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

R12: Similarity Trigonometry

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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: Similarity Trigonometry

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

3 quick questions to activate prior knowledge about similarity trigonometry. Use mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read script below. Prepare materials. Pre-teach key vocab: R12.
If stuck: See Troubleshooting below. Break into smaller steps.
Extension: See Stretch & Challenge section.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: Similarity Trigonometry. Goal: by the end you can y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2, using the method shown step by step. Connects to direct proportion (R7/R10) because you'll reuse its method throughout today's work."
Mins 5-15 - Direct Instruction: 1) Define: "Similarity Trigonometry is..." 2) Key principle: "Most important:..." 3) Draw diagram, label live. 4) Real example: "Like y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2. - 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 is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.

  1. What is asked? y
  2. Recall knowledge: Inverse: y = k/x · inverse square: y = k/x² · find k from ONE data pair; Inverse intuition: as x grows, y SHRINKS proportionally
  3. Apply: y = k/x → 12 = k/5 → k = 60 y = 60/2 = 30
  4. Check: Check the units named in the question and sense-check the size of your answer.
  5. Answer: 30

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

Worked Example 2: Multi-step

Q: d is inversely proportional to s². When s = 2, d = 45. Find d when s = 3.

  1. Break down: Separate parts?
  2. Solve each: d = k/s² → 45 = k/4 → k = 180 d = 180/9 = 20
  3. Combine: The parts chain: each answer feeds the next step of the question.
  4. Final answer: M1 for k = 180, A1 for 20. Squared inverse: tripling s divides d by NINE — the exam's favourite twist.
Worked Example 3: Data/Application

Q: d is inversely proportional to s². When s = 2, d = 45. Find d when s = 3.

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

Common Questions & Troubleshooting

Q: "Why does Inverse: y = k/x · inverse square: y = k/x² · find k from ON work this way?"
A: Analogy: "y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.". 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 Similarity Trigonometry 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 is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.

  • 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 R11-compound-measures / R13-equations-proportion
  • 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 similarity trigonometry, summarise 100 words

Plenary (5 mins)

Label the Diagram

Unlabelled diagram - write all parts in 1 min

Assessment Criteria

Lesson 2: Core Concepts: Similarity Trigonometry

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

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

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read script below. Prepare materials. Pre-teach key vocab: R12.
If stuck: See Troubleshooting below. Break into smaller steps.
Extension: See Stretch & Challenge section.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: Similarity Trigonometry. Goal: by the end you can y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2, using the method shown step by step. Connects to direct proportion (R7/R10) because you'll reuse its method throughout today's work."
Mins 5-15 - Direct Instruction: 1) Define: "Similarity Trigonometry is..." 2) Key principle: "Most important:..." 3) Draw diagram, label live. 4) Real example: "Like y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2. - 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 is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.

  1. What is asked? y
  2. Recall knowledge: Inverse: y = k/x · inverse square: y = k/x² · find k from ONE data pair; Inverse intuition: as x grows, y SHRINKS proportionally
  3. Apply: y = k/x → 12 = k/5 → k = 60 y = 60/2 = 30
  4. Check: Check the units named in the question and sense-check the size of your answer.
  5. Answer: 30

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

Worked Example 2: Multi-step

Q: d is inversely proportional to s². When s = 2, d = 45. Find d when s = 3.

  1. Break down: Separate parts?
  2. Solve each: d = k/s² → 45 = k/4 → k = 180 d = 180/9 = 20
  3. Combine: The parts chain: each answer feeds the next step of the question.
  4. Final answer: M1 for k = 180, A1 for 20. Squared inverse: tripling s divides d by NINE — the exam's favourite twist.
Worked Example 3: Data/Application

Q: d is inversely proportional to s². When s = 2, d = 45. Find d when s = 3.

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

Common Questions & Troubleshooting

Q: "Why does Inverse: y = k/x · inverse square: y = k/x² · find k from ON work this way?"
A: Analogy: "y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.". 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 Similarity Trigonometry 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 is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.

