Edexcel · GCSE Maths · 1MA1 · Foundation and Higher
M17 · Algebraic arguments and proof
Revision notes, worked examples and methods for algebraic arguments and proof.
Notes and quizzes ready · 50 questions · Video coming soon.
Foundation shows shared notes and questions. Higher includes the shared content and labelled Higher extensions. Changing tier starts the notes and a new quiz set.
Revise the key ideas
Equations, identities and arguments
An equation is true for particular values: 2x + 1 = 7 holds when x = 3. An identity holds for every allowed value, for example 2(x + 3) ≡ 2x + 6.
To show expressions are equivalent, expand or factorise and collect terms until they have the same form. Testing a single value cannot prove an identity.
Worked example: 3(x + 2) − x = 3x + 6 − x = 2x + 6 = 2(x + 3), so the two expressions are equivalent for every x.
A counterexample disproves a universal claim. The statement “all prime numbers are odd” is false because 2 is prime and even.
Use precise mathematical reasons in an argument. A diagram that looks correct or a few numerical examples are evidence to investigate, not a proof for every case.
Consecutive integers can be written n, n + 1, n + 2; an even integer as 2n; an odd integer as 2n + 1, with n an integer.
Higher — algebraic proofs
To prove a divisibility statement, express the result as the divisor times an integer. Define the integer variables and show each step clearly.
Worked example: The sum of two odd integers is (2m + 1) + (2n + 1) = 2(m + n + 1). Since m + n + 1 is an integer, the sum is even.
Worked example: The difference of consecutive squares is (n + 1)2 − n2 = 2n + 1, which is odd for every integer n.
Worked example: Three consecutive integers sum to n + (n + 1) + (n + 2) = 3(n + 1), so their sum is a multiple of 3.
For an odd square, (2n + 1)2 = 4n2 + 4n + 1 = 4n(n + 1) + 1. Hence its remainder on division by 4 is 1.
Distinguish a proof from solving an equation. Proving an identity must not impose a special value of x; every operation must be valid across the stated domain.
To disprove “x2 is always greater than x”, use x = 12: x2 = 14 < 12. Consider values beyond positive integers when a claim concerns real numbers.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
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