GCSE Maths specification checklist

FoundationEvery student

• 1.1Order positive and negative integers, decimals and fractions, and use the symbols =, ≠, <, >, ≤ and ≥ to compare them.
• 1.2Apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper) and mixed numbers, positive and negative, and use place value when working with very large or very small numbers and with decimals.
• 1.3Use relationships between operations, including inverse operations, and use the conventional priority of operations, including brackets, powers, roots and reciprocals.
• 1.4Use the vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor and lowest common multiple; express a number as a product of its prime factors using product notation, and know that this factorisation is unique.
• 1.5Apply systematic listing strategies, including the product rule for counting, to enumerate outcomes.
• 1.6Use positive integer powers and associated real roots (square, cube and higher), and recognise powers of 2, 3, 4 and 5.

HigherHigher tier only

• 1.7Estimate powers and roots of any given positive number, for example by comparing with known cubes and squares, and judge whether an estimate is an over- or underestimate. (Higher)

FoundationEvery student

• 2.1Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7⁄2, or 0.375 and 3⁄8): convert in both directions and order mixed lists.
• 2.3Interpret fractions and percentages as operators: a fraction of an amount, and a percentage increase or decrease as a single multiplier.
• 2.4Define percentage as “number of parts per hundred”; interpret percentages and percentage changes as fractions or decimals, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100%.
• 2.5Solve problems involving percentage increase and decrease and original-value (reverse percentage) problems, including simple interest in financial mathematics.

HigherHigher tier only

• 2.2Change recurring decimals into their corresponding fractions and vice versa, using algebra to convert. (Higher)
• 2.6Solve problems involving repeated percentage change and compound interest, interpreting the repeated multiplier and comparing it with simple interest. (Higher)

FoundationEvery student

• 3.1Use the laws of indices with positive integer powers: multiply powers with the same base by adding indices, divide by subtracting, raise a power to a power by multiplying, and use the zero index a⁰ = 1.
• 3.4Calculate exactly with fractions, surds and multiples of π, giving exact answers in place of rounded decimals where a question demands them.
• 3.5Calculate with and interpret standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer: convert in both directions, and multiply and divide numbers in standard form.

HigherHigher tier only

• 3.2Calculate with integer and fractional indices: use negative indices as reciprocals and fractional indices as roots, including a^(1/n) and a^(m/n). (Higher)
• 3.3Simplify surd expressions involving squares (such as √12 = √4 × √3 = 2√3), add and multiply simple surds, and rationalise denominators. (Higher)

FoundationEvery student

• 4.1Round numbers and measures to a given number of decimal places or significant figures, choosing an appropriate degree of accuracy.
• 4.2Estimate answers to calculations by rounding each number to one significant figure, and use approximation and estimation to check calculations, including answers produced by a calculator.
• 4.3Use inequality notation to specify simple error intervals for values that have been rounded or truncated.
• 4.4Use standard units of mass, length, time, money and other measures, using decimal quantities where appropriate.

HigherHigher tier only

• 4.5Apply and interpret limits of accuracy: find the upper and lower bounds of rounded quantities and of calculations using them, such as perimeters, areas and speeds. (Higher)

FoundationEvery student

• 5.1Use and interpret algebraic notation (ab in place of a × b, 3y in place of y + y + y, a² in place of a × a, a²b, a⁄b, fractional coefficients and brackets), and understand the vocabulary of expressions, equations, formulae, identities, inequalities, terms and factors.
• 5.2Simplify expressions by collecting like terms, and multiply a single term over a bracket, including with negative coefficients.
• 5.3Expand products of two or more binomials; factorise expressions by taking out common factors, and factorise quadratic expressions of the form x² + bx + c, including the difference of two squares.

HigherHigher tier only

• 5.4Factorise quadratic expressions of the form ax² + bx + c, and simplify and manipulate algebraic fractions. (Higher)
• 5.5Simplify expressions involving sums, products and powers using the laws of indices, and manipulate expressions involving surds. (Higher)

FoundationEvery student

• 6.1Substitute numerical values into formulae and expressions, including scientific formulae such as v = u + at, using both positive and negative values.
• 6.2Understand and use standard mathematical formulae; rearrange formulae to change the subject, including cases where the subject appears in a bracket, under a root or as a power.

