Math guides

☰ Contents

Forty-seven guides to school mathematics, free and without an account. Each one answers one question: why the method works, not just how to run it. Each comes with a figure you can drag and practice questions that mark themselves.

They are grouped below by subject. The same grouping runs down the left of every guide, so you can move between them without coming back here.

Mental math

Arithmetic in your head: what to practice, and the shortcuts worth knowing.

How to Learn Mental Math

What mental arithmetic is for, the four habits it rests on, and a 60-second drill drawn from the lessons.

Why a shortcut you understand outlasts one you memorized
Five minutes a day, mixed rather than recited
A drill that scores you, and where each question comes from

How to Do Math in Your Head

The shortcuts that turn a hard-looking sum into an easy one before you calculate anything.

Adding in your head: work left to right, round and adjust, make ten
Multiplying fast: ×11, ×5, doubling and halving, squaring numbers ending in 5
Subtracting by counting up, and by sliding both numbers
Percentages: the flip (x% of y = y% of x) and building everything from 10%

How to Learn Times Tables

Start here if multiplication facts aren't automatic yet. Nearly every other shortcut assumes they are.

Why only about six of the 144 facts are genuinely hard
The trick for each table — ×9 on your fingers, ×5 by halving, ×4 by doubling twice
A grid with the hard facts marked, and a four-week practice plan

All 36 Magic Tricks

The complete list, each with a worked example and the identity underneath it. A reference rather than a read-through.

Multiplication and squaring: ×11, ×25, near-100, ends-in-5
Division and the digit-sum tests for 3, 9 and 11
Series: the first N odd numbers, Gauss's pairing, telescoping sums
Number theory: last digits of huge powers, and why they cycle

Numbers & Counting

Divisibility Rules from 2 to 12

A rule for every number from 2 to 12, and the reason each one works.

The table: the test and a worked example for each of the eleven
Why the last digits settle 2, 5 and 10, the last two settle 4, the last three settle 8
Why digit sums test 3 and 9, and why 12 is not "divides by 2 and by 6"

Estimation and Rounding

Know the size of an answer before you work it out, and a slipped decimal point has nowhere to hide.

Rounding to ten, hundred, thousand and decimal places — one digit decides
38 × 52 ≈ 40 × 50: round one up, one down, and the errors cancel
Estimating as a sanity check, and rounding-and-adjusting for exact answers

Factors, Multiples and Number Theory

Which divisions come out exactly — and where that question leads.

Divisibility tests, the sieve, prime factorization, HCF and LCM
Euclid's proof that the primes never run out
Why 0.999… is exactly 1, and why root two cannot be a fraction

Whole Numbers and Place Value

Counting to a million, rounding, significant figures, and every rule for numbers below zero.

Why ten digits are enough to write any whole number
Rounding to a place, and when significant figures are the better measure
Why subtracting a negative adds, and why two negatives multiply to a positive

Applications: whole numbers (13), HCF and LCM (14), patterns (10), rounding (10), thousands, angles and time (10), place value (10), counting and comparing (10), significant figures and estimation (10), number theory (10). The number is the question types on the page.

The Four Operations

Arithmetic Methods

Mental strategies and written algorithms for all four operations, and how to choose between them.

Bridging through ten, rounding and adjusting, the same difference
Carrying and borrowing as place value in motion — including past a zero
Long multiplication, long division, remainders and the order of operations

Division

Every division is a multiplication read backwards — from head methods to long division, step by step.

÷4 is halve twice; ÷7 is a times-table fact in reverse
Short division digit by digit, and long division without fear
What a remainder is, and the divide-without-dividing tests

Order of Operations

Why 2 + 3 × 4 is 14, not 20 — and why that was never a matter of opinion.

The rule as three steps: brackets, then × and ÷, then + and −
Why multiplication comes first: 3 × 4 was always a package
The left-to-right trap: 8 ÷ 2 × 4 is 16, and 8 − 2 + 5 is 11

Applications: order of operations (10), multiplying and dividing in your head (10), adding and subtracting in your head (10), written addition and subtraction (10), equal groups and sharing (10), adding and subtracting within 20 (10). The number is the question types on the page.

Fractions

Equivalent Fractions and Mixed Numbers

The fraction ladder from equal parts to dividing one fraction by another.

What makes two fractions equivalent, and how to compare any two
Improper fractions and mixed numbers, converting both ways
A fraction of an amount, and why "flip and multiply" divides

Fractions

Why the rules are what they are, starting from the one idea that makes them obvious: the bottom number is a unit, not a quantity.

