Subject · GCSE and A level
Mathematics
Which schools in England make Mathematics a big part of the curriculum, and where results are strongest. Entries and grades from the 2025 performance tables.
What the data says
Mathematics GCSE was taken in 4,410 schools in England in 2025, with 619,319 entries — about 88% of the Year 11 cohort.
Nationally 22% of entries achieved grade 7 or above. The tables below show the schools where the largest share of pupils take the subject, and where grades are highest (schools with at least 20 entries).
At William Ellis School (Camden) 100% of the year group took Mathematics.
At A level, Mathematics was offered in 2,536 schools and colleges, with 93,746 entries — roughly 31% of A level students.
GCSE: schools with the highest share of grades 7–9
Schools with at least 20 entries. England: 22% of entries graded 7–9. High grades often reflect who is admitted (selective and independent schools) as much as teaching.
A level: schools and colleges where the largest share of students take Mathematics
Entries as a share of A level students (at least 30 A level students and 10 entries). England: 31% of A level students.
What pupils learn in Mathematics, stage by stage
England has no national year-by-year syllabus: exam boards and schools set the order within each stage. These are the statutory blocks from the Department for Education, so a parent can see what a child meets in Years 7–9, what every GCSE candidate must cover, and what waits in the sixth form.
Years 7–9Key Stage 3
98learning outcomes · National Curriculum, taught in every maintained schoolYears 10–11GCSE
188learning outcomes · statutory GCSE subject content; Higher-tier-only outcomes are markedYear 12A level, typically first year
111learning outcomes · content also examined at ASYear 13A level, second year
79learning outcomes · A level-only contentWhat Key Stage 3 Mathematics teaches · Years 7–9
98 learning outcomes from the Department for Education’s statutory subject content, grouped by topic. Each outcome belongs to one of twelve fields of knowledge used across startfrom.today: Mathematics & Statistics (98). Exam boards add detail on top of these; the outcomes themselves are the same whichever board a school uses.
Number26
- apply place value to decimals, measures and integers of any size — place value
- order positive and negative integers, decimals and fractions — decimal, fraction, negative integers, positive integers
- use the number line to order real numbers — number line, real numbers
- use equality and inequality symbols to compare numbers — equality symbols, inequality symbols
- use the concepts and vocabulary of prime numbers, factors and multiples — factors, multiples, prime number
- use common factors and common multiples of a set of numbers — common factors, common multiples
- use prime factorisation, highest common factor and lowest common multiple — highest common factor, lowest common multiple, prime factorisation
- use product notation and the unique factorisation property — product notation, unique factorisation
- apply the four operations to integers, decimals and fractions using formal written methods — formal written methods, four operations, fraction
- use conventional notation for order of operations, brackets, powers and reciprocals — brackets, order of operations, power, reciprocal
- use inverse operations to simplify calculations — inverse operations
- use positive integer powers and associated real roots — power, root
- recognise powers of 2, 3, 4 and 5 — cubes, power, squares
- distinguish between exact roots and their decimal approximations — decimal approximation, exact roots
- interpret and compare numbers in standard form — standard form
- convert between terminating decimals and their corresponding fractions — fraction, terminating decimals
- define percentage as number of parts per hundred — percentage
- interpret percentages and percentage changes as a fraction or decimal — percentage, percentage change
- express one quantity as a percentage of another and compare quantities using percentages — percentage comparison
- calculate percentages greater than 100% — percentage
- interpret fractions and percentages as operators — fraction, operators, percentage
- use standard units of mass, length, time and money, including decimal quantities — decimal quantities, standard units
- round numbers and measures to an appropriate degree of accuracy — degree of accuracy, rounding
- estimate answers and calculate possible errors using inequality notation — error bound, estimation, inequality notation
- use a calculator and other technologies to calculate and interpret results — calculator, technology
- explain the infinite nature of integers, real numbers and rational numbers — infinite sets, integer, rational numbers, real numbers
Algebra18
- use algebraic notation for products, powers and coefficients — algebraic notation, coefficient
- substitute numerical values into formulae and expressions, including scientific formulae — formulae, substitution
- use the vocabulary of expressions, equations, inequalities, terms and factors — equation, expression, inequality, terms and factors
- simplify expressions by collecting like terms and multiplying a term over a bracket — expanding brackets, like terms
- simplify expressions by taking out common factors and expanding products of binomials — expanding binomials, factorising
- rearrange a formula to change its subject — formulae, subject of a formula
- model a situation or procedure using algebraic expressions, formulae or graphs — mathematical modelling
- solve linear equations in one variable, including those requiring rearrangement — linear equations
- plot coordinates in all four quadrants — coordinate, quadrants
- sketch and produce graphs of linear and quadratic functions with appropriate scaling — graph, linear functions, quadratic function
- interpret mathematical relationships algebraically and graphically — algebraic representation, graphical representation
- reduce a linear equation in two variables to the form y = mx + c — linear equations, standard form
- calculate and interpret gradients and intercepts of straight-line graphs — gradient, intercept
- use linear and quadratic graphs to estimate values and solve simultaneous equations — graphical solution, simultaneous equation
- find approximate solutions to problems from piecewise, exponential and reciprocal graphs — exponential function, piecewise functions, reciprocal function
- generate terms of a sequence from a term-to-term or position-to-term rule — sequence, term-to-term rule