HigherHigher tier only

• 6.3Know the difference between an equation and an identity; argue mathematically to show that algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs. (Higher)
• 6.4Interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the inverse function and the succession of two functions as a composite function, using f(x), f⁻¹(x) and fg(x) notation. (Higher)
• 6.5Find approximate solutions to equations numerically using iteration, rearranging an equation into the form x = g(x) and iterating from a given starting value. (Higher)

FoundationEvery student

• 7.1Solve linear equations in one unknown algebraically, including those with brackets, fractions and the unknown on both sides.
• 7.2Plot graphs of equations that correspond to straight lines, and identify and interpret gradients and intercepts of linear functions graphically and algebraically, using the form y = mx + c.
• 7.4Find the equation of a straight line through two given points, or through one point with a given gradient.
• 7.5Find approximate solutions to linear equations using a graph, and interpret straight-line graphs in real contexts such as fixed charge plus rate.

HigherHigher tier only

• 7.3Use y = mx + c to identify parallel lines, and use the negative-reciprocal gradient rule to identify and construct perpendicular lines. (Higher for perpendicular)

FoundationEvery student

• 8.1Solve quadratic equations of the form x² + bx + c = 0 by factorising, including differences of two squares and equations that need rearranging first.
• 8.4Identify and interpret roots, intercepts and turning points of quadratic functions from their graphs, and use the symmetry of a parabola.
• 8.5Find approximate solutions to quadratic equations using a graph.

HigherHigher tier only

• 8.2Solve quadratic equations using the quadratic formula x = (−b ± √(b² − 4ac))⁄2a, which must be memorised, including equations of the form ax² + bx + c = 0 and those needing rearrangement. (Higher)
• 8.3Solve quadratic equations by completing the square, and use the completed-square form to deduce the turning point of a quadratic function. (Higher)

FoundationEvery student

• 9.1Solve two linear simultaneous equations in two variables algebraically, by elimination or substitution, and check the solution in both original equations.
• 9.3Solve linear inequalities in one variable and represent the solution set on a number line, using the correct open and closed endpoints.
• 9.5Find approximate solutions to simultaneous equations using their graphs.

HigherHigher tier only

• 9.2Solve two simultaneous equations where one is linear and one is quadratic, using substitution, and interpret the pairs of solutions. (Higher)
• 9.4Solve quadratic inequalities in one variable, and represent linear inequalities in two variables as regions on a graph, using set notation. (Higher)

FoundationEvery student

• 10.1Generate terms of a sequence from either a term-to-term rule or a position-to-term rule.
• 10.2Recognise and use arithmetic (linear) sequences, deduce expressions for their nth term, and use the nth term to calculate terms and decide whether a number belongs to the sequence.
• 10.3Recognise and use the sequences of triangular, square and cube numbers, and simple alternating or growing patterns built from them.

HigherHigher tier only

• 10.4Recognise Fibonacci-type sequences and simple geometric progressions (rⁿ where n is a position and r is rational and positive, or a surd), and deduce nth terms of quadratic sequences. (Higher)
• 10.5Continue and describe more complex patterns, explaining rules in words and with algebra. (Higher)

FoundationEvery student

• 11.1Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions and the reciprocal function y = 1/x (x ≠ 0), including intercepts and shape.

