GCSE Higher · grades 4–9 · ages 14-16
GCSE Higher maths practice
Higher tier awards grades 4 to 9 and covers everything on Foundation plus the material reserved for Higher: surds and rationalising denominators, algebraic fractions, completing the square and the quadratic formula, iteration, circle theorems, the sine and cosine rules, vectors as proof, histograms with unequal class widths, and conditional probability. The exam structure is identical to Foundation — three papers, each 1 hour 30 minutes and 80 marks, equally weighted, with Paper 1 non-calculator and Papers 2 and 3 calculator. The weighting is not identical: algebra rises to 30%, the largest single content area on the paper, with ratio at 20%, geometry and measures at 20%, number at 15%, and probability and statistics together at 15%. The assessment objectives also shift, to 40% AO1 with 30% each for reasoning (AO2) and problem solving (AO3), which is why Higher questions so often arrive with two steps hidden inside them rather than one technique to apply.
Content areas and their weighting
What you will practise
- Number: indices including fractional and negative, surds and rationalising denominators, recurring decimals to fractions, upper and lower bounds
- Algebra: factorising ax² + bx + c, completing the square, the quadratic formula, algebraic fractions, simultaneous equations including linear/quadratic, quadratic inequalities, inverse and composite functions, iteration, graph transformations
- Ratio, proportion and rates of change: proportion equations, compound growth and decay, area and volume scale factors, gradients of chords and tangents as rates of change
- Geometry and measures: circle theorems and their proofs, similarity with areas and volumes, the sine rule, the cosine rule, area = ½ab sin C, trigonometry in three dimensions, vector proof
- Probability: tree and Venn diagrams for dependent events, conditional probability from two-way tables and expected frequencies
- Statistics: histograms with equal and unequal class intervals, cumulative frequency, box plots, quartiles and the interquartile range
The papers
1.5 h 30 min · 80 marks
1.5 h 30 min · 80 marks
1.5 h 30 min · 80 marks
AO1 / AO2 / AO3 at Higher: 40% / 30% / 30%. A formulae sheet is provided in the exam.
Who this is for
Year 10 and Year 11 students entered for Higher tier, students aiming at grades 6 to 9, anyone who needs a grade 7 or above for A level maths, and students sitting Higher who want to make certain of a secure grade 4 or 5 first.
How to use it
- Give algebra the time its 30% deserves — quadratics, simultaneous equations and algebraic fractions repay practice more than any other area on a Higher paper.
- Work the Higher-only topics separately at first. Circle theorems, the sine and cosine rules and conditional probability each have their own habits of thought.
- Practise the non-calculator versions of surds, indices and exact trigonometric values; Paper 1 asks for them in exact form and a decimal answer scores nothing.
- For multi-step questions, write down what you have found at each stage. Method marks exist and they are won by legible working, not by a bare answer.
- Before a mock, do a mixed set across all six content areas so that topic-switching is not itself a surprise.
Questions people ask
Which tier should I be entered for?
Higher is the right entry if you are already working securely at grade 5 and the papers you sit in class come back with marks spread across the whole booklet. If you are mostly picking up marks in the first third of a Higher mock, a Foundation entry usually produces a better grade, because Foundation gives you a full paper you can actually attempt. Your school makes the entry decision; ask them what your mocks suggest.
What is the lowest grade Higher can award?
Grade 4. If a Higher entry falls just short of grade 4 there is a narrow allowed grade 3, but it is not something to plan around. That floor is the real risk of a Higher entry made too optimistically, and it is why the tier decision matters more than most students think.
Is Paper 1 really without a calculator?
Yes, and on Higher this bites harder. Paper 1 asks for surds left in exact form, exact values of sin, cos and tan at 30°, 45° and 60°, index laws and arithmetic with fractions — all of it by hand, for the same 80 marks as each calculator paper.
What's on the formulae sheet?
The exam provides a formulae sheet for the lifetime of the current specifications. It gives the volume and surface area of a sphere and a cone and the kinematics formulae. On Higher you still need to know the quadratic formula, the circumference and area of a circle, Pythagoras' theorem, the three trigonometric ratios, the sine rule, the cosine rule and area = ½ab sin C. Knowing which side of that line a formula falls on is itself worth revising.
Which topics are Higher only?
The whole-statement Higher topics are graph transformations, gradients and areas under curves, the equation of a circle and its tangent, iteration, combinations of transformations, circle theorems, the sine and cosine rules, area = ½ab sin C, instantaneous rates of change, conditional probability, and histograms and cumulative frequency. Several shared topics also have a Higher-only slice — factorising ax² + bx + c, fractional indices, surds, recurring decimals to fractions, upper and lower bounds, area and volume scale factors, and vector proof.
Do I need a grade 7 for A level maths?
Most sixth forms ask for a grade 6 or 7 in GCSE maths for A level maths, and some ask for higher. Requirements are set by each school or college, so check the one you are applying to rather than assuming a national rule.
Are these past paper questions?
No. Every question is original, written to the DfE content statements and checked before publication. We do not host past papers or mark schemes — those are the awarding organisations' own material.