  • 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 R11-compound-measures / R13-equations-proportion
  • 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 similarity trigonometry, summarise 100 words

Plenary (5 mins)

Label the Diagram

Unlabelled diagram - write all parts in 1 min

Assessment Criteria

Lesson 3: Application: Similarity Trigonometry

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

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

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read script below. Prepare materials. Pre-teach key vocab: R12.
If stuck: See Troubleshooting below. Break into smaller steps.
Extension: See Stretch & Challenge section.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: Similarity Trigonometry. Goal: by the end you can y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2, using the method shown step by step. Connects to direct proportion (R7/R10) because you'll reuse its method throughout today's work."
Mins 5-15 - Direct Instruction: 1) Define: "Similarity Trigonometry is..." 2) Key principle: "Most important:..." 3) Draw diagram, label live. 4) Real example: "Like y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2. - 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 is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.

  1. What is asked? y
  2. Recall knowledge: Inverse: y = k/x · inverse square: y = k/x² · find k from ONE data pair; Inverse intuition: as x grows, y SHRINKS proportionally
  3. Apply: y = k/x → 12 = k/5 → k = 60 y = 60/2 = 30
  4. Check: Check the units named in the question and sense-check the size of your answer.
  5. Answer: 30

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

Worked Example 2: Multi-step

Q: d is inversely proportional to s². When s = 2, d = 45. Find d when s = 3.

  1. Break down: Separate parts?
  2. Solve each: d = k/s² → 45 = k/4 → k = 180 d = 180/9 = 20
  3. Combine: The parts chain: each answer feeds the next step of the question.
  4. Final answer: M1 for k = 180, A1 for 20. Squared inverse: tripling s divides d by NINE — the exam's favourite twist.
Worked Example 3: Data/Application

Q: d is inversely proportional to s². When s = 2, d = 45. Find d when s = 3.

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

Common Questions & Troubleshooting

Q: "Why does Inverse: y = k/x · inverse square: y = k/x² · find k from ON work this way?"
A: Analogy: "y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.". 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 Similarity Trigonometry 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 is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.

  • 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 R11-compound-measures / R13-equations-proportion
  • 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 similarity trigonometry, summarise 100 words

Plenary (5 mins)

Exam Question Speed Round

5 quick questions, 1 min each. Mark together.

Assessment Criteria

Lesson 4: Exam Practice: Similarity Trigonometry

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

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

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read script below. Prepare materials. Pre-teach key vocab: R12.
If stuck: See Troubleshooting below. Break into smaller steps.
Extension: See Stretch & Challenge section.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: Similarity Trigonometry. Goal: by the end you can y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2, using the method shown step by step. Connects to direct proportion (R7/R10) because you'll reuse its method throughout today's work."
Mins 5-15 - Direct Instruction: 1) Define: "Similarity Trigonometry is..." 2) Key principle: "Most important:..." 3) Draw diagram, label live. 4) Real example: "Like y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2. - 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 is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.

  1. What is asked? y
  2. Recall knowledge: Inverse: y = k/x · inverse square: y = k/x² · find k from ONE data pair; Inverse intuition: as x grows, y SHRINKS proportionally
  3. Apply: y = k/x → 12 = k/5 → k = 60 y = 60/2 = 30
  4. Check: Check the units named in the question and sense-check the size of your answer.
  5. Answer: 30

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

Worked Example 2: Multi-step

Q: d is inversely proportional to s². When s = 2, d = 45. Find d when s = 3.

  1. Break down: Separate parts?
  2. Solve each: d = k/s² → 45 = k/4 → k = 180 d = 180/9 = 20
  3. Combine: The parts chain: each answer feeds the next step of the question.
  4. Final answer: M1 for k = 180, A1 for 20. Squared inverse: tripling s divides d by NINE — the exam's favourite twist.
Worked Example 3: Data/Application

Q: d is inversely proportional to s². When s = 2, d = 45. Find d when s = 3.

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

Common Questions & Troubleshooting

Q: "Why does Inverse: y = k/x · inverse square: y = k/x² · find k from ON work this way?"
A: Analogy: "y is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.". 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 Similarity Trigonometry 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 is inversely proportional to x. When x = 5, y = 12. Find y when x = 2.

  • 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 R11-compound-measures / R13-equations-proportion
  • 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 similarity trigonometry, 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 R12: 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 R12 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: R12-similarity-trigonometry 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)