GCSE maths foundation tier: topics, papers and how to prepare

What is the GCSE maths foundation tier?

The GCSE maths foundation tier is the one of the two tiers (levels of exam paper) that gives grades 1 to 5. The other is the higher tier, which gives grades 4 to 9. Every student sits all their maths papers at one tier, in the same exam series, so your child takes either foundation or higher, never a mix.

GCSE grades run from 9, the highest, down to 1. The Department for Education counts grade 4 and above as a standard pass. That matters in maths: young people aged 16 to 19 who haven't reached grade 4 in GCSE maths get support to keep working on their maths in their next course, and those with a grade 3 study towards the GCSE again.

Most children take GCSEs at the end of key stage 4, which is Years 10 and 11. The course doesn't say which topics are taught in Year 10 and which in Year 11, so your child's school sets the order. It also builds on KS3 maths, so a lot of what's on the foundation tier will feel familiar.

What's on GCSE foundation maths? The topics by area

The foundation tier covers all six content areas of GCSE maths. The DfE's GCSE subject content marks some topics in bold, and only the more highly attaining students are assessed on those, on the higher tier. Everything else is on the foundation tier, and it's on the higher tier too. Ofqual, the exams regulator, sets roughly how much of the foundation tier comes from each area, and it's the same for every exam board:

Content area	Share of the foundation tier
Number	about 25%
Algebra	about 20%
Ratio, proportion and rates of change	about 25%
Geometry and measures	about 15%
Probability and statistics	about 15%

So half the foundation tier is number, ratio and proportion. Being quick and accurate with fractions, decimals, percentages and ratio pays off across the whole exam.

Number

On the foundation tier, number means calculating with whole numbers, decimals, fractions and negative numbers, with and without a calculator. Students order numbers using the symbols <, >, ≤ and ≥, use the order of operations, and work with prime factors, highest common factors and lowest common multiples. They use powers and roots, standard form and the laws of indices, give answers in terms of π, round to decimal places and significant figures, and write simple error intervals for rounded numbers.

Example: to multiply 3 × 10⁴ by 2 × 10³, multiply the number parts, 3 × 2 = 6, and add the powers of 10, 4 + 3 = 7. The answer is 6 × 10⁷.

Algebra

Foundation algebra goes from simplifying expressions to quadratics. Students expand brackets, factorise, rearrange formulae, and solve linear equations, including ones with the unknown on both sides. They factorise quadratics of the form x² + bx + c (including the difference of two squares), solve quadratic equations by factorising, solve two linear simultaneous equations, and solve linear inequalities. Graphs and sequences are here too: straight-line graphs and y = mx + c, quadratic and simple cubic graphs, and the nth term of linear sequences.

Example: to factorise x² + 7x + 12, look for two numbers that multiply to make 12 and add to make 7. 3 × 4 = 12 and 3 + 4 = 7, so x² + 7x + 12 factorises to (x + 3)(x + 4).

Example: to solve x + y = 10 and x − y = 4 together, add the two equations. The y terms cancel, leaving 2x = 14, so x = 7. Then 7 + y = 10, so y = 3.

Ratio, proportion and rates of change

This area takes ratio, proportion and percentages into real problems: sharing in a ratio, scale drawings and maps, best buys, speed and density, percentage change and original value problems, simple and compound interest, and direct and inverse proportion.

Example: £200 is saved at 5% compound interest a year. Each year the amount is multiplied by 1.05, because 100% + 5% = 105%. After one year, 200 × 1.05 = 210. After two years, 210 × 1.05 = 220.5, so the savings are £220.50.

Example: 4 people take 6 hours to paint a fence. The job takes 4 × 6 = 24 hours of work, so 8 people working at the same rate take 24 ÷ 8 = 3 hours. That's inverse proportion: twice the people, half the time.

Geometry and measures

Foundation geometry covers angle facts and polygons, constructions and loci, congruence and similarity, transformations, circles and the area and volume of shapes and solids. It includes arc lengths and sector areas, the surface area and volume of spheres, pyramids and cones, bearings, plans and elevations, vectors, Pythagoras' theorem and trigonometry in right-angled triangles, and the exact values of sine, cosine and tangent for some angles.

