Every GCSE Maths term, explained in plain English.

Calculate

Work out an exact value using the given numbers. Show your method.

Describe

Write a complete sentence describing what was asked — usually a transformation needs every component.

Estimate

Round every value to 1 significant figure, then calculate. Do not use the exact values.

Explain

A short written statement showing why your answer is correct, using mathematical language.

Find

Determine the value or solution. Working not always required, but always shown by competent students.

Give reasons / Give a reason

In angle and circle questions, name the theorem or rule that justifies your step.

Hence

Use the previous part of the question to answer this part. You must connect to what you have already found.

Hence or otherwise

"Hence" gives the easy path. "Otherwise" lets you use any valid method. You earn the same marks either way — but easier with hence.

Not drawn accurately

A warning that the diagram is approximate. Do NOT measure with a ruler — use the given values only.

Prove

Demonstrate that a statement is always true, using general (often algebraic) reasoning, not specific numbers.

Show that

You must demonstrate the result starting from given values — the answer is provided, your job is the working.

Sketch vs Draw vs Plot

"Plot" needs accurate axes and exact points. "Draw" is accurate but freer. "Sketch" shows the shape and key features only.

State

One-word or one-line answer, with no working required.

Verify

Substitute the given value into the equation and show that it satisfies it.

Work out

Show how you reached the answer. Synonymous with "calculate" but emphasises the working.

Decimal place

The position after the decimal point. "To 2 d.p." means two digits after the decimal point.

Factor

A number that divides another exactly, with no remainder.

HCF (Highest Common Factor)

The largest number that divides exactly into two or more numbers.

Integer

A whole number — positive, negative or zero. No fractions or decimals.

Irrational number

A number that cannot be written as a fraction of two integers — its decimal goes on forever without repeating.

LCM (Lowest Common Multiple)

The smallest positive number that is a multiple of two or more numbers.

Lower bound

The smallest value a rounded measurement could be. For 12 cm to the nearest cm, the lower bound is 11.5 cm.

Multiple

A number found by multiplying another by an integer. Multiples of 5: 5, 10, 15, 20, ...

Prime number

A whole number greater than 1 whose only factors are 1 and itself.

Rational number

A number you can write as a fraction p/q where p and q are integers and q ≠ 0.

Reciprocal

The reciprocal of a number x is 1/x. Multiplying a number by its reciprocal gives 1.

Significant figure

A digit that contributes to the precision of a number. Leading zeros do not count.

Standard form

A way of writing very large or very small numbers as a × 10ⁿ where 1 ≤ a < 10 and n is an integer.

Surd

An irrational root that cannot be simplified to a rational number, e.g. √2, √15.

Upper bound

The largest value a rounded measurement could be. For a length given as 12 cm to the nearest cm, the upper bound is 12.5 cm.

Coefficient

The number multiplying a variable, e.g. in 4x the coefficient is 4.

Discriminant

The quantity b² − 4ac inside the quadratic formula. Determines how many real roots a quadratic has.

Equation

A mathematical statement with an equals sign — true for some specific value(s) of the unknown.

Expand

Multiply out brackets so that no brackets remain.

Expression

A combination of numbers, letters and operations with no equals sign, e.g. 3x + 5.

Factorise

Rewrite an expression as a product of its factors — the opposite of expanding.

Function

A rule that takes an input x and produces a single output f(x).

Gradient

The steepness of a line. Calculated as (change in y) ÷ (change in x), often written m.

Identity

A statement that is true for every value of the variable. Uses the ≡ symbol.

Inequality

A statement using <, >, ≤ or ≥ instead of =, expressing a range of valid values.

Iteration

A repeated process where each step uses the previous result to approximate the solution of an equation.

nth term

A formula in terms of n that gives any term of a sequence.

y-intercept

The point where a line or curve crosses the y-axis. Found by setting x = 0.

Arc

A portion of the circumference of a circle.

Bearing

A direction measured clockwise from north, expressed as a three-figure number from 000° to 360°.

Chord

A straight line whose endpoints both lie on the circumference of a circle.

Congruent

Two shapes are congruent if they are identical in size and shape — one can be mapped to the other by translation, rotation or reflection.

Hypotenuse

The longest side of a right-angled triangle — the side opposite the right angle.

Locus (plural: loci)

The set of all points satisfying a given condition. The locus of points 3 cm from O is a circle of radius 3 cm.

Parallel

Two lines that never meet and have the same gradient.

Perpendicular

Two lines that meet at a right angle (90°).

Sector

The "pizza slice" region of a circle bounded by two radii and an arc.

Segment

The region of a circle enclosed by a chord and an arc.

Similar

Two shapes are similar if they have the same shape but possibly different sizes — one is an enlargement of the other.

Tangent

A line that touches a circle (or curve) at exactly one point without crossing it.

Vector

A quantity with both magnitude (size) and direction. Often written as a column or in bold.

Conditional probability

The probability of one event given that another has occurred. Written P(A | B).

Correlation

The relationship between two variables on a scatter graph. Can be positive, negative or none.

Frequency density

Frequency divided by class width. Used as the y-axis on a histogram.

Independent events

Events where one occurring does not change the probability of the other. P(A and B) = P(A) × P(B).

Interquartile range (IQR)

A measure of spread: Q3 − Q1. Less sensitive to outliers than the range.

Mean

The arithmetic average. Sum of values divided by the number of values.

Median

The middle value when the data is sorted in order. For an even count, average the two middle values.

Mode

The value that appears most often. A data set can have one mode, more than one (bimodal), or no mode.

Mutually exclusive

Two events that cannot both happen at the same time. P(A or B) = P(A) + P(B).

Quartile

The values that split sorted data into four equal parts. Q1 (25%), Q2 (median, 50%), Q3 (75%).

Range

A measure of spread: largest value minus smallest value.

Tree diagram

A branching diagram showing all possible outcomes of two or more events and their probabilities.

Venn diagram

A diagram showing sets as overlapping circles. Use set notation to describe regions.