elasticity
          • Total revenue and elasticity
          • More on total revenue and elasticity
          • Cross elasticity of demand
          • Elasticity of supply
          • Elasticity and strange percent changes
      • Consumer and producer surplus
        • Consumer and producer surplus
          • Demand curve as marginal benefit curve
          • Consumer surplus introduction
          • Total consumer surplus as area
          • Producer surplus
        • Deadweight loss
          • Rent control and deadweight loss
          • Minimum wage and price floors
          • Taxation and dead weight loss
          • Percentage tax on hamburgers
          • Taxes and perfectly inelastic demand
          • Taxes and perfectly elastic demand
        • Public goods and externalities
          • Negative externalities
          • Taxes for factoring in negative externalities
          • Positive externalities
          • Tragedy of the commons
        • Bonus articles: Pollution as a negative externality
        • Bonus articles: Innovation as a positive externality
      • Scarcity, possibilities, preferences and opportunity cost
        • Production possibilities frontier
          • Production possibilities curve
          • Opportunity cost
          • Increasing opportunity cost
          • Allocative efficiency and marginal benefit
          • Economic growth through investment
        • Comparative advantage and gains from trade
          • Comparative advantage, specialization, and gains from trade
          • Comparative advantage and absolute advantage
        • Marginal utility and budget lines
          • Marginal utility and total utility
          • Utility maximization: equalizing marginal utility per dollar
          • Deriving demand curve from tweaking marginal utility per dollar
          • Budget line
          • Indifference curves and marginal rate of substitution
          • Optimal point on budget line
          • Types of indifference curves
      • Production decisions and economic profit
        • Economic profit and opportunity cost
          • Economic profit vs accounting profit
          • Depreciation and opportunity cost of capital
        • Average costs (ATC, MC) and marginal revenue (MR)
          • Marginal cost and average total cost
          • Marginal revenue and marginal cost
          • Marginal revenue below average total cost
          • Long term supply curve and economic profit
        • Average fixed, variable and marginal costs
          • Fixed, variable, and marginal cost
          • Visualizing average costs and marginal costs as slope
        • Labor and marginal product revenue
          • A firm's marginal product revenue curve
          • How many people to hire given the MPR curve
          • Adding demand curves
        • Price discrimination
          • First degree price discrimination
      • Forms of competition
        • Perfect competition
          • Perfect competition
        • Monopoly
          • Monopoly basics
          • Monopolist optimizing price: Total revenue
          • Monopolist optimizing price: Marginal revenue
          • Monopolist optimizing price: Dead weight loss
          • Optional calculus proof to show that MR has twice slope of demand
          • Review of revenue and cost graphs for a monopoly
        • Between perfect competition and monopoly
          • Oligopolies and monopolistic competition
          • Monopolistic competition and economic profit
          • Oligopolies, duopolies, collusion, and cartels
      • Game theory and Nash equilibrium
        • Nash equilibrium
          • Prisoners' dilemma and Nash equilibrium
          • More on Nash equilibrium
        • Why parties in a cartel will cheat
          • Why parties to cartels cheat
          • Game theory of cheating firms
    • Macroeconomics
      • GDP: Measuring national income
        • Introduction to economics
          • Introduction to economics
        • GDP and the circular flow of income and expenditures
          • Circular flow of income and expenditures
          • Parsing gross domestic product
          • More on final and intermediate GDP contributions
        • Components of GDP
          • Investment and consumption
          • Income and expenditure views of GDP
          • Components of GDP
          • Examples of accounting for GDP
        • Real and nominal GDP
          • Real GDP and nominal GDP
          • GDP deflator
          • Example calculating real GDP with a deflator
        • National income and inequality
          • Capital by Thomas Piketty
          • Difference between wealth and income
          • What is capital?
          • Piketty's two drivers of divergence
          • Is rising inequality necessarily bad?
          • Convergence on macro scale
          • Education as a force of convergence
          • Gilded Age versus Silicon Valley
          • Inverse relationship between capital price and returns
          • Connecting income to capital growth and potential inequality
          • r greater than g but less inequality
          • Return on capital and economic growth
          • Critically looking at data on ROC and economic growth over millenia
          • Simple model to understand r and g relationship
      • Inflation - measuring the cost of living
        • Measuring cost of living – inflation and the consumer price index
          • Introduction to inflation
          • Actual CPI-U basket of goods
          • Inflation data
        • Real and nominal return
          • Real and nominal return
          • Calculating real return in last year dollars
          • Relation between nominal and real returns and inflation
        • Deflation
          • Deflation
          • Velocity of money rather than quantity driving prices
          • Deflation despite increases in money supply
        • Inflationary and deflationary scenarios
          • Moderate inflation in a good economy
          • Stagflation
          • Deflationary spiral
          • Hyperinflation
        • The Phillips curve: Inflation and unemployment
          • Unemployment rate primer
          • Phillips curve
      • Aggregate demand and aggregate supply
        • Aggregate demand and aggregate supply
          • Aggregate demand
          • Shifts in aggregate demand
          • Long-run aggregate supply
          • Short run aggregate supply
        • Historical circumstances explained by AD/AS
          • Demand-pull inflation under Johnson
          • Real GDP driving price
          • Cost-push inflation
        • The business cycle
          • The business cycle
        • Monetary and fiscal policy
          • Monetary and fiscal policy
          • Tax lever of fiscal policy
        • Keynesian thinking
          • Keynesian economics
          • Risks of Keynesian thinking
      • The monetary system
        • Fractional reserve banking
          • Overview of fractional reserve banking
          • Weaknesses of fractional reserve lending
          • Full reserve banking
        • Money supply
          • Money supply: M0, M1, and M2
        • Fractional reserve accounting
          • Simple fractional reserve accounting (part 1)
          • Simple fractional reserve accounting (part 2)
        • Interest as the price of money
          • Interest as rent for money
          • Money supply and demand impacting interest rates
        • Functions of money
          • Functions of money
          • When the functions of money break down: Hyperinflation
          • Standard of deferred payment and legal tender
          • Commodity money vs. Fiat money
      • Income and expenditure: Keynesian cross and IS-LM model
        • Marginal propensity to consume (MPC)
          • MPC and multiplier
          • Mathy version of MPC and multiplier (optional)
        • Consumption function
          • Consumption function basics
          • Generalized linear consumption function
          • Consumption function with income dependent taxes
        • Keynesian cross
          • Keynesian cross
          • Details on shifting aggregate planned expenditures
          • Keynesian cross and the multiplier
        • IS-LM model
          • Investment and real interest rates
          • Connecting the keynesian cross to the IS curve
          • Loanable funds interpretation of IS curve
          • LM part of the IS-LM model
          • Government spending and the IS-LM model
        • Articles on the Keynesian perspective
      • Foreign exchange and trade
        • Balance of payments- current account and capital account
          • Balance of payments: Current account
          • Balance of payments: Capital account
          • Why current and capital accounts net out
        • Currency reserves
          • Accumulating foreign currency reserves
          • Using reserves to stabilize currency
          • Speculative attack on a currency
          • Financial crisis in Thailand caused by speculative attack
          • Math mechanics of Thai banking crisis
    • AP Microeconomics
      • Basic Economic Concepts
        • Introduction to economics
          • Introduction to economics
          • Scarcity
          • Economic models
        • Production possibilities frontier
          • Production possibilities curve
          • Opportunity cost
          • Increasing opportunity cost
          • Allocative efficiency and marginal benefit
          • Economic growth through investment
        • Comparative advantage and gains from trade
          • Comparative advantage, specialization, and gains from trade
          • Comparative advantage and absolute advantage
      • Supply, demand, and market equilibrium
        • The demand curve
          • Law of demand
          • Price of related products and demand
          • Change in expected future prices and demand
          • Changes in income, population, or preferences
          • Normal and inferior goods
          • Inferior goods clarification
        • The supply curve
          • Law of supply
          • Factors affecting supply
          • Long term supply curve
        • Market equilibrium
          • Market equilibrium
          • Changes in market equilibrium
        • What drives oil prices
          • Breakdown of gas prices
          • Short-run oil prices
        • Price elasticity of demand
          • Price elasticity of demand
          • More on elasticity of demand
          • Perfect inelasticity and perfect elasticity of demand
          • Constant unit elasticity
        • Elasticity and total revenue
          • Total revenue and elasticity
          • More on total revenue and elasticity
        • Cross-price elasticity of demand
          • Cross elasticity of demand
        • Price elasticity of supply
          • Elasticity of supply
          • Elasticity and strange percent changes
        • Consumer and producer surplus
          • Demand curve as marginal benefit curve
          • Consumer surplus introduction
          • Total consumer surplus as area
          • Producer surplus
        • Deadweight loss
          • Rent control and deadweight loss
          • Minimum wage and price floors
          • Taxation and dead weight loss
          • Percentage tax on hamburgers
          • Taxes and perfectly inelastic demand
          • Taxes and perfectly elastic demand
      • Theory of consumer choice
        • Total utility and marginal utility
          • Marginal utility and total utility
          • Utility maximization: equalizing marginal utility per dollar
          • Deriving demand curve from tweaking marginal utility per dollar
        • Income and substitution effects
          • Budget line
          • Indifference curves and marginal rate of substitution
          • Optimal point on budget line
          • Types of indifference curves
      • Production and costs
        • Average costs (ATC, MC) and marginal revenue (MR)
          • Marginal cost and average total cost
          • Marginal revenue and marginal cost
          • Marginal revenue below average total cost
        • Average fixed, variable and marginal costs
          • Fixed, variable, and marginal cost
          • Visualizing average costs and marginal costs as slope
        • Price discrimination
          • First degree price discrimination
      • Firm behavior and market structure
        • Economic profit and opportunity cost
          • Economic profit vs accounting profit
          • Depreciation and opportunity cost of capital
          • Long term supply curve and economic profit
        • Perfect competition
          • Perfect competition
        • Monopoly
          • Monopoly basics
          • Monopolist optimizing price: Total revenue
          • Monopolist optimizing price: Marginal revenue
          • Monopolist optimizing price: Dead weight loss
          • Optional calculus proof to show that MR has twice slope of demand
          • Review of revenue and cost graphs for a monopoly
        • Between perfect competition and monopoly
          • Oligopolies and monopolistic competition
          • Monopolistic competition and economic profit
          • Oligopolies, duopolies, collusion, and cartels
        • Nash equilibrium
          • Prisoners' dilemma and Nash equilibrium
          • More on Nash equilibrium
        • Why parties in a cartel will cheat
          • Why parties to cartels cheat
          • Game theory of cheating firms
      • Factor markets
        • Labor and marginal product revenue
          • A firm's marginal product revenue curve
          • How many people to hire given the MPR curve
          • Adding demand curves
      • Market failure and the role of government
        • Public goods and externalities
          • Negative externalities
          • Taxes for factoring in negative externalities
          • Positive externalities
          • Tragedy of the commons
        • Bonus articles: Pollution as a negative externality
        • Bonus articles: Innovation as a positive externality
    • AP Macroeconomics
      • Basic economic concepts
        • Introduction to economics
          • Introduction to economics
          • Scarcity
          • Economic models
        • Production possibilities frontier
          • Production possibilities curve
          • Opportunity cost
          • Increasing opportunity cost
          • Allocative efficiency and marginal benefit
          • Economic growth through investment
        • Comparative advantage and gains from trade
          • Comparative advantage, specialization, and gains from trade
          • Comparative advantage and absolute advantage
        • The demand curve
          • Law of demand
          • Price of related products and demand
          • Change in expected future prices and demand
          • Changes in income, population, or preferences
        • The supply curve
          • Law of supply
          • Factors affecting supply
        • Market equilibrium
          • Market equilibrium
          • Changes in market equilibrium
        • Macroeconomic issues: business cycle, unemployment, inflation, growth
          • The business cycle
          • Unemployment rate primer (v2)
          • Introduction to inflation
          • Actual CPI-U basket of goods
      • Measurement of economic performance
        • GDP and the circular flow of income and expenditures
          • Circular flow of income and expenditures
          • Parsing gross domestic product
          • More on final and intermediate GDP contributions
        • Components of GDP
          • Investment and consumption
          • Income and expenditure views of GDP
          • Components of GDP
          • Examples of accounting for GDP
        • Real and nominal GDP
          • Real GDP and nominal GDP
          • GDP deflator
          • Example calculating real GDP with a deflator
        • Measuring cost of living – inflation and the consumer price index
          • Introduction to inflation
          • Actual CPI-U basket of goods
          • Inflation data
        • Real and nominal return
          • Real and nominal return
          • Calculating real return in last year dollars
          • Relation between nominal and real returns and inflation
        • Deflation
          • Deflation
          • Velocity of money rather than quantity driving prices
          • Deflation despite increases in money supply
        • Inflationary and deflationary scenarios
          • Moderate inflation in a good economy
          • Stagflation
          • Deflationary spiral
          • Hyperinflation
        • Unemployment
          • Unemployment rate primer
          • Phillips curve
      • National income and price determination
        • Aggregate demand and aggregate supply
          • Aggregate demand
          • Shifts in aggregate demand
          • Long-run aggregate supply
          • Short run aggregate supply
        • Historical circumstances explained by AD/AS
          • Demand-pull inflation under Johnson
          • Real GDP driving price
          • Cost-push inflation
        • The business cycle
          • The business cycle
        • Monetary and fiscal policy
          • Monetary and fiscal policy
          • Tax lever of fiscal policy
        • Keynesian thinking
          • Keynesian economics
          • Risks of Keynesian thinking
      • Financial sector
        • Definition of financial assets: money, stocks, bonds
          • Functions of money
          • What it means to buy a company's stock
          • Introduction to bonds
        • Time value of money (present and future value)
          • Time value of money
          • Introduction to present value
          • Present value 2
          • Present value 3
          • Present value 4 (and discounted cash flow)
        • Real and nominal return
          • Real and nominal return
          • Calculating real return in last year dollars
          • Relation between nominal and real returns and inflation
        • Fractional reserve banking
          • Overview of fractional reserve banking
          • Weaknesses of fractional reserve lending
          • Full reserve banking
        • Money supply
          • Money supply: M0, M1, and M2
        • Fractional reserve accounting
          • Simple fractional reserve accounting (part 1)
          • Simple fractional reserve accounting (part 2)
        • Interest as the price of money
          • Interest as rent for money
          • Money supply and demand impacting interest rates
        • Central bank and control of the money supply
          • Fed open market operations
          • Quantitative easing
      • Stabilization policies
        • Marginal propensity to consume (MPC)
          • MPC and multiplier
          • Mathy version of MPC and multiplier (optional)
        • Consumption function
          • Consumption function basics
          • Generalized linear consumption function
          • Consumption function with income dependent taxes
        • Articles on the Keynesian perspective
      • Open Economy: International Trade and Finance
        • Balance of payments- current account and capital account
          • Balance of payments: Current account
          • Balance of payments: Capital account
          • Why current and capital accounts net out
        • Foreign exchange market
          • Currency exchange introduction
          • Currency effect on trade
          • Currency effect on trade review
          • Pegging the yuan
          • Chinese Central Bank buying treasuries
          • American-Chinese debt loop
          • Debt loops rationale and effects
          • China keeps peg but diversifies holdings
          • Carry trade basics
        • Currency reserves
          • Accumulating foreign currency reserves
          • Using reserves to stabilize currency
          • Speculative attack on a currency
          • Financial crisis in Thailand caused by speculative attack
          • Math mechanics of Thai banking crisis
    • Finance and capital markets
      • Interest and debt
        • Compound interest basics
          • Compound interest introduction
          • The rule of 72 for compound interest
        • Interest basics
          • Introduction to interest
          • Interest (part 2)
        • Credit cards and loans
          • Annual percentage rate (APR) and effective APR
          • Institutional roles in issuing and processing credit cards
          • Payday loans
        • Continuous compound interest and e
          • 𝑒 and compound interest
          • 𝑒 as a limit
          • Formula for continuously compounding interest
        • Present value
          • Time value of money
          • Introduction to present value
          • Present value 2
          • Present value 3
          • Present value 4 (and discounted cash flow)
        • Personal bankruptcy
          • Personal bankruptcy: Chapters 7 and 13
      • Housing
        • Home equity and personal balance sheets
          • Introduction to balance sheets
          • More on balance sheets and equity
          • Home equity loans
        • Renting vs. buying a home
          • Renting versus buying a home
          • Is buying a home always better?
          • What happens when housing depreciates
          • Renting vs. buying (detailed analysis)
        • Mortgages
          • Introduction to mortgage loans
          • Mortgage interest rates
          • Short sale basics
          • Adjustable rate mortgages ARMs
          • Hybrid ARM
          • Balloon payment mortgage
          • Finite geometric series word problem: mortgage
        • Home buying process
          • Titles and deeds in real estate
          • Title insurance
          • Making an offer on a home
          • Escrow
          • Types of escrow in real estate
      • Inflation
        • Inflation basics
          • Inflation overview
          • What is inflation
          • Inflation data
          • CPI index
        • Inflation scenarios
          • Moderate inflation in a good economy
          • Stagflation
          • Hyperinflation
        • Real and nominal return
          • Real and nominal return
          • Calculating real return in last year dollars
          • Relation between nominal and real returns and inflation
        • Capacity utilization and inflation
          • Economics of a cupcake factory
          • Cupcake economics 2
          • Cupcake economics 3
          • Inflation, deflation, and capacity utilization
          • Inflation, deflation, and capacity utilization 2
          • Inflation and deflation 3: Obama stimulus plan
        • Deflation
          • Deflation
          • Velocity of money rather than quantity driving prices
          • Deflation despite increases in money supply
          • Deflationary spiral
      • Taxes
        • Personal taxes
          • Basics of US income tax rate schedule
          • Tax deductions introduction
          • AMT overview
          • Alternative minimum tax
          • Estate tax introduction
          • Tax brackets and progressive taxation
          • Calculating federal taxes and take home pay
          • Calculating state taxes and take home pay
          • Marriage penalty
          • Married taxes clarification
        • Corporate taxation
          • Corporations and limited liability
          • Is limited liability or double taxation fair?
          • Transfer pricing and tax havens
      • Accounting and financial statements
        • Cash versus accrual accounting
          • Cash accounting
          • Accrual basis of accounting
          • Comparing accrual and cash accounting
        • Three core financial statements
          • Balance sheet and income statement relationship
          • Basic cash flow statement
          • Doing the example with accounts payable growing
          • Fair value accounting
        • Depreciation and amortization
          • Expensing a truck leads to inconsistent performance
          • Depreciating the truck
          • Depreciation in cash flow
          • Amortization and depreciation
      • Stocks and bonds
        • Introduction to stocks
          • What it means to buy a company's stock
          • Bonds vs. stocks
        • Shorting stock
          • Basic shorting
          • Shorting stock
          • Shorting stock 2
          • Is short selling bad?
        • Understanding company statements and capital structure
          • Gross and operating profit
          • Basic capital structure differences
          • Market capitalization
          • Market value of assets
        • Corporate metrics and valuation
          • Price and market capitalization
          • Introduction to the income statement
          • Earnings and EPS
          • Introduction to the price-to-earnings ratio
          • P/E discussion
          • ROA discussion 1
          • ROA discussion 2
          • Depreciation
          • Amortization
          • P/E conundrum
          • Enterprise value
          • EBITDA
        • Life of a company--from birth to death
          • Raising money for a startup
          • Getting a seed round from a VC
          • Going back to the till: Series B
          • An IPO
          • More on IPOs
          • Equity vs. debt
          • Bonds vs. stocks
          • Chapter 7: Bankruptcy liquidation
          • Chapter 11: Bankruptcy restructuring
        • Dilution
          • Stock dilution
        • Mergers and acquisitions
          • Acquisitions with shares
          • Price behavior after announced acquisition
          • Simple merger arbitrage with share acquisition
        • Leveraged buy-outs
          • Basic leveraged buyout (LBO)
        • Bonds
          • Corporate debt versus traditional mortgages
          • Introduction to bonds
          • Introduction to the yield curve
          • Relationship between bond prices and interest rates
          • Treasury bond prices and yields
          • Annual interest varying with debt maturity
          • The yield curve
        • Corporate bankruptcy
          • Chapter 7: Bankruptcy liquidation
          • Chapter 11: Bankruptcy restructuring
      • Investment vehicles, insurance, and retirement
        • Mutual funds and ETFs
          • Open-ended mutual fund (part 1)
          • Open-end mutual fund redemptions (part 2)
          • Closed-end mutual funds
          • Exchange traded funds (ETFs)
          • Ponzi schemes
        • Retirement accounts: IRAs and 401ks
          • Traditional IRAs
          • Roth IRAs
          • 401(k)s
        • Life insurance
          • Term life insurance and death probability
        • Hedge funds
          • Hedge funds intro
          • Hedge fund structure and fees
          • Are hedge funds bad?
          • Hedge funds, venture capital, and private equity
          • Hedge fund strategies: Long short 1
          • Hedge fund strategies: Long short 2
          • Hedge fund strategies: Merger arbitrage 1
        • Investment and consumption
          • Risk and reward introduction
          • Human capital
          • Return on capital
          • Investment vs. consumption 1
          • Investment vs. consumption 2
          • Wealth destruction 1
          • Wealth destruction 2
          • Back-of-envelope office space conundrum
      • Money, banking and central banks
        • Banking and money
          • Banking 1
          • Banking 2: A bank's income statement
          • Banking 3: Fractional reserve banking
          • Banking 4: Multiplier effect and the money supply
          • Banking 5: Introduction to bank notes
          • Banking 6: Bank notes and checks
          • Banking 7: Giving out loans without giving out gold
          • Banking 8: Reserve ratios
          • Banking 9: More on reserve ratios (bad sound)
          • Banking 10: Introduction to leverage (bad sound)
          • Banking 11: A reserve bank
          • Banking 12: Treasuries (government debt)
          • Banking 13: Open market operations
          • Banking 14: Fed funds rate
          • Banking 15: More on the Fed funds rate
          • Banking 16: Why target rates vs. money supply
          • Banking 17: What happened to the gold?
          • Banking 18: Big picture discussion
          • The discount rate
          • Repurchase agreements (repo transactions)
          • Federal Reserve balance sheet
          • Fractional Reserve banking commentary 1
          • FRB commentary 2: Deposit insurance
          • FRB commentary 3: Big picture
          • LIBOR
        • Quantitative easing
          • Fed open market operations
          • Quantitative easing
          • More on quantitative easing (and credit easing)
          • Open market operations and quantitative easing overview
          • Another quantitative easing video
          • US and Japanese quantitative easing
        • 2008 bank bailout
          • Bailout 1: Liquidity vs. solvency
          • Bailout 2: Book value
          • Bailout 3: Book value vs. market value
          • Bailout 4: Mark-to-model vs. mark-to-market
          • Bailout 5: Paying off the debt
          • Bailout 6: Getting an equity infusion
          • Bailout 7: Bank goes into bankruptcy
          • Bailout 8: Systemic risk
          • Bailout 9: Paulson's plan
          • Bailout 10: Moral hazard
          • Bailout 11: Why these CDOs could be worth nothing
          • Bailout 12: Lone Star transaction
          • Bailout 13: Does the bailout have a chance of working?
          • Bailout 14: Possible solution
          • Bailout 15: More on the solution
          • CNN: Understanding the crisis
        • Geithner plan
          • Geithner plan 1
          • Geithner plan 2
          • Geithner plan 3
          • Geithner plan 4
          • Geithner plan 5
          • Geithner plan 6: A better solution
        • Foreign exchange and trade
          • Currency exchange introduction
          • Currency effect on trade
          • Currency effect on trade review
          • Pegging the yuan
          • Chinese Central Bank buying treasuries
          • American-Chinese debt loop
          • Debt loops rationale and effects
          • China keeps peg but diversifies holdings
          • Carry trade basics
        • Chinese currency and U.S. debt
          • Floating exchange resolving trade imbalance
          • China pegs to dollar to keep trade imbalance
          • China buys US bonds
          • Review of China US currency situation
          • Data on Chinese M1 increase in 2010
          • Data on Chinese foreign assets increase in 2010
          • Data on Chinese US balance of payments
          • Chinese inflation
          • Floating exchange effect on China
          • Floating exchange effect on US
        • 2011-2012 Greek debt crisis
          • Greek debt recession and austerity (part 1)
          • Greek financial crisis (part 2)
          • How and why Greece would leave the Euro (part 3)
          • Why Europe is worried about Greece
        • Bitcoin
          • Bitcoin: What is it?
          • Bitcoin: Overview
          • Bitcoin: Cryptographic hash functions
          • Bitcoin: Digital signatures
          • Bitcoin: Transaction records
          • Bitcoin: Proof of work
          • Bitcoin: Transaction block chains
          • Bitcoin: The money supply
          • Bitcoin: The security of transaction block chains
      • Options, swaps, futures, MBSs, CDOs, and other derivatives
        • Put and call options
          • American call options
          • Basic shorting
          • American put options
          • Call option as leverage
          • Put vs. short and leverage
          • Call payoff diagram
          • Put payoff diagram
          • Put as insurance
          • Put-call parity
          • Long straddle
          • Put writer payoff diagrams
          • Call writer payoff diagram
          • Arbitrage basics
          • Put-call parity arbitrage I
          • Put-call parity arbitrage II
          • Put-call parity clarification
          • Actual option quotes
          • Option expiration and price
        • Forward and futures contracts
          • Forward contract introduction
          • Futures introduction
          • Motivation for the futures exchange
          • Futures margin mechanics
          • Verifying hedge with futures margin mechanics
          • Futures and forward curves
          • Contango from trader perspective
          • Severe contango generally bearish
          • Backwardation bullish or bearish
          • Futures curves II
          • Contango
          • Backwardation
          • Contango and backwardation review
          • Upper bound on forward settlement price
          • Lower bound on forward settlement price
          • Arbitraging futures contract
          • Arbitraging futures contracts II
          • Futures fair value in the pre-market
          • Interpreting futures fair value in the premarket
        • Mortgage-backed securities
          • Mortgage-backed security overview
          • Mortgage-backed securities I
          • Mortgage-backed securities II
          • Mortgage-backed securities III
        • Collateralized debt obligations
          • Collateralized debt obligation overview
          • Collateralized debt obligation (CDO)
        • Credit default swaps
          • Credit default swaps (CDS) intro
          • Credit default swaps
          • Credit default swaps 2
          • Use cases for credit default swaps
          • Financial weapons of mass destruction
        • Interest rate swaps
          • Interest rate swap 1
          • Interest rate swap 2
        • Black-Scholes formula
          • Introduction to the Black-Scholes formula
          • Implied volatility
      • Current economics
        • Unemployment
          • Unemployment
          • Unemployment rate primer (v2)
          • Unemployment rate primer
          • Simple analysis of cost per job saved from stimulus
        • Housing price conundrum
          • The housing price conundrum
          • Housing price conundrum (part 2)
          • Housing price conundrum (part 3)
          • Housing conundrum (part 4)
        • Credit crisis
          • The housing price conundrum
          • Housing price conundrum (part 2)
          • Housing price conundrum (part 3)
          • Housing conundrum (part 4)
          • Mortgage-backed securities I
          • Mortgage-backed securities II
          • Mortgage-backed securities III
          • Collateralized debt obligation (CDO)
          • Credit default swaps
          • Credit default swaps 2
          • Wealth destruction 1
          • Wealth destruction 2
        • Paulson bailout
          • CNN: Understanding the crisis
          • Bailout 1: Liquidity vs. solvency
          • Bailout 2: Book value
          • Bailout 3: Book value vs. market value
          • Bailout 4: Mark-to-model vs. mark-to-market
          • Bailout 5: Paying off the debt
          • Bailout 6: Getting an equity infusion
          • Bailout 7: Bank goes into bankruptcy
          • Bailout 8: Systemic risk
          • Bailout 9: Paulson's plan
          • Bailout 10: Moral hazard
          • Bailout 11: Why these CDOs could be worth nothing
          • Bailout 12: Lone Star transaction
          • Bailout 13: Does the bailout have a chance of working?
          • Bailout 14: Possible solution
          • Bailout 15: More on the solution
        • European Union
          • Brexit and European Union primer
          • United Kingdom vs Great Britain vs England primer
          • Greek debt recession and austerity (part 1)
          • Greek financial crisis (part 2)
          • How and why Greece would leave the Euro (part 3)
          • Why Europe is worried about Greece
  • Arts and humanities
    • US history
      • The historian's toolkit
        • How to think like a historian
          • Thinking like a historian
          • How to read a document: source identification
          • How to read a document: analyzing a historical text
          • Avoiding common mistakes in historical essays
      • Worlds collide (1491-1607)
        • Before contact
          • Native American societies before contact
        • Old and new worlds collide
          • Origins of European exploration in the Americas
          • Christopher Columbus
          • Consequences of Columbus's voyage on the Tainos and Europe
          • The Columbian Exchange
        • Spanish colonization
          • Spanish colonization
      • Colonial America (1607-1754)
        • Early colonization projects
          • French and Dutch colonization
          • England in the Age of Exploration
          • Motivations for English colonization
          • The Lost Colony of Roanoke - background and first attempts
          • The Lost Colony of Roanoke - settlement and disappearance
        • Chesapeake and Southern colonies
          • Early English settlements - Jamestown
          • Jamestown - John Smith and Pocahontas
          • Jamestown - the impact of tobacco
          • Jamestown - life and labor in the Chesapeake
          • Jamestown - Bacon's Rebellion
          • The West Indies and the Southern colonies
        • Colonial North America
          • Society and religion in the New England colonies
          • Politics and native relations in the New England colonies
          • The Middle colonies
      • The road to revolution (1754-1800)
        • The American Revolution
          • The Seven Years' War: background and combatants
          • The Seven Years' War: battles and legacy
          • The Declaration of Independence
        • Creating a nation
          • The Articles of Confederation
          • The Constitutional Convention
          • The US Constitution
      • The early republic (1800-1848)
        • Politics and society in the early nineteenth century
          • The Monroe Doctrine
          • The Market Revolution - textile mills and the cotton gin
          • The Market Revolution - communication and transportation
          • The Market Revolution - impact and significance
        • The age of Jackson
          • Jacksonian Democracy - background and introduction
          • Jacksonian Democracy - the "corrupt bargain" and the election of 1824
          • Jacksonian Democracy - mudslinging and the election of 1828
          • Jacksonian Democracy - spoils system, Bank War, and Trail of Tears
        • Culture and reform in the early nineteenth century
          • The Second Great Awakening - origins and major ideas
          • The Second Great Awakening - influence of the Market Revolution
          • The Second Great Awakening - reform and religious movements
          • The early temperance movement - origins
          • The early temperance movement - spread and temporary decline
      • The Civil War era (1844-1877)
        • Sectional tension in the 1850s
          • Uncle Tom's Cabin - influence of the Fugitive Slave Act
          • Uncle Tom's Cabin - reception and significance
          • Uncle Tom's Cabin - plot and analysis
        • The Civil War
          • Slavery and the Missouri Compromise
          • Increasing political battles over slavery in the mid-1800s
          • Start of the Civil War - secession and Fort Sumter
          • Strategy of the Civil War
          • Early phases of Civil War and Antietam
          • The Emancipation Proclamation
          • Significance of the battle of Antietam
          • The battle of Gettysburg
          • The Gettysburg Address - setting and context
          • The Gettysburg Address - full text and analysis
          • Later stages of the Civil War - 1863
          • Later stages of the Civil War - the election of 1864 and Sherman's March
          • Later stages of the Civil War - Appomattox and Lincoln's assassination
          • Big takeaways from the Civil War
        • Reconstruction
          • An overview and the 13th Amendment
          • The 14th Amendment
          • The 15th Amendment
      • The Gilded Age (1865-1898)
        • The Gilded Age
          • Introduction to the Gilded Age
          • The Gilded Age and the Second Industrial Revolution
          • Misunderstanding evolution: a biologist's perspective on Social Darwinism
          • Misunderstanding evolution: a historian's perspective on Social Darwinism
        • The South after the Civil War
          • The origins of Jim Crow - introduction
          • Origins of Jim Crow - the Black Codes and Reconstruction
          • Origins of Jim Crow - the Fourteenth and Fifteenth Amendments
          • Origins of Jim Crow - Compromise of 1877 and Plessy v. Ferguson
        • The American West
      • Rise to world power (1890-1945)
        • The age of empire
          • The age of empire
        • The United States in World War I
          • Blockades, u-boats and sinking of the Lusitania
          • Zimmermann Telegram
          • United States enters World War I
          • Woodrow Wilson's Fourteen Points
          • Paris Peace Conference and Treaty of Versailles
          • More detail on the Treaty of Versailles and Germany
        • 1920s America
        • The Great Depression
        • World War II
          • Beginning of World War II
          • 1940 - Axis gains momentum in World War II
          • 1941 Axis momentum accelerates in WW2
          • 1942 Tide turning in World War II in Europe
          • World War II in the Pacific in 1942
          • 1943 Axis losing in Europe
          • American progress in the Pacific in 1944
          • 1944 - Allies advance further in Europe
          • 1945 - End of World War II
      • The postwar era (1945-1980)
        • The postwar era, 1945-1950
          • Origins of the Cold War
          • Communism
        • 1950s America
          • Korean War overview
        • The Civil Rights Movement
        • 1960s America
          • Bay of Pigs Invasion
          • Cuban Missile Crisis
          • Vietnam War
        • 1970s America
      • The modern era (1980-present)
        • 1980s America
        • 1990s America
        • The United States after 2000
      • American civics
        • American civics
          • PPACA or "Obamacare"
          • The fiscal cliff
          • More fiscal cliff analysis
          • Electoral college
          • Sal teaches Grover about the electoral college
          • Primaries and caucuses
          • Deficit and debt ceiling
          • Government's financial condition
          • Social security intro
          • FICA tax
          • Medicare sustainability
          • SOPA and PIPA
          • Pension obligations
          • Illinois pension obligations
          • Introduction to the FAFSA
          • History of the Democratic Party
          • History of the Republican Party
          • Constitutional powers of the president
          • Presidential precedents of George Washington
          • The President as Commander-in-Chief
          • Expansion of presidential power
      • Surveys of history
        • United States history overview
          • US History Overview 1: Jamestown to the Civil War
          • US History overview 2: Reconstruction to the Great Depression
          • US History overview 3: WWII to Vietnam
          • Appomattox and Lincoln's assassination
          • When Capitalism is great and not-so-great
          • 20th century US capitalism and regulation
    • AP US History
      • AP US History skills and test strategies
        • Thinking like a historian
          • Meet Kim, one of the creators of Khan Academy's AP US History lessons
          • Thinking like a historian
          • How to read a document: source identification
          • How to read a document: analyzing a historical text
          • Avoiding common mistakes in historical essays
        • AP US History examples
          • AP US History multiple choice example 1
          • AP US History multiple choice example 2
          • AP US History short answer example 1
          • AP US History short answer example 2
          • AP US History DBQ example 1
          • AP US History DBQ example 2
          • AP US History DBQ example 3
          • AP US History DBQ example 4
          • AP US History long essay example 1
          • AP US History long essay example 2
          • AP US History long essay example 3
          • Preparing for the AP US History Exam (5/4/2016)
          • AP US History Exam Prep Session (5/1/2017)
      • Period 1: 1491-1607
        • Before contact
          • Native American societies before contact
        • Old and new worlds collide
          • Origins of European exploration in the Americas
          • Christopher Columbus
          • Consequences of Columbus's voyage on the Tainos and Europe
          • The Columbian Exchange
        • Spanish colonization
          • Spanish colonization
      • Period 2: 1607-1754
        • Early colonization projects
          • French and Dutch colonization
          • England in the Age of Exploration
          • Motivations for English colonization
          • The Lost Colony of Roanoke - background and first attempts
          • The Lost Colony of Roanoke - settlement and disappearance
          • Early English settlements - Jamestown
          • Jamestown - John Smith and Pocahontas
          • Jamestown - the impact of tobacco
          • Jamestown - life and labor in the Chesapeake
        • Colonial North America
          • Society and religion in the New England colonies
          • Politics and native relations in the New England colonies
          • The Middle colonies
          • The West Indies and the Southern colonies
      • Period 3: 1754-1800
        • The American Revolution
          • The Seven Years' War: background and combatants
          • The Seven Years' War: battles and legacy
          • The Declaration of Independence
        • Creating a nation
          • The Articles of Confederation
          • The Constitutional Convention
          • The US Constitution
      • Period 4: 1800-1848
        • Politics and society in the early nineteenth century
          • The Monroe Doctrine
          • The Market Revolution - textile mills and the cotton gin
          • The Market Revolution - communication and transportation
          • The Market Revolution - impact and significance
        • The age of Jackson
          • Jacksonian Democracy - background and introduction
          • Jacksonian Democracy - the "corrupt bargain" and the election of 1824
          • Jacksonian Democracy - mudslinging and the election of 1828
          • Jacksonian Democracy - spoils system, Bank War, and Trail of Tears
        • Culture and reform
          • The Second Great Awakening - origins and major ideas
          • The Second Great Awakening - influence of the Market Revolution
          • The Second Great Awakening - reform and religious movements
          • The early temperance movement - origins
          • The early temperance movement - spread and temporary decline
      • Period 5: 1844-1877
        • Sectional tension in the 1850s
          • Uncle Tom's Cabin - influence of the Fugitive Slave Act
          • Uncle Tom's Cabin - plot and analysis
          • Uncle Tom's Cabin - reception and significance
        • The Civil War
          • Slavery and the Missouri Compromise
          • Increasing political battles over slavery in the mid-1800s
          • Start of the Civil War - secession and Fort Sumter
          • Strategy of the Civil War
          • Early phases of Civil War and Antietam
          • The Emancipation Proclamation
          • Significance of the battle of Antietam
          • The battle of Gettysburg
          • The Gettysburg Address - setting and context
          • The Gettysburg Address - full text and analysis
          • Later stages of the Civil War - 1863
          • Later stages of the Civil War - the election of 1864 and Sherman's March
          • Later stages of the Civil War - Appomattox and Lincoln's assassination
          • Big takeaways from the Civil War
        • Reconstruction
          • An overview and the 13th Amendment
          • The 14th Amendment
          • The 15th Amendment
      • Period 6: 1865-1898
        • The Gilded Age
          • Introduction to the Gilded Age
          • The Gilded Age and the Second Industrial Revolution
          • Misunderstanding evolution: a biologist's perspective on Social Darwinism
          • Misunderstanding evolution: a historian's perspective on Social Darwinism
        • The South after the Civil War
          • The origins of Jim Crow - introduction
          • Origins of Jim Crow - the Black Codes and Reconstruction
          • Origins of Jim Crow - the Fourteenth and Fifteenth Amendments
          • Origins of Jim Crow - Compromise of 1877 and Plessy v. Ferguson
        • The American West
      • Period 7: 1890-1945
        • The age of empire
          • The age of empire
        • The United States in World War I
          • Blockades, u-boats and sinking of the Lusitania
          • Zimmermann Telegram
          • United States enters World War I
          • Woodrow Wilson's Fourteen Points
          • Paris Peace Conference and Treaty of Versailles
          • More detail on the Treaty of Versailles and Germany
        • 1920s America
        • The Great Depression
        • World War II
          • Beginning of World War II
          • 1940 - Axis gains momentum in World War II
          • 1941 Axis momentum accelerates in WW2
          • 1942 Tide turning in World War II in Europe
          • World War II in the Pacific in 1942
          • 1943 Axis losing in Europe
          • American progress in the Pacific in 1944
          • 1944 - Allies advance further in Europe
          • 1945 - End of World War II
      • Period 8: 1945-1980
        • The postwar era, 1945-1950
          • Origins of the Cold War
          • Communism
        • 1950s America
          • Korean War overview
        • The Civil Rights Movement
        • 1960s America
          • Bay of Pigs Invasion
          • Cuban Missile Crisis
          • Vietnam War
        • The 1970s
      • Period 9: 1980-present
        • 1980s America
        • 1990s America
        • The United States after 2000
    • World history
      • Beginnings - 600 BCE
        • The origin of humans and early human societies
          • History and prehistory
          • Peopling the earth
          • Organizing paleolithic societies
        • The Neolithic Revolution and the birth of agriculture
          • The Neolithic Revolution and early agriculture
          • The spread of agriculture
        • Ancient Mesopotamia
          • Ancient Mesopotamia
          • Ancient Mesopotamia and the Hebrew Bible
        • Ancient Egypt
          • Ancient Egypt and the Nile River Valley
          • The Hittite Empire and the Battle of Kadesh
        • Ancient art and artifacts
          • The Standard of Ur
          • The Rosetta Stone
        • Ancient India
          • Indus River Valley civilizations
          • The Vedic Period
          • Sanskrit connections to English
          • Hinduism: core ideas of Brahman, Atman, Samsara and Moksha.
        • Shang China
          • Shang Dynasty in Ancient China
        • Ancient Americas
          • Pre-contact Americas
        • Human innovation and the environment
          • Development of agriculture and writing
          • Collective learning
          • Agriculture and human population
          • Firestick farming in early Australia
      • 600 BCE - 600 CE Second-Wave Civilizations
        • Ancient Persia
          • Ancient Persia
          • Cyrus the Great and the Achaemenid Empire
          • Zoroastrianism
        • Classical Greece
          • Classical Greece
          • Greco Persian Wars
          • Second Persian Invasion
          • Classical Greek Society and Culture
          • Philosophy: Socrates, Plato and Aristotle
          • Prelude to the Peloponnesian War
          • The Peloponnesian War
        • The rise and fall of empires
          • Comparison: Rise of empires
          • Comparison: Fall of empires
        • Empire of Alexander the Great
          • Philip of Macedon unifies Greece
          • Alexander the Great takes power
          • Alexander the Great conquers Persia
          • Diadochi and the Hellenistic Period
        • Rise of Rome
          • An Overview of Ancient Rome
          • Roman social and political structures
          • Exchange between Rome, Carthage, and the Phoenicians
          • Punic Wars between Rome and Carthage
        • From Roman Republic to Roman Empire
          • Rise of Julius Caesar
          • Caesar, Cleopatra and the Ides of March
          • Ides of March spark a civil war
          • Augustus and the Roman Empire
        • The Roman Empire
          • Emperors of Pax Romana
          • Fall of the Roman Empire
          • Ara Pacis
          • Visualizing Imperial Rome
        • Ancient and Imperial China
          • Zhou, Qin and Han Dynasties
          • Confucius and the Hundred Schools of Thought
        • Early Judaism
          • Early Judaism, part 1
          • Early Judaism, part 2
        • Early Christianity
          • Early Christianity
          • The spread of Christianity
          • Christianity in the Roman Empire
          • The Council of Nicaea
        • Early Americas
          • Mayan civilization
          • Chavin, Nazca, Moche, Huari and Tiwanaku civilizations
        • Empires in India
          • The rise of empires in India
          • Gupta Dynasty
        • Early Hinduism
          • The Vedic Period
          • Sanskrit connections to English
          • Hinduism: core ideas of Brahman, Atman, Samsara and Moksha.
