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Subject Content - GCSE Maths

Subject Content

The subject content of this specification matches that set out in the Department for Education's Mathematics GCSE subject content and assessment objectives document. This content is common to all exam boards.

The content has been organised into broad topic areas and given a reference as follows:

All content can be assessed on any of the three question papers. As such, some questions will draw together elements of maths from different topic areas.

The weighting of the topic areas has been prescribed by Ofqual and is common to all exam boards. The table below shows the approximate weightings of the topic areas for the overall tier of assessment, not for each individual question paper.

Topic Area Coding References

  • Number references start with N
  • Algebra references start with A
  • Ratio, proportion and rates of change references start with R
  • Geometry and measures references start with G
  • Probability references start with P
  • Statistics references start with S

GCSE Maths Topic Area Weightings

Topic AreaFoundation Tier (%)Higher Tier (%)
Number2515
Algebra2030
Ratio2520
Geometry1520
Probability and statistics (combined)1515

3.1 Number

Within this specification, the assessment will reflect the key concepts of the subject as articulated in the DfE subject content and assessment objectives document.

In line with the requirements set by the Department for Education, the expectation is that all students will develop confidence and competence with the basic foundation content and additional foundation content, while higher tier students will also be assessed on higher content only.

3.1.1 Structure and Calculation

Notes are added to exemplify some of the specification references. In addition, students should be able to recall, select and apply mathematical formulae.

N1

Basic foundation contentAdditional foundation contentHigher content only
order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ≤, ≥ Notes: including use of a number line. See also A22

N2

Basic foundation contentAdditional foundation contentHigher content only
apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative; understand and use place value (eg when working with very large or very small numbers, and when calculating with decimals) Notes: including questions set in context. Knowledge and understanding of terms used in household finance, for example profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT and interest rate. See also R9

N3

Basic foundation contentAdditional foundation contentHigher content only
recognise and use relationships between operations, including inverse operations (eg cancellation to simplify calculations and expressions); use conventional notation for priority of operations, including brackets, powers, roots and reciprocals

N4

Basic foundation contentAdditional foundation contentHigher content only
use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem Notes: prime factor decomposition including product of prime factors written in index form.

N5

Basic foundation contentAdditional foundation contentHigher content only
apply systematic listing strategies Notes: including using lists, tables and diagrams.including use of the product rule for counting

N6

Basic foundation contentAdditional foundation contentHigher content only
use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 Notes: including square numbers up to 15 × 15. Students should know that 1000 = 10³ and 1 million = 10⁶estimate powers and roots of any given positive number

N7

Basic foundation contentAdditional foundation contentHigher content only
calculate with roots, and with integer indicescalculate with fractional indices

N8

Basic foundation contentAdditional foundation contentHigher content only
calculate exactly with fractionscalculate exactly with multiples of π Notes: see also G17 and G18calculate exactly with surds; simplify surd expressions (eg √12 = √(4 × 3) = √4 × √3 = 2√3) and rationalise denominators

N9

Basic foundation contentAdditional foundation contentHigher content only
calculate with and interpret standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer Notes: with and without a calculator. Interpret calculator displays.

3.1.2 Fractions, Decimals and Percentages

N10

Basic foundation contentAdditional foundation contentHigher content only
work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8) Notes: including ordering.change recurring decimals into their corresponding fractions and vice versa

N11

Basic foundation contentAdditional foundation contentHigher content only
identify and work with fractions in ratio problems Notes: see also R8

N12

Basic foundation contentAdditional foundation contentHigher content only
interpret fractions and percentages as operators Notes: including interpreting percentage problems using a multiplier. See also R9

3.1.3 Measures and Accuracy

N13

Basic foundation contentAdditional foundation contentHigher content only
use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate Notes: know and use metric conversion factors for length, area, volume and capacity. Imperial/metric conversions will be given in the question.

