GCSE Maths Algebraic Proof: How to Start
GCSE algebraic proof asks you to show that a mathematical statement is always true, not merely that it works for the numbers in one example. The safest way to start is to represent unknown whole numbers with letters, build expressions for the relevant quantities and simplify until the intended structure is visible. These worked examples explain how to lay out a proof clearly and avoid the common traps that cost marks.
What counts as a proof rather than an example?
Checking that 2 + 4 + 6 = 12 shows a result for three particular even numbers. It does not establish the general claim about all even numbers. To prove a statement about an arbitrary integer, let n represent any integer and show the required result for that n. For instance, a general even integer can be written as 2n, while an odd integer can be written as 2n + 1. The essential difference is that the algebra represents every possible case within the claim, rather than a handful of convenient choices.
GCSE algebraic proof: showing a number is always even
Prove that the sum of two consecutive even integers is even. Let the first even integer be 2n, where n is an integer. The next consecutive even integer is 2n + 2. Adding gives 2n + (2n + 2) = 4n + 2 = 2(2n + 1). Because 2n + 1 is an integer, the result is twice an integer and therefore even. Finish by stating the reason in words. Simply writing an answer such as 14 + 16 = 30 would not prove the result generally.
Prove that three consecutive integers sum to a multiple of three
Let n be any integer. Three consecutive integers are n, n + 1 and n + 2. Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). As n + 1 is an integer, the total is a multiple of 3. This proof follows a reliable pattern: introduce a general integer, represent the required neighbouring integers, simplify and factorise to reveal a multiple of the claimed number. You do not need to test each possible value separately.
Why the difference of consecutive square numbers is odd
Take the squares of n and n + 1. Their difference is (n + 1)² − n². Expanding the first bracket gives n² + 2n + 1 − n², which simplifies to 2n + 1. The result has the form of an odd integer for every integer n, so the statement is proved. Be especially careful when expanding (n + 1)²: it equals n² + 2n + 1, not n² + 1. That missing middle term is a very common error in algebraic proof questions.
Disproving a claim requires only one counterexample
Sometimes an exam question asks whether a statement is true for all cases. A single valid counterexample is enough to show it is false. For example, the claim “all prime numbers are odd” is disproved by the prime number 2. But a handful of successful examples cannot establish a universally true statement. Understand this asymmetry: one counterexample disproves an all-cases claim, while a proof must establish why every permitted case works. Read words such as always, every, some and never carefully.
Present your proof so an examiner can follow it
Begin by defining the variable and its domain, for example “Let n be an integer.” Write each transformation on a new line, preserve equality signs and show the factorisation that establishes the property. Conclude with a short sentence such as “This is twice an integer, so it is even.” If you get stuck, try a small numerical case to discover the pattern, but return to a general expression for the final proof. A tutor can help by questioning why a step is valid rather than merely correcting the final algebra.
Frequently asked questions
Can I use numbers to begin an algebraic proof?
Yes, examples can reveal a pattern, but a general proof must establish the result for every relevant value.
What should I revise before GCSE algebraic proof?
Expand brackets, factorise common factors and understand the algebraic forms of even, odd and consecutive integers.
More GCSE learning resources
Explore these existing guides when a prerequisite skill or another aspect of your GCSE preparation needs attention. Use the links selectively to find the explanation that matches your current problem.
Book a free online introduction
If the worked examples reveal a persistent difficulty, MasterMaths offers online one-to-one support tailored to the student. The free 20-minute introduction is an opportunity to discuss the learning goal and ask how Ryan teaches. Share the student’s year group, exam board and one problem they have tried independently. No tutor can promise a particular exam grade; the aim is more secure understanding.
Discuss support: Book a free maths introduction session
Comments