Maths lessons
141 skills, each with a written lesson, worked examples, practice problems with full solutions, and the mistakes people actually make. Free to read, with nothing to sign up for.
Starter Village
The last village before the Fog.
- Counting & ComparingRead, order and compare whole numbers, and place them on a number line. This is the foundation every later skill stands on.
- AdditionAdd whole numbers fluently, from single-digit facts to multi-digit sums with regrouping. Addition means combining amounts.
- SubtractionSubtract whole numbers, including borrowing across zeros, and recognise the three stories subtraction tells: take away, compare, and find the missing part.
- MultiplicationMultiply whole numbers, understanding multiplication as equal groups and as the area of a rectangle. Facts through 12 x 12 become automatic.
- DivisionDivide whole numbers, with and without remainders, seeing division as sharing equally and as repeated subtraction.
Number Kingdom
A forest that counts in both directions.
- Place Value & RoundingName the value of every digit in a number and round to any place. Place value is the engine that makes all written arithmetic work.
- Negative NumbersExtend the number line to the left of zero and compute with signed numbers. Negatives measure debt, cold, and depth below sea level.
- Order of OperationsEvaluate expressions in the agreed order: parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right.
- Factors, Multiples & PrimesTake numbers apart into their factors and build them up as multiples. Primes are the atoms every whole number is made from.
- ExponentsUse exponents as shorthand for repeated multiplication, and apply the product, quotient, zero and negative exponent rules.
- Square & Cube RootsUndo a square or a cube: find the root of a perfect square or cube, solve x^2 = p and x^3 = p, and estimate the roots that are not whole numbers.
- Scientific NotationWrite very large and very small numbers as a number between 1 and 10 times a power of ten, compare them by their exponents, and multiply and divide them.
- Simplifying RadicalsWrite a square root in simplest form by pulling out perfect-square factors, then add, multiply and divide radicals, and clear a root from a denominator.
Fraction Isles
Pieces of one whole, waiting to be put back.
- Equivalent & Simplifying FractionsRead a fraction as a division and as a part of a whole, generate equivalent fractions, and reduce to lowest terms.
- Comparing FractionsDecide which of two fractions is larger using common denominators, benchmarks, or cross-multiplication, and order a set of fractions.
- Adding & Subtracting FractionsAdd and subtract fractions and mixed numbers by rewriting them with a common denominator.
- Multiplying & Dividing FractionsMultiply fractions directly and divide by multiplying by the reciprocal, understanding why the rules work.
- DecimalsRead, compare and compute with decimals, and convert between decimals and fractions.
- PercentsTreat a percent as a fraction out of 100 and solve percent-of, percent-change and discount problems.
- Ratios & ProportionsCompare quantities with ratios, find unit rates, and solve proportions that scale a relationship up or down.
- Rational & Irrational NumbersSort the real numbers: every fraction is a decimal that stops or repeats, every other decimal is irrational, and both kinds have a place on the number line.
Algebra Kingdom
Name the unknown and it has to answer.
- Variables & ExpressionsUse letters to stand for unknown or changing quantities, evaluate expressions, and combine like terms.
- One-Step EquationsSolve equations that need a single inverse operation, keeping the balance by doing the same thing to both sides.
- Two-Step EquationsSolve equations of the form ax + b = c by undoing operations in reverse order.
- Multi-Step EquationsSolve equations that need distributing, combining like terms, or moving variables to one side.
- Equations with FractionsClear fractions from an equation by multiplying through by the least common denominator, then solve as usual.
- Linear InequalitiesSolve inequalities the way you solve equations, with one extra rule: multiplying or dividing by a negative flips the sign.
- FactoringRewrite a polynomial as a product: pull out the greatest common factor, factor trinomials, and recognise the difference of squares.
- Quadratic EquationsSolve equations containing a squared term by factoring, by taking square roots, and with the quadratic formula.
- PolynomialsName a polynomial by its degree and number of terms, add and subtract polynomials, multiply a monomial or a binomial through, and recognise the special products.