N14

Basic foundation contentAdditional foundation contentHigher content only
estimate answers; check calculations using approximation and estimation, including answers obtained using technology Notes: including evaluation of results obtained. See also N15

N15

Basic foundation contentAdditional foundation contentHigher content only
round numbers and measures to an appropriate degree of accuracy (eg to a specified number of decimal places or significant figures) Notes: including appropriate rounding for questions set in context. Students should know not to round values during intermediate steps of a calculation. See also N14use inequality notation to specify simple error intervals due to truncation or rounding

N16

Basic foundation contentAdditional foundation contentHigher content only
apply and interpret limits of accuracyincluding upper and lower bounds

3.2 Algebra

Algebra is a core component of GCSE Mathematics. Students will learn notation, graphing, and modeling expressions.

3.2.1 Notation, vocabulary and manipulation

A1

Basic foundation contentAdditional foundation contentHigher content only
use and interpret algebraic notation, including: • ab in place of a × b • 3y in place of y + y + y and 3 × y • a² in place of a × a, a³ in place of a × a × a, a²b in place of a × a × b • a/b in place of a ÷ b • coefficients written as fractions rather than as decimals • brackets Notes: it is expected that answers will be given in their simplest form without an explicit instruction to do so.

A2

Basic foundation contentAdditional foundation contentHigher content only
substitute numerical values into formulae and expressions, including scientific formulae Notes: unfamiliar formulae will be given in the question. See the Appendix for a full list of the prescribed formulae. See also A5

A3

Basic foundation contentAdditional foundation contentHigher content only
understand and use the concepts and vocabulary of expressions, equations, formulae, inequalities, terms and factors Notes: this will be implicitly and explicitly assessed.to include identities

A4

Basic foundation contentAdditional foundation contentHigher content only
simplify and manipulate algebraic expressions by: • collecting like terms • multiplying a single term over a bracket • taking out common factors • simplifying expressions involving sums, products and powers, including the laws of indicessimplify and manipulate algebraic expressions (including those involving surds) by: • expanding products of two binomials • factorising quadratic expressions of the form x² + bx + c, including the difference of two squaressimplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by: • expanding products of two or more binomials • factorising quadratic expressions of the form ax² + bx + c

A5

Basic foundation contentAdditional foundation contentHigher content only
understand and use standard mathematical formulae; rearrange formulae to change the subject Notes: including use of formulae from other subjects in words and using symbols. See the Appendix for a full list of the prescribed formulae. See also A2

A6

Basic foundation contentAdditional foundation contentHigher content only
know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct argumentsto include proofs

A7

Basic foundation contentAdditional foundation contentHigher content only
where appropriate, interpret simple expressions as functions with inputs and outputsinterpret the reverse process as the 'inverse function'; interpret the succession of two functions as a 'composite function' Notes: understanding and use of f(x), fg(x) and f⁻¹(x) notation is expected at Higher tier.

3.2.2 Graphs

A8

Basic foundation contentAdditional foundation contentHigher content only
work with coordinates in all four quadrants

A9

Basic foundation contentAdditional foundation contentHigher content only
plot graphs of equations that correspond to straight-line graphs in the coordinate planeuse the form y = mx + c to identify parallel lines; find the equation of the line through two given points, or through one point with a given gradientuse the form y = mx + c to identify perpendicular lines

A10

Basic foundation contentAdditional foundation contentHigher content only
identify and interpret gradients and intercepts of linear functions graphically and algebraically

A11

Basic foundation contentAdditional foundation contentHigher content only
identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraicallydeduce turning points by completing the square Notes: including the symmetrical property of a quadratic. See also A18

A12

Basic foundation contentAdditional foundation contentHigher content only
recognise, sketch and interpret graphs of linear functions and quadratic functionsincluding simple cubic functions and the reciprocal function y = 1/x with x ≠ 0including exponential functions y = k^x for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size Notes: see also G21

A13

Basic foundation contentAdditional foundation contentHigher content only
sketch translations and reflections of a given function