          • Hindu gods overview
          • Hindu scripture overview
          • Varnas and the Caste System
          • Bhakti movement
        • Early Buddhism
          • Hinduism: core ideas of Brahman, Atman, Samsara and Moksha.
          • Early Buddhism
          • Core spiritual ideas of Buddhism
          • Buddhism: context and comparison
          • Theravada and Mahayana Buddhism
        • Syncretism
          • Cultural Syncretism in Central Asia
        • Women and families
          • Comparative roles of women in Rome and Han China
        • Transregional Trade: the Silk Road
          • The Silk Road
        • Survey of second-wave civilizations
          • Survey from Neo Babylonians to Persians
          • Ancient Greeks and Persians
          • Rome becomes dominant
      • 600 - 1450 Regional and interregional interactions
        • Byzantine Empire
          • Justinian and the Byzantine Empire
          • Comparing Roman and Byzantine Empires
        • European Middle Ages: feudalism and serfdom
          • Overview of the Middle Ages
          • Feudal system during the Middle Ages
          • Serfs and manorialism
          • Peasant revolts
          • An overview of the Crusades (part 2)
        • Origins of Islam
          • Introduction to Islam
          • Beginnings of Islam: The life of Muhammad
          • Beginnings of Islam: The Hijra to Medina and the conversion of Mecca
          • Contextualization of the development of Islam
        • Spread of Islam
          • The spread of Islam
          • Spread of Islamic Culture
        • Sunni and Shia Islam
          • Sunni and Shia Islam part 1
          • Sunni and Shia Islam part 2
        • Golden Age of Islam
          • Golden age of Islam
        • The Great Schism
          • Christianity in the Roman Empire
          • The Council of Nicaea
          • Great Schism or East-West Schism part 1
          • Great Schism or East-West Schism part 2
        • The Crusades
          • Introduction to the Crusades
          • An overview of the Crusades (part 2)
          • Impact of the crusades
          • Knights Templar | World History | Khan Academy
        • The Mongols
          • Genghis Khan and the Mongol Empire
          • Why were the Mongols so effective?
          • Timur
          • Ottoman, Safavid and Mughal Empires
        • Song China
          • Chinese Imperial Dynasties
          • Neo-Confucianism and Zhu Xi
          • Prosperity in Song China (960-1279)
        • Medieval Japan
          • Classical Japan during the Heian Period | World History | Khan Academy
          • Shoguns, samurai and the Japanese Middle Ages
        • Maya, Aztec, and Inca
          • Aztec Empire
          • Inca Empire overview
        • Environment and trade
          • Early medieval trade
        • Human migration
          • Causes and effects of human migration
        • Development of new trading cities
        • Cultural interactions along trade routes
        • Development of financial institutions
          • Knights Templar | World History | Khan Academy
        • Disease and demography
        • Social institutions in the Islamic world
      • 1450 - 1750 Renaissance and Reformation
        • Scientific Revolution and Enlightenment
          • The Scientific Revolution and the Age of Enlightenment | World History | Khan Academy
        • Mughal rule in India
          • Mughal rule in India
          • Ottoman, Safavid and Mughal Empires
        • Sikhism
          • Sikhism introduction
          • Continuity-Sikhism connections to Hinduism and Islam
        • The Protestant Reformation
          • Introduction to the Protestant Reformation: Setting the stage
          • Introduction to the Protestant Reformation: Martin Luther
          • Introduction to the Protestant Reformation: Varieties of Protestantism
          • Introduction to the Protestant Reformation: The Counter-Reformation
        • The Russian Empire
          • How did Russia begin?
      • 1750 -1900 Enlightenment and Revolution
        • American Revolution
          • The Seven Years' War: background and combatants
          • The Seven Years' War: battles and legacy
          • Background and introduction to the United States Declaration of Independence
          • Birth of the US Constitution
        • French Revolution
          • French Revolution (part 1)
          • French Revolution (part 2)
          • French Revolution (part 3) - Reign of Terror
          • French Revolution (part 4) - The Rise of Napoleon Bonaparte
        • Napoleon Bonaparte
          • French Revolution (part 4) - The Rise of Napoleon Bonaparte
          • Napoleon and the Wars of the First and Second Coalitions
          • Napoleon and the War of the Third Coalition
          • Napoleon and the War of the Fourth Coalition
          • Napoleon's Peninsular Campaigns
          • French invasion of Russia
          • Napoleon forced to abdicate
          • Hundred days and Waterloo
        • France's many revolutions and republics
          • Les Miserables and France's many revolutions
        • Haitian Revolution
          • Haitian Revolution (Part 1)
          • Haitian Revolution (Part 2)
        • Latin American independence
          • Latin American Independence movements
        • Imperialism
          • Industrialization and imperialism
          • Opium Wars | World History | Khan Academy
          • Japanese Imperialism | World History | Khan Academy
      • The 20th century
        • Beginning of World War I
          • Empires before World War I
          • German and Italian Empires in 1914
          • Alliances leading to World War I
          • Language and religion of the former Yugoslavia
          • Assassination of Franz Ferdinand
          • The Great War begins
        • Other fronts of World War I
          • Serbian and Macedonian fronts
          • Serbian losses in World War I
          • Italy backs out of Triple Alliance
          • Italian front in World War I
          • Japan in World War I
        • Western and Eastern fronts of World War I
          • Schlieffen Plan and the First Battle of the Marne
          • Comparing the Eastern and Western fronts in WWI
          • World War I Eastern front
          • Battles of Verdun, Somme and the Hindenburg Line
          • Closing stages of World War I
          • Technology in World War I
        • Blockades and American entry
          • Blockades, u-boats and sinking of the Lusitania
          • Zimmermann Telegram
          • United States enters World War I
        • World War I shapes the Middle East
          • Theodor Herzl and the birth of political Zionism
          • Sinai, Palestine and Mesopotamia campaigns
          • Sykes-Picot Agreement and the Balfour Declaration
          • Arabia after World War I
        • Aftermath of World War I
          • Deaths in World War I
          • Woodrow Wilson's Fourteen Points
          • Paris Peace Conference and Treaty of Versailles
          • More detail on the Treaty of Versailles and Germany
          • Arabia after World War I
        • World War I Quiz
        • Rise of Hitler and the Nazis
          • Initial rise of Hitler and the Nazis
          • Hitler's Beer Hall Putsch
          • Hitler and the Nazis come to power
          • Night of the Long Knives
          • Nazi aggression and appeasement
        • Rise of Mussolini and Fascism
          • Fascism and Mussolini
          • Mussolini becomes Prime Minister
          • Mussolini becomes absolute dictator (Il Duce)
          • Mussolini aligns with Hitler
        • Overview of Chinese history 1911-1949
          • Overview of Chinese history 1911 - 1949
        • Overview of World War II
          • Beginning of World War II
          • 1940 - Axis gains momentum in World War II
          • 1941 Axis momentum accelerates in WW2
          • World War II in the Pacific in 1942
          • 1942 Tide turning in World War II in Europe
          • 1943 Axis losing in Europe
          • American progress in the Pacific in 1944
          • 1944 - Allies advance further in Europe
          • 1945 - End of World War II
        • The Cold War
          • Communism
          • Korean War overview
          • Bay of Pigs Invasion
          • Cuban Missile Crisis
          • Vietnam War
          • Allende and Pinochet in Chile
          • Pattern of US Cold War interventions
        • Human rights
          • The Holocaust
          • International Human Rights
          • Independence movements in the 20th Century
    • AP World History
      • Beginnings - 600 BCE
        • What is World History?
        • The origin of humans and early human societies
          • History and prehistory
          • Peopling the earth
          • Organizing paleolithic societies
        • The Neolithic Revolution and the birth of agriculture
          • The Neolithic Revolution and early agriculture
          • The spread of agriculture
        • Ancient Mesopotamia
          • Ancient Mesopotamia
          • Ancient Mesopotamia and the Hebrew Bible
        • Ancient Egypt
          • Ancient Egypt and the Nile River Valley
          • The Hittite Empire and the Battle of Kadesh
        • Ancient India
          • Indus River Valley civilizations
          • The Vedic Period
          • Sanskrit connections to English
        • Shang China
          • Shang Dynasty in Ancient China
        • Ancient art and artifacts
          • The Standard of Ur
          • The Rosetta Stone
        • Human innovation and the environment
          • Development of agriculture and writing
          • Collective learning
          • Agriculture and human population
          • Firestick farming in early Australia
      • 600 BCE - 600 CE Second-Wave Civilizations
        • Ancient Persia
          • Ancient Persia
          • Zoroastrianism
          • Cyrus the Great and the Achaemenid Empire
        • Classical Greece
          • Classical Greece
          • Classical Greek Society and Culture
          • Philosophy: Socrates, Plato and Aristotle
          • Greco Persian Wars
          • Second Persian Invasion
          • The Peloponnesian War
        • The Rise and Fall of Empires
          • Alexander the Great takes power
          • Alexander the Great conquers Persia
          • Comparison: Rise of empires
          • Diadochi and the Hellenistic Period
          • Comparison: Fall of empires
        • Ancient Rome
          • An Overview of Ancient Rome
          • Roman social and political structures
          • Exchange between Rome, Carthage, and the Phoenicians
          • Rise of Julius Caesar
          • Caesar, Cleopatra and the Ides of March
          • Augustus and the Roman Empire
          • Fall of the Roman Empire
        • Ancient and Imperial China
          • Zhou, Qin and Han Dynasties
          • Confucius and the Hundred Schools of Thought
        • Early Judaism
          • Early Judaism, part 1
          • Early Judaism, part 2
        • Early Christianity
          • Early Christianity
          • The spread of Christianity
          • Christianity in the Roman Empire
          • The Council of Nicaea
        • Early Americas
          • Pre-contact Americas
          • Mayan civilization
          • Chavin, Nazca, Moche, Huari and Tiwanaku civilizations
        • Empires in India
          • The rise of empires in India
          • Gupta Dynasty
        • Early Hinduism
          • The Vedic Period
          • Hinduism: core ideas of Brahman, Atman, Samsara and Moksha.
          • Hindu gods overview
          • Hindu scripture overview
          • Varnas and the Caste System
          • Bhakti movement
        • Early Buddhism
          • Early Buddhism
          • Core spiritual ideas of Buddhism
          • Theravada and Mahayana Buddhism
          • Buddhism: context and comparison
        • Syncretism
          • Cultural Syncretism in Central Asia
        • Women and families
          • Comparative roles of women in Rome and Han China
        • Transregional trade: the Silk Road
          • The Silk Road
      • 600 - 1450 Regional and interregional interactions
        • Byzantine Empire
          • Justinian and the Byzantine Empire
          • Comparing Roman and Byzantine Empires
        • European Middle Ages: feudalism and serfdom
          • Overview of the Middle Ages
          • Feudal system during the Middle Ages
          • Serfs and manorialism
          • Peasant revolts
          • An overview of the Crusades (part 2)
        • Origins of Islam
          • Introduction to Islam
          • Beginnings of Islam: The life of Muhammad
          • Beginnings of Islam: The Hijra to Medina and the conversion of Mecca
          • Contextualization of the development of Islam
        • Spread of Islam
          • The spread of Islam
          • Spread of Islamic Culture
        • Sunni and Shia Islam
          • Sunni and Shia Islam part 1
          • Sunni and Shia Islam part 2
        • Golden Age of Islam
          • Golden age of Islam
        • The Great Schism
          • Christianity in the Roman Empire
          • The Council of Nicaea
          • Great Schism or East-West Schism part 1
          • Great Schism or East-West Schism part 2
        • The Crusades
          • Introduction to the Crusades
          • An overview of the Crusades (part 2)
          • Impact of the crusades
          • Knights Templar | World History | Khan Academy
        • The Mongols
          • Genghis Khan and the Mongol Empire
          • Why were the Mongols so effective?
          • Timur
          • Ottoman, Safavid and Mughal Empires
        • Song China
          • Chinese Imperial Dynasties
          • Neo-Confucianism and Zhu Xi
          • Prosperity in Song China (960-1279)
        • Medieval Japan
          • Classical Japan during the Heian Period | World History | Khan Academy
          • Shoguns, samurai and the Japanese Middle Ages
        • Maya, Aztec, and Inca
          • Aztec Empire
          • Inca Empire overview
        • Environment and trade
          • Early medieval trade
        • Human migration
          • Causes and effects of human migration
        • Development of new trading cities
        • Cultural interactions along trade routes
        • Development of financial institutions
          • Knights Templar | World History | Khan Academy
        • Disease and demography
        • Social institutions in the Islamic world
      • 1450 - 1750 Renaissance and Reformation
        • Scientific Revolution and Enlightenment
          • The Scientific Revolution and the Age of Enlightenment | World History | Khan Academy
        • Mughal rule in India
          • Mughal rule in India
          • Ottoman, Safavid and Mughal Empires
        • Sikhism
          • Sikhism introduction
          • Continuity-Sikhism connections to Hinduism and Islam
        • The Protestant Reformation
          • Introduction to the Protestant Reformation: Setting the stage
          • Introduction to the Protestant Reformation: Martin Luther
          • Introduction to the Protestant Reformation: Varieties of Protestantism
          • Introduction to the Protestant Reformation: The Counter-Reformation
        • The Russian Empire
          • How did Russia begin?
      • 1750 - 1900 Enlightenment and Revolution
        • American Revolution
          • The Seven Years' War: background and combatants
          • The Seven Years' War: battles and legacy
          • Background and introduction to the United States Declaration of Independence
          • Birth of the US Constitution
        • French Revolution
          • French Revolution (part 1)
          • French Revolution (part 2)
          • French Revolution (part 3) - Reign of Terror
          • French Revolution (part 4) - The Rise of Napoleon Bonaparte
        • Napoleon Bonaparte
          • French Revolution (part 4) - The Rise of Napoleon Bonaparte
          • Napoleon and the Wars of the First and Second Coalitions
          • Napoleon and the War of the Third Coalition
          • Napoleon and the War of the Fourth Coalition
          • Napoleon's Peninsular Campaigns
          • French invasion of Russia
          • Napoleon forced to abdicate
          • Hundred days and Waterloo
        • France's many revolutions and republics
          • Les Miserables and France's many revolutions
        • Haitian Revolution
          • Haitian Revolution (Part 1)
          • Haitian Revolution (Part 2)
        • Latin American independence
          • Latin American Independence movements
        • Imperialism
          • Industrialization and imperialism
          • Opium Wars | World History | Khan Academy
          • Japanese Imperialism | World History | Khan Academy
      • The 20th century
        • Beginning of World War I
          • Empires before World War I
          • German and Italian Empires in 1914
          • Alliances leading to World War I
          • Language and religion of the former Yugoslavia
          • Assassination of Franz Ferdinand
          • The Great War begins
        • Other fronts of World War I
          • Serbian and Macedonian fronts
          • Serbian losses in World War I
          • Italy backs out of Triple Alliance
          • Italian front in World War I
          • Japan in World War I
        • Western and Eastern fronts of World War I
          • Schlieffen Plan and the First Battle of the Marne
          • Comparing the Eastern and Western fronts in WWI
          • World War I Eastern front
          • Battles of Verdun, Somme and the Hindenburg Line
          • Closing stages of World War I
          • Technology in World War I
        • Blockades and American entry
          • Blockades, u-boats and sinking of the Lusitania
          • Zimmermann Telegram
          • United States enters World War I
        • World War I shapes the Middle East
          • Theodor Herzl and the birth of political Zionism
          • Sinai, Palestine and Mesopotamia campaigns
          • Sykes-Picot Agreement and the Balfour Declaration
          • Arabia after World War I
        • Aftermath of World War I
          • Deaths in World War I
          • Woodrow Wilson's Fourteen Points
          • Paris Peace Conference and Treaty of Versailles
          • More detail on the Treaty of Versailles and Germany
          • Arabia after World War I
        • Rise of Hitler and the Nazis
          • Initial rise of Hitler and the Nazis
          • Hitler's Beer Hall Putsch
          • Hitler and the Nazis come to power
          • Night of the Long Knives
          • Nazi aggression and appeasement
        • Rise of Mussolini and Fascism
          • Fascism and Mussolini
          • Mussolini becomes Prime Minister
          • Mussolini becomes absolute dictator (Il Duce)
          • Mussolini aligns with Hitler
        • Overview of Chinese history 1911-1949
          • Overview of Chinese history 1911 - 1949
        • Overview of World War II
          • Beginning of World War II
          • 1940 - Axis gains momentum in World War II
          • 1941 Axis momentum accelerates in WW2
          • World War II in the Pacific in 1942
          • 1942 Tide turning in World War II in Europe
          • 1943 Axis losing in Europe
          • American progress in the Pacific in 1944
          • 1944 - Allies advance further in Europe
          • 1945 - End of World War II
        • The Cold War
          • Communism
          • Korean War overview
          • Bay of Pigs Invasion
          • Cuban Missile Crisis
          • Vietnam War
          • Allende and Pinochet in Chile
          • Pattern of US Cold War interventions
        • Human rights
          • The Holocaust
          • International Human Rights
          • Independence movements in the 20th Century
    • Art history
    • AP Art History
      • Start here
        • Why look at art?
        • How to do visual (formal) analysis in art history
        • Art historical analysis (painting), a basic introduction using Goya's Third of May, 1808
        • A brief history of religion in art
        • Is there a difference between art and craft?
        • How art can help you analyze
      • Introduction to cultures and religions for the study of AP Art History
        • Beliefs made visible: Hindu art in South Asia
        • Beliefs made visible: Buddhist art in South Asia
        • Stories of the modern pilgrimage
      • Global prehistory: 30,000-500 B.C.E.
        • Paleolithic, Mesolithic and Neolithic
          • Bushel with ibex motifs
          • Jade Cong
          • Working jade
          • Stonehenge, Avebury and Associated Sites (UNESCO/NHK)
          • Tlatilco figurines
      • Ancient Mediterranean: 3500 B.C.E.-300 C.E.
        • Ancient Near East
          • Standing Male Worshipper from Tell Asmar
          • The Standard of Ur
          • The Law Code Stele of King Hammurabi
          • Lamassu from the citadel of Sargon II
          • Capital of a column from the audience hall of the palace of Darius I, Susa
        • Ancient Egypt
          • Seated Scribe
          • Mortuary Temple of Hatshepsut and Large Kneeling Statue, New Kingdom, Egypt
          • Ancient Thebes with its Necropolis (UNESCO/TBS)
          • Akhenaten, Nefertiti, and Three Daughters
          • Last Judgement of Hunefer, from his tomb
        • Ancient Greece
          • The classical orders
          • The Athenian Agora and the experiment in democracy
          • Anavysos Kouros
          • Peplos Kore from the Acropolis
          • Making Greek vases
          • Niobid Painter, Niobid Krater
          • Polykleitos, Doryphoros (Spear Bearer)
          • Parthenon (Acropolis)
          • East Pediment sculptures, Parthenon, including Helios, Horses and Dionysus (Heracles?)
          • Parthenon frieze
          • "Plaque of the Ergastines" fragment from the frieze on the east side of the Parthenon
          • Victory (Nike) Adjusting Her Sandal, Temple of Athena Nike (Acropolis)
          • Grave Stele of Hegeso
          • Winged Victory (Nike) of Samothrace
          • Great Altar of Zeus and Athena at Pergamon
          • Alexander Mosaic from the House of the Faun, Pompeii
          • Apollonius, Seated Boxer
        • Ancient Etruria
          • Sarcophagus of the Spouses (Rome)
          • Apulu (Apollo of Veil)
          • Etruscan Necropolises of Cerveteri and Tarquinia (from UNESCO/NHK)
        • Ancient Rome
          • Visualizing Imperial Rome
          • Digging through time
          • Veristic Male Portrait
          • Augustus of Primaporta
          • Colosseum (Flavian Amphitheater)
          • The Forum of Trajan
          • Markets of Trajan
          • Column of Trajan
          • The Pantheon
          • Ludovisi Battle Sarcophagus
      • Early Europe and Colonial Americas: 200-1750 C.E.
        • Medieval art in Europe
          • Catacomb of Priscilla, Rome
          • Basilica of Santa Sabina, Rome
          • Jacob wrestling the angel, Vienna Genesis
          • San Vitale, Ravenna
          • Hagia Sophia, Istanbul
          • Theotokos mosaic, apse, Hagia Sophia, Istanbul
          • Hagia Sophia as a mosque
          • Deësis mosaic, Hagia Sophia, Istanbul
          • The Bayeux Tapestry - Seven Ages of Britain - BBC One
          • Chartres Cathedral
          • Röttgen Pietà
          • Giotto, Arena (Scrovegni) Chapel (part 1)
          • Giotto, Arena (Scrovegni) Chapel (part 2)
          • Giotto, Arena (Scrovegni) Chapel (part 3)
          • Giotto, Arena (Scrovegni) Chapel (part 4)
        • Art of the Islamic world in the medieval era
        • Renaissance Art in Europe
          • Workshop of Campin, Annunciation Triptych (Merode Altarpiece)
          • Brunelleschi, Pazzi Chapel
          • Van Eyck, The Arnolfini Portrait
          • Donatello, David
          • Alberti, Palazzo Rucellai
          • Fra Filippo Lippi, Madonna and Child with two Angels
          • Botticelli, Birth of Venus
          • The Last Supper
          • Ceiling of the Sistine Chapel
          • Studies for the Libyan Sibyl and a small Sketch for a Seated Figure (verso)
          • Last Judgment (altar wall, Sistine Chapel)
          • Raphael, School of Athens
          • Pontormo, The Entombment of Christ
          • Titian, Venus of Urbino
        • Protestant Reformation and Catholic Counter-Reformation
          • Il Gesù, including Triumph of the Name of Jesus ceiling fresco
          • Bruegel, Hunters in the Snow (Winter)
          • Caravaggio, Calling of Saint Matthew and Inspiration of St. Matthew
          • Rubens, The Presentation of the Portrait of Marie de' Medici
          • Borromini, San Carlo alle Quattro Fontane
          • Bernini, Ecstasy of Saint Teresa
          • Velázquez, Las Meninas
          • Johannes Vermeer, Woman Holding a Balance
          • Rachel Ruysch, Fruit and Insects
          • William Hogarth, Marriage A-la-Mode (including Tête à Tête)
        • Colonial Americas
          • Screen with the Siege of Belgrade and Hunting Scene (Brooklyn Biombo)
          • Virgin of Guadalupe
      • Later Europe and Americas: 1750-1980 C.E.
        • Enlightenment and revolution
          • Fragonard, The Swing
          • David, Oath of the Horatii
          • Ingres, La Grande Odalisque
          • Delacroix, Liberty Leading the People
          • Thomas Cole, The Oxbow
          • Turner, Slave Ship
          • Charles Barry and A.W.N. Pugin, Palace of Westminster (Houses of Parliament)
        • Modern and contemporary art
          • Manet, Olympia
          • Monet, Gare St. Lazare
          • Velasco, The Valley of Mexico
          • Cézanne, Mont Sainte-Victoire
          • Picasso, Les Demoiselles d'Avignon
          • Gustav Klimt, The Kiss
          • Constantin Brancusi, The Kiss
          • Kandinsky, Improvisation 28 (second version), 1912
          • Jacob Lawrence, The Migration Series (*short version*)
          • Jacob Lawrence, The Migration Series (*long version*)
          • Duchamp, Fountain
          • Willem de Kooning, Woman, I
          • Mies van der Rohe, Seagram Building
          • Robert Smithson, Spiral Jetty
      • Indigenous Americas
        • Mesa Verde and the preservation of Ancestral Puebloan heritage
        • Unearthing the Aztec past, the destruction of the Templo Mayor
        • The Sun Stone (The Calendar Stone)
        • Coyolxauhqui Stone
        • Olmec mask (Olmec-style mask)
        • Feathered headdress
      • Africa: 1100-1980 C.E.
        • Great Zimbabwe National Monument (UNESCO/NHK)
        • Sika dwa kofi (Golden Stool), Asante people
        • Female (pwo) mask (Chokwe peoples)
        • Bundu / Sowei Helmet Mask (Mende peoples)
        • Male figure, Ikenga (Igbo Peoples)
        • Elephant Mask, Kuosi Society, Bamileke Peoples, Cameroon
        • Fang Reliquary Figure
        • Olowe of Ise, Veranda Post (Yoruba people)
      • West and central Asia: 500 B.C.E.-1980 C.E.
        • Petra: UNESCO Siq Project
        • Bamiyan: Ten years on
        • Basin (Baptistère de Saint Louis)
      • South, East, and Southeast Asia: 300 B.C.E. - 1980 C.E.
        • India
          • Beliefs made visible: Hindu art in South Asia
          • Hindu temples
          • Beliefs made visible: Buddhist art in South Asia
          • The stupa
        • China
          • Terra cotta warriors from the Mausoleum of the First Qin Emperor (UNESCO/TBS)
          • Longmen Grottoes (UNESCO/NHK)
          • The David Vases
          • The Forbidden City
        • Korea
        • Japan
          • Ancient temples of Nara Japan
          • Ryoanji
        • Cambodia
          • Angkor temple mountains
        • Indonesia
      • The Pacific
        • Voyage to the moai of Rapa Nui (Easter Island)
        • Mask (Buk), Torres Strait, Mabuiag Island
      • Global contemporary: 1980-present
        • Maya Lin, Vietnam Veterans Memorial
        • Xu Bing, Book from the Sky
        • El Anatsui, Untitled
        • Zaha Hadid, MAXXI National Museum of XXI Century Arts, Rome
        • Ai Weiwei: Sunflower Seeds
    • Grammar
      • Introduction to grammar
        • Welcome to grammar!
          • Introduction to Grammar
      • Parts of speech: the noun
        • Introduction to nouns
          • Introduction to nouns
          • Introduction to singular and plural nouns
        • Types of nouns
          • Common and proper nouns
          • Concrete and abstract nouns
        • Irregular plural nouns: base plurals and irregular endings
          • Irregular plural nouns – -f to -ves
          • Irregular plural nouns – words that end in -en
          • Irregular plural nouns – base plurals
        • Irregular plural nouns: mutant and foreign plurals
          • Irregular plural nouns – the MUTANT PLURALS
          • Irregular plural nouns – foreign plurals
          • BONUS VIDEO – Origin of the Mutant Plural
      • Parts of speech: the verb
        • Introduction to verbs
          • Introduction to verbs
          • Introduction to agreement
        • The tenses
          • Introduction to verb tense
          • The present tense
          • The past tense
          • The future tense
        • Linking and helping verbs
          • Linking verbs
          • Helping verbs
        • Irregular verbs
          • Introduction to irregular verbs
          • The funky -ed irregular verb
          • The vowel-shift irregular verb
          • The irregular verb gets taken for a ride
          • The truly irregular verbs
        • Verb aspect: simple, progressive, and perfect
          • Introduction to verb aspect
          • Simple Aspect
          • Progressive Aspect
          • Perfect Aspect
        • Verb aspect and modal verbs
          • Perfect progressive aspect
          • Modal verbs
      • Parts of speech: the pronoun
        • Introduction to pronouns
          • What is a pronoun?
          • Personal pronouns
        • Possessive and reflexive pronouns
          • Possessive pronouns
          • Reflexive pronouns
        • Relative pronouns
          • Relative pronouns
          • That versus which
          • Who versus whom
        • Subject, object, person, and number
          • Subject and object pronouns
          • Grammatical person and pronouns
          • Pronoun number
        • Indefinite pronouns, pronoun vagueness, and emphatic pronouns
          • Indefinite pronouns
          • Emphatic pronouns
          • BONUS VIDEO | Singular They
      • Parts of speech: the modifier
        • Introduction to adjectives and articles
          • Intro to adjectives
          • Intro to articles
          • Definite and indefinite articles
        • Introduction to adverbs
          • Intro to adverbs
          • Relative adverbs
        • Adjective order and commas with adjectives
          • Adjective order
          • Commas and Adjectives
        • Comparative, superlative, intensifiers, and adverbs of degree
          • Intro to the comparative and the superlative
          • Forming comparative and superlative modifiers
          • Intensifiers and adverbs of degree
      • Parts of speech: the preposition and the conjunction
        • Introduction to prepositions
          • Meet the preposition
          • Prepositions of time
          • Prepositions of space
          • Prepositions of neither space nor time
        • Types of prepositions and phrases
          • Compound prepositions
          • Prepositional phrases
          • Terminal prepositions
        • Introduction to conjunctions
          • Coordinating conjunctions
          • Subordinating conjunctions
        • Correlative conjunctions and starting sentences
          • Correlative conjunctions
          • Beginning sentences with conjunctions
          • BONUS: "FANBOYS," a mnemonic song
      • Punctuation: the comma and the apostrophe
        • Introduction to commas
          • Three ways to end a sentence
          • Meet the Comma
          • Punctuating a list
        • Commas in space and time
          • Salutations and valedictions
          • Commas in space and time
          • Commas and introductory elements
        • More ways to use commas
          • More uses for commas
          • Commas in dialogue
          • Appositives
          • BONUS: The Oxford comma
        • Apostrophes and contractions
          • Introduction to the apostrophe
          • Introduction to contractions
          • Apostrophes and plurals
        • Introduction to the possessive
          • Introduction to the possessive
          • Possession for words ending in “s”
          • Advanced (plural) possession
        • Its vs. it's
          • Choosing between its and it’s
          • BONUS: History of the apostrophe
          • BONUS: History of the possessive apostrophe
      • Punctuation: the colon, semicolon, and more
        • Introduction to colons
          • Linking function of the colon
          • The colon as a separator
        • Introduction to semicolons
          • Introduction to the semicolon
          • Semicolons and complex lists
        • Formatting styles
          • Italics and underlining
          • Using quotation marks in titles
          • Parentheses
        • Hyphens, dashes, and ellipses
          • Dashes
          • Hyphens vs. dashes
          • Ellipses
      • Syntax: sentences and clauses
        • Introduction to sentences
          • What is a sentence?
          • Three types of sentence
          • Exclamations
        • Types of sentences
          • Simple and compound sentences
          • Complex sentences
          • Compound-complex sentences
        • Subjects and predicates
          • Subjects and predicates
          • Subject, direct object, and indirect object
        • Phrases and clauses
          • Phrases and clauses
          • Dependent and independent clauses
          • Relative clauses
      • Syntax: conventions of standard English
        • Subject-verb agreement and pronoun-antecedent agreement
          • Subject-verb agreement
          • Pronoun-antecedent agreement
        • Fragments and run-ons
          • Recognizing fragments
          • Run-ons and comma splices
        • Dangling modifiers and parallel structure
          • Dangling modifiers
          • Parallel structure
      • Usage and Style
        • Frequently confused words
          • Affect and effect
          • Hear/here and accept/except
          • There, their, and they're
          • To, two, and too
          • Compliment/complement and desert/dessert
          • Less versus fewer
          • Bare/bear, allowed/aloud, advice/advise, and break/brake
        • Common expressions
        • Style!
          • The Sound of Language: Alliteration, Assonance, and Onomatopoeia
          • The scale of formality
          • Under- and overstatement
          • Irony
    • Music
      • Music basics
        • Notes and rhythm
          • Lesson 1: Note values, duration, and time signatures
          • Lesson 2: Rhythm, dotted notes, ties, and rests
          • Lesson 3: Meters in double and triple time, upbeats
          • Lesson 4: Meters in 6, 9, and 12
          • Lesson 5: Review of time signatures – Simple, compound, and complex
          • Lesson 6: Constant versus changing time, adding triplets, and duplets
        • Reading music
          • Lesson 1: Staff, names of notes, treble clef
          • Lesson 2: Ledger lines and the octave
          • Lesson 3: Bass clef, grand staff and the octave
          • Lesson 4: Reading music in treble clef and the C Major scale
          • Lesson 5: C Major scale in bass clef and reading in bass clef
          • Lesson 6: Alto and tenor clefs.
          • Lesson 7: Accidentals
          • Lesson 8: Natural sign, more on accidentals and key signature
          • Lesson 9: More on sharps and flats
          • Lesson 10: Chromatic scales and the half step
      • Masterpieces old and new
        • Ludwig van Beethoven: Symphony No. 5
          • Ludwig van Beethoven: Symphony No. 5, analysis by Gerard Schwarz (part 1)
          • Ludwig van Beethoven: Symphony No. 5 Analysis by Gerard Schwarz (part 2)
          • Ludwig van Beethoven: Symphony No. 5, analysis by Gerard Schwarz (part 3 & 4)
          • Ludwig van Beethoven: Symphony No. 5, a conducting lesson by Gerard Schwarz (1st Movement)
          • Ludwig van Beethoven: Symphony No. 5, an appreciation by Leon Botstein
        • Pyotr Ilyich Tchaikovsky: Symphony No.4
          • Pyotr Ilyich Tchaikovsky: Symphony No.4, analysis by Gerard Schwarz (part 1)
          • Pyotr Ilyich Tchaikovsky: Symphony No.4, analysis by Gerard Schwarz (part 2)
          • Pyotr Ilyich Tchaikovsky: Symphony No.4, analysis by Gerard Schwarz (part 3)
          • Pyotr Ilyich Tchaikovsky: Symphony No.4, analysis by Gerard Schwarz (part 4)
        • Robert Schumann: Symphony No. 3 "Rhenish"
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 1)
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 2)
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 3)
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 4)
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 5)
        • Johannes Brahms: Academic Festival Overture
          • Johannes Brahms: "Academic Festival Overture", analysis by Gerard Schwarz
        • Antonín Dvořák: Symphony No. 9 "From the New World"
          • Antonín Dvořák: Symphony No. 9 "From the New World," analysis by Gerard Schwarz (part 1)
          • Antonín Dvořák: Symphony No. 9 "From the New World," analysis by Gerard Schwarz (part 2)
          • Antonín Dvořák: Symphony No. 9 "From the New World," analysis by Gerard Schwarz (part 3)
          • Antonín Dvořák: Symphony No. 9 "From the New World," analysis by Gerard Schwarz (part 4)
          • Antonín Dvořák: Symphony No. 9 "From the New World", a commentary by Joseph Horowitz
        • Gustav Mahler: Symphony No. 2 – 1st Movement
          • Gustav Mahler: Symphony No. 2 – 1st Movement . Analysis by Gerard Schwarz
          • Gustav Mahler – Second Symphony. An appreciation by Gilbert Kaplan
        • Igor Stravinsky: Suite from The Firebird
          • Discovering Stravinsky's "Firebird" : The story and the music
        • Maurice Ravel: Suite No. 2 from Daphnis et Chloe
          • Maurice Ravel: "Daphnis et Chloé", analysis by Gerard Schwarz
        • Dmitri Shostakovich: Symphony No. 5
          • Dmitri Shostakovich: Symphony No. 5. Analysis by Gerard Schwarz (part 1)
          • Dmitri Shostakovich: Symphony No. 5. Analysis by Gerard Schwarz (part 2)
          • Dmitri Shostakovich: Symphony No. 5. Analysis by Gerard Schwarz (part 3)
          • Dmitri Shostakovich: Symphony No. 5. Analysis by Gerard Schwarz (part 4)
        • Introduction to Philip Glass' "Harmonium Mountain"
          • Introduction to Philip Glass' "Harmonium Mountain"
        • Bernard Rands: Adieu
          • "Adieu": The composer and his work
        • Augusta Read Thomas: Of Paradise and Light
          • "Of Paradise and Light": The composer and her work
        • Bright Sheng: "Black Swan", the composer and his work
          • Bright Sheng: "Black Swan". The composer and his work
        • Richard Danielpour: Piano Concerto No. 4, Movement III
          • Richard Danielpour. Piano Concerto No.4, Movement III: The composer and his work
        • David Stock: Blast!
          • "Blast!": Composer and his work
        • Ellen Taaffe Zwilich: Avanti!
          • "Avanti!": Composer and her work
        • Joseph Schwantner: The Poet's Hour
          • "The Poet's Hour": Composer and his work
      • Instruments of the orchestra
        • Strings
          • Harp: Interview and demonstration with principal Nancy Allen
          • Violin: Interview and demonstration with concertmaster David Kim
          • Viola: Interview and demonstration with principal Rebecca Young
          • Cello: Interview and demonstration with principal Jerry Grossman
          • Bass: Interview and demonstration with principal Alexander Hanna
        • Woodwinds
          • Piccolo: Interview and demonstration with Nadine Asin
          • Flute: Interview and demonstration with principal Jeffrey Khaner
          • E-Flat Clarinet: Interview and demonstration with Jessica Phillips Rieske
          • Clarinet: Interview and demonstration with principal Jon Manasse
          • Bass Clarinet: Interview and demonstration with James Ognibene
          • Oboe: Interview and demonstration with principal John Ferrillo
          • Bassoon: Interview and demonstration with principal Nancy Goeres
          • English Horn: Interview and demonstration with Pedro Diaz
        • Brass
          • French Horn: Interview and demonstration with principal John Cerminaro
          • Trumpet: Interview and demonstration with principal David Bilger
          • Trombone: Interview and demonstration with principal Joseph Alessi
          • Bass Trombone: Interview and demonstration with Denson Paul Pollard
          • Tuba: Interview and demonstration with Chris Olka
        • Percussion
          • Timpani: Interview and demonstration with principal Jauvon Gilliam
          • Piano (as orchestral instrument): Interview and demonstration with Kimberly Russ
          • Percussion: Interview and demonstration with principal Chris Devine and members of the percussion section
    • Special topics in art history
      • ARCHES (At Risk Cultural Heritage Education Series)
        • A beginner's guide
          • What is archaeology: understanding the archaeological record
          • End trafficking, save culture
          • Ungrounded: Portrait Bust of a Flavian Woman (Fonseca bust)
        • Endangered heritage: The Americas
          • Mesa Verde and the preservation of Ancestral Puebloan heritage
          • What the bulldozers left behind: reclaiming Sicán’s past
          • Unearthing the Aztec past, the destruction of the Templo Mayor
        • Endangered heritage: Europe and West Asia
          • From tomb to museum: the story of the Sarpedon Krater
          • Looting, collecting, and exhibiting: the Bubon bronzes
          • The importance of the archaeological findspot: The Lullingstone Busts
          • A race against time: manuscripts and digital preservation
          • Saving Venice
          • Nazi looting: Egon Schiele's Portrait of Wally
        • Endangered heritage: Southeast Asia
          • The Looting of Cambodian Antiquities
          • The scourge of looting: trafficking antiquities, from temple to museum
          • Sotheby's Returns Looted 10th Century Statue to Cambodia
        • Endangered heritage: Pacific islands
          • Voyage to the moai of Rapa Nui (Easter Island)
        • Endangered heritage: Africa
          • Lost History: the terracotta sculpture of Djenné Djenno
      • Creating & conserving
        • The conservator's eye
          • The conservator's eye: Marble statue of a wounded warrior
          • The conservator's eye: Taddeo Gaddi, Saint Julian
          • The conservator's eye: a stained glass Adoration of the Magi
          • The conservator’s eye: Rembrandt's Aristotle with a Bust of Homer
          • The conservator's eye: Madame Cézanne in the Conservatory
          • The conservator’s eye: Anselm Kiefer, Bohemia Lies by the Sea
        • What is conservation?
          • Art Conservation Behind the Scenes
          • What does a conservator do?
          • Conservation: Tools of the Trade
        • About color
          • Making Purple: The Science of Art | National Gallery, London
          • Making Green: Tempera versus Oil | National Gallery, London
          • The Alchemy of Color and Chemical Change in Medieval Manuscripts
        • Painting materials
          • Tempera paint
          • Gold-ground panel painting
          • Oil paint
          • Oil paint in Venice
          • How to stretch a large canvas
          • Almost Invisible: The Cartoon Transfer Process
        • Painting conservation
          • Conservation of paintings
          • Ghent Altarpiece project overview
          • Conserving a portrait of King Edward VI
          • Jan Gossart - Conservation Discoveries
          • Conserving Velázquez's Portrait of Philip IV
          • Conserving van Walscapelle's Flowers in a Glass Vase
          • Conserving Everhard Jabach and His Family
          • Conserving the Virgin of Guadalupe
          • Conserving a Korean Buddhist painting: Vulture Peak
          • Conserving Vincent van Gogh's Field with Irises near Arles
          • Conserving Van Gogh's "Enclosed Field with Ploughman" Under Raking Light
          • The Science of Van Gogh's Bedrooms
          • Restoring Rothko
        • How to paint like...
