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Math lesson plan · 10th grade (Year 11 in England) · 30 minutes

Use length, area and volume scale factors

Geometry

CCSS HSG.GMD.A.3CCSS 7.G.A.1England: GCSE Higher Geometry

Objective and success criteria

Students will: Use length, area and volume scale factors.

  • 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: Find missing lengths in similar triangles

Leads to: Find the volume of a frustum

Materials: Rulers and protractors, Squared or dotted paper, Shape cut-outs, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 4 min

  • Quick retrieval from earlier learning (Find missing lengths in similar triangles). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Triangles ABC and PQR are similar, with AB matching PQ and BC matching QR. AB = 6 cm, PQ = 24 cm and BC = 7 cm. Find QR. → 28Prompt: Find the scale factor from the matching pair you know.
  2. In triangle ABC, D is on AB and E is on AC with DE parallel to BC. AD = 5 cm, DB = 9 cm and DE = 8 cm. Find BC to 1 decimal place. → 22.4Prompt: Triangles ADE and ABC are similar (corresponding angles).
  3. Daniel is 1.2 m tall and casts a 2 m shadow. At the same moment a tree casts a 10 m shadow. How tall is the tree, to 1 decimal place? → 6Prompt: The sun's rays make two similar right triangles.

Hook · 3 min

  • Show a photo of a building, a tile pattern or a road sign and ask what shapes and angles students can spot.
  • Share the objective: Use length, area and volume scale factors. Read the success criteria together.

I do: model it · 7 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: Two similar shapes have lengths in the ratio 1 : 3. The smaller has area 14 cm². What is the area of the larger?
  • Step 1: 14 × 3² = 14 × 9 = 126 cm²
  • Answer: 126
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 6 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. Two similar shapes have lengths in the ratio 1 : 2. The smaller has area 11 cm². What is the area of the larger? → 44Prompt: Areas scale by the square of the length factor.
  2. A model car is built to a length scale of 1 : 2 compared with a bigger model. The small model holds 15 cm³ of paint inside its body. How much would the bigger model hold? → 120Prompt: Volumes scale by the cube of the length factor.

You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.

Exit ticket · 3 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. A model car is built to a length scale of 1 : 2 compared with a bigger model. The small model holds 12 cm³ of paint inside its body. How much would the bigger model hold? → 96Prompt: Volumes scale by the cube of the length factor.
  2. Two similar jugs have surface areas in the ratio 4 : 25. The smaller jug holds 48 ml. How much does the larger jug hold? → 750Prompt: Square-root the area ratio to get the length ratio.

Watch out for

  • Thinking a square turned on its corner is no longer a square.
  • Reading the wrong scale on a protractor.

Support and stretch

Support

  • Review first: Find missing lengths in similar triangles.
  • Use concrete materials: rulers and protractors.
  • 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 the volume of a frustum.
  • 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-10.

Free lesson plan from eurekami.com/lesson-plan-builder

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Use length, area and volume scale factors

Name

Date

Score/ 6

  1. Question 1

    Two similar shapes have lengths in the ratio 1 : 2. The smaller has area 24 cm². What is the area of the larger?

    Answer

  2. Question 2

    Two similar jugs have surface areas in the ratio 4 : 25. The smaller jug holds 24 ml. How much does the larger jug hold?

    Answer

  3. Question 3

    A model car is built to a length scale of 1 : 4 compared with a bigger model. The small model holds 28 cm³ of paint inside its body. How much would the bigger model hold?

    Answer

  4. Question 4

    Two similar shapes have lengths in the ratio 1 : 5. The smaller has area 27 cm². What is the area of the larger?

    Answer

  5. Question 5

    Two similar jugs have surface areas in the ratio 9 : 16. The smaller jug holds 54 ml. How much does the larger jug hold?

    Answer

  6. Question 6

    A model car is built to a length scale of 1 : 4 compared with a bigger model. The small model holds 26 cm³ of paint inside its body. How much would the bigger model hold?

    Answer

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Answer key: Use length, area and volume scale factors

  1. 1. 96

  2. 2. 375

  3. 3. 1792

  4. 4. 675

  5. 5. 128

  6. 6. 1664