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GCSE Higher Non-Calculator Essential Skills

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Number, ratio and proportion
Multiply decimals
\[ 0.03 \times 0.4 \]
Multiply whole numbers, then put the decimal places back.
Divide by a decimal
\[ 7.2 \div 0.8 \]
Make the divisor a whole number first.
Product of prime factors
\[ 120 = 2^3 \times 3 \times 5 \]
Keep dividing by primes, then write with powers.
HCF and LCM from prime factors
\[ \text{HCF of } 2^3 \times 3, 2 \times 3^2 \]
HCF: lowest powers of shared primes. LCM: highest powers.
Add and subtract mixed numbers
\[ 4\tfrac{5}{6} – 2\tfrac{1}{4} \]
Change to improper fractions with a common denominator.
Multiply mixed numbers
\[ 1\tfrac{1}{2} \times \tfrac{2}{5} \]
Change to improper fractions, then multiply.
Divide mixed numbers
\[ 2\tfrac{1}{4} \div 9 \]
Improper fractions, then keep, change, flip.
Recurring decimal to fraction
\[ 0.\dot{4}\dot{5} = \tfrac{5}{11} \]
Multiply by 10 or 100 so the recurring parts cancel.
Upper and lower bounds
\[ 6.4 \to 6.35 \le x < 6.45 \]
Go half a unit down and half a unit up.
Combine ratios
\[ a:b = 3:4,\ b:c = 2:5 \]
Make the shared part the same in both ratios.
Reverse percentages
\[ 120\% = \text{£}84 \]
The new amount is more or less than 100%: work back.
Direct proportion
\[ y \propto x^2 \]
Write y = kx (or kx², kx³, k√x), find k, then use it.
Inverse proportion
\[ y \propto \tfrac{1}{x} \]
Write y = k ÷ x (or k ÷ x²), find k, then use it.
Indices, surds and standard form
Index laws with numbers
\[ \tfrac{2^6 \times 2^{-2}}{2^3} \]
Add, subtract or multiply the indices, including negatives.
Negative indices
\[ 3^{-2} = \tfrac{1}{9} \]
A negative index means one over (the reciprocal).
Fractional indices
\[ 8^{\frac{2}{3}} = 4 \]
Bottom of the fraction: root. Top: power.
Negative fractional indices
\[ 27^{-\frac{2}{3}} = \tfrac{1}{9} \]
Flip, root, then power.
Algebraic indices
\[ \tfrac{12x^5y^2}{4x^2y^4} \]
Deal with the numbers, then each letter separately.
Power of a term
\[ (2x^3)^4 = 16x^{12} \]
Raise the number to the power and multiply the indices.
Multiply and divide in standard form
\[ (3 \times 10^4)(5 \times 10^6) \]
Numbers first, then powers of 10, then adjust.
Simplify surds
\[ \sqrt{48} = 4\sqrt{3} \]
Take out the biggest square factor.
Add and subtract surds
\[ \sqrt{12} + \sqrt{27} \]
Simplify first so the surds match, then collect.
Expand brackets with surds
\[ (3 + \sqrt{2})(1 – \sqrt{2}) \]
Four multiplications, and √a × √a = a.
Rationalise: single surd
\[ \tfrac{8}{\sqrt{2}} = 4\sqrt{2} \]
Multiply top and bottom by the surd.
Rationalise: conjugate
\[ \tfrac{6}{2 + \sqrt{3}} \]
Multiply by the bracket with the sign swapped.
Expressions and quadratics
Expand double brackets
\[ (3x – 2)(x + 4) \]
Four multiplications, then collect like terms.
Expand a squared bracket
\[ (2x – 3)^2 \]
Write it out twice: there is always a middle term.
Expand triple brackets
\[ (x + 1)(x + 2)(x – 3) \]
Expand two brackets, then multiply by the third.
Factorise x² + bx + c
\[ x^2 + 5x + 6 \]
Two numbers that multiply to c and add to b.
Difference of two squares
\[ 4x^2 – 25 \]
A² − B² = (A − B)(A + B).
Factorise ax² + bx + c
\[ 2x^2 + 5x – 3 \]
Split the middle term, then factorise in pairs.
Simplify algebraic fractions
\[ \tfrac{x^2 – 9}{x^2 + 5x + 6} \]
Factorise top and bottom, then cancel brackets.
