GCSE Further Maths Topics: Full AQA, OCR & Edexcel List

Feeling overwhelmed by the GCSE Further Maths topics you’ve got to get your heard around? We’ve got you covered.

Further Maths is a qualification studied at GCSE level alongside GCSE Maths by the highest attaining students. It digs deeper into maths topics and introduces new and more complex concepts.

AQA GCSE Further Maths (8365) is the most popular course of its type, and this guide details all the topics for this course. Plus, we’ll dig into the key differences for OCR and Edexcel, so you can plan your revision with no guesswork.

Key Takeaways

• GCSE Further Maths covers six main areas: Number, Algebra & Functions, Coordinate Geometry, Calculus, Matrix Transformations, and Geometry.

• The course introduces new topics such as matrices, calculus, algebraic proof, and polynomial functions, helping students develop stronger problem-solving and analytical skills.

• Both AQA Further Maths exam papers assess any topic from the specification, so students need broad revision across all areas.

GCSE Further Maths Topics at a Glance

AQA GCSE Further Maths (8365) (opens in a new tab) is organised into six topic areas:

1. Number - surds and counting principles.

2. Algebra & Functions - the largest area, covering algebraic manipulation, functions, binomial expansion, sequences and inequalities.

3. Coordinate Geometry - straight lines, the equation of a circle and tangents to circles.

4. Calculus - differentiation and its applications, including tangents, normals and stationary points.

5. Matrix Transformations - 2×2 matrix multiplication and geometric transformations.

6. Geometry - geometric proof, advanced trigonometry, trig graphs and trig equations.

There are two alternative qualifications: 

1. OCR Additional Mathematics

2. Edexcel Extended Maths

AQA GCSE Further Maths Topics

Below is the full breakdown of each of the six topic areas, with the sub-topics you need to be confident on. Use these as tick-list checklists when planning revision.

1. Number

The Number topic extends content from GCSE Maths. It's a smaller area but the techniques crop up everywhere else.

• Surds: simplifying, adding and subtracting surds, and rationalising denominators.

• Counting principles: including the product rule for counting.

2. Algebra & Functions

Algebra & Functions is the largest topic area on the AQA specification and accounts for a large proportion of marks in both papers. Some content builds on GCSE Maths and some is brand new.

• Algebra toolkit: expanding double and triple brackets, factorising quadratics and cubics, algebraic long division, completing the square (for x² and ax² forms)

• Function notation: evaluating f(x), composite functions fg(x), inverse functions f⁻¹(x), domain and range, sketching functions

• Binomial expansion: using Pascal's triangle and the binomial theorem for positive integer powers

• Algebraic fractions: adding, subtracting, multiplying, dividing and solving equations involving algebraic fractions

• Rearranging formulae: including cases where the subject appears more than once

• Indices: laws of indices, fractional indices, negative indices, and solving equations involving indices and roots

• Algebraic proof: constructing proofs using algebraic manipulation.

• Polynomials & Factor Theorem: factorising and solving cubic equations using the factor theorem and algebraic long division.

• Shapes of graphs: sketching quadratics, cubics, reciprocal and exponential graphs.

• Simultaneous equations: linear-linear, linear-quadratic, and three unknowns.

• Solving inequalities: linear and quadratic inequalities, representing solutions on a number line.

• Sequences: linear and quadratic nth terms, plus limiting values of sequences.

3. Coordinate Geometry

Coordinate geometry builds on GCSE Maths, with more complex exam questions and the circle work pushed further than GCSE.

• Gradient of a line, parallel and perpendicular lines.

• Distance between two points and midpoint of a line.

• Dividing a line in a given ratio.

• Equation of a straight line.

• Equation of a circle (centre at the origin and not at the origin).

• Tangents to circles.

• Linking circle theorems to coordinate geometry.

4. Calculus

Calculus is brand new to GCSE-level students and one of the most exciting parts of the course. It's your first taste of A Level Maths.

• Differentiation from first principles: the derivative as a rate of change.

• Differentiating polynomials of the form ax^n, including after rearranging.

• Gradient of a curve at a given point.

• Equation of a tangent to a curve.

• Equation of a normal to a curve.

• Increasing and decreasing functions.

• Second derivative (d²y/dx²).

• Stationary points: finding and classifying maxima, minima and points of inflection.

• Applications of differentiation: optimisation problems and real-world rates of change.

• Curve sketching using stationary points and end behaviour.

5. Matrix Transformations

Matrices are also brand new at GCSE level. Don't let them intimidate you. This section is shorter than it looks and the techniques are easy to apply once you've practised them.

• Multiplying a matrix by a scalar.

• Multiplying matrices (2×2 by 2×1 and 2×2 by 2×2).

• The identity matrix.

• Transforming the unit square.

• Matrix transformations (reflections, rotations, enlargements).

• Combining transformations using matrix multiplication.

