Key Idea: A proof shows a statement is always true, not just true for one example. You start from what's given, justify every line with algebra, and finish with the target. It's a Paper 1, by-hand skill — and the show that method is hidden inside sequence, trig and function questions on both papers.
🔢 Name the numbers with algebra
- any integer — one letter stands for every case
- even: a multiple of 2
- odd: one more than an even number
Two independent odds need two letters: 2a + 1 and 2b + 1. Two consecutive numbers share one: n and n + 1 — they're locked one apart. Pick the tidiest layout (n − 1, n, n + 1 when there's a middle term).
✖️ Show a multiple — factor it out
≡ Identities: transform one side
Tip: Start with the messier side. Polynomial → expand every bracket, then collect like terms. Rational → put over a common denominator, combine, then simplify. Never move terms across the ≡.
✏️ IB-style worked examples
IB-style question — prove a sum is even
Prove that the sum of any two odd numbers is even.
Step by step:
Two independent odds need two letters.
Add and collect like terms.
Factor out 2.
Final answer:
2(a + b + 1) is 2 × an integer, so the sum is even. ∎
IB-style question — prove a multiple of 3
Prove that the sum of any three consecutive integers is a multiple of 3.
Step by step:
Write three consecutive integers.
Add and collect like terms.
Take out a factor of 3.
Final answer:
3(n + 1) is 3 × an integer, so the sum is a multiple of 3. ∎
IB-style question — prove an identity
Prove the identity (x + 4)(x − 1) ≡ x² + 3x − 4.
Step by step:
Start with the left side (the brackets) and multiply out.
Collect like terms.
Final answer:
This is the right-hand side, so (x + 4)(x − 1) ≡ x² + 3x − 4. ∎
Important: One example is not a proof — show that means prove it for every case, using algebra. And when the result is given, don't start from the answer. Begin with the expression you're given (or one side of an identity) and work towards the target, with a reason on every line.
Tap each card to reveal the answer.
Exam Tips
- Replace the number with algebra first: even = 2k, odd = 2k + 1, consecutive = n, n + 1, n + 2.
- Use a separate letter for each independent unknown; share one letter only for consecutive values.
- Multiple of k? Factor out k. Never a multiple? Show a constant remainder: k(…) + r.
- Spot (…)² − (…)² → use a² − b² = (a + b)(a − b) instead of expanding.
- Identity: start on the messier side and transform it into the other — never substitute one value.
- Finish with a sentence: state what you've shown. The conclusion line earns a mark.