A path through mathematics.

The shape of a quadraticMove a parabola and locate its vertex and roots.

Transforming functionsTranslate, stretch, and reflect a parent function.

Exponential growthCompare multiplicative change with additive change.

Logarithms as inversesReflect an exponential curve across the line y = x.

Systems of equationsFind the point where two constraints agree.

Linear inequalitiesUnderstand the regions that make a statement true.

Absolute valueMeasure distance from zero and solve distance constraints.

Polynomial behaviorConnect factors, roots, and end behavior.

Rational functionsExplore zeros, holes, and vertical asymptotes.

Domain & rangeDistinguish allowed inputs from possible outputs.

Inverse functionsReverse an input-output relationship.

Factoring expressionsRead multiplication backwards using area.

Completing the squareTurn a quadratic into a shifted square.

Exponent rulesExplain multiplication and division of powers.

Roots & radicalsConnect roots, powers, and real domains.

Function notationFollow an input through a mathematical machine.

Piecewise definitionsBuild one function from several rules.

Average rate of changeMeasure change across a whole interval.

Building algebraic modelsTranslate a real constraint into an equation.

Why a² + b² = c²Rearrange four triangles to reveal a timeless proof.

△Course 02

Geometry

Reason with shapes, space, and proof.

20 interactive lessonsExplore this course ↗

A suggested starting path01

The area of a triangleChange the height while keeping a base fixed.

Where πr² comes fromRearrange sectors to see a circle become an almost rectangle.

Triangle angle sumSee why three interior angles make a straight angle.

Triangle congruenceFind the measurements that determine a triangle.

Chords & arcsConnect central angles, chords, and arc lengths.

Inscribed anglesRelate angles on a circle to the arcs they intercept.

Tangents to circlesUnderstand the right angle between radius and tangent.

Polygon angle sumsBreak a polygon into triangles.

Surface areaUnfold solids into their two-dimensional nets.

Volumes of solidsCompare prisms, pyramids, cylinders, and cones.

The distance formulaUse a right triangle to measure between points.

Midpoints & segmentsLocate the point halfway between two coordinates.

RotationsTurn a shape about a chosen center.

ReflectionsFind the mirror relationship between points.

TranslationsMove every point by the same vector.

DilationsScale a figure about a fixed point.

Compass & straightedgeConstruct shapes from circles and lines.

Voronoi regionsDivide a plane by the nearest point.

The unit circleTurn a point around a circle and discover sine and cosine.

θCourse 03

Trigonometry

Circles become waves.

20 interactive lessonsExplore this course ↗

A suggested starting path01

Anatomy of a waveExplore amplitude, period, phase, and vertical shifts.

A circle full of identitiesSee why sin² θ + cos² θ is always one.

The law of cosinesExtend the Pythagorean theorem to any triangle.

RadiansMeasure angles with the arc they sweep out.

Right-triangle ratiosRelate opposite, adjacent, and hypotenuse.

The tangent functionConnect slope on a circle to the ratio sin/cos.

Inverse trig functionsRecover an angle using a restricted domain.

Reference anglesUse one acute angle to understand all four quadrants.

The law of sinesRelate side lengths to their opposite angles.

Angle additionCombine rotations to derive addition identities.

Double-angle identitiesConnect an angle to twice its size.

Half-angle identitiesFind trig values by bisecting an angle.

Trigonometric equationsFind all angles that satisfy a relationship.

Secant, cosecant & cotangentExplore the reciprocals of familiar ratios.

Simple harmonic motionProject circular motion onto a line.

Phase differenceCompare waves that peak at different times.

Beats & interferenceCombine two nearby frequencies.

Trigonometry in polar formDescribe location by radius and direction.

The ambiguous triangleExplore why SSA can produce two solutions.

Complex numbers that turnMultiply complex numbers to rotate and scale the plane.

ℂCourse 04

Precalculus

A bridge to new ways of thinking.

20 interactive lessonsExplore this course ↗

A suggested starting path01

Drawing in polar coordinatesLet an angle determine a radius.

Parametric motionLet two coordinates move together through time.

Geometric sequencesWatch repeated multiplication produce a sequence.

Function compositionUse the output of one function as the input of another.

Arithmetic sequencesBuild a pattern with a constant difference.

The binomial theoremConnect expansions to combinations.

Conic sectionsFind curves made by slicing a cone.

EllipsesKeep the sum of distances to two foci constant.

HyperbolasKeep the difference of distances constant.

Focus & directrixDefine a parabola using equal distances.

Roots in the complex planePlace roots of unity around a circle.

Euler’s formulaConnect exponentials with rotations.

Asymptotes at infinityCompare the dominant terms of rational functions.

Logarithmic modelsUnderstand scales that compress multiplicative change.

Logistic growthAdd a carrying capacity to a growth model.

The difference quotientMeasure how outputs respond to a small input change.

Joining pieces smoothlyCheck behavior at the boundary between rules.

Recursive sequencesGenerate each term using the terms before it.

Mathematical inductionBuild a proof that climbs forever.

Getting closer: limitsApproach a missing point from both sides.

∫Course 05

Calculus I

Meet change and accumulation.

20 interactive lessonsExplore this course ↗

A suggested starting path01

What is a derivative, really?Bring two points together and watch a secant become a tangent.

The chain ruleFollow a tiny change through two connected functions.

Adding up an areaBuild an integral out of thin rectangles.

The Fundamental TheoremWatch the derivative of accumulated area recover the original function.

An optimal open-top boxBalance competing dimensions to maximize volume.

ContinuityConnect limits, function values, and an unbroken graph.

The power ruleFind the derivative of a power function.

The product ruleTrack two changing factors with an area diagram.

