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Math lesson plan · 10th grade (Year 11 in England) · 45 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 · 5 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 = 8 cm, PQ = 32 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 = 6 cm, DB = 8 cm and DE = 9 cm. Find BC. → 21Prompt: Triangles ADE and ABC are similar (corresponding angles).
  3. Aisha is 1.6 m tall and casts a 2.4 m shadow. At the same moment a tree casts a 14 m shadow. How tall is the tree, to 1 decimal place? → 9.3Prompt: The sun's rays make two similar right triangles.

Hook · 5 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 · 10 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 · 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.
  1. Two similar shapes have lengths in the ratio 1 : 3. The smaller has area 10 cm². What is the area of the larger? → 90Prompt: Areas scale by the square of the length factor.
  2. A model car is built to a length scale of 1 : 3 compared with a bigger model. The small model holds 12 cm³ of paint inside its body. How much would the bigger model hold? → 324Prompt: Volumes scale by the cube of the length factor.

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.
  1. Two similar shapes have lengths in the ratio 1 : 5. The smaller has area 24 cm². What is the area of the larger? → 600Prompt: Areas scale by the square of the length factor.
  2. Two similar jugs have surface areas in the ratio 1 : 4. The smaller jug holds 6 ml. How much does the larger jug hold? → 48Prompt: 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/ 8

  1. Question 1

    Two similar shapes have lengths in the ratio 1 : 5. The smaller has area 24 cm². What is the area of the larger?

    Answer

  2. Question 2

    A model car is built to a length scale of 1 : 3 compared with a bigger model. The small model holds 9 cm³ of paint inside its body. How much would the bigger model hold?

    Answer

  3. Question 3

    Two similar jugs have surface areas in the ratio 4 : 9. The smaller jug holds 16 ml. How much does the larger jug hold?

    Answer

  4. Question 4

    Two similar shapes have lengths in the ratio 1 : 3. The smaller has area 8 cm². What is the area of the larger?

    Answer

  5. Question 5

    A model car is built to a length scale of 1 : 3 compared with a bigger model. The small model holds 21 cm³ of paint inside its body. How much would the bigger model hold?

    Answer

  6. Question 6

    Two similar jugs have surface areas in the ratio 1 : 9. The smaller jug holds 6 ml. How much does the larger jug hold?

    Answer

  7. Question 7

    Two similar shapes have lengths in the ratio 1 : 4. The smaller has area 27 cm². What is the area of the larger?

    Answer

  8. Question 8

    A model car is built to a length scale of 1 : 3 compared with a bigger model. The small model holds 22 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. 600

  2. 2. 243

  3. 3. 54

  4. 4. 72

  5. 5. 567

  6. 6. 162

  7. 7. 432

  8. 8. 594