Learn Mental Math
Free guides to doing arithmetic in your head. Each one explains why the shortcut works rather than just handing you a recipe — that is what makes a trick stick, and what tells you when it applies.
Every guide pairs with free practice in the game, because reading a shortcut and being able to use it are different skills.
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
The guides
Times Tables Practice
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
Mental Math Tricks
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%
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
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
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 and 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%
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
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
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
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
- : round one up, one down, and the errors cancel
- Estimating as a sanity check, and rounding-and-adjusting for exact answers
The catalog, branch by branch
The guides above answer one question each. The pages below follow the app's lesson ladder instead — one page per branch of the syllabus, naming and linking every lesson it covers, so you can see how a topic is built from the first rung to the last.
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
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
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
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
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
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
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
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
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
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
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
Shapes, Area and Perimeter
Every area formula comes from the rectangle. Start there and there is very little left to memorise.
- 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
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
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
Algebraic Expressions, Powers and Roots
Arithmetic with the numbers left unnamed — expanding, factorising, indices and surds.
- Like terms, expanding two brackets, and factorising quadratics
- The index laws, and why anything to the power zero is 1
- Standard form, surds and rationalising the denominator
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
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
Circle Theorems and Transformations
The circle theorems are not a list to memorise — 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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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 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
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 optimisation
Where to start
If the times tables are still slow, do those first. The shortcuts in the other guides all assume instant recall, and trying to run a trick while also working out 7 × 8 is what makes mental arithmetic feel impossible.
If recall is already solid, go straight to the shortcuts and pick two or three. Nobody uses all of them — people who are quick tend to have a handful they reach for automatically.
Fractions and percentages are the same idea in two notations, so they read well together: a percentage is a fraction whose denominator everyone has already agreed on.
Practice in the game
Math Challenge is a free mental-math game: swipe to answer, with difficulty that adapts to what you keep missing. It has 35 practice topics and 36 illustrated Magic Tricks lessons that teach the reasoning behind each shortcut. The Daily Challenge is free and needs no account.
Your turn
Three to try — tap what you get.
7 × 8
Half of 3/4
10% of 250