Identities & Algebraic Proof
SPEC A6 • ALGEBRA • FOUNDATION + HIGHER
Prove a result holds for every number — not just a few. Use general variables, manipulate algebraically, and close with a clear concluding statement.
Identity vs Equation
Before you can prove anything, you need to know exactly what you're being asked to show. The difference between an identity and an equation is fundamental.
Equation
An equation is true for only specific values of the variable. Example: 3x + 2 = 14 is only true when x = 4.
Identity
An identity is true for every value of the variable. Example: 2(x + 3) ≡ 2x + 6 is true for every x. Identities are sometimes written with the triple-bar symbol ≡ instead of =.
The Quick Test
Substitute two different values. If both work, it's probably an identity. If only one works, it's an equation. (This is a check, not a proof — proofs need general variables.)
The Proof Mindset — General, Not Specific
The word "prove" in a GCSE maths question has a strict meaning. You must show that a result holds for every possible value, not just the ones you checked.
❌ Not a Proof
1 + 2 + 3 = 6
6 is a multiple of 3.
So the sum of 3 consecutive integers is always a multiple of 3.
One example. Proves nothing. Scores zero.
✅ A Proof
Let the integers be n, n+1, n+2.
Sum = 3n + 3 = 3(n+1)
Since n+1 is an integer, 3(n+1) is a multiple of 3.
General representation. Algebra. Concluding sentence. Full marks.
Examiner Alert
AQA mark schemes explicitly reject numerical-example "proofs". The Jun 23 examiner report on Q13(a) called out students using "a specific numerical example rather than a general algebraic argument." Numerical examples can illustrate a claim — they cannot prove it.
Your Algebraic Toolkit
Almost every GCSE proof starts by writing the numbers in question as general algebraic forms. Memorise these representations — they do the heavy lifting.
To represent…Write…For any integer…
Any integernn
An even number2nn
An odd number2n + 1 (or 2n − 1)n
Two consecutive integersn, n + 1n
Three consecutive integersn, n + 1, n + 2n
Two consecutive even numbers2n, 2n + 2n
Two consecutive odd numbers2n + 1, 2n + 3n
A square numbern² (so √x = n if x = n²)n
A multiple of kk × nn
The 3-Step Proof Structure
Every GCSE proof — whether worth 2 marks or 6 — follows the same structure.
1Define. Start with "Let the integer(s) be n, n+1, …" Translate the English statement into algebra.
2Manipulate. Expand, collect, factorise until the expression is in the required form (e.g. 3 × something).
3Conclude. State explicitly what you've shown: "Since n is an integer, this is [the required form]."
How the Concluding Sentence Scores
3+ mark "prove" questions — the concluding sentence is typically the explicit A1 mark. Skipping it costs a mark.
2-mark "show that" questions — the concluding statement is often marked as "B1 implicit" (AQA mark scheme wording for Jun 23 P1 Q13(a)). Reaching the correct form implies the conclusion.
Best practice always: write the concluding sentence. It's free marks when explicit, and reinforces your thinking when implicit.
Prerequisite Skill
Proofs rely on confident expanding and factorising — skills from Lessons 2.02, 2.03 and 2.04. If expanding (n+1)² or factorising 3n + 3 is slow, revisit those lessons first.
Worked Example 1 — Consecutive Integers
01Prove algebraically [3 marks]
Prove that the sum of any three consecutive integers is a multiple of 3.
M — Method. Let the three integers be n, n+1, n+2. Add them. Factorise to show the sum is 3 × (integer).
C — Calculate.
Sum = n + (n + 1) + (n + 2)
← M1: general representation
= 3n + 3
← collect like terms
= 3(n + 1)
← M1: factorise out 3
E — Evaluate (the concluding sentence).
Since n is an integer, (n + 1) is an integer, so 3(n + 1) is a multiple of 3. Therefore the sum of any three consecutive integers is always a multiple of 3. ← A1
"Prove" vs "Show That"
AQA uses two command words for proof questions, and they're subtly different. Knowing which you're facing shapes how you structure your answer.
"Prove algebraically…"
You must define general variables
Manipulate to a required form
End with an explicit concluding sentence
Usually 3+ marks
Example: "Prove that the sum of 3 consecutive integers is a multiple of 3."
"Show that…"
The answer is given in the question
Your job is to derive it
Start with one side, manipulate to the given form
Concluding sentence often implicit (2 marks)
Example: "Show that (n+1)² − n² = 2n + 1."
