In this section, you will learn how to find limits of rational functions at points of discontinuity. Before attempting this section, you should know how to factor polynomials, such as quadratics, difference of squares, and difference of cubes.

- Graph theory
- \(x^y = y^x\)
- Conic sections
- Recreational mathematics
- Intro to radians
- Algebra 2
- Group theory
- Trigonometry
- Countable and uncountable sets
- Precalc
- Permutation of the natural numbers
- Sphinx
- Abstract algebra
- Complex analysis
- Real analysis
- Undergrad probability theory
- Linear algebra
- Elementary number theory
- Differential equations
- Generating functions
- Discrete mathematics
- Proof by induction
- Bridge course
- Logarithms
- Zero factorial
- Statistics and probability
- Tidbits on Aaron numbers
- Duodecimal
- Factoring and solving quadratics
- Completing the square
- Euler's formula and identity
- Infinite series
- Teaching algebra
- Chinese remainder theorem
- Modular arithmetic
- Intro
- Inverse functions (algebra 1)
- Polynomial long division
- Math education
- Solving right triangles
- Rationalizing the denominator
- Pythagorean theorem
- Algebra 1
- Solving linear equations
- Solving linear systems of equations in two variables by substitution
- Other resources
- Intro to irrational numbers
- Other resources
- Sum and product of roots of polynomials
- 3rd grade
- Subtracting within 1000
- Knot theory
- Linear systems of equations in two variables
- Visual proof
- Evaluating infinite series
- Continued fractions
- \(\displaystyle \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \ldots\)
- Proofs involving infinitely many digits
- Pigeonhole principle
- Eulerian cycles and paths
- Hamiltonian cycles and paths
- High school courses
- Law of cosines
- Find a length using the law of cosines
- Law of sines
- Harmonic series
- Counter-intuitive mathematics
- Magic hexagon (recreational mathematics)
- Goldbach's conjecture
- Other resources
- Other resources
- Other resources
- Calculus 1
- Resources for undergrads
- Other resources
- Solving two-step equations
- Solving multi-step equations
- Solving and graphing linear inequalities
- Solving and graphing one-step inequalities in one variable
- Solving and graphing two-step inequalities
- Other resources
- Order of operations (grade 6)
- Factorials, permutations, and combinations
- Intro to \(e\)
- High school geometry
- History of \(\pi\)
- History of mathematics
- History of trigonometry
- Math in art
- Learning mathematics
- Complex numbers
- Applications of complex numbers
- Calculus 2
- Calculus 3
- Mathematical logic
- Teaching mathematics
- Resources for graduates
- Proving Vandermonde's identity
- Ring theory
- History of calculus
- Applications of linear algebra
- Where does \(\pi\) appear?
- Where does \(e\) appear?
- Applications of statistics and probability
- Applications to criminal investigation
- Misleading statistics
- Euler characteristic
- Algebraic geometry
- Biographies
- Manipulatives
- Triangle midsegment theorem
- Cuisenaire rods
- Applications of mathematics
- Plotting integer coordinates
- Multilink Cubes
- Other resources
- Other resources
- Using the formula
- Squaring the square
- Combining like terms (abstract method)
- Active learning
- Combinatorics
- Simplifying square root expressions
- Parallelogram law
- Euler line
- Rational exponents and radicals
- Mayan mathematics
- Unsolved problems
- Inscribed square problem
- Twin prime conjecture
- Other resources
- Finding the domain and range of functions from graphs
- Properties of exponents
- Difference of cubes
- Sum of cubes
- Factoring difference of squares (abstract methods)
- Derivation
- Similarity
- \(\pi\)
- Napkin ring problem
- Pascal's rule
- Other resources
- End behavior
- Other resources
- Wilson's theorem
- Proof
- Geometric definitions
- Transformations of functions
- Absolute value function under transformations
- Graphing linear equations
- Standard form for linear equations
- Forms of linear equations
- Slope-intercept form
- Point-slope form
- Shifting functions
- Reflecting functions
- Even and odd functions
- Computer-assisted proof
- Scaling functions
- Card tricks
- Other resources
- Hyperboloid
- Si Stebbins stack
- Squaring the torus
- Mandelbrot set
- The Leaning Tower of Lire
- Strictly-convex deltahedron
- \(\pi\) tape
- Genailleâ€“Lucas rulers
- Puzzles
- Challenging exercises
- Secretary problem
- Definite integral of a polynomial function
- The monkey and the coconuts
- Riemann sums
- Midpoint rule
- Trapezoidal rule
- Other resources
- Comparing and ordering integers
- Absolute value of a number
- Adding integers
- Subtracting integers
- Subtracting decimals
- Composition of functions
- Inverse functions (algebra 2)
- Synthetic division
- Napier's bones
- Converse of the Pythagorean theorem
- 5th grade
- Adding decimals using the standard algorithm
- Decimals
- Other resources
- Other resources
- Rounding to the nearest 1000
- Other resources
- Interval notation
- Solving and graphing compound linear inequalities with AND
- Solving and graphing compound linear inequalities with OR
- Solving and graphing compound linear inequalities
- 6th grade
- Other resources
- Finding angles and side lengths in similar polygons
- Double pendulum
- 4th grade
- Multiplying one-digit numbers by multiples of \(10,\) \(100,\) and \(1000\)
- Other resources
- Other resources
- Education
- Other resources
- Kindergarten
- K-12 courses
- Exponential growth and decay
- Exponential vs. linear growth
- 1st grade
- 2nd grade
- Other resources
- 7th grade
- 8th grade
- Babylonian mathematics
- Fractions (prealgebra)
- Types of fractions
- Other resources
- Area of a trapezoid
- Simpson's paradox
- Difference between correlation and causation
- Other resources
- Other resources
- Cartesian product
- Finding the tangent or normal line through the given point
- Infinite sets
- Stable marriage problem
- Stereographic projection
- Bisection method
- Newton's method
- Sine rule for area of a triangle
- Dividing by zero
- Converting between decimals and fractions
- Other resources
- Museum
- Other resources
- Multiplying complex numbers algebraically
- Other resources
- Dividing complex numbers
- Plotting complex numbers
- Linear factorization theorem
- Solving certain cubic equations
- Simplifying fractional powers
- When is a logarithm undefined?
- Complex number logarithm
- How many ways can you arrange a deck of cards?
- Other resources
- Finding the vertex of a parabola
- Derivatives: composite, implicit, and inverse functions
- Simplifying rational expressions
- Intro to limits
- Chain rule
- First derivative using implicit differentiation
- Quotient rule
- Solving logarithmic equations
- Sum, difference, and power rules for derivatives
- Intersection of two lines in \(\mathbb{R}^3\)
- Hilbert's paradox of the Grand Hotel
- Lines in \(\mathbb{R}^3\)
- Other resources
- Minimum vertex covers
- Ternary numeral system
- \(100\) prisoners problem
- Fractals
- Paradoxes
- Fixed point
- Brachistochrone curve
- Proof by cases
- Other resources
- Perfect graph
- Interviews of teachers
- \(\LaTeX\)
- Intro to logarithms
- Futoshiki
- Biconditional
- Conditional statements
- Solving exponential equations
- Direct proof
- Proof by contrapositive
- Proof techniques
- Flows and cuts
- Inclusion–exclusion principle
- Random
- Using the theorem
- SAT math
- History of combinatorics
- Biographies (combinatorics)
- Catalan numbers
- Various topics
- SageMath
- History
- Euclid's Elements
- Undergrad geometry
- Addition rule of probability
- Multiplying two-digit numbers
- Trick for multiplying by 11
- Trick for multiplying multi-digit numbers by 11-19
- Trick for multiplying by numbers one less than a power of ten when number of digits are equal
- Logic puzzles
- Srinivasa Aiyangar Ramanujan
- Biographies
- Georg Cantor
- William Rowan Hamilton
- Évariste Galois
- Galois theory
- History
- Biographies
- Probability trees
- History of probability theory
- Biographies
- Abraham de Moivre
- Intro to probability
- Sprouts
- Chomp
- Two-player games
- Multi-digit multiplication and division
- Multi-digit multiplication
- Multi-digit division
- List all the divisors of a number
- Multiply fractions, abstract
- Other resources
- Converting between improper fractions and mixed numbers
- \(n\)-th root
- Solving and graphing absolute value inequalities
- Other resources
- Necklace splitting problem
- Intro to circles and \(\pi\)
- What is volume?
- Other resources
- Intro to integers
- Integers
- Multiplying integers
- Other resources
- Recognize fractions less than or equal to 1
- Scientific notation
- Common denominators
- Reducing fractions
- Adding and subtracting fractions with like denominators (abstract method)
- Introduction to percents
- Division (grade 4)
- Points, lines, and planes
- Other resources
- Types of quadrilaterals (grade 5)
- Intro to polynomials
- Distributive property with variables (abstract method, grade 7)
- Introduction to functions
- Metric system
- Other resources
- Archimedes's cattle problem
- Sum and product rules
- Stars and bars
- Double counting
- Kakeya set
- Ellipse
- Expected value
- Factoring trinomials using SageMath
- Extremal Principle
- Nontraditional education
- Videos for kindergarten teachers
- Teaching preschool mathematics
- Montessori education
- Education reform
- Large numbers
- Flexagon
- Ruled surface
- Sum of three cubes
- Graphs of sequences
- Frieze patterns
- Adding and subtracting polynomials
- Polygons
- Line-circle and circle-circle relationships
- Long division (two-digit divisors)
- Area of a parallelogram
- Equivalence of a decagon to a rectangle
- Equivalence of a trapezoid to a rectangle
- Equivalence of a parallelogram to a rectangle
- Equivalence of a triangle to a rectangle
- Adding within 10
- History of counting
- Other resources
- Add and subtract fractions (word problems)
- Other resources
- Subtract within 10
- Classifying triangles by angles
- Classifying triangles by side lengths
- Multiplication of one-digit by two-digit
- Multiply by 2-digit numbers (grade 4)
- Dividing fractions by whole numbers
- Adding fractions with unlike denominators
- Long division (three-digit divisors)
- Division using the stamp game
- Division using the stamp game (grade 4)
- Multiplying multi-digit numbers by one-digit numbers
- Multiplication of a multi-digit number by a two-digit number
- Multiplication of a multi-digit number by a three-digit number
- Intro to geometric series
- Partial sum of a geometric progression
- Reading travel graphs
- Unitary method
- Derivatives of exponential functions
- Counting, permutations, and combinations
- Two-way tables
- Probability
- Other resources
- Discrete random variables
- Middle school courses
- Elementary school courses
- Median
- Stem and leaf plots
- Evaluating square roots of small perfect squares
- Evaluating cube roots of small perfect cubes
- Arithmetic series
- Net of a cube
- Net of a cuboid
- Other resources
- Factoring linear expressions
- Measurement
- For teachers: Fraction talks
- Add and subtract fractions (grade 4)
- Royal arrangements
- Estimation
- Multiplying monomials by binomials
- Other resources
- Frequency tables
- Other resources
- Factoring a cubic polynomial with a double root
- Motion graphs
- Product rule
- Leibniz's derivative notation
- Constant rule
- Biographies for calc
- Perimeter of composite figures
- Proof by induction (inequalities)
- Net of a polyhedron
- Zero
- Collatz conjecture
- Adding and subtracting free vectors
- Infinitude of the primes
- Polynomial arithmetic
- Other resources
- Small-angle approximation
- Double angle identities
- Triple angle formulas
- Polynomial long division and graphing cubics
- Other resources
- Multiplying polynomials
- Transformations of the square and cube root functions
- Graphing the square root function under transformations
- Other resources
- Equation of a circle and semicircle
- Adding functions
- Subtracting functions
- Solving certain cubic inequalities
- Graphing parabolas
- Quadratic equations
- Graphing with complex numbers
- Lattice paths
- Combinations without restrictions
- Other resources
- Multi-stage events
- Greatest binomial coefficient
- Simple harmonic motion
- Integration by u-substitution (less difficult)
- Motion
- Straight line motion
- Projectile motion
- Related rates
- Graphing inverse trigonometric functions
- Applications of trigonometric integrals
- Applications of trigonometric derivatives
- Derivatives of trigonometric functions under transformations
- Definite integrals
- Intermediate value theorem
- Finding the acute angle between two lines
- Sum and difference formulas
- Finance
- Integrating exponential functions
- Integrating exponential functions
- Logarithmic functions
- Logarithmic functions
- Solids of revolution
- Area between two curves
- Antiderivatives
- Other resources
- Extrema
- Finding absolute extrema
- Primitive functions
- Oblique asymptotes
- Graphing using derivatives
- Inflection points
- Solving equations with complex numbers
- Line of intersection between two planes
- Vectors
- Arc length (calculus 2)
- Surface area of solids of revolution
- L'Hopital's rule
- Geometrical interpretation
- Simultaneous equations
- Inverse matrices
- Further integration
- 3D matrices
- Determinants
- Matrices
- Invariance
- Classifying stationary points
- Other resources
- Special points
- Deriving the quadratic formula
- Series and sequences
- Partial sums
- Quadratic formula
- Other resources
- Parametric equations and polar coordinates
- Finding the domain of radical functions
- Finding the domain and range of functions
- Parabolas
- Finding the intersection point of two graphs
- Continuity
- Linear equations
- Sketching angles in radians
- Angle measures and circles
- Degrees-minutes-seconds
- Wythoff's game
- Game of Nine
- Variance and standard deviation
- Other resources
- Multiplying and dividing in scientific notation
- Finding the inverse of discrete and linear functions
- Finding the inverse of rational functions
- Adding and subtracting rational expressions
- Rational functions
- Simplifying square roots of nonnegative integers
- Simplifying square roots of monomials with variables
- Exterior angles of regular polygons
- Circle theorems
- Other resources
- Significant figures
- Rounding to \(n\) significant figures
- Counting significant figures
- Measures of central tendency from frequency tables
- Simplifying ratios
- Dividing a quantity by a ratio
- Volume of right rectangular prisms
- Other resources
- Factoring monic quadratics
- Area of a circle
- Circumference of a circle
- Other resources
- Solving quadratic equations by factoring
- Percent change
- Graphing linear equations in slope-intercept form
- Other resources
- GCD and LCM of numbers (word problems)
- Other resources
- Other resources
- Inquiry-based learning
- Zebra Puzzle
- Wason selection task
- Dice puzzle
- Utilities puzzle
- Tidbits on Burnside's lemma
- Bridge and torch problem
- The Hardest Logic Puzzle Ever
- Hat puzzle
- Pirate game
- Power of a point
- Converting fractions to percents
- Converting between fractions and percents
- Converting percents to fractions
- Multiply any number by twelve
- Trick for dividing a number by nine
- Mental math: Subtract a near multiple of 10 from a two-digit number by adjusting
- Finger tricks for multiplication
- Non-transitive games
- Combination and permutation word problems
- Arranging in a circle
- Difference between rational and irrational numbers
- Dividing decimals
- Multiplying decimals
- Adding and subtracting mixed numbers
- Graphing the absolute value function under transformations from equations
- Converting between decimals and percents
- Converting decimals to percents
- Box plots
- Bearings
- Surface area
- Surface area of a cylinder
- Adding and subtracting square root expressions
- Disc and washer methods (circular cross sections)
- Shell method
- Geometric series
- Icosian game
- Horizontal and vertical lines
- Converting standard form to slope-intercept form
- Area of kites and rhombuses
- Area of a kite
- Expanding quadratic expressions using algebra tiles
- Solving linear systems of equations in two variables by graphing
- Other resources
- Factoring monic quadratics using algebra tiles
- Other resources
- Other resources
- Is it injective?
- Injective, surjective, and bijective
- Partial order
- Compound interest
- Simple and compound interest
- Evaluating definite integrals using geometry
- Adding mixed numbers
- Subtracting mixed numbers
- Simple interest
- Trees
- Graphing sin and cos under transformations
- Graphing the tangent function under transformations
- Is it surjective?
- Finding polynomials given complex zeros
- Finding polynomials given irrational zeros
- Finding polynomials given integer zeros
- Finding polynomials given zeros
- Functions (algebra 1)
- Linear equations and inequalities
- Intersection of circle and line
- Absolute value, piecewise, and step functions
- Evaluating piecewise functions
- Graphing piecewise functions
- Find the equation of a piecewise function from its graph
- Dividing rational expressions
- Sketching logarithmic functions under transformations
- Graphing the natural logarithm function under transformations
- Expanding logarithmic expressions
- Condensing logarithmic expressions
- Expanding and condensing logarithmic expressions
- Solving trig equations
- Solving harder trig equations by factoring
- Other resources
- Finding the inverse of exponential functions
- Evaluating natural logarithms
- Solving logarithmic equations with different bases
- Applications of logarithms
- Properties of logarithms
- Multiplying and dividing fractions
- Converting English to propositional logic
- Syntax of propositional logic
- Truth tables
- Basic propositional logic proofs using natural deduction
- Rules of inference
- Modus ponens
- Modus tollens
- Disjunctive syllogism
- Constructive dilemma
- Destructive dilemma
- Conjunction introduction and elimination
- Biconditional introduction and elimination
- Disjunction introduction
- Finding the inverse of functions
- Matrix multiplication
- Basic transformation matrices
- Division of multi-digit by one-digit
- Currency conversion
- Converting decimals to fractions
- Adding and subtracting decimals
- Other resources
- Converting percents to decimals
- Percent increase
- Percent decrease
- Converting between scientific notation and standard form
- Converting scientific notation to standard form
- Converting standard form to scientific notation
- Standard form to power of ten
- Intro to slope
- Finding slope from a graph
- Finding missing angles in right triangles
- Zero product property
- Finding missing side lengths in right triangles
- Angle bisector theorem
- Approximating definite integrals
- Find the orthocenter from coordinates
- Derivatives of inverse trigonometric functions
- Proof: Sum identities
- Limits
- Finding limits of trigonometric functions
- Other resources
- Proofs
- Triangle angle bisector theorem
- Converting vertex form to standard form
- Graphing parabolas in intercept form
- Simplifying the difference quotient
- Proving the law of cosines
- Graphing linear equations in standard form
- Find a perpendicular or parallel line
- Find the parallel line
- Find the perpendicular line
- Find the parabola from its vertex and a point
- Solving separable differential equations
- Proving the law of sines
- Linear programming
- Minimization (mathematical problems)
- Maximization (mathematical problems)
- Finding a side given an angle and a side
- Converting repeating decimals to fractions
- Volume of a cone
- Solving a quadratic by trial and improvement
- Other resources
- Normal distribution
- Binomial theorem and Pascal's triangle
- Exponential growth and decay (calculus 1)
- Functions of displacement
- Derivatives of trigonometric functions
- Derivatives of logarithmic functions
- Mechanics
- Common knowledge puzzle
- Locker Problem
- Other resources
- Standard form of ellipse equations from graphs
- Scatter plots
- Average speed from travel graph
- Finding the sum of a telescoping series
- Plotting inequalities on a number line
- Circles (conic sections)
- Other resources
- Finding or estimating the mean from a frequency table
- Binomial expansion
- Iterated function systems
- Other resources
- Binomial distribution
- Mean
- Converting terminating decimals to fractions
- Solving linear equations and expanding brackets
- Remainder theorem
- Intro to the dot product
- Inner product
- Cross product
- Eigenvalues and eigenvectors
- Diagonalization
- Area of a triangle using determinant
- Cramer's rule
- Applications of the dot product
- Orthogonal vectors
- Proof: Vector addition is commutative and associative
- Adding vectors
- Multiplying a vector by a scalar
- Length of a vector
- Limits that do not exist
- Finding limits by rationalizing
- Finding limits by factoring
- Epsilon-delta definition of limits
- Finding one-sided limits
- Knights and Knaves
- Find the inverse of a trigonometric function
- Derivative of a function from first principles
- Rolle's theorem
- Mean value theorem
- First derivative test
- Concavity
- Linear approximation
- Reverse power rule for indefinite integrals
- Properties of antiderivatives
- Other resources
- First fundamental theorem of calculus
- Using the natural log rule
- Integrating trigonometric functions
- 3D vectors
- Definition of \(\mathbb{R}^n\)
- Other resources
- Properties of vector addition
- Properties of vector addition and scalar multiplication
- Other resources
- Proof: Derivative of \(\sec(x)\)
- Proof: Derivative of \(\sin(x)\)
- Sum formulas
- Proof: Derivative of \(\tan(x)\)
- Solving linear systems of equations
- Solving linear systems of equations using matrix inverses
- Solving linear systems of equations in three variables using Gaussian elimination
- Consistency of a system of linear equations
- Inverse of a \(3{\times}3\) matrix using row operations
- Inverse of a \(4{\times}4\) matrix using row operations
- Determinant using row echelon form
- Inverse of \(3{\times}3\) matrix using determinants and cofactors
- Solving linear systems in two variables using Cramer's rule
- Solving linear systems in three variables using Cramer's rule
- Determinant using cofactors
- Determinant of a \(4{\times}4\) matrix using cofactors
- Determinant using row operations
- Determinant of a \(4{\times}4\) matrix using row operations
- Linear transformations
- Linear combinations
- Linear transformations on \(\mathbb{R}^2\)
- Basis
- Basis for a null space
- Basis for a span
- Separating rational and irrational components in a binomial expansion
- Multiplying mixed numbers
- Volume of a sphere (calculus \(2\))
- Solving perfect square quadratics by factoring
- Solving quadratics by factoring (difference of squares)
- Introduction to sets
- Set equality and subsets
- Sets
- Set operations
- Binary relations
- Vertical line test
- Intro to multivariable functions
- Two point form
- Solving linear systems of equations in three variables
- Solving linear systems of equations in three variables (word problems)
- Statistics and probability for undergraduates
- Pólya urn model
- Half angle identities
- Converting between radians and degrees
- Coterminal angles
- Product-to-sum and sum-to-product formulas
- Proof: Sum to product identity for sine
- Proofs: Product to sum identities
- Algebraic proof the cross product is distributive
- Geometric proof the cross product is distributive
- Proof the cross product is distributive
- Determinant of a \(3{\times}3\) matrix
- Determinant of a \(2{\times}2\) matrix
- Competition problems (sum of a geometric progression)
- Variation of parameters
- Factoring by grouping (one variable, quadratic)
- Factor by grouping
- Identifying and factoring perfect square trinomials
- Elementary matrices
- Matrix inverses
- Power method
- Shifted inverse power method
- Gershgorin circle theorem
- Graphing linear inequalities in point-slope form
- Finding two-variable inequalities from their graphs
- Operations with functions
- Adding and subtracting functions
- Dividing functions and finding the domain
- Find \(f \cdot g\) and its domain
- Recurrence relations
- Trig substitutions
- Partial fraction decomposition
- Solving square root equations
- Finding and plotting the domain of a function
- Graphing linear inequalities in two variables
- Credit and borrowing
- Simpson's rule
- Sketching slope fields
- Matching slope fields with equations
- Spirograph
- Harmonograph
- Overfitting
- Verifying solutions to differential equations