          • The Painting Techniques of Jackson Pollock
          • The Painting Techniques of Mark Rothko
          • The Painting Techniques of Barnett Newman
          • The Painting Techniques of Franz Kline
          • How to paint like Yayoi Kusama
          • How to paint like Willem de Kooning
          • Investigating Rothko's Technique
        • Painting terms in action
          • Art Terms in Action: Turpentine Burn
          • Art Terms in Action: Palette Knife
          • Art Terms in Action: Stain
          • Art Terms in Action: Emulsion
          • Art Terms in Action: Enamel
          • Art Terms in Action: Paint
          • Art Terms in Action: Tint, Shade, and Tone
          • Art Terms in Action: Viscosity
        • Sculpture
          • Working jade
          • Quarrying and carving marble
          • Carving marble with traditional tools
          • Casting bronze: indirect lost-wax method
          • Casting bronze: direct lost-wax casting
          • Making a Spanish polychrome sculpture
          • After the Fall: The Conservation of Tullio Lombardo's "Adam"
          • Object Conservation - Salisbury Cross
          • Contemporary Art Conservation at Smithsonian's Hirshhorn Museum
        • Printmaking
          • Introduction to printmaking
          • Introduction to relief printmaking
          • Relief process
          • Introduction to intaglio
          • Intaglio process
          • Introduction to lithography
          • Lithography process
        • Drawing and manuscripts
          • Making manuscripts
          • Drawing with charcoal: historical techniques of 19th century France
          • Conserving old master drawings: a balancing act
          • Conserving Henri Matisse's "The Swimming Pool"
        • Photographs
          • Seeing Through Photographs
          • Before Photography - Photographic Processes Series - Chapter 1 of 12
          • The Daguerreotype - Photographic Processes Series - Chapter 2 of 12
          • Talbot's Processes - Photographic Processes Series - Chapter 3 of 12
          • The Collodion - Photographic Processes Series - Chapter 5 of 12
          • The Albumen Print - Photographic Processes Series - Chapter 6 of 12
          • The Platinum Print - Photographic Processes Series - Chapter 7 of 12
          • The Pigment Processes - Photographic Processes Series - Chapter 8 of 12
          • The Woodburytype - Photographic Processes Series - Chapter 9 of 12
          • The Gelatin Silver Process - Photographic Processes Series - Chapter 10 of 12
          • Color Photography - Photographic Processes Series - Chapter 11 of 12
          • Digital Photography - Photographic Processes Series - Chapter 12 of 12
        • Ceramics and lacquer
          • Making Greek vases
          • Ancient Greek vase production and the black-figure technique
        • Mosaic, metalwork and glass
          • Glassmaking technique: free-blown glass
          • Ancient and Byzantine mosaic materials
          • Roman mold-blown glass
          • Medieval goldsmiths
          • The conservator's eye: a stained glass Adoration of the Magi
          • Enamelling
          • An Art of Attraction: The Electrotyping Process
        • Textiles
          • The Art of Making a Tapestry
          • William Morris, The Bullerswood Carpet
          • Conserving the Emperors Carpet
        • Frames
          • Framing "Christina's World" by Andrew Wyeth
          • Behind the scenes at the Framing Department | The National Gallery, London
          • Van Dyck's Self-portrait: The Frame and its Conservation
        • Architecture
          • Kasbah Taourirt: Conserving Earthen Heritage in Morocco
        • Quiz: materials and techniques
    • Art history basics
      • First things first
        • Why look at art?
        • A brief history of religion in art
        • Is there a difference between art and craft?
      • Tools for understanding art
        • How to do visual (formal) analysis in art history
        • Art historical analysis (painting), a basic introduction using Goya's Third of May, 1808
        • A brief history of representing the body in Western painting
        • A brief history of representing of the body in Western sculpture
        • What is contrapposto?
        • What is foreshortening?
        • What is chiaroscuro?
        • How one-point linear perspective works
        • What is atmospheric perspective?
        • The classical orders
        • How art can help you analyze
        • The Case for Copying
        • Looking at paintings
      • Art 1010
        • Ancient Near Eastern and ancient Egyptian art
        • Ancient Greek and Roman art
        • Medieval and Byzantine art
        • Renaissance art
        • Baroque and Rococo
        • Nineteenth-century art
        • Introduction to Twentieth-Century Art
        • Introduction to Contemporary Art
    • Prehistoric art in Europe and West Asia
      • Paleolithic art
        • Nude woman (Venus of Willendorf)
      • Neolithic art
        • Bushel with ibex motifs
        • Stonehenge, Avebury and Associated Sites (UNESCO/NHK)
      • Quiz: prehistoric art
    • Art of the ancient Mediterranean
      • Ancient Near East
        • The Ancient Near East, an introduction
        • Sumerian
          • Writing Cuneiform
          • Standing Male Worshipper from Tell Asmar
          • The Standard of Ur
          • Seated Gudea holding Temple Plan
        • Akkadian
          • Victory Stele of Naram-Sin
          • Cylinder seal and modern impression: nude bearded hero wrestling with a water buffalo; bull-man wrestling with lion
        • Babylonian
          • Visiting Babylon
          • The Babylonian mind
          • The Law Code Stele of King Hammurabi
          • Ishtar gate and Processional Way
          • Towers of Babel
        • Assyrian
          • Lamassu from the citadel of Sargon II
          • Ashurbanipal hunting lions
        • Persian
          • The Cyrus Cylinder and Ancient Persia
          • Capital of a column from the audience hall of the palace of Darius I, Susa
        • Quiz: Ancient Near Eastern art
      • Egyptian art and culture
        • A beginner's guide to ancient Egypt
        • Predynastic and Old Kingdom
          • Seated Scribe
        • Middle Kingdom and New Kingdom
          • Hippopotamus
          • Mortuary Temple of Hatshepsut and Large Kneeling Statue, New Kingdom, Egypt
          • Ancient Thebes with its Necropolis (UNESCO/TBS)
          • Akhenaten, Nefertiti, and Three Daughters
          • Portrait head of Queen Tiye with a crown of two feathers
          • Thutmose, Bust of Nefertiti
          • Head of Tutankhamun from the Amarna Period of Egypt’s New Kingdom
          • Last Judgement of Hunefer, from his tomb
          • Ancient Egyptian papyrus in the Book of the Dead Exhibition
        • Ptolemaic
          • The mummification process
          • Ancient Egyptian coffin prepared for the Book of the Dead exhibition at the British Museum
      • Aegean art
        • Cycladic
          • Harp Player, Early Cycladic period
          • Frescoes from Akrotiri, Thera
        • Mycenaean
          • The "Palace" and Grave Circle A, Mycenae
          • Mask of Agamemnon, Mycenae
          • Lion Gate, Mycenae
          • The Treasury of Atreus
      • Greek art
        • A beginner's guide to ancient Greece
          • The classical orders
          • Victorious athlete: The Vaison Daidoumenos
          • Prize amphora showing a chariot race
          • A competitor in the long jump
          • Sprinter on a vase from Rhodes and a bronze running girl
        • Pottery
          • Ancient Greek vase production and the black-figure technique
          • Dipylon Amphora
          • Terracotta Krater
          • Eleusis Amphora
          • Exekias, amphora with Ajax and Achilles playing a game
          • From tomb to museum: the story of the Sarpedon Krater
          • Exekias, Dionysos Kylix
          • Niobid Painter, Niobid Krater
        • Daedalic and Archaic
          • Lady of Auxerre
          • New York Kouros
          • Marble Statue of a kouros
          • Anavysos Kouros
          • Peplos Kore from the Acropolis
          • Ancient Greek temples at Paestum, Italy
          • Sanctuary of Apollo at Delphi
          • Siphnian Treasury, Delphi
        • Early classical
          • East and West Pediments, Temple of Aphaia, Aegina
          • Kritios Boy
          • Charioteer of Delphi
          • Artemision Zeus or Poseidon
        • Classical
          • Polykleitos, Doryphoros (Spear Bearer)
          • Myron, Discobolus (Discus Thrower), Roman copy of an ancient Greek bronze
          • The Athenian Agora and the experiment in democracy
          • Parthenon (Acropolis)
          • East Pediment sculptures, Parthenon, including Helios, Horses and Dionysus (Heracles?)
          • Parthenon frieze
          • "Plaque of the Ergastines" fragment from the frieze on the east side of the Parthenon
          • Parthenon Metopes
          • Egyptian blue on the Parthenon sculptures
          • The Erechtheion
          • Caryatid and column from the Erechtheion
          • Victory (Nike) Adjusting Her Sandal, Temple of Athena Nike (Acropolis)
          • Grave Stele of Hegeso
        • Late classical (4th century)
          • Capitoline Venus
          • Lysippos, Farnese Hercules
          • Lysippos, Apoxyomenos (Scraper)
          • Alexander Sarcophagus
        • Hellenistic
          • Barberini Faun
          • Dying Gaul
          • Bronze statue of Eros sleeping
          • Winged Victory (Nike) of Samothrace
          • Great Altar of Zeus and Athena at Pergamon
          • Apollonius, Seated Boxer
          • Alexander Mosaic from the House of the Faun, Pompeii
          • Laocoön and his sons
        • Greek art
      • Nabataean
        • Petra: UNESCO Siq Project
      • Etruscan
        • Apulu (Apollo of Veil)
        • Sarcophagus of the Spouses (Rome)
        • Etruscan Necropolises of Cerveteri and Tarquinia (from UNESCO/NHK)
      • Roman
        • A beginner's guide to ancient Rome
          • Visualizing Imperial Rome
          • Looting, collecting, and exhibiting: the Bubon bronzes
          • Digging through time
        • Wall painting
          • Empire: Painted Garden, Villa of Livia
          • Dionysiac frieze, Villa of Mysteries, Pompeii
        • Republic
          • Temple of Portunus
          • Capitoline Brutus
          • Veristic Male Portrait
        • Early empire
          • Augustus of Primaporta
          • Ara Pacis
          • The art of gem carving
          • Portrait Bust of a Flavian Woman (Fonseca Bust), part 1 of 2
          • Ungrounded: Portrait Bust of a Flavian Woman (Fonseca bust)
          • Colosseum (Flavian Amphitheater)
          • Relief from the Arch of Titus, showing The Spoils of Jerusalem being brought into Rome
          • The Forum of Trajan
          • Column of Trajan
          • Markets of Trajan
        • Middle empire
          • The Pantheon
          • Hadrian's Villa, Tivoli: A virtual tour
          • Hadrian, The imperial palace, Tivoli
          • Maritime Theatre at Hadrian's Villa, Tivoli
          • Pair of Centaurs Fighting Cats of Prey from Hadrian's Villa, Tivoli
          • Hadrian, Building the wall
          • Empire: Medea Sarcophagus
          • Equestrian Sculpture of Marcus Aurelius
          • The importance of the archaeological findspot: The Lullingstone Busts
          • Ludovisi Battle Sarcophagus
        • Late empire
          • Portraits of the Four Tetrarchs
          • Basilica of Maxentius and Constantine
          • The Colossus of Constantine
          • Arch of Constantine
        • Ancient Rome (quiz)
      • Palmyra
      • Judaism and art
        • Relief from the Arch of Titus, showing The Spoils of Jerusalem being brought into Rome
        • Sefer Musre Hafilosofim (Book of Morals of Philosophers)
    • Art of Medieval Europe
      • A beginner's guide to medieval Europe
        • Making manuscripts
        • Manuscripts: major works of art
        • The medieval calendar
        • Medieval goldsmiths
        • Sefer Musre Hafilosofim (Book of Morals of Philosophers)
        • Chivalry in the Middle Ages
      • Books in medieval Europe
        • Making the medieval book
          • Listening to the Medieval book
          • An introduction to Medieval scripts
          • A Medieval textbook
          • Making manuscripts
        • Using the medieval book
          • The medieval calendar
      • Early Christian
        • Catacomb of Priscilla, Rome
        • Santa Maria Antiqua Sarcophagus
        • Santa Maria Antiqua
        • Sarcophagus of Junius Bassus
        • Basilica of Santa Sabina, Rome
        • Santa Maria Maggiore
        • Jacob wrestling the angel, Vienna Genesis
        • The Mausoleum of Galla Placidia, Ravenna
      • Byzantine (late Roman Empire)
        • A beginner's guide to Byzantine art and culture
        • Constantinople and the East
          • Hagia Sophia, Istanbul
          • Theotokos mosaic, apse, Hagia Sophia, Istanbul
          • Deësis mosaic, Hagia Sophia, Istanbul
          • The Emperor Triumphant (Barberini Ivory)
          • Hagia Sophia as a mosque
          • Ivory panel with Archangel
          • Icon of the Triumph of Orthodoxy
        • Venice and Ravenna
          • San Vitale, Ravenna
          • Sant'Apollinare in Classe, Ravenna
          • Saint Mark's Basilica, Venice
        • Quiz: Byzantine art
      • Latin (Western) Europe
        • Early medieval
          • Sutton Hoo ship burial
        • Carolingian
          • Charlemagne: an introduction
          • Charlemagne and the Carolingian revival
          • Saint Matthew from the Ebbo Gospel
          • Lindau Gospels cover
          • Santa Prassede (Praxedes)
        • Ottonian
          • Bronze doors, Saint Michael's, Hildesheim, commissioned by Bishop Bernward, 1015
        • Romanesque
          • A look at modern veneration from the British Museum
          • The Bayeux Tapestry - Seven Ages of Britain - BBC One
          • Bayeux Tapestry
          • Durham Cathedral
          • Last Judgment Tympanum, Cathedral of St. Lazare, Autun
          • Pentecost and Mission to the Apostles Tympanum, Vézelay
          • Cistercian Abbey of Fontenay (UNESCO/NHK)
          • Historiated capitals, Church of Sant Miquel, Camarasa (Noguera)
          • Virgin from Ger
          • Wise and Foolish Virgins, Sant Quirze de Pedret
        • Gothic
          • Birth of the Gothic: Abbot Suger and the ambulatory at St. Denis
          • Chartres Cathedral
          • Reims Cathedral
          • Amiens Cathedral
          • Sainte-Chapelle, Paris
          • Salisbury Cathedral
          • Wells Cathedral
          • Lincoln Cathedral
          • Henry VII Chapel, Westminster Abbey
          • Wilton Diptych
          • Berlinghieri, St. Francis altarpiece
          • Röttgen Pietà
          • Humanizing Mary: the Virgin of Jeanne d’Evreux
    • Art of the Islamic world
      • A beginner's guide to the art of Islam
      • Early period
        • Kairouan (from UNESCO)
      • Medieval period
        • Two royal figures (Saljuq Period)
        • Coronation mantle
        • Basin (Baptistère de Saint Louis)
        • Ilkhanid Mihrab
      • Late period
        • Hagia Sophia as a mosque
        • Sinan, Süleymaniye Mosque
        • Tughra (Official Signature) of Sultan Süleiman the Magnificent from Istanbul
        • Sinan, Rüstem Pasha Mosque
        • Coins of faith and power at the British Museum
        • Two portraits, two views
        • Qa'a: The Damascus room
        • Conserving the Damascus Room at The Metropolitan Museum of Art
      • Arts of the Islamic world (quiz)
    • Renaissance & Reformation in Europe
      • Late Gothic art in Italy
        • A beginner's guide to the Late Gothic
          • Gold-ground panel painting
        • Florence, the Late Gothic
          • Cimabue, Santa Trinita Madonna
          • Giotto, The Ognissanti Madonna
          • Cimabue, Santa Trinita Madonna & Giotto's Ognissanti Madonna
          • Giotto, St. Francis Receiving the Stigmata
          • Giotto, Arena (Scrovegni) Chapel (part 1)
          • Giotto, Arena (Scrovegni) Chapel (part 2)
          • Giotto, Arena (Scrovegni) Chapel (part 3)
          • Giotto, Arena (Scrovegni) Chapel (part 4)
          • Giotto, The Entombment of Mary
          • Andrea Pisano's reliefs on the Campanile in Florence
        • Siena, the Late Gothic
          • Duccio, Maesta (front)
          • Duccio, Maesta (back)
          • Duccio, The Rucellai Madonna
          • Duccio, The Virgin and Child with Saints Dominic and Aurea
          • Ambrogio Lorenzetti, Palazzo Pubblico frescos: Allegory and effect of good and bad government
          • Lorenzetti, Presentation of Jesus in the Temple
          • Pietro Lorenzetti, Birth of the Virgin
          • Simone Martini, Annunciation
        • Pisa and Rome, the Late Gothic
          • Nicola Pisano, Pulpit, Pisa Baptistery; and Giovanni Pisano, Slaughter of the Innocents, Pulpit, Sant'Andrea church, Pistoia
          • Pietro Cavallini, The Last Judgment
      • Northern Renaissance: 1400s
        • A beginner's guide to the Northern Renaissance
        • Claus Sluter and Claus de Werve
          • Sluter, Well of Moses
          • Claus Sluter and Claus de Werve, Mourners, from the Tomb of Philip the Bold
        • The Limbourg Brothers
          • Herman, Paul, and Jean de Limbourg, The Belles Heures of Jean de France, Duc de Berry
        • Campin
          • Campin, Christ and the Virgin
          • Workshop of Campin, Annunciation Triptych (Merode Altarpiece)
        • Van Eyck
          • Van Eyck, Ghent Altarpiece (1 of 2)
          • Jan van Eyck, The Ghent Altarpiece (2 of 2)
          • Jan van Eyck, Portrait of a Man in a Red Turban (Self-Portrait?), 1433
          • Jan van Eyck, The Madonna in the Church
          • Van Eyck, The Arnolfini Portrait
        • Christus
          • Christus, Portrait of a Carthusian
          • Christus, Portrait of a Young Girl
        • Van der Weyden
          • Van der Weyden, Deposition
          • Van der Weyden, Crucifixion Triptych
          • Van der Weyden, Crucifixion, with the Virgin and Saint John
          • Rogier van der Weyden, Last Judgment
        • Van der Goes
          • Van der Goes, The Adoration of the Kings
        • Quarton (?)
          • Enguerrand Quarton(?), Pietà of Villeneuve-lès-Avignon
        • Reliquaries and tapestries
          • The Holy Thorn Reliquary of Jean, duc de Berry
          • The Unicorn in Captivity
      • Early Renaissance in Italy: 1400s
        • A beginner's guide to Renaissance Florence
          • How to recognize Italian Renaissance art
          • What is contrapposto?
          • Linear Perspective: Brunelleschi's Experiment
          • How one-point linear perspective works
          • Orsanmichele and Donatello's Saint Mark
        • Sculpture and architecture in Florence
          • Brunelleschi & Ghiberti, the Sacrifice of Isaac
          • Ghiberti, "Gates of Paradise," east doors of the Florence Baptistery
          • Brunelleschi, Old Sacristy
          • Brunelleschi, Dome of the Cathedral of Florence.
          • Brunelleschi, Pazzi Chapel
          • Brunelleschi, Santo Spirito
          • Nanni di Banco, Four Crowned Saints
          • Donatello, Saint Mark
          • Donatello, Feast of Herod
          • Donatello, Madonna of the Clouds
          • Donatello, David
          • Donatello, Equestrian Monument of Gattamelata
          • Donatello, Mary Magdalene
          • Alberti, Palazzo Rucellai
          • Alberti, Façade of Santa Maria Novella, Florence
        • Painting in Florence and Rome
          • Gentile da Fabriano, Adoration of the Magi
          • Masaccio, Virgin and Child Enthroned
          • Masaccio, Holy Trinity
          • Masaccio, The Tribute Money in the Brancacci Chapel
          • Masaccio, Expulsion of Adam and Eve from Eden
          • Fra Angelico, The Annunciation (Prado)
          • Fra Angelico, The Annunciation
          • Uccello, The Battle of San Romano
          • Fra Filippo Lippi, Madonna and Child with two Angels
          • Fra Filippo Lippi, Madonna and Child
          • Lippi, Portrait of a Man and Woman at a Casement
          • Veneziano, St. Lucy Altarpiece
          • Botticelli, Primavera
          • Botticelli, Birth of Venus
        • Central Italy
          • Piero della Francesca, The Baptism of Christ
          • Piero della Francesca, Portraits of the Duke and Duchess of Urbino
          • Martini, Architectural Veduta
      • High Renaissance: Florence and Rome
        • A beginner's guide to the High Renaissance
          • Toward the high Renaissance: Verrocchio and Leonardo
        • Leonardo da Vinci
          • Letter to the Duke of Milan
          • Leonardo: Anatomist - by Nature Video
          • Virgin of the Rocks
          • Adoration of the Magi
          • Virgin and Child with St Anne and St John the Baptist (Burlington House Cartoon)
          • The Last Supper
          • Mona Lisa
        • Michelangelo
          • Pietà (marble sculpture)
          • Moses (marble sculpture)
          • David (marble statue)
          • Carving marble with traditional tools
          • Slaves (marble sculptures)
          • Ceiling of the Sistine Chapel
          • Studies for the Libyan Sibyl and a small Sketch for a Seated Figure (verso)
          • Last Judgment (altar wall, Sistine Chapel)
          • Michelangelo, Medici Chapel (New Sacristy)
          • Laurentian Library
        • Raphael
          • Raphael, Marriage of the Virgin, 1504
          • Raphael, Madonna of the Goldfinch
          • Raphael, La belle jardinière
          • Raphael, School of Athens
          • Raphael, Alba Madonna
          • Raphael, Portrait of Pope Julius II
          • Raphael, Galatea
        • Bramante
          • Bramante, Tempietto
          • Bramante, et.al., Saint Peter's Basilica
      • The Renaissance in Venice
        • Early Renaissance in Venice
          • Oil paint in Venice
          • Bellini, St. Francis
          • Bellini, San Giobbe Altarpiece
          • Giovanni Bellini, San Zaccaria Altarpiece
          • Andrea Mantegna, San Zeno Altarpiece
          • Mantegna, Saint Sebastian
          • Mantegna, Dormition of the Virgin
          • Mantegna, Camera degli Sposi
          • Mantegna, Dead Christ
        • Late Renaissance in Venice
          • Giorgione, The Tempest
          • Giorgione, Three Philosophers
          • Giorgione, the Adoration of the Shepherds
          • Bellini and Titian, the Feast of the Gods
          • Titian, Noli me Tangere
          • Titian, Assumption of the Virgin
          • Titian, Madonna of the Pesaro Family
          • Titian, Bacchus and Ariadne
          • Titian, Venus of Urbino
          • Titian, Christ Crowned with Thorns
          • Titian, Pieta
          • Correggio, Jupiter and Io
          • Veronese, The Family of Darius Before Alexander
          • Veronese, the Dream of Saint Helena
          • Paolo Veronese. Feast in the House of Levi
          • Tintoretto, the Miracle of the Slave
          • Tintoretto, The Finding of the Body of Saint Mark
          • Tintoretto, the Origin of the Milky Way
          • Tintoretto, Last Supper
      • Reformation and Counter-Reformation
        • Introduction to the Protestant Reformation: Setting the stage
        • Introduction to the Protestant Reformation: Martin Luther
        • Introduction to the Protestant Reformation: Varieties of Protestantism
        • Introduction to the Protestant Reformation: The Counter-Reformation
      • Northern Renaissance: 1500s
        • A beginner's guide to the Northern Renaissance in the sixteenth century
        • Pacher and Grünewald
          • Pacher, St. Wolfgang Altarpiece
        • Hieronymus Bosch
          • Bosch, the Last Judgment
        • Albrecht Dürer
          • Dürer, Self-portrait, Study of a Hand and a Pillow
          • Dürer, Self-portrait (1498)
          • Dürer, Self-portrait (1500)
          • Dürer, the Large Piece of Turf
          • Albrecht Dürer, Adam and Eve
          • Dürer, Four Apostles
          • Dürer's woodcuts and engravings
        • Antwerp, Bruges and Brussels
          • David, the Virgin and Child with Saints and Donor
          • Gossaert, Saint Luke Painting the Madonna
          • Van Orley and de Pannemaker, The Last Supper
          • Bruegel, the Dutch Proverbs
          • Bruegel, Tower of Babel
          • Bruegel, Hunters in the Snow (Winter)
          • Pieter Bruegel the Elder, Peasant Wedding
          • Reliquary bust of a companion of Saint Ursula
        • Cranach and Altdorfer
          • Cranach, Adam and Eve
          • Cranach the Elder, Cupid complaining to Venus
          • Cranach the Elder, Judith with the Head of Holofernes
          • Altdorfer, the Battle of Issus
        • Hans Holbein
          • Holbein the Younger, the Ambassadors
          • Holbein the Younger, the Merchant Georg Gisze
          • Holbein the Younger, Christina of Denmark
          • Holbein the Younger, Henry VIII
        • England
          • The Harrowing of Hell
          • Room: 1540-1650
        • Quiz: The Northern Renaissance in the 16th century
      • Renaissance in Spain
        • Introduction to the Renaissance in Spain
        • Renaissance architecture in Spain
      • Late Renaissance and Mannerism
        • Pontormo, The Entombment of Christ
        • Parmigianino, Madonna of the Long Neck
        • Parmigianino, Self-Portrait in a Convex Mirror
        • Rosso Fiorentino, the Dead Christ with Angels
        • Bronzino, An Allegory with Venus and Cupid
        • Bronzino, Portrait of Eleonora of Toledo with her son Giovanni
        • Bronzino and the Mannerist Portrait
        • Cellini, Perseus
        • El Greco, Adoration of the Shepherds
    • Baroque to Neoclassical art in Europe
      • Baroque art
        • A beginner's guide to Baroque art
          • How to recognize Baroque art
        • Italy
          • Bernini, Pluto and Proserpina
          • Bernini, David
          • Bernini, Apollo and Daphne
          • Gian Lorenzo Bernini, Baldacchino
          • Bernini, Bust of Medusa
          • Bernini, Ecstasy of Saint Teresa
          • Bernini, Cathedra Petri (Chair of St. Peter)
          • Bernini, Saint Peter's Square
          • Bernini, Sant'Andrea al Quirinale
          • Borromini, San Carlo alle Quattro Fontane
          • Caravaggio, Narcissus at the Source
          • Caravaggio, Calling of Saint Matthew
          • Caravaggio, The Conversion of St. Paul (or The Conversion of Saul)
          • Caravaggio, Crucifixion of Saint Peter
          • Caravaggio, Supper at Emmaus
          • Caravaggio, Death of the Virgin
          • Gentileschi, Judith and Holofernes
          • Guercino, Saint Luke Displaying a Painting of the Virgin
          • Il Gesù, including Triumph of the Name of Jesus ceiling fresco
          • Pozzo, Saint Ignatius Chapel, Il Gesù
          • Pozzo, Glorification of Saint Ignatius, Sant'Ignazio
          • Pierre Le Gros the Younger, Stanislas Kostka on his Deathbed
        • Flanders
          • Peter Paul Rubens, Elevation of the Cross
          • Rubens, The Rape of the Daughters of Leucippus
          • Rubens, The Presentation of the Portrait of Marie de' Medici
          • Rubens, Arrival (or Disembarkation) of Marie de Medici at Marseilles, Medici Cycle
          • Rubens, the Consequences of War
          • Anthony van Dyck, Charles I at the Hunt
        • Dutch Republic
          • Hals, Singing Boy with Flute
          • Hals, Malle Babbe
          • Rembrandt, The Anatomy Lesson of Dr. Tulp
          • Rembrandt, The Night Watch
          • Rembrandt, Aristotle with a Bust of Homer
          • Rembrandt, Christ Crucified between the Two Thieves: The Three Crosses.
          • Rembrandt, Bathsheba at her Bath
          • Rembrandt, Self-Portrait
          • Judith Leyster, Self-Portrait
          • Willem Kalf, Still Life with a Silver Ewer
          • Vermeer, The Glass of Wine
          • Vermeer, Young Woman with a Water Pitcher
          • Johannes Vermeer, Woman Holding a Balance
          • Johannes Vermeer, Girl with a Pearl Earring
          • Johannes Vermeer, The Art of Painting
          • Ruisdael, View of Haarlem with Bleaching Grounds
          • Rachel Ruysch, Fruit and Insects
          • Van Huysum, Vase with Flowers
        • Spain
          • Velázquez, Los Borrachos or the Triumph of Bacchus
          • Velázquez, Vulcan's Forge
          • Velázquez, Las Meninas
          • Ribera, Martyrdom of Saint Philip
          • Jerónimo de Balbás, Altar of the Kings (Altar de los Reyes)
          • Making a Spanish polychrome sculpture
        • France
          • Nicolas Poussin, Et in Arcadia Ego
          • Poussin, Landscape with St. John
          • Antoine or Louis Le Nain, Peasant Family in an Interior
        • England
      • Rococo
        • Antoine Watteau, Pilgrimage to Cythera
        • Boucher, Madame de Pompadour
        • Vigée Le Brun, Self-Portrait with her Daughter, Julie
        • Vigée Le Brun, Madame Perregaux
        • Unlocking an 18th-century French mechanical table
        • Construction of an 18th-century French mechanical table
        • The inlay technique of marquetry
        • Fragonard, The Swing
        • Fragonard, The Meeting
      • Neo-Classicism
        • David, Oath of the Horatii
        • David, The Lictors Returning to Brutus the Bodies of His Sons
        • David, Study for The Lictors Bringing Brutus the Bodies of his Sons
        • Jacques-Louis David, The Death of Marat
        • David, The Intervention of the Sabine Women
        • Girodet, The Sleep of Endymion
      • British art in the 18th century
        • British art in the eighteenth century
          • William Hogarth, Marriage A-la-Mode (including Tête à Tête)
          • Reynolds, Lady Cockburn and Her Three Eldest Sons
          • Wren, Saint Paul's Cathedral
          • Room: 1650-1730
          • Room: 1730-1765
          • Room: 1760-1780
          • Room: 1780-1810
        • Age of revolution (quiz)
    • Art of the Americas to World War I
      • A beginner's guide to the Art of the Americas
      • N. America before European colonization
        • Prehistoric Mexico
        • Clovis culture
        • Moundbuilders
        • Tlatilco
          • Tlatilco figurines
        • Olmec
          • Olmec mask (Olmec-style mask)
          • Mask
          • Kunz Axe
        • Teotihuacan
          • Teotihuacan
        • Classic Veracruz culture
          • The Mesoamerican Ballgame and a Classic Veracruz yoke
        • Maya
          • Chakalte’, Relief with Enthroned Ruler (Maya)
          • Vessel with a mythological scene (Maya)
          • Mirror-Bearer (Maya)
        • Ancestral Puebloan (formerly Anasazi)
          • Mesa Verde and the preservation of Ancestral Puebloan heritage
        • Huastec
        • Mixtec
        • Aztec (Mexica)
          • Coatlicue
          • Unearthing the Aztec past, the destruction of the Templo Mayor
          • Tlaloc vessel
          • The Sun Stone (The Calendar Stone)
          • Coyolxauhqui Stone
          • Monolith of Tlaltecuhtli (Earth Lord)
          • Feathered headdress
      • S. America before European colonization
        • A beginner's guide
        • Cupisnique
          • Feline-Head Bottle, Cupisnique
        • Paracas and Nasca cultures
        • Chavín culture
        • Moche culture
          • Pair of Earflares, Winged Messengers (Moche Culture, Peru)
          • Moche Portrait Head Bottle
        • Chimú culture
        • Sicán
          • What the bulldozers left behind: reclaiming Sicán’s past
        • Inka
          • Inka ushnus: landscape, site and symbol in the Andes
        • Ancient Colombian chiefdoms
          • By hammer and fire: Goldmaking techniques of the ancient Colombians
          • Depletion gilding: Goldmaking techniques of the ancient Colombians
          • From wax to metal: Goldmaking techniques of the ancient Colombians
        • Amazonian peoples
          • Kayapó Headdress: a glimpse of life in the Amazon rainforest
          • Mundurukú Headdress: a glimpse of life in the Amazon rainforest
      • Spanish colonies in the Americas
        • A beginner's guide
        • Viceroyalty of New Spain
          • An introduction to New Spain
          • Atrial Cross, convento San Agustín de Acolman, mid-16th century
          • Virgin of Guadalupe
          • Baltasar de Echave Ibía, The Hermits
          • Screen with the Siege of Belgrade and Hunting Scene (Brooklyn Biombo)
          • Jerónimo de Balbás, Altar of the Kings (Altar de los Reyes)
        • Viceroyalty of Peru
          • San Pedro Apóstol de Andahuaylillas
        • Viceroyalty of New Granada
      • Native American art after 1600
        • A beginner's guide to Native American art after 1600
        • East
        • West
          • Juana Basilia Sitmelelene, Presentation Basket (Chumash)
        • Southwest
        • Northwest coast
        • The Arctic
        • Contemporary art
      • British Colonies to the Early Republic
        • Colonial period
          • Copley, Boy with a Squirrel
        • Early Republic
          • Peale, Staircase Group (Portrait of Raphaelle Peale and Titian Ramsay Peale)
      • Art of the United States: 1800s
        • Romanticism in the United States
          • Allston, Elijah in the Desert
          • Cole, Expulsion from the Garden of Eden
          • Thomas Cole, The Oxbow
          • Lane, Owl's Head, Penobscot Bay, Maine
        • Realism in the United States
          • Francis Guy, Winter Scene in Brooklyn
          • Richard Caton Woodville, War News from Mexico
          • Homer, The Life Line
          • Homer, The Fog Warning (Halibut Fishing)
          • Homer, Northeaster
        • American Aestheticism and the Gilded Age
          • Sargent, Carnation, Lily, Lily, Rose
        • 19th century American sculpture, architecture and photography
          • Mark Hopkins House Side Chair (Herter Brothers)
      • Art of Mexico in the 18th and 19th centuries
        • Velasco, The Valley of Mexico
    • Art in 19th century Europe
      • Introduction—Becoming Modern
        • Unlock Art: Where are the Women?
      • Romanticism
        • Romanticism—an introduction
        • France
          • Gros, Napoleon Bonaparte Visiting the Plague-Stricken in Jaffa
          • Ingres, La Grande Odalisque
          • Ingres, Princesse de Broglie
          • Ingres, Raphael and the Fornarina
          • Géricault, Raft of the Medusa
          • Delacroix, Scene of the Massacre at Chios
          • Delacroix, The Death of Sardanapalus
          • Delacroix, Liberty Leading the People
        • Spain
          • Goya, The Family of Charles IV
          • Art historical analysis (painting), a basic introduction using Goya's Third of May, 1808
          • Goya, Saturn Devouring His Son
        • England (Blake and Fuseli)
          • Fuseli, Titania and Bottom
          • Blake, The Parable of the Wise and Foolish Virgins
          • Room: William Blake
        • England (Constable, Turner, Martin and Nash)
          • Constable, View on the Stour near Dedham
          • Turner, Slave Ship
          • Turner, Rain, steam, and speed – the great western railway
          • Room: JMW Turner
        • Germany
          • Friedrich, Monk by the Sea
          • Friedrich, Abbey among Oak Trees
          • Friedrich, The Lone Tree
          • Friedrich, Woman at a Window
      • Early photography
        • Early photography: making daguerreotypes
        • The wet collodion process
        • P.H. Emerson's naturalistic photography
      • Victorian art and architecture
        • Early Victorian
          • Charles Barry and A.W.N. Pugin, Palace of Westminster (Houses of Parliament)
        • The Pre-Raphaelites and mid-Victorian art
          • Pre-Raphaelites: Curator's choice - Millais's Isabella
          • Sir John Everett Millais, Christ in the House of His Parents
          • Sir John Everett Millais, Ophelia
          • Millais, Mariana
          • Millais, The Vale of Rest
          • Hunt, Claudio and Isabella
          • Hunt, Our English Coasts ("Strayed Sheep")
          • Hunt, the Awakening Conscience
          • Pre-Raphaelites: Curator's choice - Ford Madox Brown's 'Work'
          • Wallis, Chatterton
          • Dyce's Pegwell Bay, Kent - a Recollection of October 5th, 1858
          • John Roddam Spencer Stanhope, Thoughts of the Past
          • Burne-Jones, The Golden Stairs
          • Burne-Jones, King Cophetua and the Beggar Maid
          • Burne-Jones, Hope
          • William Butterfield, All Saints, Margaret Street
        • Late Victorian
          • Leighton, an Athlete Wrestling with a Python
          • Alma-Tadema, Listening to Homer
          • Room: 1890-1910
      • Realism, Impressionism and Post-Impressionism
        • Art and the French state
          • Garnier, Paris Opéra
          • Couture, Romans of the Decadence
        • Realism
          • Courbet, A Burial at Ornans, 1849-50
          • Courbet, The Artist's Studio, a real allegory summing up seven years of my artistic and moral life
          • Bonheur, Plowing in the Nivernais
          • Bonheur, Sheep in the Highlands
          • Millet, L'Angelus
          • Millet, The Gleaners
          • Manet, Olympia
          • Manet, Le déjeuner sur l'herbe
          • Manet, The Railway
          • Manet, Émile Zola
          • Manet, The Balcony
          • Manet, Plum Brandy
          • Manet, In the Conservatory
          • Manet, A Bar at the Folies-Bergère
          • Manet, Corner of a Café-Concert
        • Impressionism
          • Degas, The Bellelli Family
          • Degas, At the Races in the Countryside
          • Degas, The Dance Class
          • Degas, Visit to a Museum
          • Caillebotte, The Floor Scrapers
          • Gustave Caillebotte, Paris Street; Rainy Day
          • Caillebotte, Man at his Bath
          • Morisot, The Cradle
          • Cassatt, In the Loge
          • Cassatt, Woman with a Pearl Necklace in a Loge
          • Cassatt, The Loge
          • Cassatt, The Child's Bath
          • Cassatt, Breakfast in Bed
          • Monet, The Argenteuil Bridge
          • Monet, Cliff Walk at Pourville
          • Monet, Poplars
          • Monet, Rouen Cathedral Series
          • Monet, Water Lilies
          • Renoir, La Loge
          • Renoir, The Grands Boulevards
          • Renoir, Moulin de la Galette
          • Renoir, Luncheon of the Boating Party
          • Renoir, The Large Bathers
        • Post-Impressionism
          • Seurat, Bathers at Asnières
          • Seurat, A Sunday on La Grande Jatte
          • Van Gogh, Self-Portrait Dedicated to Paul Gauguin
          • Van Gogh, The Bedroom
          • Paul Gauguin, Self-Portrait with Portrait of Émile Bernard (Les misérables)
          • Paul Gauguin, Vision after the Sermon, or Jacob Wrestling with the Angel
          • Gauguin, Nevermore
          • Gauguin, The Red Cow
          • Cézanne, Still Life with Plaster Cupid
          • Cézanne, The Red Rock
          • Cézanne, Still Life with Apples
          • Cézanne, Mont Sainte-Victoire
          • Cézanne, Bathers
          • Cezanne, The Large Bathers
          • Paul Cézanne's approach to watercolor
          • Toulouse-Lautrec, At the Moulin Rouge
        • Sculpture
          • Carpeaux, Dance
          • Rodin, The Gates of Hell
      • Symbolism & Art Nouveau
        • Böcklin, Self-Portrait with Death Playing the Fiddle
        • Gaudí, Sagrada Família
        • Khnopff, Jeanne Kéfer
        • Khnopff, I Lock My Door Upon Myself
        • Stuck, The Sin
        • Hector Guimard, Cité entrance, Paris Métropolitain
        • Gustav Klimt, Beethoven Frieze
        • Gustav Klimt, The Kiss
        • Klimt, Death and Life
        • Munch, The Storm
        • Léon Bakst, "Costume design for the ballet The Firebird"
        • Tiffany, Hair Ornament
        • Louis Comfort Tiffany, Vase
      • Russia: The Wanderers
        • The Peredvizhniki (The Wanderers)
    • Expressionism to Pop Art
      • A beginner's guide to 20th century art
        • Representation and abstraction: Millais's Ophelia and Newman's Vir Heroicus Sublimis
        • Art and context: Monet's Cliff Walk at Pourville and Malevich's White on White
      • Early abstraction: Fauvism, Expressionism & Cubism
        • Fauvism and Matisse
          • Matisse, Luxe, calme et volupté
          • Matisse, The Red Studio
          • Matisse, "The Blue Window"
          • Matisse, Piano Lesson
          • Matisse, The illustrated book, “Jazz”
          • Conserving Henri Matisse's "The Swimming Pool"
        • Expressionism
          • Kirchner, Street, Dresden
          • Ernst Ludwig Kirchner, "Street, Berlin"
          • Emil Nolde, "Young Couple," 1913
          • Jawlensky, Young Girl in a Flowered Hat
          • Schiele, Seated Male Nude (Self-Portrait)
          • Nazi looting: Egon Schiele's Portrait of Wally
          • Schiele, Hermits
          • Kandinsky, Improvisation 28 (second version), 1912
          • Vasily Kandinsky, "Klänge (Sounds)"
        • Cubism and its impact
          • The Case for Abstraction
          • Pablo Picasso, Gertrude Stein
          • Picasso, Les Demoiselles d'Avignon
          • Braque, The Viaduct at L'Estaque
          • Picasso, The Reservoir, Horta de Ebro
          • Picasso, Still Life with Chair Caning
          • Pablo Picasso, "Guitar, Glass, and Bottle"
          • Picasso, Guitar
          • Conservation | Picasso's Guitars
          • Fernand Léger, "Contrast of Forms"
          • Robert Delaunay, "Simultaneous Contrasts: Sun and Moon"
      • World War I, Futurism and Dada
        • Futurism and the Great War
          • Umberto Boccioni, "Dynamism of a Soccer Player"
          • 1913 | Schiess-Dusseldorf by Ludwig Hohlwein
          • Duchamp-Villon, Horse
          • Room: 1914 & 1915
        • Dada
          • Marcel Duchamp, The Large Glass
          • Art as concept: Duchamp, In Advance of the Broken Arm
          • Duchamp, Fountain
          • "3 Standard Stoppages," Marcel Duchamp
          • Duchamp, Boite-en-valise (the red box), series F
          • Höch, Cut with the Kitchen Knife Dada Through the Last Weimar Beer-Belly Cultural Epoch of Germany
      • Art between the wars: the avant-garde and the rise of totalitarianism
        • German art between the wars
          • Feininger, Cathedral for the Bauhaus
          • Klee, Twittering Machine
          • Moholy-Nagy, EM1, EM2, and EM3 (Telephone Pictures)
          • Moholy-Nagy, Composition A.XX
          • Moholy-Nagy, Climbing the Mast
          • Umbo, The Roving Reporter
          • Sander, Portraits
          • Dix, Portrait of the Journalist Sylvia von Harden
          • Schad, Self-Portrait
          • Troost, House of German Art and the Entartete Kunst exhibition
        • The international avant-garde
          • Giorgio de Chirico, "The Anxious Journey"
          • Modigliani, Young Woman in a Shirt
          • Constantin Brancusi, The Kiss
          • Constantin Brancusi, "Mlle Pogany"
          • Brancusi, Bird in Space
          • "Composition in Brown and Gray," Piet Mondrian
          • Mondrian, Composition No. II, with Red and Blue
          • Cartier-Bresson, Behind the Gare St. Lazare
          • Room: 1930s
          • Room: 1940s
        • Surrealism
          • The Case for Surrealism
          • Magritte, The Treachery of Images (Ceci n’est pas une pipe)
          • Conservation | René Magritte, "The Portrait," 1935
          • Dali, The Persistence of Memory
          • Dali, Metamorphosis of Narcissus
          • Giacometti, The Palace at 4am
        • Latin American Modernism
          • Frida Kahlo, Frieda and Diego Rivera
          • Diego Rivera, Man Controller of the Universe
          • Diego Rivera, Calla Lilly Vendor
        • American art to World War II
          • Bellows, Pennsylvania Station Excavation
          • O'Keeffe, The Lawrence Tree
          • Wood, American Gothic
          • Evans, Subway Passengers, New York City
          • Ansel Adams: Visualizing a Photograph
          • Jacob Lawrence, The Migration Series (*short version*)
          • Jacob Lawrence, The Migration Series (*long version*)
          • Hopper, Nighthawks
          • Calder, Mobile
          • Norman Rockwell, Rosie the Riveter
      • Figuration & abstraction in post-war Britain
        • Room: Henry Moore
        • Barbara Hepworth: Pioneering modern sculpture
        • Room: 1950s
        • The Berlin Wall and industrial England: Don McCullin's conflict photography
        • Room: 1960s
      • Art Brut
      • Abstract Expressionism & the New York School
        • Abstract Expressionism
          • Newman's Onement I, 1948
          • Barnett Newman
          • The Painting Techniques of Barnett Newman
          • Mark Rothko's No. 3/No. 13, 1949
          • Rothko, No. 210/No. 211 (Orange)
          • Mark Rothko
          • The Painting Techniques of Mark Rothko
          • Restoring Rothko
          • Why is that important? Looking at Jackson Pollock
          • Jackson Pollock
          • Paint Application Studies of Jackson Pollock's Mural
          • The Painting Techniques of Jackson Pollock
          • Conservation | Pollock, "One: Number 31, 1950"
          • Willem de Kooning, Woman, I
          • Robert Motherwell, Elegy to the Spanish Republic No. 57
          • Franz Kline
          • The Painting Techniques of Franz Kline
        • New York School
          • Jasper Johns, Flag
          • Johns, White Flag
          • Robert Rauschenberg, Bed
          • Ed Kienholz and Nancy Reddin Kienholz Useful Art #5: The Western Hotel, 1992
          • Ad Reinhardt, Abstract Painting
          • Ad Reinhardt
          • The Painting Techniques of Ad Reinhardt
      • Pop
        • Why is this art? Andy Warhol, Campbell's Soup Cans
        • The Case For Andy Warhol
        • Warhol, Gold Marilyn Monroe
        • Oldenburg, Floor Cake
        • James Rosenquist, "F-111," 1964-65
        • Lichtenstein, Rouen Cathedral Set V
      • Minimalism and Earthworks
        • The Case for Minimalism
        • Donald Judd, Untitled
        • Robert Morris: Bodyspacemotionthings at Tate Modern
        • Robert Smithson, Spiral Jetty
        • Dorothea Rockburne: Drawing Which Makes Itself
        • Richard Serra, "Intersection II"
        • Richard Serra, "Torqued Ellipse IV"
        • Richard Serra, "Band," 2006
        • Maya Lin, Vietnam Veterans Memorial
        • James Turrell, Skyscape, The Way of Color
      • Architecture and design
        • Glass Chair at the 1939 New York World's Fair
        • Frank Lloyd Wright, Solomon R. Guggenheim Museum
        • Gordon Bunshaft, Lever House
        • Mies van der Rohe, Seagram Building
        • Breuer, The Whitney Museum of American Art (now The Met Breuer)
    • Global contemporary art
      • A beginner's guide to contemporary art
        • Damien Hirst, The Physical Impossibility of Death in the Mind of Someone Living
      • Identity, the body & the subversion of Modernism
        • United States
          • Hesse, Untitled
          • Hesse, Untitled (Rope Piece)
          • Chicago, Pasadena Lifesaver, Blue Series, No. 4 & Benglis, Omega
          • Winsor, #1 Rope
          • Dorothea Rockburne: Drawing Which Makes Itself
          • Sherman, Untitled Film Still #21
          • Nan Goldin's snapshots of innocence and childhood
          • Sherrie Levine, Untitled (After Edward Weston, ca. 1925)
          • Carrie Mae Weems on her series "From Here I Saw What Happened and I Cried"
          • Art and identity in the work of Lorna Simpson
        • Asia, Africa, Europe, Latin America and the Middle East
          • Saloua Raouda Choucair: From Beirut to Tate Modern
          • Ibrahim El-Salahi
          • Gerhard Richter, Betty
          • Gerhard Richter, The Cage Paintings (1-6)
          • Gerhard Richter, September
          • Bacon, Triptych - August 1972
          • Freud, Standing by the Rags
          • Yayoi Kusama
          • Fiona Rae on paint and passion
          • Graciela Iturbide
          • Dia Al-Azzawi
          • Room: 1970s & 1980s
          • Room: 1990s
          • Conserving Chris Ofili
      • Conceptual and Performance art
        • Unlock Art: Frank Skinner on Performance Art
        • Joseph Beuys, Table with Accumulator
        • The Case for Yoko Ono
        • John Baldessari, I Will Not Make Any More Boring Art
        • Jannis Kounellis, "Da inventare sul posto (To invent on the spot)"
        • Vito Acconci
        • Juan Downey: Plato Now
        • Hans Haacke: "A Breed Apart" in South Africa
        • Hans Haacke, Seurat's 'Les Poseuses' (small version)
        • Cildo Meireles
        • An installation by Xu Bing at the British Museum
        • Xu Bing, Book from the Sky
        • Marina Abramović
        • Marina Abramović: What is performance art?