- find the nth term of an arithmetic sequence — arithmetic sequence, nth term
- recognise geometric sequences and other types of sequence — geometric sequence
Ratio, proportion and rates of change11
- convert between related standard units of time, length, area, volume and mass — unit conversion
- use scale factors, scale diagrams and maps — map, scale diagram, scale factor
- express one quantity as a fraction of another — fraction, proportional comparison
- use ratio notation, including reduction to simplest form — ratio notation, simplifying ratios
- divide a quantity into two parts in a given ratio — ratio division
- explain a multiplicative relationship between two quantities as a ratio or fraction — multiplicative relationship, ratio
- relate ratio calculations to the arithmetic of fractions and linear functions — fraction, linear functions, ratio
- solve problems involving percentage increase, decrease and original value — percentage change
- solve problems involving simple interest in financial mathematics — financial mathematics, simple interest
- solve problems involving direct and inverse proportion graphically and algebraically — direct proportion, inverse proportion
- use compound units such as speed, unit pricing and density to solve problems — compound units, density, speed
Geometry and measures24
- derive formulae for perimeter and area of triangles, parallelograms and trapezia — area, formulae, perimeter
- derive and apply formulae for the volume of cuboids and other prisms — prisms, volume
- calculate perimeters and areas of 2-D shapes, including circles and composite shapes — area, circle, perimeter
- draw and measure line segments and angles, including interpreting scale drawings — angle, line segments, scale drawing
- construct a perpendicular bisector and a perpendicular to a line using ruler and compasses — ruler and compass constructions
- bisect a given angle using ruler and compass constructions — angle bisection
- use the perpendicular distance from a point to a line as the shortest distance to the line — perpendicular distance
- draw points, parallel lines, perpendicular lines and right angles using notation — geometric notation
- draw regular polygons and polygons with reflective or rotational symmetry — regular polygon, symmetry
- use standard conventions for labelling the sides and angles of a triangle — triangle labelling
- use the criteria for congruence of triangles — congruence
- derive properties of triangles, quadrilaterals and circles using appropriate language — circle properties, quadrilateral properties, triangle properties
- describe the results of translations, rotations and reflections applied to a figure — reflection, rotation, translation
- construct similar shapes by enlargement, with and without coordinate grids — enlargement, similar shapes
- construct congruent triangles — congruent triangles
- apply angle facts at a point, on a straight line and between vertically opposite angles — angle facts
- use the relationship between parallel lines and alternate and corresponding angles — alternate angles, corresponding angles, parallel lines
- derive the sum of angles in a triangle and use it to deduce the angle sum of a polygon — angle sum, polygon
- derive properties of regular polygons using angle facts — angle properties, regular polygon
- apply angle facts, congruence and similarity to derive results and Pythagoras' theorem — congruence, pythagoras' theorem, similarity
- construct a simple proof from known geometric results — geometric proof
- use Pythagoras' theorem and trigonometric ratios in similar triangles to solve problems — pythagoras' theorem, trigonometric ratios
- use properties of faces, edges and vertices of 3-D shapes to solve problems — 3-d shapes, edges, faces, vertices
- interpret mathematical relationships algebraically and geometrically — algebraic representation, geometric representation
Probability5
- analyse the frequency of outcomes of a probability experiment — frequency, probability experiment
- explain that the probabilities of all possible outcomes sum to one — probability scale
- enumerate sets and their unions and intersections using tables, grids and Venn diagrams — set notation, venn diagram
- generate a theoretical sample space for single and combined events — combined events, sample space
- calculate theoretical probabilities for equally likely, mutually exclusive outcomes — theoretical probability
Statistics4
- compare distributions of a single variable using measures of central tendency and spread — mean, median, mode, range
- construct and interpret frequency tables, bar charts, pie charts and pictograms — bar chart, frequency table, pie chart
- construct vertical line charts for ungrouped and grouped numerical data — grouped data, line chart
- describe a relationship between two variables using a scatter graph — bivariate data, scatter graph
Working mathematically10
- select and use appropriate calculation strategies to solve increasingly complex problems — calculation strategies
- move fluently between numerical, algebraic, graphical and diagrammatic representations — fluency, representation
- test conjectures about patterns and relationships, seeking proofs or counter-examples — conjecture, counter-example, proof
- reason deductively in geometry, number and algebra using geometrical constructions — deductive reasoning
- interpret whether a problem requires additive, multiplicative or proportional reasoning — additive reasoning, multiplicative reasoning, proportional reasoning
- evaluate what can and cannot be inferred in statistical and probabilistic settings — statistical inference
- evaluate the outcomes of a multi-step problem — multi-step problems
- solve problems in financial mathematics using formal mathematical knowledge — financial mathematics
- model a situation mathematically and express results using formal representations — mathematical modelling
- select concepts, methods and techniques for unfamiliar and non-routine problems — problem-solving strategies
Source: DfE National Curriculum in England, Open Government Licence v3.0. Outcomes are split into single actions and rewritten only where the original bundled several; wording otherwise follows the DfE text. Methodology.
What GCSE Mathematics teaches · Years 10–11
188 learning outcomes from the Department for Education’s statutory subject content, grouped by topic. Each outcome belongs to one of twelve fields of knowledge used across startfrom.today: Mathematics & Statistics (188). Exam boards add detail on top of these; the outcomes themselves are the same whichever board a school uses.
