Homeschool Guide: These lesson plans are a guide for parents. Content may contain errors — always cross-reference with official exam board specifications.

R13: Equations Proportion

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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: Equations Proportion

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

3 quick questions to activate prior knowledge about equations proportion. Use mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read script below. Prepare materials. Pre-teach key vocab: R13.
If stuck: See Troubleshooting below. Break into smaller steps.
Extension: See Stretch & Challenge section.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: Equations Proportion. Goal: by the end you can y ∝ x². When x = 5, y = 75. Find y when x = 8, using the method shown step by step. Connects to proportion (R7/R10) because you'll reuse its method throughout today's work."
Mins 5-15 - Direct Instruction: 1) Define: "Equations Proportion is..." 2) Key principle: "Most important:..." 3) Draw diagram, label live. 4) Real example: "Like y ∝ x². When x = 5, y = 75. Find y when x = 8. - the same maths you already do when shopping or cooking..."
Mins 15-25 - Guided Practice: Ex1: Teacher models. Ex2: Student tries with guidance. Ex3: Student independent.
Mins 25-35 - Independent Practice: 3-5 questions. Immediate feedback.
Worked Example 1: Basic

Q: y ∝ x². When x = 5, y = 75. Find y when x = 8.

  1. What is asked? y
  2. Recall knowledge: Models: y = kx, kx², kx³, k√x, k/x, k/x²; Find k by substituting the given pair — powers FIRST
  3. Apply: y = kx²: 75 = 25k → k = 3 → y = 3 × 64 = 192
  4. Check: Check the units named in the question and sense-check the size of your answer.
  5. Answer: 192

Tip: "Show working - method marks > answer marks."

Worked Example 2: Multi-step

Q: v ∝ 1/w. When w = 4, v = 9. Find w when v = 6.

  1. Break down: Separate parts?
  2. Solve each: v = k/w: 9 = k/4 → k = 36 → 6 = 36/w → w = 6
  3. Combine: The parts chain: each answer feeds the next step of the question.
  4. Final answer: M1 for k = 36, A1 for w = 6. Reverse direction: solve the same equation for the other letter.
Worked Example 3: Data/Application

Q: v ∝ 1/w. When w = 4, v = 9. Find w when v = 6.

  1. Read data: Axes, units, scale
  2. Patterns: Trends, anomalies
  3. Explain: Why? Using equations proportion
  4. Conclude: Evidence-based

Common Questions & Troubleshooting

Q: "Why does Models: y = kx, kx², kx³, k√x, k/x, k/x² work this way?"
A: Analogy: "y ∝ x². When x = 5, y = 75. Find y when x = 8.". Try re-draw the worked example from memory, then check it against the model answer.
Q: "Wrong answers on calculations."
A: Checklist: units? formula? calculator mode? sig figs?
Q: "Can't remember key terms."
A: Flashcards (term front, def+diagram back). Spaced repetition: day 1, 3, 7.
Q: "Exam wording confuses me."
A: Commands: Describe=what you see; Explain=reasons; Evaluate=pros/cons+judgement; Calculate=working+units.
Q: "Running out of time."
A: 1 mark/min. Skip hard, return. Attempt all - partial marks count.

Assessment & Model Answers

Mastery Checklist:
  • [ ] Define Equations Proportion in own words
  • [ ] Identify features in diagram
  • [ ] Explain process step-by-step
  • [ ] Apply to unfamiliar context
  • [ ] Score 80%+ on past paper question

Model Answer: Past Paper Question

Q: y ∝ x². When x = 5, y = 75. Find y when x = 8.

  • 1 mark: [Key terminology]
  • 1 mark: [Mechanism/process]
  • 1 mark: [Application/example]
  • 1 mark: [Evaluation - Higher]

Model response: "[Complete answer with expected detail/terminology]"

Why full marks: [What makes this strong]

Stretch & Challenge (Grade 8-9):
  • Synoptic links: Connect to R12-similarity-trigonometry / R14-graphs-rates-change
  • Real-world: Research use in industry, medicine or research
  • Advanced calc: Try a more challenging variant
  • Critical: "Evaluate limitations of today's model"
  • Research: Recent discovery on equations proportion, summarise 100 words

Plenary (5 mins)