A14

Basic foundation contentAdditional foundation contentHigher content only
plot and interpret graphs, and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration Notes: including problems requiring a graphical solution. See also A15including reciprocal graphsincluding exponential graphs

A15

Basic foundation contentAdditional foundation contentHigher content only
calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts Notes: see also A14, R14 and R15

A16

Basic foundation contentAdditional foundation contentHigher content only
recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point

3.2.3 Solving equations and inequalities

A17

Basic foundation contentAdditional foundation contentHigher content only
solve linear equations in one unknown algebraically; find approximate solutions using a graph Notes: including use of brackets.including those with the unknown on both sides of the equation

A18

Basic foundation contentAdditional foundation contentHigher content only
solve quadratic equations algebraically by factorising; find approximate solutions using a graph Notes: see also A11including those that require rearrangement; including completing the square and by using the quadratic formula

A19

Basic foundation contentAdditional foundation contentHigher content only
solve two simultaneous equations in two variables (linear/linear) algebraically; find approximate solutions using a graphincluding linear/quadratic

A20

Basic foundation contentAdditional foundation contentHigher content only
find approximate solutions to equations numerically using iteration Notes: including the use of suffix notation in recursive formulae.

A21

Basic foundation contentAdditional foundation contentHigher content only
translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution Notes: including the solution of geometrical problems and problems set in context.

A22

Basic foundation contentAdditional foundation contentHigher content only
solve linear inequalities in one variable; represent the solution set on a number line Notes: students should know the conventions of an open circle on a number line for a strict inequality and a closed circle for an included boundary. See also N1. In graphical work the convention of a dashed line for strict inequalities and a solid line for an included inequality will be required.solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph

3.2.4 Sequences

A23

Basic foundation contentAdditional foundation contentHigher content only
generate terms of a sequence from either a term-to-term or a position-to-term rule Notes: including from patterns and diagrams.

A24

Basic foundation contentAdditional foundation contentHigher content only
recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressionsincluding Fibonacci-type sequences, quadratic sequences, and simple geometric progressions (rⁿ where n is an integer and r is a rational number > 0) Notes: other recursive sequences will be defined in the question.including other sequences; including where r is a surd

A25

Basic foundation contentAdditional foundation contentHigher content only
deduce expressions to calculate the nth term of linear sequencesincluding quadratic sequences

3.3 Ratio, proportion and rates of change

R1

Basic foundation contentAdditional foundation contentHigher content only
change freely between related standard units (eg time, length, area, volume/capacity, mass) and compound units (eg speed, rates of pay, prices) in numerical contextscompound units (eg density, pressure) in numerical and algebraic contexts

R2

Basic foundation contentAdditional foundation contentHigher content only
use scale factors, scale diagrams and maps Notes: including geometrical problems.

R3

Basic foundation contentAdditional foundation contentHigher content only
express one quantity as a fraction of another, where the fraction is less than 1 or greater than 1

R4

Basic foundation contentAdditional foundation contentHigher content only
use ratio notation, including reduction to simplest form

R5

Basic foundation contentAdditional foundation contentHigher content only
divide a given quantity into two parts in a given part : part or part : whole ratio; express the division of a quantity into two parts as a ratio; apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations) Notes: including better value or best-buy problems.

R6

Basic foundation contentAdditional foundation contentHigher content only
express a multiplicative relationship between two quantities as a ratio or a fraction

R7

Basic foundation contentAdditional foundation contentHigher content only
understand and use proportion as equality of ratios

R8

Basic foundation contentAdditional foundation contentHigher content only
relate ratios to fractions and to linear functions Notes: see also N11, R14

R9

Basic foundation contentAdditional foundation contentHigher content only
define percentage as 'number of parts per hundred'; interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively; express one quantity as a percentage of another; compare two quantities using percentages; work with percentages greater than 100%; solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics Notes: see also N2, N12

R10

Basic foundation contentAdditional foundation contentHigher content only
solve problems involving direct and inverse proportion, including graphical and algebraic representations

