Subject · A level
Further Mathematics
Which schools in England make Further Mathematics a big part of the curriculum, and where results are strongest. Entries and grades from the 2025 performance tables.
What the data says
At A level, Further Mathematics was offered in 1,817 schools and colleges, with 14,172 entries — roughly 5% of A level students.
A level: schools and colleges where the largest share of students take Further Mathematics
Entries as a share of A level students (at least 30 A level students and 10 entries). England: 5% of A level students.
What pupils learn in Further 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
not offered as a separate subjectYears 10–11GCSE
not offered as a separate subjectYear 12A level, typically first year
39learning outcomes · content also examined at ASYear 13A level, second year
57learning outcomes · A level-only contentWhat A level Further Mathematics teaches · Years 12–13
96 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 (92), Natural Sciences (4). 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 associated with set theoryYear 12 · AS — set notation, set theory
- use the definition of a function, including domain and rangeYear 13 — domain, function, range
- critique mathematical arguments, proofs and justifications of methods and formulaeYear 12 · AS — formula, justification, mathematical proof
Overarching themes: Mathematical problem solving6
- 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
- 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 problemsYear 12 · AS — mathematical diagram
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 appropriate to the situationYear 12 · AS — appropriateness, mathematical model
- use modelling assumptions when constructing a mathematical modelYear 12 · AS — mathematical model, modelling assumption
Proof1
- construct proofs by mathematical inductionYear 13 — divisibility, matrix power, proof by induction, series
Complex numbers16
- solve quadratic equations with real coefficients including complex rootsYear 12 · AS — complex root, quadratic equation
- solve cubic and quartic equations with real coefficients given a known root or factorYear 12 · AS — complex root, cubic equation, quartic equation
- add, subtract, multiply and divide complex numbers in the form x + iyYear 12 · AS — arithmetic operation, complex number
- use the terms real part and imaginary part of a complex numberYear 12 · AS — imaginary part, real part
- use the complex conjugate of a complex numberYear 12 · AS — complex conjugate
- state that non-real roots of a real polynomial equation occur in conjugate pairsYear 12 · AS — complex root, conjugate pair, polynomial equation
- use and interpret Argand diagrams to represent complex numbersYear 12 · AS — argand diagram, complex number
- convert a complex number between Cartesian and modulus-argument formYear 12 · AS — cartesian form, modulus-argument form
- multiply and divide complex numbers in modulus-argument formYear 12 · AS — complex number, modulus-argument form
- construct and interpret loci in the Argand diagramYear 12 · AS — argand diagram, argument, locus, modulus
- use de Moivre's theorem to derive multiple angle formulaeYear 13 — de moivre's theorem, multiple angle formula
- use de Moivre's theorem to find sums of seriesYear 13 — de moivre's theorem, series
- use the exponential form of a complex numberYear 13 — complex number, exponential form
- find the n distinct nth roots of a complex number in exponential formYear 13 — complex number, exponential form, nth root
- state that the nth roots of a complex number form a regular n-gon in the Argand diagramYear 13 — argand diagram, nth root, regular polygon
- use complex roots of unity to solve geometric problemsYear 13 — geometric problem, root of unity
Matrices14
- add, subtract and multiply conformable matricesYear 12 · AS — conformability, matrix
- multiply a matrix by a scalarYear 12 · AS — matrix, scalar multiplication
- use zero and identity matricesYear 12 · AS — identity matrix, zero matrix
- use matrices to represent linear transformations in two dimensionsYear 12 · AS — linear transformation, matrix
- use matrices to represent successive linear transformationsYear 12 · AS — composition of transformations, matrix
- use matrices to represent single linear transformations in three dimensionsYear 12 · AS — 3d transformation, matrix
- find invariant points and lines for a linear transformationYear 12 · AS — invariant line, invariant point, linear transformation
- calculate the determinant of a 2 by 2 matrixYear 12 · AS — determinant, matrix
- calculate the determinant of a 3 by 3 matrix and interpret it as a scale factorYear 13 — determinant, orientation, scale factor
- use singular and non-singular matrices and properties of inverse matricesYear 12 · AS — inverse matrix, non-singular matrix, singular matrix
- calculate and use the inverse of a non-singular 2 by 2 matrixYear 12 · AS — 2x2 matrix, inverse matrix
- calculate and use the inverse of a non-singular 3 by 3 matrixYear 13 — 3x3 matrix, inverse matrix
- solve three linear simultaneous equations using the inverse matrixYear 13 — inverse matrix, simultaneous equation
- interpret geometrically the solution or failure of solution of simultaneous equationsYear 13 — geometric interpretation, simultaneous equation
Further algebra and functions8