- Initial value problems
- Euler's method
- Separable first order differential equations
- Parabolas (conic sections)
- Graphing the sine and cosine functions under transformations
- Other topics
- Proof: Only one magic square of size three
- Motivation for learning statistics and probability
- Peano arithmetic
- Slide rules and log tables
- \(\sin(z) = 2\)
- Approximating square roots to hundredths
- Approximating square roots to tenths
- False proofs
- Graphing linear equations in point-slope form
- Graphing horizontal and vertical equations
- Complex exponentiation
- \(i^i\)
- Naive set theory
- Negation (bridge course)
- Relations
- Equivalence relations
- Tautologies and contradictions
- Functions (bridge course)
- Proving parallelograms with two column proofs
- Proof
- Heron's formula
- Area of a triangle
- Disjunction
- Arithmetic properties
- Additive identity
- Converting improper fractions to mixed numbers
- Converting degrees to radians
- Converting radians to degrees
- Converting units of measure
- Converting between minutes and seconds
- Range
- Converting mixed numbers to improper fractions
- Rearrangement proof: Pythagorean theorem
- Using the Pythagorean theorem (easier mathematical problems)
- Mode
- Adjacent angles
- Integration by parts
- Other resources
- Disjoint sets
- What is symmetric difference?
- Basic indefinite integrals
- Perimeter of composite figures
- Other resources
- Hyperbola
- Graphing ellipses
- Inverse trig functions
- Tangent half-angle substitution
- IMO problems
- Julia set
- Proving the sum and difference of cubes formulas
- Visual proof: Area of a circle
- Squeeze theorem
- Proof
- Proof: Derivative of \(\cos(x)\)
- Proof: Morrie's law
- Proof: Generalization of Morrie's law
- Proof
- Proving Morrie's law and its generalization
- Proving Mollweide's formula
- Proofs
- Proof: Sum of the interior angles of a simple polygon
- Sum of the interior angles of a simple polygon (algebra problems)
- Sum of the interior angles for simple polygons
- General chain rule
- Other resources
- Graphing sin and cos
- Graphing tan and cot
- Exponents of decimals
- Finding an angle using the law of cosines
- Solving triangles using the law of cosines
- Find a length using the law of sines
- Find an angle using the law of sines
- Solving triangles using the law of sines
- Proving the product rule
- Pythagorean identities
- Using the Pythagorean identities
- Proving the Pythagorean identities
- Proof
- Symmetric group
- Symmetries of an equilateral triangle
- Is it bijective? (infinite domain)
- Adding and subtracting matrices
- Practice
- Multiplying a matrix by a scalar
- Practice
- Area of a triangle using determinants
- Determinant of a \(3{\times}3\) matrix using expansion by minors
- Determinant of a \(3{\times}3\) matrix using the diagonal method
- Intuition: Determinants
- Intuition: Matrix multiplication
- Solving matrix equations
- Solving linear systems of equations in two variables (word problems)
- Converting circle equations between standard and general form
- Converting circle equations from standard to general form
- Converting circle equations from general to standard form
- Double factorial
- Other resources
- Solving unit rate problems
- Exterior angle theorem
- Proof
- Using the theorem
- Finding the center and radius of a circle by completing the square
- Binary
- Multiplying mixed numbers (word problems)
- Other resources
- Add within 5
- Subtract within 5
- Finger trick for multiplying 6-10 by 6-10
- Elapsed time
- Division with small divisor using base ten blocks
- Division with partial quotients
- Lattice multiplication
- Multiplying two-digit numbers using base ten blocks
- Add within 10 using fingers
- Subtract within 10 using fingers
- Multiplying integers using two color counters
- Adding integers using two color counters
- Subtracting integers using two color counters
- Area model for division
- Finding the area of composite figures by adding
- Finding the area of composite figures by subtracting
- Finding the area of composite figures (grade 3)
- Commutative property of addition (prealgebra)
- Other resources
- Other resources
- Making 10 using a tens frame (concrete)
- Making 10
- Add within 20
- Subtract within 20
- Lattice division
- Long division (grade 4)
- Multiplying two-digit numbers using partial products
- Subtraction using partial differences
- Breaking apart to add within 1000
- Multiplying multi-digit numbers by one-digit numbers using base ten blocks
- Subtracting within 1000 using base ten blocks
- Add within 1000
- Add within 1000 using base ten blocks
- Adding and subtracting fractions with like denominators using the length model
- Comparing and ordering fractions with like denominators
- Equivalent fractions using fraction sectors
- Equivalent fractions using the length model
- Adding and subtracting fractions with like denominators (concrete and visual methods)
- Multiplying fractions using the area and length models
- Converting improper fractions to mixed numbers using fraction rectangles and two color counters
- Congruent figures
- Other resources
- Ordinal numbers
- Angle Wars
- Other resources
- Other resources
- Subtracting fractions with unlike denominators
- Identity element of multiplication and division
- Adding polynomials
- Subtracting polynomials
- Subtracting mixed numbers without regrouping
- Dividing decimals by whole numbers
- Divisibility tests
- Adding within 100 without regrouping (abstract method)
- Adding within 100 without regrouping using base ten blocks
- Adding and subtracting within 1000 using place value blocks
- Special education
- Adding integers using a number line
- Dividing integers using Algeblocks
- Multiplying integers using Algeblocks
- Subtracting integers using a number line
- Multiplying linear expressions using Algeblocks
- Factoring trinomials using the box method
- Subtracting by counting up
- Adding and subtracting fractions with like denominators using the set model
- Dividing fractions using the set model
- Other resources
- Comparing and ordering fractions with unlike denominators using the length model
- Comparing and ordering fractions with like denominators using the set model
- Subtracting fractions with unlike denominators using the area model
- Dividing fractions using the area model
- Shikaku
- Comparing and ordering fractions with unlike denominators using the area model
- Converting improper fractions to mixed numbers using fraction rectangles
- Multiplying whole numbers and fractions using two color counters
- Converting mixed numbers to improper fractions using fraction rectangles
- Adding fractions with unlike denominators using the length model
- Adding fractions with unlike denominators using the area model
- Adding fractions with unlike denominators using the set model
- Subtracting fractions with unlike denominators using the set model
- Subtracting fractions with unlike denominators using the length model
- Decomposing fractions using fraction rectangles
- Multiplying whole numbers and fractions using the area model
- Multiplying fractions using the length model
- Comparing and ordering fractions with unlike denominators using the set model
- Multiplying fractions using the set model
- Dividing fractions using the length model
- Comparing and ordering fractions with like denominators using the area model
- Adding and subtracting fractions with like denominators using the area model
- Comparing fractions with unlike denominators using the set model
- Comparing and ordering fractions with like denominators using the length model
- Preschool
- Representing 1-10
- Songs about shapes
- Golden beads addition
- Division using the bead board
- Subtracting three-digit numbers using the stamp game
- Adding within 1000 (concrete and pictorial)
- Add within 1000 (abstract)
- Trapezoid midsegment theorem
- Ordering triangle sides and angles
- Finding the inverse of square root functions
- Finding the domain and range of quadratic functions
- Finding the domain and range of square root functions
- Finding the rate of change from tables
- Graphing exponential functions under transformations
- Add within \(10\) using the strip board
- Adding multi-digit numbers using the bead frame
- Adding multi-digit numbers
- Subtracting multi-digit numbers
- Subtracting three-digit numbers using the chip model
- Adding and subtracting within 1000
- Adding within 1000 using the chip model
- Adding and subtracting within 100
- Adding within 100
- Subtracting within 100
- Other resources
- Word problem strategies
- Adding and subtracting within 1000 using a number line
- Adding within 100 using base ten blocks
- Add within 20 using number bonds
- Add within 10 using counters
- Subtract within 10 using counters or two color counters
- Other resources
- Other resources
- Number pairs
- Multiplication using counters and paper plates
- Making change
- Division using manipulatives (grade 4)
- Eureka Math, grade 1
- Eureka Math, grade 1, module 1
- Eureka Math, grade 1, module 2
- Eureka Math, grade 1, module 3
- Eureka Math, grade 1, module 4
- Eureka Math, grade 1, module 5
- Eureka Math, grade 1, module 6
- Compare within 10 by pairing
- Eureka Math, grade 2
- Eureka Math, grade 2, module 1
- Eureka Math, grade 2, module 2
- Eureka Math, grade 2, module 3
- Eureka Math, grade 2, module 4
- Eureka Math, grade 2, module 5
- Eureka Math, grade 2, module 6
- Eureka Math, grade 2, module 7
- Eureka Math, grade 2, module 8
- Eureka Math, grade 3
- Eureka Math, grade 3, module 1
- Eureka Math, grade 3, module 2
- Eureka Math, grade 3, module 3
- Eureka Math, grade 3, module 4
- Eureka Math, grade 3, module 5
- Eureka Math, grade 3, module 6
- Eureka Math, grade 3, module 7
- Eureka Math, grade 4
- Eureka Math, grade 4, module 1
- Eureka Math, grade 4, module 2
- Eureka Math, grade 4, module 3
- Eureka Math, grade 4, module 4
- Eureka Math, grade 4, module 5
- Eureka Math, grade 4, module 6
- Eureka Math, grade 4, module 7
- Eureka Math, grade 5
- Multiplying whole numbers using the standard algorithm
- Eureka Math, grade 5, module 1
- Eureka Math, grade 5, module 2
- Eureka Math, grade 5, module 3
- Eureka Math, grade 5, module 4
- Eureka Math, grade 5, module 5
- Eureka Math, grade 5, module 6
- Tabular and graphical estimation of limits
- Finding limits at infinity
- Topology teasers
- Reflecting across axes
- Reflecting points across axes
- Reflecting line segments and polygons (grade 8)
- Estimating limits from tables
- Add within 1000 using the standard algorithm
- Counting by 2s starting from any number
- Other resources
- Adding and subtracting fractions with unlike denominators
- Oh Snap!
- Rounding to the nearest 10 or 100
- Multiplying 10s (grade 3)
- Subtracting three-digit numbers using the standard algorithm
- Converting between words and numbers
- Converting between whole numbers and expanded form
- Graphing linear inequalities in slope-intercept form
- Multiplying binomials
- Multiplying fractions (concrete and pictorial methods)
- Classifying real numbers
- Subtracting mixed numbers using a visual model
- Approximating square roots
- Multiplying two-digit numbers using the stamp game
- Multiplying two-digit numbers using manipulatives
- Subtracting multi-digit numbers with regrouping using the bead frame
- Infinity Street
- Multiplying multi-digit numbers by one-digit numbers using the stamp game
- Adding multi-digit numbers with regrouping using the stamp game
- Adding multi-digit numbers without regrouping using base ten blocks
- Subtracting multi-digit numbers using base ten blocks
- Adding multi-digit numbers with regrouping using base ten blocks
- Estimating multi-digit division
- Finding cube roots using units, tens, and hundreds place blocks
- Finding square roots using units, tens, and hundreds place beads
- Multiplying whole numbers using the flat bead board
- Adding multi-digit numbers with regrouping using the dot game
- Adding multi-digit numbers without regrouping using the dot game
- Multiplying multi-digit numbers by one-digit numbers using the bank game
- Finding square roots with backtracking, using units, tens, hundreds, and thousands place beads
- Non-decimal bases
- Making 10 using the addition strip board (concrete)
- Subtracting multi-digit numbers without regrouping using the bead frame
- Subtract within 100 using base ten blocks
- Money word problems
- Plotting fractions on the number line
- Unit fractions on the number line
- Decomposing fractions
- Adding three numbers in any order
- Additive identity
- Addition is commutative (without variables)
- Subtract within 20 using counters or pictures
- Subtract within 20 using bar models
- Subtract within 10 using number bonds
- Add within 10 using number bonds
- Add using place value
- Adding ones or tens using base ten blocks
- Add within 20 using ten frames
- Adding ones with regrouping
- Naming shapes (grade 2)
- Breaking apart to add within 100
- Other resources
- Counting to 10
- Other resources
- Count within 1000 by 10s starting at any number
- Count forwards or backwards by 100s starting at any number
- Other resources
- Making bar graphs
- Comparison word problems
- Multiply using counters
- Intersection of parabola and line
- Comparing numbers within 100 (concrete and visual methods)
- Bounding square roots
- Volume of a cylinder (mathematical problems)
- Volume of spheres
- Volume of cones, cylinders, and spheres
- Find a side using the law of cosines (mathematical problems)
- Finding an angle using the law of cosines (mathematical problems)
- Finding a length using the law of sines (mathematical problems)
- Finding a length using the law of sines (word problems)
- Slope-intercept equation from a graph
- Lesson
- Graphing proportional relationships
- Solving linear equations (word problems)
- Expanding and factoring
- Solving quadratic equations by factoring (word problems)
- Solving quadratic equations by factoring (mathematical problems)
- Multiplying and dividing rational expressions
- Other resources
- Other resources
- Ratios, rates, and percentages
- Buffon's needle
- Paradox: Birthday problem
- Converting terminating decimals to fractions with denominators 10 or 100
- Plotting decimals on a number line
- Other resources
- Solving systems of linear equations in two variables by substitution (word problems)
- Solving multi-step linear equations involving simple interest (word problems)
- Mixture problems
- Adding measurements with different units (word problems)
- Other resources
- Other resources
- Factoring monomials completely
- Multiplying monomials by polynomials
- Intercepts from a graph
- Understanding slope with similar triangles
- Other resources
- Translation
- Drawing quarter turn rotations about the origin
- Subtracting fractions with unlike denominators using graph paper
- Division (grade 3, concrete and visual)
- Adding a two-digit number and a multiple of ten
- Add within 10 using a number line
- Subtract within 10 using a number line
- Compare within 10 on the number line
- Other resources
- Polynomial factorization, division, and end behavior
- Graphing parabolas in standard form
- Graphing quadratics in vertex form
- Other resources
- Intercepts of a linear function from a table
- Number of solutions to a system of equations algebraically
- Graphing linear equations (word problems)
- Difference between functions and relations
- Measures of central tendency from stem and leaf plots
- Other resources
- Finding area by counting unit squares (whole number side lengths)
- Area of a rectangle with fractional side lengths
- Other resources
- Exponentiation as repeated multiplication
- Discrete vs continuous functions
- Solving linear equations with decimals
- Estimate sums within 1000
- Adding and subtracting within 100 on your fingers
- Count to 120 by ones
- Order of operations with decimals
- Equivalent fractions on a number line
- Solving two-step equations (word problems)
- Multiplying decimals by whole numbers
- Verifying inverse functions
- Other resources
- Graphing ellipses in general form
- Breaking apart to subtract within 100
- Missing number problems with multiplication and division
- Multiplying one-digit by two-digit numbers using the box method
- Breaking apart to subtract within 100 without regrouping
- Breaking apart to subtract within 100 with regrouping
- Percents (word problems)
- Multiplying trinomials
- Multiplying mixed numbers (easier mathematical problems)
- Long multiplication
- Multiplying two-digit numbers using the box method
- Vertical angles theorem
- Solving two-step equations with decimals
- Comparing fractions using the butterfly method
- Comparing fractions using manipulatives
- Multiplying multi-digit numbers with area models
- Constructing angle and segment bisectors
- Cofunction identities
- Hinge theorem and its converse
- Finding angles of elevation and depression
- Definite integration by trigonometric substitution
- Integrals involving trig functions
- Improper integrals
- Other resources
- Intersections of lines, circles, and parabolas
- Other resources
- Evaluating natural logarithms with a calculator
- Graphing hyperbolas in standard form
- Graphing tan and cot under transformations
- Graphing sec and csc under transformations
- Polar coordinates
- Solving absolute value equations and inequalities
- Solving absolute value equations
- Limit comparison test
- Ratio test
- Root test
- Verifying trigonometric identities involving the sum and difference formulas
- Verifying trig identities involving the half and double angle formulas
- Ratios and proportions (word problems)
- Solving linear systems of equations in three variables by substitution
- Other resources
- Scientific notation, word problems
- Average value over an interval
- Integrating exponential functions by u-substitution
- Converting between decimals and mixed numbers
- Converting mixed numbers to decimals
- Converting decimals to mixed numbers
- Microsoft Excel
- Fraction arithmetic
- Adding and subtracting rational expressions with like denominators
- Adding and subtracting rational expressions with unlike denominators
- Multiplying rational expressions
- Other resources
- Constant multiple rule for derivatives
- Extreme value theorem
- Gaussian elimination
- Gaussâ€“Jordan elimination
- Adding and subtracting radicals
- Other resources
- Absolute value function under transformations, inequalities
- Solving one-step absolute value equations
- Solving two-step absolute value equations
- Converting exponential form to logarithmic form
- Converting logarithmic form to exponential form
- Converting between logarithmic form and exponential form
- Other resources
- Sum of the interior angles for regular polygons
- Other resources
- Converting between polar and Cartesian form
- Converting Cartesian points to polar points
- Converting polar points to Cartesian points
- Second derivative test
- Find a side using the law of cosines (word problems)
- Solving linear systems in slope-intercept form by graphing
- Solving linear systems of equations in standard form and two variables by graphing
- Solving and graphing linear inequalities (word problems)
- Finding asymptotes of rational functions
- Solving linear systems of equations in two variables using matrix inverses
- Law of sines (word problems)
- Circular permutations
- Simplifying square roots of rational expressions
- Simplifying radical expressions
- Other resources
- Finding reference angles
- Evaluating trigonometric functions using reference angles
- Other resources
- Solving exponential equations using logarithms
- Other resources
- Factor theorem
- Derivatives of natural logarithmic functions
- Sign of the discriminant
- Using your calculator
- Solving linear equations with fractions
- Earning and managing money
- Investing money
- Taxation
- Graphing the sum of trig functions from their graphs
- Subtracting functions (linear-linear)
- Graphing ellipses at the origin
- Graphing ellipses not at the origin
- Graphing circles in general form
- Graphing circles in standard form
- Graphing the cotangent function under transformations
- Graphing csc under transformations
- Graphing circles
- Determining the type of conic from its equation
- Properties of radicals
- Finding the general form of an ellipse by completing the square
- Graphing hyperbolas in general form
- Finding the domain of logarithmic functions under transformations
- Triple integrals
- Reducing fractions using prime factorization
- Adding decimals
- Finding the domain of the sum of functions
- Holiday math
- Using the derivatives of the hyperbolic functions
- Alternating series test
- Using the theorems
- Comparison test
- Finding the other zeros given complex zeros
- Finding the inverse of a relation on a finite set
- Graphing a line given a point and a slope
- Wilcoxon rank-sum test
- Survey problems
- Work problems
- Testing solutions to equations
- Testing solutions to one-variable inequalities
- Testing solutions to linear systems of equations in two variables
- Solving linear equations involving perimeter
- Wind and current problems
- Consecutive integer problems
- Rate problems
- Solving coin and stamp problems
- Converting ellipse equations from standard form to general form
- Converting ellipse equations from general form to standard form
- Converting ellipse equations between standard form and general form
- Finding the LCD of rational expressions
- Multiplying in scientific notation
- Dividing in scientific notation
- Converting units of measurement (grade 5)
- Other resources
- Solving right triangles (word problems)
- Graphing sec under transformations
- Pythagorean theorem (word problems)
- Solving trigonometric equations using a calculator
- Adding and subtracting DMS
- Evaluating trig functions
- Evaluating trigonometric functions using a calculator
- Adding and subtracting mixed numbers with unlike denominators
- Adding and subtracting mixed numbers with like denominators
- Intersection of parabola and circle
- Lattice multiplication with decimals
- Finding the LCD of fractions
- Desmos
- Alternating harmonic series
- Finding the domain and range of piecewise functions
- Integrating logarithmic functions
- Plotting integers on a number line
- Intro to negative numbers
- Graph a line given \(x\) and \(y\) intercepts
- Other resources
- Applying derivatives to analyze functions
- Solving nonlinear systems in two variables (word problems)
- Finding missing side lengths in right triangles (word problems)
- Solving trigonometric equations using t-results
- Verifying trigonometric identities using t-formulas
- Two column proofs
- Solving trigonometric equations quadratic in form
- Add by making 10s
- Simplifying square roots of decimals and fractions
- Subtracting within 100 with regrouping
- Number of solutions to linear equations
- Number of real roots of a quadratic
- Maclaurin series
- Taylor series
- Other resources
- Practice
- Solving trigonometric equations using sum and difference formulas
- Manipulatives
- Number disks
- Adding multi-digit numbers from left to right
- Finding the whole
- Percents: Finding the percent
- Solving and graphing logarithmic inequalities
- Graphing the absolute value function
- Derivatives of inverse functions
- Finding the inverse of cube root functions
- Probability with geometry
- Subtracto Bingo
- Cryptography
- Sieve of Eratosthenes
- Mills's constant
- Caesar cipher
- Dijkstra's algorithm
- Fleury's algorithm
- Kruskal's algorithm
- Collecting data
- Logistic growth
- Apportionment
- Octal
- Hexadecimal
- Adding and subtracting in binary
- Hamilton's method
- Jefferson's method
- Webster's method
- Huntington–Hill method
- Lowndes's method
- Voting
- Weighted voting
- Scheduling
- Fair division
- Using even-odd identities to evaluate trigonometric functions
- Using the theorem
- Using the theorem
- Categorical propositions
- Propositional logic
- Predicate logic
- Informal and inductive logic
- Challenge
- Adding and subtracting rational trig expressions
- Using the theorems
- Simplifying rational trigonometric expressions
- Finding the inverse of logarithmic functions
- Solving linear equations with congruent triangles
- SAS congruence
- ASA and AAS congruence
- Dividing square roots
- Finding the domain of radical functions (monic linear functions)
- Finding the domain of radical functions (non-monic linear functions)
- Finding the domain of radical functions (quadratic functions)
- Other resources
- Partial sum of an arithmetic series
- Other resources
- Multiplying and dividing decimals by powers of ten
- Vector projection
- Proof: Vector projection formula