The quotient ruleDifferentiate a ratio of changing quantities.

Implicit differentiationFind a slope without solving for y.

Linear approximationUse the tangent line to estimate nearby values.

The Mean Value TheoremFind an instantaneous rate matching an average rate.

Concavity & inflectionRead the changing slope of a curve.

Critical pointsFind where a differentiable curve can turn.

L’Hôpital’s ruleResolve suitable indeterminate limits with derivatives.

AntiderivativesRecover a family of functions from a rate.

SubstitutionReverse the chain rule to evaluate an integral.

Net changeTranslate an accumulated rate into a total change.

One function, many polynomialsBuild a local approximation one term at a time.

ΣCourse 06

Calculus II

Add infinitely many possibilities.

20 interactive lessonsExplore this course ↗

A suggested starting path01

Approximating an integralCompare rectangles, trapezoids, and Simpson’s rule.

An infinite interval, a finite area?Explore the p-integral and its convergence threshold.

Volumes of revolutionTurn disks into a solid and add their volumes.

An infinite geometric sumSee when infinitely many terms add to a finite number.

Integration by partsReverse the product rule to integrate a product.

Partial fractionsSplit a rational function into simpler pieces.

Trigonometric substitutionUse a triangle to simplify a radical.

Arc lengthApproximate a curve by tiny straight segments.

Surface area of revolutionAdd the areas of thin frustums.

Limits of sequencesFind what a discrete sequence approaches.

The term testRecognize a necessary condition for series convergence.

Comparison testsCompare an unfamiliar series to a familiar one.

The ratio testUse consecutive terms to test convergence.

The root testUse nth roots to measure term growth.

Alternating seriesTrack cancellation and bound the remainder.

Power seriesFind a series’ radius and interval of convergence.

Taylor error boundsPut a limit on approximation error.

Area in polar coordinatesAdd tiny circular sectors.

Calculus of parametric curvesDifferentiate and integrate curves defined by time.

A new dimensionRotate a surface and inspect its height.

∇Course 07

Calculus III

Take your intuition into space.

20 interactive lessonsExplore this course ↗

A suggested starting path01

One direction at a timeSlice a surface to reveal its partial derivatives.

Follow the gradientFind the direction of steepest increase.

Choose a directionProject the gradient onto a unit direction.

A plane close to a surfaceApproximate a surface near a single point.

Volume under a surfaceAdd signed columns over a rectangular region.

Vectors in three dimensionsDescribe direction and length in space.

The dot productMeasure how much two vectors point together.

The cross productFind an oriented area vector.

Lines & planesDescribe straight paths and flat surfaces in space.

Level curvesRead a three-dimensional surface from a contour map.

Multivariable limitsCompare different paths toward the same point.

Lagrange multipliersOptimize while staying on a constraint.

Triple integralsAccumulate density throughout a solid.

The JacobianMeasure local area and volume changes.

Vector fieldsAssign a direction and magnitude at every point.

Line integralsAccumulate work or density along a path.

DivergenceMeasure a vector field’s local net outflow.

CurlMeasure local circulation in a vector field.

Stokes & divergence theoremsConnect interior behavior to a boundary.

What does a matrix do?Move an entire grid by changing four numbers.

↗Course 08

Linear Algebra

See what a matrix really does.

20 interactive lessonsExplore this course ↗

A suggested starting path01

The determinant is an area scaleSee signed area change under a linear map.

The directions that stayFind vectors that remain on their own line.

Vector projectionFind the part of one vector along another.

When vectors make a planeDiscover whether two vectors span a plane or a line.

Undoing a matrixSee when a linear transformation can be reversed.

Vector additionAdd displacements tip to tail.

SpanExplore every linear combination of a set of vectors.

Composing matricesApply one transformation after another.

Row reductionSimplify a system without changing its solutions.

RankCount independent directions in a matrix’s image.

Null spaceFind vectors that collapse to zero.

Column spaceExplore every possible matrix output.

Basis & coordinatesDescribe the same vector using different building blocks.

DimensionCount the vectors in a basis.

OrthogonalityUnderstand perpendicular vectors and subspaces.

Gram–SchmidtBuild a perpendicular basis one vector at a time.

DiagonalizationDescribe a transformation in its eigenvector basis.

Least squaresFind the closest possible solution to an inconsistent system.

Singular value decompositionRead a matrix as orthogonal maps and axis stretches.

Order from randomnessFlip a biased coin and watch a running proportion.

∞Course 09

Curiosities & Applications

Follow a question somewhere unexpected.

20 interactive lessonsExplore this course ↗

A suggested starting path01

The door worth switching toTest the Monty Hall problem over many games.

Fibonacci & the golden ratioWatch consecutive ratios approach a special number.

Building a wave from wavesAdd odd harmonics to approach a square wave.

A window into fractalsExplore Mandelbrot orbits and the growing edge of a Koch curve.

The birthday paradoxFind why shared birthdays appear sooner than expected.

A sieve for primesRemove multiples to discover prime numbers.

Sizes of infinityPair infinite sets and find where pairing fails.

Pascal’s triangleDiscover combinations, patterns, and powers.

The normal distributionExplore the geometry of the bell curve.

The central limit theoremWatch sample averages form a familiar shape.

Bayes’ theoremUpdate a belief when new evidence arrives.

Random walksFollow repeated random steps.

Paths & networksModel connections with vertices and edges.

Walking every edgeFind when a network can be traced without lifting a pen.

Differential equationsDescribe a curve by the rate at which it changes.

The logistic mapFind complexity inside a simple recurrence.

The Möbius stripExplore a surface with one side.

Estimating π with needlesTurn a random experiment into a geometric estimate.

Moving things efficientlyFind the least costly way to move a distribution.