England · Foundation & Higher
Both tiers Foundation and Higher. Higher only
Read, compare and order whole numbers and decimals; use place value and comparison symbols.
Number essentials · PDFs
DfE subject content: N1, N2. Sources
Calculate with whole numbers and decimals; use brackets, inverse operations and the order of operations.
Number essentials · PDFs
DfE subject content: N2, N3. Sources
Order and calculate with positive and negative numbers, including temperatures and bank balances.
Number essentials · PDFs
DfE subject content: N1, N2. Sources
Find HCF and LCM, write numbers as products of prime factors and solve divisibility problems.
Factors, multiples and primes · PDFs
DfE subject content: N4. Sources
Compare and simplify fractions; calculate with proper fractions, improper fractions and mixed numbers, including negatives.
Fractions, decimals and recurring decimals · PDFs
DfE subject content: N1, N2, N3, N8. Sources
Convert terminating decimals, fractions and percentages; interpret a fraction or percentage as a multiplier.
Fractions, decimals and recurring decimals · PDFs
DfE subject content: N10, N12. Sources
Use positive whole-number powers, square roots, cube roots and familiar square and cube numbers.
Estimating arbitrary powers and roots is Higher-tier content; see Fractional indices and root estimates.
Number essentials · PDFs
DfE subject content: N3, N6. Sources
Calculate with integer powers, including zero and negative powers, and simplify products and quotients of powers.
Indices and surds · PDFs
DfE subject content: N7, A4. Sources
Connect fractional powers to roots and estimate unfamiliar positive powers and roots.
Indices and surds · PDFs
DfE subject content: N6, N7. Sources
Write, compare and calculate with very large or small numbers using powers of ten; interpret calculator displays.
Standard form · PDFs
DfE subject content: N9. Sources
Simplify and calculate with surds, preserve exact answers and rationalise denominators.
Fractions and exact multiples of π are also on Foundation; simplifying surds and rationalising denominators are Higher only.
Indices and surds · PDFs
DfE subject content: N8. Sources
Round to decimal places and significant figures; estimate calculations and judge whether an answer is reasonable.
Number essentials · PDFs
DfE subject content: N14, N15. Sources
Describe the possible original values of a rounded or truncated quantity using inequalities.
Calculations using upper and lower bounds are Higher only.
Bounds and error intervals · PDFs
DfE subject content: N15, N16. Sources
Choose upper and lower values to bound a calculation and justify a suitable final accuracy.
Bounds and error intervals · PDFs
DfE subject content: N16. Sources
Convert recurring decimals into exact fractions and explain the algebraic method.
Fractions, decimals and recurring decimals · PDFs
DfE subject content: N10. Sources
Organise combinations in lists, tables or diagrams without missing or repeating an outcome.
Systematic listing and counting · PDFs
DfE subject content: N5. Sources
Multiply numbers of choices across stages, accounting for any restrictions on later choices.
Systematic listing and counting · PDFs
DfE subject content: N5. Sources
Read terms, coefficients, expressions, equations and inequalities; distinguish a variable from its value.
Expressions and substitution · PDFs
DfE subject content: A1, A3. Sources
Substitute numbers into expressions and formulae, including negative and fractional values.
Expressions and substitution · PDFs
DfE subject content: A2, A5. Sources
Collect like terms and simplify products, quotients and powers of algebraic terms.
Expressions and substitution · PDFs
DfE subject content: A4. Sources
Expand a single bracket or a pair of binomials and take out a common factor.
Products of more than two binomials are Higher-tier content in Advanced factorising and expansion.
Expanding and factorising · PDFs
DfE subject content: A4. Sources
Factorise quadratics of the form x² + bx + c and differences of two squares.
Expanding and factorising · PDFs
DfE subject content: A4. Sources
Factorise ax² + bx + c with a ≠ 1 and expand products of more than two binomials.
Expanding and factorising · PDFs
DfE subject content: A4. Sources
Solve equations, including brackets and unknowns on both sides, and check by substitution.
Linear equations and modelling · PDFs
DfE subject content: A17. Sources
Change the subject of a formula, applying the same operation to both sides.
Rearranging formulae · PDFs
DfE subject content: A5. Sources
Simplify and calculate with algebraic fractions, identifying values of the variable that cannot be used.
Algebraic fractions · PDFs
DfE subject content: A4. Sources
Distinguish an equation from an identity and explain why expressions are equivalent.
Formal algebraic proof is Higher only.
Algebraic proof · PDFs
DfE subject content: A3, A6. Sources
Represent general integers or quantities and construct a proof, rather than checking selected examples.