- Applications of vectors
- Proving De Moivre's formula by induction
- Powers of complex numbers using De Moivre's formula
- First order differential equations
- First order homogeneous differential equations
- Using the theorems
- Using the power reducing identities to evaluate or simplify expressions
- Comparing fractions with the same numerators or denominators
- Multiplying fractions and mixed numbers using a visual model
- Second order differential equations
- Exact first order differential equations
- Reduction of order
- Factoring quadratic equations by graphing
- Finding the eigenvalues and eigenvectors of \(2{\times}2\) matrices
- Finding the eigenvalues and eigenvectors of \(3{\times}3\) matrices
- Finding the eigenvalues and eigenvectors of \(4{\times}4\) matrices
- Other resources
- Largest rectangle in a semicircle
- Apportionment paradox
- Finding one-sided limits from graphs
- Sum and difference formulas for sine
- Sum and difference formulas for tangent
- Sum formula for cosine
- Sum formula for tangent
- Proof
- Proof: Sum formula for tangent
- Using the sum formula for tangent
- Difference formula for tangent
- Proof 1
- Using the difference formula for tangent
- Parallel vectors
- Test preparation
- ACT math
- Other resources
- Find the value of \(c\) that completes the square
- Converting standard form to vertex form without completing the square
- Finding the domain of rational functions
- Finding the domain of rational functions involving radicals
- Finding extrema graphically
- Using the theorem
- Using the difference formula for cosine
- Converting between DMS and degrees
- Converting between radicals and rational exponents
- Converting radicals to rational exponents
- Converting rational exponents to radicals
- Classifying polynomials
- Finding and classifying discontinuities
- Is the sequence arithmetic, geometric, or neither?
- Are the vectors parallel, orthogonal, or neither?
- Finding the domain of rational functions (linear denominator)
- Finding the quadrant of an angle in degrees
- Finding the quadrant of an angle in radians
- Finding the domain of rational functions (quadratic denominator)
- Pythagorean inequality theorem
- Evaluating trigonometric functions using the unit circle (negative angles)
- Graphing exponential functions (reflection and horizontal shift)
- Graphing exponential functions (stretch and compress)
- Graphing basic exponential functions by plotting points
- Evaluating trigonometric functions of angles in degrees using reference angles
- Evaluating trigonometric functions of angles in radians using reference angles
- Characteristics of functions
- Differentiability of a function
- Collinear points
- Coplanar points and lines
- Find the number of sides for a regular polygon given one interior angle
- Find the number of sides for a regular polygon given one exterior angle
- Find the number of sides for a simple polygon given the sum of interior angles
- Is the equation linear?
- Domain of linear functions
- Finding the domain and range of the absolute value function under transformations
- Finding the domain and range of logarithmic functions under transformations
- Evaluating the six basic trig functions given a point
- Finding the domain and range of functions from their graphs
- Finding the domain and range of functions (finite set of ordered pairs)
- Find the first \(n\) terms of a geometric progression
- Perpendicular bisector theorem and its converse
- Finding critical values using derivatives
- Unit vectors
- Finding the unit vector
- Finding the domain and range of functions from tables
- Practice
- Lesson
- Find the angle given the value of a reciprocal trigonometric function
- What is a relation?
- Graphing linear equations from tables
- Finding basic limits
- Finding limits of rational functions
- Finding limits by simplifying complex fractions
- Find one exterior angle given the number of sides for a regular polygon
- Evaluating trigonometric functions using coterminal angles
- Evaluating the tangent of angles in degrees using the sum formula for tangent
- Evaluating the tangent of angles in radians using the sum formula for tangent
- Evaluating the tangent of angles in degrees using the difference formula for tangent
- Evaluating the tangent of angles in radians using the difference formula for tangent
- Area and perimeter of parallelograms
- Finding the \(n\)
^{th}term of a binomial expansion - Graphing exponential functions (vertical shift)
- Using the Pythagorean theorem to find the missing side of a right triangle
- Finding missing side lengths in special right triangles
- Sketching the inverse of a function from its graph
- Finding the inverse of quadratic functions
- Other resources
- Graphing the secant function under transformations (phase and vertical shift)
- Graphing the secant function under transformations (period and amplitude)
- Higher order derivatives
- Convergent and divergent geometric series
- Finding the \(x\) and \(y\) intercepts of rational functions
- Volume and surface area
- Derivatives of transcendental functions
- Volume of prisms and cylinders
- Proof
- Intro
- Evaluating finite sets using Venn diagrams
- Set union
- Set intersection
- Using the difference formula for cosine when given two right triangles
- Using the difference formula for tangent when given two right triangles
- Solving quintic polynomials
- Solving quartic polynomials
- Other resources
- Graphing hyperbolas centered at the origin
- Graphing vertical parabolas not at the origin
- Graphing the tangent function under transformations (period and amplitude)
- Graphing the tangent function under transformations (phase and vertical shift)
- Solving multi-step absolute value equations
- Solving exponential equations using the one-to-one property
- Other resources
- Sketching logarithmic functions under transformations (level 2)
- Systems of inequalities, horizontal and vertical lines
- Adding and subtracting vectors
- Finding the rate of change (word problems)
- Graphing exponential functions with base \(e\)
- Graphing horizontal ellipses at the origin
- Graphing vertical ellipses at the origin
- Graphing horizontal parabolas at the origin
- Graphing horizontal parabolas not at the origin
- Graphing vertical ellipses in general form
- Graphing vertical parabolas at the origin
- Graphing the cube root function under transformations
- Solving exponential equations with fractional powers
- Other resources
- Graphing inequalities in one variable
- Other resources
- Solving polynomials (difference of two squares)
- Other resources
- Other resources
- Solving logarithmic equations using the quadratic formula
- Multiplying by 5 (Trachtenberg system)
- Other resources
- Other resources
- Evaluating the six basic trig functions given a triangle
- Evaluating trigonometric functions using special right triangles
- Evaluating logarithmic expressions using the one-to-one property
- Solving linear systems of equations in two variables by elimination (basic)
- Solving trigonometric equations by factoring out the GCD (no angle constraint)
- Solving trigonometric equations by factoring out the GCD (angle constraint)
- Finding the value that makes the function continuous
- Second fundamental theorem of calculus
- Fundamental theorem of calculus
- Geometric mean
- Graphing horizontal parabolas in general form
- Graphing horizontal and vertical inequalities
- Graphing logarithmic functions
- Sketching logarithmic functions under transformations (level 3)
- Sketching logarithmic functions under transformations (level 1)
- Graphing parabolas in general form
- Graphing vertical parabolas in general form
- Systems of inequalities, slope-intercept form
- Systems of inequalities, standard form
- Graphing horizontal ellipses in general form
- Other resources
- Solving monic quadratics by factoring
- Other resources
- Other resources
- Using the zero product property (polynomials)
- Solving polynomials (non-factorable)
- Finding the other zeros
- Finding the other zeros given rational zeros
- Finding the other zeros given irrational zeros
- Other resources
- Graphing sec and csc
- Congruent polygons (algebra problems)
- Other resources
- Finding polynomials given rational zeros
- Finding parabolas given three points
- Finding linear equations from tables using first differences
- Functions and relations
- Eliminating the parameter
- Rational functions
- Other resources
- Evaluating expressions with multiple variables
- Other resources
- Hypotenuse-leg congruence theorem
- Simplifying radicals of monomials
- Finding the domain of composite functions
- Adding and subtracting cube root expressions
- Testing ordered pair solutions to linear equations
- Solving and graphing linear inequalities in two variables
- Finding the GCD and LCM of polynomials
- Finding the domain and range of relations
- Negative bases
- Distance between two numbers
- Evaluating expressions with a single variable
- Other resources
- Multiplying perfect squares
- Multiplying binomials by polynomials
- Multiplying binomials by polynomials using the box method
- Find the number of sides for a regular polygon given the sum of interior angles
- Alternate interior angles theorem
- Alternate exterior angles theorem
- What is a radicand?
- Multiplying binomials with square roots
- Derivation: Area of a kite
- Using the area of a triangle formula
- Derivation: Area of a right triangle
- Lesson
- Intro to solving and graphing linear inequalities
- Finding one-variable inequalities from graphs
- Method 1
- Lesson
- What is a Saccheri quadrilateral?
- Product of powers rule for exponents
- Area of an equilateral triangle
- Derivation
- Isosceles triangle theorem and its converse
- Find the parabola from its focus and directrix
- Temperature conversion
- Area of a segment
- Reciprocals of mixed numbers
- Zero matrix
- Other resources
- Set-builder notation
- Area of an ellipse
- Quotient of powers rule for exponents
- Set complement
- Lesson
- Lesson
- Integrating hyperbolic functions
- Simplifying factorial expressions
- Proving the absorption laws for sets
- Proof: Intersection distributes over union
- Proof: Composite of surjections is surjection
- Book reviews
- Compact sets
- Competition problems (calculus 2)
- Evaluating fractions raised to whole number powers
- Cyclic quadrilaterals
- Lesson
- Solving exponential equations in quadratic form using u-substitution
- Sustainability
- Negative powers with variables
- Negative powers without variables
- Adding rational expressions (word problems)
- Subtracting rational expressions (word problems)
- Distance between a point and a line
- Dividing polynomials with area models
- Linear inequalities in one variable
- Proof by induction (divisibility)
- Proof by induction (matrices)
- Proof by induction (recurrences)
- Adding and subtracting rational expressions (word problems)
- Lesson: Adding and subtracting rational expressions
- Spherical geometry
- List of math software
- Lesson
- List of math encyclopedias
- Power of a product rule for exponents
- What is a Lambert quadrilateral?
- Non-Euclidean geometry
- Diagonal length of a right rectangular prism
- Geometric constructions
- Proving the perpendicular bisector theorem
- Proving the converse of the perpendicular bisector theorem
- Proof: Sum of the exterior angles of a triangle
- Ceva's theorem
- Proving lines are parallel
- Intercepts from a quadratic equation
- Diagonals of a parallelogram bisect each other
- Finding the diagonal length of a rectangle
- Chord properties
- Find the missing side given the perimeter
- Subtract within 1000 (word problems)
- Relate division to multiplication word problems
- Time word problems
- Line plots (grade 3)
- Lesson
- Multiplication using the array model
- Breaking apart to subtract within 1000
- Inquiry-based learning (kindergarten)
- Inquiry-based learning (grade 1)
- Inquiry-based learning (grade 2)
- Inquiry-based learning (grade 3)
- Inquiry-based learning (grade 4)
- Inquiry-based learning (grade 5)
- Inquiry-based learning (grade 6)
- Inquiry-based learning (grade 7)
- Inquiry-based learning (grade 8)
- Inquiry-based learning (grade 9)
- Inquiry-based learning (grade 10)
- Inquiry-based learning (grade 11)
- Inquiry-based learning (grade 12)
- Using the theorem
- Other resources
- Using the theorem
- Adding multi-digit numbers with regrouping using the standard algorithm
- Division (grade 4 word problems)
- Multiplying one-digit by three-digits
- Multiplying whole numbers and fractions (word problems)
- Estimating quotients
- Multiplying 10s (grade 4)
- Comparing fractions using tape diagrams (visual)
- Subtracting multi-digit numbers with regrouping using the standard algorithm
- Comparing decimals using a number line
- Comparing fractions using a number line
- Comparing decimals using fraction rectangles
- Multiplying fractions (word problems)
- Adding fractions with power of 10 denominators
- Multi-step problems, properties of exponents
- Identifying functions from ordered pairs and tables
- Factoring difference of squares (concrete and visual methods)
- Using the theorem
- Geometric transformations
- Finding the volume and surface area of composite figures
- Comparing numbers in scientific notation
- Recursive to explicit formula for arithmetic series
- Finding the scale factor for similar polygons
- Explicit to recursive formula for arithmetic series
- Recursive to explicit formula for geometric sequences
- Volume of a triangular prism
- Volume of a trapezoidal prism
- Surface area of a triangular prism
- Visual proof: Derivative of \(\sin(x)\)
- Using the intermediate value theorem
- Direct proof (divisibility)
- What is the Cartesian product?
- Evaluating integers raised to whole number powers
- Multiplying fractions by cross-simplifying
- Adding multiples of 10 on a hundreds chart
- Count forwards or backwards by 1000s starting at any number
- Counting forwards by 50s starting at any number
- Mental math: Adding near double decimals by adjusting
- Adding within 1000 using a number line
- Adding and subtracting within 100 using a hundreds chart
- Adding within 100 using a hundreds chart
- Subtracting within 100 using a hundreds chart
- Division (grade 4)
- Counting forward in 100s starting at any number
- Adding multi-digit numbers using place value disks
- Counting forward in 1000s starting at any number
- Missing number problems for adding and subtracting within 1000
- Lesson
- Mental math: Add a near multiple of 100 to a three-digit number by adjusting
- Mental math: Subtract a near multiple of 100 from a three-digit number by adjusting
- Whole number addition pyramids
- Partitioning numbers in different ways
- Other resources
- Other resources
- Other resources
- Trick for subtracting from powers of ten
- Missing number problems for equivalent fractions
- Converting whole numbers to fractions
- Adding fractions with like denominators using the area model without simplifying
- Adding fractions with like denominators using the area model with simplifying
- Mental math: Multiplying teen numbers
- Trick for multiplying two-digit numbers
- Trick for multiplying whole numbers
- Trick for multiplying a number by a number ending in five
- Crisscross method for multiplying numbers
- Other resources
- Add and subtract fractions with like denominators (word problems)
- Estimating volume (cups, pints, quarts, and gallons)
- Whole number arithmetic (word problems)
- Other resources
- Elapsed time for analog clocks
- Multiplying numbers or binomials by crossing lines
- Multiplying multi-digit numbers by one-digit numbers with area models
- Equivalent fractions with denominators of 10 or 100
- Multiplying whole numbers by powers of 10
- Adding and subtracting mixed numbers with like denominators (word problems)
- Identifying hundredths on a number line
- Other resources
- Choosing the appropriate unit of measure (inches, feet, yards, miles)
- Rounding decimals (word problems)
- Converting US customary units (word problems)
- Rules for linear equations (graphs)
- Fraction multiplication as scaling
- Estimating the sum of fractions with unlike denominators (word problems)
- Estimating the difference of decimals
- Dividing whole numbers by decimals
- Estimating decimal sums
- Rules for linear equations (tables)
- Volume of right rectangular prisms (word problems)
- Multiplying whole numbers by decimals less than 1
- Adding decimals using a visual model
- Multiplying by taking out factors of 10
- Powers of 10
- Lesson
- Dividing whole numbers by unit fractions
- Adding decimals with tenths
- Adding decimals with hundredths
- Estimating decimal products
- Subtracting fractions with unlike denominators (word problems)
- Converting centimeters to meters
- Multiplying decimals with tenths
- Estimating the product of multi-digit numbers
- Dividing whole numbers to get a decimal
- Add and subtract fractions with unlike denominators (word problems)
- Rounding decimals using a number line
- Finding the volume of composite rectangular prisms
- Adding and subtracting mixed numbers with unlike denominators (no regrouping)
- Adding and subtracting mixed numbers with unlike denominators (regrouping)
- Multiplying fractions using a number line
- Converting yards to inches
- Adding decimals with tenths using shaded rectangles
- Adding fractions with unlike denominators (word problems)
- Division with two-digit divisors by trial-and-error
- Comparing decimals in different forms
- Dividing whole numbers by 0.1 and 0.01
- Subtracting decimals with hundredths
- Multiplying whole numbers and fractions using a number line
- Finding the volume of composite rectangular prisms (word problems)
- Equivalent fractions with denominators of 10 or 100 using shaded rectangles
- Comparing decimals using visual models
- Factoring the GCD out of linear expressions using algebra tiles
- Practice: Volume of right rectangular prisms (mathematical problems)
- Theoretical and experimental probability
- Constants of proportionality
- Other resources
- Other resources
- Probability models
- Lesson
- Finding the circumference from the area of a circle
- Other resources
- Construction of the integers
- Construction of the rationals
- Construction of the reals
- Binomial probability
- Solving permutation equations
- Associative
- Commutative
- Proof: Symmetric difference is associative
- Closed binary operations
- Binary operations and relations
- Is the table a commutative semigroup?
- Klein four-group
- Proof: Inverse of group inverse
- Proof: If \(y\) is a left or right inverse for \(x\) in a group, then \(y\) is the inverse of \(x\)
- Proof: Symmetric difference on power set forms group
- Order of an element in a group
- Proof: Order of group element equals order of inverse
- Cyclic groups
- Proof: Inverse of generator of cyclic group is generator
- Subgroups
- Proof: Centralizer of group element is subgroup
- Proof: Center of group is subgroup
- Normal subgroups
- Proof: The conjugation map is an automorphism
- Other resources
- Law of total probability
- Section formula
- Simplifying trigonometric expressions by factoring
- Simplifying trigonometric expressions using the even and odd identities
- Reciprocal and quotient identities
- Intersection of parabola and line (word problems)
- Solving equations with radicals and rational exponents
- Is it even, odd, or neither? (given an equation)
- Rationalization
- Finding one-sided limits of piecewise functions
- Finding limits of exponential functions
- Limits
- Mutually exclusive events
- Discrete compound interest
- Continuous compound interest
- Adding and subtracting square root expressions with variables
- Rationalizing the numerator
- Modeling piecewise functions (word problems)
- Solving rational equations (word problems)
- 3D vector operations
- Finding the LCM of polynomials
- Properties of binary operations
- Proof: Order of element divides order of finite group
- Finding the equation of a parabola from its graph
- Finding the inverse of quadratic functions (word problems)
- Using the theorem
- Derivative at a point from first principles
- Verifying trig identities
- Direct product of finite cyclic groups
- Adding and subtracting square root expressions without variables (hard)
- Adding and subtracting square root expressions without variables (easy)
- Adding and subtracting square root expressions without variables
- Comparing numbers within 100 (abstract method)
- Trick for dividing a number by five
- Trick for squaring a two-digit number
- Paradox: Bertrand's box paradox
- Trick for multiplying three-digit numbers
- Trick for multiplying four-digit numbers
- Trick for multiplying by numbers one less than a power of ten when number of digits are not equal
- Trick for multiplying numbers near the same power of ten when deviations have the same sign
- Trick for multiplying numbers near the same power of ten when deviations have different signs
- Trick for squaring a number ending in five
- Adding and subtracting fractions by cross-multiplying (no common factor in the denominators)
- Adding and subtracting fractions by cross-multiplying (common factor in the denominators)
- First trick for finding the square root of perfect squares within \(120^2\)
- Trick for multiplying 2-4 digit numbers
- Trick for multiplying numbers near the same multiple of a power of ten
- Trick for multiplying two numbers equidistant from a third number
- Trick for multiplying by 5
- Finger trick for multiplying 1-10 by 9
- Trick for dividing a number by twenty five
- Trick for dividing a number by one hundred twenty five
- Trick for squaring a number near a power of ten
- Trick for squaring a number near a multiple of a power of ten
- Trick for squaring a number near fifty
- Trick for squaring a three-digit number
- Second trick for finding the square root of perfect squares within \(120^2\)
- Trick for cubing a two-digit number
- Trick for cubing a two-digit number near a power of ten
- Trick for finding the cube root of 4-6 digit perfect cubes
- Check your addition by casting out nines
- Check your multiplication by casting out nines
- Trick for percent of a number when percent is a multiple of ten
- Trick for percent of a number
- First trick for percent of a number (percent not always a multiple of ten)
- Multiplying by 6 (Trachtenberg system)
- Trick for finding all primes within one hundred
- Trick for squaring repeating ones, three, or nines
- Trick for listing the multiples of a two digit number
- Trick for doubling numbers
- Trick for multiplying by 2, 4, 8
- Trick for dividing by two, four, and eight
- Trick for multiplying by 5, 25, 50, 500
- Trick for dividing by particular numbers
- Speedy math
- Multiplying by 7 (Trachtenberg system)
- Trachtenberg system
- Multiplying by 8 (Trachtenberg system)
- Multiplying by 2 (Trachtenberg system)
- Multiplying by 3 (Trachtenberg system)
- Multiplying by 4 (Trachtenberg system)
- Multiplying by 9 (Trachtenberg system)
- Multiplying by 11 (Trachtenberg system)
- Multiplying by 12 (Trachtenberg system)
- Proving the altitude of an isosceles triangle cuts two congruent right triangles
- Compare within 10 by counting
- Lesson
- Compare length
- Other resources
- Vampire numbers
- Slide method of multiplication
- Tricks for multiplication
- Tricks for division
- Tricks for roots
- Math of the 15 puzzle
- Computer assisted math
- Missing dollar riddle
- Nim-like games
- Factoring non-monic quadratics using algebra tiles
- Coin problem
- Balance puzzles
- Weighing 9 balls puzzle
- Weighing 10 bags puzzle
- Weighing 12 balls puzzle
- Magic Caterpillars
- Plum Tree
- Instant Insanity
- Overarch 1
- Overarch 2
- Proving properties of even and odd functions (calculus 1)
- Isosceles Reduction (NRICH activity)
- Challenge
- Isosceles (NRICH activity)
- Parabolic Patterns (NRICH activity)
- Pent (NRICH activity)
- Challenge
- Challenge
- Pond Runners (NRICH activity)
- NRICH activities (geometry for undergraduates)
- Challenge
- Nine Colours (NRICH activity)
- Proof
- Challenge
- Other resources
- Partly Circles (NRICH activity)
- Counting Binary Ops (NRICH activity)
- Venn diagrams (combinatorics)
- Challenge
- Challenge
- Laps (NRICH activity)
- Constructing a square inside a triangle
- British flag theorem
- Polyominoes
- Challenge
- Comparing and ordering exponents
- Twisting and Turning (NRICH activity)
- Speeding Boats (NRICH activity)
- Postage (NRICH activity)
- Latin squares
- Euler squares
- Challenge
- Menelaus's theorem
- 18 or 20 trees?
- Proof: Conway's soldiers cannot reach row five
- Conway's Soldiers
- Other resources
- Isosceles Meld (NRICH activity)
