GCSE Maths Topics Explained: Foundation, Higher and Exam Boards

Knowing what to revise is one of the hardest parts of GCSE Maths. You might recognise topics such as percentages, algebra and trigonometry, but still be unsure which skills your exam requires—or how much harder they become at Higher tier.

GCSE Maths covers six main areas: Number, Algebra, Ratio and Proportion, Geometry and Measures, Probability, and Statistics. For England’s GCSE qualifications, Edexcel, AQA, OCR and Eduqas follow the same national subject content. Their papers look different, but the underlying mathematics is shared.

The more important distinction is your tier. Foundation covers grades 1–5, while Higher targets grades 4–9, combining Foundation content with additional techniques and more demanding problems.

This guide explains what each area includes, how Foundation and Higher differ, and how to turn the syllabus into a revision plan that addresses the marks you are losing.

02Are GCSE Maths Topics Different Between Exam Boards?

If you compare two GCSE Maths specifications, they can look like different courses. One board might group everything under “Geometry and Measures”, while another separates basic geometry, similarity and mensuration.

Usually, this is a difference in organisation. Mensuration, for example, simply means calculating measurements such as perimeter, area and volume.

Edexcel GCSE Maths 1MA1 and AQA GCSE Maths 8300 both use three papers at each tier. Paper 1 is non-calculator, and Papers 2 and 3 allow a calculator. Each paper lasts 1 hour 30 minutes.

OCR GCSE Maths J560 also has three papers at each tier, but its non-calculator paper comes in the middle. Foundation students take Papers 1–3, with Paper 2 non-calculator. Higher students take Papers 4–6, with Paper 5 non-calculator.

Eduqas GCSE Maths uses two papers: one non-calculator and one calculator. Each lasts 2 hours 15 minutes, making sustained concentration particularly important.

For learning an individual method, resources from another board can be useful. When practising under timed conditions, use your own board’s papers so that you become familiar with its instructions, question presentation and mark schemes.

This guide covers GCSE Maths in England. International GCSE qualifications, including Cambridge IGCSE and Edexcel International GCSE Maths, have their own specifications.

03Foundation and Higher Maths: What Is the Difference?

Foundation and Higher overlap substantially. Both assess arithmetic, percentages, ratio, equations, graphs, geometry, probability and statistics.

Higher adds techniques such as the quadratic formula, circle theorems and conditional probability. It also requires students to connect familiar skills in less obvious ways. A difficult Higher question might combine ratio, algebra and geometry without telling you which method to use first.

Foundation is therefore more demanding than its name sometimes suggests. Students aiming for grade 5 need to handle topics such as simultaneous equations, simple quadratics, Pythagoras’ theorem and right-angled trigonometry.

Think of the tiers as different levels of depth within a shared course. Learning a topic name is only the beginning; what matters is knowing which questions you must be able to answer.

04GCSE Maths Topics: What You Need to Learn

Number

Number underpins almost every other part of GCSE Maths. It includes calculating with positive and negative numbers, fractions and decimals; using factors, multiples and prime numbers; and working with powers, roots and standard form.

You also need to round answers appropriately, estimate calculations and recognise when a calculator result is unreasonable. These skills matter beyond number questions: an accurate algebraic method can still produce a wrong answer if you mishandle a negative sign.

At Foundation, you should be comfortable converting between fractions, decimals and percentages, finding fractions of amounts, applying index laws and interpreting error intervals.

For example, a length recorded as 8.4 cm to the nearest 0.1 cm could be anywhere from 8.35 cm up to, but not including, 8.45 cm. Understanding that interval is different from simply rounding a number.

Higher extends this area through fractional indices, calculations with surds, rationalising denominators, recurring decimals expressed as fractions and calculations involving upper and lower bounds.

Algebra and Graphs

Algebra turns relationships into expressions, equations and graphs. You need to simplify expressions, substitute values, expand brackets, factorise, rearrange formulae and solve equations.

Foundation includes linear equations, linear inequalities and simultaneous linear equations. It also includes expanding two brackets and solving suitable quadratic equations by factorising.

For example:

x² + 5x + 6 = 0

Factorising gives:

(x + 2)(x + 3) = 0

So x = −2 or x = −3. This type of quadratic is not exclusively Higher content.

Graphs are closely connected to algebra. Foundation students need to work with coordinates, gradients, intercepts and straight-line equations, as well as recognise and interpret quadratic, simple cubic and reciprocal graphs. Sequences include finding the nth term of a linear sequence and recognising other patterns.

Higher adds more demanding factorisation, algebraic fractions, the quadratic formula, completing the square, iteration and simultaneous equations involving a line and a quadratic. It also extends into composite and inverse functions, graph transformations, quadratic nth terms and equations of circles.

Ratio, Proportion and Rates of Change

Ratio and proportion connect mathematics to situations such as sharing money, comparing prices, adjusting recipes and calculating journeys.