The Golden Rule for "Show That"
Never use the given answer in your working. Work forwards from a starting expression to the answer — not backwards from it. Mark schemes reject proofs where the student assumes what they are trying to show.
Worked Example 2 — "Show That" Identity
02Show that [2 marks]
Show that (n + 1)² − n² ≡ 2n + 1.
M — Method. Start with the left-hand side. Expand (n+1)², then subtract n². Simplify.
C — Calculate.
LHS = (n + 1)² − n²
← starting point
= (n² + 2n + 1) − n²
← M1: expand the bracket
= 2n + 1
← A1: n² terms cancel
E — Evaluate.LHS = 2n + 1 = RHS, as required.
What This Identity Tells You
The difference between consecutive square numbers is always odd. Try it: 4² − 3² = 7, 10² − 9² = 19. Both sides hold for every value of n — that's what makes it an identity, written with ≡.
Worked Example 3 — "Simplifies to an Integer"
A recurring AQA pattern: expand, collect, and show every variable term vanishes, leaving a constant integer. Based on AQA Nov 22 Paper 1, Q15.
03Show that [3 marks]
Show that (3x + 4)(2x − 5) − 11x(x − 2) + 5(x² − 3x − 1) simplifies to an integer.
M — Method. Expand each product separately, then collect like terms. The x² and x terms should all cancel, leaving a constant.
C — Calculate.
(3x + 4)(2x − 5) = 6x² − 7x − 20
← M1: expand each product
−11x(x − 2) = −11x² + 22x
← watch the signs
5(x² − 3x − 1) = 5x² − 15x − 5
← distribute carefully
Sum of x² terms: 6 − 11 + 5 = 0
← M1: collect
Sum of x terms: −7 + 22 − 15 = 0
← x terms cancel
Constants: −20 − 5 = −25
← A1: the integer
E — Evaluate.The expression simplifies to −25, which is an integer, as required.
Why the Variable Terms Vanish
These questions are designed so that the x² and x coefficients sum to zero. If yours don't, you've made an expansion error — most commonly a sign slip on the −11x term. Recheck the signs before assuming the question is wrong.
Worked Example 4 — "Always a Square" (Past Paper)
This is based on AQA Jun 23 Paper 3, Q19.
04Prove algebraically [3 marks]
Two integers have a difference of 6. The integers are multiplied together. 9 is then added. Prove algebraically that the result is always a square number.
M — Method. Let the smaller integer be n. Then the other is n + 6. Multiply them, add 9, and factorise to show the result is (something)².
C — Calculate.
Let the integers be n and n + 6.
← M1: general representation
Product + 9 = n(n + 6) + 9
← translate to algebra
= n² + 6n + 9
← expand
= (n + 3)²
← M1: factorise as perfect square
E — Evaluate.Since n is an integer, n + 3 is an integer, so (n + 3)² is a square number. Therefore the result is always a square number. ← A1
Worked Example 5 — Next Square Number (Past Paper)
Based on AQA Jun 24 Paper 1, Q24(b). The algebra requires turning "x is a square number" into something you can manipulate. The mark scheme accepts three different methods — any valid approach scores.
05Show that [2 marks]
x is a square number. Show that the next square number is x + 2√x + 1.
M — Method. Write x = n² (so √x = n). The next square number is (n+1)². Expand and substitute back.
C — Calculate.
Let x = n², so √x = n
← M1: general representation
Next square = (n + 1)²
← next integer squared
= n² + 2n + 1
← expand
= x + 2√x + 1
← A1: substitute x = n², n = √x
E — Evaluate.Therefore the next square number is x + 2√x + 1, as required.
Alternative Approach (also full marks)
The mark scheme also accepts: expand (√x + 1)² directly — since the next square number after x must be (√x + 1)². Then (√x + 1)² = x + 2√x + 1. This is actually shorter and works without introducing n. Either approach is fully credited.
Worked Example 6 — Function Behaviour
A recurring AQA question type: prove something about how a function changes when its input changes. Based on AQA Jun 23 Paper 1, Q13(a).
06Show that [2 marks]
A function f(x) = ax + b, where a and b are constants. Show that if x increases by 2, f(x) increases by 2a.
M — Method. Compute f(x) and f(x+2). Subtract to find the change.
C — Calculate.
f(x) = ax + b
← starting point
f(x + 2) = a(x + 2) + b
← M1: substitute x + 2
= ax + 2a + b
← expand
f(x + 2) − f(x) = 2a
← A1: show the increase is 2a
E — Evaluate.So f(x) increases by 2a when x increases by 2, as required.