- One Golden Equation (Flammable Maths)
- Challenge: Surviving The Deadliest Game
- Combinations with restrictions
- Solving factorial equations
- Basic combinatorics proofs
- Challenge
- Proof
- STEP
- Puzzle ~ Packing Eight Numerals Into Eight Squares (Crystal Clear Maths)
- Using the Pythagorean theorem to find the hypotenuse of a right triangle
- Using the Pythagorean theorem to find a leg of a right triangle
- Solving 30-60-90 triangles
- Solving 45-45-90 triangles
- Menseki Meiro puzzles
- Other resources
- Sigma notation and the binomial theorem
- Properties of sigma notation
- Challenge
- Coffin problems
- Other resources
- Careful counting 5
- Careful counting 6
- Solving linear congruence equations
- Solving quadratic congruence equations
- Finding the line of reflection (grade 8)
- Tangent theorem
- Diagonals of a rhombus are perpendicular bisectors
- Proof: Rhombus diagonals are perpendicular bisectors (linear algebra)
- Proving that opposite angles of a parallelogram are congruent
- Diagonals of a kite are perpendicular
- Opposite sides of a parallelogram are congruent
- A number theory proof (blackpenredpen)
- Area of a rhombus from diagonals
- Other resources
- GMAT math
- Centroid
- Putnam exam
- Newton–Pepys problem
- Potato paradox
- Simplifying rational expressions by factoring and cancelling
- Two envelopes problem
- Careful counting 1
- Challenge
- Other resources
- Direct proof (floor and ceiling)
- Other resources
- Systems of inequalities from a graph
- Fifth root trick
- Multiplying decimals using base ten blocks
- Power of a quotient rule for exponents
- Zero exponents
- Simplifying rational expressions by cancelling
- Find the equation of a square root function under transformations from a graph
- Graphing the cube root function under transformations (level 2)
- Factoring by grouping (multiple variables)
- Factoring by grouping (one variable, not quadratic)
- Solving trigonometric equations linear in form
- Solving trigonometric equations quadratic in form by taking square roots
- One and Three (NRICH activity)
- Challenge
- Centered hexagonal number
- One Basket or Group Photo (NRICH activity)
- Other resources
- Challenge
- Solving linear homogeneous recurrence relations
- Characteristic equation and characteristic roots of linear homogeneous recurrence relations
- Tiling the strip with dominos and monominos
- Arithmagons
- Deriving formulas for figurate numbers
- Visual proof for the sum of the first \(n\) odd natural numbers
- Geometric vanishing puzzles
- What portion is shaded? 2
- Cut into identical pieces 1
- Cut into identical pieces 2
- What portion is shaded? 1
- What portion is shaded? 3
- What portion is shaded? 4
- What portion is shaded? 5
- Four fours
- Ten coins in five rows of four
- Careful counting 2
- Challenge 32: Dots and Boxes (Science Mom)
- What portion is shaded? 6
- Challenge 19: A Chicken and a Half (Science Mom)
- Challenge 20: Timing 9 Minutes (Science Mom)
- What portion is shaded? 7
- Challenge 23: Sum to 100 (Science Mom)
- Careful counting 3
- What portion is shaded? 8
- River crossing puzzle 1
- River crossing puzzle 2
- River crossing puzzle 3
- What portion is shaded? 9
- Careful counting 4
- Careful counting
- What portion is shaded?
- Strimko
- Handshaking couples
- Frogs 1
- How To Solve Amazon's Pair Of Socks Interview Puzzle (MindYourDecisions)
- Picking socks in the dark
- Multiplayer games
- Seven coins in six rows of three
- Challenge 40: Lazy Reflection (Science Mom)
- Careful counting 7
- Challenge 44: Mystery Boxes (Science Mom)
- Challenge 45: Grid Shading Puzzle (Science Mom)
- Challenge 43: Path Counter (Science Mom)
- Proof: Varignon's theorem (linear algebra)
- Billiard Ball Problem (Math for Love)
- Proof: Law of cosines using the dot product
- Geometric proofs using vectors
- Totality (NRICH activity)
- Subtracting decimals with tenths
- Subtracting decimals with thousandths
- Factoring quadratics with difference of squares
- Factoring non-monic quadratics with difference of squares (all resources)
- Other resources
- Factoring non-monic quadratics with difference of squares using algebra tiles
- Other resources
- Adding decimals with thousandths
- Practice: Find a length using the law of sines (mathematical problems)
- Evaluate a function from its graph
- Proof: Thales's theorem using the dot product
- Quinary
- Quinary multiplication
- Proof: Cauchy–Schwarz inequality
- Number of ways to triangulate a regular polygon
- Young tableau
- Determinant of a \(3 \times 3\) matrix using cofactors
- Solving linear systems of equations in two variables using Gauss–Jordan elimination
- Using the theorem
- Other resources
- Basic modular arithmetic proofs
- Set builder notation
- Vacuous truth
- Proof using binomial theorem: Cardinality of power set of finite set
- Proof by induction: Cardinality of power set of finite set
- Cardinality of power set of finite set
- Converting polar equations to Cartesian equations
- Converting Cartesian equations to polar equations
- NRICH activities (grade 7)
- NRICH activities (grade 8)
- Using the binomial theorem
- Proving the binomial theorem by induction
- Making rectangles with a given area from unit squares
- Combinatorial proof: Binomial theorem
- Challenge
- Other resources
- NRICH activities (grade 6)
- Challenge problems (geometry for high schoolers)
- NRICH activities (geometry for high schoolers)
- Lambert W function
- Gabriel's Horn
- Add using number disks
- Direct product of groups
- NRICH activities (combinatorics for high schoolers)
- Group homomorphisms
- Sum of exterior angles (algebra problems)
- Epsilon-delta proofs for limits with one variable
- Proof: Pythagorean theorem
- Proving the inverse Pythagorean theorem
- Similarity proof: Pythagorean theorem
- Epsilon-delta proofs for limits with two variables
- Jack Oliver's proof: Pythagorean theorem
- Proofs: Pythagorean theorem
- Survivorship bias
- Derivation: Area of a non-obtuse triangle
- James Garfield's proof: Pythagorean theorem
- Bhaskara's proof: Pythagorean theorem
- Epsilon-delta proof: Sum of limits
- Number tricks
- Bayes's theorem and false positives
- Proving properties of absolute value
- Proving the triangle inequality
- Proving the reverse triangle inequality
- Finding delta given epsilon
- Proof: 1089 trick
- Subsets
- Supremum and infimum
- Interior point of a set
- Proof: Cantor set is uncountable
- Disproof by counterexample
- Dihedral group
- Proof: One-step subgroup test
- Simple groups
- Solvable groups
- Quotient groups
- Proof: Union of countably many countable sets is countable
- Proof: Cardinality of Cartesian product
- Proof: Group abelian iff cross cancellation property
- Alternating groups
- Proof: First isomorphism theorem for groups
- Proof: Second isomorphism theorem for groups
- Proof: Fundamental homomorphism theorem
- Proof: Special linear group is subgroup of general linear group
- Proof: Two-step subgroup test
- Proof: Finite subgroup test
- Group automorphisms
- Kernel of a group homomorphism
- Proving group homomorphisms
- Group isomorphisms
- Compositions of group morphisms
- Other resources
- Basic group theory proofs
- Multiplying permutations in cycle notation
- Multiplying permutations in array notation
- Add within 10 (concrete or pictorial)
- Subtraction word problems within 10
- Addition word problems within 10
- Other resources
- Cosets of a subgroup
- Cosets and Lagrange's theorem
- Proving equivalence relations
- Equivalence classes
- Number of edges in a complete graph
- Other resources
- Derivative using u-substitution
- Size of tree is one less than order
- Properties of relations
- Representing relations using digraphs
- Basic set theory proofs using Venn diagrams
- Basic set theory proofs using predicate logic
- What are De Morgan's laws?
- Inverse of composite relation
- Composition of relations
- Converting English to predicate logic
- Combining relations
- Representing relations using matrices
- Intersection of equivalence relations
- Set is subset of intersection of supersets
- Intersection is largest subset
- Union is smallest superset
- Supersets
- Image and preimage of a set under a function
- Intro to continuity
- Proving Cauchy sequences
- Proving divergence using proof by contradiction
- Proving divergence using subsequences
- Every convergent sequence is a Cauchy sequence
- Sum rule for Cauchy sequences
- Archimedean principle
- Using the Archimedean principle
- Rationals are dense in the reals
- Proving properties of supremum
- What is the least upper bound property?
- Rates and proportional relationships
- Epsilon-delta proof: Product of limits
- Proof
- What is the cardinality of a set?
- Proving properties of the empty set
- Multiplying perfect squares using algebra tiles
- Finding the smallest multiplier to produce a perfect square
- Sum of squares of binomial coefficients
- What is a vector space?
- Proving properties of Cartesian product
- Properties of the power set
- Combinatorial proof: Cardinality of power set of finite set
- Proof: Least upper bound property
- What are upper and lower bounds?
- Proving limits of sequences are unique
- Euclid's lemma
- Sufficient conditions for a group to be abelian
- Other resources
- Proof: Cantor's theorem
- Construct a rhombus with given side
- Construct a rhombus with given side and angle
- Copy an angle
- What is a complete graph?
- What is a spanning subgraph?
- What is an edge-induced subgraph?
- Proof: Derivative of \(\cot(x)\)
- Other resources
- Isomorphic graphs
- What is a vertex-induced subgraph?
- What is a cubic graph?
- What is an irregular graph?
- Proving isosceles trapezoids
- Converting between array notation and cycle notation
- Converting cycle notation to array notation
- Converting array notation to cycle notation
- What is cycle notation?
- What is an equivalence relation?
- What is a relation? (for undergraduates)
- Verifying equivalence relations on groups
- Using the sum identities
- Epsilon-delta proofs of uniform continuity
- Properties of scalar multiplication
- Proof: Scalar multiplication is associative
- Proof: Dot product formula
- Proof: Cross product distributes over addition
- Number patterns (grade 4)
- What are decimal numbers?
- Multiplying one-digit by two-digit numbers using partial products
- Multiplying three one-digit numbers
- Proof
- Sophie Germain's identity
- Fermat's little theorem
- Proving irrationality
- Basic set theory proofs using membership tables
- Basic propositional logic proofs using truth tables
- What is hypothetical syllogism?
- Converse, inverse, and contrapositive statements
- Converting a Venn diagram to a set expression
- Converting a set expression to a Venn diagram
- Converting between set expressions and Venn diagrams
- Intro to integral ideals
- Other resources
- Testing validity of syllogisms by drawing Venn diagrams
- Square of opposition
- Proof by induction (simplifying sums)
- Shading Venn diagrams with two sets
- Shading Venn diagrams with three sets
- Difference of squares
- Basic ring theory proofs
- More ring theory proofs
- Proving limits using the squeeze theorem
- What is the squeeze theorem?
- Solving trigonometric equations using the quadratic formula
- Adding within 1000 (word problems)
- Add within 20 (word problems)
- Missing number for subtraction within 1000
- Derivation
- Finding area of regular polygon given apothem or perimeter
- Percents: Finding the whole (word problems)
- Other resources
- Graphing linear equations by plotting points
- Is it a ring?
- What is a subring?
- Proving properties of injective, surjective, and bijective functions
- Finding absolute extrema on a closed interval
- Using the theorem
- NRICH activities (probability theory for high schoolers)
- NRICH activities (algebra 2)
- Method of differences
- Other resources
- Intro to even and odd functions
- Proof: Power reducing identities
- Power reducing trig identities
- Proofs
- Pascal's identity
- Connection between Pascal's identity and Pascal's triangle
- Properties of Pascal's triangle
- Explanation for powers of eleven in Pascal's triangle
- Using the power reducing identities to graph
- Partial fraction decomposition (linear factors)
- Partial fraction decomposition (repeated linear factors)
- Partial fraction decomposition (quadratic factors)
- Partial fraction decomposition (repeated quadratic factors)
- Partial fraction decomposition (improper fractions)
- The cover-up method and why it works
- False proofs by induction
- Finding absolute value transformations from graphs
- Intro to bipartite graphs
- Maximization (word problems)
- Minimization (word problems)
- Write a system of linear inequalities given a graph
- Finding square root transformations from graphs
- Using the sine rule for area of a triangle
- Using the theorem
- Finding equations of sine and cosine functions from graphs
- Finding transformations of sine and cosine functions from equations
- Intro to graphing sine and cosine under transformations
- Converting between English and propositional logic
- Converting propositional logic to English
- Making truth tables for expressions
- Simplifying expressions in propositional logic
- Converting between set-builder and set-roster notation
- Converting set-builder notation to set-roster notation
- Converting set-roster notation to set-builder notation
- Operations with set-builder notation
- Testing validity of simple statements in predicate logic
- Converting set theory expressions to predicate logic expressions
- Basic predicate logic proofs
- Union and intersection of a family of sets
- Direct proof (inequalities)
- Intro
- Adding and subtracting linear and quadratic functions
- Adding functions (linear-quadratic)
- Adding functions (piecewise linear)
- Adding functions (other)
- Subtracting functions (quadratic-linear)
- Square root of a complex number
- Finding volume using unit cubes
- Other resources
- Roots of complex numbers using De Moivre's theorem
- Interactive: Sine and cosine functions under transformations
- Measuring angles using a protractor
- Graphically finding points of inflection
- Using the second derivative to find points of inflection
- Intro to digraphs
- Other resources
- False proofs for calculus 2
- Derivatives of hyperbolic and inverse hyperbolic functions
- Proving the derivatives of the hyperbolic functions
- Other resources
- Other resources
- Hyperbolic and inverse hyperbolic functions
- Proving the derivatives of the inverse hyperbolic functions
- Arc length formula
- Lesson
- Practice
- Practice: Simpson's rule
- NRICH activities (algebra 1)
- Multiplying fractions (mathematical problems)
- What is a quadrilateral?
- Factoring quadratics by trial and error
- Deriving point-slope form
- What is point-slope form?
- Converting point-slope form to slope-intercept form
- Basic integration problems
- Definite integrals by u-substitution
- Reverse power rule for definite integrals
- Convert to vertex form by completing the square (using algebra tiles)
- Solve by completing the square
- Factoring by completing the square
- Find the missing side of the rectangle given the area
- Integration by u-substitution (more difficult)
- Integration by u-substitution
- Proving properties of the subset relation
- Counting (kindergarten)
- Making 10 with written numerals
- Add within 20 using fingers
- Subtract within 5 using counters (concrete and pictorial)
- Subtract within 5 (abstract)
- Subtract within 20 (concrete and pictorial methods)
- Subtract within 20 using fingers
- Counting by 1s (1 to 20)
- Lesson
- Practice
- Lesson
- Practice
- Practice
- Practice
- Lesson
- Practice
- Lesson
- Practice
- Probability vs odds
- Other resources
- Other resources
- Lesson
- Practice
- Lesson
- Practice
- Practice
- Intro to functions
- Radius and tangent theorem
- What is a maximal clique?
- Subgraphs
- Alternate segments theorem
- Subtraction and place value
- Lesson: Adding and subtracting within 1000 using a number line
- Practice: Adding and subtracting within 1000 using a number line
- Adding or subtracting by adjusting
- Lesson
- Practice
- Paths and cycles
- Planar graphs
- Coloring
- Basics of graph theory
- Intro to vertex colorings
- What is an orthogonal set?
- Other resources
- Other resources
- Other resources
- Subtracting (grade 1)
- Other resources
- Lesson
- Practice
- Lesson
- Practice
- Measurement (grade 2)
- Geometry (grade 2)
- Rewriting before integrating
- Practice
- Other resources
- Counting by 1s (1 to 50 and 50 to 1)
- Counting by 1s (1 to 100)
- Other resources
- Counting to 5
- Counting by 5s (5 to 100)
- Counting by 2s (2 to 100)
- Subtract within 1000 using a number line
- Understand fractions
- Adding and subtracting (grade 3)
- Area (grade 3)
- Intro to division
- Lesson
- Practice
- Plane figures (grade 4)
- Units of measure (grade 4)
- Understand decimals (grade 4)
- Factors, multiples, and patterns (grade 4)
- Place value (grade 4)
- Properties of shapes (grade 5)
- Lesson
- Practice
- Lesson: Dividing decimals by whole numbers
- Practice: Dividing decimals by whole numbers
- Find the mistake in one-step equations
- Lesson: Find the mistake in one-step equations
- Practice: Find the mistake in one-step equations
- Lesson
- Practice
- Intro to equations and inequalities (grade 6)
- Solving multi-step equations (word problems)
- Lesson
- Practice
- Geometry (grade 6)
- Geometry
- Geometry (grade 7)
- Intro to proof by induction
- Proof: Integers are countable
- Lesson: What is the image of a set under a function?
- Proofs: Image and preimage of a set under a function
- Lesson: What is the preimage of a set under a function?
- Line integrals
- Partial derivatives
- Conservative vector fields
- Curl and divergence of vector fields
- Vector magnitude
- Double integrals over rectangular regions
- Directional derivatives
- Total derivatives
- Lesson: Converting decimals to percents
- Practice: Converting decimals to percents
- Polygons on the coordinate plane
- Practice
- Lesson
- Area and perimeter of polygons on the coordinate plane
- Drawing polygons on the coordinate plane
- Lesson: Drawing polygons on the coordinate plane
- Practice: Drawing polygons on the coordinate plane
- Dividing fractions (word problems)
- Other resources
- Lesson: Volume of right rectangular prisms (word problems)
- Practice: Volume of right rectangular prisms (word problems)
- Lesson: Volume of right rectangular prisms (mathematical problems)
- Volume of right rectangular prisms (mathematical problems)
- Adding and subtracting decimals (word problems)
- Other resources
- Area of a triangle on the coordinate plane
- Find the missing vertex of the parallelogram
- Lesson: Find the missing vertex of the parallelogram
- Practice: Find the missing vertex of the parallelogram
- Is it even, odd, or neither? (given a graph)
- Practice
- Practice
- Finding the inverse of quadratic functions (mathematical problems)
- Determining whether two functions are inverses of each other
- Deriving the t-formulas
- Disproving trigonometric identities by counterexample
- Lesson: Evaluating trigonometric functions using the unit circle
- Practice: Evaluating trigonometric functions using the unit circle
- Lesson
- Practice
- Practice
- Comparing rates
- Lesson: Comparing rates
- Practice: Comparing rates
- Lesson
- Practice
- Lesson: Solving and graphing absolute value inequalities (word problems)
- Practice: Solving and graphing absolute value inequalities (word problems)
- Solving and graphing absolute value inequalities (mathematical problems)
- Solving and graphing absolute value inequalities (word problems)
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Practice
- Solving systems of linear equations in two variables by substitution (mathematical problems)
- Lesson
- Practice
- Solving linear systems of equations in two variables by elimination (harder mathematical problems)
- Solving linear systems of equations in two variables by elimination (word problems)
- Lesson
- Practice
- Practice
- Graphing quadratic equations from tables
- Lesson
- Practice
- Cardinality from Venn diagrams
- Example of a nonconstructive proof
- Lesson
- Other resources
- Lesson
- Practice
- Order of operations (grade 3)
- Lesson
- Practice
- Comparing and ordering exponent expressions
- Derivation: Area of a parallelogram
- Recognizing common 3D shapes
- Lesson: Recognizing common 3D shapes
- Practice: Recognizing common 3D shapes
- Long division (one-digit divisors)
- Lesson
- Practice
- Naming angles
- Other resources
- Lesson: Multiplying 10s
- Practice: Multiplying 10s
- Unit fractions (grade 3)
- Lesson: Plotting fractions on the number line
- Practice: Plotting fractions on the number line
- Number patterns (grade 5)
- Other resources
- Area of a rectangle with whole number side lengths
- Lesson
- Practice
- Lesson
- Practice
- Lesson: Dividing fractions by whole numbers
- Practice: Dividing fractions by whole numbers
- Lesson
- Practice
- Dividing unit fractions by whole numbers
- Lesson
- Practice
- Rounding mixed numbers
- Rounding improper fractions
- Lesson
- Practice
- Lesson: Multiplying whole numbers by powers of 10
- Practice: Multiplying whole numbers by powers of 10
- Place value (grade 5)
- Lines, line segments, and rays
- Order of operations (grade 5)
- Adding and subtracting fractions with unlike denominators (word problems)
- Converting between words and decimals
- Subtracting mixed numbers by converting to improper fractions
- Subtracting mixed numbers with regrouping
- Subtracting mixed numbers (word problems)
- Adding mixed numbers with regrouping
- For teachers: Add within 20 using number bonds
- Counting with tally marks
- Practice
- Lesson
- Practice
- Rounding to the nearest 10 or 100 (abstract)
- Rounding to the nearest 10 or 100 (concrete and pictorial)
- Line plots (grade 5)
- Lesson
- Practice
- Decomposing angles
- Lesson
- Practice
- Other resources
- Lesson
- Practice
- Adding and subtracting (grade 2, concrete and pictorial)
- Adding and subtracting (grade 2, abstract)
- Lesson
- Practice
- Subtract by compensating
- Lesson: Comparing fractions with the same numerators or denominators
- Practice: Comparing fractions with the same numerators or denominators
- Measuring to the nearest quarter inch
- Measure lengths (cm, m)
- Measurement (grade 3)
- Lesson
- Practice
- Measuring lengths in different units
- Other resources
- Lesson
- Practice
- Other resources
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Practice
- Telling time to the nearest 15 minutes
- Lesson
- Practice
- Adding within 100 (mental)
- Lesson
- Practice
- Adding within 100 (concrete and pictorial methods)
- Estimating mass (grams and kilograms)
- Lesson
- Practice
- Solving two-step equations (concrete and pictorial)
- Solving two-step equations (abstract)
- Find the mistake in two-step equations
- Solving one-step equations (concrete and pictorial)
- Solving one-step equations (abstract)
- Hard factoring problems
- Other resources
- Lesson
- Practice
- Practice
- Solving and graphing linear inequalities by clearing fractions
- Solving linear equations using reciprocals
- Subtract within 20 using ten frames
- Culmination: Multiplying and dividing fractions
- Lesson
- Practice
- Intro
- Other resources
- Recognize fractions greater than 1
- Recognize fractions
- Lesson
- Practice
- Lesson
- Practice
- Solving quadratic equations (word problems)
- Adding integers (concrete and pictorial)
- Adding integers (abstract)
- Other resources
- Solving and graphing two-step inequalities in one variable (word problems)
- Solving and graphing two-step inequalities in one variable (mathematical problems)
- Find the explicit formula for a geometric series given terms
- Intro to parallel and perpendicular lines
- Lesson
- Practice
- Recognize ratios
- Ratio tables
- Equivalent ratios (mathematical problems)
- Equivalent ratios (word problems)
- Ratios with tape diagrams
- Lesson: Ratios with tape diagrams
- Practice: Ratios with tape diagrams
- Lesson: Ratio tables
- Practice: Ratio tables
- Ratios and units of measurement
- Lesson
- Practice
- Part-to-whole ratios
- Lesson: Adding integers using a number line
- Practice: Adding integers using a number line
- Lesson: Inverse property of addition
- Dependent and independent variables
- Lesson
- Practice
- Lesson: Common denominators
- Practice: Common denominators
- Representative samples
- Graphing parabolas by plotting points
- Number opposites
- Lesson: Number opposites
- Practice: Number opposites
- Lesson
- Practice: Plotting integers on a number line
- Plotting rational coordinates
- Lesson
- Practice
- Plotting whole number coordinates
