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similarity and trigonometry

FoundationHigherAll Boards

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 and trigonometry

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Write down everything you already know about similarity and trigonometry. Then check against the key terms: Inverse Proportion. Use a mini-whiteboard or paper.

Main Content (35 minutes)

Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Inverse Proportion.
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: similarity and trigonometry. 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: y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10.

Plenary (5 minutes)

Check Out

Your student states one thing they learned and one question they still have about similarity and trigonometry.

Lesson 2: Core Concepts: similarity and trigonometry

Duration: 50 minutes

Starter Activity (5 minutes)

Review Previous Lesson

Quick recap: write 3 key points from Lesson 1 on similarity and trigonometry. Check them against the notes below.

Main Content (35 minutes)

Inverse Proportion: X is inversely proportional to Y means X = k/Y, which is the same as X being proportional to 1/Y.
TermMeaningExample
y126

Practice (10 minutes)

Q: y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10.

Answer: y = 6

Plenary (5 minutes)

Explain Back

Your student teaches the key points back to you without looking. Fill any gaps immediately.

Lesson 3: Application: similarity and trigonometry

Duration: 50 minutes

Starter Activity (5 minutes)

Quick Recall

Recall the key terms: Inverse Proportion. 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: y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10.

Answer: y = 6

Q2: P ∝ 1/Q. When Q = 6, P = 5. Find the constant k.

Answer: k = 30

Q3: The time to paint a fence is inversely proportional to the number of painters. 3 painters take 8 hours. How long for 4 painters?

Answer: 6 hours

Q4: a is inversely proportional to b². When b = 2, a = 50. Find a when b = 5.

Answer: a = 8 (a = 200/b²)

Q5: Verify: If x doubles and y is inversely proportional to x, then y halves.

Answer: Original: y = k/x. New: y' = k/(2x) = y/2 ✓

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: similarity and trigonometry

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: Two similar pyramids have base areas of 20 cm² and 45 cm². The height of the smaller pyramid is 8 cm. (a) Find the height of the larger pyramid. (b) The smaller pyramid has volume 64 cm³. Find the volume of the larger pyramid. (c) Show that the volume scale factor equals the linear scale factor cubed. <div class="

(a) Area ratio = 45/20 = 2.25. Linear scale factor = √2.25 = 1.5. Height = 8 × 1.5 = 12 cm. (b) Volume scale factor = 1.5³ = 3.375. Volume = 64 × 3.375 = 216 cm³. (c) Linear SF = 1.5. Linear SF³ = 1.5³ = 3.375. Volume ratio = 216/64 = 3.375. They are equal ✓ Mark scheme: M1 area ratio, M1 square root for linear SF, A1 height = 12cm, M1 volume SF, A1 216cm³, A1 verification shown

Exam Tips: Inverse proportion: one increases as other decreases | Use XY = k to find the constant quickly | If x doubles, y halves (and vice versa) | Graph is a curve, not a straight line | Check: product of values should be constant
Common Errors: Watch Out! 1. Wrong: Using the linear scale factor for area: if lengths double, area doubles too Correct: If lengths double (scale factor 2), area multiplies by 2² = 4 and volume by 2³ = 8. 2. Wrong: Mixing up opposite and adjacent sides when using trigonometry Correct: Label sides relative to the angle: opposite is across from the angle, adjacent is next to it (not the hypotenuse). 3. Wrong: Using the area scale factor to find a length Correct: To find a length from an area ratio, take the square root. If area ratio = 9, length factor = √9 = 3.
AO3 - Reasoning & Interpretation: Reasoning and Interpretation A model car is built at 1:24 scale. The real car is 4.2 m long. (a) How long is the model car in cm? (b) The model has a surface area of 350 cm². Estimate the real car's surface area in m². (c) A student says "If the model weighs 0.8 kg, the real car weighs 0.8 × 24 = 19.2 kg." Why is this wrong? Answers: (a) 4.2 m ÷ 24 = 0.175 m = 17.5 cm. (b) Area SF = 24² = 576. Real area = 350 × 576 = 201,600 cm² = 20.16 m². (c) Weight depends on volume, not length. Volume SF = 24³ = 13,824. Real weight ≈ 0.8 × 13,824 = 11,059 kg ≈ 11 tonnes. The student used the linear SF instead of the volume SF.
Stretch & Challenge (Grade 8-9):
  • Synoptic links: explain how similarity and trigonometry connects to another Mathematics topic you have studied
  • Real-world: research one real-world use or example of similarity and trigonometry
  • Critical: "What are the limitations of the models used in similarity and trigonometry?"

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

Recommended Resources

🎓 Smart Lesson (Guided)