The Classic Trap
"Increases by 2" means x + 2, NOT 2x. Many students misread "increases by 2" as "doubles" and write f(2x) = 2ax + b. The examiner report for this specific question called this out as the dominant error. Read the words carefully: "increases by 2" → add 2.
Weak vs. Strong — Proof by Example?
The single biggest marker of a weak proof is reliance on specific numbers. Compare these two attempts at "Prove that the sum of two odd numbers is always even."
❌ Weak Student
3 + 5 = 8 ✓7 + 1 = 8 ✓11 + 13 = 24 ✓
Marks: 0 / 3
Checks three cases. Not a proof — there are infinitely many pairs of odd numbers, and three examples prove nothing.
✅ Strong Student
Let two odd numbers be 2m+1 and 2n+1.
Sum = 2m + 1 + 2n + 1
= 2m + 2n + 2= 2(m + n + 1)
which is of the form 2k, so even.
Marks: 3 / 3
The Rule
Examples illustrate — they do not prove. A proof must use general algebraic variables.
Common Errors to Avoid
Students Often WriteCorrect VersionWhy It Matters
"Let the numbers be 1, 2, 3…""Let n be any integer. The three consecutive integers are n, n+1, n+2."Specific numbers are examples, not proofs
Stop at "3n + 3"Factorise: 3n + 3 = 3(n+1), then concludeForm must be 3 × (integer) to show multiple of 3
No concluding sentence"Therefore X is always a multiple of Y, as required"A1 on 3-mark questions; implicit B1 on 2-mark
"x increases by 2" → use 2x"x increases by 2" → use x + 2Addition, not multiplication (Jun 23 P1 Q13a trap)
"Increases by 2a" → compute f(2a)Compare f(x+2) with f(x): show the difference is 2aChange in output means new minus old
Work backwards from the targetWork forwards from a starting expressionAssuming what you're proving invalidates the proof
Use the same letter for different integersUse distinct letters (e.g. 2m+1 and 2n+1 for two odd numbers)Using n for both forces them to be equal
Sign errors on products like −11x(x − 2)−11x(x − 2) = −11x² + 22x (two sign flips)Most common expansion error in "simplifies to integer" Qs
Examiner Alerts — The Big Ones
Proof by Example
What students write: "1 + 2 + 3 = 6, which is a multiple of 3. So the sum of three consecutive integers is always a multiple of 3."
Why it loses marks: one example proves nothing. The statement "always" requires a general argument covering every integer.
What to write instead: let the integers be n, n+1, n+2. Sum algebraically. Factorise. Conclude.
Missing Concluding Sentence
What students write: they get to "Sum = 3(n+1)" and stop.
Why it loses marks: on 3-mark "prove" questions, the concluding sentence is typically the explicit A1. On 2-mark "show that" questions it's implicit but still expected.
What to write instead: always end with a sentence naming the property you've shown.
The "Increases by 2" → "2x" Trap
What students write: on Jun 23 P1 Q13(a), many substituted 2x (doubling) instead of x + 2 (adding 2). The examiner report specifically called this out as the biggest single error.
What to write instead: "increases by k" always means + k. "Doubles" or "is multiplied by 2" means × 2. Read the verb carefully.
The MCE Framework
Proofs use a slightly adapted MCE: Method (define variables), Calculate (manipulate), Evaluate (conclude).
MMethod. Define your variables with "Let the integer be n…" Translate the statement into algebra.
CCalculate. Expand, collect, factorise. One step per line. End with the form the question requires (e.g. k × bracket).
EEvaluate. Write the concluding sentence linking algebra to the stated property.
Typical Mark Allocations (from AQA Mark Schemes)
2-mark "show that" identity: M1 (expand/manipulate correctly) + A1 (shown in required form). Concluding sentence marked B1 implicit.
3-mark prove-always: M1 (general representation) + M1 (algebraic manipulation) + A1 (concluding statement, explicit)
3-mark "simplifies to integer": M1 (expand all brackets) + M1 (collect all terms) + A1 (final constant value identified)
4-mark proof: adds an extra M1 for multi-step expansion or factorisation
How Often Does This Come Up?