Label the Diagram

Unlabelled diagram - write all parts in 1 min

Assessment Criteria

Lesson 2: Core Concepts: Equations Proportion

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from last lesson on equations proportion. Use mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read script below. Prepare materials. Pre-teach key vocab: R13.
If stuck: See Troubleshooting below. Break into smaller steps.
Extension: See Stretch & Challenge section.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: Equations Proportion. Goal: by the end you can y ∝ x². When x = 5, y = 75. Find y when x = 8, using the method shown step by step. Connects to proportion (R7/R10) because you'll reuse its method throughout today's work."
Mins 5-15 - Direct Instruction: 1) Define: "Equations Proportion is..." 2) Key principle: "Most important:..." 3) Draw diagram, label live. 4) Real example: "Like y ∝ x². When x = 5, y = 75. Find y when x = 8. - the same maths you already do when shopping or cooking..."
Mins 15-25 - Guided Practice: Ex1: Teacher models. Ex2: Student tries with guidance. Ex3: Student independent.
Mins 25-35 - Independent Practice: 3-5 questions. Immediate feedback.
Worked Example 1: Basic

Q: y ∝ x². When x = 5, y = 75. Find y when x = 8.

  1. What is asked? y
  2. Recall knowledge: Models: y = kx, kx², kx³, k√x, k/x, k/x²; Find k by substituting the given pair — powers FIRST
  3. Apply: y = kx²: 75 = 25k → k = 3 → y = 3 × 64 = 192
  4. Check: Check the units named in the question and sense-check the size of your answer.
  5. Answer: 192

Tip: "Show working - method marks > answer marks."

Worked Example 2: Multi-step

Q: v ∝ 1/w. When w = 4, v = 9. Find w when v = 6.

  1. Break down: Separate parts?
  2. Solve each: v = k/w: 9 = k/4 → k = 36 → 6 = 36/w → w = 6
  3. Combine: The parts chain: each answer feeds the next step of the question.
  4. Final answer: M1 for k = 36, A1 for w = 6. Reverse direction: solve the same equation for the other letter.
Worked Example 3: Data/Application

Q: v ∝ 1/w. When w = 4, v = 9. Find w when v = 6.

  1. Read data: Axes, units, scale
  2. Patterns: Trends, anomalies
  3. Explain: Why? Using equations proportion
  4. Conclude: Evidence-based

Common Questions & Troubleshooting

Q: "Why does Models: y = kx, kx², kx³, k√x, k/x, k/x² work this way?"
A: Analogy: "y ∝ x². When x = 5, y = 75. Find y when x = 8.". Try re-draw the worked example from memory, then check it against the model answer.
Q: "Wrong answers on calculations."
A: Checklist: units? formula? calculator mode? sig figs?
Q: "Can't remember key terms."
A: Flashcards (term front, def+diagram back). Spaced repetition: day 1, 3, 7.
Q: "Exam wording confuses me."
A: Commands: Describe=what you see; Explain=reasons; Evaluate=pros/cons+judgement; Calculate=working+units.
Q: "Running out of time."
A: 1 mark/min. Skip hard, return. Attempt all - partial marks count.

Assessment & Model Answers

Mastery Checklist:
  • [ ] Define Equations Proportion in own words
  • [ ] Identify features in diagram
  • [ ] Explain process step-by-step
  • [ ] Apply to unfamiliar context
  • [ ] Score 80%+ on past paper question

Model Answer: Past Paper Question

Q: y ∝ x². When x = 5, y = 75. Find y when x = 8.

  • 1 mark: [Key terminology]
  • 1 mark: [Mechanism/process]
  • 1 mark: [Application/example]
  • 1 mark: [Evaluation - Higher]

Model response: "[Complete answer with expected detail/terminology]"

Why full marks: [What makes this strong]

Stretch & Challenge (Grade 8-9):
  • Synoptic links: Connect to R12-similarity-trigonometry / R14-graphs-rates-change
  • Real-world: Research use in industry, medicine or research
  • Advanced calc: Try a more challenging variant
  • Critical: "Evaluate limitations of today's model"
  • Research: Recent discovery on equations proportion, summarise 100 words

Plenary (5 mins)