        • Marina Abramović: The Body as medium
      • Global modernisms in the 21st century
        • Shirazeh Houshiary
        • Mona Hatoum's self-contradictory objects
        • El Anatsui, Untitled
        • Doris Salcedo's "Shibboleth"
        • Zarina Hashmi
        • Traces of illegal migration in Mexico: Mark Ruwedel
        • Shai Kremer in an Israeli military training stage
        • Gabriel Orozco
        • Zaha Hadid, MAXXI National Museum of XXI Century Arts, Rome
        • East-West Divan at the Venice Biennale
        • Doug Fishbone: Elmina
        • Hrair Sarkissian: Syria's "Execution Squares"
        • The Case for Ai Weiwei
        • Ai Weiwei: Sunflower Seeds
        • Isaac Julien, Ten Thousand Waves | MoMA
        • Nawa, PixCell-Deer#24
        • Wang Peng: Performance as politics in China and beyond
        • Michael Rakowitz on Star Wars and Saddam Hussein
        • Wolfgang Laib, "Pollen from Hazelnut"
        • Meschac Gaba
    • Art of Asia
      • A beginner's guide to Asian art and culture
        • Hindu art and culture, an introduction
          • Hindu temples
          • The Hindu deity Shiva
          • The Hindu deities Vishnu and Krishna
          • The Hindu deity Ganesha
          • Festival of the Goddess Durga
        • Buddhist art and culture, an introduction
          • Zen Buddhism
          • The stupa
          • Bodh Gaya: center of the Buddhist world
          • Varanasi: sacred city
          • Four Buddhas at the American Museum of Natural History
          • The Buddha triumphing over Mara
      • China
        • Neolithic art in China
          • Working jade
          • Jade Cong
        • A beginner's guide to Imperial China
          • Discoveries in Chinese archaeology
          • Buddhist Temples at Wutaishan
          • Decoding Chinese calligraphy
          • Appreciating Chinese calligraphy
          • Ascending the Heron Tower written in cursive script
        • Shang dynasty (c. 1600-1050 B.C.E.)
          • Oracle Bone, Shang Dynasty
          • Ritual vessel in the shape of a rhinoceros
        • Zhou dynasty (c. 1046 B.C.E.-256 B.C.E.)
          • Da Ke Ding
        • Qin dynasty (221-206 B.C.E.)
        • Han dynasty (206 B.C.E–220 C.E.)
        • Later Zhao (319-351)
          • Seated Buddha dated 338
        • Northern Wei dynasty (386 - 534)
          • Longmen Grottoes (UNESCO/NHK)
        • Northern Qi dynasty (550 - 577)
          • Bodhisattva
        • Sui dynasty (589-618)
        • Tang dynasty (618-907)
          • Chinese Buddhist cave shrines
        • Liao Dynasty (907–1125)
          • Arhat (Luohan)
        • Song dynasty (960-1279)
          • Master of the (Fishing) Nets Garden
          • Liang Kai, Poet Strolling by a Marshy Bank
        • Yuan dynasty (1271-1368)
          • Buddha of Medicine Bhaishajyaguru (Yaoshi fo)
          • The David Vases
        • Ming dynasty (1368–1644)
          • The Forbidden City
          • Classical gardens of Suzhou
          • Congyi, Cloudy Mountains
      • Korea
        • A beginner's guide to Korean art
          • Korean Buddhist art
          • Guardian King of the West
        • Proto-Three Kingdoms (300 B.C.E.-300 C.E.)
        • Three Kingdoms Period (57 B.C.E.-676 C.E.)
        • Goryeo dynasty (918-1392)
          • Reviving traditional Korean celadons
          • Ewer with lid
        • Joseon dynasty (1392–1910)
          • Royal palaces of Seoul
          • Confucian scholar's house
          • Nine Cloud Dream
      • Japan
        • Beginner’s guide to Japanese Art
          • Zen Buddhism
          • Shinto
        • Kofun period (300-552 C.E.)
          • Haniwa warrior in keiko armor
        • Nara period (710–794)
          • Standing Brahma and standing Indra
        • Heian period (794–1185)
          • The Buddhist guardian deity: Fudo Myoo
          • Fudō Myōō (Achala-vidyaraja)
        • Kamakura period (1185-1333)
          • A brief history of samurai armor
          • From castle to palace: samurai architecture
          • Tale of the Heike
        • Muromachi period (1392–1573)
          • Ryoanji
          • The Way of Tea
          • Teahouse at the Asian Art Museum
          • Muromachi to Momoyama period Negoro ware ewer
        • Edo period (1615–1868)
          • Genji Ukifune
          • Hunting for fireflies
          • Street scene in the pleasure quarter of Edo Japan
          • Courtesan playing with a cat
          • Courtesans of the South Station
          • The actor Ichikawa Danzo IV in a Shibaraku role
          • Conserving the Gan Ku Tiger scroll painting at the British Museum
      • South Asia
        • South Asia, an introduction
        • Hindu art in South Asia
          • Beliefs made visible: Hindu art in South Asia
          • The making and worship of Ganesha statues in Maharashtra
        • Buddhist art in South Asia
          • Beliefs made visible: Buddhist art in South Asia
          • Bodh Gaya: center of the Buddhist world
          • Varanasi: sacred city
          • Mahakala, Protector of the Tent
        • Art in Afghanistan
          • Crossroads of the Ancient World
          • Afghanistan: The lasting legacy of Alexander the Great
          • Hidden treasures revealed
      • Southeast Asia
        • A beginner's guide to Southeast Asian art
        • Cambodia
          • Angkor (UNESCO/TBS)
          • The Looting of Cambodian Antiquities
          • The scourge of looting: trafficking antiquities, from temple to museum
          • Sotheby's Returns Looted 10th Century Statue to Cambodia
          • The Hindu deities Shiva and Parvati
        • Myanmar
        • Indonesia
          • Indonesian rod puppets (wayang golek)
        • Bali
          • Balinese shadow theater
          • Balinese dance, costumes, music
          • Symbolism in Balinese dance costumes
          • Balinese boar Barong dance performance
          • Repairing sacred Balinese masks
          • Balinese mask carver
        • Thailand
      • The Himalayas
        • Nepal
          • Secrets of the stupa
        • Tibet
          • Sacred arts of Tibet
          • Views of Tibet
          • The Buddhist deity Simhavaktra Dakini
    • Art of Africa
      • African art, an introduction
      • East Africa and the Horn of Africa
        • Ethiopia
        • Tanzania
          • Kilwa Kisiwani, Tanzania
      • West Africa
        • Burkina Faso
        • Cameroon
        • Ghana
          • Sika dwa kofi (Golden Stool), Asante people
          • El Anatsui, Untitled
        • Guinea
        • Ivory Coast
        • Mali
          • Seated Figure (Djenné peoples)
          • Lost History: the terracotta sculpture of Djenné Djenno
          • Couple (Dogon peoples)
          • The Artist Project: Willie Cole
        • Nigeria
          • Ife uncovered
          • Ife remembered
          • Olowe of Ise, Veranda Post (Yoruba people)
          • Male figure, Ikenga (Igbo Peoples)
        • Republic of Benin
        • Sierra Leone
          • Bundu / Sowei Helmet Mask (Mende peoples)
          • Sowei mask: Spirit of Sierra Leone
      • North Africa
        • Algeria and Libya
          • Rock-Art Sites of Tadrart Acacus (UNESCO/NHK)
      • Southern Africa
        • Namibia
        • South Africa
        • Zimbabwe
          • Great Zimbabwe National Monument (UNESCO/NHK)
      • Central Africa
        • Cameroon
          • Elephant Mask, Kuosi Society, Bamileke Peoples, Cameroon
        • Democratic Republic of the Congo
          • Female (pwo) mask (Chokwe peoples)
          • Power Figure: Male (Nkisi)
        • Gabon
          • Fang Reliquary Figure
    • Art of Oceania
      • Lapita Culture
      • Polynesia
        • Voyage to the moai of Rapa Nui (Easter Island)
        • Fly Whisk (Tahiri), Austral Islands
        • Necklace (Lei Niho Palaoa), Hawai'i
        • A welcome to a Maori meeting house
      • Melanesia
        • Mask (Buk), Torres Strait, Mabuiag Island
        • Bis Poles at The Metropolitan Museum of Art
        • Slit Gong (Atingting kon)
      • Micronesia
        • Navigation Chart, Marshall Islands
    • For teachers
      • Course materials
        • Art history (Smarthistory) on Khanacademy.org
  • Computing
    • Computer programming
      • Intro to JS: Drawing & Animation
        • Test Yourself: Intro to JS: Drawing & Animation
        • Intro to programming
          • What is Programming?
        • Drawing basics
        • Coloring
          • The Power of the Docs
        • Variables
        • Animation basics
        • Interactive programs
        • Bonus: Resizing with variables
        • Text and strings
        • Functions
        • Logic and if Statements
        • Debugging programs
        • Looping
        • Writing clean code
        • Arrays
        • Objects
        • Object-Oriented Design
        • Becoming a better programmer
          • Help others learn!
      • Intro to HTML/CSS: Making webpages
        • Intro to HTML
          • Welcome to the web!
        • Intro to CSS
        • More HTML tags
        • CSS text properties
          • CSS Zen Garden
        • Web development tools
        • CSS layout
          • CSS in the wild: Google Maps
        • More CSS selectors
        • Other ways to embed CSS
        • Further learning
          • HTML validation
      • Intro to SQL: Querying and managing data
        • SQL basics
          • Welcome to SQL
          • S-Q-L or SEQUEL?
        • More advanced SQL queries
        • Relational queries in SQL
        • Modifying databases with SQL
        • Further learning in SQL
      • Advanced JS: Games & Visualizations
        • Intro to Games & Visualizations
          • Intro to Games and Visualizations
        • Scene management
        • Buttons
        • Transformations
        • 3D Shapes
        • Making a Side Scroller: Hoppy Beaver
        • Making a Memory Game
      • Advanced JS: Natural Simulations
        • Intro to Natural Simulations
          • Welcome to Natural Simulations
        • Randomness
        • Noise
        • Vectors
        • Forces
        • Angular Movement
        • Oscillations
        • Particle Systems
      • HTML/JS: Making webpages interactive
        • Test Yourself: HTML/JS: Making webpages interactive
        • Get ready to make your webpages interactive
          • Welcome to making webpages interactive
        • JS and the DOM
          • Debugging webpages with the browser console
        • DOM access methods
        • DOM modification
        • DOM events
          • Making webpages interactive with events
        • DOM animation
        • Using JS libraries in your webpage
        • Further learning
      • HTML/JS: Making webpages interactive with jQuery
        • Welcome to jQuery
          • From the inventor: Welcome!
          • What is jQuery?
        • DOM access with jQuery
          • Debugging webpages with the browser console
        • DOM modification with jQuery
        • jQuery techniques
        • DOM events with jQuery
        • Form processing with jQuery
        • DOM animation & effects with jQuery
        • Further learning
      • Meet the professional
        • Welcome to "Meet the professional"!
      • Browse Projects
      • Project Feedback
      • Processing.js Documentation
      • Webpage Documentation
      • SQL Documentation
    • Computer science
      • Algorithms
        • Intro to algorithms
          • What is an algorithm and why should you care?
        • Binary search
        • Asymptotic notation
        • Selection sort
        • Insertion sort
        • Recursive algorithms
        • Towers of Hanoi
        • Merge sort
        • Quick sort
        • Graph representation
        • Breadth-first search
        • Further learning
      • Journey into cryptography
        • Ancient cryptography
          • What is cryptography?
          • The Caesar cipher
          • Polyalphabetic cipher
          • The one-time pad
          • Frequency stability property short film
          • The Enigma encryption machine
          • Perfect secrecy
          • Pseudorandom number generators
        • Ciphers
        • Cryptography challenge 101
        • Modern cryptography
          • The fundamental theorem of arithmetic
          • Public key cryptography: What is it?
          • The discrete logarithm problem
          • Diffie-hellman key exchange
          • RSA encryption: Step 1
          • RSA encryption: Step 2
          • RSA encryption: Step 3
          • Euler's totient function
          • RSA encryption: Step 4
          • What should we learn next?
        • Modular arithmetic
        • Primality test
          • Primality test challenge
          • What is computer memory?
          • Algorithmic efficiency
          • Sieve of Eratosthenes
          • Primality test with sieve
          • The prime number theorem
          • Time space tradeoff
          • Summary (what's next?)
        • Randomized algorithms
          • Randomized algorithms (intro)
          • Conditional probability explained visually
          • Random primality test (warm up)
          • Fermat's little theorem
          • Fermat primality test
      • Journey into information theory
        • Ancient information theory
          • What is information theory?
          • Origins of written language
          • History of the alphabet
          • The Rosetta Stone
          • Source encoding
          • Visual telegraphs (case study)
          • Electrostatic telegraphs (case study)
          • The battery and electromagnetism
          • Morse code and the information age
        • Modern information theory
          • Symbol rate
          • Introduction to channel capacity
          • Measuring information
          • Origin of Markov chains
          • A mathematical theory of communication
          • Information entropy
          • Compression codes
          • Error correction
          • The search for extraterrestrial intelligence
      • Internet 101
        • How the internet works
          • Wires, cables, and WiFi
          • IP addresses and DNS
          • Packet, routers, and reliability
          • HTTP and HTML
          • Encryption and public keys
          • Cybersecurity and crime
    • Hour of Code
      • Drawing with code
        • Hour of Code - WORLDWIDE
        • Welcome to our Hour of Code™!
      • Creating webpages
        • Welcome to the web!
      • Creating SQL databases
        • Welcome to SQL
      • Hour of Code (for teachers and parents)
        • Hour of Code - WORLDWIDE
      • Drawing with code
        • Welcome to our Hour of Code™!
  • Test prep
    • SAT
      • Full-length SAT
        • Online SAT Tests
        • Paper SAT Tests
      • SAT Math practice
        • Heart of algebra
          • Solving linear equations and linear inequalities — Basic example
          • Solving linear equations and linear inequalities — Harder example
          • Interpreting linear functions — Basic example
          • Interpreting linear functions — Harder example
          • Linear equation word problems — Basic example
          • Linear inequality word problems — Basic example
          • Linear inequality word problems — Harder example
          • Graphing linear equations — Basic example
          • Graphing linear equations — Harder example
          • Linear function word problems — Basic example
          • Linear function word problems — Harder example
          • Systems of linear inequalities word problems — Basic example
          • Systems of linear inequalities word problems — Harder example
          • Solving systems of linear equations — Basic example
          • Solving systems of linear equations — Harder example
          • Systems of linear equations word problems — Harder example
        • Passport to advanced mathematics
          • Solving quadratic equations — Basic example
          • Solving quadratic equations — Harder example
          • Interpreting nonlinear expressions — Basic example
          • Interpreting nonlinear expressions — Harder example
          • Quadratic and exponential word problems — Basic example
          • Quadratic and exponential word problems — Harder example
          • Manipulating quadratic and exponential expressions — Basic example
          • Manipulating quadratic and exponential expressions — Harder example
          • Radicals and rational exponents — Basic example
          • Radicals and rational exponents — Harder example
          • Radical and rational equations — Basic example
          • Radical and rational equations — Harder example
          • Operations with rational expressions — Basic example
          • Operations with rational expressions — Harder example
          • Operations with polynomials — Basic example
          • Operations with polynomials — Harder example
          • Polynomial factors and graphs — Basic example
          • Polynomial factors and graphs — Harder example
          • Nonlinear equation graphs — Basic example
          • Nonlinear equation graphs — Harder example
          • Linear and quadratic systems — Basic example
          • Linear and quadratic systems — Harder example
          • Structure in expressions — Basic example
          • Structure in expressions — Harder example
          • Isolating quantities — Basic example
          • Isolating quantities — Harder example
          • Function notation — Basic example
          • Function notation — Harder example
        • Problem solving and data analysis
          • Ratios, rates, and proportions — Basic example
          • Ratios, rates, and proportions — Harder example
          • Percents — Basic example
          • Percents — Harder example
          • Units — Basic example
          • Units — Harder example
          • Table data — Basic example
          • Table data — Harder example
          • Scatterplots — Basic example
          • Scatterplots — Harder example
          • Key features of graphs — Basic example
          • Linear and exponential growth — Basic example
          • Linear and exponential growth — Harder example
          • Data inferences — Basic example
          • Data inferences — Harder example
          • Center, spread, and shape of distributions — Basic example
          • Center, spread, and shape of distributions — Harder example
          • Data collection and conclusions — Basic example
          • Data collection and conclusions — Harder example
        • Additional topics in math
          • Volume word problems — Basic example
          • Volume word problems — Harder example
          • Right triangle word problems — Basic example
          • Right triangle word problems — Harder example
          • Congruence and similarity — Basic example
          • Congruence and similarity — Harder example
          • Right triangle trigonometry — Basic example
          • Right triangle trigonometry — Harder example
          • Angles, arc lengths, and trig functions — Basic example
          • Angles, arc lengths, and trig functions — Harder example
          • Circle theorems — Basic example
          • Circle theorems — Harder example
          • Circle equations — Basic example
          • Circle equations — Harder example
          • Complex numbers — Basic example
          • Complex numbers — Harder example
      • SAT Reading and Writing practice
        • Reading
          • Reading: Science — How-to Part 1
          • Reading: Science — How-to Part 2
          • Reading: Literature — How-to Part 1
          • Reading: Literature — How-to Part 2
          • Reading: History and social studies — How-to Part 1
          • Reading: History and social studies — How-to Part 2
          • Reading: Social science — How-to Part 1
          • Reading: Social science — How-to Part 2
        • Writing: Passages
          • Writing: Argument — How-to example
          • Writing: Informative — How-to example
          • Writing: Narrative — How-to example
        • Writing: Grammar
          • Writing: Precision — Basic example
          • Writing: Precision — Harder example
          • Writing: Concision, style, and tone — Basic example 1
          • Writing: Concision, style, and tone — Harder example 1
          • Writing: Concision, style, and tone — Basic example 2
          • Writing: Concision, style, and tone — Harder example 2
          • Writing: Syntax — Basic example
          • Writing: Sentence boundaries — Harder example
          • Writing: Subordination and coordination — Basic example
          • Writing: Subordination and coordination — Harder example
          • Writing: Parallel structure — Basic example
          • Writing: Parallel structure — Harder example
          • Writing: Modifier placement — Basic example
          • Writing: Modifier placement — Harder example
          • Writing: Shift in verb, tense, and mood — Basic example
          • Writing: Shift in verb, tense, and mood — Harder example
          • Writing: Pronoun person and number — Basic example
          • Writing: Pronoun clarity — Basic example
          • Writing: Pronoun-antecedent agreement — Basic example
          • Writing: Pronoun-antecedent agreement — Harder example
          • Writing: Possessive determiners — Basic example
          • Writing: Possessive determiners — Harder example
          • Writing: Subject-verb agreement — Basic example
          • Writing: Subject-verb agreement — Harder example
          • Writing: Noun agreement — Basic example
          • Writing: Noun agreement — Harder example
          • Writing: Frequently confused words — Basic example
          • Writing: Frequently confused words — Harder example
          • Writing: Conventional expression — Basic example
          • Writing: Conventional expression — Harder example
          • Writing: Logical comparison — Harder example
          • Writing: End-of-sentence punctuation — Basic example
          • Writing: Within-sentence punctuation — Harder example 1
          • Writing: Within-sentence punctuation — Basic example
          • Writing: Within-sentence punctuation — Harder example 2
          • Writing: Possessive pronouns — Harder example
          • Writing: Items in a series — Basic example
          • Writing: Items in a series — Harder example
          • Writing: Nonrestrictive and parenthetical elements — Basic example
          • Writing: Nonrestrictive and parenthetical elements — Harder example
      • SAT Tips & Strategies
        • About the SAT
          • Content changes to the new SAT
        • About Official SAT Practice on Khan Academy
          • Overview of Official SAT Practice
        • General SAT Strategies
          • Dave answers student questions about the SAT
          • Effective strategies for each SAT section
          • PSAT/NMSQT inside scoop: Khan Academy interviews the College Board
        • About the SAT Reading Test
          • The SAT Reading Test: What to expect
          • Introduction to the SAT Reading test
        • SAT Reading Test Strategies
        • About the SAT Writing and Language Test
          • The SAT Writing and Language Test: What to expect
          • Introduction to the SAT Writing & Language test
        • SAT Writing and Language Test Strategies
        • About the SAT Math Test
          • The SAT Math Test: What to expect
          • SAT Math Test inside scoop: Meet the Maker
        • SAT Math Test Strategies
        • About the SAT Essay
          • The SAT Essay: What to expect
        • SAT Essay Strategies
        • How to Prep for the SAT
        • SAT Coach Tools
        • SAT Reading & Writing lesson plans for teachers, by teachers
        • SAT Math lesson plans for teachers, by teachers
    • MCAT
      • Critical analysis and reasoning skills practice questions
        • Critical analysis and reasoning skills (CARS) practice questions
          • CARS overview
          • Foundations of comprehension
          • Reasoning within the text
          • Reasoning beyond the text
          • Worked example: Living in a rational society
          • Worked example: The happy American
          • Worked example: Seeing color through Homer's eyes
          • Worked example: Physical education in the UK
          • Worked example: The honest truth about dishonesty
      • Biological and Biochemical Foundations of Living Systems Passages
        • Biological sciences practice passage questions
      • Chemical and Physical Foundations of Biological Systems Passages
        • Chemical and physical sciences practice passage questions
      • Psychological, Social, and Biological Foundations of Behaviors Passages
        • Social sciences practice passage questions
      • Biomolecules
        • Amino acids and proteins
          • Central dogma of molecular biology
          • Central dogma - revisited
          • Peptide bonds: Formation and cleavage
          • Special cases: Histidine, proline, glycine, cysteine
          • Amino acid structure
          • Isoelectric point and zwitterions
          • Classification of amino acids
          • Four levels of protein structure
          • Conformational stability: Protein folding and denaturation
        • Enzyme structure and function
          • Introduction to enzymes and catalysis
          • Enzymes and activation energy
          • Induced fit model of enzyme catalysis
          • Six types of enzymes
          • Co-factors, co-enzymes, and vitamins
          • Enzymes and their local environment
        • Enzyme kinetics
          • An introduction to enzyme kinetics
          • Steady states and the Michaelis Menten equation
          • Enzymatic inhibition and Lineweaver Burk plots
          • Cooperativity
          • Allosteric regulation and feedback loops
          • Non-enzymatic protein function
          • Covalent modifications to enzymes
        • DNA
          • DNA
          • Molecular structure of DNA
          • Antiparallel structure of DNA strands
          • Telomeres and single copy DNA vs repetitive DNA
          • Leading and lagging strands in DNA replication
          • Transcription and mRNA processing
          • Speed and precision of DNA replication
          • Translation (mRNA to protein)
          • Differences in translation between prokaryotes and eukaryotes
          • DNA repair 1
          • DNA repair 2
          • Semi conservative replication
          • Protein modifications
          • Jacob Monod lac operon
        • Gene control
          • Jacob-Monod: The Lac operon
          • DNA and chromatin regulation
          • Regulation of transcription
          • Post-transcriptional regulation
          • Non-coding RNA (ncRNA)
          • Oncogenes
          • Tumor suppressors
        • Genetic mutations
          • An introduction to genetic mutations
          • The different types of mutations
          • The causes of genetic mutations
          • Mutagens and carcinogens
          • The effects of mutations
        • Mendelian genetics
          • An Introduction to Mendelian Genetics
          • Co-dominance and Incomplete Dominance
          • Worked example: Punnett squares
          • Hardy-Weinberg equation
          • Applying the Hardy-Weinberg equation
        • DNA technology
          • Gel electrophoresis
          • Polymerase chain reaction (PCR)
          • DNA libraries & generating cDNA
          • DNA cloning and recombinant DNA
          • Hybridization (microarray)
          • Expressing cloned genes
          • Southern blot
          • DNA sequencing
          • Gene expression and function
          • Applications of DNA technologies
          • Safety and ethics of DNA technologies
        • Chromosomal inheritance
          • Evidence that DNA is genetic material 1
          • Evidence that DNA is genetic material 2
          • Sex-linked traits
          • Worked example: Punnett squares
          • Genetic recombination 1
          • Genetic recombination 2
          • Gene mapping
          • Extranuclear inheritance 1
          • Extranuclear inheritance 2
        • Evolution and population dynamics
          • Evolution and natural selection
          • Fitness and fecundity
          • Alternative selection
          • Genetic drift, bottleneck effect, and founder effect
          • Inbreeding
          • Reproductive isolation
        • Principles of bioenergetics
          • Gibbs free energy introduction
          • An analogy for Gibbs free energy
          • Heat transfer
          • Enthalpy
          • Le Chatelier's principle
          • Thermodynamics vs kinetics
        • Overview of metabolism
          • Overview of metabolism: Anabolism and catabolism
          • ATP: Adenosine triphosphate
          • ATP hydrolysis: Gibbs free energy
          • ATP hydrolysis: Transfer of a phosphate group
          • Oxidation and reduction review from biological point-of-view
          • Oxidation and reduction in metabolism
          • Electron carrier molecules
          • ATP hydrolysis mechanism
        • Carbohydrates
          • Carbohydrates - naming and classification
          • Carbohydrates - absolute configuration, epimers, common names
          • Carbohydrates - Cyclic structures and anomers
          • Carbohydrates- di and polysaccharides
          • Keto-enol tautomerization (by Jay)
        • Carbohydrate Metabolism
          • Cellular respiration introduction
          • Overview of glycolysis
          • Gluconeogenesis: the big picture
          • Gluconeogenesis: unique reactions
          • Regulation of glycolysis and gluconeogenesis
          • Pentose phosphate pathway
        • Krebs (citric acid) cycle and oxidative phosphorylation
          • Krebs / citric acid cycle
          • Regulation of pyruvate dehydrogenase
          • Regulation of Krebs-TCA cycle
          • Electron transport chain
          • Oxidative phosphorylation and chemiosmosis
          • Regulation of oxidative phosphorylation
          • Mitochondria, apoptosis, and oxidative stress
          • Calculating ATP produced in cellular respiration
        • Fat and protein metabolism
          • Introduction to energy storage
          • Digestion, Mobilization, and Transport of Fats - Part I
          • Fatty Acid Synthesis - Part I
          • Digestion, Mobilization, and Transport of Fats - Part II
          • Fatty Acid Synthesis - Part II
          • Overview of Fatty Acid Oxidation
          • Fatty Acid Oxidation - Part I
          • Fatty Acid Oxidation - Part II
          • How does the body adapt to starvation?
          • Overview of Amino Acid Metabolism
        • Endocrine system
          • Endocrine gland hormone review
          • The hypothalamus and pituitary gland
          • Hormone concentration metabolism and negative feedback
          • Types of hormones
          • Cellular mechanism of hormone action
          • From terpenes to steroids part 1: Terpenes
          • From terpenes to steroids part 2: Squalene, cholesterol, and steroids
        • Hormonal regulation
          • Endocrine gland hormone review
          • Hormone concentration metabolism and negative feedback
          • Types of hormones
          • Overview of metabolism: Anabolism and catabolism
          • How does the body adapt to starvation?
          • Tissue specific metabolism and the metabolic states
          • Insulin and glucagon
          • Glucose insulin and diabetes
          • Production of insulin and glucagon
          • Hormone control of hunger
          • Hormones, body mass, and obesity
      • Cells
        • Cell membrane overview
          • Cell membrane introduction
          • Phospholipid structure
          • Cell membrane overview and fluid mosaic model
          • Cell membrane proteins
          • Cell membrane fluidity
          • Membrane dynamics
        • Cell-cell interactions
          • Cell Junctions
          • Membrane Receptors
          • Ligand Gated Ion Channels
          • G Protein Coupled Receptors
          • Enzyme Linked Receptors
        • Transport across a cell membrane
          • How do things move across a cell membrane?
          • Passive Transport by Facilitated Diffusion
          • Diffusion and osmosis
          • Glomerular filtration in the nephron
          • Sodium-potassium pump
          • Secondary active transport in the nephron
          • Exocytosis
          • Membrane potentials - part 2
          • Membrane potentials - part 1
          • Permeability and membrane potentials
        • Cell theory
          • History and development of cell theory
        • Eukaryotic cells
          • Characteristics of eukaryotic cells
          • The nucleus
          • Mitochondria
          • Endoplasmic reticulum and golgi apparatus
          • Lysosomes and peroxisomes
          • Epithelial and connective tissue
        • Cytoskeleton
          • Introduction to cytoskeleton
          • Microfilaments and intermediate filaments
          • Microtubules
        • Prokaryotes/bacteria
          • Overview of Archaea, Protista, and Bacteria
          • Bacterial characteristics - Gram staining
          • Bacterial genetic recombination
        • Viruses
          • Virus structure and classification
          • Viral replication: lytic vs lysogenic
          • Retroviruses
          • Subviral particles: viroids and prions
        • Cellular division
          • Cell cycle phases
          • Cell cycle control
          • Loss of cell cycle control in cancer
          • Fertilization terminology: gametes, zygotes, haploid, diploid
          • Zygote differentiating into somatic and germ cells
          • Interphase
          • Mitosis
          • Comparing mitosis and meiosis
          • Phases of meiosis I
          • Phases of meiosis II
          • Embryonic stem cells
          • Cancer
        • Cellular development
          • Stem cells
          • Cellular communication
          • Mitochondria, apoptosis, and oxidative stress
          • Cellular specialization (differentiation)
          • Telomeres and cell senescence
          • Cellular movement
        • Embryology
          • Egg, sperm, and fertilization
          • Early embryogenesis - Cleavage, blastulation, gastrulation, and neurulation
          • Implantation
          • Gestation
          • Germ layer derivatives
      • Organ systems
        • Biological basis of behavior: The nervous system
          • Structure of the nervous system
          • Functions of the nervous system
          • Motor unit
          • Peripheral somatosensation
          • Muscle stretch reflex
          • Autonomic nervous system
          • Gray and white matter
          • Upper motor neurons
          • Somatosensory tracts
          • Cerebellum
          • Brainstem
          • Subcortical cerebrum
          • Cerebral cortex
          • Neurotransmitter anatomy
          • Early methods of studying the brain
          • Lesion studies and experimental ablation
          • Modern ways of studying the brain
        • Neural cells
          • Introduction to neural cell types
          • Overview of neuron structure
          • Overview of neuron function
          • Astrocytes
          • Microglia
          • Ependymal cells
          • Oligodendrocytes
          • Schwann cells
        • Neuron membrane potentials
          • Neuron graded potential description
          • Neuron resting potential description
          • Neuron resting potential mechanism
          • Neuron graded potential mechanism
          • Neuron action potential description
          • Neuron action potential mechanism
          • Effects of axon diameter and myelination
          • Action potential patterns
        • Neuronal synapses
          • Synapse structure
          • Neurotransmitter release
          • Types of neurotransmitters
          • Types of neurotransmitter receptors
          • Neurotransmitter removal
          • Neuroplasticity
        • Biosignaling
          • Membrane Receptors
          • Ligand Gated Ion Channels
          • G Protein Coupled Receptors
          • Enzyme Linked Receptors
        • Endocrine system
          • Endocrine gland hormone review
          • The hypothalamus and pituitary gland
          • Hormone concentration metabolism and negative feedback
          • Types of hormones
          • Cellular mechanism of hormone action
          • From terpenes to steroids part 1: Terpenes
          • From terpenes to steroids part 2: Squalene, cholesterol, and steroids
        • Circulatory system
          • Meet the heart!
          • Layers of the heart
          • Flow through the heart
          • Two circulations in the body
          • Lub dub
          • Layers of a blood vessel
          • Arteries vs. veins - what's the difference?
          • Resistance in a tube
          • Putting it all together: Pressure, flow, and resistance
          • Thermoregulation in the circulatory system
        • Hematologic system
          • What's inside of blood?
          • Hemoglobin moves O2 and CO2
          • Bohr effect vs. Haldane effect
          • Blood types
          • How do we make blood clots?
          • Coagulation cascade
          • Life and times of RBCs and platelets
          • Blood cell lineages
        • Respiratory system
          • Meet the lungs
          • Inhaling and exhaling
          • How does lung volume change?
          • O2 and CO2 solubility
          • Henry's law
          • Fick's law of diffusion
          • Oxygen movement from alveoli to capillaries
          • The respiratory center
          • Thermoregulation in the lungs
        • Lymphatic system
          • Why we need a lymphatic system
          • How lymphatic vessels move fluid
          • The lymphatic system's role in immunity
          • Lipid and protein transport in the lymphatic system
          • What is actually in lymph
        • Immune system
          • Role of phagocytes in innate or nonspecific immunity
          • Types of immune responses: Innate and adaptive, humoral vs. cell-mediated
          • B lymphocytes (B cells)
          • Professional antigen presenting cells (APC) and MHC II complexes
          • Helper T cells
          • Cytotoxic T cells
          • Review of B cells, CD4+ T cells and CD8+ T cells
          • Clonal selection
          • Self vs. non-self immunity
          • How white blood cells move around
          • Blood cell lineages
        • Renal system
          • Meet the kidneys!
          • Kidney function and anatomy
          • Glomerular filtration in the nephron
          • Changing glomerular filtration rate
          • Secondary active transport in the nephron
          • Countercurrent multiplication in the kidney
          • Urination
          • The kidney and nephron
        • Renal regulation of blood pressure
          • General overview of the RAAS system: Cells and hormones
          • Renin production in the kidneys
          • Activating angiotensin 2
          • Angiotensin 2 raises blood pressure
          • Aldosterone raises blood pressure and lowers potassium
          • Aldosterone removes acid from the blood
          • ADH secretion
          • ADH effects on blood pressure
          • Aldosterone and ADH
        • Gastrointestinal system
          • Meet the gastrointestinal tract!
          • Mouth
          • Teeth
          • Esophagus
          • Stomach
          • Small intestine 1: Structure
          • Small intestine 2: Digestion
          • Small intestine 3: Absorption
          • Liver
          • Hepatic lobule
          • Biliary tree
          • Exocrine pancreas
          • Endocrine pancreas
          • Colon, rectum, and anus
          • Control of the GI tract
        • Muscular system
          • Myosin and actin
          • How tropomyosin and troponin regulate muscle contraction
          • Role of the sarcoplasmic reticulum in muscle cells
          • Anatomy of a skeletal muscle cell
          • Three types of muscle
          • Motor neurons
          • Neuromuscular junction, motor end-plate
          • Type 1 and type 2 muscle fibers
          • Calcium puts myosin to work
          • Muscle innervation
          • Autonomic vs somatic nervous system
          • Thermoregulation by muscles
        • Skeletal system
          • Skeletal structure and function
          • Microscopic structure of bone - the Haversian system
          • Cellular structure of bone
          • Skeletal endocrine control
          • Cartilage
          • Ligaments, tendons, and joints
        • Integumentary system
          • Meet the skin! (Overview)
          • What is skin? (Epidermis)
          • What lies beneath the epidermis? (Dermis and Hypodermis)
          • Where do our nails and hair come from?
          • What's in sweat? (Holocrine, Apocrine, Merocrine Glands)
          • LeBron Asks: Why does sweating cool you down?
          • Overview of Sensation and Meissner's Corpuscle
          • Pacinian's Corpuscle and Merkel's Disk
          • Ruffini's Ending and Hair Follicle Receptor
          • Pain and temperature
          • Thermoregulation by muscles
        • Reproductive system
          • Welcome to the reproductive system
          • Anatomy of the male reproductive system
          • Transport of sperm via erection and ejaculation
          • Spermatogenesis
          • Testosterone
          • Basics of egg development
          • The ovarian cycle
          • Meet the placenta!