- Lesson
- Practice
- Other resources
- Lesson
- Practice
- Writing numerical inequalities
- Lesson
- Practice
- Rates with fractions
- Relating whole number division to multiplication (mathematical problems)
- Relating whole number division to multiplication (word problems)
- Relating fraction division to multiplication
- Lesson
- Practice
- Subtraction as adding the opposite
- Lesson
- Practice
- Writing proportional equations
- Writing proportional equations from tables
- Writing proportional equations (word problems)
- Lesson: Writing proportional equations (word problems)
- Practice: Writing proportional equations (word problems)
- Lesson: Writing proportional equations from tables
- Practice: Writing proportional equations from tables
- Find the center and scale factor
- Lesson
- Practice
- Comparing distributions
- Statistics and probability (grade 7)
- Comparing areas and perimeters of rectangles
- Lesson
- Practice
- Other resources
- Sampling methods
- Lesson
- Practice
- Comparing and ordering absolute values
- Lesson
- Practice
- Modeling with linear functions
- Lesson
- Practice
- Reflecting figures
- Lesson: Reflecting line segments and polygons (grade 8)
- Practice: Reflecting line segments and polygons (grade 8)
- Finding the line of reflection (geometry for high schoolers)
- Lesson
- Practice
- Trends in categorical data
- Lesson
- Practice
- Perimeter (grade 3)
- Perimeter (grade 3, mathematical problems)
- Perimeter (grade 3, word problems)
- Uniform and non-uniform probability models
- Finding the volume of composite rectangular prisms (mathematical problems)
- Lesson
- Practice
- Factor with the distributive property (no variables)
- Lesson
- Practice
- Measuring and comparing lengths (grade 2)
- Other resources
- Lesson
- Practice
- Lesson
- Practice
- Other resources
- Lesson
- Practice
- Other resources
- Subtracting multiples of 10 within 100
- Other resources
- Other resources
- Missing digit problems for adding and subtracting within 1000
- Describing measurable attributes of objects
- Converting decimals between standard form, expanded form, and number names
- Equal sign
- Other resources
- Lesson: Representing numbers using base ten blocks (grade 4)
- Practice: Representing numbers using base ten blocks (grade 4)
- Adding three numbers in any order (mathematical problems)
- Adding three numbers in any order (word problems)
- Other resources
- Count in order
- Other resources
- Other resources
- Making and reading bar and picture graphs (grade 2)
- Reading bar graphs (grade 2)
- Making picture graphs
- Reading picture graphs
- Lesson: Reading bar graphs (grade 2)
- Practice: Reading bar graphs (grade 2)
- Lesson: Making bar graphs
- Practice: Making bar graphs
- Lesson
- Practice
- Lesson
- Practice
- Two-step word problems (grade 2)
- Lesson
- Practice
- Other resources
- Negative signs in fractions
- Fractions of shapes
- Lesson
- Practice
- Other resources
- Other resources
- Representing decimals using shaded rectangles
- Lesson
- Practice
- Other resources
- Comparing numbers within 100
- Estimate lengths
- Lesson
- Practice
- Using different strategies to add and subtract (grade 2)
- Other resources
- Length word problems
- Other resources
- Lesson
- Practice
- Drawing angles in degrees
- Other resources
- Lesson
- Practice
- Subtracting within 100 (concrete and pictorial methods)
- Add and subtract within 100 on the number line
- Adding and subtracting within 100 (abstract methods)
- Patterns in arithmetic
- Patterns in sequences
- Patterns with even and odd
- Patterns in hundreds chart
- Patterns in multiplication tables
- Lesson: Patterns in hundreds chart
- Practice: Patterns in hundreds chart
- Lesson: Patterns in multiplication tables
- Practice: Patterns in multiplication tables
- Lesson: Patterns in sequences
- Practice: Patterns in sequences
- Lesson
- Practice
- Other resources
- Commutative property of multiplication (grade 3)
- Lesson
- Practice
- Using the distributive property to multiply (grade 3)
- Using the associative property of multiplication (grade 3)
- Lesson
- Practice
- Other resources
- Lesson
- Two-step word problems (grade 3)
- Lesson
- Practice
- Other resources
- Lesson
- Practice
- Area and the distributive property
- Lesson
- Practice
- Translating to multiplication (mathematical problems)
- Lesson
- Practice
- Adding and subtracting mixed numbers with like denominators (mathematical problems)
- Multiplying unit fractions and whole numbers
- Lesson: Multiplying unit fractions and whole numbers
- Practice: Multiplying unit fractions and whole numbers
- Area and perimeter of a rectangle (grade 4)
- Lesson
- Practice
- Representing 1 as a fraction
- Lesson: Representing 1 as a fraction
- Practice: Representing 1 as a fraction
- Representing any whole number as a fraction
- Lesson
- Practice
- Music videos: Multiplication (grade 4)
- Multiplying whole numbers and fractions (concrete and pictorial methods)
- Multiplying fractions using the area model
- Lesson
- Practice
- Other resources
- Classifying shapes by line and angle types
- Equivalent fraction and whole number multiplication expressions
- Lesson
- Practice
- Other resources
- Lesson
- Practice
- Other resources
- Change in median and mean after shifting, adding, or removing a data point
- Lesson
- Practice
- Fractions as division
- Dividing with unit fractions
- Identify coordinates
- Lesson
- Practice
- Testing solutions to two-variable inequalities
- Lesson
- Practice
- Modeling one-variable inequalities
- Lesson
- Practice
- Challenge problems (statistics for high schoolers)
- Statistical and non-statistical questions
- Lesson
- Practice
- Line plots (grade 2)
- Find the sequence of rigid transformations
- Other resources
- Find measures using rigid transformations
- Preserved properties of rigid transformations
- Reading box plots
- Interpreting quartiles
- Identifying outliers from box plots
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Practice
- Analyzing graphs of step functions
- Interpreting linear expressions
- Lesson
- Practice
- Approximating with powers of 10
- Lesson
- Practice
- Expression value intuition
- Lesson
- Practice
- Addition snake game (part 1)
- Addition snake game (part 2)
- Negative snake game
- Estimating volume (milliliters and liters)
- One-step word problems with volume
- Lesson
- Practice
- One-step word problems with masses
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Practice
- Reading dot plots and frequency tables
- Making dot plots and frequency tables
- Dot plots and frequency tables
- Shape of distributions
- Identifying symmetric and skewed distributions
- Clusters, gaps, peaks, and outliers
- Lesson
- Practice
- Lesson
- Practice
- Practice
- Comparing linear functions
- Comparing linear functions (mathematical problems)
- Comparing linear functions (word problems)
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Practice
- Other resources
- Dividing unit fractions by whole numbers (concrete and pictorial methods)
- Dividing unit fractions by whole numbers (abstract methods)
- Lesson
- Practice
- Dividing whole numbers by unit fractions (concrete and pictorial methods)
- Dividing whole numbers by unit fractions (abstract methods)
- Lesson
- Practice
- Other resources
- Lesson
- Practice
- Other resources
- Place value blocks within 1,000
- Lesson
- Practice
- Place value tables within 1,000
- Histograms
- Making histograms
- Reading histograms
- Lesson
- Practice
- Other resources
- Counting by 1s starting at any number
- Number of observations from line plots and histograms
- Converting complex numbers between polar and rectangular form
- Complex number conjugates
- Practice
- Word problems with multiple units
- Components of vectors
- Lesson
- Practice
- Area of a parallelogram using determinant
- Subtracting vectors
- Other resources
- Lesson
- Practice
- Simple hypothesis testing
- Lesson
- Practice
- Magnitude and direction form of vectors
- Lesson
- Practice
- Properties of matrix multiplication
- Solving quadratics by u-substitution
- Other resources
- Equivalent vectors
- Other resources
- Make a dilation given the center and scale factor
- Dilating lines
- Sum of interior angles of a simple polygon (using the formula)
- Sum of interior angles of a simple polygon (rearranging the formula)
- Adding within 100 (abstract method)
- Multiplying integers (concrete and pictorial methods)
- Multiplying integers (abstract methods)
- Compare within 5 by pairing
- Book reviews (discrete math)
- Bridge course books
- Abstract algebra books for undergrads
- Book reviews (linear algebra)
- Book reviews (calculus)
- Book reviews (complex analysis)
- Book reviews (field theory)
- Book reviews (number theory)
- Abstract algebra books for graduates
- Topology books
- Book reviews (differential equations)
- Compare within 5 by counting
- Compare within 5
- Compare within 10
- Long division using the box method
- Advice from Eddie Woo
- Making residual plots
- Lesson
- Practice
- Hypothesis testing in experiments
- Lesson
- Practice
- Double negation introduction and elimination
- Advice
- Advice from The Math Sorcerer
- Surface area using nets (word problems)
- Surface area using nets (mathematical problems)
- Missing value given the mean
- Lesson
- Practice
- Age problems (challenge problem)
- Lesson
- Book reviews (naive set theory)
- Book reviews (graph theory)
- Book reviews (probability theory)
- Add within 20 using a number line
- Subtract within 20 by counting forward on the number line
- Subtract within 20 by counting backward on the number line
- Subtract within 20 (abstract methods)
- Add by making 100s
- Lesson
- Practice
- Data (grade 2)
- Addition, subtraction, and estimation
- More with multiplication and division
- Data (grade 3)
- Arithmetic patterns and problem solving
- Lesson
- Practice
- Multiply fractions
- Measuring angles
- Add and subtract fractions (grade 5)
- Coordinate plane
- Coordinate plane word problems
- Lesson
- Practice
- Algebraic thinking
- Dividing fractions
- Multiplying fractions
- Negative numbers
- Arithmetic operations
- Data and statistics (grade 6)
- Expressions, equations, and inequalities
- Fractions, decimals, and percentages
- Travel graphs
- Numbers and operations
- Linear functions and equations
- Data and modeling
- Algebra word problems
- Simplifying square roots of monomials with real variables
- Simplifying square roots of monomials with nonnegative real variables
- Lesson
- Practice
- Lesson
- Practice
- Adding and subtracting radical expressions
- Lesson
- Practice
- Practice
- Transpose and its properties
- Other resources
- Practice
- Practice
- Practice
- Linear pair theorem
- Using the formula: Area of a parallelogram
- Practice
- Lesson
- Congruence
- Other resources
- Expressions representing consecutive terms
- Other resources
- Practice
- Practice
- Normal distributions and the empirical rule
- Probability density functions
- Median of a discrete random variable
- Practice
- Mode of a continuous random variable from a probability density function
- Practice
- Practice
- Practice
- Practice
- Squaring large numbers
- Practice
- Practice
- Practice
- Practice
- Book reviews (real analysis)
- Other resources
- Intro
- Distance and midpoint of complex numbers
- Angles standing on the same arc are congruent
- Thales's theorem and its converse
- Theorem 2
- Using the theorems
- Using the theorem
- Using the theorem
- Proof
- Intro
- Using the theorem
- Intro
- Proof
- Even and odd integers
- Proofs
- Intro
- Proof
- Using the formula
- Find an absolute value equation from its solutions
- Solving absolute value equations by graphing
- Lesson
- Practice
- Solving first-order linear differential equations
- Working with units
- Intro
- Solving multi-step equations (geometry)
- Determining if a discrete function is invertible
- Restrict the domain then find the inverse
- Slope-intercept form from word problems
- Solving systems of differential equations
- Proof
- Intro
- Lesson
- Practice
- Other resources
- Parity proofs using modular arithmetic
- Intro to modular arithmetic
- Congruence in modular arithmetic
- Solving and graphing rational inequalities using a sign chart
- Division in modular arithmetic
- Bezout's identity
- Addition and subtraction in modular arithmetic
- Intro
- Proof
- Euclid's GCD algorithm
- Finding Bezout coefficients
- Equations and inequalities
- Graceful labeling
- Proving the difference of squares formula
- Lesson
- Practice
- Proof
- Using the theorem
- Intro to naive set theory
- Intro
- Proofs
- Using the theorems
- Solving literal equations using properties of logarithms
- Practice
- Intro
- Proof
- Using the theorem
- Intro
- Using the theorem
- Definite integral of an exponential function
- Definite integral of a piecewise polynomial function
- Definite integral of a piecewise trigonometric function
- Definite integral of a trigonometric function
- Functions
- Other resources
- Identify the value of the digit (grade 4)
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Practice
- Practice
- Intro
- Using the theorem
- Lesson
- Practice
- Lesson
- Practice
- Other resources
- Intro to double integrals
- Changing the order of integration
- Solving equations by factoring
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Using the theorem
- Proof
- Using the theorem
- Intro to metric spaces
- Finding the tangent or normal line through the given point using implicit differentiation
- Implicit differentiation
- Second derivative using implicit differentiation
- Find the absolute value equation from its intercepts
- Find the absolute value equation from a description
- Finding \(x\) and \(y\) intercepts of absolute value equations algebraically
- Functional equations
- Cauchy's functional equation
- Proving a function satisfies a functional equation
- Lesson
- Practice
- Transforming a point
- Composition of transformations
- Identifying functions algebraically
- Challenge problems
- Volume of a cone (calculus 2)
- Other resources
- Proving the volume of a sphere formula by integrating
- False proof that all triangles are equilateral
- Interviews of mathematicians
- Challenge
- Other resources
- Other resources
- Art featuring mathematicians or mathematical objects
- Challenge
- Challenge
- Challenge
- Division (mathematical problems, concrete and visual)
- Division (word problems, concrete and visual)
- Checkerboard proofs
- Challenge
- Challenge
- Alternative quadratic formula
- Solving linear Diophantine equations in two variables
- Challenge
- Intro
- Midsegment and proportionality theorems
- Intro
- Proof
- Using the theorem
- Converse of the triangle proportionality theorem
- Proof
- Using the theorem
- Proof 1
- Proof 2
- Using the theorem
- Intro to scientific notation
- Intro to triangle congruence
- Converse of the corresponding angles postulate
- Converse of the alternate interior angles theorem
- Converse of the alternate exterior angles theorem
- Triangle proportionality theorem
- Proof
- Finding relative extrema from the derivative
- Maximizing profit (optimization problems)
- Challenge
- Challenge
- Challenge
- Challenge
- Challenge
- Pythagorean triples
- Challenge
- Challenge
- Challenge
- Combining ratios (word problems)
- Combining ratios (mathematical problems)
- Combining ratios
- Lesson
- Lesson
- Practice
- Square It
- First to 22
- Stop the Clock
- Dilating polygons
- Lesson
- Intro to dilating polygons
- Lesson
- Practice
- Intro to calc 2
- Intro to trigonometry
- Solving linear systems of equations in two variables by elimination (easier mathematical problems)
- GCD and LCM by listing factors and multiples
- GCD and LCM from prime factorizations
- Practice
- Distance-speed-time problems
- Multiplying mixed numbers (harder mathematical problems)
- Lesson
- Practice
- Lesson
- Practice
- Fourth proof of the sum and difference identities
- First proof of the sum and difference identities
- Second proof of the sum and difference identities
- A pattern in Pascal's triangle
- Third proof of the sum and difference identities
- Rationalizing the denominator (type 1)
- Intro
- Inverse function theorem
- Complex numbers in polar form
- Powers of complex numbers using De Moivre's theorem
- Trigonometric identities
- Solving trigonometric equations using the auxiliary angle method
- Auxiliary angle method
- Simplifying the sum of two wave functions using the auxiliary angle method
- Challenge
- Challenge
- Odds and Threes
- Challenge
- Dicey Operations in Line
- Lesson
- Practice
- Challenge
- Determine whether linear, quadratic, or neither, using second differences
- Challenge
- Challenge
- Lesson
- Practice
- Other resources
- Solving proportions using inverse operations
- Lesson
- Practice
- Solving rational equations (mathematical problems)
- Lesson
- Practice
- Challenge
- Using the Pythagorean theorem (harder mathematical problems)
- Lesson
- Practice
- Finding quadratic equations from tables using second differences
- Other resources
- Lesson
- Practice
- Challenge
- One-player games
- Quadrilaterals
- Lesson
- Practice
- Lesson
- Practice
- Method 2
- Challenge
- Other resources
- Combining like terms (concrete and visual method)
- Lesson
- Practice
- Challenge
- Trick for percents
- Second trick for percent of a number (percent not always a multiple of ten)
- Challenge
- Challenge
- Challenge
- Challenge
- Challenge
- Challenge
- Challenge: Divisible by 13
- Lesson
- Practice
- American Billions
- Lesson
- Practice
- Lesson
- Practice
- Sum and product of roots of quadratics
- Proving the sum formula for sine and cosine using Euler's formula
- Intro
- Proof
- Angles around a point
- Lesson
- Intro to tessellations
- Proving properties of trees
- Intro to trees
- Challenge
- Challenge
- Proof
- Using the theorem
- Proof
- Diagonals of a rectangle are congruent
- Proof 2
- Using the theorem
- Consecutive interior angles theorem
- Proof 1
- Proof
- Converse of the consecutive interior angles theorem
- Intro
- Using the theorem
- Challenge
- Using the theorem
- Proof
- Challenge
- Challenge
- Challenge
- Challenge
- Tidbits
- Tidbits on knot theory
- Intro
- Lesson
- Practice
- Lesson
- Challenge
- Challenge
- Integral ideals
- Proving multiples form an integral ideal
- Additive property of angles
- Lesson
- Practice
- Lesson
- Practice
- Intro
- Proving the product rule from first principles
- Intro
- Intro
- Intro
- Proof
- Proving properties of even and odd functions (algebra 2)
- Lesson
- Practice
- Other resources
- Kernel of a group homomorphism
- Intro to group homomorphisms
- Abelian groups
- Lesson
- Practice
- Challenge
- Lesson
- Practice
- Spiralling Decimals
- Lesson
- Practice
- Finding critical values from graphs
- Lesson
- Practice
- Lesson
- Practice
- Proof 1
- Group actions
- Lesson
- Practice
- Tidbits on integer sequences
- Principle of explosion
- Deductive, inductive, and abductive reasoning
- Lesson
- Practice
- Proof
- Intro
- Using the theorem
- Lesson
- Practice
- Intro to solving separable differential equations
- Solving separable differential equations with growth and decay
- Solving more complicated separable differential equations
- Volumes of solids with known cross sections
- Work done by a variable force
- Center of mass (calculus 2)
- Center of mass for one and two-dimensional systems
- Center of mass for planar lamina
- Pappus's theorem
- Hydrostatic force
- Intro to vectors
- Proving equivalence of the dot product definitions
- Proving a property of the cross product
- Relating the cross product to the area of a parallelogram
- Vector functions
- Multiple integrals
- Iterated integrals
- Midpoint rule for double integrals
- Riemann sums for double integrals
- Average value of a double integral
- Orbit stabilizer theorem
- Double integrals over general regions
- Finding equations of lines and planes
- Green's theorem
- Surface integrals
- Higher order partial derivatives
- First order partial derivatives
- Intro to partial derivatives
- Curl and divergence
- Finding limits using u-substitution
- Proving the derivative of the natural logarithm
- Lesson
- Tidbits on the Riemann hypothesis
- Tidbits on Goldbach's conjecture
- Proofs
- Integration techniques
- Lesson
- Practice
- Integrating quadratics by completing the square
- Lesson
- Practice
- Sum, difference, and reverse power rules for indefinite integrals
- Practice
- Applications of integrals
- Using the formula
- Undetermined coefficients
- Other resources
- Intro to differential equations
- Sketching solution curves in slope fields
- Intervals of validity
- Intro to initial value problems
- Solving initial value problems
- Real eigenvalues
- Equilibrium solutions
- Intro to equilibrium solutions
- Basic concepts of differential equations
- Laplace transforms
- Partial differential equations
- Systems of differential equations
- Bernoulli differential equations
- Po-Shen Loh method
- Logistic growth
- 3-dimensional space
- Cylindrical coordinates
- Monotonic intervals and functions
- Expressions vs equations
- Practice
- Lesson
- Exponents and radicals
- Inequalities and systems of inequalities
- Triangular numbers
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Lesson
- Constants
- Predicates
- Intro to calc 1
- Critical thinking
- Informal fallacies
- Applications of partial derivatives
- Lagrange multipliers
- Lesson
- Lesson
- Triangle and reverse triangle inequalities
- Lesson
- Practice
- Intro
- Other resources
- Quadrants
- Lesson
- Practice
- Lesson
- Practice
- Memorizing the Pythagorean identities
- Right triangles and trig
- Using the theorem
- Unit circle
- Non-right triangles and trig
- Memorizing how the six basic trig functions relate
- Lesson
- Intro to the six basic trig functions
- Practice
- Lesson
- Practice
- Why rationalize the denominator?
- Rationalizing the denominator (type 2)
- Rationalizing the denominator (type 3)
- Lesson
- Practice
- Distributive property with variables (visual method, grade 7)
- Intro
- Proof
- Intro
- Proof
- Using the theorem
- Proof
- Using the theorem
- Intro
- Proof
- Incenter theorem
- Proving differentiable implies continuous
- Subtracting integers (abstract method)
- Subtracting integers (concrete and visual methods)
- Lesson
- Practice
- Subtracting integers (word problems)
- Converting units of time
- Lesson
- Practice
- Lesson
- Practice
- Combinatorics and graph theory
- Intro to graphs
- Connected graphs
- Graph components
- Walks, paths, and cycles
- Lesson
- Reciprocal polynomials
- Adjacency lists, adjacency matrices, and incidence matrices
- Ordered field
- Maximum and minimum of a set
- Cardinality
- Intro to real analysis
- Intro to categorical statements
- Regular graphs
- Proof
- Monotone convergence theorem
- Intro
- Using the theorem
- Monotonic sequences
- Squeeze theorem for sequences
- Lesson
- Practice
- Variations of proof by induction
- Intro
- Proofs
- Proving inequalities
- Proof 2
- Double quantifiers
- Lesson
- Practice
- Bolzano-Weirstrass theorem
- Proving limit laws
- Lesson
- Practice
- Finite calculus
- Bell numbers
- Lesson
- Practice
- Stirling numbers
- Intro to the Ackermann function
- Using the theorem
- Proof
- Generating functions from recurrence relations
- Generating functions (differencing)
- Boolean SAT
- Comparison tests for convergence
- Absolute convergence, conditional convergence, and divergence
- Lesson
- Practice
- Intro to generating functions
- Equating coefficients
- Riemann's rearrangement theorem
- Lesson
- Practice
- Other simple planar graphs
- Minimum spanning tree
- Other resources
- Fermat's infinite descent
- Riemann integrability
- The real numbers
- Sequences
- Topology of the real numbers
- epsilon-N definition of convergence
- Lesson
- Practice
- Pointwise and uniform convergence
- Lesson
- Practice
- Lesson
- Practice
- Lesson
- Practice
- Fermat's two squares theorem
- Follow-up to Euler's formula
- Using the theorem
- Automorphisms of a graph
- Challenge
- Reference
- Recursive function theory
- Female mathematicians
- Inverse function theorem (calc 3)
- Intro to calc 3
- Presburger arithmetic
- Automated theorem provers
- Quantifiers
- Book reviews (popular math)
- Book reviews (recursive function theory)
- Tiling
- Root-finding algorithms
- Algorithms for evaluating polynomials
- Perfect numbers
- Ramanujan's infinite nested roots
- Graph isomorphism problem
- Division algorithm
- Counting from pictures, oral response
- Write numbers
- Representing numbers with drawings
- Understanding zero
- Compare written numerals
- Counting to 100 by ones
- Counting to 100 by tens
- Counting objects arranged in different ways