Algebraic proof · PDFs
DfE subject content: A6. Sources
Plot points in four quadrants, find midpoints and solve coordinate problems about shapes.
Coordinates and column vectors · PDFs
DfE subject content: A8, G11. Sources
Plot and interpret y = mx + c, find gradients and intercepts, identify parallel lines and find a line equation.
Straight-line graphs · PDFs
DfE subject content: A9, A10, R14. Sources
Use the negative-reciprocal relationship for perpendicular non-vertical lines, treating horizontal and vertical lines separately.
Straight-line graphs · PDFs
DfE subject content: A9. Sources
Plot and interpret parabolas; read roots, intercepts, symmetry and turning points, and estimate solutions graphically.
Quadratic and other curved graphs · PDFs
DfE subject content: A11, A12, A18. Sources
Recognise and sketch simple cubic graphs and y = 1/x. Identify their shapes and any values of x for which the function is undefined.
Quadratic and other curved graphs · PDFs
DfE subject content: A12, A14. Sources
Recognise and interpret y = kˣ for positive k, including examples of growth and decay.
Quadratic and other curved graphs · PDFs
DfE subject content: A12, A14. Sources
Recognise and sketch sine, cosine and tangent graphs in degrees and interpret angles beyond a right triangle.
Trigonometric graphs and transformations · PDFs
DfE subject content: A12. Sources
Translate and reflect the graph of a given function, keeping track of changes to inputs and outputs.
Trigonometric graphs and transformations · PDFs
DfE subject content: A13. Sources
Read and draw conversion graphs, travel graphs and graphs of other real situations. Use gradients to find constant rates and intersections to solve problems.
Graphs in context · PDFs
DfE subject content: A14, R14. Sources
Estimate tangent and chord gradients and areas under curves; interpret rates, velocity and displacement in context.
Graphs in context · PDFs
DfE subject content: A15, R15. Sources
Use x² + y² = r² for a circle centred at the origin and find a tangent equation at a point on it.
Circle equations and tangents · PDFs
DfE subject content: A16. Sources
Solve quadratics by factorising: if two factors multiply to zero, at least one must be zero. Connect the solutions to where the graph crosses or touches the x-axis.
Quadratic equations · PDFs
DfE subject content: A18, A11. Sources
Solve general quadratic equations, complete the square, identify turning points and retain exact solutions where needed.
Quadratic equations · PDFs
DfE subject content: A18, A11. Sources
Solve a pair of linear equations by elimination or substitution. On a graph, the solution is where the two lines intersect.
Simultaneous equations · PDFs
DfE subject content: A19. Sources
Solve a linear and a quadratic equation together and interpret where their graphs intersect.
Simultaneous equations · PDFs
DfE subject content: A19. Sources
Use repeated substitution to approximate a solution and check whether the values approach a limit. Use repeated calculations to model change.
Iteration · PDFs
DfE subject content: A20, R16. Sources
Choose variables and write equations for a problem. Solve them, interpret the answers and consider the assumptions made.
Linear equations and modelling · PDFs
DfE subject content: A21. Sources
Solve a one-variable linear inequality and represent its solution on a number line.
Inequalities · PDFs
DfE subject content: A22. Sources
Represent linear inequalities in two variables as regions; solve quadratic inequalities and express solution sets.
Inequality regions · PDFs
DfE subject content: A22. Sources
Generate terms from rules; recognise arithmetic, Fibonacci-type, square, cube, triangular and simple geometric sequences; find linear nth terms.
Quadratic nth terms and geometric sequences involving surd ratios are Higher-tier content.
Sequences · PDFs
DfE subject content: A23, A24, A25. Sources
Find a quadratic nth term using second differences and work with geometric sequences including surd ratios.
Sequences · PDFs
DfE subject content: A24, A25. Sources
Interpret an expression as a rule connecting an input to an output and reverse simple operations.
Inverse and composite function notation is Higher only.
DfE subject content: A7. Sources
Use function notation, combine functions in the correct order and find inverses with appropriate input restrictions.
Functions · PDFs
DfE subject content: A7. Sources
Choose and convert units of length, area, volume, capacity, mass, time and money, including compound-unit conversions.
Units, rates and compound measures · PDFs
DfE subject content: N13, R1, G14. Sources
Simplify ratios, use them to compare how many times one quantity is another, and connect equivalent ratios to fractions.
Ratio problems · PDFs
DfE subject content: R3, R4, R6, R7, N11. Sources
Share a total in a given ratio, find the total from one part and solve mixing problems. Convert between ratios of two parts and ratios of a part to the whole.