Add algebraic fractions
\[ \tfrac{2}{x} + \tfrac{3}{x + 1} \]
Common denominator: multiply the denominators.
Complete the square
\[ (x + 4)^2 – 6 \]
Halve the x coefficient, then subtract its square.
Simplify for a proof
\[ (n + 3)^2 – (n + 1)^2 \]
Expand fully, then collect what is left.
Equations, graphs and functions
Linear equations with fractions
\[ \tfrac{2x + 1}{3} = \tfrac{x – 4}{2} \]
Clear the fractions by multiplying, then solve.
Change the subject: powers and roots
\[ A = \pi r^2,\ r = ? \]
Undo in reverse order; the power or root comes last.
Change the subject: appears twice
\[ y = \tfrac{x + 2}{x – 3} \]
Collect the subject on one side and factorise it out.
Solve a quadratic by factorising
\[ x^2 – 5x – 14 = 0 \]
Factorise, then set each bracket equal to 0.
Simultaneous equations
\[ 2x + 3y = 12 \]
Match one coefficient, then add or subtract.
Equations with algebraic fractions
\[ \tfrac{4}{x} + \tfrac{3}{x + 1} = 2 \]
Multiply by every denominator, then solve the quadratic.
Linear inequalities
\[ 5 – 2x < 11 \]
Solve like an equation; flip if you divide by a negative.
Quadratic inequalities
\[ x^2 – 4x – 5 < 0 \]
Find the roots, sketch, then pick the region.
nth term of a quadratic sequence
\[ 4, 9, 18, 31, \ldots \]
Half the second difference goes in front of n².
Equation of a line through two points
\[ (1, 5) \text{ and } (3, 11) \]
Gradient first, then substitute a point to find c.
Perpendicular gradient
\[ m = 3 \to -\tfrac{1}{3} \]
Flip the gradient and change its sign.
Turning point from completed square
\[ y = (x – 3)^2 + 5 \]
y = (x − a)² + b turns at (a, b).
Composite functions
\[ fg(3) \]
Do the function next to x first.
Inverse functions
\[ f^{-1}(x) \]
Write y = f(x) and make x the subject.
Geometry and trigonometry
Exact trig values
\[ \sin 60^\circ = \tfrac{\sqrt{3}}{2} \]
Use the 30-60-90 and 45-45-90 triangles.
Solve trig equations, 0° to 360°
\[ \cos x = 0.5 \]
Find one angle, then use the graph for the rest.
Angles in polygons
\[ \text{Regular 12-gon} \]
Exterior angles add to 360°; interior sum (n − 2) × 180°.
Circle theorems: centre and same segment
\[ \text{Centre} = 2 \times \text{edge} \]
Centre angle is double; same segment angles are equal.
Circle theorems: semicircle and cyclic quadrilateral
\[ \text{Opposite angles: } 180^\circ \]
Semicircle angle is 90°; opposite angles add to 180°.
Circle theorems: tangents
\[ \text{Tangent} \perp \text{radius} \]
Tangent ⊥ radius; alternate segment theorem.
Arc length and sector area
\[ \tfrac{\theta}{360} \times \pi r^2 \]
Fraction of the circle × whole circle; leave π in.
Surface area of a cylinder
\[ 2\pi r^2 + 2\pi rh \]
Two circles plus the curved rectangle.
Area and volume scale factors
\[ \times 3 \to \times 9 \to \times 27 \]
Area: square the length factor. Volume: cube it.
Column vectors
\[ 2\mathbf{a} – \mathbf{b} \]
Work with the top and bottom numbers separately.
Rotate a point about a centre
\[ 90^\circ \text{ cw about } (1, 0) \]
Find the movement from the centre, then turn it.
Statistics and probability
Venn diagram probability
\[ P(A \cap B’) \]
∩ is AND, ∪ is OR, a dash means NOT.
Conditional probability
\[ P(A \mid B) \]
Given that: only look inside that part.
Tree diagrams
\[ P(\text{exactly one}) \]
Multiply along branches, add between routes.
Relative frequency
\[ P \approx \tfrac{12}{50} \]
Times it happened ÷ trials; expected = P × trials.
Histograms
\[ \text{FD} \times \text{width} \]
Frequency = frequency density × class width.
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