Algebraic proof questions appear regularly on Higher papers — usually 2–3 marks, sometimes 4. Here's recent verified AQA evidence:
PaperQuestionMarksType
Nov 22 P1H Q15Show (3x+4)(2x−5) − 11x(x−2) + 5(x²−3x−1) simplifies to an integer3Simplifies to integer
Jun 23 P1H Q13(a)f(x) = ax + b; show f increases by 2a when x increases by 22Function behaviour
Jun 23 P2H Q25Show algebraic fraction simplifies to given form3"Show that" identity
Jun 23 P3H Q19Two integers differ by 6; product + 9 = square number3Prove-always
Jun 24 P1H Q24(b)x is a square number; show next square is x + 2√x + 12"Show that" identity
Jun 24 P2H Q19(a)Show algebraic expression simplifies to an integer3Simplifies to integer
Patterns to Expect
Every Higher paper contains at least one algebraic proof question. Three dominant patterns: simplifies to an integer (expand & cancel), prove-always (consecutive integers, squares, odd/even), and function behaviour (f(x+k) comparison). Know your algebraic toolkit.
Exam Technique
Top Exam Tip
Before you write anything, translate the English statement into algebra. "Three consecutive integers" → n, n+1, n+2. "An odd number" → 2n+1. "A square number" → n². Once you have your variables defined, the rest is standard algebra.
Checklist for Every Proof Question
Did I define my variables? "Let n be an integer." or "Let x = n²."
Did I use general variables, not specific numbers?
Did I show every algebraic step? One operation per line.
Did I arrive at the required form? (multiple of 3 = 3×bracket; square = (something)²; odd = 2k+1; integer = all variable terms vanished)
Did I write a concluding sentence? "Therefore X is always Y, as required."
Common Distractor-Sentences That Examiners Reject
"This works because…" (vague; state the algebraic reason)
"As you can see from the example above…" (examples aren't proofs)
"Obviously it's a multiple of 3." (explain why the algebraic form makes it so)
Practice Question
Standard AQA pattern — "prove always" [3 marks]
Prove algebraically that the difference between the squares of any two consecutive odd numbers is always a multiple of 8.
M
Let two consecutive odd numbers be 2n + 1 and 2n + 3. Take the difference of their squares and factorise to show the result is 8 × (integer).
C
Difference = (2n + 3)² − (2n + 1)² (M1 — general representation)
C
Expand: (4n² + 12n + 9) − (4n² + 4n + 1)
C
= 12n + 9 − 4n − 1 = 8n + 8
C
= 8(n + 1) (M1 — factorise out 8)
E
Since n is an integer, n + 1 is an integer, so 8(n + 1) is a multiple of 8. Therefore the difference between the squares of any two consecutive odd numbers is always a multiple of 8.(A1 — concluding sentence)
Quick-Fire Checklist
Before you move on, you should be able to do all of these without pausing:
Can You...
Write an odd number, an even number, and a multiple of 5 in general algebraic form? (2n+1, 2n, 5n)
Represent three consecutive integers starting with any integer? (n, n+1, n+2)
Prove the sum of three consecutive integers is a multiple of 3?
Show that (n+1)² − n² = 2n + 1?
Translate "x is a square number" into x = n²?
Recognise "x increases by 2" as x + 2 (not 2x)?
Expand three products and show they simplify to an integer? (Nov 22 P1 Q15 pattern)
Finish every proof with a concluding sentence?
One-Line Takeaway
Define variables generally, manipulate algebraically, conclude with a sentence. Never use a numerical example as the body of a proof — it's illustration, not evidence.
Summary
Key Points
Identity — true for every value of the variable (uses ≡). Equation — true only for specific values.
"Prove" vs "Show that" — "prove" expects an explicit concluding sentence; "show that" accepts implicit conclusion once the form is reached.
Algebraic toolkit — odd: 2n+1, even: 2n, consecutive: n, n+1, n+2, square: n², multiple of k: kn.
3-step structure — define variables → manipulate algebraically → conclude with a sentence.
Three question patterns — simplifies-to-integer (all variables cancel), prove-always (multiples, squares, parity), function behaviour (compare f(x+k) with f(x)).
Work forwards, not backwards from the given answer. Assuming the target invalidates the proof.
Concluding sentence — A1 on 3-mark questions; B1 implicit on 2-mark questions; write it either way.
MCE Framework Recap
M — Method: define your variables; translate the statement into algebra.
C — Calculate: expand, collect, factorise. One step per line. End in the form the question requires.
E — Evaluate: state the concluding sentence linking the algebra to the stated property.
Exam Tip
Every AQA proof question has the same structure: translate the English to algebra → manipulate → conclude. The marks are evenly distributed: general representation (M1), manipulation (M1), concluding sentence (A1). Never skip the concluding sentence — it's one-third of the question.
Next: Identity vs Equation