Example: a sector with a right angle at the centre is a quarter of a circle. If the radius is 4 cm, the whole circle has an area of π × 4², which is 16π cm². A quarter of that is 4π cm², or about 12.6 cm².

Probability

Foundation probability uses frequency trees, tables, Venn diagrams and tree diagrams to find the probability of combined events, both independent (one doesn't affect the other) and dependent (the first changes the chances for the second). Students also learn that the probabilities of all the possible outcomes add up to 1, and that bigger samples tend to give results closer to the theoretical probability.

Example: a bag has 3 red counters and 2 blue counters. Two are taken out without putting the first back. The chance the first is red is ³⁄₅. That leaves 2 red out of 4, so the chance the second is red too is ²⁄₄. Multiply along the branches of the tree diagram: ³⁄₅ × ²⁄₄ = ³⁄₁₀.

Statistics

Foundation statistics is about reading, drawing and comparing data: frequency tables, bar charts, pie charts, line graphs for data over time, the mean, median, mode and range, sampling and its limits, and scatter graphs with lines of best fit, knowing that correlation doesn't prove that one thing causes another.

Example: two classes sit a quiz. Class A's scores have a median of 7 and a range of 9 (from 3 to 12, and 12 − 3 = 9). Class B's median is 7 too, but its range is 4. The averages are the same, but class B's scores are more consistent.

The GCSE maths higher tier page lists the extra topics that only come up on higher papers, such as surds, the quadratic formula, circle theorems and histograms.

How the GCSE foundation maths papers work

Every exam board's foundation papers test the same subject content, but the boards set their papers out differently. Your child's school chooses the board, so ask which one your child is entered for.

Exam board	Papers	Calculator	Time and marks
AQA (8300)	3	Paper 1 non-calculator; Papers 2 and 3 calculator	1 hour 30 minutes and 80 marks each
Pearson Edexcel (1MA1)	3	Paper 1 non-calculator; Papers 2 and 3 calculator	1 hour 30 minutes and 80 marks each
OCR (J560)	3 (Papers 1, 2 and 3)	Paper 2 non-calculator; Papers 1 and 3 calculator	1 hour 30 minutes and 100 marks each
WJEC Eduqas (C300)	2 (Components 1 and 2)	Component 1 non-calculator; Component 2 calculator	2 hours 15 minutes and 120 marks each

So with every board there's 4 hours 30 minutes of exams in total, and a third or a half of the marks are on a paper without a calculator. Any topic can come up on any paper, and questions often join two topics together, so a paper doesn't map neatly onto a list of chapters.

The papers also test three kinds of skill, called assessment objectives. On the foundation tier, 50% of the marks are for using standard techniques (AO1), 25% for reasoning and communicating (AO2) and 25% for solving problems (AO3). So half the marks need more than a remembered method: explaining, interpreting and working out what to do. Showing clear working is how that reasoning gets seen.

Grades on the foundation tier

The foundation tier gives grades 1 to 5. A student who doesn't reach the mark for grade 1 gets a U (unclassified) and no certificate for the subject. Grade 5 is the highest grade available on foundation, whatever the mark.

The marks needed for each grade are called grade boundaries. They're set after every exam series, so they change from year to year and from board to board. Our GCSE maths grade boundaries page explains how they work and where each board publishes them.

How does the school choose foundation or higher?

Your child's school makes the exam entries, so it decides which tier your child sits. A student can only be entered for one tier in each exam series. Each school has its own way of deciding and its own timing, so ask your child's maths teacher how and when the choice is made.

The choice is about which papers suit your child best. Foundation papers only test foundation content, and grades 4 and 5 can be reached on either tier. If you're not sure the tier is right, the teacher can explain the reasons and what would change their view.

How to prepare for GCSE foundation maths

The revision that helps most is regular, mixed practice, with mistakes looked at straight away.

