GCSE MATHEMATICS — HIGHER TIER
Printable revision guide • Core facts, methods and formulae
Name: ____________________________________ Date: ____________________
How to use this sheet
Revise one topic at a time. Cover the method column and try to recall it, then complete exam questions without looking. Show every
step, keep exact values until the final line, and check whether your answer is sensible.
Higher Tier note: This guide covers the main content shared by the major GCSE exam boards. Your exact specification may include
additional detail, so use it alongside your teacher’s topic list.
1. Number
Topic Key facts / method Example or check
Fractions Add/subtract: use a common denominator. Multiply: top × top, bottom × 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
bottom. Divide: multiply by the reciprocal.
Percentages Multiplier: increase by p% → 1 + p/100; decrease → 1 − p/100. Reverse £80 reduced by 15%: 80 × 0.85 = £68.
percentage: original = final ÷ multiplier.
Ratio Simplify by dividing by the HCF. Sharing: add the parts, divide the total by £60 in 2:3 → 60 ÷ 5 = 12; shares £24 and £36.
this sum, then multiply by each part.
Standard form Write as a × 10■, where 1 ≤ a < 10. Positive n for large numbers; negative n 0.00072 = 7.2 × 10■■.
for small numbers.
Bounds Rounded to nearest unit u: lower bound = stated − u/2; upper bound = 4.6 cm to nearest 0.1: 4.55 ≤ x < 4.65.
stated + u/2 (upper bound is not included).
Indices a■ × a■ = a■■■; a■ ÷ a■ = a■■■; (a■)■ = a■■; a■ = 1; a■■ = 1/a■. x³ × x■ = x■; 2■³ = 1/8.
Surds Simplify by taking out square factors. Rationalise a denominator by √50 = 5√2; 3/√5 = 3√5/5.
multiplying top and bottom by the surd (or conjugate).
Recurring decimals Let x equal the recurring decimal; multiply to line up recurring digits; For 0.333…: 10x=3.333…; 9x=3; x=1/3.
subtract to eliminate them; solve.
2. Algebra
Topic Key facts / method Example
Expand Multiply every term in one bracket by every term in the other. Be careful with (x + 3)(x − 2) = x² + x − 6.
negative signs.
Factorise Take out the highest common factor. For quadratics, find two numbers x² + 7x + 12 = (x + 3)(x + 4).
whose product is c and sum is b.
Solve equations Do the same operation to both sides. Collect x terms on one side and 3x + 5 = 17 → 3x=12 → x=4.
constants on the other.
Rearrange formulae Undo operations in reverse order. To make x the subject, isolate x. y=3x+4 → x=(y−4)/3.
Quadratics Solve by factorising, completing the square, or the quadratic formula: x = For ax²+bx+c=0, identify a, b, c carefully.
(−b ± √(b²−4ac))/(2a).
Sequences Linear: constant first difference; nth term an+b. Quadratic: constant second 5, 8, 11, … has nth term 3n+2.
difference. Geometric: multiply by a constant ratio.
Inequalities Solve like equations, but reverse the inequality when multiplying or dividing −2x > 6 → x < −3.
by a negative number. Show regions on a number line.
Simultaneous Elimination: make one variable’s coefficients equal, then add/subtract. Check the pair works in both original equations.
equations Substitution: replace one variable using the other equation.
Algebraic fractions Factorise first, state restrictions where denominators are zero, then cancel (x²−9)/(x−3) = x+3, where x≠3.
common factors. For adding, use a common denominator.
GCSE Mathematics • Higher Tier revision guide Page 1
Printable revision guide • Core facts, methods and formulae
Name: ____________________________________ Date: ____________________
How to use this sheet
Revise one topic at a time. Cover the method column and try to recall it, then complete exam questions without looking. Show every
step, keep exact values until the final line, and check whether your answer is sensible.
Higher Tier note: This guide covers the main content shared by the major GCSE exam boards. Your exact specification may include
additional detail, so use it alongside your teacher’s topic list.
1. Number
Topic Key facts / method Example or check
Fractions Add/subtract: use a common denominator. Multiply: top × top, bottom × 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
bottom. Divide: multiply by the reciprocal.
Percentages Multiplier: increase by p% → 1 + p/100; decrease → 1 − p/100. Reverse £80 reduced by 15%: 80 × 0.85 = £68.
percentage: original = final ÷ multiplier.
Ratio Simplify by dividing by the HCF. Sharing: add the parts, divide the total by £60 in 2:3 → 60 ÷ 5 = 12; shares £24 and £36.
this sum, then multiply by each part.
Standard form Write as a × 10■, where 1 ≤ a < 10. Positive n for large numbers; negative n 0.00072 = 7.2 × 10■■.
for small numbers.
Bounds Rounded to nearest unit u: lower bound = stated − u/2; upper bound = 4.6 cm to nearest 0.1: 4.55 ≤ x < 4.65.
stated + u/2 (upper bound is not included).
Indices a■ × a■ = a■■■; a■ ÷ a■ = a■■■; (a■)■ = a■■; a■ = 1; a■■ = 1/a■. x³ × x■ = x■; 2■³ = 1/8.
Surds Simplify by taking out square factors. Rationalise a denominator by √50 = 5√2; 3/√5 = 3√5/5.
multiplying top and bottom by the surd (or conjugate).
Recurring decimals Let x equal the recurring decimal; multiply to line up recurring digits; For 0.333…: 10x=3.333…; 9x=3; x=1/3.
subtract to eliminate them; solve.
2. Algebra
Topic Key facts / method Example
Expand Multiply every term in one bracket by every term in the other. Be careful with (x + 3)(x − 2) = x² + x − 6.
negative signs.
Factorise Take out the highest common factor. For quadratics, find two numbers x² + 7x + 12 = (x + 3)(x + 4).
whose product is c and sum is b.
Solve equations Do the same operation to both sides. Collect x terms on one side and 3x + 5 = 17 → 3x=12 → x=4.
constants on the other.
Rearrange formulae Undo operations in reverse order. To make x the subject, isolate x. y=3x+4 → x=(y−4)/3.
Quadratics Solve by factorising, completing the square, or the quadratic formula: x = For ax²+bx+c=0, identify a, b, c carefully.
(−b ± √(b²−4ac))/(2a).
Sequences Linear: constant first difference; nth term an+b. Quadratic: constant second 5, 8, 11, … has nth term 3n+2.
difference. Geometric: multiply by a constant ratio.
Inequalities Solve like equations, but reverse the inequality when multiplying or dividing −2x > 6 → x < −3.
by a negative number. Show regions on a number line.
Simultaneous Elimination: make one variable’s coefficients equal, then add/subtract. Check the pair works in both original equations.
equations Substitution: replace one variable using the other equation.
Algebraic fractions Factorise first, state restrictions where denominators are zero, then cancel (x²−9)/(x−3) = x+3, where x≠3.
common factors. For adding, use a common denominator.
GCSE Mathematics • Higher Tier revision guide Page 1