Label the Diagram

Unlabelled diagram - write all parts in 1 min

Assessment Criteria

Lesson 3: Application: Equations Proportion

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from last lesson on equations proportion. Use mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read script below. Prepare materials. Pre-teach key vocab: R13.
If stuck: See Troubleshooting below. Break into smaller steps.
Extension: See Stretch & Challenge section.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: Equations Proportion. Goal: by the end you can y ∝ x². When x = 5, y = 75. Find y when x = 8, using the method shown step by step. Connects to proportion (R7/R10) because you'll reuse its method throughout today's work."
Mins 5-15 - Direct Instruction: 1) Define: "Equations Proportion is..." 2) Key principle: "Most important:..." 3) Draw diagram, label live. 4) Real example: "Like y ∝ x². When x = 5, y = 75. Find y when x = 8. - the same maths you already do when shopping or cooking..."
Mins 15-25 - Guided Practice: Ex1: Teacher models. Ex2: Student tries with guidance. Ex3: Student independent.
Mins 25-35 - Independent Practice: 3-5 questions. Immediate feedback.
Worked Example 1: Basic

Q: y ∝ x². When x = 5, y = 75. Find y when x = 8.

  1. What is asked? y
  2. Recall knowledge: Models: y = kx, kx², kx³, k√x, k/x, k/x²; Find k by substituting the given pair — powers FIRST
  3. Apply: y = kx²: 75 = 25k → k = 3 → y = 3 × 64 = 192
  4. Check: Check the units named in the question and sense-check the size of your answer.
  5. Answer: 192

Tip: "Show working - method marks > answer marks."

Worked Example 2: Multi-step

Q: v ∝ 1/w. When w = 4, v = 9. Find w when v = 6.

  1. Break down: Separate parts?
  2. Solve each: v = k/w: 9 = k/4 → k = 36 → 6 = 36/w → w = 6
  3. Combine: The parts chain: each answer feeds the next step of the question.
  4. Final answer: M1 for k = 36, A1 for w = 6. Reverse direction: solve the same equation for the other letter.
Worked Example 3: Data/Application

Q: v ∝ 1/w. When w = 4, v = 9. Find w when v = 6.

  1. Read data: Axes, units, scale
  2. Patterns: Trends, anomalies
  3. Explain: Why? Using equations proportion
  4. Conclude: Evidence-based

Common Questions & Troubleshooting

Q: "Why does Models: y = kx, kx², kx³, k√x, k/x, k/x² work this way?"
A: Analogy: "y ∝ x². When x = 5, y = 75. Find y when x = 8.". Try re-draw the worked example from memory, then check it against the model answer.
Q: "Wrong answers on calculations."
A: Checklist: units? formula? calculator mode? sig figs?
Q: "Can't remember key terms."
A: Flashcards (term front, def+diagram back). Spaced repetition: day 1, 3, 7.
Q: "Exam wording confuses me."
A: Commands: Describe=what you see; Explain=reasons; Evaluate=pros/cons+judgement; Calculate=working+units.
Q: "Running out of time."
A: 1 mark/min. Skip hard, return. Attempt all - partial marks count.

Assessment & Model Answers

Mastery Checklist:
  • [ ] Define Equations Proportion in own words
  • [ ] Identify features in diagram
  • [ ] Explain process step-by-step
  • [ ] Apply to unfamiliar context
  • [ ] Score 80%+ on past paper question

Model Answer: Past Paper Question

Q: y ∝ x². When x = 5, y = 75. Find y when x = 8.

  • 1 mark: [Key terminology]
  • 1 mark: [Mechanism/process]
  • 1 mark: [Application/example]
  • 1 mark: [Evaluation - Higher]

Model response: "[Complete answer with expected detail/terminology]"

Why full marks: [What makes this strong]

Stretch & Challenge (Grade 8-9):
  • Synoptic links: Connect to R12-similarity-trigonometry / R14-graphs-rates-change
  • Real-world: Research use in industry, medicine or research
  • Advanced calc: Try a more challenging variant
  • Critical: "Evaluate limitations of today's model"
  • Research: Recent discovery on equations proportion, summarise 100 words

Plenary (5 mins)

Exam Question Speed Round

5 quick questions, 1 min each. Mark together.

