Math lesson plan · 11th grade (Year 12 in England) · 45 minutes
Find a side with the cosine rule
Trigonometry
CCSS HSG.SRT.D.11England: GCSE Higher Geometry
Objective and success criteria
Students will: Find a side with the cosine rule.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: The cosine rule (law of cosines)
Leads to: Find an angle with the cosine rule; Decide which rule solves a triangle; Solve bearings problems with trigonometry
Materials: Scientific calculators, Protractors and rulers, Right-triangle diagrams, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (The cosine rule (law of cosines)). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- In triangle ABC, b = 20 cm, c = 11 cm and angle A = 93°. Find side a to 1 decimal place. → 23.3Prompt: Cosine rule: a² = b² + c² − 2bc cos A.
- A triangle has sides a = 12, b = 5 and c = 11. Find angle C (opposite side c) to 1 decimal place. → 66.4Prompt: Rearranged cosine rule: cos C = (a² + b² − c²) / 2ab.
- In triangle ABC, b = 11 cm, c = 11 cm and angle A = 140°. Find side a to 1 decimal place. → 20.7Prompt: Cosine rule: a² = b² + c² − 2bc cos A.
Hook · 5 min
- Show a ramp or a ladder against a wall and ask how steep it is.
- Share the objective: Find a side with the cosine rule. Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: In triangle ABC, b = 17 cm, c = 5 cm and A = 82°. Find a to 1 decimal place.
- Step 1: a² = 289 + 25 − 2 × 17 × 5 × cos 82° ≈ 290.34
- Step 2: a ≈ 17 cm
- Answer: 17
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- In triangle ABC, b = 9 cm, c = 7 cm and A = 87°. Find a to 1 decimal place. → 11.1Prompt: a² = b² + c² − 2bc cos A.
- Zara and Ethan cycle away from the same café along two straight paths with an angle of 47° between them. Zara rides 13 km and Ethan rides 8 km. How far apart are they? Give 1 decimal place. → 9.5Prompt: Sketch a triangle: two known sides and the angle between them.
You do: independent practice · 10 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- In triangle ABC, b = 13 cm, c = 7 cm and A = 52°. Find a to 1 decimal place. → 10.3Prompt: a² = b² + c² − 2bc cos A.
- Elena and Aisha cycle away from the same café along two straight paths with an angle of 124° between them. Elena rides 11 km and Aisha rides 18 km. How far apart are they? Give 1 decimal place. → 25.8Prompt: Sketch a triangle: two known sides and the angle between them.
Watch out for
- Mixing up opposite and adjacent sides.
- Calculator in radians instead of degrees.
Support and stretch
Support
- Review first: The cosine rule (law of cosines).
- Use concrete materials: scientific calculators.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Find an angle with the cosine rule; Decide which rule solves a triangle; Solve bearings problems with trigonometry.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-11.
Free lesson plan from eurekami.com/lesson-plan-builder