Pass an Exam Four operations
Practise adding, subtracting, multiplying and dividing integers and decimals, with clear place-value explanations and fully worked GCSE questions.
Learn · Practise · Check
GCSE Maths study guides
Choose an area of maths, then open a guide for clear explanations, worked examples and exam-style questions. Follow the learning stages within each area, starting with the basics before linked applications.
This is a suggested route, not a compulsory course. Skip skills you already know; some guides include later Higher extensions. For cross-topic prerequisites, use the links inside each guide.
105 guides availableFree to useEdexcel · AQA · OCR
Number
Fractions, percentages, powers and roots: build confidence with calculations.
1Start with calculations
Foundation & Higher6 worked examples10 questionsOpen guide
Negative numbers
Understand negative numbers using a number line, then order and calculate with signed integers and decimals. GCSE questions with complete worked answers.
BIDMAS
Understand BIDMAS and equal-priority operations, brackets, powers and fraction bars, with six worked examples and ten GCSE practice questions.
Prime Factors, HCF and LCM
Understand factors, multiples, prime factorisation, HCF and LCM through grouping and repeated events, with six examples and ten GCSE questions.
Fractions
Choose the correct fraction operation, work exactly and practise mixed numbers, fractions of amounts and multi-step contexts.
2Use percentages and money
Percentages
Identify the 100% base, calculate percentage change and use multipliers for reverse and repeated percentage problems.
Money problems
Solve GCSE money questions on change, wages, budgets, profit and currency conversion. Clear explanations and original questions with fully worked solutions.
3Understand powers and scale
Indices
Understand index notation and why each index law works, then practise zero, negative, fractional and combined powers.
Standard form
Convert, compare and calculate with powers of 10, interpret calculator notation and practise 15 original questions with worked answers.
4Round, estimate and check accuracy
Rounding
Round to decimal places and significant figures, understand place value and avoid double rounding. Six GCSE examples and ten questions with worked answers.
Estimation
Estimate calculations and square or cube roots using sensible rounding and neighbouring powers, with fully worked GCSE questions.
Upper and lower bounds
Understand error intervals and calculate upper and lower bounds for lengths, areas and rates, with six GCSE examples and ten fully worked questions.
5Extend roots and exact values
Fractional indices
Understand why fractional powers mean roots, then calculate positive and negative fractional indices. Six GCSE Higher examples and ten worked answers.
Surds
Simplify and calculate with exact roots, expand surd brackets and rationalise single and binomial denominators.
Recurring Decimals
Read recurring decimal notation and understand why multiplying and subtracting gives exact fractions, with six examples and ten worked GCSE questions.
Algebra
Find unknown values, solve equations and explore patterns and graphs.
1Understand and rewrite expressions
Expressions, equations and identities
Understand expressions, equations, formulae, identities and inequalities through everyday examples and fully worked GCSE questions.
Substitution
Substitute numbers into GCSE expressions and formulae, including negatives, powers and fractions. Six examples and ten questions with complete working.
Expanding brackets
Learn what expanding brackets means, from equal groups to two brackets and negative signs, with worked answers and GCSE practice questions.
Factorising
Understand factorising as making equal groups. Practise common factors, quadratic brackets and difference of two squares with fully worked answers.
2Solve linear problems
Linear equations
Solve GCSE linear equations by keeping both sides equal, including brackets, fractions and unknowns on both sides. Includes ten fully solved practice questions.
Rearranging formulae
Learn to change the subject of a formula using equal operations, including fractions, powers and a repeated subject. Worked GCSE examples and practice answers.
Linear inequalities
Read limits and number lines, solve inequalities and explain why negative division reverses order.
3Read lines and real-life graphs
Straight-line graphs
Understand coordinates, gradient and y-intercept through a constant-rate example. Plot lines, find equations and practise GCSE questions with worked answers.
Distance–time graphs
Read and draw distance–time graphs, calculate speeds and interpret rests and return journeys. GCSE examples and practice questions with full worked answers.
Velocity–time graphs
Read GCSE velocity–time graphs and connect gradient to acceleration and area to displacement, with six worked examples and ten questions with answers.
4Build up to quadratic problems
Quadratic equations
Solve quadratic equations by factorising, taking square roots, completing the square and using the formula. Understand both roots and practise with worked answers.
Completing the square
Rewrite quadratics in completed-square form, then use that form to find turning points and solve equations exactly.
Quadratic formula
Identify signed coefficients, substitute safely and solve exact, decimal, graphical and contextual quadratic problems.
Quadratic graphs
Plot and interpret GCSE quadratic graphs using tables, roots, intercepts and turning points. Six worked examples and ten questions with solutions.
Simultaneous equations
Choose elimination, substitution or a graphical method, then practise forming equations and Higher linear–quadratic pairs.
Quadratic inequalities
Solve GCSE quadratic inequalities using roots, graphs and sign tests. Understand intervals and endpoints with six examples and ten worked practice answers.
5Explore sequences
Sequences
Move from term-to-term rules to linear nth terms, membership tests and familiar arithmetic, geometric and quadratic patterns.