          • Reproductive cycle graph - Follicular phase
          • Reproductive cycle graph - Luteal phase
          • Estrogen
          • Maternal changes in pregnancy
          • Labor (parturition)
          • Breast anatomy and lactation
      • Physical processes
        • Vectors and scalars
          • Intro to vectors & scalars
          • Visualizing vectors in 2 dimensions
          • Unit vector notation
          • Unit vector notation (part 2)
        • Speed and velocity
          • Calculating average speed and velocity edited
          • Solving for time
          • Displacement from time and velocity example
          • Instantaneous speed and velocity
        • Acceleration
          • Acceleration
          • Airbus A380 take-off time
          • Airbus A380 take-off distance
          • Why distance is area under velocity-time line
          • Average velocity for constant acceleration
        • Newton's laws and equilibrium
          • More on Newton's first law of motion
          • Newton's first law of motion concepts
          • Newton's second law of motion
          • Unbalanced forces and motion
          • Newton's third law of motion
          • Center of mass
          • Introduction to torque
        • Normal forces
          • Balanced and unbalanced forces
          • Normal force and contact force
          • Normal force in an elevator
          • Slow sock on Lubricon VI
          • Normal forces on Lubricon VI
        • Force of tension
          • Introduction to tension
          • Introduction to tension (part 2)
          • Tension in an accelerating system and pie in the face
        • Forces on inclined planes
          • Inclined plane force components
          • Force of friction keeping the block stationary
          • Ice accelerating down an incline
          • Correction to force of friction keeping the block stationary
          • Force of friction keeping velocity constant
          • Intuition on static and kinetic friction comparisons
          • Static and kinetic friction example
        • Work and energy
          • Introduction to work and energy
          • Work and energy (part 2)
          • Work and the work-energy principle
          • Work example problems
          • Conservation of energy
          • Work/energy problem with friction
          • Intro to springs and Hooke's law
          • Potential energy stored in a spring
          • Spring potential energy example (mistake in math)
          • Work as the transfer of energy
          • Conservative forces
          • Power
          • Introduction to mechanical advantage
        • Fluids at rest
          • Pressure and Pascal's principle (part 1)
          • Pressure and Pascal's principle (part 2)
          • Pressure at a depth in a fluid
          • Finding height of fluid in a barometer
          • Archimedes principle and buoyant force
          • Buoyant force example problems edited
          • Specific gravity
        • Fluids in motion
          • Volume flow rate and equation of continuity
          • Bernoulli's equation (part 1)
          • Bernoulli's equation (part 2)
          • Bernoulli's equation (part 3)
          • Bernoulli's equation (part 4)
          • Bernoulli's example problem
          • Viscosity and Poiseuille flow
          • Turbulence at high velocities and Reynold's number
          • Surface Tension and Adhesion
          • Venturi effect and Pitot tubes
          • Two circulations in the body
          • Arteries vs. veins - what's the difference?
          • Resistance in a tube
          • Putting it all together: Pressure, flow, and resistance
        • Gas phase
          • Absolute temperature and the kelvin scale
          • Pressure and the simple mercury barometer
          • Definition of an ideal gas, ideal gas law
          • Derivation of gas constants using molar volume and STP
          • Boyle's law
          • Charles's law
          • Avogadro's law
          • Van der Waals equation
          • Partial pressure
        • Kinetic molecular theory of gases
          • Boltzmann's constant
          • Heat capacity at constant volume and pressure
          • Kinetic molecular theory of gases
        • Electrostatics
          • Triboelectric effect and charge
          • Coulomb's Law
          • Conservation of Charge
          • Conductors and Insulators
          • Electric field
          • Electric potential energy
          • Voltage
          • Electric potential at a point in space
        • Current and resistance
          • Introduction to circuits and Ohm's law
          • Resistors in series
          • Resistors in parallel
          • Example: Analyzing a more complex resistor circuit
          • Resistivity and conductivity
          • Electrolytic conductivity
          • Voltmeters and Ammeters
        • Capacitors
          • Capacitors and capacitance
          • Capacitance
          • Energy of a capacitor
          • Capacitors in series
          • Capacitors in parallel
          • Dielectrics in capacitors
        • Magnetism
          • Magnetism - Part 1
          • Magnetism - Part 2
          • Magnetism - Part 3
          • Magnetism - Part 4
        • Electrochemistry
          • Redox reaction from dissolving zinc in copper sulfate
          • Introduction to galvanic/voltaic cells
          • Electrodes and voltage of Galvanic cell
          • Shorthand notation for galvanic/voltaic cells
          • Free energy and cell potential
          • Standard reduction potentials
          • Voltage as an intensive property
          • Using reduction potentials
          • Spontaneity and redox reactions
          • Standard cell potential and the equilibrium constant
          • Calculating the equilibrium constant from the standard cell potential edited
          • Nernst equation
          • Using the Nernst equation
          • Concentration cell
          • Introduction to electrolysis
          • Quantiative electrolysis
          • Electrolysis of molten sodium chloride edited
          • Lead storage battery
          • Nickel-cadmium battery
        • Sound
          • Production of sound
          • Sound Properties (Amplitude, Period, Frequency, Wavelength)
          • Speed of Sound
          • Relative speed of sound in solids, liquids, and gases
          • Decibel Scale
          • Why do sounds get softer?
          • Ultrasound medical imaging
          • Standing waves in open tubes
          • Standing waves in closed tubes
          • Doppler effect introduction
          • Doppler effect formula for observed frequency
          • Doppler effect formula when source is moving away
          • When the source and the wave move at the same velocity
          • Doppler effect for a moving observer
          • Doppler effect: reflection off a moving object
        • Light and electromagnetic radiation
          • Electromagnetic waves and the electromagnetic spectrum
          • Polarization of light, linear and circular
          • Wave interference
          • Young's double slit introduction
          • Young's double slit equation
          • Young's double slit problem solving
          • Diffraction grating
          • Single slit interference
          • More on single slit interference
          • Thin Film Interference part 1
          • Thin Film Interference part 2
          • Photon Energy
        • Infrared and Ultraviolet/Visible spectroscopy
          • Introduction to infrared spectroscopy
          • Bonds as springs
          • Signal characteristics - wavenumber
          • IR spectra for hydrocarbons
          • Signal characteristics - intensity
          • Signal characteristics - shape
          • Symmetric and asymmetric stretching
          • IR signals for carbonyl compounds
          • IR spectra practice
          • UV/Vis spectroscopy
          • Absorption in the visible region
          • Conjugation and color
        • Proton nuclear magnetic resonance
          • Introduction to proton NMR
          • Nuclear shielding
          • Chemical equivalence
          • Chemical shift
          • Electronegativity and chemical shift
          • Diamagnetic anisotropy
          • Integration
          • Spin-spin splitting (coupling)
          • Multiplicity: n + 1 rule
          • Coupling constant
          • Complex splitting
          • Hydrogen deficiency index
          • Proton NMR practice 1
          • Proton NMR practice 2
          • Proton NMR practice 3
        • Thin lenses
          • Convex lenses
          • Convex lens examples
          • Concave lenses
          • Object image and focal distance relationship (proof of formula)
          • Object image height and distance relationship
          • Thin lens equation and problem solving
          • Multiple lens systems
          • Diopters, Aberration, and the Human Eye
        • Spherical mirrors
          • Virtual image
          • Parabolic mirrors and real images
          • Parabolic mirrors 2
          • Convex parabolic mirrors
        • Reflection and refraction
          • Specular and diffuse reflection
          • Specular and diffuse reflection 2
          • Refraction and Snell's law
          • Refraction in water
          • Snell's law example 1
          • Snell's law example 2
          • Total internal reflection
          • Dispersion
        • Atomic nucleus
          • Atomic number, mass number, and isotopes
          • Atomic mass
          • Mass defect and binding energy
          • Nuclear stability and nuclear equations
          • Writing nuclear equations for alpha, beta, and gamma decay
          • Types of decay
          • Half-life and carbon dating
          • Half-life plot
          • Exponential decay formula proof (can skip, involves calculus)
          • Introduction to exponential decay
          • Exponential decay and semi-log plots
          • More exponential decay examples
          • Mass spectrometer
        • Electronic structure
          • Photoelectric effect
          • Bohr model radii (derivation using physics)
          • Bohr model radii
          • Bohr model energy levels (derivation using physics)
          • Bohr model energy levels
          • Absorption and emission
          • Emission spectrum of hydrogen
          • Heisenberg uncertainty principle
          • Quantum numbers
          • Quantum numbers for the first four shells
          • Electron configurations for the first period
          • Electron configurations for the second period
          • Electron configurations for the third and fourth periods
          • Electron configurations in the 3d orbitals
          • Paramagnetism and diamagnetism
        • Periodic table
          • Electronegativity and bonding
          • The periodic table - classification of elements
          • The periodic table - transition metals
          • Counting valence electrons for main group elements
          • Atomic and ionic radii
          • Group trend for ionization energy
          • Period trend for ionization energy
          • First and second ionization energy
          • Electron affinity: period trend
        • Stoichiometry
          • Empirical formula from mass composition edited
          • Molecular and empirical formulas
          • The mole and Avogadro's number
          • Stoichiometry example problem 1
          • Stoichiometry
          • Stoichiometry: Limiting reagent
          • Limiting reactant example problem 1 edited
          • Specific gravity
        • Balancing chemical equations
          • Balancing chemical equations
          • Balancing more complex chemical equations
          • Visually understanding balancing chemical equations
          • Balancing another combustion reaction
          • Balancing chemical equation with substitution
        • Redox reactions
          • Oxidizing and reducing agents
          • Disproportionation
          • Balancing redox reactions in acid
          • Balancing redox reactions in base
      • Chemical processes
        • Acid/base equilibria
          • Acid-base definitions
          • Ka and acid strength
          • Autoionization of water
          • Definition of pH
          • Strong acids and strong bases
          • Weak acid equilibrium
          • Weak base equilibrium
          • Relationship between Ka and Kb
          • Acid-base properties of salts
          • pH of salt solutions
          • Common ion effect and buffers
          • Buffer solutions
          • Buffer solution pH calculations
        • Titrations
          • Titration introduction
          • Titration calculation example
          • Titration of a strong acid with a strong base
          • Titration of a strong acid with a strong base (continued)
          • Titration of a weak acid with a strong base
          • Titration of a weak acid with a strong base (continued)
          • Titration of a weak base with a strong acid
          • Titration of a weak base with a strong acid (continued)
          • Titration curves and acid-base indicators
          • Redox titration
        • Solubility equilibria
          • Common polyatomic ions
          • Dissolution and precipitation
          • Introduction to solubility and solubility product constant
          • Solubility from the solubility product constant
          • Solubility and the common-ion effect
          • Solubility and the pH of the solution
          • Solubility and complex ion formation
        • Dot structures
          • Drawing dot structures
          • Formal charge and dot structures
          • Resonance and dot structures
          • VSEPR for 2 electron clouds
          • VSEPR for 3 electron clouds
          • VSEPR for 4 electron clouds
          • VSEPR for 5 electron clouds
          • VSEPR for 5 electron clouds (continued)
          • VSEPR for 6 electron clouds
        • Stereochemistry
          • Structural (constitutional) isomers
          • Chiral vs achiral
          • Stereoisomers, enantiomers, and chirality centers
          • Identifying chirality centers
          • R,S system
          • R,S system practice
          • Optical activity
          • Enantiomers and diastereomers
          • Cis-trans isomerism
          • E-Z system
          • Conformations of ethane
          • Conformational analysis of butane
        • Covalent bonds
          • Electronegativity and bonding
          • Intermolecular forces
          • sp³ hybridization
          • Steric number and sp3 hybridization
          • Dipole moment
          • sp² hybridization
          • Sp hybridization
          • Organic hybridization practice
          • Acid-base definitions
        • Separations and purifications
          • Simple and fractional distillations
          • Extractions
          • Basics of chromatography
          • Thin layer chromatography (TLC)
          • Calculating retention factors for TLC
          • Column chromatography
          • Gas chromatography
          • Gel electrophoresis
          • Resolution of enantiomers
        • Nucleic acids, lipids, and carbohydrates
          • Nucleic acid structure 1
          • Antiparallel structure of DNA strands
          • Saponification - Base promoted ester hydrolysis
          • Lipids - Structure in cell membranes
          • Lipids as cofactors and signaling molecules
          • Carbohydrates - Naming and classification
          • Fischer projections
          • Carbohydrates - Epimers, common names
          • Carbohydrates - Cyclic structures and anomers
          • Carbohydrate - Glycoside formation hydrolysis
          • Keto-enol tautomerization (by Sal)
          • Disaccharides and polysaccharides
        • Amino acids, peptides, proteins
          • Central dogma of molecular biology
          • Central dogma - revisited
          • Peptide bonds: Formation and cleavage
          • Special cases: Histidine, proline, glycine, cysteine
          • Amino acid structure
          • Isoelectric point and zwitterions
          • Classification of amino acids
          • Four levels of protein structure
          • Conformational stability: Protein folding and denaturation
        • Proteins
          • Amino acid structure
          • Alpha amino acid synthesis
          • Classification of amino acids
          • Peptide bonds: Formation and cleavage
          • Four levels of protein structure
          • Conformational stability: Protein folding and denaturation
          • Non-enzymatic protein function
        • Carbohydrates
          • Carbohydrates- di and polysaccharides
          • Carbohydrates - cyclic structures and anomers
          • Carbohydrates - absolute configuration, epimers, common names
          • Carbohydrates - naming and classification
          • Keto-enol tautomerization (by Jay)
        • Alpha-carbon chemistry
          • Keto-enol tautomerization (by Jay)
          • Enolate formation from aldehydes
          • Enolate formation from ketones
          • Kinetic and thermodynamic enolates
          • Aldol condensation
          • Mixed (crossed) aldol condensation
          • Mixed (crossed) aldol condensation using a lithium enolate
          • Retro-aldol and retrosynthesis
          • Intramolecular aldol condensation
        • Aldehydes and ketones
          • Nomenclature of aldehydes and ketones
          • Physical properties of aldehydes and ketones
          • Reactivity of aldehydes and ketones
          • Formation of hydrates
          • Formation of hemiacetals and hemiketals
          • Acid and base catalyzed formation of hydrates and hemiacetals
          • Formation of acetals
          • Acetals as protecting groups and thioacetals
          • Formation of imines and enamines
          • Formation of oximes and hydrazones
          • Addition of carbon nucleophiles to aldehydes and ketones
          • Formation of alcohols using hydride reducing agents
          • Oxidation of aldehydes using Tollens' reagent
        • Alcohols and phenols
          • Alcohol nomenclature
          • Properties of alcohols
          • Oxidation of alcohols
          • Oxidation of alcohols (examples)
          • Protection of alcohols
          • Preparation of mesylates and tosylates
          • SN1 and SN2 reactions of alcohols
          • Biological redox reactions of alcohols and phenols
          • Aromatic stability of benzene
          • Aromatic heterocycles
        • Carboxylic acids
          • Carboxylic acid nomenclature and properties
          • Reduction of carboxylic acids
          • Preparation of esters via Fischer esterification
          • Preparation of acyl (acid) chlorides
          • Preparation of acid anhydrides
          • Preparation of amides using DCC
          • Decarboxylation
          • Alpha-substitution of carboxylic acids
        • Carboxylic acid derivatives
          • Nomenclature and properties of acyl (acid) halides and acid anhydrides
          • Nomenclature and properties of esters
          • Nomenclature and properties of amides
          • Reactivity of carboxylic acid derivatives
          • Nucleophilic acyl substitution
          • Acid-catalyzed ester hydrolysis
          • Acid and base-catalyzed hydrolysis of amides
          • Beta-lactam antibiotics
        • Enzymes
          • Introduction to enzymes and catalysis
          • Induced fit model of enzyme catalysis
          • Six types of enzymes
          • An introduction to enzyme kinetics
          • Allosteric regulation and feedback loops
        • Kinetics
          • Rate of reaction
          • Rate law and reaction order
          • Experimental determination of rate laws
          • First-order reaction (with calculus)
          • Plotting data for a first-order reaction
          • Half-life of a first-order reaction
          • Plotting data for a second-order reaction
          • First-order reaction example
          • Second-order reaction (with calculus)
          • Half-life of a second-order reaction
          • Second-order reaction example
          • Zero-order reaction (with calculus)
          • Collision theory
          • Arrhenius equation
          • Forms of the Arrhenius equation
          • Using the Arrhenius equation
          • Elementary rate laws
          • Mechanisms and the rate-determining step
          • Catalysts
          • Kinetic and thermodynamic enolates
        • Equilibrium
          • Reactions in equilibrium
          • Le Chatelier's principle
          • Changes in free energy and the reaction quotient
          • Standard change in free energy and the equilibrium constant
          • Galvanic cells and changes in free energy
        • Bioenergetics
          • An analogy for Gibbs free energy
          • Bioenergetics: The transformation of free energy in living systems
          • Why we need metabolism?
          • Insulin and glucagon
          • Tissue specific metabolism and the metabolic states
        • Thermodynamics
          • Specific heat and latent heat of fusion and vaporization
          • Zeroth law of thermodynamics
          • First law of thermodynamics
          • First law of thermodynamics problem solving
          • PV diagrams - part 1: Work and isobaric processes
          • PV diagrams - part 2: Isothermal, isometric, adiabatic processes
          • Second law of thermodynamics
        • Thermochemistry
          • Phase diagrams
          • Enthalpy
          • Heat of formation
          • Hess's law and reaction enthalpy change
          • Gibbs free energy and spontaneity
          • Gibbs free energy example
          • More rigorous Gibbs free energy / spontaneity relationship
          • A look at a seductive but wrong Gibbs spontaneity proof
      • Processing the environment
        • Sensory perception
          • Visual cues
          • Sensory adaptation
          • Weber's law and thresholds
          • Absolute threshold of sensation
          • Somatosensation
          • The vestibular system, balance, and dizziness
          • Signal detection theory - part 1
          • Signal detection theory - part 2
          • Bottom-up vs. top-down processing
          • Gestalt principles
        • Sight (vision)
          • The structure of the eye
          • Visual sensory information
          • The phototransduction cascade
          • Photoreceptors (rods vs cones)
          • Photoreceptor distribution in the fovea
          • Visual field processing
          • Feature detection and parallel processing
        • Sound (Audition)
          • Auditory structure - part 1
          • Auditory structure - part 2
          • Auditory processing
          • Cochlear implants
        • Somatosensation
          • Somatosensation
          • Sensory adaptation and amplification
          • Somatosensory homunculus
          • Proprioception and kinesthesia
          • Pain and temperature
        • Taste (gustation) and smell (olfaction)
          • Pheromones
          • Olfaction - structure and function
          • Gustation - structure and function
        • Sleep and consciousness
          • States of consciousness
          • Sleep stages and circadian rhythms
          • Dreaming
          • Dream theories Freud, activation synthesis hypothesis
          • Sleep disorders
          • Breathing related sleep disorders
          • Hypnosis and meditation
        • Drug dependence
          • Overview of psychoactive drugs
          • Psychoactive drugs: Depressants and opiates
          • Psychoactive drugs: Stimulants
          • Psychoactive drugs: Hallucinogens
          • Drug dependence and homeostasis
          • Routes of drug entry
          • Reward pathway in the brain
          • Tolerance and withdrawal
          • Substance use disorders
          • Treatments and triggers for drug dependence
        • Attention
          • Divided attention, selective attention, inattentional blindness, & change blindness
          • Theories of selective attention
          • The spotlight model of attention and our ability to multitask
        • Memory
          • Information processing model: Sensory, working, and long term memory
          • Encoding strategies
          • Retrieval cues
          • Retrieval: Free recall, cued recall, and recognition
          • Memory reconstruction, source monitoring, and emotional memories
          • Long term potentiation and synaptic plasticity
          • Decay and interference
          • Aging and cognitive abilities
          • Alzheimer's disease and Korsakoff's syndrome
          • Semantic networks and spreading activation
        • Cognition
          • Piaget's stages of cognitive development
          • Schemas, assimilation, and accommodation
          • Problem solving
          • Decision making
          • Semantic networks and spreading activation
          • Intelligence
          • Theories of intelligence
          • Aging and cognitive abilities
          • Cognitive dissonance
          • Information processing model: Sensory, working, and long term memory
        • Language
          • Language and the brain: Aphasia and split-brain patients
          • Theories of language and cognition
          • Theories of language development: Nativist, learning, interactionist
        • Emotion
          • Emotions: limbic system
          • Emotions: cerebral hemispheres and prefrontal cortex
          • Autonomic nervous system (ANS) and physiologic markers of emotion
          • Three components of emotion and the universal emotions
          • Theories of Emotion
        • Stress
          • What is stress?
          • Stressors
          • Responding to stress
          • Physical effects of stress
          • Behavioral effects of stress
          • Stress management
      • Behavior
        • Biological basis of behavior: Nervous system
          • Structure of the nervous system
          • Functions of the nervous system
          • Motor unit
          • Peripheral somatosensation
          • Muscle stretch reflex
          • Autonomic nervous system
          • Gray and white matter
          • Upper motor neurons
          • Somatosensory tracts
          • Overview of the functions of the cerebral cortex
          • Hemispheric differences and hemispheric dominance
          • The old brain
          • Cerebellum
          • Brainstem
          • Subcortical cerebrum
          • Cerebral cortex
          • Neurotransmitter anatomy
          • Early methods of studying the brain
          • Lesion studies and experimental ablation
          • Modern ways of studying the brain
          • Endocrine system and influence on behavior - Part 1
          • Endocrine system and influence on behavior - Part 2
        • Human development
          • Egg, sperm, and fertilization
          • Early embryogenesis - Cleavage, blastulation, gastrulation, and neurulation
          • Implantation
          • Germ layer derivatives
          • Gestation
          • Major motor milestones
          • Motor development
          • Neonatal reflexes
          • Physical development in adolescence
          • Brain changes during adolescence
        • Behavior and genetics
          • Temperament, heredity, and genes
          • Twin studies and adoption studies
          • Heritability
          • Regulatory genes
          • Gene environment interaction
          • Adaptive value of behavioral traits
        • Motivation and attitudes
          • Physiological concept of positive and negative feedback
          • Instincts, Arousal, Needs, Drives: Drive-Reduction and Cognitive Theories
          • Maslow's hierarchy of needs
          • Incentive theory
          • Biological and Sociocultural Factors Food, Sex, and Drugs
          • Components of attitudes
          • Attitude influences behavior
          • Behavior influences attitude
          • Cognitive dissonance
          • Situational approach
        • Theories of personality
          • Situational approach
          • Psychoanalytic theory
          • Maslow's hierarchy of needs
          • Humanistic theory
          • Biological theory
          • Behavioral theory
          • Trait theory
          • Observational learning: Bobo doll experiment and social cognitive theory
          • Defense mechanisms
          • Freud - Death drive, reality principle, and pleasure principle
        • Psychological disorders
          • Introduction to mental disorders
          • Categories of mental disorders
          • Schizophrenia
          • Biological basis of schizophrenia
          • Biological basis of depression
          • Anxiety disorders and obsessive compulsive disorder
          • Dissociative identity disorder
          • Somatic symptom disorder and other disorders
          • Personality disorders
          • Sleep disorders
          • Sleep wake disorders breathing related sleep disorders
          • Reward pathway in the brain
          • Drug dependence and homeostasis
          • Tolerance and withdrawal
          • Substance use disorders
          • Biological basis of parkinson's disease
          • Depression and major depressive disorder
          • Depression and bipolar disorder
        • Social psychology
          • Conformity and groupthink
          • Conformity and obedience
          • Asch conformity studies (Asch line studies)
          • Events that inspired the Milgram studies on obedience
          • Milgram experiment on obedience
          • What can we learn from the Milgram experiment
          • Zimbardo prison study The Stanford prison experiment
          • A closer look at the Stanford prison experiment
          • Factors that influence obedience and conformity
          • Bystander effect
          • Social facilitation and social loafing
          • Agents of socialization
        • Normative and non-normative behavior
          • What is normal? Exploring folkways, mores, and taboos
          • Perspectives on deviance: Differential association, labeling theory, and strain theory
          • Aspects of Collective Behavior: Fads, Mass Hysteria, and Riots
        • Learning
          • Classical conditioning: Neutral, conditioned, and unconditioned stimuli and responses
          • Classical conditioning: Extinction, spontaneous recovery, generalization, discrimination
          • Operant conditioning: Positive-and-negative reinforcement and punishment
          • Operant conditioning: Shaping
          • Operant conditioning: Schedules of reinforcement
          • Operant conditioning: Innate vs learned behaviors
          • Operant conditioning: Escape and avoidance learning
          • Observational learning: Bobo doll experiment and social cognitive theory
          • Long term potentiation and synaptic plasticity
          • Non associative learning
          • Biological constraints on learning
        • Theories of attitude and behavior change
          • Components of attitudes
          • Attitude influences behavior
          • Behavior influences attitude
          • Persuasion, attitude change, and the elaboration likelihood model
          • Reciprocal determinism
          • Locus of control, learned helplessness, and the tyranny of choice
          • Self control
      • Individuals and society
        • Self-identity
          • Self concept, self identity, and social identity
          • Self esteem, self efficacy, and locus of control
          • Overview of theories of development
          • Freud's psychosexual development
          • Erikson's psychosocial development
          • Vygotsky sociocultural development
          • Kohlberg moral development
          • Social influences
          • George Herbert Mead- The I and the Me
          • Charles Cooley- Looking glass self
        • Perception, prejudice, and bias
          • Attribution Theory - Basic covariation
          • Attribution theory - Attribution error and culture
          • Stereotypes stereotype threat and self fulfilling prophecies
          • Emotion and cognition in prejudice
          • Prejudice and discrimination based on race, ethnicity, power, social class, and prestige
          • Stigma - Social and self
          • Social perception - Primacy recency
          • Social perception - The Halo Effect
          • Social perception - The Just World Hypothesis
          • Ethnocentrism and cultural relativism in group and out group
        • Attributing behavior to persons or situations
          • Self esteem, self efficacy, and locus of control
          • Self concept, self identity, and social identity
          • Social influences
          • Locus of control, learned helplessness, and the tyranny of choice
        • Social behavior
          • Proximity and the mere exposure effect
          • Physical attraction
          • Similarity
          • Harlow monkey experiments
          • Secure and insecure attachment
          • Aggression
          • Altruism
          • Social support
        • Social interactions
          • Status
          • Role strain and role conflict
          • Primary and secondary groups
          • Ethnocentrism and cultural relativism in group and out group
          • Dramaturgical approach
          • Impression management
          • Aggression
          • Harlow monkey experiments
          • Altruism
          • Discrimination individual vs institutional
          • Prejudice vs discrimination
          • Prejudice and discrimination based on race, ethnicity, power, social class, and prestige
          • Organizations and bureaucratization
          • Characteristics of an ideal bureaucracy
          • Social support
        • Self-presentation and interacting with others
          • Charles Cooley- Looking glass self
          • George Herbert Mead- The I and the Me
          • Three components of emotion and universal emotions
        • Biological explanations of social behavior in animals
          • Animal behavior: foraging
          • Animal communication
          • Types of animal communication
          • Mating behavior and inclusive fitness
          • Evolutionary game theory
        • Discrimination
          • Discrimination individual vs institutional
          • Prejudice and discrimination based on race, ethnicity, power, social class, and prestige
          • Stereotypes stereotype threat, and self fulfilling prophecy
      • Society and culture
        • Social structures
          • Macrosociology vs microsociology
          • Social institutions
          • Social institutions - education, family, and religion
          • Social institutions - government, economy, health and medicine
          • Functionalism
          • Conflict theory
          • Social constructionism
          • Symbolic interactionism
          • Rational choice-exchange theory
          • Social theories overview (part 1)
          • Social theories overview (part 2)
          • Relating social theories to medicine
        • Demographics
          • Demographic structure of society - age
          • Demographic structure of society - race and ethnicity
          • Demographic structure of society - immigration
          • Demographic structure of society - sex, gender, and sexual orientation
          • Demographic structure of society overview
          • Urbanization
          • Population dynamics
          • Demographic transition
          • Globalization theories
          • Globalization- trade and transnational corporations
          • Social movements
          • Overview of demographics
        • Culture
          • Culture and society
          • Overview of culture
          • Subculture vs counterculture
          • Jim goes to college subculture
          • Culture lag and culture shock
          • Diffusion
          • Mass media
          • Evolution and human culture
      • Social inequality
        • Social inequality
          • Overview of social inequality
          • Upward and downward mobility, meritocracy
          • Intergenerational and intragenerational mobility social mobility
          • Absolute and relative poverty
          • Social reproduction
          • Social exclusion (segregation and social isolation)
          • Environmental justice
          • Residential segregation
          • Global inequality
          • Prejudice and discrimination based on race, ethnicity, power, social class, and prestige
          • Health and healthcare disparities in the US
          • Intersectionality
          • Class consciousness and false consciousness
    • NCLEX-RN
      • NCLEX-RN practice questions
        • NCLEX-RN questions
      • Circulatory system physiology
        • Circulatory system introduction
          • Meet the heart!
          • Flow through the heart
          • Two circulations in the body
          • Lub dub
          • Layers of a blood vessel
          • Arteries vs. veins - what's the difference?
          • Resistance in a tube
          • Putting it all together: Pressure, flow, and resistance
          • Thermoregulation in the circulatory system
        • Blood pressure
          • What is blood pressure?
          • Learn how a stethoscope can help determine blood pressure
          • Resistance in a tube
          • Adding up resistance in series and in parallel
          • Adding up resistance problem
          • Flow and perfusion
          • Putting it all together: Pressure, flow, and resistance
          • Blood pressure changes over time
        • Blood pressure control
          • Regulation of blood pressure with baroreceptors
          • Parts of a nephron
          • General overview of the RAAS system: Cells and hormones
          • Renin production in the kidneys
          • Activating angiotensin 2
          • Angiotensin 2 raises blood pressure
          • Aldosterone raises blood pressure and lowers potassium
          • Aldosterone removes acid from the blood
          • ADH secretion
          • ADH effects on blood pressure
          • Aldosterone and ADH
        • Fetal circulation
          • Meet the placenta!
          • Umbilical vessels and the ductus venosus
          • Flow through the heart
          • Hypoxic pulmonary vasoconstriction
          • Foramen ovale and ductus arteriosus
          • Fetal hemoglobin and hematocrit
          • Double Bohr effect
          • Fetal circulation right before birth
          • Baby circulation right after birth
          • Fetal structures in an adult
        • Blood vessels
          • Layers of a blood vessel
          • Three types of capillaries
          • Pre-capillary sphincters
        • Arterial stiffness
          • Compliance and elastance
          • Bernoulli's equation of total energy
          • Stored elastic energy in large and middle sized arteries
          • Compliance - decreased blood pressure
          • Compliance - increased blood flow
        • Heart muscle contraction
          • How tropomyosin and troponin regulate muscle contraction
          • Three types of muscle
          • Heart cells up close!
          • Myosin and actin
          • Calcium puts myosin to work
          • Sympathetic nerves affect myosin activity
          • Why doesn't the heart rip?
        • Heart depolarization
          • Membrane potentials - part 1
          • Membrane potentials - part 2
          • Permeability and membrane potentials
          • Action potentials in pacemaker cells
          • Action potentials in cardiac myocytes
          • Resetting cardiac concentration gradients
          • Depolarization waves flowing through the heart
          • Electrical system of the heart
          • A race to keep pace!
          • Thinking about heartbeats
          • New perspective on the heart
        • Nerve regulation of the heart
          • Changing the AV node delay - chronotropic effect
          • Changing the heart rate - chronotropic effect
          • Increasing ventricular contractility - inotropic effect
          • Autonomic nervous system effects on the heart
          • Getting a new heart
        • Preload and afterload
          • Why doesn't the heart rip?
          • What is preload?
          • Preload and pressure
          • Preload stretches out the heart cells
          • Frank-Starling mechanism
          • Sarcomere length-tension relationship
          • Active contraction vs. passive recoil
          • What is afterload?
          • Increasing the heart's force of contraction
        • Pressure volume loops
          • Pressure in the left heart - part 1
          • Pressure in the left heart - part 2
          • Pressure in the left heart - part 3
          • Left ventricular pressure vs. time
          • Left ventricular volume vs. time
          • Drawing a pressure-volume loop
          • Understanding the pressure-volume loop
          • End diastolic pressure-volume relationship (EDPVR)
          • End systolic pressure-volume relationship (ESPVR)
          • Reimagine the pressure volume relationship
          • What is preload?
          • Why doesn't the heart rip?
        • Changing the PV loop
          • What is preload?
          • What is afterload?
          • Increasing the heart's force of contraction
          • Reimagine the pressure volume relationship
          • What is contractility?
          • Getting Ea (arterial elastance) from the PV loop
          • Arterial elastance (Ea) and afterload
          • Arterial elastance (Ea) and preload
          • Stroke work in PV loops and boxes
          • Contractility, Ea, and preload effects on PV boxes
      • Circulatory system diseases
        • Hypertension
          • What is hypertension
          • Stages of hypertension
          • Primary hypertension
          • Hypertension and heart damage
          • Hypertension and blood vessel damage
          • Complications of hypertension
          • Secondary hypertension
          • Diagnosis of hypertension
          • Treatment of hypertension
          • Hypertensive crisis
        • Coronary artery disease
          • What is coronary artery disease?
          • Risk factors for coronary artery disease
          • Atherosclerosis
          • Heart attack (myocardial infarction) pathophysiology
          • Heart attack (myocardial infarct) diagnosis
          • Heart attack (myocardial infarct) medications
          • Heart attack (myocardial infarction) interventions and treatment
          • Healing after a heart attack (myocardial infarction)
          • Complications after a heart attack (myocardial infarction)
        • Stroke
          • Cerebral blood supply: Part 1
          • Cerebral blood supply: Part 2
          • What is a stroke?
          • Risk factors for stroke
          • Ischemic stroke
          • Hemorrhagic strokes
          • Ischemic core and penumbra
          • The ischemic cascade in stroke
          • Blood brain barrier and vasogenic edema
          • Post stroke inflammation
          • How do you know if someone is having a stroke: Think FAST!
          • Common stroke signs and symptoms
          • Diagnosing strokes by history and physical exam
          • Diagnosing strokes with imaging CT, MRI, and Angiography
          • Diagnosing strokes with lab tests
          • Acute treatment of stroke with medications
          • Treatment of stroke with interventions
          • Preventing further strokes
        • Heart failure
          • What is heart failure?
          • Systolic heart failure pathophysiology
          • Diastolic heart failure pathophysiology
          • Compensation and decompensation in heart failure
          • Symptoms of left sided heart failure
          • Symptoms of right sided heart failure
          • Heart failure diagnosis
          • Heart failure treatment - Early stages
          • Heart failure treatment - Late stages
          • Heart failure treatment - Devices and surgery
        • Cardiac dysrhythmias and tachycardias
          • Electrical conduction in heart cells
          • Normal sinus rhythm on an EKG
          • Supraventricular tachycardia (SVT)
          • Atrial fibrillation (Afib)
          • Atrial flutter (AFL)
          • Multifocal atrial tachycardia (MAT)
          • Atrioventricular reentrant tachycardia (AVRT) & AV nodal reentrant tachycardia (AVNRT)
          • Ventricular tachycardia (Vtach)
          • Torsades de pointes
          • What is ventricle fibrillation (Vfib)?
          • Pulseless electrical activity (PEA) and asystole
          • Electrocardioversion
          • Pacemakers
          • Antiarrhythmics
          • Ablation
        • Acyanotic heart diseases
          • What is acyanotic heart disease?
          • Acyanotic heart disease diagnosis
          • What is Eisenmenger syndrome?
          • Atrial septal defect
          • Ventricular septal defect
          • Patent ductus arteriosus
        • Cyanotic heart diseases
          • What is cyanotic heart disease
          • Shunting in the heart
          • Einsenmenger coarctation of aorta
          • Tetralogy of fallot
          • Truncus arteriosus
          • Total anomalous pulmonary venous return
          • Tricuspid atresia
          • Transposition of great arteries
          • Ebstein's anomaly
          • Hypoplastic left heart syndrome and norwood glenn fontan
          • Cyanotic heart diseases - Diagnosis and treatment
        • Vasculitis
          • What is vasculitis?
          • Vasculitis pathophysiology
          • Temporal arteritis
          • Takayasu arteritis
          • Polyarteritis nodosa
          • What is Kawasaki disease?
          • Kawasaki disease: diagnosis and treatment
          • Buerger disease
          • Wegener’s granulomatosis and microscopic polyangiitis
          • Churg-Strauss syndrome
          • Henoch-Schonlein purpura
        • Cardiomyopathy
          • What is cardiomyopathy?
          • Cardiomyopathy signs and symptoms
          • Dilated cardiomyopathy: Pathophysiology and diagnosis
          • Hypertrophic cardiomyopathy: Pathophysiology and diagnosis
          • Restrictive cardiomyopathy: Pathophysiology and diagnosis
          • Cardiomyopathy treatment
        • Myocarditis and pericarditis
          • What is myocarditis and pericarditis?
          • Causes of myocarditis
          • Causes of pericarditis
          • Diagnosis of myocarditis and pericarditis (part 1)
          • Diagnosis of myocarditis and pericarditis (part 2)
          • Myocarditis and pericarditis treatment
        • Aortic dissection and aneurysm
          • What is an aortic dissection?
          • Thoracic aortic aneurysms
          • Abdominal aortic aneurysms
          • Aortic dissection treatment
          • Aneurysm treatment
        • Heart valve diseases
          • What is valvular heart disease?
          • Valvular heart disease causes
          • How to identify murmurs
          • Systolic murmurs, diastolic murmurs, and extra heart sounds - Part 1
          • Systolic murmurs, diastolic murmurs, and extra heart sounds - Part 2
          • Aortic stenosis and aortic regurgitation
          • Mitral valve regurgitation and mitral valve prolapse
          • Mitral stenosis
          • Valvular heart disease diagnosis and treatment
        • Shock
          • What is shock?
          • Shock - hemodynamics
          • Shock - diagnosis and treatment
          • Shock - oxygen delivery and metabolism
          • Cardiogenic shock
          • Sepsis: Systemic inflammatory response syndrome (SIRS) to multiple organ dysfunction syndrome (MODS)
          • Septic shock - pathophysiology and symptoms
          • Septic shock: Diagnosis and treatment
          • Hypovolemic shock
          • Neurogenic shock
          • Obstructive shock
          • Anaphylactic shock
          • Dissociative shock
          • Differentiating shock
      • Respiratory system physiology
        • Respiratory system introduction
          • Meet the lungs
          • Inhaling and exhaling
          • How does lung volume change?
          • O2 and CO2 solubility
          • Henry's law
          • Fick's law of diffusion
          • Oxygen movement from alveoli to capillaries
          • The respiratory center
          • Thermoregulation in the lungs
        • Gas Exchange
          • Alveolar gas equation - part 1
          • Alveolar gas equation - part 2
          • Henry's law
          • O2 and CO2 solubility
          • Graham's law of diffusion
          • Fick's law of diffusion
          • Oxygen movement from alveoli to capillaries
        • Breathing Control
          • Peripheral chemoreceptors
          • Central chemoreceptors
          • The respiratory center
          • Sneeze, cough, and hiccup
      • Respiratory system diseases
        • Introduction to pulmonary diseases
          • Types of pulmonary diseases
          • Respiratory distress
          • Streptococcus pneumoniae and flu vaccines
        • Asthma
          • What is asthma?
          • Asthma pathophysiology
          • Asthma diagnosis
          • Asthma severity
          • Asthma shortterm treatments
          • Asthma longterm treatment
        • Emphysema (COPD)
          • What is emphysema?
          • Emphysema pathophysiology
          • Emphysema diagnosis
          • Diffusing capacity of the lung for carbon monoxide (DLCO)
          • Centriacinar emphysema vs panacinar emphysema
          • Emphysema treatment
        • Chronic bronchitis (COPD)
          • What is chronic bronchitis?
          • Chronic bronchitis pathophysiology
          • Chronic bronchitis treatment
        • Cystic fibrosis
          • What is cystic fibrosis?
          • Cystic fibrosis pathophysiology
          • Cystic fibrosis diagnosis
          • Cystic fibrosis treatments
          • Cystic fibrosis complications
        • Bronchiolitis
          • What is bronchiolitis?
          • Bronchiolitis pathophysiology
          • Bronchiolitis diagnosis
          • Bronchiolitis treatment
        • Lung cancer
          • What is lung cancer?
          • Lung cancer diagnosis
          • Lung cancer types
          • Lung cancer complications
          • Lung cancer metastasis
          • Lung cancer staging
          • Lung cancer treatment
        • Pneumonia
          • What is pneumonia?
          • Pneumonia diagnosis and treatment
          • Streptococcus pneumoniae and flu vaccines
          • Pulmonary abscess
          • Pneumonia vs. pneumonitis
        • Pleural effusion
          • What is a pleural effusion?
          • Pleural effusion diagnosis and treatment
          • Transudate vs exudate
          • Neurovascular bundle
        • Nose, sinus, and upper respiratory track infections
          • What is an upper respiratory infection (URI)?
          • What is allergic rhinitis?
          • Allergic rhinitis diagnosis and treatment
          • Nasal polyps
          • What is sinusitis?
          • Sinusitis diagnosis and treatment
        • Laryngeal conditions
          • What is epiglottitis?
          • Epiglottitis diagnosis and treatment
          • Hib vaccine
          • What is laryngitis ?
          • Laryngitis diagnosis, treatment, and prevention
          • Laryngomalacia
          • Vocal cord growths
        • Occupational lung diseases
          • Occupational asthma and mesothelioma
          • Asbestosis, silicosis, sick building syndrome
        • Pulmonary hypertension
          • What is pulmonary hypertension?
          • Pulmonary hypertension symptoms and causes
          • Hypoxic vasoconstriction
          • Pulmonary hypertension diagnosis Swan Ganz catheterization
          • Pulmonary hypertension diagnosis
          • Pulmonary hypertension treatment
        • Throat conditions
          • Tonsillitis
          • Pharyngitis
          • Tonsillopharyngitis - Complications, diagnosis, and treatment
          • What is a tonsillectomy?
        • Pertussis
          • What is pertussis?