Ratio problems · PDFs
DfE subject content: R5, R8, N11. Sources
Find a percentage of an amount and express one quantity as a percentage of another, including over 100%.
Percentages · PDFs
DfE subject content: N12, R9. Sources
Calculate and compare percentage increases and decreases, including simple interest.
Percentages · PDFs
DfE subject content: R9. Sources
Find the original amount before a percentage increase or decrease.
Percentages · PDFs
DfE subject content: R9. Sources
Use repeated percentage multipliers to calculate compound interest, depreciation and other growth or decay.
General iterative processes are Higher content; see Iteration and numerical solutions.
Percentages · PDFs
DfE subject content: R16. Sources
Solve numerical direct and inverse proportion problems; recognise their graphs and interpret given proportional equations.
Forming proportional equations is Higher only.
Direct and inverse proportion · PDFs
DfE subject content: R7, R8, R10, R13, R14. Sources
Construct direct and inverse proportion equations, including powers and roots, and find the constant from given values.
Direct and inverse proportion · PDFs
DfE subject content: R10, R13. Sources
Solve problems involving speed, density, pressure, pay rates and unit prices; compare rates and keep units consistent.
Units, rates and compound measures · PDFs
DfE subject content: R1, R11. Sources
Use a scale factor to connect a drawing or map with actual lengths and distances.
Scale drawings and bearings · PDFs
DfE subject content: R2, G15. Sources
Recognise and describe triangles, quadrilaterals and polygons; use geometric labels, parallel and perpendicular lines, and symmetry.
Angles, shapes and circles · PDFs
DfE subject content: G1, G4. Sources
Use angles at a point and on a line, vertically opposite angles, parallel-line facts and polygon angle sums.
Angles, shapes and circles · PDFs
DfE subject content: G3, G4. Sources
Use ruler and compass to construct bisectors and perpendiculars; combine loci to locate possible positions.
Constructions and loci · PDFs
DfE subject content: G2. Sources
Use triangle congruence criteria and angle or shape facts to justify conclusions and construct simple geometric proofs.
Congruence and geometric proof · PDFs
DfE subject content: G5, G6. Sources
Recognise similar shapes and use a linear scale factor to find corresponding lengths.
Similarity · PDFs
DfE subject content: G7, G19, R12. Sources
Use k² and k³ for areas and volumes of similar figures and work backwards to the length scale factor.
Similarity · PDFs
DfE subject content: G19, R12. Sources
Draw and describe a single transformation, specifying a mirror line, centre and angle, or translation vector.
Transformations and enlargement · PDFs
DfE subject content: G7, G24. Sources
Draw and describe an enlargement with a positive whole-number or fractional scale factor and a given centre.
Negative scale factors are Higher only.
Transformations and enlargement · PDFs
DfE subject content: G7, R2. Sources
Enlarge with a negative scale factor. Combine rotations, reflections and translations, describing what changes and what stays the same.
Transformations and enlargement · PDFs
DfE subject content: G7, G8. Sources
Identify radii, diameters, chords, tangents, arcs, sectors and segments.
Angles, shapes and circles · PDFs
DfE subject content: G9. Sources
Use and prove circle angle, chord and tangent results, explaining each step in a geometric argument.
Circle theorems · PDFs
DfE subject content: G10. Sources
Describe faces, edges and vertices; interpret and construct plans and elevations of solids.
Volume and surface area · PDFs
DfE subject content: G12, G13. Sources
Measure lengths and angles; draw and interpret three-figure bearings and use parallel north lines.
Scale drawings and bearings · PDFs
DfE subject content: G15. Sources
Find perimeters and areas of triangles, quadrilaterals and composite shapes; distinguish boundary length from area.
Perimeter and area · PDFs
DfE subject content: G16, G17. Sources
Calculate circumference, circle area, arc length and sector area; give exact answers in terms of π where appropriate.
Arcs and sectors · PDFs
DfE subject content: G17, G18, N8. Sources
Calculate volume and surface area for prisms, cylinders, pyramids, cones, spheres and composite solids.
Volume and surface area · PDFs
DfE subject content: G16, G17. Sources
Identify the hypotenuse and find an unknown side of a right-angled triangle, including distances between coordinates.
Pythagoras and trigonometry · PDFs
DfE subject content: G20, G11. Sources
Use sine, cosine and tangent to find sides or angles in 2D right-angled triangles. Use the exact values at 0°, 30°, 45°, 60° and 90° (tangent is undefined at 90°).
Pythagoras and trigonometry · PDFs
DfE subject content: G20, G21. Sources
Choose and apply the sine rule, cosine rule or ½ab sin C for a non-right triangle; interpret possible angles.