R11

Basic foundation contentAdditional foundation contentHigher content only
use compound units such as speed, rates of pay, unit pricing Notes: including making comparisons.use compound units such as density and pressure

R12

Basic foundation contentAdditional foundation contentHigher content only
compare lengths, areas and volumes using ratio notation; scale factors Notes: see also G19, G20make links to similarity (including trigonometric ratios)

R13

Basic foundation contentAdditional foundation contentHigher content only
understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; interpret equations that describe direct and inverse proportionconstruct and interpret equations that describe direct and inverse proportion

R14

Basic foundation contentAdditional foundation contentHigher content only
interpret the gradient of a straight-line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion Notes: see also A15, R8

R15

Basic foundation contentAdditional foundation contentHigher content only
interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts Notes: see also A15

R16

Basic foundation contentAdditional foundation contentHigher content only
set up, solve and interpret the answers in growth and decay problems, including compound interestand work with general iterative processes

3.4 Geometry and measures

Geometry teaches properties of space, angles, shapes, and structural designs.

3.4.1 Properties and constructions

G1

Basic foundation contentAdditional foundation contentHigher content only
use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries; use the standard conventions for labelling and referring to the sides and angles of triangles; draw diagrams from written description

G2

Basic foundation contentAdditional foundation contentHigher content only
use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line Notes: including constructing an angle of 60°.

G3

Basic foundation contentAdditional foundation contentHigher content only
apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles; understand and use alternate and corresponding angles on parallel lines; derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons) Notes: colloquial terms such as Z angles are not acceptable and should not be used.

G4

Basic foundation contentAdditional foundation contentHigher content only
derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus and triangles and other plane figures using appropriate language Notes: including knowing names and properties of isosceles, equilateral, scalene, right-angled, acute-angled, obtuse-angled triangles. Including knowing names and using the polygons: pentagon, hexagon, octagon and decagon.

G5

Basic foundation contentAdditional foundation contentHigher content only
use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)

G6

Basic foundation contentAdditional foundation contentHigher content only
apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras' theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

G7

Basic foundation contentAdditional foundation contentHigher content only
identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargementincluding fractional scale factorsincluding negative scale factors

G8

Basic foundation contentAdditional foundation contentHigher content only
describe the changes and invariance achieved by combinations of rotations, reflections and translations Notes: including using column vector notation for translations. See also G24

G9

Basic foundation contentAdditional foundation contentHigher content only
identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumferenceincluding: tangent, arc, sector and segment

G10

Basic foundation contentAdditional foundation contentHigher content only
apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results

Notes: including angle subtended by an arc at the centre is equal to twice the angle subtended at any point on the circumference, angle subtended at the circumference by a semicircle is 90°, angles in the same segment are equal, opposite angles in a cyclic quadrilateral sum to 180°, tangent at any point on a circle is perpendicular to the radius at that point, tangents from an external point are equal in length, the perpendicular from the centre to a chord bisects the chord, alternate segment theorem.

G11

Basic foundation contentAdditional foundation contentHigher content only
solve geometrical problems on coordinate axes

G12

Basic foundation contentAdditional foundation contentHigher content only
identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres

G13

Basic foundation contentAdditional foundation contentHigher content only
interpret plans and elevations of 3D shapesconstruct and interpret plans and elevations of 3D shapes

3.4.2 Mensuration and calculation

G14

Basic foundation contentAdditional foundation contentHigher content only
use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money etc.)

G15

Basic foundation contentAdditional foundation contentHigher content only
measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings

Notes: including the eight compass point bearings and three-figure bearings.