- Literal Equations & FormulasRearrange a formula to isolate any one of its letters, using the same moves as a numeric equation, and use the rearranged form to solve for the quantity you actually need.
- Absolute Value & Compound InequalitiesSolve equations with an absolute value by splitting them into two cases, and solve compound inequalities joined by "and" or "or", writing each solution as an interval on the number line.
- Polynomial FunctionsDivide one polynomial by another, use the remainder and factor theorems to find zeros, and read the end behaviour and shape of a polynomial graph from its degree and leading coefficient.
- Complex NumbersExtend the number system with i = sqrt(-1): simplify powers of i, add, subtract, multiply and divide complex numbers, and solve the quadratics whose discriminant is negative.
- Rational Expressions & EquationsTreat a fraction of polynomials the way you treat a fraction of numbers: simplify it by factoring, state the values it excludes, multiply, divide, add and subtract, and solve equations that contain one.
- Rational Exponents & Radical EquationsRead a fractional exponent as a root, move between radical and exponent form, simplify with the exponent rules, and solve equations with a variable under a root, checking for solutions the squaring step invents.
- The Complex Plane & Polar FormSee a complex number as a point and an arrow: plot a + bi, read its modulus as a distance and its argument as an angle, convert between rectangular and polar form, multiply and divide in polar form, raise to a power with De Moivre, and find every nth root.
Geometry Kingdom
The kingdom where you build instead of fight.
- AnglesMeasure and classify angles, and use complementary, supplementary, vertical and parallel-line relationships to find unknown angles.
- TrianglesClassify triangles, use the 180-degree angle sum, and apply the triangle inequality and exterior angle theorem.
- Perimeter & AreaCompute perimeter and area for rectangles, triangles, parallelograms, trapezoids and composite figures.
- Pythagorean TheoremRelate the three sides of a right triangle with a^2 + b^2 = c^2, and use it to find distances.
- CirclesUse radius, diameter, circumference and area, and work with arcs and sectors.
- Similar TrianglesRecognise similar figures, write proportions between corresponding sides, and use them to find unknown lengths.
- Volume & Surface AreaCompute volume and surface area of prisms, cylinders, pyramids, cones and spheres.
- The Coordinate PlaneName every point on the plane with an ordered pair, read and plot points in all four quadrants, and measure along the grid lines between them.
- Transformations & CongruenceSlide, flip, turn and scale figures on the coordinate plane, and use the moves that preserve length and angle to decide when two figures are congruent or similar.
- Logic & ProofRead a conditional statement, write its converse, inverse and contrapositive, test a claim with a counterexample, and give the reason for every step of a proof.
- Parallel Lines & TransversalsName the angle pairs a transversal makes with two lines, use them to find unknown angles when the lines are parallel, and use them backwards to prove that two lines are parallel.
- Coordinate GeometryUse the distance, midpoint and slope formulas to measure and classify figures drawn on the coordinate plane, and to prove that lines are parallel or perpendicular.
- Triangle CongruenceDecide when two triangles are congruent from the SSS, SAS, ASA, AAS and HL criteria, match corresponding parts, and supply the missing statement or reason in a congruence proof.
- Polygons & QuadrilateralsFind the angle sums of any polygon, use the properties of parallelograms, rectangles, rhombi, squares, trapezoids and kites, and classify a quadrilateral from what is known about it.
- Angles & Segments in CirclesRelate the angles and segments inside and around a circle: central and inscribed angles, chords, tangents and secants, and the products of the segments they cut.
- Equations of CirclesWrite the equation of a circle from its centre and radius, read the centre and radius back from an equation, complete the square to find them, and test whether a point lies on the circle.
- Special Right TrianglesUse the 45-45-90 and 30-60-90 side ratios to find exact side lengths written with radicals, and recognise the two triangles inside squares, equilateral triangles and hexagons.
- Similar Solids & Composite FiguresScale the area and volume of similar figures by the square and cube of the scale factor, find the area of regular polygons, and measure composite figures and cross-sections of solids.
