GCSE maths higher tier: the extra topics, papers and grades

What's on the GCSE maths higher tier?

The GCSE maths higher tier tests all of the GCSE maths content: everything on the foundation tier, plus the topics the DfE's subject content marks in bold, which only the more highly attaining students are assessed on. Every student sits all their maths papers at one tier, so your child takes either higher or foundation, never a mix.

The higher tier has a different balance from foundation. Ofqual, the exams regulator, sets roughly how much of each tier comes from each content area, and it's the same for every exam board:

Content area	Share of the higher tier
Number	about 15%
Algebra	about 30%
Ratio, proportion and rates of change	about 20%
Geometry and measures	about 20%
Probability and statistics	about 15%

Algebra is the biggest single area on higher, half as much again as on foundation. If your child is aiming for the top grades, algebra that's quick and accurate matters most.

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 which year, so your child's school sets the order. The GCSE maths foundation page goes through the content on both tiers, area by area.

Higher maths GCSE topics: what's only on higher

These are the topics in bold in the GCSE subject content, so they only come up on higher papers. Each area also uses the foundation content in harder, multi-step questions.

Number

Higher number adds surds (exact square roots like √3) and rationalising denominators, fractional indices, changing recurring decimals into fractions, upper and lower bounds, estimating powers and roots, and the product rule for counting.

Example: to simplify √12, look for a square number that's a factor of 12. 12 = 4 × 3, and the square root of 4 is 2, so √12 simplifies to 2√3.

Example: to write 0.7 recurring (0.777...) as a fraction, call it x. Then 10x is 7.777..., and taking x away leaves 9x = 7, so x is ⁷⁄₉.

Algebra

Higher algebra adds the harder work with quadratics and graphs. Students factorise quadratics like 2x² + 7x + 3, solve quadratic equations by completing the square and with the quadratic formula, and solve simultaneous equations where one is a quadratic. They simplify algebraic fractions, find the nth term of quadratic sequences, use inverse and composite functions, solve quadratic inequalities, find approximate solutions by iteration, and write algebraic proofs. Graph work adds perpendicular lines, exponential and trigonometric graphs, translating and reflecting graphs, gradients and areas under curves, and the equation of a circle with its centre at the origin.

Example: completing the square on x² + 6x + 2 means writing it as (x + 3)² − 7. Halve the 6 to get the 3, then take away 9 because (x + 3)² gives x² + 6x + 9, and 9 − 7 = 2. It shows the turning point of the graph is at (−3, −7).

Ratio, proportion and rates of change

Higher adds constructing equations for direct and inverse proportion (such as y = kx²), reading the gradient of a curve as an instantaneous rate of change, and general iterative processes, like a population that grows by the same percentage every year.

Example: y is directly proportional to x², and y = 50 when x = 5. So y = kx², and 50 ÷ 25 = 2 gives k = 2. When x = 3, y is 2 × 9, which is 18.

Geometry and measures

Higher geometry adds proving and using the circle theorems, the sine rule and cosine rule, the area of any triangle as ¹⁄₂ab sin C, Pythagoras and trigonometry in 3D shapes, the areas and volumes of similar shapes, negative scale factors, combinations of transformations, and using vectors in proofs.

Example: a triangle has sides of 5 cm and 8 cm with an angle of 60° between them. The cosine rule finds the third side, a. 5² + 8² is 25 + 64, which is 89. Then 2 × 5 × 8 = 80, and cos 60° is a half, so take away half of 80, which is 40. 89 − 40 = 49, so a² is 49 and a is 7 cm.

Example: one circle theorem says the angle at the centre is twice the angle at the edge (the circumference) standing on the same arc. If the angle at the edge is 40°, the angle at the centre is 80°, because 2 × 40 = 80.

Probability

Higher probability adds conditional probability: the chance of something given that something else has happened, worked out from two-way tables, tree diagrams and Venn diagrams.

Example: 30 students choose languages. 12 take French, and 3 of those also take Spanish. Picking one of the French students at random, the probability they take Spanish is ³⁄₁₂, which simplifies to ¹⁄₄. The question limits the choice to the French students, so the denominator is 12, not 30.

Statistics

Higher statistics adds histograms with equal and unequal class widths, cumulative frequency graphs, box plots, quartiles and the inter-quartile range.