G16

Basic foundation contentAdditional foundation contentHigher content only
know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders)

G17

Basic foundation contentAdditional foundation contentHigher content only
know the formulae: circumference of a circle = 2πr = πd area of a circle = πr² calculate perimeters of 2D shapes, including circles areas of circles and composite shapessurface area and volume of spheres, pyramids, cones and composite solids

Notes: including frustums. Solutions in terms of π may be asked for. See also N8, G18

G18

Basic foundation contentAdditional foundation contentHigher content only
calculate arc lengths, angles and areas of sectors of circles

Notes: see also N8, G17

G19

Basic foundation contentAdditional foundation contentHigher content only
apply the concepts of congruence and similarity, including the relationships between lengths in similar figuresincluding the relationships between lengths, areas and volumes in similar figures

Notes: see also R12

G20

Basic foundation contentAdditional foundation contentHigher content only
know the formulae for: Pythagoras' theorem, a² + b² = c² and the trigonometric ratios, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent; apply them to find angles and lengths in right-angled triangles in two dimensional figuresapply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures

Notes: see also R12

G21

Basic foundation contentAdditional foundation contentHigher content only
know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45°, 60°

Notes: see also A12

G22

Basic foundation contentAdditional foundation contentHigher content only
know and apply the sine rule, a/sin A = b/sin B = c/sin C and cosine rule, a² = b² + c² - 2bc cos A to find unknown lengths and angles

G23

Basic foundation contentAdditional foundation contentHigher content only
know and apply Area = 1/2 ab sin C to calculate the area, sides or angles of any triangle

3.4.3 Vectors

G24

Basic foundation contentAdditional foundation contentHigher content only
describe translations as 2D vectors

Notes: see also G8

G25

Basic foundation contentAdditional foundation contentHigher content only
apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectorsuse vectors to construct geometric arguments and proofs

3.5 Probability

P1

Basic foundation contentAdditional foundation contentHigher content only
record, describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees

Notes: probabilities should be written as fractions, decimals or percentages.

P2

Basic foundation contentAdditional foundation contentHigher content only
apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments

P3

Basic foundation contentAdditional foundation contentHigher content only
relate relative expected frequencies to theoretical probability, using appropriate language and the 0 to 1 probability scale

P4

Basic foundation contentAdditional foundation contentHigher content only
apply the property that the probabilities of an exhaustive set of outcomes sum to 1; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1

P5

Basic foundation contentAdditional foundation contentHigher content only
understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size

P6

Basic foundation contentAdditional foundation contentHigher content only
enumerate sets and combinations of sets systematically, using tables, grids, Venn diagramsincluding using tree diagrams

P7

Basic foundation contentAdditional foundation contentHigher content only
construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities

P8

Basic foundation contentAdditional foundation contentHigher content only
calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions

Notes: including knowing when to add and when to multiply two or more probabilities.

P9

Basic foundation contentAdditional foundation contentHigher content only
calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams

3.6 Statistics

S1

Basic foundation contentAdditional foundation contentHigher content only
infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling

S2

Basic foundation contentAdditional foundation contentHigher content only
interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data, and know their appropriate useincluding tables and line graphs for time series data

Notes: including choosing suitable statistical diagrams.

S3

Basic foundation contentAdditional foundation contentHigher content only
construct and interpret diagrams for grouped discrete data and continuous data, ie histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use

S4

Basic foundation contentAdditional foundation contentHigher content only
interpret, analyse and compare the distributions of data sets from univariate empirical distributions through: • appropriate graphical representation involving discrete, continuous and grouped data• including box plots
• appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers)• including quartiles and inter-quartile range

Notes: students should know and understand the terms: primary data, secondary data, discrete data and continuous data.

S5

Basic foundation contentAdditional foundation contentHigher content only
apply statistics to describe a population

S6

Basic foundation contentAdditional foundation contentHigher content only
use and interpret scatter graphs of bivariate data
recognise correlationknow that it does not indicate causation draw estimated lines of best fit make predictions interpolate and extrapolate apparent trends whilst knowing the dangers of so doing

Notes: students should know and understand the terms: positive correlation, negative correlation, no correlation, weak correlation and strong correlation.