GCSE MATHS • HIGHER TIER
Algebraic Proof Made Simple
Learn how to use variables and algebraic reasoning to prove mathematical statements are always true.
Clear method
Exam practice
Worked proofs
Proof in Four Moves
1
Define
Choose a variable to represent.
2
Express
Write the expressions using that variable.
3
Simplify
Simplify to a single algebraic form.
4
Conclude
State why the result proves the claim.
What You’ll Learn
Represent numbers
Use variables to represent even, odd and consecutive numbers.
Build expressions
Form algebraic expressions using starting values and operations.
Simplify algebra
Simplify by expanding, factorising and collecting like terms.
Write conclusions
Explain clearly why your result proves the statement.
Key Representations
Even number
2n
Odd number
2n + 1
Consecutive numbers
n, n + 1
Consecutive even numbers
2n, 2n + 2
Worked Proof: Consecutive Numbers
1. Define
Let the numbers be n and n + 1.
2. Express
Write the sum.
3. Simplify
Simplify the expression.
4. Conclude
State what this proves.
More Worked Examples
1
Three consecutive numbers
n + (n + 1) + (n + 2) = 3(n + 1)
Step 2: Factorise 3 from the expression.
2
Even number squared
(2n)² = 4n² = 2(2n²)
Step 2: Factorise 2 from the expression.
3
Product of consecutive numbers
n(n + 1)
Step 2: One factor must be even.
4
Consecutive squares
(n + 1)² − n² = 2n + 1
Step 2: Simplify by subtraction.
- Common GCSE Mistakes
- Using specific numbers instead of variables
- Not defining variables at the start
- Missing algebraic steps
- No final conclusion
- Expansion or sign errors
- Exam Tips
- Use standard forms for even and odd numbers
- Show every step clearly
- Factorise when proving divisibility
- Explain your conclusion in words
- Check signs and arithmetic carefully
Skills Used in Algebraic Proof
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Algebraic Proof Tutorial
Step-by-step walkthroughs of algebraic proofs with clear methods and exam tips.
Frequently Asked Questions
Algebraic proof uses variables and algebraic manipulation to demonstrate that a mathematical statement is always true.
Write an even number as 2n and an odd number as 2n + 1, where n is an integer.
Examples only test selected values. Algebra proves that the statement works for every valid value.
Explain in words why the final algebraic form proves the original statement.
