algebraic proof
4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.
Lesson Overview
Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Learning Objectives
- Explain the key ideas of algebraic proof
- Apply algebraic proof to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for algebraic proof first (GCSE Revise summary)
- Key vocab to pre-teach: Equation vs Identity, Algebraic Proof
- Basic skills: reading the summary notes and answering the practice questions there
Materials & Equipment
- Exercise book, pencil, ruler
- Scientific calculator
- Pair of compasses and protractor (geometry topics)
- Graph paper
- Printed revision notes (link below)
Lesson 1: Introduction: algebraic proof
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about algebraic proof. Then check against the key terms: Equation vs Identity, Algebraic Proof. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Equation vs Identity, Algebraic Proof.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Before lesson: Read the script below. Pre-teach key vocab: Equation vs Identity, Algebraic Proof.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: algebraic proof. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
Mins 0-5 - Hook: "Today: algebraic proof. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Mathematics because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
First Look
Start with the revision notes summary, then attempt: Prove that (x + 4)² - (x + 2)² ≡ 4(2x + 3)
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about algebraic proof.
Lesson 2: Core Concepts: algebraic proof
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on algebraic proof. Check them against the notes below.
Main Content (35 minutes)
Equation vs Identity: Equation: True for specific values (e.g., 2x = 6 is true only when x = 3) Identity: True for ALL values, shown with ≡ symbol (e.g., 2(x + 3) ≡ 2x + 6)
Algebraic Proof: Using algebra to show that a mathematical statement is always true.
| Item | Detail |
|---|---|
| Any integer n | n |
| Even number | 2n |
| Odd number | 2n + 1 or 2n - 1 |
| Consecutive integers | n, n + 1, n + 2... |
| Consecutive even numbers | 2n, 2n + 2, 2n + 4... |
| Consecutive odd numbers | 2n + 1, 2n + 3, 2n + 5... |
| Multiple of 3 | 3n |
| Square of n | n² |
Practice (10 minutes)
Q: Prove that (x + 4)² - (x + 2)² ≡ 4(2x + 3)
Answer: LHS = x² + 8x + 16 - (x² + 4x + 4) = 4x + 12 = 4(2x + 3) = RHS ✓
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: algebraic proof
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: Equation vs Identity, Algebraic Proof. Define each in one sentence.
Main Content (35 minutes)
Parent/Teacher Guide: Let your student attempt each question alone first, then compare with the model answer. Award method marks for correct working even if the final answer is wrong.
Q1: Prove that (x + 4)² - (x + 2)² ≡ 4(2x + 3)
Answer: LHS = x² + 8x + 16 - (x² + 4x + 4) = 4x + 12 = 4(2x + 3) = RHS ✓
Q2: Write an expression for two consecutive odd numbers and find their sum.
Answer: 2n + 1 and 2n + 3, Sum = 4n + 4
Q3: Prove that the sum of three consecutive even numbers is divisible by 6.
Answer: Let numbers be 2n, 2n + 2, 2n + 4. Sum = 6n + 6 = 6(n + 1) ✓
Q4: Prove that (2n + 1)² - (2n - 1)² = 8n
Answer: LHS = (4n² + 4n + 1) - (4n² - 4n + 1) = 8n = RHS ✓
Q5: Find a counter-example to disprove: "All prime numbers are odd."
Answer: 2 is prime and 2 is even
Q6: Prove that the product of two consecutive integers is always even.
Answer: Let numbers be n and n + 1. One must be even (if n is odd, n + 1 is even; if n is even, n is even). Product of any even number is even ✓
Plenary (5 minutes)
Error Review
Review any questions answered incorrectly. Identify whether the error was knowledge, method, or reading the question.
Lesson 4: Exam Practice: algebraic proof
Duration: 50 minutes
Starter Activity (5 minutes)
Command Words
Review what these command words require: state (one point), describe (say what happens), explain (say why), compare (both sides), evaluate (judgement).
Main Content (35 minutes)
Extended Answer
Extended question: Extended Answer 6 marks: Prove that the difference between the squares of any two consecutive even numbers is always a multiple of 4. <div class="
Let the consecutive even numbers be 2n and 2n + 2. Their squares are (2n)² = 4n² and (2n + 2)² = 4n² + 8n + 4 Difference = (2n + 2)² - (2n)² = 4n² + 8n + 4 - 4n² = 8n + 4 = 4(2n + 1) Since 4(2n + 1) has a factor of 4 for any integer n, the difference is always a multiple of 4. ∎ Mark scheme: 1 mark for correct expressions, 1 mark for squaring, 2 marks for correct subtraction, 1 mark for factorising, 1 mark for conclusion.
Exam Tips: Use ≡ for identities, = for equations | Define your variables clearly (Let n be...) | Show every step of your working | End proofs with QED or ∎ | To disprove, find ONE counter-example | Know standard expressions: 2n (even), 2n+1 (odd)
Common Errors: Watch Out! 1. Wrong: Using n and n + 1 for consecutive odd numbers Correct: Consecutive odd numbers are 2n + 1 and 2n + 3 2. Wrong: Proving an identity by substituting one value Correct: Must show both sides are algebraically identical, or test multiple values 3. Wrong: Saying "it works for n = 1, 2, 3 so it's proven" Correct: This only shows a pattern — a proof must work for ALL values using algebra
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A student claims: "The sum of any three consecutive integers is always even." (a) Write an expression for the sum of n, n + 1 and n + 2. (b) Prove or disprove the claim. (c) What type of number is the sum always a multiple of? Answers: (a) 3n + 3 = 3(n + 1). (b) Disproved: when n = 1, sum = 6 (even); when n = 2, sum = 9 (odd). The claim is false. (c) The sum is always a multiple of 3, not always even.
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how algebraic proof connects to another Mathematics topic you have studied
- Real-world: research one real-world use or example of algebraic proof
- Critical: "What are the limitations of the models used in algebraic proof?"
Plenary (5 minutes)
Assessment Criteria
- Got it: Confident explanation + correct worked examples
- Getting there: Main points OK, needs support with detail
- Not yet: Confused on key concepts - re-run Lesson 2
Homework & Consolidation
- Consolidation: Re-answer any Lesson 3 practice questions answered incorrectly (20 mins)
- Retrieval: Write flashcards for the key terms: Equation vs Identity, Algebraic Proof (10 mins)
- Exam practice: One past-paper question on algebraic proof from the board websites (15 mins)
- Extension: Explain algebraic proof to someone else in your own words (10 mins)