Number: Structure and calculation18
- order positive and negative integers, decimals and fractions — decimal, fraction, negative integers, positive integers
- use equality and inequality symbols to compare numbers — equality symbols, inequality symbols
- apply the four operations to integers, decimals and fractions — decimal, four operations, fraction, integer
- use place value when working with large or small numbers — place value
- use inverse operations to simplify calculations and expressions — inverse operations, simplification
- use conventional notation for order of operations — brackets, order of operations, power, root
- use prime factorisation, highest common factor and lowest common multiple — highest common factor, lowest common multiple, prime factorisation
- use product notation and the unique factorisation theorem — product notation, unique factorisation theorem
- apply systematic listing strategies for counting — product rule for counting, systematic listing
- use positive integer powers and associated roots — power, root
- recognise powers of 2, 3, 4 and 5 — cubes, power, squares
- estimate powers and roots of a given positive number — estimation, power, root
- calculate with roots and fractional indices — fractional indices, integer indices, root
- calculate exactly with fractions, surds and multiples of piHigher tier — exact calculation, surd
- simplify surd expressions involving squaresHigher tier — simplification, surd
- rationalise denominators involving surdsHigher tier — rationalising denominators, surd
- calculate with standard form — standard form
- interpret standard form — standard form
Number: Fractions, decimals and percentages5
- convert between terminating decimals and fractions — fraction, terminating decimals
- convert recurring decimals into fractions and vice versa — fraction, recurring decimals
- identify fractions in ratio problems — fraction, ratio
- use fractions to solve ratio problems — fraction, ratio problems
- interpret fractions and percentages as operators — fraction, operators, percentage
Number: Measures and accuracy7
- use standard units of measure including compound measures — compound measure, standard units
- estimate answers to calculations — estimation
- check calculations using approximation and estimation — approximation, estimation
- round numbers and measures to a given degree of accuracy — decimal places, rounding, significant figures
- use inequality notation to specify error intervals — error intervals, rounding, truncation
- apply limits of accuracy using upper and lower bounds — limits of accuracy, lower bound, upper bound
- interpret limits of accuracy using upper and lower bounds — limits of accuracy, lower bound, upper bound
Algebra: Notation, vocabulary and manipulation18
- use algebraic notation — algebraic notation
- interpret algebraic notation — algebraic notation
- substitute numerical values into formulae and expressions — expression, formulae, substitution
- use the vocabulary of expressions, equations, formulae and identities — equation, expression, formulae, identity
- collect like terms in algebraic expressions — like terms
- multiply a single term over a bracket — expanding brackets
- factorise algebraic expressions by taking out common factors — common factors, factorisation
- expand products of two or more binomials — binomials, expansion
- factorise quadratic expressions including the difference of two squares — difference of two squares, quadratic expressions
- simplify expressions involving surds and algebraic fractions using indicesHigher tier — algebraic fractions, laws of indices, surd
- use standard mathematical formulae — standard formulae
- rearrange formulae to change the subject — formulae, subject of a formula
- distinguish between an equation and an identity — equation, identity
- argue mathematically to show algebraic expressions are equivalentHigher tier — algebraic proof, equivalence
- construct algebraic arguments and proofsHigher tier — algebraic proof
- interpret expressions as functions with inputs and outputs — function, inputs and outputs
- interpret the reverse process as an inverse function — inverse function
- interpret the succession of two functions as a composite function — composite function
Algebra: Graphs20
- use coordinates in all four quadrants — coordinate, quadrants
- plot straight-line graphs from equations — coordinate plane, straight-line graphs
- use y = mx + c to identify parallel and perpendicular lines — gradient, parallel lines, perpendicular lines, y-intercept
- find the equation of a straight line through given points or gradient — equation of a line, gradient
- identify gradients and intercepts of linear functions — gradient, intercept, linear functions
- interpret gradients and intercepts of linear functions — gradient, intercept, linear functions
- identify roots, intercepts and turning points of quadratic functions — intercept, quadratic function, root, turning point
- deduce roots of a quadratic function algebraically — quadratic equation, root
- find turning points of a quadratic function by completing the square — completing the square, turning point
- recognise graphs of standard function types — cubic function, exponential function, reciprocal function, trigonometric function
- sketch graphs of standard function types — cubic function, exponential function, reciprocal function, trigonometric function
- interpret graphs of standard function types — cubic function, exponential function, reciprocal function, trigonometric function
- sketch translations and reflections of a function — function graphs, reflection, translation
- plot graphs of non-standard functions in real-life contexts — non-standard functions, real-life contexts
- interpret graphs of non-standard functions in real-life contexts — non-standard functions, real-life contexts
- find approximate solutions to kinematic problems from graphs — acceleration, distance, kinematics, speed
- calculate or estimate gradients and areas under graphs — area under a graph, gradient, non-linear graphs
- interpret gradients and areas under distance-time and financial graphs — distance-time graph, financial contexts, velocity-time graph
- use the equation of a circle centred at the originHigher tier — coordinate geometry, equation of a circle
- find the equation of a tangent to a circle at a given pointHigher tier — circle, coordinate geometry, tangent
Algebra: Solving equations and inequalities12
- solve linear equations in one unknown algebraically — linear equations
- find approximate solutions to linear equations using a graph — graphical solution, linear equations