- Counting on by ones or tens
- Compare by matching or counting
- Relative positions
- Naming shapes in the real-world
- Addition and subtraction, concrete and visual
- Add and subtract within 10
- Decompose in multiple ways
- Addition, missing number word problems
- Add and subtract within 5
- Count, represent and write teen numbers
- Compare measurements without tools
- More, most, fewer, fewest
- Name shapes regardless of size
- Identify shapes and solids
- Compare attributes of shapes and solids
- Build solids
- Compose 2D shapes
- Add and subtract within 20, word problems
- Add within 20, word problems involving three addends
- Use one addition problem to solve another
- Add by making tens
- Subtract by adding
- Add and subtract by counting
- Add and subtract within 20, abstract
- Understand the equals sign
- Add and subtract within 20, missing number problems
- Count and write numbers within 120
- Place value with tens
- Compare two-digit numbers
- Add within 100
- 10 more or 10 less
- Subtract multiples of ten from multiples of ten
- Order without tools
- Measure given a unit
- Tell and write time, hour and half-hours
- Bar and picture graphs having three categories of data
- Definitions of shapes
- Compose 2D and 3D shapes
- Halves and quarters
- Two-step word problems, add and subtract within 100
- Mentally add and subtract within 20
- Intro to even and odd
- Add to count rectangular arrays
- Place value with hundreds
- Counting and skip-counting within 1000
- Convert between various representations, numbers within 1000
- Compare three-digit numbers
- Add multiple two-digit numbers
- Add and subtract within 1,000
- Mentally add or subtract powers of ten
- Add and subtract within 100, abstract
- Compare lengths by measuring
- Estimate lengths
- Add and subtract within 100 to solve word problems involving lengths
- Add and subtract within 100 on the number line
- Tell and write time, five minute increments
- Money word problems, including dollars
- Plot measurements on a line plot
- Draw picture and bar graphs
- Draw shapes with certain attributes
- Area of a rectangle by counting unit squares
- Partition circles and rectangles into equal shares
- Multiplication using rectangular arrays and equal groups, concrete and visual
- Divide using equal groups
- Relating multiplication to division
- Missing number problems, multiplication and division
- Use properties to multiply and divide
- Multiply, abstract
- Two-step word problems
- Patterns in addition and multiplication
- Round to the nearest 10 or 100
- Add and subtract within 1000, abstract
- Multiply one-digit numbers by multiples of 10
- Equal shares of shapes and unit fractions
- Plot fractions on the number line
- Comparing fractions with like numerators, denominators, or both
- Tell and write time, minute increments
- Measure and estimate volumes and masses
- Scaled bar and picture graphs
- Line plots with fractions
- Area and units of measure
- Area of composite rectangles by counting unit squares
- Area as multiplication
- Area, perimeter, and how they're related
- Shape heirarchy
- Solve multiplication and division word problems
- Multi-step word problems
- Prime and composite numbers
- Number patterns
- Represent numbers within 1000 using place value blocks
- Multiply and divide by 10
- Convert between written forms of numbers
- Round whole numbers
- Multiplication
- Quotients and remainders
- Equivalent fractions
- Compare fractions with unlike numerators, denominators, or both
- Add and subtract fractions with like denominators
- Multiply fractions by whole numbers
- Add fractions with power of ten denominators
- Convert fractions to decimals, power of ten denominators
- Compare decimals to hundredths
- Convert larger units to smaller ones
- Word problems involving measures
- Represent and reason with line plots
- Intro to angles
- Adjacent angles and the additive property of angles
- Geometric definitions
- Lines of symmetry
- Intro to parentheses
- Convert between words and arithmetic expressions
- Linear equations, tables
- Add and subtract fractions, abstract
- Add and subtract fractions, word problems
- Fraction quotients, word problems
- Areas and perimeters, fractional side lengths
- Multiplication as scaling
- Dividing whole numbers by unit fractions
- Squaring the circle, visual proof
- Geometric mean theorem
- Sum of an infinite geometric series
- Hashi puzzles
- Counting and place value
- Addition and subtraction
- Measurement and geometry
- Exploration: Bicycle math
- Place value
- Addition and subtraction
- Measurement, data, and geometry
- Third angle theorem
- Add and subtract within 20
- Place value
- Add and subtract within 100
- Add and subtract within 1,000
- Money and time
- Measurement
- Data
- Geometry
- Divide, abstract
- Multiplication
- Addition, subtraction, estimation
- Division
- Fractions
- Relating multiplication and division
- Arithmetic patterns and problem solving
- Geometry
- Time, measurement, and data
- Place value of whole numbers
- Multiplication and division
- Factors, multiples, and patterns
- Equivalent fractions and comparing fractions
- Fraction arithmetic
- Understand decimals
- Geometry
- Units of measurement
- Decimal place value
- Multiplying whole numbers by powers of 10
- Decimals to thousandths
- Rounding decimals
- Multiplication, partial products method
- Long division
- Decimal arithmetic
- Plot coordinates
- Plot coordinates from word problems
- Classifying shapes
- Unit conversion
- Volume by counting unit cubes
- Volume of right rectangular prisms by multiplying
- Multiply fractions, word problems
- Decimal place value
- Multi-digit multiplication and division
- Decimal arithmetic
- Add and subtract fractions
- Multiply fractions
- Divide fractions
- Volume
- Coordinate plane
- Algebraic thinking
- Measurement and data
- Properties of shapes
- Raising numbers to whole number powers
- Tower of Hanoi
- Convert phrases to algebraic expressions
- Identify parts of expressions
- Distributive property with variables
- Determining equivalence of algebraic expressions
- Understanding solutions to equations and inequalities
- Area
- Intro to ratios
- Unit rates
- Equivalent ratios
- Volume of right rectangular prisms with fractional lengths
- Coordinate plane
- Nets and surface area
- Divide fractions
- One-step equations
- Long division
- Decimal arithmetic
- Congruent supplements and complements theorems
- GCD and LCM
- Negative numbers in the real-world
- Negative numbers on the number line
- Absolute value of rational numbers
- Plotting points in all four quadrants
- Identifying statistical questions
- Wishful thinking strategy
- Circumference and area of a circle
- Proving de Moivre's theorem
- Vertical and horizontal line tests
- Addition rule
- Expected value
- Difference of squares
- Dependent and independent variables
- Solving one-step equations
- Modeling one-variable inequalities
- Analyzing distributions
- Plotting data
- Ratios, rates, and percentages
- Arithmetic operations
- Negative numbers
- Factors and multiples
- Variables and expressions
- Equations and inequalities introduction
- Geometry
- Data and statistics
- Interpreting graphs of proportional relationships
- Percents
- Simplifying complex fractions without variables
- Scale drawings
- Basic probability proofs
- Intro to the triangle inequality
- Cross sections
- Distributing, factoring, and combining like terms
- Convert word problems to linear expressions
- Multi-step problems
- Testing polar equations for symmetry
- Intro to arithmetic sequences
- Area, volume, and surface area
- Surface area of rectangular prisms
- Proving the divisibility rules for 7 and 13
- Remainders of large powers
- Percentages
- Round fractions
- Negative exponents
- Challenge: Skipped Counts
- Solving linear systems of equations in two variables by any method
- Special right triangles
- Higher order roots
- Multiplication as repeated addition
- Division using repeated subtraction
- Intro to the Fibonacci sequence
- Pythagorean theorem
- Other resources
- Lesson
- Symmetry
- Other resources
- Other resources
- Factor pairs
- Factoring and solving monic quadratics
- Prime factorization
- Intro to polyominoes
- Sketching polynomials using stationary points
- Simple events
- Conditional probability
- Compound probability of independent events
- Converting fractions to decimals
- Other resources
- Comparing and ordering fractions with unlike denominators
- Dividing whole numbers by fractions
- Multiplying whole numbers and fractions
- Finding the part
- Solving linear equations by clearing decimals and fractions
- Solving linear systems of equations in two variables by elimination
- Algebraic expansion with area
- Solving rational equations
- Remainders of Fibonacci numbers
- Counting vertices, edges, and faces
- Dividing integers
- Rep-tiles and irrep-tiles
- River crossing puzzles
- Sum of the interior angles of a triangle
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Solving equations and inequalities
- Linear equations and graphs
- Polynomials
- Irrational numbers
- Solving literal linear equations
- Solving and graphing multi-step inequalities in one-variable
- Solving for a variable given a linear equation in standard form
- Other resources
- Transformations
- Solving one-step equations (geometry)
- Complementary and supplementary angles
- Using the circle theorems
- Sum of interior angles for simple polygons
- Sum of exterior angles
- Additional topics
- Other resources
- Derivatives: definition and basic derivative rules
- Epsilon-delta proofs
- Other resources
- Integrals
- Applications of integrals
- Other resources
- Comparison test for series
- Proving the formula for the area of a circle
- Vectors
- Other resources
- Tangent planes and linear approximations
- Special points of multivariable functions
- Other resources
- Proof by strong induction
- Proof by contradiction
- Other resources
- Counting in bases 5 and 2
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Factorials
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Binomial theorem
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Rounding decimals (unsorted)
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Adding and subtracting with negative numbers
- Multiplying and dividing with negative numbers
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Circles
- Other resources
- Other resources
- Remainder and factor theorems
- Other resources
- Solving polynomials
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Permutations
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Direct and inverse variation
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Area of a sector formula
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Solving proportions
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Multiplying and dividing functions
- Other resources
- Other resources
- Integration by partial fractions
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Solving trig equations by factoring
- Other resources
- Using the law of cosines
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Polynomials (algebra 1)
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Congruent triangles
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Calculus
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Evaluating inverse trig functions
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Imaginary numbers
- Other resources
- Other resources
- Other resources
- Other resources
- Angle intro
- Other resources
- Other resources
- Polygon angle sum theorems
- Other resources
- Other resources
- Other resources
- Other resources
- Volume of a pyramid
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Polynomial factorization
- Other resources
- Finding the GCD of polynomials
- Other resources
- Multiplying binomials by polynomials using the abstract method
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Finding the domain and graphing rational functions
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Intro to conic sections
- Arc length and sector formulas
- Intro to radians
- Other resources
- Telling time to the nearest 5 minutes
- Other resources
- Distance formula
- Irrational numbers
- Other resources
- Solving literal rational equations
- Other resources
- Relatively prime
- Central angle theorem
- Visually determining antiderivative
- Solving non-monic quadratics by factoring
- Gauss's trick
- Dividing mixed numbers
- Changing linear dimensions
- Volume
- Other resources
- Finding and verifying the inverse of a nonlinear function
- Other resources
- Other resources
- Analyzing categorical data
- Displaying and comparing quantitative data
- Summarizing quantitative data
- Modeling data distributions
- Exploring bivariate numerical data
- Study design
- Random variables
- Sampling distributions
- Confidence intervals
- Significance tests (hypothesis testing)
- Two-sample inference for the difference between groups
- Inference for categorical data (chi-square tests)
- Advanced regression (inference and transforming)
- Analysis of variance (ANOVA)
- Power set
- False position method
- Rigid transformations
- Permutations and factorials
- Combinations
- Midpoint formula
- Proving properties of even and odd numbers
- Sums of squares
- Solving proportions by cross multiplying
- Evaluating and graphing the floor function
- Evaluating piecewise functions
- Other resources
- Basic statistics
- Other resources
- Other resources
- Multiplying complex numbers
- Multiplying and dividing complex numbers in polar form
- Other resources
- Proving the divisibility tests
- Triangles
- Area of a regular polygon
- Powers of complex numbers
- Find the angle of rotation
- Divisibility with sums and differences
- Logarithmic scale
- Finding exponential functions from tables, graphs, and points
- Other resources
- Other resources
- Age problems
- Length of a vector
- Adding and subtracting mixed numbers with regrouping
- Short division
- Other resources
- Other resources
- Evaluating logarithms
- Platonic solids
- Logarithms with negative bases and arguments
- Other resources
- Water pouring puzzles
- Multiplication principle
- Classifying quadrilaterals from four points
- Other resources
- Other resources
- Shifting, scaling, and reflecting
- Adding vectors in polar form
- Side splitter theorem
- Other resources
- Koch curve
- Other resources
- Set operations
- Vector equations of lines
- Adjacency and degrees
- Complex conjugate root theorem
- Other resources
- Quadratic equations with complex solutions
- Evaluating trig functions and solving trig equations using the unit circle
- Tournament graphs
- Intro to groups
- Finding the quadrant of an angle
- Center of a group
- Other resources
- Determining derivatives from graphs
- Basics of geometry
- Golden ratio
- Determining quadratics from their roots
- Solving quadratic equations by taking square roots
- Finding the radius given the area
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Area of a regular polygon
- Other resources
- Evaluating the six basic trig functions
- Other resources
- Evaluate the remaining basic trig functions
- Other resources
- Other resources
- Estimating the line of best fit
- Graphing trig functions
- Reciprocal identities
- Other resources
- Finding parabolas given three points
- Solving linear systems of equations in three variables by elimination
- Basic set theory proofs using set identities
- Other resources
- Other resources
- Solving equations with rational exponents
- Similar triangles
- Other resources
- Other resources
- SSS congruence
- Other resources
- Other resources
- Graphing harder absolute value equations
- Other resources
- Other resources
- Factoring non-monic quadratics by factoring out the GCD
- Equivalent expressions with negative numbers
- Comparing and ordering fractions
- Finding the equation of a line
- Other resources
- Solving proportions using equivalent fractions
- Factoring monic quadratics using algebra tiles
- Other resources
- Counting objects by property
- Other resources
- Converse of the corresponding angle theorems
- Other resources
- Intro to similarity
- Finding cube roots of small perfect cubes
- Other resources
- Other resources
- Negative Fibonacci numbers
- Proof by induction (Fibonacci numbers)
- Other resources
- Other resources
- Volume of extruded figures
- Using the theorems
- Corresponding angle theorems
- Method 1
- Method 2
- Three squares puzzle
- Other resources
- Other resources
- Combining means
- Simple hypothesis testing
- Hypothesis testing and p-values
- Other resources
- Other resources
- Other resources
- Other resources
- Finding the midpoint on the number line
- Missing square puzzle
- Other resources
- Evaluating trig functions using coterminal and reference angles
- Measuring and drawing angles
- Other resources
- Adding and subtracting in scientific notation
- Other resources
- Other resources
- Other resources
- Solving proportions using inverse operations and by cross multiplying
- Other resources
- Making and describing scatter plots
- Other resources
- Other resources
- Other resources
- Other resources
- Power of a power rule for exponents
- Other resources
- Other resources
- Slope-intercept equation from two points or a table
- Classifying angles
- Other resources
- Other resources
- Other resources
- Other resources
- Reflecting a point over any non-horizontal line
- Cavalieri's principle
- Other resources
- More on binary
- Other resources
- Other resources
- Other resources
- Degree of a polynomial
- Powers of 10
- Other resources
- Measurement with an unknown unit of length
- Other resources
- Intercepts from a linear equation
- Other resources
- Congruent polygons
- Other resources
- Working with powers of 10
- Other resources
- Theorem 1
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Polynomial identities
- Other resources
- Volume and surface area of a sphere
- Other resources
- Scaling along an axis
- Graphing reciprocal functions
- Solving hyperbolic equations
- Proving hyperbolic identities
- Intro to hyperbolic functions
- Integral recurrence relations
- Integration by long division
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Intro to variables
- Rate conversion
- Other resources
- Intro to the triangular numbers
- Classifying triangles
- Classifying polygons
- Other resources
- Graphing linear inequalities in standard form
- Graphing systems of linear inequalities in two variables
- Other resources
- Evaluating expressions with multiple variables
- Other resources
- Other resources
- Find the formula of an arithmetic sequence from two terms
- Find the explicit formula for an arithmetic series given terms
- Other resources
- Arithmetic series
- Univariate expressions representing consecutive terms
- Explicit formula from the first few terms
- Explicit formula from two terms
- Other resources
- Graphs of piecewise functions
- Other resources
- Other resources
- Solving floor and ceiling equations
- Product property for square roots
- The Cambridge Puzzle
- Intro to Pascal's triangle
- Other resources
- Other resources
- Simplifying cube roots of integers
- Simplifying nested square roots
- Other resources
- Exponential growth and decay
- Converting quadratics between standard form and vertex form
- Other resources
- Factoring non-monic quadratics using algebra tiles
- Factoring non-monic quadratics
- Other resources
- Other resources
- Ratios of area, surface area, and volume, for similar shapes
- Other resources
- Other resources
- Other resources
- Reducing to a linear equation
- Intro to vertex form for quadratics
- Other resources
- Determining quadratics from their roots
- First and second differences
- Other resources
- Sums and products of rational and irrational numbers
- Intro to the six basic trig functions
- Graphing sin and cos
- Graphing tan and cot
- Other resources
- Graphing sec and csc
- Other resources
- Other resources
- Finding a sin or cos equation from a graph
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Centroid and orthocenter
- Other resources
- Descartes's rule of signs
- Difficult factoring problems
- Center and radii of an ellipse
- Foci of an ellipse
- Other resources
- Other resources
- Solving equations in quadratic form using u-substitution
- Other resources
- Intro to hyperbolas
- Other resources
- Foci of a hyperbola
- Other resources
- Hyperbolas not centered at the origin
- Infinite consecutive independent events
- Other resources
- Graphically finding the increasing, decreasing, and constant intervals
- Other resources
- Complementary events
- Solving logarithmic equations using the one-to-one property
- Sum and difference of cubes
- Proofs using the unit circle
- Signs of trig functions
- Finding the point on the unit circle
- Composition of trig functions
- Evaluating inverse trig functions using the unit circle
- Evaluating trig functions using a calculator
- Evaluating inverse trig functions using a calculator
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Proving inverse hyperbolic identities
- Mentally raising small numbers to small powers
- Rational root theorem
- Higher-order polynomials
- Adding and subtracting multi-digit numbers
- Distance and midpoint in 3D
- Other resources
- Distributive property without variables
- Congruent tangents theorem
- Other resources
- Radii inside right triangles
- Circumradii of equilateral and isosceles triangles
- Circumcircles and incircles
- Double and half angle formulas
- Mentally solving multi-step equations
- Converting improper algebraic fractions to mixed algebraic fractions
- t-results
- Other resources
- Other resources
- Other resources
- Other resources
- Solving trigonometric equations by factoring out the GCD
- Other resources
- Plotting polar points
- Graphing polar equations
- Other resources
- Other resources
- Even-odd trig identities
- Other resources
- Asymptotes and intercepts of rational functions
- Other resources
- Linear factorization
- Other resources
- Multiplying a complex number by a real number
- Adding and subtracting complex numbers
- Powers of the imaginary unit
- Simplifying square roots of negative integers
- Solving and graphing polynomial inequalities using a sign chart
- Factoring and solving polynomials by graphing
- Intro to complex numbers
- Proving properties of the complex modulus
- Other resources
- Inverses of exponential and logarithmic functions
- Other resources
- Inverse variation
- Direct variation
- Other resources
- Other resources
- Modulus and argument of a complex number
- Other resources
- Dividing a polynomial by a monomial
- Dividing quadratics by linear expressions
- Other resources
- Solving trigonometric equations by graphing
- Finding general solutions to trig equations
- Graphing rational functions
- Other resources
- Solving logarithmic equations using the properties of logarithms
- Verifying logarithmic identities
- Simplifying complex fractions with variables
- Other resources
- Intro to Pascal's triangle
- Converting sums between sigma notation and expanded form
- Proving inverse trig identities
- Solving equations with radicals and rational exponents
- Simplifying expressions with radicals and rational exponents
- Other resources
- Other resources
- Other resources
- Other resources
- Other resources
- Solving literal equations
- Other resources
- Simplifying trig expressions
- Rewrite exponential expressions
- Solving trig equations by factoring trinomials
- Other resources
- Fractal dimension
- Other resources
- Other resources
- Estimating sums and differences by rounding
- Estimating products by rounding
- Other resources
- Other resources
- Other resources
- Other resources
- Mentally add within 1000
- Other resources
- Incenter and circumcenter of a triangle
- Other resources
- Intro to the incenter and circumcenter of a triangle
- Circumcenter theorem
- Area of composite figures with semicircles
- Crossed ladders theorem
- Standard equation of a circle
- Dividing unit fractions by whole numbers
- The four fundamental spaces
- Column space
- Orthogonal complements
- Other resources
- Visualizing a projection onto a plane
- Linear subspace
- Characteristics of functions
- Sketching a polynomial given characteristics
- Orthogonal projection
- Nullspace, left nullspace, and row space
- Other resources
- Other resources
- Other resources
- Other resources
- Adding multiple two-digit numbers
- Other resources
- Other resources
- One-step word problems, add and subtract within 100
- Partition rectangles into rows and columns
- Other resources