Example: in a histogram, the area of each bar shows the frequency, so the height is the frequency density: frequency ÷ class width. If 20 people took between 10 and 15 minutes, the class width is 5 and the bar's height is 20 ÷ 5 = 4.

How the GCSE higher maths papers work

The higher papers have the same layout as the foundation ones with each board, with different questions. Your child's school chooses the board, so ask which one your child is entered for.

Exam board	Higher papers	Calculator	Time and marks
AQA (8300H)	Papers 1, 2 and 3	Paper 1 non-calculator; Papers 2 and 3 calculator	1 hour 30 minutes and 80 marks each
Pearson Edexcel (1MA1)	Papers 1H, 2H and 3H	Paper 1 non-calculator; Papers 2 and 3 calculator	1 hour 30 minutes and 80 marks each
OCR (J560)	Papers 4, 5 and 6	Paper 5 non-calculator; Papers 4 and 6 calculator	1 hour 30 minutes and 100 marks each
WJEC Eduqas (C300PH)	Components 1 and 2	Component 1 non-calculator; Component 2 calculator	2 hours 15 minutes and 120 marks each

Any topic can come up on any paper. With AQA, for example, the questions get harder as each paper goes on. The marks from all the papers are added up and the grade comes from the total, so a hard paper can be balanced by a better one.

What the higher papers ask students to do

Every question tests one or more of three assessment objectives, and the higher tier puts more weight on reasoning and problem solving than foundation does:

• AO1, use and apply standard techniques (40% on higher): recalling facts and definitions, using notation and carrying out routine methods, including multi-step ones. "Solve x² + 5x + 6 = 0" is a typical AO1 question.
• AO2, reason, interpret and communicate (30% on higher): drawing conclusions, building a chain of reasoning, presenting arguments and proofs, and judging whether someone else's argument is valid. "Sam says 2x² is always bigger than 2x. Is he right? Explain." is a typical AO2 question.
• AO3, solve problems (30% on higher): turning a problem, often in words or a real setting, into maths, linking different topics, and checking whether the answer and any assumptions make sense.

So 60% of the higher tier marks need more than a remembered method. Clear working, with each step written down, is how the reasoning gets seen and marked.

Grades on the GCSE maths higher tier

The higher tier gives grades 4 to 9, where 9 is the highest grade. A student on higher who just misses grade 4, by a small number of marks, is given an allowed grade 3. Below that, the result is a U (unclassified), with no certificate for the subject. Grade 3 is the lowest grade available on higher, while foundation goes down to grade 1.

For comparison with the old letter grades, the bottom of grade 7 lines up with the bottom of the old grade A, and the bottom of grade 4 with the bottom of the old grade C. The Department for Education counts grade 4 and above as a standard pass.

The marks needed for each grade change every exam series and differ between boards. Our GCSE maths grade boundaries page explains how they work and where each board publishes them.

Is the higher tier right for your child?

Your child's school makes the exam entries, so it decides the tier, and a student can only sit 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 higher tier is the only route to grades 6, 7, 8 and 9. Grades 4 and 5 can be reached on either tier. If you're unsure, ask the teacher what grade range they expect and why they've chosen the tier, and what would change their view.

Formulae for the higher tier

Students are given a formula sheet in every GCSE maths exam until the current GCSEs are replaced. On higher it adds the quadratic formula, the sine rule, the cosine rule and area = ¹⁄₂ab sin C to the foundation formulae. Some formulae still have to be learned, such as the area of a triangle, arc lengths and sector areas, and the exact trig values. Our GCSE maths formulae page lists what's on the sheet and what to learn.

Example: solve x² + 4x + 1 = 0, to 2 decimal places. It doesn't factorise, so use the quadratic formula with a = 1, b = 4 and c = 1. First, b² − 4ac is 16 − 4, which is 12, and the square root of 12 is about 3.464. So x is (−4 + 3.464) ÷ 2 or (−4 − 3.464) ÷ 2, which gives x = −0.27 or x = −3.73 to 2 decimal places.

GCSE higher maths revision that helps

Good higher tier revision is mixed, regular and honest about mistakes, with plenty of longer, multi-step questions.