          • Pertussis complications
          • Pathophysiology of pertussis
          • Pertussis diagnosis and treatment
          • Pertussis vaccine
      • Hematologic system diseases
        • Anemia
          • What is anemia?
          • Anemia pathophysiology
          • Microcytic, normocytic, and macrocytic anemias
        • Leukemia
          • Hematopoiesis
          • What is leukemia?
          • Leukemia pathophysiology
          • Leukemia diagnosis
          • Leukemia classifications
          • Acute leukemia
          • Chronic leukemia
          • Myelodysplastic syndrome
          • Leukemia treatment
        • Plasma cell dyscrasias
          • What are plasma cell dyscrasias?
          • What is Monoclonal gammopathy of undetermined significance (MGUS)?
          • What is multiple myeloma?
          • What is waldenstrom macroglobulinemia?
        • Iron deficiency anemia and anemia of chronic disease
          • What is iron deficiency?
          • Microcytic anemia
          • Iron deficency anemia diagnosis
          • Anemia of chronic disease
          • Chronic disease vs iron deficiency anemia
          • Iron deficiency and anemia of chronic disease treatment
        • Myeloproliferative disorders
          • What are myeloproliferative disorders?
          • What is polycythemia vera?
          • What is essential thrombocythemia?
          • What is chronic myelogenous leukemia?
          • What is primary myelofibrosis?
      • Endocrine system physiology
        • Endocrine system introduction
          • Endocrine gland hormone review
          • The hypothalamus and pituitary gland
          • Hormone concentration metabolism and negative feedback
          • Types of hormones
          • Cellular mechanism of hormone action
          • From terpenes to steroids part 1: Terpenes
          • From terpenes to steroids part 2: Squalene, cholesterol, and steroids
      • Endocrine system diseases
        • Diabetes
          • What is diabetes mellitus?
          • Breaking down diabetes
          • Types of diabetes
          • Pathophysiology - Type I diabetes
          • Pathophysiology - Type II diabetes
          • Diagnosing diabetes
          • Treating type I diabetes
          • Treating type II diabetes - Pharmacology
          • Treating type II diabetes - A practical approach
          • Acute complications of diabetes - Diabetic ketoacidosis
          • Acute complications of diabetes - Hyperosmolar hyperglycemic nonketotic state
          • Diabetic nephropathy - Mechanisms
          • Diabetic nephropathy - Clinical presentation & treatment
          • Diabetic retinopathy
      • Lymphatic system physiology
        • Lymphatic system
          • Why we need a lymphatic system
          • How lymphatic vessels move fluid
          • The lymphatic system's role in immunity
          • Lipid and protein transport in the lymphatic system
          • What is actually in lymph
      • Immune system physiology
        • Immune system
          • Role of phagocytes in innate or nonspecific immunity
          • Types of immune responses: Innate and adaptive, humoral vs. cell-mediated
          • B lymphocytes (B cells)
          • Professional antigen presenting cells (APC) and MHC II complexes
          • Helper T cells
          • Cytotoxic T cells
          • Review of B cells, CD4+ T cells and CD8+ T cells
          • Clonal selection
          • Self vs. non-self immunity
          • How white blood cells move around
          • Blood cell lineages
      • Renal system physiology
        • Renal system introduction
          • Meet the kidneys!
          • Kidney function and anatomy
          • Glomerular filtration in the nephron
          • Changing glomerular filtration rate
          • Countercurrent multiplication in the kidney
          • Urination
          • The kidney and nephron
          • Secondary active transport in the nephron
        • Renal regulation of blood pressure
          • General overview of the RAAS system: Cells and hormones
          • Renin production in the kidneys
          • Activating angiotensin 2
          • Angiotensin 2 raises blood pressure
          • Aldosterone raises blood pressure and lowers potassium
          • Aldosterone removes acid from the blood
          • ADH secretion
          • ADH effects on blood pressure
          • Aldosterone and ADH
      • Gastrointestinal system physiology
        • Gastrointestinal system
          • Meet the gastrointestinal tract!
          • Mouth
          • Teeth
          • Esophagus
          • Stomach
          • Small intestine 1: Structure
          • Small intestine 2: Digestion
          • Small intestine 3: Absorption
          • Liver
          • Hepatic lobule
          • Biliary tree
          • Exocrine pancreas
          • Endocrine pancreas
          • Colon, rectum, and anus
          • Control of the GI tract
      • Gastrointestinal system diseases
        • Gastroenteritis
          • What is gastroenteritis?
          • What is viral gastroenteritis?
          • What is staphylococcal enteritis?
          • What is cholera?
          • What is campylobacteriosis?
          • What is shigellosis?
          • What is salmonella poisoning?
          • What is traveler's diarrhea?
        • Inflammatory bowel disease
          • Inflammatory bowel disease vs irritable bowel syndrome
          • Inflammatory bowel disease: Epidemiology & pathology
          • Inflammatory bowel disease: Diagnosis
          • Inflammatory bowel disease: Treatment
        • Celiac disease
          • What is gluten?
          • What is celiac disease
      • Muscular-skeletal system physiology
        • Muscular system
          • Myosin and actin
          • How tropomyosin and troponin regulate muscle contraction
          • Role of the sarcoplasmic reticulum in muscle cells
          • Anatomy of a skeletal muscle cell
          • Three types of muscle
          • Motor neurons
          • Neuromuscular junction, motor end-plate
          • Type 1 and type 2 muscle fibers
          • Calcium puts myosin to work
          • Muscle innervation
          • Autonomic vs somatic nervous system
          • Thermoregulation by muscles
        • Skeletal system
          • Skeletal structure and function
          • Microscopic structure of bone - the Haversian system
          • Cellular structure of bone
          • Skeletal endocrine control
          • Cartilage
          • Ligaments, tendons, and joints
      • Muscular-skeletal diseases
        • Arthritis and rheumatoid arthritis
          • What is arthritis?
          • Osteoarthritis vs rheumatoid arthritis symptoms
          • Osteoarthritis vs rheumatoid arthritis pathophysiology
          • Osteoarthritis vs rheumatoid arthritis treatments
          • Ankylosing spondylitis
          • Reiter's syndrome
          • Infectious arthritis
          • Gout and pseudogout
          • Gout pathophysiology
      • Nervous system physiology
        • Neural cells
          • Overview of neuron structure
          • Overview of neuron function
          • Astrocytes
          • Microglia
          • Ependymal cells
          • Oligodendrocytes
          • Schwann cells
        • Neuron membrane potentials
          • Neuron resting potential description
          • Neuron resting potential mechanism
          • Neuron graded potential description
          • Neuron graded potential mechanism
          • Neuron action potential description
          • Neuron action potential mechanism
          • Effects of axon diameter and myelination
          • Action potential patterns
        • Neuronal synapses
          • Synapse structure
          • Neurotransmitter release
          • Types of neurotransmitters
          • Types of neurotransmitter receptors
          • Neurotransmitter removal
          • Neuroplasticity
        • Biosignaling
          • Membrane Receptors
          • Ligand Gated Ion Channels
          • G Protein Coupled Receptors
          • Enzyme Linked Receptors
      • Nervous system diseases
        • Parkinson's disease
          • What is Parkinson's disease?
          • Movement signs and symptoms of Parkinson's disease
          • Non-movement symptoms of Parkinson's disease
          • The basal ganglia - The direct pathway
          • The basal ganglia - Concepts of the indirect pathway
          • Putting it all together - Pathophysiology of Parkinson's disease
          • The basal ganglia - Details of the indirect pathway
          • Genetics and Parkinson's disease
          • Diagnosing Parkinson's disease
          • Managing Parkinson's disease with medications
          • Managing Parkinson's disease with surgery
        • Multiple sclerosis
          • What is multiple sclerosis?
          • Multiple sclerosis signs and symptoms
          • Clinical progression of multiple sclerosis
          • Multiple sclerosis pathophysiology
          • Multiple sclerosis risk factors
          • Multiple sclerosis diagnosis
          • Multiple sclerosis treatment
      • Integumentary system physiology
        • Integumentary system
          • Meet the skin! (Overview)
          • What is skin? (Epidermis)
          • What lies beneath the epidermis? (Dermis and Hypodermis)
          • Where do our nails and hair come from?
          • What's in sweat? (Holocrine, Apocrine, Merocrine Glands)
          • LeBron Asks: Why does sweating cool you down?
          • Overview of Sensation and Meissner's Corpuscle
          • Pacinian's Corpuscle and Merkel's Disk
          • Ruffini's Ending and Hair Follicle Receptor
          • Pain and temperature
          • Thermoregulation by muscles
      • Reproductive system physiology
        • Reproductive system
          • Welcome to the reproductive system
          • Anatomy of the male reproductive system
          • Transport of sperm via erection and ejaculation
          • Spermatogenesis
          • Testosterone
          • Basics of egg development
          • The ovarian cycle
          • Meet the placenta!
          • Reproductive cycle graph - Follicular phase
          • Reproductive cycle graph - Luteal phase
          • Estrogen
          • Maternal changes in pregnancy
          • Labor (parturition)
          • Breast anatomy and lactation
        • Pregnancy and pregnancy complications
          • Diagnosis of pregnancy
          • Pregnancy physiology I
          • Labor
          • Postpartum physiology
          • Preeclampsia
          • Placenta previa
          • Placental abruption
          • Placenta accreta
          • UTIs in pregnancy
          • Blood conditions in pregnancy
          • Sheehan syndrome
          • Postpartum hemorrhage
          • Uterine inversion
          • Diabetes in pregnancy
          • Preterm labor
          • C-section
      • Infectious diseases
        • Sexually transmitted diseases
          • What is gonorrhea?
          • Pathophysiology, diagnosis, treatment, and prevention of gonorrhea
          • What is chlamydia?
          • Pathophysiology of chlamydia
          • Diagnosis, treatment, and prevention of chlamydia
          • What is syphilis?
          • What is tertiary syphilis?
          • What is congenital syphilis?
          • Diagnosis, treatment, and prevention of syphilis
          • What is chancroid?
          • What is trichomoniasis?
          • What is herpes?
          • Pathophysiology of herpes
          • What is neonatal herpes?
          • Diagnosis, treatment, and prevention of herpes
          • What are warts?
          • What is bacterial vaginosis?
        • Polio
          • What is polio?
          • Poliomyelitis pathophysiology
          • Polio epidemiology
          • Poliomyelitis diagnosis and treatment
          • Polio vaccinations
        • HIV and AIDS
          • What is HIV and AIDS?
          • Transmission of HIV
          • How HIV infects us: Mucous membranes, dendritic cells, and lymph nodes
          • How HIV infects us: CD4 (T-helper) lymphocyte infection
          • How HIV kills so many CD4 T cells
          • Diagnosing HIV - Concepts and tests
          • Treating HIV: Antiretroviral drugs
          • HAART treatment for HIV - Who, what, why, when, and how
          • Defining AIDS and AIDS defining illnesses
          • Immune reconstitution inflammatory syndrome (IRIS) in AIDS
          • Preventing an HIV infection
        • Malaria
          • The basics of malaria
          • Diagnosing malaria
          • Treatment of malaria
          • Preventing malaria
        • Tuberculosis
          • What is TB?
          • TB epidemiology
          • TB pathogenesis
          • Primary and Secondary TB
          • Pulmonary TB
          • Extrapulmonary TB (part 1)
          • Extrapulmonary TB (Part 2)
          • Mantoux test (aka. PPD or TST)
          • Interpreting the PPD
          • Diagnosing active TB
          • Preventing TB transmission
          • Preventing TB using the "4 I's"
          • Treatment of Active TB
          • Drug-resistant TB
          • TB and HIV
      • Mental health
        • Drug abuse and drug addictions
          • Overview of psychoactive drugs
          • Psychoactive drugs: Depressants and opiates
          • Psychoactive drugs: Stimulants
          • Psychoactive drugs: Hallucinogens
          • Routes of drug entry
          • Drug dependence and homeostasis
          • Reward pathway in the brain
          • Tolerance and withdrawal
          • Risk factors for drug use and drug abuse
          • Substance use disorders
          • The development of substance use - Why do people use legal and illegal substances?
          • Why do some people but not others develop substance use problems?
          • Treatments and triggers for drug dependence
          • How does substance use develop into substance abuse
          • Drug use prevention - school programming and protective factors
        • Depression and related disorders
          • Introduction to psychology - Depression and major depressive disorder
          • Diagnosing depression
          • Introduction to psychology - Depression and bipolar disorder
          • Diagnosing bipolar disorder
          • Types of depression and bipolar disorder in the DSM5
          • Biological basis of depression
          • Risk factors for bipolar disorder
          • Treating depression with antidepressants
          • Treatments for depression - Psychological therapies
          • Treatments for bipolar disorder
        • Anxiety
          • Generalized anxiety disorder
          • Phobias
          • Panic disorder
          • Obsessive compulsive disorder
          • Post traumatic stress disorder
        • Psychotic disorders
          • Psychosis
          • Schizophrenia symptoms
          • Schizophrenia diagnosis
          • Schizophrenia treatment
          • Schizoaffective disorder
          • Schizotypal personality disorder
          • Delusional disorder
        • Dementia, delirium, and Alzheimer's
          • What are dementia and Alzheimer's
          • Alzheimer's disease: Plaques and tangles
          • Other types of dementia
          • Risk factors for dementia
          • Stages of dementia and Alzheimer's disease
          • Diagnosis of dementia and Alzheimer's disease
          • Treatment of dementia and Alzheimer's disease
          • What is delirium?
        • Neurodevelopmental disorders
          • Intro to neurodevelopment milestones
          • Neurodevelopmental disorders: Sufficient and necessary causes
          • What is cerebral palsy and what causes it?
          • Types of cerebral palsy part 1 - Spastic
          • Types of cerebral palsy part 2 - Dyskinetic and ataxic
          • Diagnosing cerebral palsy
          • Managing cerebral palsy
          • What is autism spectrum disorder?
          • What is Tourette's?
          • Diagnosing autism spectrum disorder
          • Managing autism spectrum disorder
          • What is asperger syndrome?
    • GMAT
      • Problem solving
        • GMAT: Math 1
        • GMAT: Math 2
        • GMAT: Math 3
        • GMAT: Math 4
        • GMAT: Math 5
        • GMAT: Math 6
        • GMAT: Math 7
        • GMAT: Math 8
        • GMAT: Math 9
        • GMAT: Math 10
        • GMAT: Math 11
        • GMAT: Math 12
        • GMAT: Math 13
        • GMAT: Math 14
        • GMAT: Math 15
        • GMAT: Math 16
        • GMAT: Math 17
        • GMAT: Math 18
        • GMAT: Math 19
        • GMAT: Math 20
        • GMAT: Math 21
        • GMAT: Math 22
        • GMAT: Math 23
        • GMAT: Math 24
        • GMAT: Math 25
        • GMAT: Math 26
        • GMAT: Math 27
        • GMAT: Math 28
        • GMAT: Math 29
        • GMAT: Math 30
        • GMAT: Math 31
        • GMAT: Math 32
        • GMAT: Math 33
        • GMAT: Math 34
        • GMAT: Math 35
        • GMAT: Math 36
        • GMAT: Math 37
        • GMAT: Math 38
        • GMAT: Math 39
        • GMAT: Math 40
        • GMAT: Math 41
        • GMAT: Math 42
        • GMAT: Math 43
        • GMAT: Math 44
        • GMAT: Math 45
        • GMAT: Math 46
        • GMAT: Math 47
        • GMAT: Math 48
        • GMAT: Math 49
        • GMAT: Math 50
        • GMAT: Math 51
        • GMAT: Math 52
        • GMAT: Math 53
        • GMAT: Math 54
      • Data sufficiency
        • GMAT: Data sufficiency 1
        • GMAT: Data sufficiency 2
        • GMAT: Data sufficiency 3
        • GMAT: Data sufficiency 4
        • GMAT: Data sufficiency 5
        • GMAT: Data sufficiency 6
        • GMAT: Data sufficiency 7
        • GMAT: Data sufficiency 8
        • GMAT: Data sufficiency 9
        • GMAT: Data sufficiency 10
        • GMAT: Data sufficiency 11
        • GMAT: Data sufficiency 12
        • GMAT: Data sufficiency 13
        • GMAT: Data sufficiency 14
        • GMAT: Data sufficiency 15
        • GMAT: Data sufficiency 16
        • GMAT: Data sufficiency 17
        • GMAT: Data sufficiency 18
        • GMAT: Data sufficiency 19
        • GMAT: Data sufficiency 20
        • GMAT: Data sufficiency 21
        • GMAT: Data sufficiency 21 (correction)
        • GMAT: Data sufficiency 22
        • GMAT: Data sufficiency 23
        • GMAT: Data sufficiency 24
        • GMAT: Data sufficiency 25
        • GMAT: Data sufficiency 26
        • GMAT: Data sufficiency 27
        • GMAT: Data sufficiency 28
        • GMAT: Data sufficiency 29
        • GMAT: Data sufficiency 30
        • GMAT: Data sufficiency 31
        • GMAT: Data sufficiency 32
        • GMAT: Data sufficiency 33
        • GMAT: Data sufficiency 34
        • GMAT: Data sufficiency 35
        • GMAT: Data sufficiency 36
        • GMAT: Data sufficiency 37
        • GMAT: Data sufficiency 38
        • GMAT: Data sufficiency 39
        • GMAT: Data sufficiency 40
        • GMAT: Data sufficiency 41
    • CAHSEE
      • CAHSEE
        • CAHSEE practice: Problems 1-3
        • CAHSEE practice: Problems 4-9
        • CAHSEE practice: Problems 10-12
        • CAHSEE practice: Problems 13-14
        • CAHSEE practice: Problems 15-16
        • CAHSEE practice: Problems 17-19
        • CAHSEE practice: Problems 20-22
        • CAHSEE practice: Problems 23-27
        • CAHSEE practice: Problems 28-31
        • CAHSEE practice: Problems 32-34
        • CAHSEE practice: Problems 35-37
        • CAHSEE practice: Problems 38-42
        • CAHSEE practice: Problems 43-46
        • CAHSEE practice: Problems 47-51
        • CAHSEE practice: Problems 52-53
    • IIT JEE
      • IIT JEE
        • Trig challenge problem: arithmetic progression
        • IIT JEE perpendicular planes (part 1)
        • IIT JEE perpendicular plane (part 2)
        • IIT JEE complex root probability (part 1)
        • IIT JEE complex root probability (part 2)
        • IIT JEE position vectors
        • IIT JEE integral limit
        • IIT JEE algebraic manipulation
        • IIT JEE function maxima
        • IIT JEE diameter slope
        • IIT JEE hairy trig and algebra (part 1)
        • IIT JEE hairy trig and algebra (part 2)
        • IIT JEE hairy trig and algebra (part 3)
        • Challenging complex numbers problem (1 of 3)
        • Challenging complex numbers problem (2 of 3)
        • Challenging complex numbers problem (3 of 3)
        • IIT JEE differentiability and boundedness
        • IIT JEE integral with binomial expansion
        • IIT JEE symmetric and skew-symmetric matrices
        • IIT JEE trace and determinant
        • IIT JEE divisible determinants
        • Intersection of circle & hyperbola
        • Common tangent of circle & hyperbola (1 of 5)
        • Common tangent of circle & hyperbola (2 of 5)
        • Common tangent of circle & hyperbola (3 of 5)
        • Common tangent of circle & hyperbola (4 of 5)
        • Common tangent of circle & hyperbola (5 of 5)
        • Trig challenge problem: multiple constraints
        • Trig challenge problem: maximum value
        • Vector triple product expansion (very optional)
        • IIT JEE lagrange's formula
        • Representing a line tangent to a hyperbola
        • 2010 IIT JEE Paper 1 Problem 50: Hyperbola eccentricity
        • Normal vector from plane equation
        • Point distance to plane
        • Distance between planes
        • Challenging complex numbers problem: complex determinant
        • Series sum example
        • Trig challenge problem: system of equations
        • Simple differential equation example
  • KA Educator
  • Partner content
    • The Metropolitan Museum of Art
      • 82nd & Fifth
        • Body and mind
          • Marble Statue of a kouros
          • Arhat (Luohan)
          • Seated Figure (Djenné peoples)
          • Mask
          • Studies for the Libyan Sibyl and a small Sketch for a Seated Figure (verso)
          • Bronze statue of Eros sleeping
          • Fudō Myōō (Achala-vidyaraja)
          • Albrecht Dürer, Adam and Eve
          • Degas’s Little Girl Practicing at the Bar
        • Books, music, and literature
          • Hermann Hauser, Guitar
          • Goto Teijo, Koto
          • Delacroix, The Abduction of Rebecca
          • Cristofori, Grand Piano
          • Rembrandt, Aristotle with a Bust of Homer
          • Liang Kai, Poet Strolling by a Marshy Bank
          • Herman, Paul, and Jean de Limbourg, The Belles Heures of Jean de France, Duc de Berry
          • Shitao, Returning Home
        • Devotion
          • Rossellino, Madonna and Child with Angels
          • Pensive Bodhisattva
          • Curtain of the Tabernacle
          • Berlinghiero, Madonna and Child
          • Plaque with the Journey to Emmaus and Noli Me Tangere
          • Rembrandt, The Last Supper, after Leonardo da Vinci
          • Bassano’s The Baptism of Christ
          • Relief of the Betrayal and Arrest of Jesus
          • Cult Image of the God Ptah
          • Plaque with Censing Angels
          • Mahakala, Protector of the Tent
          • Blake, The Parable of the Wise and Foolish Virgins
        • Dining
          • Silver spoon and fork
          • Covered Broth Bowl and Stand
          • Statue of an Offering Bearer
          • Wine Cup
        • Family and community
          • Figure from a Reliquary Ensemble: Seated Female
          • Carpeaux, Le Trait d’Union
          • Wright, Living room from the Little House, Wayzata, Minnesota
          • Bellini, Madonna and Child
          • Slit Gong (Atingting kon)
          • David, Study for The Lictors Bringing Brutus the Bodies of his Sons
          • Statue of two men and a boy that served as a domestic icon
        • Intersections
          • Ewer
          • Figure of a Seated Court Lady
          • Cylinder seal and modern impression: nude bearded hero wrestling with a water buffalo; bull-man wrestling with lion
          • Marble capital and finial in the form of a sphinx
          • Hunters in a Landscape
          • Card table
          • Hunt, The Great Hall
        • Nature
          • Tiffany, Hair Ornament
          • Hippopotamus
          • Palissy, Pilgrim Flask
          • El Greco, View of Toledo
          • Fenton, Landscape with Clouds
          • Homer, Northeaster
          • Gifford, A Gorge in the Mountains (Kauterskill Clove)
          • Congyi, Cloudy Mountains
        • Observations
          • Emmoser, Celestial globe with clockwork
          • Rembrandt, Christ Crucified between the Two Thieves: The Three Crosses.
          • "Krishna Holds Up Mount Govardhan to Shelter the Villagers of Braj", Folio from a Harivamsa (The Legend of Hari (Krishna))
          • Rochford’s Girls I Have Known
          • Dürer, Self-portrait, Study of a Hand and a Pillow
          • Groom and Rider
          • Van Eyck, The Crucifixion; The Last Judgment
          • Evans, Subway Passengers, New York City
        • Power
          • Tughra (Official Signature) of Sultan Süleiman the Magnificent from Istanbul
          • Tiffany & Co., The Adams Vase
          • Head of Tutankhamun from the Amarna Period of Egypt’s New Kingdom
          • Pectoral and Necklace of Sithathoryunet with the Name of Senwosret II
          • Relief panel from Assyria
          • Two Panels with striding lions from Babylon
        • Puzzles and surprises
          • Crossbow of Ulrich V of Württemberg
          • Goa Stone and Gold Case from Goa, India
          • Alexander Danilovich Menshikov, an 18th-century Russian portrait bust
          • Canopic Jar with a Lid in the Shape of a Royal Woman’s Head
          • Johns, White Flag
        • The art of dress
          • Helmschmid, Portions of a costume armor
          • Saddle (gser sga) from Derge, Tibet
          • Clasp with an eagle and its prey dating from the Parthian Empire
          • McQueen, spring/summer 1999 collection, "No. 13"
          • Shawl, designed by Deneirouse and Boisglavy
          • Poiret, Paris
          • Nose Ornament with Spiders from ancient Peru
          • James, Evening Dress
        • Transformations
          • Nawa, PixCell-Deer#24
          • Calder, Mobile
          • Mascarade à la Grecque; Suite des Vases; and [Cheminées], designed by Petitot
          • Van Orley and de Pannemaker, The Last Supper
          • Oeben, Mechanical table
          • Statue of Dionysos leaning on a female figure (“Hope Dionysos”)
        • War and conquest
          • Negroli, Burgonet
          • Tiepolo, The Triumph of Marius
          • Matisse, The illustrated book, “Jazz”
          • Byzantine Plate with the Battle of David and Goliath
          • Trumbull, The Sortie Made by the Garrison of Gibraltar
          • Master of Belmonte, Saint Michael
          • Remington, The Old Dragoons of 1850
      • Extravagant Inventions
        • Demonstration of David Roentgen's Automaton of Queen Marie Antoinette, The Dulcimer Player
        • Demonstration of the Roentgens' Dressing Table (Poudreuse)
        • Rolltop Desk by David Roentgen (demonstration)
        • Rolltop Desk by David Roentgen (animation)
        • Unfolding an 18th-century Game Table
      • Making, finding, and conserving
        • After the Fall: The Conservation of Tullio Lombardo's "Adam"
        • Conserving Tullio Lombardo's "Adam": Time-Lapse
        • Conserving Tullio Lombardo's "Adam": Order of Assembly
        • An Art of Attraction: The Electrotyping Process
        • Conserving the Emperors Carpet
        • Conserving the Damascus Room at The Metropolitan Museum of Art
        • Craftsman Allan Breed Turns and Carves a Duncan Phyfe Bedpost
        • Craftsman Allan Breed Carves a Bow Knot and Thunderbolt Crest
        • Jan Gossart - Conservation Discoveries
      • #MetKids: Made for, with, and by kids
        • Q&A
          • Why does design matter in arms and armor?
          • Why is writing so important in Islamic art?
          • What's the story behind the world's oldest piano?
          • Were there superheroes in the ancient world?
          • What's at the Met for sports fans like me?
          • What's that artist making in the galleries?
          • Is there more than one way to see a work of art?
          • Can I learn about Greek mythology at the Met?
          • How were mummies made in Ancient Egypt?
          • How did they get all this stuff into the museum?
          • How can I recognize ancient Greek architecture?
          • How does the Met decide how and where to hang the art?
          • Can doodles be art?
          • What's special about these galleries?
          • How does the museum take care of all the armor?
        • Celebrate
          • ¡Fiesta! Celebrate Latin America / Celebren América Latina
          • Lunar New Year Festival 2015
          • Museum Mile 2015
          • Assyria to Iberia: A journey through land and sea
        • Made by kids
          • Animation: Pieter Breugel
          • Animation: Edgar Degas
          • Animation: Ancient Greek art
          • Animation: Ancient Egyptian tomb
          • Animation: Nicolas Poussin
          • Animation lab 2014: Behind the scenes
          • Animation lab 2015: Behind the scenes
          • Animation: Tapestries
          • Animation: Renaissance portrait
          • Animation: A Japanese protector
          • Animation: Whale transformation mask
          • Animation: Armor
        • Create
          • Weave on a mini loom
          • Draw and ink comic panels
          • Create a flip book
          • Trace and cut paper silhouettes
          • Pose like a sculpture
          • Make a symmetrical print
          • Make a photogram
          • Make a mixed-up beings book
          • Draw with an eraser
          • Create scratch art
          • Create an optical toy: Thaumatrope
          • Create a collage
          • Design a fashion accessory: Fancy cuffs
    • Tate
      • Get started and The Big Question
        • Welcome to Tate on Khan Academy
          • Why Is Art Important?
      • Conflict and Contradiction in Art
        • Activities
        • Art and Conflict
        • Documenting Conflict
          • Conflict, time, and photography
          • The Berlin Wall and industrial England: Don McCullin's conflict photography
          • Kurt Schwitters' wartime portraits
          • Traces of illegal migration in Mexico: Mark Ruwedel
          • Shai Kremer in an Israeli military training stage
          • Burke and Norfolk: Photographs from the war In Afghanistan
          • Hrair Sarkissian: Syria's "Execution Squares"
        • Challenging the State
          • Hans Haacke: "A Breed Apart" in South Africa
          • Doris Salcedo's "Shibboleth"
          • Mounted police and crowd control at Tate Modern
          • "Immigrant Movement International": Nationalism and immigration in the gallery
          • Wang Peng: Performance as politics in China and beyond
        • Creating Contradiction
          • Michael Rakowitz on Star Wars and Saddam Hussein
          • Peter Kennard's political photomontages
          • Personal memory versus political history in Eastern Europe
          • Mona Hatoum's self-contradictory objects
      • Women in Art and Art History
        • Activities
        • A Brief History of Women in Art
          • Unlock Art: Where are the Women?
        • Women as Artists
          • Barbara Hepworth: Pioneering modern sculpture
          • More than Picasso's muse: Françoise Gilot
          • Biddy Peppin on the female Vorticists
          • Art and identity in the work of Lorna Simpson
          • Fiona Rae on paint and passion
          • Nan Goldin's snapshots of innocence and childhood
        • Women's Issues in Art
          • Margaret Harrison: Feminism, irony, and women's rights
          • The Folkestone Mermaid and the model figure
          • Lily van der Stokker's "non-shouting feminism"
          • Barbara Hammer: Pushing the boundaries of cinema and the body
          • Keren Cytter: Radical gender reversals and theatrical performance
      • Global Modernisms
        • Pop art around the world
          • What is Pop art?
          • Roy Lichtenstein
          • Roy Lichtenstein: Diagram of an Artist
          • Ed Ruscha
          • Patrick Caulfield
          • Peter Blake: Studio visit
          • Sigmar Polke
          • Nicola L’s Red Coat
          • Parviz Tanavoli
          • Niki de Saint Phalle
        • Africa
          • Ibrahim El-Salahi
          • Meschac Gaba
          • Doug Fishbone: Elmina
          • Contested Terrains, Part 1
          • Contested Terrains, Part 2
        • Latin America
          • Cildo Meireles
          • Graciela Iturbide
          • Gabriel Orozco
          • Melanie Smith
        • Asia
          • Ai Weiwei: Sunflower Seeds
          • The Real Thing: Contemporary art from China
          • Yayoi Kusama
          • Amrita Sher-Gil
          • Zarina Hashmi
        • The Middle East
          • Saloua Raouda Choucair: From Beirut to Tate Modern
          • Dia Al-Azzawi
          • Shirazeh Houshiary
          • East-West Divan at the Venice Biennale
      • Meet 500 years of British art
        • Welcome to 500 years of British art
          • Director's Highlights: Penelope Curtis
        • 16th to 18th century: Dynasty, revolution, and the Classical Age
          • Room: 1540-1650
          • Room: 1650-1730
          • Room: 1730-1765
          • Room: 1760-1780
          • Room: 1780-1810
        • 19th century: Romanticism and the Victorian Era
          • Room: 1810-1840
          • Room: JMW Turner
          • Room: William Blake
          • Room: 1840-1890
          • Room: 1890-1910
        • Early 20th century: War and dynamism
          • Room: 1910-1914
          • Room: 1914 & 1915
          • Room: 1930s
          • Room: 1940s
        • Mid-century to today: Modernism and its legacy
          • Room: 1950s
          • Room: Henry Moore
          • Room: 1960s
          • Room: 1970s & 1980s
          • Room: 1990s
          • Room: 2000s and beyond
      • Meet the artist: JMW Turner
        • Activities
        • Turner: The Artist and His Work
          • "Modern" art in Victorian times: the late Turner
          • Capturing Turner's colours: Olafur Eliasson's colour experiments
        • Visit Turner's Gallery
          • Recreating Turner's gallery for the big screen
        • Learn to paint like Turner
          • Line: Drafting, sketching, and composing
          • Tone: Telling stories with light and dark
          • Colour: Delineating shapes and capturing atmosphere
      • Archives, Memory, and Conservation
        • Activities
        • What is an archive?
          • Animating the Archives: An Introduction
          • A Juke Box at Tate Britain: Listening to and making audio archives
          • Josef Herman's personal sketches of Wales
          • Graham Sutherland's hands-on sketchbooks
          • How do we document and archive performance? A live art salon
          • Archive conservation volunteers at work
          • Kurt Schwitters' archives and artists in captivity
          • The archives of Eileen Agar, a British surrealist
          • Time and Influence: A fashion project with University of the Arts London
        • Art and Memory
          • Michael Landy and Jean Tinguely: Art that breaks down
          • Cyprien Gaillard: Memory, nostalgia, and anachronism
          • Katie Paterson: A map of every dead star in the Universe
          • Do Ho Suh's memory of a staircase
          • Johno Verity, avalanche survivor, on "An Avalanche in the Alps"
        • Conservation: Behind the scenes
          • Restoring Rothko
          • Conserving the work of Choucair
          • Conserving Chris Ofili
      • Performance and Participation
        • Performance art
          • Unlock Art: Frank Skinner on Performance Art
          • Actions and Interruptions at Tate Modern
          • Marina Abramović
          • Vito Acconci
          • Martin Creed
        • Collaboration
          • Unlock Art: Great Double Acts
          • Harrison and Wood
          • Daido Moriyama
          • Michelangelo Pistoletto
          • Ei Arakawa
        • Participation
          • Suzanne Lacy: Silver Action
          • Thomas Hirschhorn: Flamme Éternelle
          • Juan Downey: Plato Now
          • Ryan Gander: Locked Room Scenario
          • Robert Morris: Bodyspacemotionthings at Tate Modern
      • The Art of Inspiration
        • Film meets art
          • Film meets art: Mike Leigh inspired by JMW Turner
          • Film meets art: Chris Nolan inspired by Francis Bacon
          • Film meets art: Ken Loach inspired by William Hogarth
        • Fashion meets art
          • Fashion meets art: Jonathan Saunders inspired by Anthony Caro
          • Fashion meets art: Preen inspired by David Hockney
          • Fashion meets art: Simone Rocha inspired by Louise Bourgeois
        • Literature meets art
          • Literature meets art: Maurice Sendak inspired by William Blake
          • Poetry meets art: George the Poet inspired by Paul Graham
        • Art and appropriation
          • AK Dolven on Edvard Munch
          • Storm Thorgerson on René Magritte
          • John Myatt on Glenn Brown
    • The Museum of Modern Art
      • Printmaking
        • Introduction to printmaking
        • Introduction to relief printmaking
        • Relief process
        • Introduction to intaglio
        • Intaglio process
        • Introduction to lithography
        • Lithography process
        • How to make a monotype
      • 1913 Centennial Celebration
        • Pablo Picasso, "Guitar, Glass, and Bottle"
        • Umberto Boccioni, "Dynamism of a Soccer Player"
        • Louis Comfort Tiffany, Vase
        • Vasily Kandinsky, "Klänge (Sounds)"
        • Fernand Léger, "Contrast of Forms"
        • Lois Weber and Phillips Smalley, "Suspense"
        • Giorgio de Chirico, "The Anxious Journey"
        • Olga Rozanova, "A Little Duck's Nest... of Bad Words"
        • Léon Bakst, "Costume design for the ballet The Firebird"
        • Constantin Brancusi, "Mlle Pogany"
        • Robert Delaunay, "Simultaneous Contrasts: Sun and Moon"
        • D. W. Griffith, "The Mothering Heart"
        • Emil Nolde, "Young Couple," 1913
        • Léopold Survage, "Colored Rhythm: Study for the Film"
        • Ludwig Hohlwein, "Kaffee Hag"
        • Mack Sennett, "Barney Oldfield's Race for a Life"
        • Louis Raemaekers, "Tegen de Tariefwet, Vliegt niet in't Web!"
        • "Composition in Brown and Gray," Piet Mondrian
        • "3 Standard Stoppages," Marcel Duchamp
        • 1913 | Schiess-Dusseldorf by Ludwig Hohlwein
        • Matisse, "The Blue Window"
        • Ernst Ludwig Kirchner, "Street, Berlin"
        • Frank Lloyd Wright, Midway Gardens
      • Behind the scenes at MoMA
        • Conserving Henri Matisse's "The Swimming Pool"
        • Framing "Christina's World" by Andrew Wyeth
        • Installation of Richard Serra's sculptures at MoMA
        • Conservation | Pollock, "One: Number 31, 1950"
        • Conservation | Picasso's Guitars
        • Conservation | Paik, "Untitled," 1993
        • Conservation | René Magritte, "The Portrait," 1935
        • Conservation | René Magritte, "The False Mirror," 1928
        • Conservation | René Magritte, "The Menaced Assassin," 1927
        • Conservation | René Magritte, "The Palace of Curtains, III," 1928-29
      • Abstract Expressionism
        • AB EX NY: MoMA and Abstract Expressionism
        • Jackson Pollock
        • The Painting Techniques of Jackson Pollock
        • Mark Rothko
        • The Painting Techniques of Mark Rothko
        • Franz Kline
        • The Painting Techniques of Franz Kline
        • Barnett Newman
        • The Painting Techniques of Barnett Newman
        • Ad Reinhardt
        • The Painting Techniques of Ad Reinhardt
      • Art terms in action
        • Art Terms in Action: Viscosity
        • Art Terms in Action: Turpentine Burn
        • Art Terms in Action: Tint, Shade, and Tone
        • Art Terms in Action: Stain
        • Art Terms in Action: Palette Knife
        • Art Terms in Action: Paint
        • Art Terms in Action: Enamel
        • Art Terms in Action: Emulsion
      • Artist interviews
        • Andrés Jaque: COSMO | Young Architects Program 2015
        • Gilbert & George: The Early Years
        • Cai Guo-Qiang | Borrowing Your Enemy's Arrows
        • Richard Serra | Equal
        • "Weaving the Courtyard" by Escobedo Soliz | Young Architects Program 2016
        • Artists Experiment 2014 | MoMA
        • THIS IS ISA GENZKEN | MoMA
        • Isaac Julien, Ten Thousand Waves | MoMA
        • James Rosenquist, "F-111," 1964-65
        • Lee Quinones on graffiti
        • Studio Tour: Thomas Lanigan-Schmidt
        • Richard Serra, "Intersection II"
        • Richard Serra, "Torqued Ellipse IV"
        • Richard Serra, "Band," 2006
        • Wolfgang Laib, "Pollen from Hazelnut"
        • Gabriel Byrne revisiting "The Quiet Man"
        • Carolee Schneemann, "Up to and Including Her Limits"
        • Dorothea Rockburne: Drawing Which Makes Itself
      • Performance Art
        • Maria Hassabi | PLASTIC
        • Yvonne Rainer: "The Concept of Dust, or How do you look when there's nothing left to move?"
        • Jannis Kounellis, "Da inventare sul posto (To invent on the spot)"
        • Marina Abramović
        • Marina Abramović: What is performance art?
        • Marina Abramović: Marina's first performance
        • Marina Abramović: The Body as medium
        • Marina Abramović: Documenting performance
        • Marina Abramović: Teaching the next generation
        • Marina Abramović: "Cleaning the House" workshop
        • Marina Abramović: Performance vs. acting
      • MoMA Learning
        • "Collective Suicide" by David Alfaro Siqueiros, 1936 | MoMA Education
        • "Girl Before a Mirror" by Pablo Picasso, 1932 | MoMA Education
        • "To Lift" by Richard Serra, 1967 | MoMA Education
        • "Self-Portrait with Cropped Hair" by Frida Kahlo, 1940 | MoMA Education
        • Five Tips for Teaching with Works of Art | MoMA Education
        • "Dive Bomber and Tank" by Jose Clemente Orozco, 1940 | MoMA Education
        • "Wall Drawing #1144, Broken Bands of Color in Four Directions" by Sol LeWitt, 2004 | MoMA Education
        • "Bicycle Wheel" by Marcel Duchamp, 1951 | MoMA Education
        • "One: Number 31, 1950" by Jackson Pollock, 1950 | MoMA Education
        • "Map" by Jasper Johns, 1961 | MoMA Education
        • Meet Me | Art Discussion Programs for Individuals with Dementia
        • Meet Me | Art Making Programs for Individuals with Dementia
      • Modern Art & Ideas
        • Modern Art & Ideas
          • Introduction to Modern Art & Ideas
          • Art and Identity | Modern Art & Ideas
          • Places and Spaces | Modern Art & Ideas
          • Transforming Everyday Objects | Modern Art & Ideas
          • Art and Society | Modern Art & Ideas
      • Seeing Through Photographs
        • Seeing Through Photographs
          • Seeing Through Photographs
          • Nicholas Nixon | The Brown Sisters
          • Hank Willis Thomas | Unbranded
          • Katy Grannan | Boulevard
          • Vik Muniz | Equivalents (The Museum of Modern Art)
          • Marvin Heiferman | Seeing Through Photographs
          • Sarah Meister | Seeing Through Photographs
          • Lucas Blalock | Strawberries (Fresh Forever), Strawberries (Forever Fresh)
          • David Horvitz | Mood Disorder
          • Anouk Kruithof | Subconscious Travelling
          • Ilit Azoulay | Shifting Degrees of Certainty
    • The J. Paul Getty Museum
      • Antiquities
        • Ancient glassmaking
          • Glassmaking technique: mold-blown glass
          • Roman mold-blown glass
          • Glassmaking technique: core-formed glass
          • Glassmaking technique: free-blown glass
          • Glassmaking technique: mosaic glass
          • Glassmaking technique: gold glass
          • Glassmaking technique: cameo glass
        • Ancient Greek vases
          • Making Greek vases
          • Making a molded Athenian vase
        • The Roman silversmith's craft
          • The making of a Roman silver cup
        • Gem carving in antiquity
          • The art of gem carving
        • Marble from antiquity to modern times
          • Carving marble with traditional tools
        • Inspiration from the classical past
          • Jim Dine's "Poet Singing (The Flowering Sheets)": a documentary
      • Manuscripts
        • Manuscripts: major works of art
        • Making manuscripts
        • The medieval calendar
        • Chivalry in the Middle Ages
      • Art conservation
        • The mummification process
        • Conservation of antiquities
        • Conserving bronze: "The Lamp with Erotes from Vani"
        • Conservation of mosaics in Roman North Africa
        • Tracing the colors of ancient sculpture
        • "Statue of an Emperor:" A conservation partnership
        • The conservation of the "Apollo Saettante" at the Getty Villa
        • Protecting art in an earthquake: seismic isolator technology
        • Conservation of sculpture and decorative arts
        • A Renaissance cabinet rediscovered
        • Conservation of works on paper
        • Conserving old master drawings: a balancing act
        • Conservation of paintings
        • Conservation of Jackson Pollock's Mural
        • Paint Application Studies of Jackson Pollock's Mural
        • Paint Analysis of Jackson Pollock's Mural
        • De Wain Valentine's "Gray Column" and the Getty Conservation Institute
        • De Wain Valentine: surface matters
        • De Wain Valentine and polyester resin
      • Paintings
        • Paintings
        • Looking at paintings
        • Gold-ground panel painting
        • Orazio Gentileschi's paintings of women and divine love
      • Decorative arts
        • Unlocking an 18th-century French mechanical table
        • Construction of an 18th-century French mechanical table
        • The inlay technique of marquetry
        • Making maiolica
      • Sculpture
        • Making a Spanish polychrome sculpture
        • Casting bronze: direct lost-wax casting
        • Carving marble with traditional tools
      • Photographs
        • Introducing photography
          • Photographs: a closer look
        • Photographic processes
          • Early photography: making daguerreotypes
          • The wet collodion process
          • The development of photography and the railroad
        • Photographers
          • P.H. Emerson's naturalistic photography
          • John Humble's photographs of Los Angeles
          • Camilo Vergara documents the changing urban landscape
          • David Hockney's "Pearblossom Hwy"
          • Eileen Cowin on her series "I See What You're Saying"
          • Carrie Mae Weems on her series "From Here I Saw What Happened and I Cried"
      • Drawings
        • Drawings
        • Jean-Baptiste Greuze's drawings
        • Taddeo Zuccaro's artistic training: drawings by Federico Zuccaro
        • Henri Toulouse-Lautrec's circus drawings
        • Paul Cézanne's approach to watercolor
      • For kids
        • Bored angel and twin
        • Snake-haired monster
        • Blue gangster demon
        • Ferocious but cute sea monster
        • Pair of chests, attributed to Antonio Maffei
        • "A Hare in the Forest," Hans Hoffmann
        • "An Allegory of Passion," Hans Holbein the Younger
        • "The Adoration of the Magi with Saint Anthony Abbott," Franco-Flemish master
        • Horned monster
        • Sad angels
        • Mighty mover angel
        • Tongue-tied baby dragon
        • St. Michael the archangel, sassy devil and crew
    • California Academy of Sciences
      • Welcome to the world of biodiversity!
