Prove \(f : \mathbb{Z}^+ \longrightarrow \mathbb{Z}\) given by \(f(n) = (-1)^n \cdot n\) is injective.

Prove that \(f : [0, \infty) \longrightarrow \mathbb{R}\) defined by \(f(x) = x^2\) is injective.

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- Solving linear systems in two variables using Cramer's rule
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- Determinant using cofactors
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- Linear transformations
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- Basis
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- Introduction to sets
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- Factor by grouping
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- 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
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- Initial value problems
- Euler's method
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- 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
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- \(\sin(z) = 2\)
- Approximating square roots to hundredths
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- Complex exponentiation
- \(i^i\)
- Naive set theory
- Negation (bridge course)
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- Proving parallelograms with two column proofs
- Proof
- Heron's formula
- Area of a triangle
- Disjunction
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- Additive identity
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- Converting degrees to radians
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- Mode
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- 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
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- 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 and range 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
- Other resources
- 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
- 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
- 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
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- 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
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- 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
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- Other resources
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- 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
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- 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
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- 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
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- Other resources
- Other resources
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- 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
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- Other resources
- Other resources
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- Other resources
- Other resources
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- 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
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- Polynomials (algebra 1)
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- Congruent triangles
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- Calculus
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- Evaluating inverse trig functions
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- Imaginary numbers
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- Angle intro
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- Polygon angle sum theorems
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- Volume of a pyramid
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- Polynomial factorization
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- Finding the GCD of polynomials
- Other resources
- Multiplying binomials by polynomials using the abstract method
- Other resources
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- Finding the domain and graphing rational functions
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- 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
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- Power of a power rule for exponents
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- Slope-intercept equation from two points or a table
- Classifying angles
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- Reflecting a point over any non-horizontal line
- Cavalieri's principle
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- More on binary
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- Degree of a polynomial
- Powers of 10
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- Measurement with an unknown unit of length
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- Intercepts from a linear equation
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- Congruent polygons
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- Working with powers of 10
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- Theorem 1
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- Polynomial identities
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- Volume and surface area of a sphere
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- Scaling along an axis
- Graphing reciprocal functions
- Solving hyperbolic equations
- Proving hyperbolic identities
- Intro to hyperbolic functions
- Integral recurrence relations
- Integration by long division
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- Intro to variables
- Rate conversion
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- Intro to the triangular numbers
- Classifying triangles
- Classifying polygons
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- Graphing linear inequalities in standard form
- Graphing systems of linear inequalities in two variables
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- Evaluating expressions with multiple variables
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- Find the formula of an arithmetic sequence from two terms
- Find the explicit formula for an arithmetic series given terms
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- Arithmetic series
- Univariate expressions representing consecutive terms
- Explicit formula from the first few terms
- Explicit formula from two terms
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- Graphs of piecewise functions
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- Solving floor and ceiling equations
- Product property for square roots
- The Cambridge Puzzle
- Intro to Pascal's triangle
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- Simplifying cube roots of integers
- Simplifying nested square roots
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- Exponential growth and decay
- Converting quadratics between standard form and vertex form
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- Factoring non-monic quadratics using algebra tiles
- Factoring non-monic quadratics
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- Ratios of area, surface area, and volume, for similar shapes
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- Reducing to a linear equation
- Intro to vertex form for quadratics
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- Determining quadratics from their roots
- First and second differences
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- Sums and products of rational and irrational numbers
- Intro to the six basic trig functions
- Graphing sin and cos
- Graphing tan and cot
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- Graphing sec and csc
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- Finding a sin or cos equation from a graph
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- Centroid and orthocenter
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- Descartes's rule of signs
- Difficult factoring problems
- Center and radii of an ellipse
- Foci of an ellipse
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- Solving equations in quadratic form using u-substitution
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- Intro to hyperbolas
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- Foci of a hyperbola
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- Hyperbolas not centered at the origin
- Infinite consecutive independent events
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- Graphically finding the increasing, decreasing, and constant intervals
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- 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
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- 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
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- Distributive property without variables
- Congruent tangents theorem
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- 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
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- Solving trigonometric equations by factoring out the GCD
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- Plotting polar points
- Graphing polar equations
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- Even-odd trig identities
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- Asymptotes and intercepts of rational functions
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- Linear factorization
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- 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
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- Inverses of exponential and logarithmic functions
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- Inverse variation
- Direct variation
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- Modulus and argument of a complex number
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- Dividing a polynomial by a monomial
- Dividing quadratics by linear expressions
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- Solving trigonometric equations by graphing
- Finding general solutions to trig equations
- Graphing rational functions
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- Solving logarithmic equations using the properties of logarithms
- Verifying logarithmic identities
- Simplifying complex fractions with variables
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- 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
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- Solving literal equations
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- Simplifying trig expressions
- Rewrite exponential expressions
- Solving trig equations by factoring trinomials
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- Fractal dimension
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- Estimating sums and differences by rounding
- Estimating products by rounding
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- Mentally add within 1000
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- Incenter and circumcenter of a triangle
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- 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
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- Visualizing a projection onto a plane
- Linear subspace
- Characteristics of functions
- Sketching a polynomial given characteristics
- Orthogonal projection
- Nullspace, left nullspace, and row space
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- Adding multiple two-digit numbers
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- One-step word problems, add and subtract within 100
- Partition rectangles into rows and columns
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- Rotational symmetry
- Number of diagonals in a convex polygon
- Other resources
- z-scores

Learn how to prove a function is injective, or alternatively, is not injective.

Prove \(f : \mathbb{Z}^+ \longrightarrow \mathbb{Z}\) given by \(f(n) = (-1)^n \cdot n\) is injective.

45565Prove the function f:Z^+ → Z given by f(n) = (-1)^n * n is Injective(one-to-one)

The Math Sorcerer

Prove that \(f : [0, \infty) \longrightarrow \mathbb{R}\) defined by \(f(x) = x^2\) is injective.