- solve quadratic equations by factorising, completing the square or using the formula — completing the square, factorising, quadratic equation, quadratic formula
- find approximate solutions to quadratic equations using a graph — graphical solution, quadratic equation
- solve simultaneous equations in two variables algebraically — simultaneous equation
- find approximate solutions to simultaneous equations using a graph — graphical solution, simultaneous equation
- find approximate solutions to equations numerically using iterationHigher tier — iteration, numerical method
- translate situations or procedures into algebraic expressions or formulae — algebraic modelling
- derive an equation or simultaneous equations from a problem — equation, problem modelling
- interpret the solution of an equation derived from a problem — equation, solution interpretation
- solve linear and quadratic inequalities in one or two variables — linear inequality, quadratic inequality
- represent the solution set of an inequality on a number line or graph — number line, set notation, solution set
Algebra: Sequences4
- generate terms of a sequence from a term-to-term or position-to-term rule — position-to-term rule, sequence, term-to-term rule
- recognise triangular, square, cube, arithmetic and geometric sequences — arithmetic progression, geometric progression, quadratic sequence, triangular numbers
- use triangular, square, cube, arithmetic and geometric sequences — arithmetic progression, geometric progression, quadratic sequence
- deduce the nth term of linear and quadratic sequences — linear sequence, nth term, quadratic sequence
Ratio, proportion and rates of change29
- convert between related standard and compound units — compound units, standard units
- use scale factors, scale diagrams and maps — map, scale diagram, scale factor
- express one quantity as a fraction of another — fraction, proportion
- use ratio notation including simplifying ratios — ratio notation, simplifying ratios
- divide a quantity into two parts in a given ratio — part:part, part:whole, ratio
- express the division of a quantity into two parts as a ratio — ratio
- apply ratio to real-life contexts and problems — ratio, real-life contexts
- express a multiplicative relationship as a ratio or fraction — fraction, multiplicative relationship, ratio
- use proportion as equality of ratios — equality of ratios, proportion
- relate ratios to fractions and linear functions — fraction, linear functions, ratio
- define percentage as number of parts per hundred — percentage
- interpret percentages and percentage changes as a fraction or decimal — percentage, percentage change
- express one quantity as a percentage of another — percentage
- compare two quantities using percentages — comparison, percentage
- use percentages greater than 100 percent — percentage
- solve problems involving percentage increase, decrease and original value — original value, percentage change
- solve problems involving simple interest in financial contexts — financial mathematics, simple interest
- solve problems involving direct and inverse proportion — direct proportion, inverse proportion
- use compound units such as speed, density and pressure — compound units, density, pressure, speed
- compare lengths, areas and volumes using ratio notation — area, length, ratio notation, volume
- relate similarity and scale factors to trigonometric ratios — scale factor, similarity, trigonometric ratios
- explain that inverse proportion is equivalent to direct proportion with a reciprocal — direct proportion, inverse proportion, reciprocal
- construct and interpret equations describing direct and inverse proportion — direct proportion, equation, inverse proportion
- interpret the gradient of a straight-line graph as a rate of change — gradient, rate of change
- recognise and interpret graphs of direct and inverse proportion — direct proportion, graph, inverse proportion
- interpret the gradient at a point on a curve as an instantaneous rate of changeHigher tier — gradient of a curve, instantaneous rate of change
- apply average and instantaneous rate of change using chords and tangentsHigher tier — average rate of change, chord, tangent
- set up and solve growth and decay problems including compound interest — compound interest, growth and decay
- interpret answers to growth and decay problems using iterative processes — growth and decay, iteration
Geometry and measures: Properties and constructions24
- use conventional geometric terms and notation — line, point, polygon, vertices
- use standard conventions for labelling triangles — angle, sides, triangle labelling
- draw diagrams from a written description — diagram, geometric description
- use ruler and compass constructions for bisectors and perpendiculars — angle bisector, construction, perpendicular bisector
- construct given figures and solve loci problems — construction, loci
- explain that perpendicular distance is the shortest distance to a line — perpendicular distance
- apply angle properties at a point and on a straight line — angles at a point, vertically opposite angles
- use alternate and corresponding angles on parallel lines — alternate angles, corresponding angles, parallel lines
- derive and use the angle sum of a triangle and polygons — angle sum, polygon, triangle
- derive and apply the properties of quadrilaterals and triangles — plane figures, quadrilaterals, triangle
- use congruence criteria for triangles — asa, congruence, rhs, sas, sss
- derive results about angles and sides using congruence and similarity — angle facts, congruence, similarity
- use known results to construct simple geometric proofs — geometric proof, pythagoras' theorem
- identify congruent and similar shapes — congruent shapes, similar shapes
- construct congruent and similar shapes using transformations — enlargement, reflection, rotation, translation
- describe the changes and invariance from combined transformations — invariance, reflection, rotation, translation
- identify circle definitions and properties — centre, chord, radius, tangent
- apply circle definitions and properties — arc, circumference, sector, segment
- apply the standard circle theoremsHigher tier — circle theorem
- prove results using circle theoremsHigher tier — circle theorem, proof
- solve geometrical problems on coordinate axes — coordinate geometry
- identify properties of 3D shapes — 3d shapes, edges, faces, vertices
- construct plans and elevations of 3D shapes — elevations, plans
- interpret plans and elevations of 3D shapes — elevations, plans
Geometry and measures: Mensuration and calculation17
- use standard units of measure for length, area, volume and mass — area, length, mass, volume