Function Kingdom
Every machine here runs on a rule, and every rule has stopped.
- Function NotationRead f(x) as a machine that turns an input into exactly one output, and evaluate and interpret function notation.
- Proportional RelationshipsRecognise when two quantities grow in lockstep, find the constant of proportionality from a table, a graph or an equation, and compare unit rates across representations.
- Linear FunctionsWork with straight lines: slope as a rate of change, intercepts, and the equation y = mx + b.
- Graphs, Tables & EquationsMove fluently between the four faces of a function: equation, table, graph and words.
- Systems of EquationsSolve two equations at once by graphing, substitution or elimination, and interpret the intersection point.
- Exponential FunctionsModel repeated growth and decay with y = a * b^x, and contrast constant differences with constant ratios.
- Function TransformationsShift, stretch and reflect a parent graph, and read the transformation directly off the equation.
- Forms of a LineWrite the equation of a line in point-slope, slope-intercept and standard form, convert between the three, read intercepts off standard form, and build parallel and perpendicular lines through a point.
- Systems of InequalitiesGraph a linear inequality in two variables as a half-plane with a solid or dashed boundary, test whether a point satisfies a system of inequalities, and write a system from the constraints of a situation.
- Quadratic Functions & ParabolasRead a parabola from its equation: find the vertex, the axis of symmetry and the zeros of y = ax^2 + bx + c, convert to vertex form by completing the square, and use the vertex to answer maximum and minimum questions.
- Rational FunctionsFind where a rational function is undefined and what its graph does there: vertical and horizontal asymptotes, holes, intercepts and end behaviour, and use them to sketch and read the graph.
- Compound Interest & Exponential ModelsTurn a percent rate into a growth factor, compound it monthly, daily or continuously with e, and build the exponential model behind interest, population and half-life problems.
- LogarithmsRead a logarithm as the exponent that answers "b to what power gives x", convert between exponential and logarithmic form, evaluate exact logs, and expand or condense with the product, quotient and power laws.
- Exponential & Logarithmic EquationsSolve for an exponent by taking a logarithm and for a logarithm by rewriting in exponential form, check every answer against the domain, and use the method on doubling times, pH, decibels and radioactive dating.
- Arithmetic & Geometric SequencesSee a list of numbers as a function of its position: tell an arithmetic sequence from a geometric one, write explicit and recursive formulas, and find any term without listing the ones before it.
- Series & Sigma NotationAdd the terms of a sequence without adding them one by one: the arithmetic series formula, the finite geometric series formula, sigma notation, and the infinite geometric series that settles on a value.
- Composite FunctionsFeed one function into another: evaluate (f ∘ g)(a) from rules and tables, write f(g(x)) as one expression, find the domain of a composition, and take a composite function apart again.
- Inverse FunctionsRun a function backwards: find the inverse of a linear, rational, radical, cubic, exponential or logarithmic function, verify two functions are inverses by composing them, read an inverse off a table or a graph, and decide when an inverse is a function at all.
- Transformations of Any FunctionMove past the parabola: stretch a graph horizontally with f(kx), reflect it in the y-axis, apply several transformations in the right order, track where a single point goes, and do it all to square roots, cube roots, reciprocals, cubics and exponentials.
- Piecewise Functions & ModellingBuild the function a situation needs: evaluate and write piecewise and step functions, find the average rate of change on an interval, turn a box, a fence or a price into a polynomial model, choose the right family, and use direct, inverse and joint variation.
Trigonometry Peaks
Everything here comes back around.
- Right Triangle TrigonometryUse sine, cosine and tangent as ratios of sides in a right triangle to find unknown sides and angles.
- Radians & DegreesMeasure angles by arc length instead of by 360ths of a turn, and convert between radians and degrees.
- The Unit CircleExtend sine and cosine to all angles using coordinates on the unit circle, and read exact values off it.
