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iteration

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4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.

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Lesson Overview

Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA

Learning Objectives

Prerequisites

Materials & Equipment

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.
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.
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

✅ Success Criteria

You are ready for iteration - GCSE Mathematics Lessons when you can:

  • [ ] Define the key terms in your own words
  • [ ] Explain the core concept with an example
  • [ ] Apply the method to an unfamiliar context
  • [ ] Score 80%+ on past-paper questions

Tick each item honestly — if any box is unticked, re-read the relevant section above.

Board Notes & Revision Links

This lesson is examined by: AQA 8300 | Edexcel 1MA1 | OCR J560 | Eduqas C300 | CCEA M-units. Tier and paper details vary - always check your board's current specification.

Follow-up revision: A20-iteration on GCSE Revise (notes, flashcards and auto-graded target tests for this exact topic).

📐 Reference Diagram

3 parts (£30) 5 parts (£50) Total: 8 parts = £80 → 1 part = £10 Ratio 3 : 5 → Amy gets £30, Ben gets £50

✏️ Draw-Along Instructions

Parent/teacher: Draw a bar divided into parts proportional to the ratio. Label each part with its value. Write the total above. The ratio tells you how many parts each person gets.

Display the reference diagram (or draw it live) while pupils copy. Check each label as it is added.

Homework & Consolidation

Recommended Resources

🎓 Smart Lesson (Guided)