- measure line segments and angles in geometric figures — angle, line segments
- interpret maps, scale drawings and bearings — bearings, map, scale drawing
- calculate the area of triangles, parallelograms and trapezia — area, trapezia, triangle
- calculate the volume of cuboids, prisms and cylinders — cylinders, prisms, volume
- calculate the circumference and area of a circle — area of a circle, circumference
- calculate perimeters of 2D shapes including composite shapes — composite shapes, perimeter
- calculate the surface area and volume of spheres, pyramids and cones — cones, spheres, surface area, volume
- calculate arc lengths, angles and areas of sectors — arc length, circle, sector
- apply congruence and similarity to lengths, areas and volumes — congruence, scale factor, similarity
- use Pythagoras' theorem to find lengths in right-angled triangles — pythagoras' theorem
- use trigonometric ratios to find angles and lengths in right-angled triangles — cosine, right-angled triangles, sine, tangent
- apply Pythagoras' theorem and trigonometry in two and three dimensionsHigher tier — 3d figures, pythagoras' theorem, trigonometry
- recall exact trigonometric values for standard anglesHigher tier — exact trigonometric value, standard angles
- apply the sine rule to find unknown lengths and anglesHigher tier — sine rule
- apply the cosine rule to find unknown lengths and anglesHigher tier — cosine rule
- apply the sine area formula to solve triangle problemsHigher tier — area of a triangle, sine formula
Geometry and measures: Vectors3
- describe translations as 2D vectorsHigher tier — translation, vector
- apply addition, subtraction and scalar multiplication of vectorsHigher tier — scalar multiplication, vector addition
- use vectors to construct geometric arguments and proofsHigher tier — geometric proof, vector
Probability13
- record and describe the frequency of outcomes using tables and frequency trees — frequency, frequency trees, probability experiment
- analyse the frequency of outcomes of probability experiments — frequency, probability experiment
- calculate expected outcomes using randomness and equally likely events — expected outcomes, fairness, randomness
- relate expected frequency to theoretical probability using the probability scale — probability scale, relative frequency, theoretical probability
- apply the property that probabilities of exhaustive outcomes sum to one — exhaustive outcomes, probability
- apply the property that mutually exclusive probabilities sum to one — mutually exclusive events, probability
- explain that larger samples approach theoretical probability — sample size, theoretical probability
- enumerate sets and combinations of sets using tables and diagrams — sets, tree diagram, venn diagram
- construct theoretical possibility spaces for single and combined experiments — equally likely outcomes, possibility space
- calculate theoretical probabilities from a possibility space — possibility space, theoretical probability
- calculate the probability of independent and dependent combined events — dependent events, independent events, tree diagram
- calculate conditional probability using two-way tables and tree diagrams — conditional probability, two-way table
- interpret conditional probability using expected frequencies — conditional probability, expected frequency
Statistics18
- infer properties of a population from a sample — distribution, population, sampling
- explain the limitations of sampling — bias, sampling
- construct frequency tables, bar charts, pie charts and pictograms — bar chart, categorical data, frequency table, pie chart
- construct line graphs for time series data — line graph, time series
- interpret tables, charts and diagrams for categorical and time series data — categorical data, chart, time series
- construct histograms with equal class intervals — grouped data, histogram
- construct histograms with unequal class intervalsHigher tier — histogram, unequal class width
- construct cumulative frequency graphs — cumulative frequency
- interpret histograms and cumulative frequency graphs — cumulative frequency, histogram
- represent distributions using box plots and other graphical representations — box plot, distribution
- calculate the median, mean and mode of a data set — mean, median, modal class, mode
- calculate the range, quartiles and inter-quartile range of a data set — inter-quartile range, outlier, quartile, range
- compare distributions using measures of central tendency and spread — central tendency, distribution, spread
- apply statistics to describe a population — descriptive statistics, population
- interpret scatter graphs of bivariate data — bivariate data, scatter graph
- recognise correlation and distinguish it from causation — causation, correlation
- draw a line of best fit on a scatter graph — line of best fit, scatter graph
- interpolate and extrapolate trends to make predictions — extrapolation, interpolation, prediction
Source: DfE GCSE mathematics subject content (2013), Open Government Licence v3.0. Outcomes are split into single actions and rewritten only where the original bundled several; wording otherwise follows the DfE text. Methodology.
What A level Mathematics teaches · Years 12–13
190 learning outcomes from the Department for Education’s statutory subject content, grouped by topic. Each outcome belongs to one of twelve fields of knowledge used across startfrom.today: Mathematics & Statistics (166), Natural Sciences (24). Exam boards add detail on top of these; the outcomes themselves are the same whichever board a school uses.
Overarching themes: Mathematical argument, language and proof6
- construct mathematical arguments using diagrams, sketches and logical deductionYear 12 · AS — diagram, logical deduction, mathematical argument
- use correct mathematical symbols and terminology in written argumentsYear 12 · AS — mathematical notation, symbol, term
- use mathematical language and syntax accuratelyYear 12 · AS — mathematical language, syntax
- use set notation and language to express solutions of inequalities and probabilityYear 12 · AS — inequality, probability, set notation
- use the definition of a function, including domain and rangeYear 13 — domain, function, range
- critique mathematical arguments, proofs and justificationsYear 12 · AS — justification, mathematical proof
Overarching themes: Mathematical problem solving9
- simplify and abstract a problem to reveal its underlying mathematical structureYear 12 · AS — abstraction, mathematical structure
- construct extended arguments to solve unstructured problemsYear 12 · AS — extended argument, unstructured problem