In this section, you will learn how to find limits of rational functions at points of discontinuity. Before attempting this section, you should know how to factor polynomials, such as quadratics, difference of squares, and difference of cubes.

\(\displaystyle \lim_{x \to -2} \dfrac{x^2 - 4}{x + 2}\)

$$\begin{align}
& \lim_{x \to -2} \dfrac{x^2 - 4}{x + 2} \\[1.2em]
=\ & \lim_{x \to -2} \dfrac{(x - 2)(x + 2)}{x + 2} \\[1.2em]
=\ & \lim_{x \to -2} x - 2 \\[1.2em]
=\ & -4
\end{align}$$

\(\displaystyle \lim_{x \to -6^-} \dfrac{x^2 + 4x - 12}{x^2 - 36}\)

$$\begin{align}
& \lim_{x \to -6^-} \dfrac{x^2 + 4x - 12}{x^2 - 36} \\[0.85em]
=\ & \lim_{x \to -6^-} \dfrac{(x + 6)(x - 2)}{(x - 6)(x + 6)} \\[0.85em]
=\ & \lim_{x \to -6^-} \dfrac{x - 2}{x - 6} \\[0.85em]
=\ & \dfrac{-6 - 2}{-6 - 6} \\[0.85em]
=\ & \dfrac{-8}{-12} \\[0.85em]
=\ & \dfrac{2}{3}
\end{align}$$

