QunoMath: interactive maths, from arithmetic to machine learning

QunoMath is a free, interactive maths course with no account required. Every idea is built from first principles, shown on a graph you can move, then checked with a quiz. It spans mental-math tricks, Foundations and quantitative reasoning, GCSE, A-Level pure, statistics and mechanics, and the maths behind machine learning: 143 connected topics.

Calculator · Mental math · Curriculum · Applied and modelling

Full topic list

Mental math

  • Fast addition (round and adjust): Add the easy round number, then correct. Speed from algebra.
  • Multiplication shortcuts: Times eleven, squaring numbers ending in five, difference of squares.
  • Index laws at a glance: Powers are counting copies: add, subtract, multiply the exponents.
  • Speed drill: Lock the tricks in under light time pressure.

Number

  • Place value & number sense: What digits actually mean; reading and comparing magnitudes.
  • Adding & subtracting integers: Arithmetic as motion on the number line.
  • Multiplication as area: Why a×b = b×a, and the grid model.
  • Division & remainders: Sharing vs grouping; quotient and remainder.
  • Negative numbers: Direction as well as size; sign rules.
  • Order of operations: Why BIDMAS/PEMDAS removes ambiguity.
  • Factors, multiples, primes: The building blocks of every integer.
  • Fractions: Equal parts; why you need a common denominator to add.
  • Decimals: Fractions in base ten; place value past the point.
  • Percentages: Per hundred; increase, decrease, reverse.
  • Ratio & proportion: Comparing quantities; scaling recipes and maps.
  • Rates & unit conversion: Speed, density, price-per-unit; cancelling units.
  • Powers & roots: Repeated multiplication and its inverse.
  • Index laws: Why aᵐ·aⁿ = aᵐ⁺ⁿ, and zero/negative powers.
  • Standard form: Taming very large and very small numbers.
  • Estimation & sanity checks: Rounding to catch wrong answers fast.
  • Rounding & significant figures: Keeping the digits that matter, and error bounds.
  • Bounds & error intervals: Upper and lower bounds of a rounded measurement.
  • HCF & LCM: Prime factor trees to share or synchronise.
  • Four operations with fractions: Add, subtract, multiply, divide without a calculator.
  • Recurring decimals to fractions: Algebra turns a repeating decimal into an exact fraction.
  • Rational & irrational numbers: Why root two cannot be a fraction.

Reasoning

  • Translating word problems: Turning English into equations — the core skill.
  • Data interpretation: Reading tables and charts without being fooled.
  • Mathematical proof: Why something is true for all cases.
  • Logic, sets & proof by contradiction: Assume the opposite and break it.
  • Proof by induction: Topple the first domino, then every domino.
  • Algorithms & complexity: Sorting, searching and why big O matters.
  • Modular arithmetic: Clock arithmetic behind cryptography.

Algebra

  • Variables & expressions: Letters as unknown numbers; collecting like terms.
  • Linear equations: Balance: do the same to both sides.
  • Rearranging formulae: Make any letter the subject.
  • Expanding brackets: Distribution; double brackets.
  • Factorising: Expanding in reverse — the key to solving.
  • Quadratic equations: Factor, complete the square, or use the formula.
  • Simultaneous equations: Two facts, two unknowns; lines crossing.
  • Inequalities: Ranges of solutions; flipping the sign.
  • Surds: Exact irrational roots; rationalising.
  • Sequences: nth terms; arithmetic and geometric.
  • Series & sigma notation: Summing sequences in closed form.
  • Binomial expansion: Pascal’s triangle and (a+b)ⁿ.
  • Partial fractions: Splitting fractions to integrate them.
  • The quadratic formula: Completing the square once, for every quadratic at once.
  • Completing the square: Rewrite to read off the vertex and solve.
  • Algebraic fractions: Simplify, add and solve fractions with letters.
  • Domain, range & graphs of functions: What goes in, what can come out.
  • Iteration & numerical roots: Repeat a formula to close in on a solution.
  • Parametric equations: Describe a curve by a third variable, time.

Graphs

  • Coordinates & the plane: Locating points; the language of graphs.
  • Straight lines: y = mx + c: Gradient and intercept — the most useful model in maths.
  • Quadratic graphs: Parabolas: roots, vertex, symmetry.
  • Functions & mappings: Inputs to outputs; domain and range.
  • Graph transformations: Shift, stretch, reflect a curve.
  • Inverse & composite functions: Undoing and chaining functions.
  • Exponentials & logarithms: Growth, decay, and the inverse of powers.
  • Exponential growth & decay: Compound interest, half-life, populations.

