iteration
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 iteration
- Apply iteration to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for iteration first (GCSE Revise summary)
- Key vocab to pre-teach: Iteration, Recursive formula
- 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: iteration
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about iteration. Then check against the key terms: Iteration, Recursive formula. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Iteration, Recursive formula.
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: Iteration, Recursive formula.
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: iteration. 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: iteration. 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: Rearrange x² + 2x - 4 = 0 into the form x = √(4 - 2x).
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about iteration.
Lesson 2: Core Concepts: iteration
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on iteration. Check them against the notes below.
Main Content (35 minutes)
Iteration: A method for finding approximate solutions to equations by repeatedly applying a formula.
Recursive formula: Uses previous values to find new values. Written with suffix notation: x₁, x₂, x₃, etc.
Practice (10 minutes)
Q: Rearrange x² + 2x - 4 = 0 into the form x = √(4 - 2x).
Answer: x² + 2x = 4 → x² = 4 - 2x → x = √(4 - 2x) (when x > 0)
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: iteration
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: Iteration, Recursive formula. 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: Rearrange x² + 2x - 4 = 0 into the form x = √(4 - 2x).
Answer: x² + 2x = 4 → x² = 4 - 2x → x = √(4 - 2x) (when x > 0)
Q2: Use x n+1 = 4 ⁄ x n - 2 with x₁ = 1. Find x₂ and x₃.
Answer: x₂ = 2, x₃ = 0
Q3: Use x n+1 = √(x n + 2) with x₁ = 2. Find x₂, x₃, x₄.
Answer: x₂ = 2, x₃ = 2, x₄ = 2 (converges)
Q4: Find a root of x³ - 2x - 3 = 0 using x n+1 = ³√(2x n + 3), x₁ = 2, to 2 d.p.
Answer: x ≈ 1.89
Q5: Write down an iterative formula from x = 5 - x³.
Answer: x n+1 = 5 - x n ³
Q6: A sequence converges to 1.618. What equation might it be solving?
Answer: x² = x + 1 (golden ratio) or similar
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: iteration
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: The equation x² + 2x - 5 = 0 has a root between 1 and 2. (a) Show that x = √(5 - 2x) is a rearrangement. (b) Use the iterative formula x n+1 = √(5 - 2x n ) with x₁ = 1.5 to find x₂, x₃, x₄, x₅. (c) Find the root correct to 2 decimal places. <div class="
(a) x² + 2x - 5 = 0 → x² = 5 - 2x → x = √(5 - 2x) ✓ (positive root between 1 and 2) (b) x₂ = √(5 - 3) = √2 = 1.414... x₃ = √(5 - 2.828) = √2.172 = 1.474... x₄ = √(5 - 2.948) = √2.052 = 1.433... x₅ = √(5 - 2.866) = √2.134 = 1.461... (c) Continuing: the values converge to approximately x = 1.45 (2 d.p.) Mark scheme: (a) 1 mark. (b) 3 marks for iterations (allow small rounding differences). (c) 2 marks for correct root.
Exam Tips: Always show your working with each iteration step | Use ANS button on calculator for accuracy | Stop when values are the same to required accuracy | If diverging, try a different rearrangement | Check your answer in the original equation
Common Errors: Watch Out! 1. Wrong: Rounding intermediate values instead of using full calculator display Correct: Use the ANS button to keep full accuracy between iterations 2. Wrong: Stopping too early before values have converged Correct: Continue until consecutive values are the same to the required decimal places 3. Wrong: Assuming all rearrangements will converge Correct: Some rearrangements diverge — if values oscillate wildly, try a different form
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A population is modelled by P n+1 = 0.8P n + 100, where P n is population in year n. (a) If P₁ = 200, find P₂, P₃, P₄. (b) What does the 100 represent in context? (c) The population approaches a stable value. Find this value. Answers: (a) P₂ = 0.8(200) + 100 = 260. P₃ = 0.8(260) + 100 = 308. P₄ = 0.8(308) + 100 = 346.4. (b) 100 represents annual immigration/new arrivals. (c) When P n+1 = P n = P: P = 0.8P + 100 → 0.2P = 100 → P = 500.
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how iteration connects to another Mathematics topic you have studied
- Real-world: research one real-world use or example of iteration
- Critical: "What are the limitations of the models used in iteration?"
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: Iteration, Recursive formula (10 mins)
- Exam practice: One past-paper question on iteration from the board websites (15 mins)
- Extension: Explain iteration to someone else in your own words (10 mins)