• Keep the algebra fluent. Algebra is about 30% of the higher tier and turns up inside geometry and ratio questions too. A few expanding, factorising and rearranging questions most days keep it quick.
• Learn the formulae from memory. Cover the list, write it out, check. Do the same with the circle theorems and the exact sine, cosine and tangent values.
• Do the long questions. The last questions on a higher paper often link two or three topics. Practise reading them slowly, writing down what's known and what's needed, and starting somewhere even when the whole route isn't clear.
• Practise proofs and explanations. Saying "show that" and "explain why" answers out loud, then writing them in full sentences, builds AO2 marks.
• Work on the mistakes. Keep a short list of "things I keep getting wrong" and try a similar question the next day.
• Estimate and check. Round to 1 significant figure for a quick estimate, and put solutions back into the equation.
• Try full papers under time. A paper from a previous year, done in the real time and then marked with the mark scheme, shows where the marks are being lost.

The GCSE is also meant to give a strong foundation for studying maths after 16. If your child is thinking about studying maths after 16, the higher tier algebra is the part most worth making secure.

Papers from previous years for GCSE maths higher

Each exam board publishes its own papers from previous years and mark schemes on its website. Check the front cover says "Higher tier"; AQA and Pearson Edexcel codes end in H.

• 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

Your child's teacher may have more, and can say which suit the work your child is doing now.

Maths words for the higher tier

Surd

A square root that can't be written exactly as a whole number or fraction, like √2. Leaving it as a surd keeps the answer exact.

Rationalise the denominator

Rewrite a fraction so there's no surd on the bottom.

Upper and lower bounds

The largest and smallest values a rounded measurement could have been. 8 cm to the nearest cm has a lower bound of 7.5 cm and an upper bound of 8.5 cm.

Completing the square

Rewriting a quadratic like x² + 6x + 2 as (x + 3)² − 7, which shows its turning point and helps to solve it.

Quadratic formula

A formula that solves any quadratic equation, including ones that don't factorise.

Composite function

One function followed by another, like "double it, then add 3".

Iteration

Using a formula over and over, putting each answer back in, to get closer and closer to a solution.

Circle theorems

Facts about angles, radii, chords and tangents in circles, such as the angle in a semicircle being a right angle.

Sine rule and cosine rule

Formulae that link the sides and angles of any triangle, not just right-angled ones.

Conditional probability

The probability of something happening given that something else has already happened.

Histogram

A chart for grouped data where the area of each bar, not its height, shows the frequency.

Inter-quartile range

The spread of the middle half of the data: the upper quartile minus the lower quartile.

Proof

A step-by-step argument that shows something is always true, not just in the examples tried.

Higher maths GCSE questions to try

1. Simplify √50.
2. Write 0.4 recurring (0.444...) as a fraction.
3. Write x² + 8x + 5 in the form (x + p)² + q.
4. Solve x² + 2x − 4 = 0 with the quadratic formula. Give your answers to 2 decimal places.
5. A triangle has sides of 3 cm and 8 cm with an angle of 60° between them. How long is the third side?
6. 40 students walk or take the bus. 25 walk, and 10 of the walkers are in Year 10. A walker is picked at random. What's the probability they're in Year 10?
7. A length is 8 cm, to the nearest centimetre. Write down its lower and upper bounds.
8. Make t the subject of v = u + at.

Answers

1. 5√2 (50 = 25 × 2, and the square root of 25 is 5)
2. ⁴⁄₉ (if x is 0.444..., then 10x − x leaves 9x = 4)
3. (x + 4)² − 11 (halve 8 to get 4; (x + 4)² gives x² + 8x + 16, and 16 − 11 = 5)
4. x = 1.24 or x = −3.24 (b² − 4ac is 4 + 16, which is 20, and the square root of 20 is about 4.472)
5. 7 cm (9 + 64 = 73, half of 2 × 3 × 8 is 24, and 73 − 24 = 49)
6. ²⁄₅ (it's 10 out of the 25 walkers, and ¹⁰⁄₂₅ = ²⁄₅)
7. Lower bound 7.5 cm, upper bound 8.5 cm
8. t = (v − u) ÷ a

How to help with GCSE maths higher 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.
• Test the formulae. Read out "the cosine rule" or "the quadratic formula" and ask for it from memory. It takes two minutes, and saying them out loud helps them stick.
• Ask for the reasoning. "How do you know?" and "is there another way?" are good practice for AO2 and AO3, and you don't need to know the answer yourself to ask.
• Help plan the time. Short, regular, mixed sessions work better than long ones on one topic. Leave room for rest and for other subjects.
• 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