        • Welcome
          • California Academy of Sciences
      • Why is biodiversity important?
        • Biodiversity and ecosystem function
          • Ecosystems and ecological networks
        • Biodiversity and ecosystem services
          • Ecosystem services
      • Where is biodiversity found?
        • Biodiversity Hotspots
          • What is a biodiversity hotspot?
          • Biodiversity hotspot case study: California
          • Biodiversity hotspot case study: Galapagos
          • Biodiversity hotspot case study: Gaoligongshan
          • Biodiversity hotspot case study: Madagascar
          • Biodiversity hotspot case study: Mesoamerica
          • Biodiversity hotspot case study: Philippines
          • Biodiversity hotspot case study: Sao Tomé
        • Biodiversity distribution patterns
          • How biodiversity is distributed globally
          • Why biodiversity is distributed unevenly
      • How is biodiversity studied?
        • Biodiversity fieldwork
          • Biodiversity Expeditions Past and Present
          • Field Methods for Documenting Biodiversity
        • Biodiversity analyses and unknowns
          • Studying biodiversity in the lab
          • How much biodiversity do we really know?
      • Why is biodiversity threatened?
        • Local threats to biodiversity
          • Human activities that threaten biodiversity
          • Introduced species and biodiversity
        • Global threats to biodiversity
          • How does climate change affect biodiversity?
          • Demystifying ocean acidification and biodiversity impacts
      • How can biodiversity be protected?
        • Harnessing science and education for biodiversity conservation
          • Science and education join forces to protect biodiversity
          • Conservation and the race to save biodiversity
        • Biodiversity champions
          • Protecting biodiversity: the power of the individual
          • Protecting biodiversity: local and global policies
      • Where does biodiversity come from?
        • The evolutionary causes of biodiversity
          • Biodiversity and natural selection
          • Genetic variation, gene flow, and new species
        • Biodiversity patterns of speciation and extinction
          • New localities lead to new biodiversity
          • Biodiversity and extinction, then and now
      • What are the different levels of biodiversity?
        • Genetic and ecosystem biodiversity
          • Genes and biodiversity
          • Ecosystem biodiversity
        • Biodiversity and the tree of life
          • Discovering the tree of life
          • Understanding the past to preserve the future
      • Exploring ecosystems
        • Coral reef symbiosis
        • Coastal food webs
        • Tropical rainforest diversity
    • Exploratorium
      • Sound
        • Sound Introduction
        • Vocal Visualizer
          • Vocal Visualizer: Introduction
          • Build your own Vocal Visualizer: Materials, tools, and steps
          • What’s going on: Transforming vibrating air molecules into patterns you can see
          • Challenge: Get loud!
      • Light and color
        • Light and Color Introduction
        • Soap Film on a Can
          • Introduction: Why do we see colors in oily water and soap bubbles?
          • Create your soap film viewer: Materials, tools, and steps
          • To do and notice: Horizontal bands of color
          • What’s going on: Reflections, interference, and … a water sandwich?
        • Soap Film Interference Model
          • Do the experiment: Part 1
          • Do the experiment: Part 2
          • What's going on? Wave drawings, light, and color
        • CD Spectroscope
          • Introduction: Turn an old CD into a spectroscope to analyze light and color
          • Build your own CD spectroscope: Materials, tools, and steps
          • To do and notice: See the true colors
          • What’s going on: Spreading light into a spectrum
        • Colored Shadows
          • Colored shadows: Introduction
          • Build your own colored shadows: Materials & steps
          • What's going on: Light travels in straight lines
          • Yellow and cyan challenge
          • Yellow and cyan solution
      • Chemistry
        • Chemistry introduction
        • Penny Battery
          • Penny battery introduction
          • Build your own penny battery: Materials & steps
          • What's going on: Chemical energy into electrical energy
          • Challenge: Design a more powerful battery
          • Challenge solution
        • Ocean Acidification in a Cup
          • Introduction: This simple experiment illustrates ocean acidification ... in a cup!
          • Set up your experiment: Tools, materials, and assembly
          • What’s going on: Creating carbon dioxide gas that diffuses into liquid
          • Challenge: Can you prove that diffusion happens both ways?
          • Experiment tips
        • Indicating Electrolysis
          • Introduction: Breaking up water isn't hard to do
          • Do the experiment: Tools, materials, and assembly
          • What's going on: Use electricity to break up water
          • Experiment tips
      • Electromagnetism
        • Electromagnetism Introduction
        • Flying Tinsel
          • Flying Tinsel Introduction
          • Do the experiment
          • How it works
          • Teaching tips & going further
        • Turn Light into Sound
          • Light into sound introduction
          • Build and test the receiver
          • Build the transmitter
          • Construct the parallel circuit, put it all together
    • American Museum of Natural History
      • Dinosaurs
        • What is a dinosaur?
          • What is and is not a dinosaur?
          • How do dinosaurs get their names?
          • How do we know which kinds of dinosaurs were most closely related?
        • How do scientists study dinosaurs?
          • Where in the world did dinosaurs live?
          • Did dinosaurs travel in herds or packs?
          • How fast were dinosaurs?
          • Were dinosaurs warm-blooded?
          • How fast did dinosaurs grow, and how long did they live?
          • What was dinosaur skin like?
          • What color were extinct dinosaurs?
          • What were the biggest and smallest dinosaurs?
          • Did dinosaurs fight?
          • How did dinosaurs reproduce?
          • How intelligent were dinosaurs?
          • New research points to dinosaurs' colorful past
          • New dinosaur research: Microraptor's feather color revealed
        • Dinosaur extinction
          • Were dinosaurs undergoing long-term decline before mass extinction?
          • Are any dinosaurs still alive today?
          • How did all dinosaurs except birds go extinct?
        • Dinosaur fossils
          • What Is a fossil?
          • How do we know where to look for dinosaur fossils?
          • How are dinosaur fossils discovered and collected?
          • How are dinosaur fossils prepared in the laboratory?
          • Preparing dinosaur fossils
          • Inside the Collections: Paleontology and the Big Bone Room
          • Can we clone extinct dinosaurs from DNA preserved in their fossils?
          • Barnum Brown: The man who discovered Tyrannosaurus rex
      • The Universe
        • Planets
          • Curiosity: Searching for carbon
          • Seeing planets like never before
        • Stars
          • Space weather: Storms from the Sun
          • Interferometry: Sizing up the Stars
          • Neil deGrasse Tyson on Finding Krypton
        • Galaxies
          • SALT—Imaging the southern sky
        • Universe
          • The known Universe
          • Exploring the dark Universe: Dark matter
          • Exploring the dark universe: Dark energy
          • The cosmic microwave background: A new view from the South Pole
          • Our expanding Universe
      • Human Evolution
        • Darwin and evolution by natural selection (American Museum of Natural History)
        • Human evolution: The evidence (American Museum of Natural History)
          • Human evolution
          • H. neanderthalensis, H. sapiens, and the peopling of the world
          • Expedition Rusinga—uncovering our adaptive origins
        • Evolution and human health (American Museum of Natural History)
          • Genes and health—moving beyond race
          • Malaria—natural selection and new medicine
          • SciCafe: the story of the human body with Daniel Lieberman
      • Earthquakes and Volcanoes
        • Plate Tectonics
          • Computer Model of Mantle Convection
          • Science Bulletins: Mapping the Heat Beneath
        • Volcanoes
          • Monitoring Mount Etna: Magma on the Move
          • Science Bulletins: Yellowstone—Monitoring the Fire Below
          • Scientists at Work: Hawaii
        • Earthquakes
          • On Shaky Ground: Building a Safer Future in Haiti
          • Science Bulletins: Tsunami Science—Reducing the Risk
          • Scientists at Work: San Andres Fault
    • Asian Art Museum
      • Overview
        • Overview
          • Teahouse at the Asian Art Museum
      • South Asia
        • South Asia
          • Beliefs made visible: Buddhist art in South Asia
          • Beliefs made visible: Hindu art in South Asia
          • Raja Sanai Hari Singh hunting
      • Southeast Asia
        • Cambodia and Myanmar (Burma)
          • Angkor temple mountains
          • The Hindu deities Shiva and Parvati
        • Indonesia
          • Indonesian rod puppets (wayang golek)
          • Batik of Java: A Centuries Old Tradition
        • Bali
          • Balinese shadow theater
          • Balinese dance, costumes, music
          • Symbolism in Balinese dance costumes
          • Balinese boar Barong dance performance
          • Repairing sacred Balinese masks
          • Balinese mask carver
          • The Making of Balinese Double Ikat Textiles
      • The Himalayas and the Tibetan Buddhist World
        • The Himalayas and the Tibetan Buddhist World
          • Sacred arts of Tibet
          • Views of Tibet
          • The Buddhist deity Simhavaktra Dakini
          • Secrets of the stupa
      • China
        • China
          • Discoveries in Chinese archaeology
          • Working jade
          • Ritual vessel in the shape of a rhinoceros
          • Seated Buddha dated 338
          • Chinese Buddhist cave shrines
          • Buddhist Temples at Wutaishan
          • Classical gardens of Suzhou
          • Appreciating Chinese calligraphy
          • Decoding Chinese calligraphy
          • Climbing Huangshan (Yellow Mountain)
          • The Forbidden City
      • Korea
        • Korea
          • Guardian King of the West
          • Ewer with lid
          • Reviving traditional Korean celadons
          • Nine Cloud Dream
          • Confucian scholar's house
          • Royal palaces of Seoul
          • Making mother-of-pearl lacquerware
      • Japan
        • Japan
          • Shinto
          • Ancient temples of Nara Japan
          • Standing Brahma and standing Indra
          • The Buddhist guardian deity: Fudo Myoo
          • The Way of Tea
          • Tale of the Heike
          • A brief history of samurai armor
          • From castle to palace: samurai architecture
          • Street scene in the pleasure quarter of Edo Japan
          • Courtesans of the South Station
          • Courtesan playing with a cat
          • Hunting for fireflies
          • The actor Ichikawa Danzo IV in a Shibaraku role
          • Genji Ukifune
          • How Japan Inspired Monet, Van Gogh, and Other Western Artists
      • Hinduism
        • Hinduism
          • Hindu temples
          • The Hindu deity Shiva
          • The Hindu deities Vishnu and Krishna
          • Festival of the Goddess Durga
          • The Hindu deity Ganesha
          • Varanasi: sacred city
      • Buddhism
        • Buddhism
          • How a prince became the Buddha
          • Bodh Gaya: center of the Buddhist world
          • The stupa
          • The Buddha triumphing over Mara
          • Korean Buddhist art
          • Zen Buddhism
      • Art conservation
        • Art Conservation
          • Conservation and installation of cosmological painting
          • Conservation: copper paint deterioration
          • Mounting batik textiles
          • Reconsecrating the stupa
          • Conservation of mother-of-pearl lacquerware
    • The British Museum
      • Africa
        • Introduction to Africa
        • Ancient Egypt
          • Ancient Egyptian coffin prepared for the Book of the Dead exhibition at the British Museum
          • Ancient Egyptian coffin mask conserved for the Book of the Dead exhibition at the British Museum
          • An ancient Egyptian scribal palette in the Book of the Dead exhibition at the British Museum
          • Ancient Egyptian baboon deity conserved for the Book of the Dead exhibition at the British Museum
          • Ancient Egyptian papyrus in the Book of the Dead Exhibition
          • Ancient Egyptian coffin panel prepared for the Book of the Dead exhibition at the British Museum
          • Gebelein Man: virtual autopsy, exploring a natural mummy from early Egypt
        • Aksum
        • Ife
          • Opening of Kingdom of Ife exhibition
          • Ife uncovered
          • Ife remembered
        • Benin
        • Sierra Leone
          • Sowei mask: Spirit of Sierra Leone
      • The Americas
        • South America
          • Inka ushnus: landscape, site and symbol in the Andes
          • From wax to metal: Goldmaking techniques of the ancient Colombians
          • By hammer and fire: Goldmaking techniques of the ancient Colombians
          • Depletion gilding: Goldmaking techniques of the ancient Colombians
        • Mesoamerica and Central America
        • Native North America
      • Asia
        • China
          • Contemporary Chinese Seals by Li Lanqing
          • An installation by Xu Bing at the British Museum
        • Japan
          • Conserving the Gan Ku Tiger scroll painting at the British Museum
        • India
          • The making and worship of Ganesha statues in Maharashtra
        • Afghanistan
          • Crossroads of the Ancient World
          • Hidden treasures revealed
          • Afghanistan: Afghan crown arrives at the British Museum
          • Afghanistan: The lasting legacy of Alexander the Great
          • Afghanistan: returning ivories
        • Thailand
        • Myanmar
        • Indonesia
        • Iran
          • Coins of faith and power at the British Museum
          • Two portraits, two views
          • An English ambassador
          • Takhti: a modern Iranian hero, exhibition at the British Museum
      • Europe
        • Ancient Greece
          • Victorious athlete: The Vaison Daidoumenos
          • Prize amphora showing a chariot race
          • A competitor in the long jump
          • Sprinter on a vase from Rhodes and a bronze running girl
          • Egyptian blue on the Parthenon sculptures
          • Bonnie Greer on the Parthenon sculptures at the British Museum
        • Ancient Rome
          • Hadrian, The imperial palace, Tivoli
          • Hadrian, The power of image
          • Hadrian, Building the wall
        • Medieval era
          • Installation of the St Baudime reliquary in the Treasures of Heaven exhibition at the British Museum
          • Medieval goldsmiths
        • Renaissance
          • The Holy Thorn Reliquary of Jean, duc de Berry
          • Reliquary bust of a companion of Saint Ursula
      • The Middle East
        • Ancient Near East
          • The Babylonian mind
          • Towers of Babel
          • The Cyrus Cylinder and Ancient Persia
        • Islamic Middle East
          • Stories of the modern pilgrimage
          • The Mahmal
      • Oceania
        • Polynesia
          • A welcome to a Maori meeting house
        • Melanesia
        • Micronesia
      • Contemporary Sculptors at the British Museum
        • Marc Quinn's Siren at the British Museum (Kate Moss)
        • Tim Noble and Sue Webster, Dark Stuff, at the British Museum
        • Damien Hirst's Cornucopia at the British Museum
        • Antony Gormley's Case for an Angel I at the British Museum
    • All-Star Orchestra
      • Music basics
        • Notes and rhythm
          • Lesson 1: Note values, duration, and time signatures
          • Lesson 2: Rhythm, dotted notes, ties, and rests
          • Lesson 3: Meters in double and triple time, upbeats
          • Lesson 4: Meters in 6, 9, and 12
          • Lesson 5: Review of time signatures – Simple, compound, and complex
          • Lesson 6: Constant versus changing time, adding triplets, and duplets
        • Reading music
          • Lesson 1: Staff, names of notes, treble clef
          • Lesson 2: Ledger lines and the octave
          • Lesson 3: Bass clef, grand staff and the octave
          • Lesson 4: Reading music in treble clef and the C Major scale
          • Lesson 5: C Major scale in bass clef and reading in bass clef
          • Lesson 6: Alto and tenor clefs.
          • Lesson 7: Accidentals
          • Lesson 8: Natural sign, more on accidentals and key signature
          • Lesson 9: More on sharps and flats
          • Lesson 10: Chromatic scales and the half step
      • Masterpieces old and new
        • Ludwig van Beethoven: Symphony No. 5
          • Ludwig van Beethoven: Symphony No. 5, analysis by Gerard Schwarz (part 1)
          • Ludwig van Beethoven: Symphony No. 5 Analysis by Gerard Schwarz (part 2)
          • Ludwig van Beethoven: Symphony No. 5, analysis by Gerard Schwarz (parts 3 - 4 )
          • Ludwig van Beethoven: Symphony No. 5, a conducting lesson by Gerard Schwarz (1st Movement)
          • Ludwig van Beethoven: Symphony No. 5, an appreciation by Leon Botstein
        • Pyotr Ilyich Tchaikovsky: Symphony No.4
          • Pyotr Ilyich Tchaikovsky: Symphony No.4, analysis by Gerard Schwarz (part 1)
          • Pyotr Ilyich Tchaikovsky: Symphony No.4, analysis by Gerard Schwarz (part 2)
          • Pyotr Ilyich Tchaikovsky: Symphony No.4, analysis by Gerard Schwarz (part 3)
          • Pyotr Ilyich Tchaikovsky: Symphony No.4, analysis by Gerard Schwarz (part 4)
        • Robert Schumann: Symphony No. 3 "Rhenish"
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 1)
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 2)
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 3)
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 4)
          • Robert Schumann: Symphony No. 3, "Rhenish". Analysis by Gerard Schwarz (part 5)
        • Johannes Brahms: Academic Festival Overture
          • Johannes Brahms: "Academic Festival Overture", analysis by Gerard Schwarz
        • Antonín Dvořák: Symphony No. 9 "From the New World"
          • Antonín Dvořák: Symphony No. 9 "From the New World," analysis by Gerard Schwarz (part 1)
          • Antonín Dvořák: Symphony No. 9 "From the New World," analysis by Gerard Schwarz (part 2)
          • Antonín Dvořák: Symphony No. 9 "From the New World," analysis by Gerard Schwarz (part 3)
          • Antonín Dvořák: Symphony No. 9 "From the New World," analysis by Gerard Schwarz (part 4)
          • Antonín Dvořák: Symphony No. 9 "From the New World", a commentary by Joseph Horowitz
        • Gustav Mahler: Symphony No. 2 – 1st Movement
          • Gustav Mahler: Symphony No. 2 – 1st Movement . Analysis by Gerard Schwarz
          • Gustav Mahler – Second Symphony. An appreciation by Gilbert Kaplan
        • Igor Stravinsky: Suite from The Firebird
          • Discovering Stravinsky's "Firebird" : The story and the music
        • Maurice Ravel: Suite No. 2 from Daphnis et Chloe
          • Maurice Ravel: "Daphnis et Chloé", analysis by Gerard Schwarz
        • Dmitri Shostakovich: Symphony No. 5
          • Dmitri Shostakovich: Symphony No. 5. Analysis by Gerard Schwarz (part 1)
          • Dmitri Shostakovich: Symphony No. 5. Analysis by Gerard Schwarz (part 2)
          • Dmitri Shostakovich: Symphony No. 5. Analysis by Gerard Schwarz (part 3)
          • Dmitri Shostakovich: Symphony No. 5. Analysis by Gerard Schwarz (part 4)
        • Introduction to Philip Glass' "Harmonium Mountain"
          • Introduction to Philip Glass' "Harmonium Mountain"
        • Bernard Rands: Adieu
          • "Adieu": The composer and his work
        • Augusta Read Thomas: Of Paradise and Light
          • "Of Paradise and Light": The composer and her work
        • Bright Sheng: Black Swan
          • Bright Sheng: "Black Swan". The composer and his work
        • Richard Danielpour: Piano Concerto No. 4, Movement III
          • Richard Danielpour. Piano Concerto No.4, Movement III: The composer and his work
        • David Stock: Blast!
          • "Blast!": Composer and his work
        • Ellen Taaffe Zwilich: Avanti!
          • "Avanti!": Composer and her work
        • Joseph Schwantner: The Poet's Hour
          • "The Poet's Hour": Composer and his work
      • Instruments of the orchestra
        • Strings
          • Harp: Interview and demonstration with principal Nancy Allen
          • Violin: Interview and demonstration with concertmaster David Kim
          • Viola: Interview and demonstration with principal Rebecca Young
          • Cello: Interview and demonstration with principal Jerry Grossman
          • Bass: Interview and demonstration with principal Alexander Hanna
        • Woodwinds
          • Piccolo: Interview and demonstration with Nadine Asin
          • Flute: Interview and demonstration with principal Jeffrey Khaner
          • E-Flat Clarinet: Interview and demonstration with Jessica Phillips Rieske
          • Clarinet: Interview and demonstration with principal Jon Manasse
          • Bass Clarinet: Interview and demonstration with James Ognibene
          • Oboe: Interview and demonstration with principal John Ferrillo
          • English Horn: Interview and demonstration with Pedro Diaz
          • Bassoon: Interview and demonstration with principal Nancy Goeres
        • Brass
          • French Horn: Interview and demonstration with principal John Cerminaro
          • Trumpet: Interview and demonstration with principal David Bilger
          • Trombone: Interview and demonstration with principal Joseph Alessi
          • Bass Trombone: Interview and demonstration with Denson Paul Pollard
          • Tuba: Interview and demonstration with Chris Olka
        • Percussion
          • Timpani: Interview and demonstration with principal Jauvon Gilliam
          • Piano (as orchestral instrument): Interview and demonstration with Kimberly Russ
          • Percussion: Interview and demonstration with principal Chris Devine and members of the percussion section
    • Stanford School of Medicine
      • Tuberculosis
        • What is TB?
        • TB epidemiology
        • TB pathogenesis
        • Primary and Secondary TB
        • Pulmonary TB
        • Extrapulmonary TB (part 1)
        • Extrapulmonary TB (Part 2)
        • Mantoux test (aka. PPD or TST)
        • Interpreting the PPD
        • Diagnosing active TB
        • Preventing TB transmission
        • Preventing TB using the "4 I's"
        • Treatment of Active TB
        • Drug-resistant TB
        • TB and HIV
      • Influenza
        • Flu Symptoms and Diagnosis
        • Flu Epidemiology
        • Flu Pathophysiology
        • Genetic Shift and Drift
        • 2 Types of Flu Vaccine
        • Flu Vaccine Risks and Benefits
        • Flu Vaccine Excuses
      • Growth and Metabolism
        • Basics of Metabolism
        • Digesting Food
        • Growth Curves
        • Cycle of Malnutrition
      • Asthma
        • Breathing Basics
        • Asthma
      • Breastfeeding
        • Breast Anatomy
        • Breastfeeding - Letdown Reflex
    • Wireless Philosophy
      • Critical thinking
        • Fundamentals
          • Fundamentals: Introduction to Critical Thinking
          • Fundamentals: Deductive Arguments
          • Fundamentals: Abductive Arguments
          • Fundamentals: Necessary and Sufficient Conditions
          • Fundamentals: More About Necessary and Sufficient Conditions
          • Fundamentals: Intrinsic vs. Instrumental Value
          • Fundamentals: Implicit Premise
          • Fundamentals: Justification and Explanation
          • Fundamentals: Normative and Descriptive Claims
          • Fundamentals: Validity
          • Fundamentals: Truth and Validity
          • Fundamentals: Soundness
          • Fundamentals: Bayes' Theorem
        • Fallacies
          • Fallacies: Formal and Informal Fallacies
          • Fallacies: Fallacy of Composition
          • Fallacies: Fallacy of Division
          • Fallacies: Introduction to Ad Hominem
          • Fallacies: Ad Hominem
          • Fallacies: Affirming the Consequent
          • Fallacies: Denying the Antecedent
          • Fallacies: Post Hoc Ergo Propter Hoc
          • Fallacies: Appeal to the People
          • Fallacies: Begging the Question
          • Fallacies: Equivocation
          • Fallacies: Straw Man Fallacy
        • Cognitive biases
          • Cognitive Biases: Alief
          • Cognitive Biases: Anchoring
          • Cognitive Biases: Pricing Biases
          • Cognitive Biases: Reference Dependence and Loss Aversion
          • Cognitive Biases: Mental Accounting
          • Cognitive Biases: Peak-End Effect
          • Cognitive Biases: The GI Joe Fallacy
      • Metaphysics and epistemology
        • Metaphysics
          • Metaphysics: Ship of Theseus
          • Metaphysics: Problem of Free Will
          • Metaphysics: The Grandfather Paradox
          • Metaphysics: Sizes of Infinity, Part 1 (Hilbert's Hotel)
          • Metaphysics: Sizes of Infinity, Part 2 (Getting Real)
        • Mind
          • Mind: Mind-Body Dualism
          • Mind: Personal Identity (The True Self)
          • Mind: Personal Identity (The Essential Moral Self)
          • Mind: Personal Identity (The Narrative Self)
        • Language
          • Language: Meaning and Language
          • Language: Gricean Pragmatics
          • Language: Conditionals, Part 1
          • Language: Conventional Implicature
          • Language: Conditionals, Part 2
          • Language: Conditionals, Part 3
          • Language: Conditionals, Part 4
        • Theory of knowledge
          • Theory of Knowledge: Introduction to Theory of Knowledge
          • Theory of Knowledge: The Problem of Skepticism
          • Theory of Knowledge: Three Responses to Skepticism
          • Theory of Knowledge: New Responses to Skepticism
          • Theory of Knowledge: Analyzing Knowledge #1 (The Gettier Problem)
          • Theory of Knowledge: Analyzing Knowledge #2 (No-False-Lemma and No-Defeater Approaches)
          • Theory of Knowledge: Analyzing Knowledge #3 (Causal and Reliabilist Theories)
          • Theory of Knowledge: Analyzing Knowledge #4 (Tracking Theories)
          • Theory of Knowledge: 'Knowledge First' Epistemology
          • Theory of Knowledge: Virtue Epistemology
        • Epistemology
          • Epistemology: Argument and Evidence
          • Epistemology: Science, Can It Teach Us Everything?
          • Epistemology: The Will to Believe
          • Epistemology: Reason and Faith
          • Epistemology: Sleeping Beauty
          • Epistemology: The Monty Hall Problem
          • Epistemology: Rationality
          • Epistemology: Paradoxes of Perception #1 (Argument from Illusion)
          • Epistemology: Paradoxes of Perception #2 (Argument from Hallucination)
          • Epistemology: The Paradox of the Ravens
          • Epistemology: The Puzzle of Grue
        • Religion
          • Religion: Cosmological Argument, Part 1
          • Religion: Cosmological Argument, Part 2
          • Religion: Classical Theism, Part 1 (Two Conceptions of God)
          • Religion: Classical Theism, Part 2 (In Favor of Classical Theism)
          • Religion: Classical Theism, Part 3 (God's Omnipotence)
          • Religion: Classical Theism, Part 4 (God's Omniscience)
          • Religion: Classical Theism, Part 5 (God’s Goodness and Justice)
          • Religion: Classical Theism, Part 6 (Evil and Goodness in the World)
          • Religion: Classical Theism, Part 7 (Atheistic Arguments from Evil)
      • Value theory
        • Ethics
          • Ethics: The Problem of Evil
          • Ethics: Problem of Evil, Part 1
          • Ethics: Problem of Evil, Part 2
          • Ethics: Problem of Evil, Part 3
          • Ethics: God and Morality, Part 1
          • Ethics: God and Morality, Part 2
          • Ethics: Moral Status
          • Ethics: Killing Animals for Food
          • Ethics: Hedonism and The Experience Machine
          • Ethics: Consequentialism
          • Ethics: Utilitarianism, Part 1
          • Ethics: Utilitarianism, Part 2
          • Ethics: Utilitarianism, Part 3
          • Ethics: The Problem of Moral Luck
          • Ethics: The Nonidentity Problem
          • Ethics: The Nonidentity Problem, Part 2
        • Political
          • Political: Original Position
          • Political: Race and the Carceral State
          • Political: Race and Racist Institutions
          • Political: Racial Ontology
          • Political: Naturalist Theories of Race
          • Political: Government and Marriage (Government's Role)
          • Political: The Prisoner's Dilemma
          • Political: Tragedy of the Commons
          • Political: Collective Action Problems
          • Political: What are Public Goods?
          • Political: Government and Marriage (Minimal Marriage)
          • Political: Government and Marriage (Friends with Legal Benefits)
          • Political: Government and Marriage (Polyamory)
          • Political: Government and Marriage (Just Care)
        • The good life
          • The Good Life: Plato
          • The Good Life: Aristotle
          • The Good Life: Kant
          • The Good Life: Nietzsche
      • History of philosophy
        • Ancient
          • Ancient: Aristotle on the Purpose of Life
          • Ancient: Epicurus’ Cure for Unhappiness
          • Ancient: Mengzi (Mencius) on Human Nature
          • Ancient: Xunzi on Human Nature
        • Early modern
          • Early Modern: Locke on Personal Identity, Part 1
          • Early Modern: Locke on Personal Identity, Part 2
          • Early Modern: Locke on Personal Identity, Part 3
          • Early Modern: Descartes' Cogito Argument
          • Early Modern: Émilie du Châtelet, Part 1
          • Early Modern: Émilie du Châtelet, Part 2
          • Early Modern: Margaret Cavendish, Part 1
          • Early Modern: Margaret Cavendish, Part 2
        • Hume
          • Hume: Skepticism and Induction, Part 1
          • Hume: Skepticism and Induction, Part 2
        • Kant
          • Kant: On Metaphysical Knowledge
          • Kant: On Space, Part 1
          • Kant: On Space, Part 2
        • Contemporary
          • Contemporary: Experimental Philosophy
    • MIT+K12
      • Chemistry and biology
        • Why we fart
        • Solving biology's mysteries with plants
        • Why do we have snot?
        • How do braces work?
        • Squid skin with a mind of its own
        • Why can we regrow a liver (but not a limb)?
        • MIT's choose-your-own: Chemistry adventure
        • The food chain
        • Homeostasis
        • Bread mold kills bacteria
      • Engineering and electronics
        • What Is a semiconductor?
        • Trash to treasure
        • Growing nanotube forests
        • Humanoid robot brains
        • Engineering engines
        • How computers compute
        • MIT explains: How does virtual reality work?
        • How do ships float?
        • MIT explains: How to make a video game
      • Physics
        • The physics of skydiving
        • The physics of invisibility cloaks
        • The science of bouncing
        • How do ships float?
        • Thomas Young's double slit experiment
        • Newton's prism experiment
        • Bridge design and destruction! (part 1)
        • Bridge design and destruction! (part 2)
        • Shifts in equilibrium
        • The Marangoni effect: How to make a soap propelled boat!
        • The invention of the battery
        • The forces on an airplane
        • Bouncing droplets: Superhydrophobic and superhydrophilic surfaces
        • A crash course on indoor flying robots
        • Heat transfer
      • Earth and space
        • How to discover a new planet
        • The science of warp drives
        • Glaciers with chocolate
        • Rock cycle
      • Math
        • What is a fractal (and what are they good for)?
        • Trippy shapes
      • Materials
        • What Is a semiconductor?
        • Trash to treasure
        • Growing nanotube forests
        • What's all the matter? Atoms and molecules
        • What is soap?
        • Flocculation: Making clean water
        • Manufacturing processes: Hands on to hands off
      • Measurement
        • Under pressure
        • Earth's tilt 1: The reason for the seasons
        • Earth's tilt 2: Land of the midnight sun
        • 2D equilibrium -- balancing games
    • LeBron asks
      • LeBron asks
        • LeBron Asks: What muscles do we use when shooting a basket?
        • LeBron Asks: How does shooting a basketball illustrate Newton's 3rd Law?
        • LeBron Asks: If Earth's history were a basketball game, when did humans appear?
        • LeBron Asks: Why does sweating cool you down?
        • LeBron Asks: Why does humidity make it feel hotter?
        • Free-throw probability
        • Three-pointer vs free-throw probability
        • Monte Carlo simulation to answer LeBron's question
    • The Brookings Institution
      • Introduction to health care in the U.S.
        • Introduction to the U.S. health care system
        • Medicare
        • Medicaid
        • The U.S. uninsured population
        • Paying for medicines: copays and deductibles
        • Paying for medicines: tiering and formularies
        • Understanding your medical bill
        • Health care costs
        • Private health insurance
        • Health care for the chronically ill
    • Pixar in a Box
      • Orientation
        • Start here
          • Overview of Pixar in a box
      • The art of storytelling
        • 1. We are all storytellers
          • Introduction to storytelling
          • Your unique perspective
          • Your favorite stories
          • What if...
          • World & character
          • Advice
        • 2. Character
          • Introduction to character
          • Internal vs. external features
          • Wants vs. needs
          • Obstacles
          • Character arc
          • Stakes
          • Advice
        • 3. Story structure
          • Introduction to structure
          • Story spine
          • Theme
          • Act 1
          • Act 2
          • Act 3
          • Advice
        • 4. Visual language
          • Introduction to visual language
          • Line
          • Shape
          • Space
          • Motion
          • Tone
          • Color
          • Advice
        • 5. Film grammar
          • Introduction to film grammar
          • Major vs. minor beats
          • Basic shot types
          • Extreme shots
          • Dynamics shots
          • Storyboarding
          • Advice
        • 6. Pitching and feedback
          • Introduction to pitching and feedback
          • Pitching
          • Giving feedback
          • Digesting feedback
          • Storyreels
          • Advice
          • A thank you from John Lasseter
      • Simulation
        • Hair simulation 101
          • Hair simulation overview
          • Modeling hair
          • Spring-mass system
          • Stiffness and damping
          • Support springs
          • Case study: The Good Dinosaur
          • Getting to know Hayley Iben
        • Code your own simulation
          • Starting with particles
          • 1D spring-mass system
          • Damping
          • 2D spring-mass system
          • Multiple spring-mass system
      • Color science
        • Introduction to color
          • Color Science
          • Spectrum of light
          • RGB color model
          • HSL color model
          • Color contrast
          • Color correction
          • Getting to know Dominic Glynn
        • Color spaces
          • Introduction to color space
          • HSL color space
          • CIE chromaticity diagram
          • Introduction to Gamuts
          • Color grading
      • Virtual cameras
        • How virtual cameras work
          • Introduction to virtual cameras
          • What is a pinhole camera?
          • Focal distance
          • Camera lenses
          • Depth of field
          • Storytelling with camera
          • Getting to know Eben Ostby
        • Mathematics of depth of field
          • Mathematics of pinhole cameras
          • Thin lens approximation
          • Simple lens law
          • Out of focus objects
          • Algebra of focus
      • Effects
        • Introduction to particle systems
          • Effects overview
          • Introduction to particle systems
          • Simulating water
          • Smooth collisions
          • Create water surface using particles
          • What else can you make with particle systems?
          • Getting to know Matt Wong
        • The physics of particle systems
          • Graphing motion over time
          • Vector addition
          • Understanding net forces
          • Force and acceleration
          • Particle collisions
          • Animating particles
      • Patterns
        • Geometry of dinosaur skin
          • Introduction to Patterns
          • What are shading packets?
          • Voronoi Partition
          • Distributing sites randomly
          • Modeling dino skin
          • Getting to know Ana
          • Getting to know Beth
        • Painting with randomness
          • Looking at different resolutions
          • One dimensional noise
          • Perlin noise (1D)
          • Perlin noise (2D)
          • Painting your dino skin
      • Rigging
        • Introduction to rigging
          • Welcome to rigging
          • 1. Rotate deformers
          • 2. Translation deformers
          • 3. Scale deformers
          • 4. Putting everything together
          • 5. Make a face
          • Getting to know Brian Green
        • Code a character
          • 1. Creating objects
          • 3. Putting shapes together
          • 4. Functions
      • Animation
        • Introduction to animation
          • Math meets artistry
          • 1. Straight ahead animation
          • 2. Linear interpolation
          • 3. Bezier curves
          • 4. Squash and stretch
          • Getting to know Rob Jensen
        • Mathematics of animation curves
          • 1. Mathematics of linear interpolation
          • 2. Repeated linear interpolation
          • 3. De Casteljau's algorithm
          • 4. What degree are these curves?
      • Environment modeling
        • Modeling grass with parabolas
          • Introduction to parabolic arcs
          • 1. String art
          • 2. Midpoint formula
          • 3. Parabolic arcs
          • 4. Modeling grass
          • 5. Animating grass
          • Getting to know Tony DeRose
        • Calculating parabolas
          • 1. Weighted average of two points
          • 2. Where is the touching point?
          • 3. Compute the touching point
          • 4. How can we prove this?
          • Bonus: Completing the proof
      • Character modeling
        • Modeling with subdivision surfaces
          • Introduction to subdivision surfaces
          • 1. Split vs. average
          • 2. Subdivide operation
          • 3. Subdividing your own designs
          • 4. Subdivision in 3D
          • Getting to know Alonso Martinez
        • Mathematics of subdivision
          • 1. Weighted average of three points
          • 2. Weighted subdivision
          • 3. Fun with weights
      • Crowds
        • Building crowds
          • Introduction to combinatorics
          • 1. Counting with tables
          • 2. Robot combinations
          • 3. Tree challenge
          • 4. Counting with trees
          • 5. Casting challenge
          • Getting to know Fran Kalal
        • Counting crowds
          • 1. Two headed robots
          • 2. Snake bots
          • 3. Calculating factorials
          • 4. Casting problem
          • 5. Does order matter?
          • 6. Binomial coefficient
      • Sets & staging
        • Geometric transformations
          • Introduction to geometric transformations
          • 1. Coordinate plane
          • 2. Translation
          • 3. Scaling
          • 4. Commutativity
          • 5. Rotation
          • 6. Composite transformations
          • Getting to know Fran Kalal
        • Mathematics of rotation
          • 1. How do we rotate points?
          • 2. Geometry of rotation
          • 3. Completing the proof
      • Rendering
        • Rendering 101
          • Introduction to rendering
          • 1. What is ray tracing?
          • 2. Light reflection
          • 3. Light rays
          • 4. Rendering Mike Wazowski
          • Getting to know Susan Fong
        • Mathematics of rendering
          • 1. Ray tracing intuition
          • 2. Parametric form of a ray
          • 3. Calculate intersection point
          • 4. Using the line equation
          • 5. 3D ray tracing part 1
          • 6. 3D ray tracing part 2
    • The Aspen Institute
      • The founding documents of the United States of America
        • The Declaration of Independence
          • The American Revolution: 1775
          • Background and introduction to the United States Declaration of Independence
          • All men are created equal
          • Life, liberty and the pursuit of happiness
          • Tyranny and despotism
          • First draft of the Declaration of Independence
          • Birth of the US Constitution
        • The Articles of Confederation
          • Articles of Confederation
        • The Constitution of the United States
          • The Constitution and the Constitutional Convention
          • The Constitution and democracy
          • The Constitution and the role of the President
          • The Constitution and slavery part 1
          • The Constitution and slavery part 2
          • The Constitution and proportional representation
          • The Constitution and "We, the People of the United States"
          • The Constitution and the Bill of Rights: Amendments 1-3
          • The Constitution and the Bill of Rights: Amendments 4, 5, 10
      • The founding fathers of the United States of America
        • George Washington
          • The founding fathers of the United States of America: An overview
          • An introduction to George Washington
        • Benjamin Franklin
          • An introduction to Benjamin Franklin
          • Benjamin Franklin becomes a writer
          • Benjamin Franklin and Poor Richard's Almanac
          • Benjamin Franklin the civic leader
          • Benjamin Franklin the inventor
          • Benjamin Franklin as diplomat
          • Takeaways from Benjamin Franklin's life
        • James Madison
          • An introduction to James Madison by historian Joe Ellis
          • Overview: James Madison
          • Madison's role in the Constitutional Convention in 1787
          • The personal aspects of James Madison
          • Dolley Madison
          • Madison and Thomas Jefferson: Their friendship, commonalities, and differences
          • The Federalist Papers
          • Political parties
          • Madison, foreign policy, and declaring war
          • Madison and the Louisiana Purchase
          • Madison and the War of 1812
        • Alexander Hamilton
          • An introduction to Alexander Hamilton
      • The founding mothers of the United States of America
        • Women in early America
          • Women in colonial America
          • Women during the American Revolution
          • Women as builders of the social safety net
          • Women and Thomas Jefferson
        • An introduction to the founding mothers
          • The founding mothers of the United States of America: An overview
          • Martha Washington
          • Deborah Franklin
          • Mercy Otis Warren
        • Women and the expansion of America
          • Women and the expansion of America
      • Civil liberties, privacy, and the Constitution
        • The modern U.S. Supreme Court on privacy and civil liberties
          • How the U.S. Supreme Court approaches questions involving privacy and technology
          • Translating the 4th Amendment into a new technological age: United States v. Jones
          • Digital Privacy: “Third Party Doctrine”, Smith v. Maryland, Riley v. California, People v. Schutter.