- interpret and communicate solutions in the context of the original problemYear 12 · AS — context, solution interpretation
- explain why numerical methods are needed when problems cannot be solved analyticallyYear 13 — analytical solution, numerical method
- evaluate the accuracy and limitations of a mathematical solutionYear 12 · AS — accuracy, limitation
- evaluate the accuracy of solutions obtained using numerical methodsYear 13 — accuracy, numerical method
- apply the mathematical problem-solving cycle of specifying, collecting and interpretingYear 12 · AS — problem-solving cycle
- extract information from mathematical diagrams to solve problemsYear 12 · AS — mathematical diagram
- construct mathematical diagrams to solve problems, including in mechanicsYear 12 · AS — mathematical diagram, mechanics
Overarching themes: Mathematical modelling6
- translate a real-world situation into a mathematical modelYear 12 · AS — mathematical model, simplifying assumption
- use a mathematical model with given inputs to explore a situationYear 12 · AS — mathematical model
- interpret the outputs of a mathematical model in contextYear 12 · AS — mathematical model, output
- refine a mathematical model by reviewing its outputs and assumptionsYear 12 · AS — assumption, mathematical model
- evaluate whether a mathematical model is appropriateYear 12 · AS — mathematical model
- state and use modelling assumptionsYear 12 · AS — modelling assumption
Pure mathematics: Proof4
- construct a mathematical proof using a logical sequence of steps from assumptionsYear 12 · AS — logical deduction, mathematical proof
- use proof by deduction and proof by exhaustionYear 12 · AS — proof by deduction, proof by exhaustion
- use disproof by counter-exampleYear 12 · AS — counter-example
- use proof by contradiction, including for the irrationality of √2 and infinity of primesYear 13 — irrational number, prime number, proof by contradiction
Pure mathematics: Algebra and functions23
- use the laws of indices for rational exponentsYear 12 · AS — laws of indices, rational exponent
- manipulate surds, including rationalising the denominatorYear 12 · AS — rationalising the denominator, surd
- sketch and interpret graphs of quadratic functionsYear 12 · AS — graph, quadratic function
- use the discriminant to determine the nature of roots of a quadraticYear 12 · AS — discriminant, root
- solve quadratic equations, including by completing the squareYear 12 · AS — completing the square, quadratic equation
- solve simultaneous equations in two variables, including one linear and one quadraticYear 12 · AS — simultaneous equation
- solve linear and quadratic inequalities in a single variableYear 12 · AS — linear inequality, quadratic inequality
- express solutions of inequalities using set notationYear 12 · AS — inequality, set notation
- represent linear and quadratic inequalities graphicallyYear 12 · AS — graph, inequality
- manipulate polynomials by expanding, factorising and simplifyingYear 12 · AS — factorisation, polynomial
- use the factor theoremYear 12 · AS — factor theorem
- simplify rational expressions by factorising, cancelling and algebraic divisionYear 13 — algebraic division, rational expression
- sketch curves of polynomial functionsYear 12 · AS — graph, polynomial function
- sketch the graph of the modulus of a linear functionYear 13 — graph, modulus function
- sketch reciprocal graphs y=a/x and y=a/x², including asymptotesYear 12 · AS — asymptote, reciprocal function
- use intersection points of graphs to solve equationsYear 12 · AS — equation, graph intersection
- use proportional relationships and their graphsYear 12 · AS — graph, proportional relationship
- form and use composite functionsYear 13 — composite function
- find and sketch inverse functions and their graphsYear 13 — graph, inverse function
- sketch graphs of simple transformations of y=f(x)Year 12 · AS — function transformation, graph
- sketch graphs of combined transformations of y=f(x)Year 13 — function transformation, graph
- decompose rational functions into partial fractionsYear 13 — partial fraction
- use functions to model real-world situations, including their limitationsYear 13 — function, mathematical model
Pure mathematics: Coordinate geometry7
- use the equation of a straight line in different formsYear 12 · AS — gradient, straight line equation
- use gradient conditions for parallel and perpendicular linesYear 12 · AS — gradient, parallel lines, perpendicular lines
- use straight-line models in contextYear 12 · AS — straight line model
- use the equation of a circle, including finding centre and radius by completing the squareYear 12 · AS — centre, circle equation, radius
- use circle theorems relating tangents, chords and semicirclesYear 12 · AS — chord, circle theorem, tangent
- convert between parametric and Cartesian equations of curvesYear 13 — cartesian equation, parametric equation
- use parametric equations to model real-world contextsYear 13 — mathematical model, parametric equation
Pure mathematics: Sequences and series10
- use the binomial expansion of (a+bx)ⁿ for positive integer nYear 12 · AS — binomial expansion
- link the binomial expansion to binomial probabilitiesYear 12 · AS — binomial expansion, binomial probability
- use the binomial expansion for rational n to approximate valuesYear 13 — approximation, binomial expansion
- generate sequences from an nth term formula or a recurrence relationYear 13 — recurrence relation, sequence
- classify sequences as increasing, decreasing or periodicYear 13 — periodicity, sequence
- use sigma notation to represent sums of seriesYear 13 — series, sigma notation
- use the formulae for the nth term and sum of an arithmetic seriesYear 13 — arithmetic sequence, arithmetic series
- use the formulae for the nth term and sum of a geometric seriesYear 13 — geometric sequence, geometric series
- find the sum to infinity of a convergent geometric seriesYear 13 — convergence, geometric series
- use sequences and series to model real-world situationsYear 13 — mathematical model, sequence, series
Pure mathematics: Trigonometry18
- use the definitions of sine, cosine and tangent for all argumentsYear 12 · AS — cosine, sine, tangent
- use the sine rule and cosine rule to solve trianglesYear 12 · AS — cosine rule, sine rule
- use the formula ½ab sinC for the area of a triangleYear 12 · AS — area of a triangle
- use radian measure to find arc length and sector areaYear 13 — arc length, radian, sector area
- use the small-angle approximations for sine, cosine and tangentYear 13 — small-angle approximation
- sketch and interpret graphs of sine, cosine and tangent functionsYear 12 · AS — periodicity, symmetry, trigonometric graph