Geometry

  • Angles & parallel lines: Angle facts that unlock everything else.
  • Triangles & polygons: Angle sums, congruence, similarity.
  • Pythagoras’ theorem: a² + b² = c², proved by area.
  • Trigonometry (SOH-CAH-TOA): Sides from angles and back.
  • The unit circle: Why sin and cos are waves.
  • Trig identities & equations: Provable relationships; solving over a range.
  • Circle theorems: Angles in circles follow strict rules.
  • Area, surface area & volume: Measuring 2D and 3D space.
  • Transformations & symmetry: Translate, rotate, reflect, enlarge.
  • Vectors: Magnitude and direction; adding tip-to-tail.
  • Similarity & scale factors: Length, area and volume scale at different rates.
  • Constructions & loci: Compass and ruler define exact paths.
  • Bearings: Navigation angles measured clockwise from north.
  • Sine & cosine rules: Triangles that are not right angled.
  • Radians, arcs & sectors: Angle measured by arc length, the natural unit for calculus.
  • Small angle approximations: For tiny x, sin x is about x. Why, from the series.
  • Vector geometry & proof: Position vectors prove lines collinear and ratios.

Calculus

  • Limits & continuity: What a function approaches — the idea behind everything in calculus.
  • The derivative from first principles: Slope as a vanishing secant; the limit definition.
  • Differentiation rules: Power, chain, product, quotient — shortcuts that follow from the definition.
  • Applications of differentiation: Maxima, minima, rates of change.
  • Integration as area: Accumulation; the fundamental theorem.
  • Integration methods: Substitution, by parts, partial fractions.
  • Taylor series: Any smooth curve is a polynomial up close.
  • Differential equations: Equations about rates of change.
  • Partial derivatives & gradient: Slopes in many directions; the gradient vector.
  • Chain rule: Differentiate a function inside a function.
  • Product & quotient rules: Derivatives of things multiplied or divided.
  • Implicit differentiation: Differentiate equations not solved for y.
  • Definite integrals & area under a curve: Net signed area between curve and axis.
  • Volumes of revolution: Spin a curve, integrate discs into a solid.
  • Maclaurin & power series: Build e^x, sin and cos from their derivatives at zero.

Probability

  • Basic probability: Chance as a fraction of outcomes.
  • Counting & combinatorics: Permutations and combinations.
  • Tree diagrams & Venn: Combined and conditional events.
  • Conditional probability & Bayes: Updating belief on evidence.
  • Random variables & expectation: Average outcome of a process.
  • Binomial & normal distributions: The shapes chance takes.
  • Law of large numbers & CLT: Why averages become normal.
  • Addition & multiplication rules: And, or, independent, mutually exclusive.
  • Discrete random variables: Probability distributions you can tabulate.
  • The Poisson distribution: Rare events over a fixed interval.
  • Continuous distributions & density: Area under a density is probability.
  • Markov chains: The next state depends only on the current one.

Statistics

  • Averages & spread: Mean, median, mode; variance and σ.
  • Charts & sampling: Representing data; bias in samples.
  • Sampling distributions & SE: How sure a sample lets you be.
  • Hypothesis testing: Is this effect real or noise?
  • Correlation & regression: Lines through data; what r really means.
  • Box plots, histograms & cumulative frequency: Shapes of data: spread, skew, quartiles.
  • Correlation coefficient (PMCC): Putting a single number on linear association.
  • Using a regression line: Interpolation, extrapolation and its danger.
  • Confidence intervals: A range that probably holds the true value.
  • Maximum likelihood estimation: Pick the parameter that makes the data most likely.

Linear algebra

  • Vectors in n-D: Lists of numbers as arrows; dot product.
  • Matrices as transformations: A matrix moves all of space at once.
  • Determinant: Signed area/volume scaling.
  • Eigenvectors & eigenvalues: Directions that only stretch — the basis of PCA.
  • Solving linear systems: Gaussian elimination; existence of solutions.
  • Complex numbers: A plane where multiplication rotates.

Machine learning

  • Loss functions: Turning "wrong" into a number to minimise.
  • Gradient descent: Walk downhill on the loss surface.
  • Linear regression (ML view): The same line, learned by optimisation.
  • Logistic regression: Bending a line into a probability.
  • Overfitting & regularisation: Fitting signal, not noise.
  • k-means clustering: Finding groups with no labels.
  • Neural networks: Stacked bends learn any boundary.

Applied & modelling

  • Game theory & Nash equilibrium: When the rational choice for each makes everyone worse off.
  • Graph theory & shortest paths: Nodes and edges: maps, networks, the internet.
  • Supply, demand & utility: Price as an intersection; marginal thinking.
  • Population & epidemic models: Logistic growth and predator vs prey from rates of change.
  • Finance: return and risk: Risk is the standard deviation of returns. Learn sigma, then you can price risk.
  • Optimisation problems: Biggest volume, least cost: turn a worded goal into a maximum.

Mechanics

  • Kinematics (SUVAT): Position, velocity and acceleration under constant a.
  • Forces & Newton laws: Resultant force equals mass times acceleration.
  • Projectiles: Independent horizontal and vertical motion.
  • Moments & equilibrium: Turning effect; when a rigid body balances.
  • Calculus in kinematics: Differentiate and integrate to switch between s, v, a.

Notation & language

  • Set notation: Bags of things: membership, builder form, union and intersection.
  • Intervals & brackets: Round excludes, square includes; infinity is always round.
  • Logic & quantifiers: And, or, not, if-then, for-all and there-exists.
  • Symmetry: from a butterfly to groups: Start with folding paper, end at the structure behind physics.

Practice

  • Random mixed practice: Press once for six questions pulled at random across every level.

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