You need to simplify ratios, divide quantities in a given ratio, work with scale drawings and solve direct and inverse proportion problems. Percentages include increases, decreases, reverse percentages, interest and depreciation.

Reverse percentages are particularly easy to misunderstand. If a jacket costs £48 after a 20% reduction, £48 represents 80% of its original price. The original price is therefore:

£48 ÷ 0.8 = £60

Adding 20% to £48 would not reverse the original reduction, because the percentage would be calculated from a different amount.

This area also includes speed, density, pressure and unit conversions. Higher develops proportion through constructing equations, analysing rates of change and working with more general growth and decay processes.

Geometry and Measures

Geometry includes the properties of shapes and the relationships between angles, lengths, areas and volumes.

At Foundation, you need angle facts for triangles, parallel lines and polygons; bearings; constructions and loci; and transformations such as reflection, rotation, translation and enlargement.

You also calculate perimeter, area, circumference, surface area and volume. Questions may involve composite shapes, so you need to decide how to divide a diagram into manageable parts.

Pythagoras’ theorem and right-angled trigonometry are both Foundation topics. Knowing a formula is not enough: you must identify the hypotenuse, recognise the relevant sides and decide whether you are finding an angle or a length.

Higher extends geometry through circle theorems, negative enlargements, area and volume scale factors, the sine and cosine rules, and calculations in three dimensions. Vector reasoning and proof also become more demanding.

Probability

Probability describes how likely an event is to happen. It begins with the probability scale, equally likely outcomes and the fact that the probabilities of all possible outcomes total 1.

Foundation includes relative frequency, expected frequency, sample spaces, Venn diagrams and probability trees. It also includes dependent events, where one outcome changes the probability of the next.

Suppose a bag contains three red counters and two blue counters. If two counters are drawn without replacement, the probability that both are red is:

3/5 × 2/4 = 3/10

The second fraction changes because one red counter has already been removed.

Higher adds conditional probability: finding the probability of an event when you know another event has occurred.

Statistics

Statistics involves presenting, analysing and interpreting data. You need to calculate averages and range, use frequency tables and interpret charts such as bar charts, pie charts, time-series graphs and scatter graphs.

Foundation also includes estimating a mean from grouped data, drawing lines of best fit and understanding the limitations of samples and predictions.

When comparing two groups, an average alone rarely tells the whole story. One group might have a higher mean but a much wider spread of results. Your comparison should explain what the statistics suggest in context.

Higher adds histograms, frequency density, cumulative frequency, quartiles, interquartile range and box plots.

05Which GCSE Maths Topics Should You Revise First?

Your weakest topics should shape your revision, but some skills deserve early attention because they support many others.

Fractions, percentages, ratio, equation solving and rearranging formulae are useful starting points. Weaknesses here can make later topics difficult even when you understand the new method.

The assessment weightings also show the overall balance of the course:

These are approximate weightings across the assessment as a whole. They do not predict the contents of a particular paper.

For Foundation, Number and Ratio together account for around half the assessment. However, algebra, geometry and data questions remain essential to a strong result.

At Higher, Algebra has the largest share. Build confidence with manipulation and equations before moving into algebraic fractions, functions and more demanding proofs.

06How to Turn the Topic List into a Revision Plan

Start with a past paper for your board and tier. Attempt it independently, mark it carefully and identify the specific reason behind each lost mark.

“Revise geometry” is too broad to guide your next session. “I used the diameter instead of the radius” gives you a precise mistake to correct.

Separate problems into three categories:

• Knowledge: You did not know the method.
• Application: You knew the method but did not recognise when to use it.
• Accuracy: You chose the correct method but made an arithmetic, rounding or notation error.

Each requires a different response. A knowledge gap needs explanation and focused practice. An application gap needs varied questions where you choose the method yourself. An accuracy problem needs careful checking and practice with the underlying skill.

After revising, answer questions without notes. Return to the topic after a few days, then test it in a mixed set. This checks whether you can remember the method and recognise when it applies.

Keep a short error log recording the topic, mistake, correction and date of your next retest. Use that record to choose your next revision session.

07Can Any Topic Appear on Any GCSE Maths Paper?

You should prepare for any eligible topic to appear on a paper in a form appropriate to its calculator rules.

There is no dependable list of “Paper 1 topics”, and a topic appearing on an earlier paper does not prevent it from appearing again. A later question may assess a different technique or combine it with another area.

Predicted papers can provide additional practice, but they should not decide which parts of the syllabus you leave out.

08How Do You Know When You Have Revised a Topic Properly?

A topic is becoming secure when you can answer questions without notes, explain your method and apply it again after a gap.

The final test is a mixed question or past paper. Here, the topic is not always labelled, and you must decide which approach to use.

Choose one skill you lost marks on recently. Review the method, practise it independently and schedule a retest. As those individual skills improve, use full papers to develop timing, reasoning and the ability to connect topics.

A GCSE Maths topic list tells you what you need to learn. Your answers—and the mistakes you correct—tell you what to do next.