HigherHigher tier only

• 11.2Recognise and sketch exponential functions y = kˣ for positive k, and the trigonometric functions y = sin x, y = cos x and y = tan x for angles of any size. (Higher)
• 11.3Sketch translations and reflections of a given function, understanding how y = f(x) + a, y = f(x + a), y = −f(x) and y = f(−x) transform the graph. (Higher)
• 11.4Plot and interpret graphs in real contexts, including distance–time graphs, and calculate or estimate gradients of curves and areas under graphs, interpreting them as rates and distances. (Higher for curved graphs)
• 11.5Recognise and use the equation of a circle with centre at the origin, x² + y² = r², and find the equation of the tangent to a circle at a given point. (Higher)

FoundationEvery student

• 12.1Use ratio notation, reduce ratios to simplest form using common factors, and express one quantity as a fraction of another where the fraction is less than or greater than 1.
• 12.2Divide a given quantity into two parts in a given part:part ratio, and express the division of a quantity into two parts as a ratio.
• 12.3Apply ratio to real contexts and problems, such as those involving conversion, comparison, scaling, mixing and concentrations.
• 12.4Use scale factors, scale diagrams and maps, including 1:n scales, converting between diagram measurements and real lengths with correct units.

HigherHigher tier only

• 12.5Relate the language of ratio to fractions and to linear functions: the multiplier linking two proportional quantities appears as the gradient of the line y = kx.

FoundationEvery student

• 13.1Change freely between related standard units and compound units (speed, rates of pay, unit pricing, density and pressure) in numerical and algebraic contexts.
• 13.2Use direct proportion: solve problems with the unitary method or a single multiplier, and recognise direct proportion graphically as a straight line through the origin.
• 13.3Use inverse proportion in context: as one quantity increases the other decreases so that their product stays constant (workers and days, speed and time for a fixed journey).
• 13.5Interpret the gradient of a real-life graph as a rate of change, such as a cost per unit, a filling rate in litres per minute or a speed from a distance–time graph.

HigherHigher tier only

• 13.4Represent direct and inverse proportion algebraically: y = kx for direct and y = k/x for inverse proportion, finding k from one known pair and using the equation to predict further values.
• 13.6Estimate the size of a population using the capture–recapture method and comment on the assumptions the estimate depends on.

FoundationEvery student

• 14.1Use conventional terms and notation for points, lines, vertices, edges, right angles, parallel and perpendicular lines, and apply the angle facts: angles at a point sum to 360°, angles on a straight line sum to 180°, and vertically opposite angles are equal.
• 14.2Apply the parallel-line angle rules: corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°.
• 14.3Apply triangle and quadrilateral angle facts: the interior angles of a triangle sum to 180°, the exterior angle equals the sum of the two opposite interior angles, base angles of isosceles triangles are equal, and the interior angles of a quadrilateral sum to 360°.
• 14.4Calculate interior and exterior angles of polygons: the interior angles sum to (n − 2) × 180°, the exterior angles sum to 360°, and regular polygons split both sums equally.
• 14.5Construct figures exactly with ruler and compasses: the perpendicular bisector of a line segment, the perpendicular to a line from or at a point, and the bisector of an angle.

HigherHigher tier only

• 14.6Describe and shade regions defined by loci, and build formal chains of geometric reasoning that cite the angle fact used at every step.

FoundationEvery student

• 15.1Identify congruent shapes and justify congruence with the criteria SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the corresponding side) and RHS (right angle, hypotenuse, side).
• 15.2Identify similar shapes — equal matching angles, sides in the same ratio — and calculate missing lengths using a length scale factor.
• 15.4Describe and carry out rotations (centre, angle, direction), reflections (mirror line), translations (vector) and enlargements (centre and scale factor), including fractional scale factors.
• 15.6Represent translations with column vectors, and add and subtract vectors and multiply them by scalars.

HigherHigher tier only

• 15.3Use the relationships between similar figures: areas scale as k² and volumes as k³, where k is the length scale factor.
• 15.5Perform and describe enlargements with negative scale factors, combine transformations, and identify invariant points.
• 15.7Construct geometric arguments and proofs using vectors, for example showing that diagonals of a parallelogram bisect each other.

FoundationEvery student

• 16.1Use Pythagoras’ theorem (a² + b² = c²) in right-angled triangles to find the hypotenuse or a shorter side.
• 16.2Use the trigonometric ratios in right-angled triangles — sin θ = opposite⁄hypotenuse, cos θ = adjacent⁄hypotenuse, tan θ = opposite⁄adjacent — to find unknown sides and, using inverse functions, unknown angles.
• 16.3Solve problems in context by choosing and combining Pythagoras’ theorem and trigonometry in right-angled triangles, in two dimensions.