- Trigonometric IdentitiesUse the Pythagorean and quotient identities to simplify expressions and verify equations.
- Law of Sines & CosinesSolve triangles that are not right-angled using the law of sines and the law of cosines.
- Graphs of Sine & CosineUnroll the unit circle into a wave: read the amplitude, period, midline and phase shift of y = a sin(b(x - h)) + k off its equation and its graph, and write the equation of a wave you are shown.
- Trigonometric EquationsFind every angle that makes a trigonometric equation true: isolate the ratio, read the reference angle off the unit circle, place it in the right quadrants, and use an identity or a substitution when the equation is quadratic.
- Half-Angle & Product IdentitiesFinish the identity toolkit: the half-angle formulas and their signs, the power-reducing formulas, product-to-sum and sum-to-product, tan(t/2), and verifying an identity with all of them.
- Inverse Trigonometric FunctionsGo from a ratio back to an angle, carefully: exact values of arcsin, arccos and arctan, the restricted ranges that make them functions, compositions such as sin(arccos x), calculator values, and the angle a situation asks for.
- VectorsA quantity with a size and a direction: write a vector in component form, find its magnitude and direction angle, add, subtract and scale vectors, build one from a magnitude and an angle, and find the unit vector that points the same way.
- Dot Product & Vector ApplicationsPut vectors to work: the dot product, the angle between two vectors, when they are perpendicular, the projection of one onto another, work done by a force, and the resultant of forces, winds and currents.
Calculus Kingdom
The isle that never quite arrives.
- LimitsDescribe what a function approaches near a point, even where it is undefined, using tables, graphs and algebra.
- Derivatives & the Power RuleDefine the derivative as an instantaneous rate of change and compute it for polynomials with the power rule.
- Derivative RulesDifferentiate products, quotients and compositions with the product, quotient and chain rules.
- Applications of DerivativesUse derivatives to find tangent lines, velocities, and maximum and minimum values.
- IntegralsReverse differentiation to find antiderivatives, use definite integrals to measure accumulated area, and pin down the one antiderivative through a given point.
- Continuity & DiscontinuitiesDecide whether a function is continuous at a point with the three conditions, name the kind of break where it is not, choose the constant that mends a piecewise rule, and use the Intermediate Value Theorem to promise a root.
- Infinite Limits & AsymptotesRead a vertical asymptote as a one-sided limit that grows without bound, a horizontal asymptote as a limit at infinity, and tell an asymptote from a hole — for rational, radical and exponential forms.
- The Definition of the DerivativeBuild the derivative from scratch: an average rate over an interval, the difference quotient, and the limit that turns a secant into a tangent — then estimate it from a table or a graph, and see where a graph has no derivative at all.
- Implicit DifferentiationDifferentiate a relation that is not solved for y: treat y as a function of x, apply the chain rule to every y, and solve for dy/dx — then find the slope and the tangent at a point on a circle, an ellipse or a hyperbola.
- Concavity & Curve SketchingUse the second derivative to read the bend of a graph: concave up or down, inflection points, the second derivative test, and the shape that the signs of f′ and f″ together force — then assemble the features of a curve before drawing it.
- Related RatesTwo quantities tied by an equation change together: differentiate the equation with respect to time, substitute what is known at the instant in question, and solve for the rate that is asked — ladders, balloons, cones, shadows and things moving apart.
- Increasing, Decreasing & ExtremaRead the sign of the derivative as the direction of the graph: critical numbers, intervals of increase and decrease, the first derivative test for local extrema, absolute extrema on a closed interval, and the Mean Value Theorem.
- Riemann Sums & Approximating AreaApproximate the area under a curve with rectangles and trapezoids, say whether the estimate is too big or too small, write the limit of the sum as a definite integral, and evaluate an integral from geometry and from its properties.
- Applications of IntegralsPut the definite integral to work: the area between two curves, the average value of a function, the derivative of an accumulation function, displacement against distance, substitution for a composite integrand, and total change from a rate in context.