Quadratic sequences
Use first and second differences to continue quadratic patterns, derive nth terms and test whether values appear.
6Connect functions and further graphs
Functions
Follow inputs through algebraic rules, then build to Higher-tier inverse and composite functions with fully worked practice.
Inverse functions
Learn how inverse functions undo an operation, find inverse formulae and handle restricted inputs. GCSE Higher examples and ten fully answered questions.
Composite functions
Make sense of composite functions by following two number machines in order. GCSE Higher worked examples, equations and ten questions with solutions.
Cubic and reciprocal graphs
Plot and recognise cubic and reciprocal graphs for GCSE Maths. Learn tables, intercepts and asymptotes through six examples and ten answered questions.
Exponential graphs
Understand exponential graphs through repeated multiplication. Sketch growth and decay, find equations and practise GCSE Higher questions with worked answers.
Trigonometric graphs
Sketch sine, cosine and tangent graphs in degrees, recognise key points and find all solutions in an interval. GCSE Higher questions with worked answers.
Graph transformations
Understand GCSE graph transformations through coordinates: translations and reflections of y = f(x), with six worked examples and ten practice questions.
Equation of a circle
Connect coordinates to radius with Pythagoras, test points and find tangent equations from perpendicular gradients.
Graphical inequalities
Learn to shade GCSE inequality regions, draw solid and dashed boundaries and test coordinates. Six Higher-tier examples and ten fully worked questions.
Iteration
Understand repeated calculations, read subscripts and estimate roots with accuracy checks.
7Extend algebra and prove results
Algebraic fractions
Build from ordinary fractions to cancellation, operations, excluded values and equations.
Algebraic proof
Learn GCSE algebraic proof through even and odd numbers, consecutive integers and counterexamples. Six worked proofs and ten questions with full reasoning.
Ratio & proportion
Compare quantities, share amounts and work with repeated percentage changes.
1Compare and scale quantities
Ratio
Keep quantities in order, find one ratio part and practise sharing, recipes, maps, linked ratios and changing groups.
Unit conversions
Convert GCSE length, mass, capacity, area, volume and time units by understanding the unit relationship. Six examples and ten questions with worked answers.
Scale drawings
Read map scales, find real and drawing distances, and handle area scale factors. Six GCSE examples and ten original questions with fully worked answers.
Best buys
Compare unit prices, pack sizes and special offers in GCSE best-buy questions. Learn when best value differs from lowest total cost, with worked answers.
2Model proportional relationships
Direct proportion
Understand direct proportion through a constant price per item, then use unit rates, scale factors and proportional equations. GCSE questions with worked solutions.
Inverse proportion
Understand inverse proportion through a fixed job shared between workers. Use constant products, interpret assumptions and solve GCSE proportional equations.
3Work with rates and compound measures
Speed, distance and time
Solve speed, distance and time GCSE questions, including unit conversions, journey times and average speed. Six examples and ten fully worked answers.
Density, mass and volume
Understand density as mass per unit volume. Solve GCSE mass, volume, unit conversion and mixture questions with six examples and ten worked answers.
Pressure, force and area
Understand pressure as force spread over contact area. Learn P = F/A with units, six GCSE examples and ten original questions with worked solutions.
4Follow percentage changes over time
Reverse percentages
Find original amounts before percentage increases or discounts. Understand the original 100%, then practise six GCSE examples and ten fully worked questions.
Simple interest
Understand simple interest as equal additions based on the original amount. GCSE questions on interest, balance, time and rate, with full worked answers.
Compound interest
Understand changing balances and percentage multipliers, then practise annual, non-annual and reverse compound interest problems.
Growth and decay
Model repeated percentage change, then Higher iterative processes with deposits, repayments and target crossings.
Geometry & measures
Work with shapes, angles, lengths, directions and movement.
1Know shapes, angles and coordinates
Symmetry and properties of shapes
Recognise line and rotational symmetry and use triangle and quadrilateral properties. GCSE questions with diagrams, reasons and fully worked answers.
Angle rules
Understand angles as turns, then use straight lines, triangles, parallel lines and isosceles triangles to find missing angles with clear GCSE reasons.
Angles in polygons
Find GCSE polygon angle sums, missing angles and sides of regular polygons. Understand triangle splitting and exterior turns with six examples and ten answers.
Coordinate geometry
Understand coordinates, midpoints, distances and gradients in GCSE Maths. Six worked examples and ten original questions with fully explained answers.
2Measure and represent shapes
Area and perimeter
Tell area from perimeter and solve GCSE rectangle, triangle, trapezium, circle and compound-shape questions with six examples and ten worked answers.
Volume
Find GCSE volumes of cuboids, prisms, cylinders, cones, pyramids and spheres. Six worked examples and eleven questions explain units and each calculation.
Surface area
Calculate GCSE surface area using nets and exposed faces, including open boxes, cylinders, cones and spheres. Six examples and ten worked questions.
Plans and elevations
See how 3D shapes become plans, front and side elevations. Learn hidden stacks and view direction with six examples and ten fully explained GCSE questions.
Arc length and sector area
Find GCSE arc lengths, sector areas, perimeters and missing angles using fractions of a circle. Six worked examples and ten questions with answers.