- recall exact trigonometric values for standard anglesYear 13 — exact trigonometric value
- use the definitions of secant, cosecant and cotangentYear 13 — cosecant, cotangent, secant
- use the inverse trigonometric functions arcsin, arccos and arctanYear 13 — domain, inverse trigonometric function, range
- use the identity tanθ = sinθ/cosθYear 12 · AS — trigonometric identity
- use the identity sin²θ+cos²θ=1Year 12 · AS — trigonometric identity
- use the identities sec²θ=1+tan²θ and cosec²θ=1+cot²θYear 13 — trigonometric identity
- use double angle formulae and the addition formulae for sin, cos and tanYear 13 — addition formula, double angle formula
- construct geometrical proofs of the trigonometric addition formulaeYear 13 — addition formula, geometrical proof
- express acosθ+bsinθ in the form rcos(θ±α) or rsin(θ±α)Year 13 — amplitude-phase form, trigonometric identity
- solve trigonometric equations in a given intervalYear 12 · AS — trigonometric equation
- construct proofs involving trigonometric identitiesYear 13 — proof, trigonometric identity
- use trigonometric functions to solve problems in contextYear 13 — force, kinematics, trigonometric function, vector
Pure mathematics: Exponentials and logarithms10
- use the function aˣ and its graphYear 12 · AS — exponential function, graph
- use the function eˣ and its graphYear 12 · AS — exponential function, graph
- explain why the gradient property of eᵏˣ makes it suitable for modellingYear 12 · AS — exponential function, gradient
- use the definition of log_a x as the inverse of aˣYear 12 · AS — inverse function, logarithm
- use the function ln x and its graph as the inverse of eˣYear 12 · AS — inverse function, natural logarithm
- use the laws of logarithms to simplify expressionsYear 12 · AS — laws of logarithms
- solve equations of the form aˣ=bYear 12 · AS — exponential equation
- use logarithmic graphs to estimate parameters in power and exponential relationshipsYear 12 · AS — logarithmic graph, parameter estimation
- use exponential growth and decay models in contextYear 12 · AS — exponential decay, exponential growth
- evaluate the limitations of exponential modelsYear 12 · AS — exponential model, limitation
Pure mathematics: Differentiation16
- interpret the derivative as the gradient of a tangent and as a rate of changeYear 12 · AS — derivative, rate of change
- sketch the gradient function of a given curveYear 12 · AS — gradient function
- differentiate from first principles for small positive integer powers of xYear 12 · AS — differentiation from first principles
- differentiate sin x and cos x from first principlesYear 13 — differentiation from first principles
- use the second derivative as the rate of change of gradientYear 12 · AS — second derivative
- identify convex and concave sections of a curve and points of inflectionYear 13 — concavity, point of inflection, second derivative
- differentiate xⁿ for rational n, and sums, differences and multiplesYear 12 · AS — differentiation, power rule
- differentiate exponential and trigonometric functions, including sin kx and cos kxYear 13 — differentiation, exponential function, trigonometric function
- use the derivative of ln xYear 13 — differentiation, natural logarithm
- use differentiation to find gradients, tangents, normals and stationary pointsYear 12 · AS — normal, stationary point, tangent
- use differentiation to identify points of inflectionYear 13 — point of inflection
- determine where a function is increasing or decreasing using its derivativeYear 12 · AS — derivative, monotonicity
- differentiate using the product rule, quotient rule and chain ruleYear 13 — chain rule, product rule, quotient rule
- solve connected rates of change problems using differentiationYear 13 — connected rates of change
- differentiate functions defined implicitly or parametricallyYear 13 — implicit differentiation, parametric differentiation
- construct simple differential equations to model a given contextYear 13 — differential equation, mathematical model
Pure mathematics: Integration11
- use the Fundamental Theorem of CalculusYear 12 · AS — fundamental theorem of calculus
- integrate xⁿ (n≠-1) and related sums, differences and multiplesYear 12 · AS — integration, power rule
- integrate eᵏˣ, 1/x, sin kx and cos kx and related expressionsYear 13 — exponential function, integration, trigonometric function
- evaluate definite integrals and use them to find the area under a curveYear 12 · AS — area under a curve, definite integral
- use definite integrals to find the area between two curvesYear 13 — area between curves, definite integral
- use integration as the limit of a sumYear 13 — integration, limit of a sum
- integrate by substitutionYear 13 — integration by substitution
- integrate by partsYear 13 — integration by parts
- integrate using partial fractions with linear denominatorsYear 13 — integration, partial fraction
- solve first order differential equations with separable variablesYear 13 — differential equation, separation of variables
- interpret the solution of a differential equation in context, including its limitationsYear 13 — differential equation, limitation
Pure mathematics: Numerical methods9
- locate roots of an equation by considering changes of signYear 13 — change of sign, root location
- explain how change-of-sign methods for locating roots can failYear 13 — change of sign, numerical method
- solve equations approximately using simple iterative methodsYear 13 — iterative method
- draw cobweb and staircase diagrams for iterative methodsYear 13 — cobweb diagram, staircase diagram
- solve equations using the Newton-Raphson methodYear 13 — newton-raphson method
- explain how iterative and Newton-Raphson methods can failYear 13 — iterative method, newton-raphson method
- use the trapezium rule to estimate the area under a curveYear 13 — numerical integration, trapezium rule
- estimate the bounds within which the true area under a curve must lieYear 13 — error bound, numerical integration
- use numerical methods to solve problems in contextYear 13 — numerical method
Pure mathematics: Vectors8
- use vectors in two dimensionsYear 12 · AS — vector
- use vectors in three dimensionsYear 13 — vector
- calculate the magnitude and direction of a vector and convert between formsYear 12 · AS — vector direction, vector magnitude
- add vectors diagrammatically and algebraicallyYear 12 · AS — vector addition
- multiply a vector by a scalar and interpret the result geometricallyYear 12 · AS — scalar multiplication, vector
- use position vectors to calculate the distance between two pointsYear 12 · AS — distance, position vector
- use vectors to solve problems in pure mathematics and in mechanics involving forcesYear 12 · AS — force, vector