- Limit Laws & the Formal DefinitionWork with limits as objects: combine known limits with the limit laws, rationalise a radical quotient, use the two special trigonometric limits and the squeeze theorem, and read the epsilon-delta definition on a straight line.
- Transcendental DerivativesDifferentiate the functions that are not powers of x: exponentials of any base, logarithms, tangent and secant, their compositions and products, and the inverse of a function at a point.
- Higher-Order DerivativesDifferentiate again and again: second, third and nth derivatives of polynomials, exponentials, sines and reciprocals, the patterns they fall into, and what each order means.
- OptimizationTurn a word problem into a function of one variable and find its best value: two numbers, a fenced field, an open box, a can of least material, the closest point, the best price.
- Motion Along a LineRead a moving particle from its position, velocity or acceleration: when it stops and turns, when it speeds up or slows down, how far it travels against how far it ends up, and where it is from where it started.
- Linear Approximation & DifferentialsReplace a curve by its tangent line where you stand: write the linearization, use it to estimate a root or a power, work with differentials and propagated error, and say whether the estimate is high or low.
- Newton's MethodApproximate a root by sliding down tangent lines: take one step from a starting guess, write the iteration formula, run it to a given precision, and recognise the starts where it gets stuck.
- Integration by SubstitutionUndo the chain rule: choose u, adjust the constant, integrate powers of an inside function, trigonometric, exponential and logarithm composites and radicals, change the limits of a definite integral, and put x back in terms of u when one is left over.
- Volumes: Cross-Sections & Solids of RevolutionAdd up thin slices to measure a solid: discs and washers spun about an axis or a line, solids built on a region from square, semicircular or triangular cross-sections, and the cone and sphere recovered from an integral.
Probability Marsh
Nothing here is certain, and everything is measured.
- Outcomes & ProbabilityMeasure how likely something is on a scale from 0 to 1 by comparing favourable outcomes with all the outcomes there are.
- Complements, Or & AndCombine probabilities: the complement rule for "not", the addition rule for "or", and the multiplication rule for independent events.
- Experimental ProbabilityEstimate a probability by running the experiment and counting, then use that estimate to predict what a longer run will do.
- Counting PrinciplesCount arrangements and selections without listing them, using the multiplication principle, factorials, permutations and combinations.
- Compound EventsWork out probabilities for experiments with more than one stage, and keep track of whether the first stage changes the second.
- Expected ValueWork out what a chance situation is worth in the long run by weighting every payout by how likely it is.
Statistics Steppe
One number never describes a crowd.
- Reading Data DisplaysPull exact answers out of tables, bar charts, pictographs, two-way tables and line graphs without being misled by the picture.
- Mean, Median & ModeSummarise a data set with a single number, and choose the summary that tells the truth about the data you actually have.
- Range & SpreadMeasure how spread out a data set is with the range, the quartiles and the interquartile range, and use the spread to spot outliers.
- Scatter Plots & TrendsPlot two measurements against each other, describe the association you see, and fit a straight line you can predict from.
- Samples & InferenceUse a sample to say something about a population you cannot count, and be honest about how much the answer could be out by.
- Standard Deviation & z-scoresMeasure spread with the average squared distance from the mean, and use z-scores to compare values from different data sets.
- Two-Way TablesCross two yes-or-no questions in one table, turn the counts into relative frequencies by row or column, and say whether the answers to one question lean on the other.
- Histograms, Box Plots & ShapeRead a dot plot, a histogram and a box plot, build the five-number summary a box plot is drawn from, describe the shape of a distribution as symmetric or skewed, and compare two distributions by centre and spread.
- Correlation & RegressionUse a fitted line to predict and to measure how far each point misses it, read the strength and direction of a relationship from the correlation coefficient r, and keep correlation apart from causation.
- The Normal DistributionUse the bell curve to turn a value into a proportion and back: the empirical rule, z-scores with a table, percentiles, the value behind a percentage, and comparing scores from different distributions.
Linear Algebra Kingdom
Nowhere is anywhere until you can say how to get there.