Assessment Criteria

Lesson 4: Exam Practice: Equations Proportion

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from last lesson on equations proportion. Use mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read script below. Prepare materials. Pre-teach key vocab: R13.
If stuck: See Troubleshooting below. Break into smaller steps.
Extension: See Stretch & Challenge section.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: Equations Proportion. Goal: by the end you can y ∝ x². When x = 5, y = 75. Find y when x = 8, using the method shown step by step. Connects to proportion (R7/R10) because you'll reuse its method throughout today's work."
Mins 5-15 - Direct Instruction: 1) Define: "Equations Proportion is..." 2) Key principle: "Most important:..." 3) Draw diagram, label live. 4) Real example: "Like y ∝ x². When x = 5, y = 75. Find y when x = 8. - the same maths you already do when shopping or cooking..."
Mins 15-25 - Guided Practice: Ex1: Teacher models. Ex2: Student tries with guidance. Ex3: Student independent.
Mins 25-35 - Independent Practice: 3-5 questions. Immediate feedback.
Worked Example 1: Basic

Q: y ∝ x². When x = 5, y = 75. Find y when x = 8.

  1. What is asked? y
  2. Recall knowledge: Models: y = kx, kx², kx³, k√x, k/x, k/x²; Find k by substituting the given pair — powers FIRST
  3. Apply: y = kx²: 75 = 25k → k = 3 → y = 3 × 64 = 192
  4. Check: Check the units named in the question and sense-check the size of your answer.
  5. Answer: 192

Tip: "Show working - method marks > answer marks."

Worked Example 2: Multi-step

Q: v ∝ 1/w. When w = 4, v = 9. Find w when v = 6.

  1. Break down: Separate parts?
  2. Solve each: v = k/w: 9 = k/4 → k = 36 → 6 = 36/w → w = 6
  3. Combine: The parts chain: each answer feeds the next step of the question.
  4. Final answer: M1 for k = 36, A1 for w = 6. Reverse direction: solve the same equation for the other letter.
Worked Example 3: Data/Application

Q: v ∝ 1/w. When w = 4, v = 9. Find w when v = 6.

  1. Read data: Axes, units, scale
  2. Patterns: Trends, anomalies
  3. Explain: Why? Using equations proportion
  4. Conclude: Evidence-based

Common Questions & Troubleshooting

Q: "Why does Models: y = kx, kx², kx³, k√x, k/x, k/x² work this way?"
A: Analogy: "y ∝ x². When x = 5, y = 75. Find y when x = 8.". Try re-draw the worked example from memory, then check it against the model answer.
Q: "Wrong answers on calculations."
A: Checklist: units? formula? calculator mode? sig figs?
Q: "Can't remember key terms."
A: Flashcards (term front, def+diagram back). Spaced repetition: day 1, 3, 7.
Q: "Exam wording confuses me."
A: Commands: Describe=what you see; Explain=reasons; Evaluate=pros/cons+judgement; Calculate=working+units.
Q: "Running out of time."
A: 1 mark/min. Skip hard, return. Attempt all - partial marks count.

Assessment & Model Answers

Mastery Checklist:
  • [ ] Define Equations Proportion in own words
  • [ ] Identify features in diagram
  • [ ] Explain process step-by-step
  • [ ] Apply to unfamiliar context
  • [ ] Score 80%+ on past paper question

Model Answer: Past Paper Question

Q: y ∝ x². When x = 5, y = 75. Find y when x = 8.

  • 1 mark: [Key terminology]
  • 1 mark: [Mechanism/process]
  • 1 mark: [Application/example]
  • 1 mark: [Evaluation - Higher]

Model response: "[Complete answer with expected detail/terminology]"

Why full marks: [What makes this strong]

Stretch & Challenge (Grade 8-9):
  • Synoptic links: Connect to R12-similarity-trigonometry / R14-graphs-rates-change
  • Real-world: Research use in industry, medicine or research
  • Advanced calc: Try a more challenging variant
  • Critical: "Evaluate limitations of today's model"
  • Research: Recent discovery on equations proportion, summarise 100 words

Plenary (5 mins)

Exam Question Speed Round

5 quick questions, 1 min each. Mark together.

Assessment Criteria

📝 Mark Scheme Guidance

How examiners mark R13: Read the command word carefully — State = one word/phrase (1 mark), Describe = pattern or detail (2 marks), Explain = reason with "because" (2–3 marks), Compare = similarities AND differences, Evaluate = both sides + conclusion. Method marks are awarded for correct working even if the final answer is wrong. Always show your working — unexplained answers score zero for AO2.

Success on R13 questions: Underline the command word, count the marks (one mark per point expected), and answer in the format the command word demands.

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: R7-proportion 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)