GCSE Maths revision (AQA, Edexcel, OCR)

Order every kind of number on the line, calculate with confidence using written methods and the priority of operations, break numbers into primes for HCF and LCM, count outcomes with the product rule, and handle powers and roots — including the Higher-only estimates.

Move fluently between fractions, decimals and percentages, treat both as operators on quantities, and solve the big-money GCSE questions: percentage change, original values, simple interest and — at Higher — recurring decimals and compound interest.

Master the index laws, then extend them to negative and fractional powers, simplify and rationalise surds for the exact-answer non-calculator paper, and handle very large and very small numbers in standard form.

Round confidently to decimal places and significant figures, estimate answers before you calculate, write error intervals in inequality notation, and — at Higher — track the largest and smallest possible results of a calculation with upper and lower bounds.

Speak the language of algebra fluently: collect like terms, expand single and double brackets, factorise into brackets — including quadratics and the difference of two squares — and, at Higher, simplify algebraic fractions and handle indices and surds in algebra.

Substitute into the formulas scientists use, rearrange any equation to change its subject, and then move into Higher territory: inverse and composite functions, constructing algebraic proofs, and hunting equation roots with iteration.

Solve any linear equation — brackets, fractions and unknowns on both sides — then master y = mx + c: reading gradients and intercepts, drawing lines, writing equations through two points, and the Higher perpendicular rule.

Solve quadratic equations three ways — factorising, the formula and completing the square — then read the same information off the parabola: roots, y-intercept and the turning point hiding in the completed-square form.

Solve paired equations by elimination and substitution — including the Higher linear/quadratic pairs — then handle inequalities on number lines, quadratic inequality ranges and shaded graph regions.

Generate and describe sequences, master the nth term of linear sequences for finding and testing terms, meet the famous special sequences, and — at Higher — handle Fibonacci-type and geometric progressions and quadratic nth terms.

Recognise every function family by its shape, move graphs with translations and reflections, read rates from curved graphs with gradients and areas, and finish with the Higher circle equation and its tangent.

Write and simplify ratios, split amounts in part:part and part:whole ratios, scale recipes and maps, and judge the best buy — the everyday half of the heaviest-weighted GCSE topic area.

Recognise when two quantities rise together or trade off, write y = kx and y = k/x, work with speed, density, pressure and exchange rates, and estimate populations by capture–recapture.

Chain angle facts together with reasons — point, line, parallel and polygon rules — then build exact constructions and shade loci regions without ever measuring an angle.

Prove triangles congruent or similar, scale lengths, areas and volumes, carry out the four transformations with full instructions, and finish with column-vector arithmetic.

Find any side or angle of a right-angled triangle with Pythagoras and SOH CAH TOA, memorise the exact trig values, then extend to any triangle with the sine and cosine rules.

Measure every shape the GCSE can throw at you — rectilinear floors, trapezia, circles, sectors, prisms, cylinders and (Higher) cones and spheres — with the formulas memorised, not given.

Learn the seven circle theorems with the isosceles-triangle proofs behind them, then hunt angles round any diagram of chords, tangents and cyclic quadrilaterals.

Price events between 0 and 1, test fairness with relative frequency, organise samples with Venn diagrams and frequency trees, then multiply through tree diagrams — conditionally at Higher.

Sample fairly, draw the right diagram, boil a data set down to an average and a spread, then read histograms, box plots and scatter lines — with the Higher-only machinery clearly tagged.