Equivalent fractions, and why multiplying top and bottom changes nothing
Adding and subtracting: what a common denominator is actually for
Comparing by cross-multiplying, and why that shortcut is legitimate
Simplifying, mixed numbers, and why "flip and multiply" divides

Applications: fractions (18), multiplying and dividing fractions (10), adding and subtracting fractions (10), comparing fractions (10), halves and quarters (10). The number is the question types on the page.

Decimals

A decimal is a fraction whose bottom number is ten — hold that idea and the rules become obvious.

Why 0.9 beats 0.35: more digits does not mean bigger
Line up the point to add; count the places to multiply
Dividing by a decimal: scale both up until the divisor is whole

Applications: decimals (14). The number is the question types on the page.

Percent

Decimals, Percentages and Interest

One subject in three notations, followed all the way to loans and annuities.

Percentage change, reverse percentages and depreciation
Simple against compound interest, and what compounding more often buys
Inflation, the time value of money, and how a loan repayment splits

Percentages

You are never really calculating a percentage — you are building it out of a few you already know.

Every percentage assembled from 10%, 5% and 1%
Percentage change, and why 40 → 50 and 50 → 40 are not the same size
Discounts in one step with a multiplier, and reverse percentages
Why 20% off then 10% off is 28% off, not 30%

Applications: percentage (16), simple and compound interest (10), financial mathematics (10). The number is the question types on the page.

Ratio, Rate & Proportion

Ratio, Rates and Proportion

The topic where the arithmetic is easy and the reading is hard.

Sharing in a ratio, bar models and the unitary method
Speed, density, exchange rates and compound-unit conversion
Direct and inverse proportion, and before-and-after ratio problems

The Bar Model Method

Word problems drawn as bars, also called tape diagrams, so that equal amounts are equal lengths.

Part-whole, comparison and before-and-after models
Fractions of a remainder and percentages as bars
When algebra is quicker than the drawing

Applications: ratio (20), speed (15), rate and work (14), ratio basics (10). The number is the question types on the page.

Algebra

Algebraic Expressions, Powers and Roots

Arithmetic with the numbers left unnamed — expanding, factoring, indices and surds.

Like terms, expanding two brackets, and factoring quadratics
The index laws, and why anything to the power zero is 1
Standard form, surds and rationalising the denominator

Equations and Inequalities

One permission — do the same to both sides — applied everywhere it reaches.

Brackets, fractions, the unknown on both sides, and changing the subject
Simultaneous equations by substitution, elimination and scaling
Inequality regions, and when the sign flips

Functions, Logarithms and Rational Functions

What f(x) means, why an inverse reflects in y = x, and why some graphs have holes in them.

Domain and range, piecewise functions, composites and inverses
Graph transformations, and why the inside of the bracket runs backwards
Exponential growth and decay, the laws of logarithms, asymptotes and holes

Graphs and Coordinate Geometry

What gradient really measures, and why a shape with an equation is a shape you can reason about.

y = mx + c, lines through two points, parallel and perpendicular gradients
Circles by completing the square, tangents, and where a line meets a curve
Ellipses, hyperbolas, and reading distance-time and speed-time graphs

Matrices and Markov Chains

A matrix is a machine, not a table of numbers — which is why AB is not BA and why some matrices cannot be undone.

The matrix product, the identity, and why order is part of the meaning
Determinants as area factors, inverses, and solving equations with them
Eigenvalues, diagonalization, and why a Markov chain settles down

Quadratics and Polynomials

Three ways to solve a quadratic, what the discriminant predicts before you solve, and the binomial theorem.

Factoring, completing the square, and where the formula comes from
The discriminant, the sum and product of the roots, quadratic inequalities
The factor theorem, cubics, and solving by iteration when nothing else works

Sequences and Series

From spotting that 5, 8, 11 goes up in threes to proving an infinite sum has a finite value.

nth terms, quadratic and geometric sequences, sigma notation
Gauss's pairing trick, the sum to infinity, and proof by induction
The convergence tests, and Taylor series with an error bound

Applications: negative numbers and linear equations (10), powers and roots (10), algebraic expressions (14), equations and inequalities (16), quadratic equations (13), graphs of equations (15), quadratic graphs and coordinate geometry (10), functions (17), rational functions (10), matrix arithmetic (10), determinants and inverses (10), eigenvalues and eigenvectors (10), transition matrices and Markov chains (10), polynomials and the binomial theorem (10), motion in two dimensions (10), series and convergence (10), sequences and series (10). The number is the question types on the page.