3Move, compare and construct shapes
Transformations
Translate, reflect, rotate and enlarge shapes with precise grids and complete descriptions.
Enlargement
Learn GCSE enlargement from a centre, including fractional and negative scale factors. Six worked examples, accurate diagrams and ten answered questions.
Congruent triangles
Understand triangle congruence before using SSS, SAS, ASA and RHS. GCSE proof examples, matching sides and ten practice questions with complete reasoning.
Similar shapes
Solve GCSE similar-shape questions using corresponding sides and length, area and volume scale factors. Six examples and ten fully worked questions.
Constructions and loci
Construct perpendiculars from and at a point, bisect angles and solve loci tasks with clear diagrams and worked answers.
4Find triangle lengths and angles
Pythagoras' theorem
Understand Pythagoras using square areas, identify the hypotenuse and find either missing side. GCSE worked examples, coordinate distances and a 3D extension.
Trigonometry
Use SOHCAHTOA to find sides and angles, then practise contextual and Higher 3D problems with fully worked answers.
Exact trig values
Build exact sine, cosine and tangent values from special triangles and use them in non-calculator calculations.
Bearings
Read, draw and calculate three-figure bearings using north lines, angle facts and right-angled trigonometry.
Sine rule
Match opposite side–angle pairs, find missing lengths and angles and check ambiguous cases.
Cosine rule
Match sides with opposite angles, then use SAS or SSS information to find missing sides and angles in any triangle.
Area of a triangle using sine
Learn why triangle area is ½ab sin C, choose the included angle and find missing sides or angles. GCSE Higher examples and questions with worked solutions.
5Develop geometric reasoning
Circle theorems
Learn the relationships created by centres, chords and tangents, then solve angle, length and proof questions.
Vectors
Understand vector notation, column vectors and geometric routes, then practise with worked examples and 13 original questions.
Probability
Use diagrams and possible outcomes to work out the chance of an event.
1Explore chance and possible outcomes
Relative frequency
Use GCSE relative frequency to estimate probability and expected outcomes. Understand trials, changing samples and reliability with six examples and ten answers.
Sample space diagrams
List every possible outcome without missing pairs. GCSE sample-space tables for coins, dice and spinners, with six worked examples and ten full answers.
2Organise events and repeated trials
Venn diagrams
Learn set notation, fill and read two- and three-set diagrams, then calculate probabilities with 16 fully worked questions.
Frequency trees
Read frequency trees as counts of people or items. Complete missing branches and find probabilities with six GCSE examples and ten fully answered questions.
Tree diagrams
Read probability trees as sequences of outcomes. Understand multiplying along paths, adding separate routes and changing probabilities without replacement, with GCSE practice.
3Use information to update probability
Statistics
Read charts and graphs, summarise data and compare groups.
1Organise and display data
Frequency tables
Read and construct GCSE frequency tables, find averages and estimate grouped means. Six examples and ten questions with complete worked solutions.
Pictograms
Read pictogram keys, interpret half symbols and draw GCSE pictograms from counts. Six examples and ten original questions with complete worked solutions.
Bar charts
Read and construct GCSE bar charts, choose sensible scales and compare categories without misleading axes. Six examples and ten questions with worked answers.
Pie charts
Connect category frequency to the same share of 360°, then read, construct and compare pie charts accurately.
Stem-and-leaf diagrams
Read keys, order leaves and compare back-to-back stem-and-leaf diagrams. GCSE questions on construction, median, mode and range with worked answers.
2Summarise and compare distributions
Averages
Understand GCSE mean, median, mode and range, including missing values, combined groups and comparisons. Six examples and ten fully worked questions.
Frequency polygons
Draw frequency polygons from class midpoints and frequencies, read distributions and compare groups. GCSE examples and questions with full worked answers.
Histograms
Understand why bar area represents frequency, then calculate density and draw, read and complete histograms.
Cumulative frequency
Build running totals, plot cumulative-frequency curves and estimate quartiles, IQR and interval frequencies.
Box plots
Read and draw five-number summaries, calculate IQR and compare typical values and spread in context.
3Interpret relationships and trends
Scatter graphs
Plot and interpret GCSE scatter graphs, correlation and lines of best fit. Practise interpolation, outliers and reliability with six examples and ten answers.
Time series graphs
Read and draw time series graphs, distinguish trend from seasonal variation, and judge forecasts. Six GCSE examples and ten questions with explained answers.
4Collect representative evidence
Revision & exam preparation
Make a revision plan, practise exam skills and understand your papers.
1Plan, practise and prepare for your papers
How to revise for GCSE Maths
Use a diagnostic-first study loop, realistic timetable, error log and past-paper routine to turn gaps into measurable progress.
How to pass GCSE Maths
Build a realistic grade 4 strategy from diagnostic evidence, connected core skills, a four-week plan and exam routines.
Non-calculator preparation
Practise written calculations, exact answers and mixed questions for a non-calculator paper.
Edexcel Higher revision hub
Understand the Edexcel Higher papers, plan your revision and find the topic guides you need.