HigherHigher tier only

• 16.4Know and use the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°, and of tan θ for θ = 0°, 30°, 45° and 60°.
• 16.5Apply the sine rule (a⁄sin A = b⁄sin B), the cosine rule (a² = b² + c² − 2bc cos A) and the area formula Area = ½ab sin C in non-right-angled triangles.
• 16.6Apply Pythagoras’ theorem and trigonometric ratios in three-dimensional problems, using right-angled triangles found inside solids.

FoundationEvery student

• 17.1Use standard units of measure and convert between related units of length, area, volume, capacity and mass (mm–cm–m–km, mm²–cm²–m², mm³–cm³–m³, litres–cm³).
• 17.2Find perimeters and areas of 2D shapes: rectangles, triangles, parallelograms, trapezia (Area = ½(a + b)h) and composite shapes built from them.
• 17.3Calculate the circumference (2πr = πd) and area (πr²) of a circle, and the arc length and area of a sector as the appropriate fraction of the whole circle.
• 17.4Calculate the volume and surface area of cuboids, right prisms (cross-section × length) and cylinders (V = πr²h).

HigherHigher tier only

• 17.5Calculate the surface area and volume of spheres, pyramids, cones and composite solids, using the formulae given in the question where applicable.

FoundationEvery student

• 18.1Identify and name the parts of a circle: centre, radius, diameter, chord, tangent, arc, sector and segment.

HigherHigher tier only

• 18.2Apply and derive the angle at the centre theorem: the angle subtended at the centre is twice the angle subtended at the circumference by the same arc.
• 18.3Apply and derive the results that the angle in a semicircle is 90° and that angles subtended by the same chord in the same segment are equal.
• 18.4Apply the property that opposite angles of a cyclic quadrilateral sum to 180°.
• 18.5Apply the tangent properties: a tangent meets the radius at 90°, and the two tangents drawn from an external point have equal lengths.
• 18.6Apply the alternate segment theorem — the angle between a tangent and a chord equals the angle in the alternate segment — and use radii to build isosceles-triangle arguments and short proofs.

FoundationEvery student

• 19.1Record and use the probability scale from 0 (impossible) to 1 (certain), applying theoretical probability to equally likely outcomes as favourable outcomes over total outcomes.
• 19.2Use relative frequency from experiments, compare it with theoretical probability to judge fairness, and compute expected frequency as probability × number of trials.
• 19.3Apply the addition rule for mutually exclusive events and the property that probabilities of an exhaustive set sum to 1, including P(not A) = 1 − P(A).
• 19.4Enumerate combined events systematically with sample-space tables and grids, Venn diagrams and frequency trees.
• 19.5Use tree diagrams for combined events, multiplying along branches and adding across branches, distinguishing independent from dependent events.

HigherHigher tier only

• 19.6Apply conditional probability from tree diagrams, two-way tables and Venn diagrams, and use set notation (∪, ∩, ∈, ∉, ∅, n(A), A′) to describe events.

FoundationEvery student

• 20.1Infer properties of a population from a sample, understanding that conclusions are only as good as the sample: it must be representative, of a sensible size, and free of bias.
• 20.2Construct and interpret frequency tables, bar charts, pie charts, pictograms and vertical line charts for ungrouped discrete data, and time-series graphs for data collected over time.
• 20.3Calculate and interpret the mean, mode and median of a data set, the mean and mode from a frequency table, the modal class from grouped data, and the range as a measure of spread.
• 20.4Compare two distributions using a measure of centre and a measure of spread together, commenting on consistency as well as typical values.
• 20.6Draw and interpret scatter graphs, recognise correlation and know it does not establish causation, draw a line of best fit by eye, and distinguish interpolation (reliable) from extrapolation (speculative).

HigherHigher tier only

• 20.5Construct and interpret histograms with equal and unequal class intervals (frequency density = frequency ÷ class width), cumulative frequency graphs, quartiles, the interquartile range and box plots; identify outliers and know their effect.