A mathematical proof is a logical argument that shows a statement is always true — not just true for a few examples. GCSE proof questions are usually algebraic, using expressions that represent any number.
The key idea: Examples prove nothing on their own. Showing $3 + 5 = 8$ doesn't prove that the sum of two odd numbers is always even — you need algebra.
Common proof types at GCSE:
Prove a statement about integers: Use $n$, $2n$, $2n+1$ etc. to represent general numbers. See Reasoning for the full toolkit.
Prove two expressions are equal: Start with one side and manipulate it until you reach the other. Don't work on both sides at once.
Disprove a statement (counterexample): One single example that doesn't work is enough to disprove a statement. Find it and state it clearly.
Geometric proof: Use angle facts and theorems as named reasons. Every step needs a justification. See Circle theorems and Angle facts.
Structure for a written proof:
- State what you're representing algebraically
- Work through each step clearly
- Interpret your final line — explain why it shows what you set out to prove