• Little and often, mixed up. A short session most days on a mix of topics keeps everything in use. Exam questions don't say which topic they're from, so mixed practice is practice at choosing the method too.
• Make number work automatic. Times tables, fractions, decimals, percentages and ratio come up across the foundation papers. Quick recall of the times tables makes the non-calculator paper much less tiring.
• Practise without a calculator. With every board, at least a third of the marks are on a non-calculator paper. Written methods such as long multiplication and long division are worth keeping sharp.
• Work on the mistakes. When an answer is wrong, find the exact step where it went wrong and try a similar question the next day.
• Estimate and check. The GCSE asks students to check answers using estimation, including calculator answers. Rounding each number to 1 significant figure first catches slips with the decimal point.
• Learn the formulae. Some formulae aren't given in the exam. Our GCSE maths formulae page lists which to learn and which are given.
• Try full papers under time. Papers from previous years show the layout and the kinds of question. Doing one in the real time, then marking it with the mark scheme, shows which topics need more work.

Papers from previous years for GCSE foundation maths

Each exam board publishes its own papers from previous years, with mark schemes, on its website. Your child's teacher may have more for practice, and knows which board your child's papers come from.

• AQA GCSE Mathematics (8300) assessment resources
• Pearson Edexcel GCSE Mathematics (1MA1) course materials
• OCR GCSE Mathematics (J560) assessment materials
• WJEC Eduqas papers and mark schemes

Check the tier on the front cover: foundation papers say "Foundation tier", and AQA and Pearson Edexcel codes end in F.

Maths words for the foundation tier

Tier

The level of GCSE paper a student is entered for: foundation or higher.

Standard form

A way of writing very large or very small numbers, like 6 × 10⁷.

Significant figures

The digits that give a number its size, counted from the first digit that isn't zero.

Error interval

The range of values a rounded number could have been. 8 rounded to the nearest whole number could have been anything from 7.5 up to (but not including) 8.5.

Factorise

Write an expression as a product of its factors, often using brackets. It's the reverse of expanding.

Quadratic expression

An expression where the highest power of the letter is 2, like x² + 7x + 12.

Gradient

How steep a line is: how far it goes up for every 1 across.

Inversely proportional

When one amount doubles, the other halves, so the two always multiply to give the same number.

Compound interest

Interest paid on the amount saved plus the interest already earned.

Bearing

A direction given as an angle measured clockwise from north, written with three figures, like 060°.

Independent events

Events where one happening doesn't change the chance of the other.

Correlation

A link between two sets of data, shown by the pattern on a scatter graph. It doesn't prove one causes the other.

GCSE foundation maths questions to try

1. Factorise x² + 8x + 15.
2. Solve x + y = 12 and x − y = 2 together.
3. Multiply 4 × 10⁵ by 2 × 10³. Give your answer in standard form.
4. Share £40 in the ratio 3 : 5.
5. £500 is saved at 2% compound interest a year. How much is there after 2 years?
6. A circle has a radius of 6 cm. Give its area in terms of π.
7. A bag has 4 green counters and 1 yellow counter. Two are taken out without putting the first back. What's the probability both are green?
8. B is on a bearing of 060° from A. What is the bearing of A from B?

Answers

1. (x + 3)(x + 5), because 3 × 5 = 15 and 3 + 5 = 8
2. x = 7 and y = 5 (adding the equations gives 2x = 14)
3. 8 × 10⁸ (4 × 2 = 8, and 5 + 3 = 8 for the power)
4. £15 and £25 (there are 8 parts and 40 ÷ 8 = 5, so 3 parts are £15 and 5 parts are £25)
5. £520.20 (500 × 1.02 = 510, then 510 × 1.02 = 520.2)
6. 36π cm²
7. ³⁄₅ (the working is ⁴⁄₅ × ³⁄₄ = ³⁄₅)
8. 240° (turn round: 60 + 180 = 240)

How to help with GCSE foundation maths at home

• Know the board, the tier and the dates. Ask the school which exam board and tier your child is entered for, and when the mocks and the real exams are. That tells you which papers to look at and when.
• Help build a revision timetable. Short, regular slots that mix topics work better than cramming one topic at a time. Leave room for rest and for other subjects.
• Talk through real numbers. Best-buy prices, savings interest, recipes in a ratio, journey times and sports statistics are all foundation tier maths. Asking "how would you work that out?" costs nothing.
• Ask for the reasoning. "How do you know?" and "is there another way?" are good practice for the reasoning and problem solving marks, and you don't need to know the answer yourself to ask.
• Keep it calm. Every child moves at their own pace, and steady work through the year matters more than a last-minute push. If something is worrying your child, their maths teacher is the best person to talk to.

See all our maths guides, from Year 1 to GCSE