\(\displaystyle \lim_{x \to -1} \dfrac{x^2 + 4x + 3}{x + 1}\)

\(2\)

\(\displaystyle \lim_{x \to 2^-} \dfrac{x + 5}{x^2 - 3x + 2}\)

Does not exist.

\(\displaystyle \lim_{h \to 0} \dfrac{(a + h)^2 - a^2}{h}\)

\(2a\)

\(\displaystyle \lim_{x \to 1} \dfrac{x - 1}{\sqrt{x} - 1}\)

\(2\)

\(\displaystyle \lim_{x \to 1} \dfrac{x^3 - 1}{x - 1}\)

\(3\)

\(\displaystyle \lim_{x \to -1} \dfrac{x^2 - 5x - 6}{x + 1}\)

\(-7\)

\(\displaystyle \lim_{x \to 25} \dfrac{\sqrt{x} - 5}{x - 25}\)

\(\dfrac{1}{10}\)

\(\displaystyle \lim_{x \to -1} \dfrac{x^2 - 4x - 5}{x + 1}\)

\(-6\)

\(\displaystyle \lim_{x \to 1} \dfrac{x^2 - 5x + 4}{x - 1}\)

\(-3\)

\(\displaystyle \lim_{x \to -3} \dfrac{x^2 - 9}{x + 3}\)

\(-6\)

\(\displaystyle \lim_{x \to 7} \dfrac{x^2 - 5x + 4}{x^2 - 5x - 14}\)

Does not exist.