          • DNA Surveillance and Maryland v. King
          • NSA Surveillance: Klayman v Obama, Smith v. Maryland
          • U.S. Supreme Court Justice Anthony Kennedy on civil liberties and privacy
          • U.S. Supreme Court Justice Ruth Bader Ginsburg on civil liberties and privacy
        • How the Constitution deals with civil liberties and privacy in an age of technological change
          • How the Constitution deals with civil liberties and privacy in an age of technological change
          • The Preamble of the U.S. Constitution
          • Article I of the U.S. Constitution
          • Article II of the U.S. Constitution
          • Article III of the U.S. Constitution
          • First Amendment to the U.S. Constitution
          • The Fourth Amendment of the U.S. Constitution
        • The Reconstruction Amendments
          • An overview and the 13th Amendment
          • The 14th Amendment
          • The 15th Amendment
      • The American Diplomatic Toolbox
        • Madeleine Albright, former Secretary of State
          • Madeleine Albright on the diplomatic toolbox and foreign policy
          • Albright on Syncopation & the diplomatic toolbox
          • Albright on the diplomatic toolbox and the Balkans I
          • Albright on strategic interests & humanitarian interests, the Balkans II
          • American diplomacy throughout history
          • Albright on 21st century technology and American diplomacy
        • Joe Nye, American political scientist
          • Joseph Nye on power & American diplomacy
          • Nye on power shifts and the 21st century
          • Nye Power and the 2016 election
          • Nye on China and the use of power
        • General Colin Powell
          • General Colin Powell on the American diplomatic toolbox
          • Powell on "The Powell Doctrine"
          • Powell and the First Gulf War
          • Powell on public diplomacy and the 24-hour news cycle
          • Powell on 9/11
          • Powell on the invasion of Panama
          • Powell on the Second Gulf War
        • Alec Ross, former senior advisor for innovation of the U.S. Department of State
          • Alec Ross on defining digital diplomacy
          • Ross on challenges of digital diplomacy
          • Ross on digital tools and the Arab Spring
          • Ross on digital tools and Joseph Kony
          • Ross on the future of digital diplomacy
        • Tara Sonenshine, former Under Secretary of State for Public Diplomacy and Public Affairs
          • Sonenshine on the American diplomatic toolbox and public diplomacy
          • Sonenshine on diplomacy vs. propaganda
          • Sonenshine on the history of diplomacy
          • Sonenshine public diplomacy and the American dream narrative
          • Sonenshine on diaspora & cultural exchanges
          • Sonenshine on public diplomacy from WWI - Cold War
          • Sonenshine on 9/11 and messaging
          • Sonenshine on public diplomacy today – Ukraine
          • Sonenshine on American diplomacy in Cuba
          • Sonenshine on public diplomacy today - Ebola outbreak
          • Sonenshine on public diplomacy today - ISIS
          • Sonenshine on public diplomacy and China
          • Sonenshine on freedom of expression
          • Sonenshine on past failures of American public diplomacy
        • Strobe Talbott, former Deputy Secretary of State
          • Strobe Talbott on the beginning of the Cold War
          • Talbott on containment and the diplomatic toolbox
          • Talbott on arms control and the Cold War
          • Talbott on United States, Russia, and China
          • Talbott on the Reagan Years
          • Talbott on Boris Yeltsin and Bill Clinton
          • Talbott on expanding NATO
          • Talbott on globalization and non-state actors
    • NASA
      • Welcome to NASA Mission Control
        • Welcome
          • Khan Academy & NASA introduction
      • Measuring the Universe
        • Modeling the solar system
          • Conjunctions
          • Lunar eclipse
        • Measuring the solar system
          • Circumference of Earth
          • Scale of solar system
        • Orbital mechanics 1
          • Intro to ellipses
        • Spectroscopy
          • Visible light (1672)
          • Infrared waves (1800)
          • Ultraviolet waves (1801)
          • Spectroscopy
          • Radio waves (1888)
          • X-rays (1895)
          • Gamma waves (1900)
          • Microwaves (1930)
          • Spitzer telescope
      • Exploring the Universe
        • Mars: Modern exploration
          • Viking mission
          • Mars global surveyor
          • Dry ice experiment
          • Pathfinder
          • Mars odyssey
          • Mars express
          • Martian methane
          • Spirit & Opportunity
        • Mars: Ancient observations
          • Is Mars really red?
          • Temperature & atmosphere
        • Curiosity rover: mission briefing
          • Seeking signs of habitability
          • Destination: Gale crater
          • Rover vision
          • ChemCam
          • Curiosity's arm
          • Preparing for landing
          • Entry, descent & landing
        • Curiosity rover: discoveries
          • Curiosity has landed
          • Curiosity descent
          • Systems check
          • Curiosity's first drive
          • Discovery: Streambed
          • First CheMin results
          • Preparation for holidays
          • Calcium-rich deposits found
          • Results of first drilling
          • Mars' bygone atmosphere
          • 'Spring Break' over: commanding resumes
          • Rover ready to switch gears
          • Trek to Mt. Sharp begins
          • Dating younger rocks
          • Curiosity completes its first martian year
          • A softer trek to mount sharp
          • A taste of mount sharp
    • NOVA Labs
      • Evolution
        • Evolution 101
          • Evolution 101
          • Training trees
          • Fossils: Rocking the Earth
          • DNA spells evolution
          • Biogeography: Where life lives
      • Cybersecurity
        • Cybersecurity
          • Cybersecurity 101
          • The secret lives of hackers
          • A cyber privacy parable
          • Cyber codes
      • RNA: the wonder molecule
        • The RNA enigma
        • Protein synthesis in the cellular factory
        • The RNA origin of life
        • Virus wars
      • The Sun and solar storms
        • Anatomy of the Sun
        • The Sun's energy
        • The dynamic Sun
        • Solar wind and storms
        • Earth's magnetic shield
        • The threat to earth
        • The electromagnetic spectrum
        • Solar space telescopes
        • How can I study the Sun?
        • Introducing NOVA's Sun Lab
      • Energy
        • Growing appetites, limited resources
        • Energy defined
        • Putting energy to use
        • Never-ending supply
        • Solar power
        • Wind power
        • Solving the storage problem
        • Toward a smarter grid
      • Clouds
        • The making of a cloud
        • Why so many cloud types?
        • Clouds and weather
        • Severe storms
        • The climate wild card
        • The Coriolis effect
    • Big History Project
      • 1. What Is Big History?
        • 1.0—Welcome to Big History
          • The Big History
          • Crash Course Big History: The Big Bang
          • What Is Big History?
        • 1.1—Scale
          • Powers of Ten™ (1977)
        • 1.2—Origin Stories
        • 1.3—What Are Disciplines?
          • Ways of Knowing – Introduction to Cosmology
          • Ways of Knowing – Introduction to Astrophysics
        • 1.4—My Big History
          • Introduction to Thresholds of Increasing Complexity
        • Glossary
        • Other Materials
          • One Student of Big History
        • Resources
      • 2. The Big Bang
        • 2.0—How Did Our Understanding of the Universe Change?
          • How Did Our View of the Universe Change?
        • 2.1—The Big Bang
          • What Emerged from the Big Bang?
          • Threshold 1: The Big Bang
          • Questions About the Big Bang
        • 2.2—Claim Testing
          • How Do We Decide What to Believe?
        • Glossary
        • Other Materials
        • Resources
      • 3. Stars & Elements
        • 3.0—How Were Stars Formed?
          • How Were Stars Formed?
          • Threshold 2: Stars Light Up
        • 3.1—Creation of Complex Elements
          • Threshold 3: New Chemical Elements
          • What Did Stars Give Us?
          • Navigating the Periodic Table
          • Crash Course Big History: Stars & Galaxies
        • 3.2—Way of Knowing: Stars and Elements
          • Intro to Chemistry
          • Crash Course Chemistry: Periodic Table of Elements
        • Glossary
        • Other Materials
        • Resources
      • 4. Our Solar System & Earth
        • 4.0—Earth & the Formation of Our Solar System
          • Threshold 4: Earth & Solar System
          • How Did Earth and the Solar System Form?
        • 4.1—What Was Young Earth Like?
          • What Was The Young Earth Like?
          • Earth and the Early Atmosphere
        • 4.2—Why Is Plate Tectonics Important?
          • Crash Course Big History: The Solar System & the Earth
          • Our Shifting Globe
        • 4.3—Ways of Knowing: Our Solar System and Earth
          • Introduction to Geology
          • Introduction to the Geologic Time Chart
        • Glossary
        • Other Materials
        • Resources
      • 5. Life
        • 5.0—What Is Life?
          • Threshold 5: Life on Earth Video
          • What is Life?
          • Crash Course Big History: The Origin of Life
        • 5.1—How Did Life Begin and Change?
          • How Did Life Begin and Change?
          • Mini Thresholds Of Life
          • Life In All Its Forms
          • Crash Course Big History: The Evolutionary Epic
        • 5.2—How Do Earth and Life Interact? ?
          • How Do Earth and Life Interact?
          • How We Proved An Asteroid Wiped Out The Dinosaurs
        • 5.3—Ways of Knowing: Life
        • Glossary
        • Other Materials
          • Introduction to Biology
          • Introduction to Conservation Science
          • Life's Evolutionary Story
          • The Carbon Cycle
          • Life in Italian Limestone
        • Resources
      • 6. Early Humans
        • 6.0—How Our Ancestors Evolved
          • Threshold 6: Humans and Collective Learning
          • Crash Course Big History: Human Evolution
        • 6.1—Ways of Knowing: Early Humans
          • Intro to Anthropology
          • Intro to Archaeology
          • Making Stone Tools
        • 6.2—Collective Learning
          • Collective Learning at Scale
          • Early Evidence of Collective Learning
        • 6.3—How Did the First Humans Live?
          • How Did The First Humans Live?
          • From Foraging to Food Shopping
          • Genealogy and Human Ancestry
        • Glossary
        • Other Materials
          • How Did Our Ancestors Evolve?
          • What Makes Humans Different?
          • Migrations and Technological Creativity
        • Resources
      • 7. Agriculture & Civilization
        • 7.0—The Rise of Agriculture
          • Threshold 7: Agriculture
          • Why Was Agriculture So Important?
        • 7.1—The First Cities and States Appear
          • Where and Why Did the First Cities and States Appear?
        • 7.2—Ways of Knowing: Agriculture and Civilization
          • Intro to History
        • Glossary
        • Other Materials
          • Transition to Agriculture
        • Resources
      • 8. Expansion & Interconnection
        • 8.0—Expansion
          • Why Did Civilizations Expand?
          • Crash Course Big History: Modernity
        • 8.1—Exploration & Interconnection
          • How Did the World Become Interconnected?
        • 8.2—Commerce & Collective Learning
          • Systems of Exchange and Trade
        • Glossary
        • Other Materials
          • Re-Creating Pangaea
        • Resources
      • 9. Acceleration
        • 9.0—Acceleration
          • Threshold 8: The Modern Revolution
          • Crash Course World History: The Industrial Revolution
          • How Did Change Accelerate?
        • 9.1—The Anthropocene
        • 9.2—Changing Economies
          • How Was the Modern World Created?
        • Glossary
        • Other Materials
          • Chemistry and Energy
        • Resources
      • 10. The Future
        • 10.0—Looking Back
          • The History of Everything—TED
        • 10.1—The Biosphere
          • Crash Course World History: Globalization II – Good or Bad?
          • The Atmosphere and Climate
        • 10.2—Looking Forward
          • Threshold 9: The Future
          • Bill Gates: Visions of the Future
        • Glossary
        • Other Materials
          • Henry Louis Gates, Jr.: Visions of the Future
          • Sanjayan: Visions of the Future
        • Resources
    • Silicon Schools Fund and Clayton Christensen Institute
      • Introduction to blended learning
        • An overview of blended learning
          • The case for blended learning
          • The definition of blended learning
        • Four different blended learning models
          • Four different blended learning models
          • The flipped classroom model
          • The station rotation model
          • The lab rotation model
          • The flex model
      • Teaching in a blended learning environment – rethinking the role of the teacher
        • An overview of the teacher experience
          • Teacher role: intro and big ideas
          • Rethinking the role of the teacher and the key shifts a teacher experiences
        • Case studies: teaching in a blended learning environment
          • Case study: teaching in a flex model at Summit Public Schools
          • Case study #2: Teaching in a Station Rotation model at KIPP LA
          • Case study #3: Teaching in Lab Rotation model at Navigator Schools
          • Helping teachers make the change
        • The moves of a blended learning teacher
          • Part 1: Tips from blended-learning teachers about the structures and systems that make blended work
          • Part 2: Tips from blended-learning teachers about the structures and systems that make blended work
      • Redesigning a school toward blended learning
        • School redesign overview
          • School redesign - intro and big ideas
          • The assumptions that limit innovation -- the school year, daily schedule, class size
        • Challenging existing assumptions
          • New ways to structure the school day and year in blended-learning settings
          • New ways to group students and organize staff
        • Leading change in blended learning
          • Leading organizational change to transform existing schools or launch new schools
          • Driving the process of innovation
      • Making aligned hardware, software, and space decisions in a blended learning environment
        • Hardware, software and facilities decisions overview
          • Hardware, software and space: intro and big ideas
        • Blended learning software
          • The different types of software used in blended learning
          • Criteria to pick software
          • The challenges to picking software
          • Resources to learn about software: the "consumer reports" of ed tech
          • Building customized software and solutions
        • Blended learning hardware and infrastructure
          • Putting the right hardware and infrastructure in place
        • Blended learning facilities and furniture
          • Making sure the facilities and furniture support your educational model
    • Dartmouth College
      • Learn Algorithms with Dartmouth College
        • What is an algorithm and why should you care?
    • Code.org
      • How the Internet Works
        • Wires, cables, and WiFi
        • IP addresses and DNS
        • Packet, routers, and reliability
        • HTTP and HTML
        • Encryption and public keys
        • Cybersecurity and crime
    • Breakthrough Junior Challenge
      • 2015 Challenge
        • 2015 Challenge — Mathematics
          • Infinitesimals and non-standard analysis
          • Trinary
          • The fourier transform
        • 2015 Challenge — Life sciences
          • Phantom limbs
          • Cell differentiation and gene expression
          • RNA interference
          • Oncolytic virotherapy
          • Epigenetics
          • Cell Communication
          • A novel form of cancer therapy: Targeting metastatic EMT
          • CRISPR Cas-9
          • CRISPR
          • Combating MRSA with echinacea
          • Optogenetics
        • 2015 Challenge — Physics
          • Special theory of relativity - 2015 Challenge Winner!
          • A visualization of special relativity
          • Chromatic aberration
          • Orbital mechanics
          • What is light?
          • Special and general relativity
          • The science of fireworks
          • Quantum tunneling
          • Antimatter
          • Special Relativity
          • What Einstein missed: The EPR paradox
          • Virtual particles and gravity
          • Interstellar and Hawking radiation
          • All about superconductors
          • Entropy and the direction of time
          • Antimatter, black holes, and my own theory
          • Magnetism, light and the magneto optic Kerr effect
          • The theory of everything
    • Science & engineering partners
    • Arts & humanities partners
    • Computing partners
    • 49ers STEAM education
      • 49ers ask
        • 49ers ask
          • Why is it good to throw a spiral?
          • To kick or not to kick?
          • How many footballs would fit in Levi’s® Stadium?
          • Can fans affect the outcome of a football game?
          • Football or pigskin?
      • Science behind the game
        • Force and motion
          • Newton's first law of motion
          • Newton's second law of motion
          • Newton's third law of motion
        • Environmental sustainability
          • Environmental sustainability
          • Solar Power
      • Technological advancements
        • How technology impacts the game
          • Technology advances
      • Gridiron engineering
        • Innovations in football equipment
          • Engineering design process
        • Stadium structural engineering
      • Art in football
        • Elements of art & design in football
      • Geometry & statistics in football
        • The use of angles in the game
          • Angles: introduction
        • Statistics in football
          • Statistics and analytics in football
          • Statistical questions
      • Related Khan Academy videos
        • Related Khan Academy videos
          • Wireless technology
          • Truss basics
          • What is art?
          • Materials in football
    • 23andMe
      • Human prehistory
        • Human prehistory 101: Prologue
        • Human prehistory 101 part 1: Out of (eastern) Africa
        • Human prehistory 101 part 2: Weathering the storm
        • Human prehistory 101 part 3: Agriculture rocks our world
        • Human prehistory 101: Epilogue
    • LearnStorm growth mindset activities
      • Activity #1: The truth about your brain
        • Get started!
      • Activity #2: Supercharge your growth
        • Get started!
      • Activity #3: Which voice will you hear?
        • Get started!
      • Activity #4: Turning mistakes into opportunities
        • Get started!
      • Activity #5: Overcoming frustration
        • Get started!
      • Activity #6: The path forward
        • Get started!
      • Growth mindset teacher resources
        • Growth mindset activity teacher resources
      • Growth mindset activity submission highlights
        • Growth mindset activity submission highlights
  • Talks and interviews
    • Conversations with Sal: Talks and presentations
      • Salman Khan: Let's use video to reinvent education | TED Talk | 2011
      • Radio interview: Sal on AirTalk talking about his new book
      • Salman Khan on Charlie Rose 2013
      • Sal Khan on Digital and Physical Learning
      • Year 2060: Education Predictions
      • The Gates Notes: Sal on Khan Academy
      • Khan Academy Computer Science Launch
      • MIT 2012 Commencement Address
      • Harvard Business School Class Day 2014
      • Salman Khan at Rice University's 2012 commencement
      • A Conversation with US Secretary of Education Arne Duncan
      • Authors@Google: Salman Khan
      • Khan Academy Vision and Social Return
      • Using Khan Academy
      • Sal Khan: Let's teach for mastery -- not test scores | TED Talk | 2015
    • Schools using Khan Academy
      • Khan Academy's Discovery Lab - Summer 2012
      • Khan Academy at Eastside College Prep (grades 6-8)
      • Khan Academy in Los Altos School District
      • Khan Academy at Summit Public Schools
      • Khan Academy at KIPP
      • Khan Academy at Oakland Unity High School
      • Khan Academy at Marlborough School
      • The Gates Notes: Administrators in Los Altos
      • The Gates Notes: Teachers in Los Altos
      • The Gates Notes: Students in Los Altos
      • The Gates Notes: Insights into students' progress
    • Khan Academy chats
      • Reid Hoffman - Founder of LinkedIn
      • Thomas Friedman - Author
      • Walter Isaacson - President and CEO of the Aspen Institute
      • Elon Musk - CEO of Tesla Motors and SpaceX
      • Davis Guggenheim - Filmmaker
      • Scott Cook - Founder and Chairman of the Executive Committee, Intuit
      • Angela Ahrendts - Former CEO of Burberry
      • Sean O'Sullivan - Founder of SOSventures
      • Drew Houston - CEO and Founder of Dropbox
      • A conversation with Eric Schmidt and Jonathan Rosenberg
      • Sal and Francis Ford Coppola fireside chat
  • College, careers, and more
    • College admissions
      • Getting started
        • Introduction: College admissions
          • Sal Khan's story: College admissions
        • Importance of college
          • The college experience
          • Student story: College increases confidence
          • Student story: College expands opportunities
          • Student story: College offers exposure to diversity
          • Student story: College brings new friends, learning, and freedom
        • Access to college
          • Who can succeed in college?
          • Student story: Overcoming cultural obstacles to college
          • Student story: Overcoming anxiety around not fitting in as an obstacle to college
          • Student story: Overcoming time management obstacles to college
          • Student story: Overcoming social obstacles to college
          • Student story: Overcoming lack of high school support as an obstacle to college
          • Student story: Overcoming immigration obstacles to college
          • Student story: Overcoming financial obstacles to college
          • Student story: Overcoming the financial aid process as an obstacle to college
          • Student story: Overcoming family obstacles to college
      • Making high school count
        • Introduction: Making high school count
          • Sal Khan's story: Making high school count
          • Overview: Making high school count
        • High school classes
          • Making the most of high school classes
          • Student story: High school classes
        • Extracurricular and leadership activities
          • Engaging in extracurricular activities
          • Student story: Extracurriculars that are leadership opportunities
          • Student story: Extracurriculars that are nontraditional
          • Student story: Extracurriculars that are self created
          • Student story: Extracurriculars that are a passion
          • Student story: Extracurriculars that carry over from high school to college
        • Standardized tests
          • Facing the standardized tests
          • Official SAT Practice on Khan Academy
          • Deciding whether to retake the SAT
          • Overview of the ACT
          • Deciding whether to retake the ACT
          • Student story: Standardized tests
      • Exploring college options
        • Introduction: Exploring college options
          • Sal Khan's story: Exploring college options
          • Overview: Exploring college options
        • College search: Type of college
          • Comparing vocational vs 2 year vs 4 year colleges
          • Comparing public vs private colleges
          • Comparing universities vs liberal arts colleges
          • Comparing nonprofit vs. for profit colleges
          • Comparing highly selective vs selective vs nonselective colleges
        • College search: Other important choices
          • Comparing colleges based on financial aid policies
          • Comparing colleges based on special focus or affiliation
          • Comparing colleges based on diversity
          • Comparing colleges based on location, size, and housing
          • Comparing colleges based on campus activities
          • Comparing colleges based on majors offered
          • Student story: Prioritizing financial aid in the college search
          • Student story: Prioritizing location in the college search
          • Student story: Prioritizing financial aid, major, and location in the college search
          • Student story: Prioritizing size, campus, major, and selectivity in the college search
          • Student story: Prioritizing size, selectivity, diversity and financial aid in the college search
        • Campus visit
          • Visiting campus
      • Applying to college
        • Introduction: Applying to college
          • Sal Khan's story: Applying to college
          • Overview: Applying to college
        • College application process
          • Deciding when to apply: Early vs regular decision
          • Applying to the right number of colleges
          • Receiving an admissions decision: Admit, deny, or waitlist
        • Admissions essays
          • Writing a strong college admissions essay
          • Avoiding common admissions essay mistakes
          • Brainstorming tips for your college essay
          • How formal should the tone of your college essay be?
          • Taking your college essay to the next level
          • Student story: Admissions essay about a formative experience
          • Student story: Admissions essay about personal identity
          • Student story: Admissions essay about community impact
          • Student story: Admissions essay about a past mistake
          • Student story: Admissions essay about a meaningful poem
        • Letters of recommendation
          • Requesting letters of recommendation
          • Elements of a strong recommendation letter
        • Admissions interviews
          • Succeeding at the college admissions interview
          • Student story: Admissions interview
        • Applying as a homeschooler
          • Student Story: Applying as a homeschooler
        • Applying as a transfer student
      • Paying for college
        • Introduction: Paying for college
          • Sal Khan's story: Paying for college
          • Overview: Paying for college
          • Best strategies for funding college
          • 4 Most Important Considerations in Analyzing College Costs
          • What to do when parents are divorced, and in situations with step-parents, foster care, etc.
          • How do I know if I qualify for need-based aid?
          • Cost of in-state vs out-of-state tuition
          • Watch out for scholarship displacement!!
          • A message to parents on paying for college
        • Financial aid application process
          • Overview of FAFSA and CSS PROFILE
          • Applying for financial aid when facing immigration challenges
        • Grants and scholarships
          • Information about athletic scholarships
          • Types of grants and scholarships
          • Searching for scholarships
          • Student story: Searching for scholarships
          • Student story: Applying for scholarships
          • Student story: Relying on scholarships when facing immigration challenges
        • Work study
          • Working during college
          • Student story: Work study as an opportunity to learn
          • Student story: Work study as a study hall
        • Loans
          • Benefits and drawbacks of college loans
          • Types of college loans
          • Managing your debt level
          • Paying back your loans
        • Financial aid packages
          • Financial aid package components: Part 1
          • Financial aid package example: Part 2
          • Financial aid package comparison: Part 3
          • Appealing a financial aid package
      • Wrapping up
        • Introduction: Wrapping up
          • Sal Khan's story: Wrapping up
        • Post college acceptance
          • Selecting your college
          • Making the college transition
        • Special thanks to contributors
      • Life after college
        • Your career and finances
    • Careers
      • Introduction
        • Careers series introduction
          • Sal Khan announces new careers and personal finance content
      • Serve your community
        • City planner
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • District representative
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Working in public service
        • Firefighter
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Police officer
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Volunteer coordinator
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Boundaries and self-care in a direct service career
      • Support health and wellness
        • Clinical nurse specialist
          • What I do and how much I make
          • How I got my job and where I'm going
          • Our budget and planning for our future
        • Population health director
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Learning from every job
      • Start a business
        • Food entrepreneur
          • What I do and how much I make
          • How I got my job and where I'm going
          • Our budget and planning for our future
        • Food truck manager and chef
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Freelance journalist and podcaster
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Salary negotiation
        • Freelance product design consultant
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Hairstylist and salon owner
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Starting and owning a salon business
        • Startup co-founder and CEO
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Advice for running a startup
      • Design, create, perform
        • Animation director
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Author
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Working with an agent
        • Chef de cuisine
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Composer
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Advice on pursuing a creative career
        • Dancer
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Freelance product design consultant
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Product designer
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Aligning career with personal values
      • Conduct research
        • Judicial staff attorney
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Lab technician
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • User experience researcher
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
      • Educate
        • Education resource specialist (special education teacher)
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Work life balance and parental leave
        • Instructional designer
          • What I do and how much I make
          • My budget and planning for the future
        • School social worker and BUILD teacher
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
      • Travel and explore
        • Commercial airline pilot
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
      • Build and fix things
        • Architectural designer
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Senior software engineer
          • What I do and how much I make
          • How I got my job and where I'm going
          • Our budget and planning for our future
        • Site reliability engineer
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Welder
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
      • Analyze and advise
        • Consultant
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • The pros and cons of traveling for work
      • Manage people and processes
        • Assistant hotel manager
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Business office manager
          • What I do and how much I make
          • How I got my job and where I'm going
        • Chef de cuisine
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Customer success manager
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • My educational choices
        • Director of products
          • What I do and how much I make
          • How I got my job and where I'm going
        • Environmental specialist
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
        • Nonprofit program director
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Paying for college without debt
        • Population health director
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
          • Learning from every job
        • Senior product manager
          • What I do and how much I make
          • How I got my job and where I'm going
          • Our budget and planning for our future
        • VP of wealth management
          • What I do and how much I make
          • How I got my job and where I'm going
          • My budget and planning for the future
    • Personal finance
      • Saving and budgeting
        • Advice for individuals
          • What is the bare minimum I need to know to avoid ruining my life?
          • Should I spend more money on experiences or things?
          • What advice do you have for new grads?
          • How much of my pay should I spend?
        • Perspectives for families
          • What does money have to do with getting married?
          • How can parents talk to kids about money?
          • How can kids talk to parents about money?
          • How can parents talk to their kids about the economy?
          • Math behind saving for college
        • Tips for tracking and saving money
          • Options for tracking and managing your spending - by Better Money Habits
          • Easy tips to save money every day - by Better Money Habits
          • Ways to save money on your monthly bills - by Better Money Habits
      • Interest and debt
        • Compound interest basics
          • Compound interest introduction
          • The rule of 72 for compound interest
        • Interest basics
          • Introduction to interest
          • Interest (part 2)
          • Why interest rates go up and down - by Better Money Habits
        • Credit basics
          • FICO scores and credit bureaus
        • Credit cards and loans
          • Annual percentage rate (APR) and effective APR
          • How credit card interest is calculated
          • Institutional roles in issuing and processing credit cards
          • Payday loans
        • Debt repayment
          • High rate vs snowball method
        • Personal bankruptcy
          • Personal bankruptcy: Chapters 7 and 13
      • Investment vehicles and retirement
        • Retirement accounts: IRAs and 401ks
          • Traditional IRAs
          • Roth IRAs
          • 401(k)s
          • Managing your 401(k) when you change jobs
          • Building a foundation for retirement - by Better Money Habits
          • The effect of time on your retirement account - by Better Money Habits
        • Mutual funds and ETFs
          • Open-ended mutual fund (part 1)
          • Open-end mutual fund redemptions (part 2)
          • Closed-end mutual funds
          • Exchange traded funds (ETFs)
          • Ponzi schemes
        • Stocks
          • What it means to buy a company's stock
          • Bonds vs. stocks
        • Life insurance
          • Term life insurance and death probability
      • Income
        • Paycheck
          • Anatomy of a paycheck
          • Intro to the W-4
          • Impact of W-4 allowances on paycheck
      • Housing
        • Renting vs. buying a home
          • Renting versus buying a home
          • The true cost of renting a place - by Better Money Habits
          • Is buying a home always better?
          • What happens when housing depreciates
        • Mortgages
          • Introduction to mortgage loans
          • Mortgage interest rates
          • Impact of credit score on interest rate
          • Short sale basics
          • Adjustable rate mortgages ARMs
          • Hybrid ARM
          • Balloon payment mortgage
          • Finite geometric series word problem: mortgage
        • Home buying process
          • Titles and deeds in real estate
          • Title insurance
          • Making an offer on a home
          • Escrow
          • Types of escrow in real estate
        • Home equity and personal balance sheets
          • Introduction to balance sheets
          • More on balance sheets and equity
          • Home equity loans
      • Auto
        • Buying vs. leasing
          • Buying vs. leasing a car
          • When leasing might come out better than buying
      • Taxes
        • Personal taxes overview
          • Tax deductions introduction
          • Standard and itemized tax deductions
          • AMT overview
          • Alternative minimum tax
          • Estate tax introduction
          • Tax brackets and progressive taxation
          • Calculating federal taxes and take home pay
          • Calculating state taxes and take home pay
          • Marriage penalty
          • Married taxes clarification
        • Tax forms
          • Intro to the W-2
          • Filling out 1040EZ
          • Preparing your tax return for the first time - by Better Money Habits
        • Protecting your personal information
          • Protecting your identity at tax time - by Better Money Habits
      • Paying for college
        • Introduction: Paying for college
          • Sal Khan's story: Paying for college
          • Overview: Paying for college
          • Best strategies for funding college
          • 4 Most Important Considerations in Analyzing College Costs
          • What to do when parents are divorced, and in situations with step-parents, foster care, etc.
          • How do I know if I qualify for need-based aid?
          • Cost of in-state vs out-of-state tuition
          • Watch out for scholarship displacement!!
          • A message to parents on paying for college
        • Financial aid application process
          • Overview of FAFSA and CSS PROFILE
          • Applying for financial aid when facing immigration challenges
        • Grants and scholarships
          • Information about athletic scholarships
          • Types of grants and scholarships
          • Searching for scholarships
          • Student story: Searching for scholarships
          • Student story: Applying for scholarships
          • Student story: Relying on scholarships when facing immigration challenges
        • Work study
          • Working during college
          • Student story: Work study as an opportunity to learn
          • Student story: Work study as a study hall
        • Loans
          • Benefits and drawbacks of college loans
          • Types of college loans
          • Managing your debt level
          • Paying back your loans
          • Student loan repayment options - by Better Money Habits
          • Consolidating student loans - by Better Money Habits
          • Income-based student loan repayment plans - by Better Money Habits
          • How loan deferment works
        • Financial aid packages
          • Financial aid package components: Part 1
          • Financial aid package example: Part 2
          • Financial aid package comparison: Part 3
          • Appealing a financial aid package
      • Keeping your information safe
        • Password strength and other strategies
          • Math behind password security
          • Protecting your identity at tax time - by Better Money Habits
    • Entrepreneurship
      • Interviews with entrepreneurs
        • Sal chats with entrepreneurs
          • Elon Musk - CEO of Tesla Motors and SpaceX
          • Reid Hoffman - Founder of LinkedIn
          • Scott Cook - Founder and Chairman of the Executive Committee, Intuit
          • Angela Ahrendts - Former CEO of Burberry
          • Sean O'Sullivan - Founder of SOSventures
          • Drew Houston - CEO and Founder of Dropbox
          • A conversation with Eric Schmidt and Jonathan Rosenberg
        • Richard Branson - Chairman of the Virgin Group
          • Richard Branson - Chairman of the Virgin Group
        • Marc Ecko - Founder of Ecko Unlimited
          • Marc Ecko - Founder of Ecko Unlimited
          • Marc Ecko - Failures are lessons for future success
        • Philip Rosedale - Founder and Chairman of Second Life
          • Philip Rosedale - Founder of Coffee and Power
          • Philip Rosedale - Founder and Chairman of Second Life
          • Philip Rosedale - Evolving ideas vs. new ideas
        • Zach Kaplan - CEO of Inventables
          • Zach Kaplan - CEO of Inventables
          • Zach Kaplan - Digital manufacturing revolution
        • T.A. McCann - Founder and CEO of Gist
          • T.A. McCann - Founder and CEO of Gist
          • T.A. McCann - Translating from ideas to execution
        • Ben Milne - CEO of Dwolla
          • Ben Milne - CEO of Dwolla
          • Ben Milne - Importance of Curiosity
        • Giles Shih - President and CEO of BioResource International
          • Giles Shih - President and CEO of BioResource International
          • Giles Shih - The importance of education
          • Giles Shih - Making an idea into a business
        • Danny O'Neill - President of The Roasterie
          • Danny O'Neill - President of The Roasterie
          • Danny O'Neill - Getting started
          • Danny O'Neill - Finding the perfect customer
          • Danny O'Neill - Differentiation in your market
        • Lara Morgan - Founder of Pacific Direct
          • Lara Morgan - Founder of Pacific Direct
        • Dave Smith - CEO & Founder of TekScape IT
          • Developing Value for the Product
          • Engineering a Solution to Every Problem
          • Shifting from Loyalty to Accountability
          • Top of the World, Ma
        • Jason Christiansen - President & CEO of Rigid Industries
          • Turning a Hobby into a Bright Business
          • The World of Copy Cat Companies
          • Baseball is Just a Game
        • Linda Jeschofnig - Co-founder of Hands-On Labs
          • Creating a Poof and Not a Giant Kaboom
          • A Passion for Science Education
        • LaKeshia Grant - CEO & Founder of Virtual Enterprise Architects
          • Being Heard and Giving Others a Voice
          • Business and Family
        • Dave Gilboa & Neil Blumenthal - Co-founders & Co-CEOs of Warby Parker
          • A fix for a broken industry
          • It was just mayhem
          • Starting and staying in business with friends
        • Calvin Carter - Founder of Bottle Rocket Apps
          • A 2 a.m. wake up call
          • Living the company’s values
          • The excitement of making something
          • Making money making apps
        • Beth Schmidt - Founder of Wishbone.org
          • Student wishes come true
          • There better be an authentic need
          • An invitation for innovation
        • Toby Rush - CEO & Founder of EyeVerify
          • University scientist meets entrepreneur
          • Staying grounded
  • Talent search
    • India talent search 2016
      • Overview
    • India talent search 2017
      • Overview
  • Resources
    • Explore new features
    • K-12 teachers
      • Getting started
      • Our content
      • Assignments
      • Implementation
      • Math teachers
      • Official SAT Practice
      • Safety, privacy, and additional resources
    • Parents and mentors
      • Khan Academy accounts
      • Helping your child
        • You can learn anything
        • See your student's experience
      • Privacy and security
      • Homeschool with Khan Academy
        • Homeschool testimonial: Tim Peters
      • Need more help?
    • Out-of-school programs
      • Khan Academy in out-of-school programs
      • Out-of-school programs: case studies
        • Boys & Girls Club of Chaffee County, CO
        • Numeric in South Africa
      • Teaching Computer Programming
        • Video tour: Teaching programming in the classroom
    • Recursos
      • Cadastrando-se como professor
      • Como criar turmas
      • Como recomendar conteúdos
  • MAP Recommended Practice
    • Geometry < 160
      • Reason with Shapes, Attributes, & Coordinate Plane
    • Geometry 161-178
      • Geometric Measurement and Relationships
      • Reason with Shapes, Attributes, & Coordinate Plane
    • Geometry 179-191
      • Geometric Measurement and Relationships
      • Reason with Shapes, Attributes, & Coordinate Plane
    • Geometry 192-202
      • Geometric Measurement and Relationships
      • Reason with Shapes, Attributes, & Coordinate Plane
    • Geometry 203-212
      • Geometric Measurement and Relationships
      • Reason with Shapes, Attributes, & Coordinate Plane
    • Geometry 213-219
      • Geometric Measurement and Relationships
      • Reason with Shapes, Attributes, & Coordinate Plane
    • Geometry 220-223
      • Geometric Measurement and Relationships
      • Reason with Shapes, Attributes, & Coordinate Plane
    • Geometry 224-227
      • Geometric Measurement and Relationships
    • Geometry 228-230
      • Congruence, Similarity, Right Triangles, & Trig
      • Geometric Measurement and Relationships
    • Geometry > 231
      • Congruence, Similarity, Right Triangles, & Trig
      • Geometric Measurement and Relationships
    • Operations and Algebra < 160
      • Represent and Solve Problems
    • Operations and Algebra 161-178
      • Represent and Solve Problems
      • Expressions and Equations
    • Operations and Algebra 179-191
      • Represent and Solve Problems
      • Expressions and Equations
    • Operations and Algebra 192-202
      • Represent and Solve Problems
      • Analyze Patterns and Relationships
      • Expressions and Equations
    • Operations and Algebra 203-212
      • Represent and Solve Problems
      • Analyze Patterns and Relationships
      • Expressions and Equations
    • Operations and Algebra 213-219
      • Represent and Solve Problems
      • Analyze Patterns and Relationships
      • Expressions and Equations
      • Use Functions to Model Relationships
    • Operations and Algebra 220-223
      • Represent and Solve Problems
      • Expressions and Equations
    • Operations and Algebra 224-227
      • Represent and Solve Problems
      • Expressions and Equations
    • Operations and Algebra 228-230
      • Expressions and Equations
      • Use Functions to Model Relationships
    • Operations and Algebra > 231
      • Represent and Solve Problems
      • Expressions and Equations
      • Use Functions to Model Relationships
    • Measurement and Data < 160
      • Represent and Interpret Data
      • Geometric Measurement and Problem Solving
    • Measurement and Data 161-178
      • Represent and Interpret Data
      • Geometric Measurement and Problem Solving
    • Measurement and Data 179-191
      • Represent and Interpret Data
      • Geometric Measurement and Problem Solving
    • Measurement and Data 192-202
      • Represent and Interpret Data
      • Geometric Measurement and Problem Solving
    • Measurement and Data 203-212
      • Represent and Interpret Data
      • Geometric Measurement and Problem Solving
    • Measurement and Data 213-219
      • Represent and Interpret Data
      • Geometric Measurement and Problem Solving
    • Measurement and Data 220-223
      • Represent and Interpret Data
      • Geometric Measurement and Problem Solving
    • Measurement and Data 224-227
      • Geometric Measurement and Problem Solving
    • Numbers and Operations < 160
      • Understand Place Value, Counting, and Cardinality
    • Numbers and Operations 161-178
      • Understand Place Value, Counting, and Cardinality
      • Base Ten
    • Numbers and Operations 179-191
      • Understand Place Value, Counting, and Cardinality
      • Base Ten
    • Numbers and Operations 192-202
      • Understand Place Value, Counting, and Cardinality
      • Base Ten
      • Fractions
    • Numbers and Operations 203-212
      • Understand Place Value, Counting, and Cardinality
      • Base Ten
      • Fractions
    • Numbers and Operations 213-219
      • Understand Place Value, Counting, and Cardinality
      • Base Ten
      • Fractions
    • Numbers and Operations 220-223
      • Base Ten
      • Fractions
    • Numbers and Operations 224-227
      • Base Ten
      • Fractions
    • Real and Complex Numbers < 160
      • Extend and Use Properties
      • Perform Operations
    • Real and Complex Numbers 161-178
      • Extend and Use Properties
      • Perform Operations
    • Real and Complex Numbers 179-191
      • Perform Operations
      • Extend and Use Properties
    • Real and Complex Numbers 192-202
      • Extend and Use Properties
      • Perform Operations
    • Real and Complex Numbers 203-212
      • Extend and Use Properties
      • Perform Operations
      • Ratios and Proportional Relationships
    • Real and Complex Numbers 213-219
      • Perform Operations
      • Extend and Use Properties
      • Ratios and Proportional Relationships
    • Real and Complex Numbers 220-223
      • Extend and Use Properties
      • Perform Operations
      • Ratios and Proportional Relationships
    • Real and Complex Numbers 224-227
      • Ratios and Proportional Relationships
      • Perform Operations
    • Real and Complex Numbers 228-230
      • Extend and Use Properties
    • Real and Complex Numbers > 231
      • Extend and Use Properties
      • Perform Operations
    • Statistics and Probability 161-178
      • Interpreting Categorical and Quantitative Data
    • Statistics and Probability 179-191
      • Interpreting Categorical and Quantitative Data
    • Statistics and Probability 192-202
      • Interpreting Categorical and Quantitative Data
    • Statistics and Probability 203-212
      • Interpreting Categorical and Quantitative Data
    • Statistics and Probability 213-219
      • Interpreting Categorical and Quantitative Data
    • Statistics and Probability 220-223
      • Interpreting Categorical and Quantitative Data
    • Statistics and Probability 224-227
      • Using Sampling and Probability to Make Decisions
    • Statistics and Probability 228-230
      • Interpreting Categorical and Quantitative Data
    • Statistics and Probability > 231
      • Interpreting Categorical and Quantitative Data
      • Using Sampling and Probability to Make Decisions