- Vectors in n DimensionsA vector stops being an arrow on a page and becomes a list of numbers of any length: add and scale vectors in three, four or n dimensions, measure their length, find the unit vector along one, take a dot product to test whether two are perpendicular, and write a line in space as a point plus a direction.
- Linear CombinationsBuild one vector out of others: evaluate a combination such as 3u - 2v, work backwards to find the weights that reach a target, decide whether a target can be reached at all, and see that a matrix times a vector is nothing more than a combination of the columns.
- SpanEverywhere a set of vectors can reach: decide whether a vector lies in a span, recognise a span as the origin, a line, a plane or the whole space, find the value that puts a vector inside one, and say how many vectors it takes to span n dimensions.
- Linear IndependenceWhether a set carries any repetition: test two or three vectors for independence, find the dependence relation when there is one, spot the value that makes a set dependent, read independence off a pivot count, and know why more than n vectors in n dimensions can never be independent.
- Vector Spaces & SubspacesWhich sets of vectors are worlds in their own right: check that a set contains the zero vector and is closed under addition and scaling, tell a plane through the origin from one that misses it, name the rule a set breaks, and give the dimension of the subspace that survives.
- Basis, Dimension & CoordinatesThe smallest set that still reaches everywhere: decide whether a set is a basis, count the dimension of a span or a subspace, write a vector in coordinates relative to a chosen basis, and turn those coordinates back into the ordinary vector.
- Matrix OperationsArithmetic on rectangles of numbers: add and scale matrices, multiply them row by column, work out which products are even defined and what size they come out, take a transpose, recognise a symmetric matrix and the identity, and see for yourself that AB and BA are usually different.
- Row Reduction & Gaussian EliminationThe one procedure the whole subject leans on: apply a row operation and say what it did, clear a column below a pivot, drive a matrix all the way to reduced row echelon form, count the pivots, and recognise when a matrix is not in that form yet and which rule it breaks.
- DeterminantsOne number that says whether a matrix flattens space: compute a 2 by 2 and a 3 by 3 determinant, use row operations and a triangular form to make it quick, apply the rules for a product, a scalar multiple and a transpose, read the determinant as an area, and solve a small system with it.
- Linear Systems in Matrix FormA system of equations rewritten as one matrix equation: build the augmented matrix, read a matrix back as a system, solve two and three unknowns by elimination, and decide from the reduced form whether there is one solution, none at all, or a whole line of them.
- Inverse MatricesThe matrix that undoes another: invert a 2 by 2 with the formula, decide from a determinant whether an inverse exists at all, find one by reducing A beside the identity, solve a system with it, and collect the many equivalent ways of saying a matrix is invertible.
- Rank & the Shape of SolutionsCounting what a matrix really pins down: read the rank off a reduced form, work out the nullity from it, count free variables, compare the rank of a matrix with the rank of its augmented form to decide consistency, and say the largest rank a given shape allows.
- Linear TransformationsA function that moves every vector at once, and the matrix that describes it: test the two rules a linear map obeys, build the standard matrix from the images of the basis vectors, recognise rotations, reflections, scalings and shears, and read a composition as a matrix product.
- Kernel, Range & Rank–NullityWhat a transformation loses and what it can reach: decide whether a vector is sent to zero, find the dimension of the kernel and a vector that spans it, decide whether a target is in the range, and use rank plus nullity to account for every dimension of the input.
- Eigenvalues & EigenvectorsThe directions a matrix leaves pointing the same way: check whether a vector is an eigenvector and read its eigenvalue, form the characteristic equation of a 2 by 2, find both eigenvalues and an eigenvector for each, and use the trace and determinant as a shortcut.
- Diagonalization & Its UsesChoosing the basis that makes a matrix simple: decide whether a matrix can be diagonalised, write down the diagonal matrix of eigenvalues, raise a matrix to a high power through its eigenbasis, and read the long run of a two-state chain or a population model off the eigenvalues.
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