Sine and cosine rules · PDFs
DfE subject content: G22, G23. Sources
Select useful triangles in a solid and find lengths or angles through several connected steps.
Pythagoras and trigonometry · PDFs
DfE subject content: G20. Sources
Read column and diagram vectors; add, subtract and multiply by a number, including reversing direction.
Coordinates and column vectors · PDFs
DfE subject content: G24, G25. Sources
Use routes and ratios to express vectors, prove that lines are parallel or points lie on a straight line, and deduce properties of shapes.
Vector ratios and intersections · PDFs
DfE subject content: G25. Sources
Use unknown fractions to describe points along lines. Express the same point by two routes, solve the resulting equations and find ratios or positions.
Vector ratios and intersections · PDFs
DfE subject content: G25, A19, A21. Sources
Use probabilities from 0 to 1, identify equally likely outcomes and use complements and mutually exclusive outcomes.
Probability and sample spaces · PDFs
DfE subject content: P2, P3, P4. Sources
List outcomes of one or more experiments using tables or grids and calculate probabilities when outcomes are equally likely.
Probability and sample spaces · PDFs
DfE subject content: P6, P7. Sources
Record experiments, estimate probability from relative frequency, predict frequencies and discuss the effect of sample size.
Probability and sample spaces · PDFs
DfE subject content: P1, P2, P3, P5. Sources
Use Venn diagrams for overlapping groups and two-way tables for data sorted by two categories. Read set notation and calculate probabilities from the counts.
Venn diagrams, two-way tables and frequency trees · PDFs
DfE subject content: P6, P7. Sources
Split counts into successive categories, fill in missing frequencies and find probabilities from the groups.
Venn diagrams, two-way tables and frequency trees · PDFs
DfE subject content: P1, P6. Sources
Calculate probabilities of independent and dependent events, including without replacement; add alternative paths and multiply along a path.
Conditional probability is Higher only.
Probability trees · PDFs
DfE subject content: P6, P8. Sources
Recalculate probabilities when information restricts the sample space, using tables, trees or Venn diagrams.
Probability trees · PDFs
DfE subject content: P9. Sources
Distinguish categorical, discrete and continuous data; organise raw or grouped data into frequency tables.
Data, charts and averages · PDFs
DfE subject content: S2, S4. Sources
Choose and interpret bar charts, pictograms, pie charts, vertical line charts and time-series graphs.
Data, charts and averages · PDFs
DfE subject content: S2. Sources
Find and compare mean, median, mode and range; use frequency tables and estimate a mean from grouped data.
Data, charts and averages · PDFs
DfE subject content: S4, S5. Sources
Distinguish a sample from a population, recognise bias and limitations, and evaluate conclusions from data.
Sampling and scatter graphs · PDFs
DfE subject content: S1, S5. Sources
Plot paired data, describe correlation, draw a line of best fit and judge interpolation or extrapolation; distinguish correlation from causation.
Sampling and scatter graphs · PDFs
DfE subject content: S6. Sources
Represent grouped data using bar area for frequency, including unequal class widths, and interpret frequency density.
Histograms · PDFs
DfE subject content: S3. Sources
Construct cumulative frequency graphs and estimate medians, quartiles and percentiles from grouped data.
Cumulative frequency and box plots · PDFs
DfE subject content: S3, S4. Sources
Find quartiles and interquartile range, draw box plots and compare distributions using their medians and spread.
Cumulative frequency and box plots · PDFs
DfE subject content: S4. Sources
OCR’s post-16 GCSE Maths resources include topic Check In tests, with Initial and Foundation practice.
Use ratio to divide a vector route, algebra to simplify it, and simultaneous equations to find an intersection.
Sharing in a ratio · Expanding and common factors · Simultaneous linear equations
Then use the multiplier backwards or repeatedly.
Substitute into the quadratic formula and simplify any surds in the answers.
For problems in solids, combine right-angle trigonometry with Pythagoras and spatial reasoning.
Pythagoras in 2D · 3D shapes, plans and elevations · Pythagoras and trigonometry in 3D
Fraction arithmetic supports each stage; tables and Venn diagrams help with restricted sample spaces.
Fractions and fraction arithmetic · Venn diagrams, sets and two-way tables
Read medians and quartiles from cumulative frequency graphs. For histograms, use bar area and frequency per unit of class width.
GCSE Mathematics (9–1) in England: AQA 8300, Pearson Edexcel 1MA1 and OCR J560.
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Specifications checked 10 September 2026.
Contains public sector information adapted under the Open Government Licence v3.0.