6. Geometry

Geometry builds on GCSE Maths trigonometry and proof. Questions combine trigonometry with algebra, so practise multi-step problems.

• Geometric proof: including circle theorems and congruence arguments.

• Trigonometry in 2D and 3D: including 3D Pythagoras and angles between lines and planes.

• Sine Rule: for sides and angles, including the ambiguous case.

• Cosine Rule: for sides and angles.

• Area of a triangle using ½ ab sin C.

• Exact trig values for 0°, 30°, 45°, 60° and 90°.

• Trigonometric graphs: sin, cos and tan, including transformations.

• Trigonometric identities: sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.

• Solving trigonometric equations, including those requiring identities or factorisation.

What is covered in each AQA GCSE Further Maths paper?

AQA's Level 2 Certificate in Further Mathematics is assessed by two equally weighted exam papers. 

Both assess content drawn from any part of the specification, so there’s no fixed "Paper 1 content" vs "Paper 2 content".

Revision Resources for AQA GCSE Further Maths

Although needing to have a sound knowledge of all content within the GCSE Further Maths course sounds daunting, we’ve got you covered with our course-specific revision resources. 

We’ve combed through the specification to make sure that our revision resources are aligned with AQA’s specification and style. Check out the links below.

• Revision Notes - teacher-written study notes aligned to the exam board.

• Exam Questions - exam-style questions with student-friendly worked solutions.

• Past Papers - real papers with mark schemes.

GCSE Further Maths Topics for OCR and Edexcel

AQA isn't the only Further Maths qualification at this level. There are two alternatives:

• OCR Level 3 Certificate in Additional Mathematics (6993)

• Edexcel Level 2 Extended Maths Certificate (7M20)

The core idea is the same - to extend GCSE Maths into A Level-style topics. But the content and assessment differ.

OCR Additional Mathematics Topics

OCR Additional Mathematics is assessed with a single 2-hour exam, rather than two papers. The topics covered overlap with AQA, but have some key differences:

• Calculus: includes differentiation and integration (AQA only has differentiation).

• Exponentials and logarithms: taught in some depth - not a core focus of AQA.

• Numerical methods: approximation methods, iteration and the trapezium rule for areas.

• No matrix transformations: unlike AQA, OCR doesn't include 2×2 matrix transformations.

Check out the official OCR Additional Mathematics specification (opens in a new tab), and our OCR past papers and mark schemes on our website.

Edexcel Extended Maths Topics

Like AQA, the Edexcel Extended Maths (opens in a new tab) qualification is assessed by two papers (non-calculator and calculator), each 1 hour 15 minutes long and equally weighted. 

The emphasis is more on extending GCSE concepts than on introducing lots of brand-new content.

• A balance of algebra, coordinate geometry and some calculus (differentiation only).

• Matrix transformations (2×2) for enlargements, reflections and rotations.

• Further algebraic techniques, polynomial factorisation, proof and some circle theorems.

• Focus on deepening existing GCSE Maths knowledge rather than introducing brand-new concepts.

Frequently Asked Questions

Is it hard to get a 9 in GCSE Further Maths?

GCSE Further Maths is challenging. To get a grade 9 is achievable, but demands practice and commitment to consistent revision.

In 2025, the mark needed to achieve a grade 9 was 141 out of 160. (opens in a new tab) That means you need to hit 88% to come away with the top grade.

Is Further Maths the hardest GCSE?

GCSE Further Maths is widely considered one of the hardest qualifications taken at GCSE level. It introduces A Level concepts such as calculus and matrices, demands strong algebraic fluency, and is taken alongside GCSE Maths - usually only by students predicted a grade 8 or 9. 

What's the hardest topic in GCSE Further Maths?

Most students find calculus and matrix transformations the hardest topics in GCSE Further Maths, because both are brand new at this level and don't appear in GCSE Maths. 

Algebraic proof, functions (especially inverse and composite functions) and trigonometric identities also come up often. They need fluent algebra and a bit of abstract reasoning.

However, topic difficulty is subjective. What one student finds difficult, another may find less challenging, and vice versa.

Hit Your Target Grade with Save My Exams

Save My Exams is here to help you achieve the best grade possible in GCSE Further Maths with examiner-written resources made specifically for your exam board. Join over 2 million students worldwide, and enjoy access to:

• Detailed, exam-aligned revision notes

• Exam-style questions with student-friendly worked solutions

• Past papers with mark schemes

Whether you want to firm up a tricky maths concept, test what you know, or sharpen your exam technique, our resources are designed to make revision more efficient - and a lot less stressful.

Explore our GCSE Further Maths resources

References

AQA GCSE Further Maths Specification (opens in a new tab)

OCR Level 3 Additional Maths Specification (opens in a new tab)

Edexcel Extended Maths Specification (opens in a new tab)

AQA AQA L2 Certificate in Further Mathematics (8365) Grade Boundaries (opens in a new tab)