- use the relationship between roots and coefficients of a polynomial equationYear 12 · AS — coefficient, polynomial equation, root
- form a polynomial equation from a linear transformation of another equation's rootsYear 12 · AS — polynomial equation, root transformation
- use formulae for the sums of integers, squares and cubesYear 13 — sum of cubes, sum of integers, sum of squares
- use standard sum formulae to sum other seriesYear 13 — series, sum formula
- use the method of differences to sum a seriesYear 13 — method of differences, partial fraction, series
- find the Maclaurin series of a function including its general termYear 13 — general term, maclaurin series
- use standard Maclaurin series for exponential, logarithmic and trigonometric functionsYear 13 — maclaurin series, standard function
- state the range of validity of a standard Maclaurin seriesYear 13 — maclaurin series, range of validity
Further calculus8
- evaluate improper integralsYear 13 — improper integral
- derive formulae for volumes of revolutionYear 13 — formula, volume of revolution
- calculate volumes of revolutionYear 13 — volume of revolution
- evaluate the mean value of a functionYear 13 — mean value of a function
- integrate using partial fractions including quadratic factors in the denominatorYear 13 — integration, partial fraction, quadratic factor
- differentiate inverse trigonometric functionsYear 13 — differentiation, inverse trigonometric function
- integrate standard functions using trigonometric substitutionYear 13 — integration, trigonometric substitution
- choose an appropriate trigonometric substitution to integrate a functionYear 13 — integration, trigonometric substitution
Further vectors9
- use the vector and Cartesian forms of the equation of a straight line in 3DYear 13 — cartesian equation, straight line, vector equation
- use the vector and Cartesian forms of the equation of a planeYear 13 — cartesian equation, plane, vector equation
- calculate the scalar product of two vectorsYear 13 — scalar product, vector
- use the scalar product to express the equation of a planeYear 13 — equation of a plane, scalar product
- use the scalar product to calculate angles between lines and planesYear 13 — angle between lines, angle between planes, scalar product
- check whether two vectors are perpendicular using the scalar productYear 13 — perpendicularity, scalar product
- find the intersection of a line and a planeYear 13 — intersection, line, plane
- calculate the perpendicular distance between two linesYear 13 — line, perpendicular distance
- calculate the perpendicular distance from a point to a line or planeYear 13 — perpendicular distance, plane, point
Polar coordinates3
- convert between polar and Cartesian coordinatesYear 13 — cartesian coordinate, polar coordinate
- sketch curves given as a function of theta in polar formYear 13 — polar curve, trigonometric function
- find the area enclosed by a polar curveYear 13 — area, polar curve
Hyperbolic functions7
- state the definitions, domains and ranges of hyperbolic functionsYear 13 — domain, hyperbolic function, range
- sketch the graphs of hyperbolic functionsYear 13 — graph, hyperbolic function
- differentiate and integrate hyperbolic functionsYear 13 — differentiation, hyperbolic function, integration
- use the definitions, domains and ranges of inverse hyperbolic functionsYear 13 — domain, inverse hyperbolic function, range
- derive and use the logarithmic forms of inverse hyperbolic functionsYear 13 — inverse hyperbolic function, logarithmic form
- integrate standard functions leading to inverse hyperbolic functionsYear 13 — integration, inverse hyperbolic function
- choose a substitution to integrate a function involving hyperbolic formsYear 13 — integration, substitution
Differential equations12
- use an integrating factor to solve a first order linear differential equationYear 13 — differential equation, integrating factor
- recognise when an integrating factor is an appropriate methodYear 13 — differential equation, integrating factor
- find general and particular solutions to a differential equationYear 13 — differential equation, general solution, particular solution
- use differential equations to model kinematics and other contextsYear 13 — differential equation, kinematics, mathematical model
- solve a homogeneous second order differential equation using the auxiliary equationYear 13 — auxiliary equation, differential equation
- solve a non-homogeneous second order differential equation using a particular integralYear 13 — complementary function, differential equation, particular integral
- relate the discriminant of the auxiliary equation to the form of the solutionYear 13 — auxiliary equation, differential equation, discriminant
- solve the simple harmonic motion equation and relate it to the motionYear 13 — differential equation, simple harmonic motion
- model damped oscillations using second order differential equationsYear 13 — damped oscillation, differential equation
- interpret solutions of a damped oscillation modelYear 13 — damped oscillation, solution
- model two dependent variables as coupled first order differential equationsYear 13 — coupled differential equation, dependent variable
- solve a pair of coupled first order simultaneous differential equationsYear 13 — coupled differential equation, simultaneous equation
Source: DfE GCE AS and A level further 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.