Complex Numbers

What happens once you stop insisting every number sits on a line.

Why multiplying by i is a quarter turn, and why that makes i² = −1 obvious
The Argand diagram, modulus and argument, and polar form
De Moivre's theorem, the roots of unity, and circles drawn by equations

Applications: complex arithmetic (10), the complex plane (10). The number is the question types on the page.

Calculus

Applications of Differentiation

What to do once you can differentiate: find the turning points, the shape, the optimum.

The first and second derivative tests, inflections and curve sketching
Optimization, motion in a straight line, and related rates
Parametric curves, L'Hôpital's rule and the Newton-Raphson method

Applications of Integration and Polar Curves

Choose what the thin slices are, and the same integral gives areas, volumes, lengths and averages.

Area between curves, volumes of revolution, arc length, improper integrals
Differential equations from separable to second-order, and oscillation
Coupled systems and phase portraits, and polar coordinates and areas

Limits and Continuity

What a limit actually is, and how it resolves the contradiction at the heart of calculus.

One-sided limits, the limit laws, and what 0/0 really means
Limits at infinity, the squeeze theorem, and the limit of sin x over x
Continuity, the three kinds of discontinuity, and the intermediate value theorem

Partial Differentiation

Almost nothing depends on one variable. Calculus on surfaces, where every direction has its own slope.

Partial derivatives, level curves and the mixed derivative theorem
The gradient vector, directional derivatives and tangent planes
Saddle points, and Lagrange multipliers for constrained optimization

Rules of Differentiation

What dy/dx means, and how to tell at a glance which rule a question is asking for.

The power, product, quotient and chain rules, and when each applies
Trigonometric, exponential and logarithmic derivatives, and the number e
Implicit differentiation, inverse functions and the hyperbolic family

Techniques of Integration

There is no algorithm for integration — only a ladder of techniques and the skill of choosing one.

Why there is always a + C, Riemann sums and the fundamental theorem
Substitution, integration by parts, partial fractions and f'/f
Trigonometric substitution, and the integrals that give inverse functions

Applications: introduction to calculus (10), solving equations numerically (10), limits (10), continuity (10), rules of differentiation (10), using differentiation (11), hyperbolic functions (10), techniques of integration (11), using integration (11), polar curves (10), coupled differential equations (10), functions of several variables (10), partial derivatives (10). The number is the question types on the page.

Geometry

Circle Theorems and Transformations

The circle theorems are not a list to memorize — one of them generates most of the others.

The congruence tests, and why SSA is deliberately missing
Every circle theorem with the isosceles triangle underneath it
Reflections, rotations and enlargements, then the same moves as matrices

Lines, Angles and Pythagoras

Where geometry stops being measurement and starts being deduction.

The angle facts, and the parallel-line rules they all rest on
Why a triangle's angles make 180 — and where that stops being true
Pythagoras proved by dissection, its converse, constructions and loci

Polygons, Solids and Surface Area

Cut it into pieces you know, or unfold it flat. Those two moves give every formula here.

Prisms, cones and spheres, and why a cone is a third of its cylinder
Surface area by unrolling, and the slant height that trips cone questions
Why doubling the lengths multiplies volume by eight, and the five Platonic solids

Shapes, Area and Perimeter

Every area formula comes from the rectangle. Start there and there is very little left to memorize.

Area versus perimeter, and how the units settle which is which
Why a triangle is half a rectangle, and where pi comes from
Volume, nets, symmetry and the first angle work

Vectors

How far and which way, carried as one object — which is what makes geometry in three dimensions manageable.

Adding, scaling, magnitude and unit vectors, and vector proofs
The dot product for angles, the cross product for areas and normals
Lines and planes in space: equations, angles, intersections, distances

Applications: angles (10), triangles (10), quadrilaterals (10), polygons (10), paper folding (10), circles (10), composite areas (10), overlaps (10), symmetry and grids (10), solids (10), building with shapes (10), shapes and where they sit (10), Pythagoras and similar shapes (10), congruence, similarity and circle theorems (12), vectors in the plane (12), planes and the vector product (10), matrices as transformations (10), volume and surface area (10), transformations (10), constructions and loci (10). The number is the question types on the page.

Measurement

Units, Money and Time

Two decimal systems and one that runs on sixties.