- use vectors to solve problems in kinematicsYear 13 — kinematics, vector
Statistics: Statistical sampling4
- distinguish between a population and a sampleYear 12 · AS — population, sample
- use a sample to make informal inferences about a populationYear 12 · AS — inference, sample
- use sampling techniques, including simple random sampling and opportunity samplingYear 12 · AS — opportunity sampling, simple random sampling
- select or critique a sampling technique for a given statistical problemYear 12 · AS — sampling technique
Statistics: Data presentation and interpretation10
- interpret diagrams for single-variable data, including histogramsYear 12 · AS — histogram, single-variable data
- connect data distributions to probability distributionsYear 12 · AS — data distribution, probability distribution
- interpret scatter diagrams and regression lines for bivariate dataYear 12 · AS — regression line, scatter diagram
- give an informal interpretation of correlation from a scatter diagramYear 12 · AS — correlation
- explain that correlation does not imply causationYear 12 · AS — causation, correlation
- interpret measures of central tendency and variation, including standard deviationYear 12 · AS — central tendency, standard deviation
- calculate standard deviation, including from summary statisticsYear 12 · AS — standard deviation, summary statistics
- recognise and interpret possible outliers in data sets and diagramsYear 12 · AS — outlier
- select or critique data presentation techniques for a statistical problemYear 12 · AS — data presentation
- clean data, including dealing with missing values, errors and outliersYear 12 · AS — data cleaning, missing data, outlier
Statistics: Probability5
- calculate probabilities using mutually exclusive and independent eventsYear 12 · AS — independent events, mutually exclusive events
- link probability calculations to discrete and continuous distributionsYear 12 · AS — continuous distribution, discrete distribution
- calculate conditional probability using tree diagrams, Venn diagrams and two-way tablesYear 13 — conditional probability, tree diagram, venn diagram
- use the conditional probability formula P(A|B) = P(A∩B)/P(B)Year 13 — conditional probability formula
- critique probability model assumptions and evaluate more realistic alternativesYear 13 — assumption, probability model
Statistics: Statistical distributions5
- use discrete probability distributions as models, including the binomial distributionYear 12 · AS — binomial distribution, discrete probability distribution
- calculate probabilities using the binomial distributionYear 12 · AS — binomial distribution
- use the Normal distribution as a model and find probabilities using itYear 13 — normal distribution
- connect the Normal distribution to histograms and points of inflectionYear 13 — normal distribution, point of inflection, standard deviation
- select an appropriate probability distribution for a context, with reasoningYear 13 — probability distribution
Statistics: Statistical hypothesis testing5
- apply the language of statistical hypothesis testing using a binomial modelYear 12 · AS — null hypothesis, significance level, test statistic
- interpret a correlation coefficient using a given p-value or critical valueYear 13 — correlation coefficient, critical value, p-value
- conduct and interpret a hypothesis test for a binomial proportionYear 12 · AS — binomial proportion, hypothesis test
- explain the significance level as the probability of wrongly rejecting the null hypothesisYear 12 · AS — null hypothesis, significance level
- conduct and interpret a hypothesis test for the mean of a Normal distributionYear 13 — hypothesis test, normal distribution
Mechanics: Quantities and units3
- use fundamental SI quantities and units: length, time and massYear 12 · AS — length, mass, si unit, time
- use derived SI quantities and units: velocity, acceleration, force and weightYear 12 · AS — acceleration, force, si unit, velocity
- use the SI unit of momentYear 13 — moment, si unit
Mechanics: Kinematics8
- use kinematics terminology: position, displacement, distance, velocity and accelerationYear 12 · AS — kinematics terminology
- interpret displacement-time graphs, including gradientYear 12 · AS — displacement-time graph, gradient
- interpret velocity-time graphs, including gradient and area under the graphYear 12 · AS — area under graph, velocity-time graph
- derive and use the constant acceleration formulae for motion in a straight lineYear 12 · AS — constant acceleration, suvat formulae
- use the constant acceleration formulae for motion in two dimensions using vectorsYear 13 — constant acceleration, vector
- use calculus to relate displacement, velocity and acceleration in a straight lineYear 12 · AS — acceleration, calculus, displacement, velocity
- use calculus with vectors for displacement, velocity and acceleration in 2DYear 13 — calculus, vector
- model projectile motion under gravity using vectorsYear 13 — projectile motion, vector
Mechanics: Forces and Newton's laws12
- use Newton's first lawYear 12 · AS — newton's first law
- use Newton's second law for motion in a straight lineYear 12 · AS — newton's second law
- resolve forces and apply Newton's second law in two dimensionsYear 13 — newton's second law, resolving forces
- use weight and gravitational acceleration g in motion under gravityYear 12 · AS — gravitational acceleration, weight
- explain that g is not a universal constant and depends on locationYear 12 · AS — gravitational acceleration
- use Newton's third law for equilibrium and motion in a straight lineYear 12 · AS — equilibrium, newton's third law
- solve problems involving smooth pulleys and connected particlesYear 12 · AS — connected particles, pulley
- resolve forces and solve equilibrium problems for a particle in two dimensionsYear 13 — coplanar forces, equilibrium, resolving forces
- find the resultant of forces acting on a particleYear 13 — resultant force
- use dynamics for motion of a particle in a planeYear 13 — dynamics, motion in a plane
- use the F≤μR model of friction, including coefficient of frictionYear 13 — coefficient of friction, friction
- solve problems involving limiting friction and statics on a rough surfaceYear 13 — limiting friction, statics
Mechanics: Moments1
- use moments to solve simple static equilibrium problemsYear 13 — equilibrium, moment
Source: DfE GCE AS and A level mathematics subject content (2016), Open Government Licence v3.0. Outcomes are split into single actions and rewritten only where the original bundled several; wording otherwise follows the DfE text. Methodology.