Metric conversions, and why area units square the factor
Reading a scale between the marks
The 24-hour clock, durations across the hour, and money as decimals

Applications: water tanks (10), measurement (10), time (10), money (10). The number is the question types on the page.

Statistics

Averages and Spread

An average without a spread is half an answer, and usually the less interesting half.

Mean, median and mode, and which one an outlier moves
Quartiles from a cumulative frequency curve, percentiles and box plots
Standard deviation, why the deviations are squared, and Simpson's paradox

Sampling and Inference

How a sample of 800 is allowed to say something about forty million.

Sampling methods, standard error, and the central limit theorem
What a p-value is actually telling you, and what it is not
Confidence intervals, type one and two errors, and the chi-squared, t and F tests

Scatter, Correlation and Regression

Two measurements per subject, and the question of whether they move together.

Reading a scatter plot before computing anything
Pearson's r, Spearman's rank, and what r² does and does not claim
The least-squares line, and why extrapolation has no evidence behind it

Statistical Charts and Displays

Which chart suits which data, and how an accurate graph still misleads.

Pictograms, bar charts, pie charts, dot diagrams and stem-and-leaf
Why a histogram carries its frequency in the area, not the height
Two-way tables, frequency trees, and the anatomy of a misleading graph

Applications: averages and charts (10), tables and line graphs (10), bar charts (10), pictograms (10), statistical displays (11), measuring data (12), scatter plots and correlation (10), sampling (10), statistical inference (10), correlation and regression (10). The number is the question types on the page.

Probability

Probability and Combined Events

The topic where a correct calculation regularly contradicts a strong intuition.

When to add and when to multiply, and the at-least-one shortcut
Tree diagrams with and without replacement, and conditional probability
Monty Hall, the birthday paradox, and why a positive test can mean healthy

Sets, Counting and Distributions

Counting outcomes without listing them, and the two distributions that model most things.

Set notation, Venn diagrams, and Cantor's bigger infinities
Permutations or combinations — the question that decides it
Expectation and variance, the binomial, and reading the normal distribution

Applications: probability with several events (12), sets and counting (10), probability distributions (10), chance (10). The number is the question types on the page.

Trigonometry

Radians and Trigonometric Identities

Why calculus insists on radians, and how identities turn an unsolvable equation into a solvable one.

Radian measure, arc length and sector area
The compound and double angle formulas, and where they come from
The R form, and writing every solution of a trigonometric equation

Trigonometry and Bearings

Why a fixed angle means fixed ratios, and how that one fact solves triangles nobody can measure.

SOHCAHTOA, exact values, elevation and depression
The sine and cosine rules, the area formula, and the ambiguous case
Trigonometric graphs and equations, and three-figure bearings

Applications: compass directions and bearings (10), triangle trigonometry (16), radians and trigonometric identities (10). The number is the question types on the page.

Logic & Proof

Logic and Proof

The part that decides whether an answer has been established rather than found.

Truth tables, the contrapositive, and why the converse proves nothing
Quantifiers, and how to negate a statement containing one
The six proof techniques, the pigeonhole principle and the named inequalities

Applications: logic and counting (10), mathematical statements (10), proof techniques (10), named inequalities (10), problem solving (10). The number is the question types on the page.

Graph Theory & Networks

Graph Theory, Networks and Voronoi

Strip anything down to what is connected to what, and the same theorems apply.

Degree, trees, adjacency matrices, and counting walks with matrix powers
Eulerian and Hamiltonian routes, Kruskal, Prim and the Chinese postman
Voronoi cells, nearest-site questions and the toxic waste dump problem

Applications: graphs, degree and trees (10), adjacency matrices (10), Eulerian and Hamiltonian routes (10), spanning trees and route problems (10), Voronoi diagrams (10). The number is the question types on the page.

Applications

111 pages, 1197 question types, each worked step by step with a figure that follows the steps. Each page sits under its subject above, and the index filters them by exam.

Syllabus maps

Which lessons cover which part of a syllabus, and which parts are not covered yet.

Every lesson

The guides above are articles. The catalog is the practice ladder they draw on, in the order the app teaches it.

Every lesson in the ladder

The full course index — 823 lessons from counting to calculus, grouped by stage and strand, each with its own page and a set of flashcards.

Where any single topic sits in the sequence, and what comes before it
A page per strand, so you can start in the middle if that is where you are

Math ChallengePractice that adapts to you, the whole lesson ladder, and your progress saved.