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Math lesson plan · 8th grade (Year 9 in England) · 45 minutes

Pythagoras in real life: ladders, screens, shortcuts and boxes

Geometry

CCSS 8.G.B.7England: KS3 Geometry and measuresEngland: GCSE Foundation Geometry and measures

Objective and success criteria

Students will: Pythagoras in real life: ladders, screens, shortcuts and boxes.

  • 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: Use Pythagoras to find a shorter side

Leads to: the next skill in this area.

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 (Use Pythagoras to find a shorter side). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. A right triangle has hypotenuse 11 cm and one leg 8 cm. Find the other leg, to 1 decimal place. → 7.5Prompt: Subtract the square of the known leg from the square of the hypotenuse.
  2. A right triangle has hypotenuse 15 cm and one leg 10 cm. Find the other leg, to 1 decimal place. → 11.2Prompt: Subtract the square of the known leg from the square of the hypotenuse.
  3. A right triangle has hypotenuse 19 cm and one leg 15 cm. Find the other leg, to 1 decimal place. → 11.7Prompt: Subtract the square of the known leg from the square of the hypotenuse.

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: Pythagoras in real life: ladders, screens, shortcuts and boxes. 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: A rectangular park is 110 m long and 67 m wide. Instead of walking along two sides, you cut across the diagonal. How many metres shorter is the shortcut? Round to the nearest metre.
  • Step 1: √(110² + 67²) ≈ 128.8 m
  • Step 2: 177 − 128.8 ≈ 48 m
  • Answer: 48
  • 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. A rectangular park is 49 m long and 72 m wide. Instead of walking along two sides, you cut across the diagonal. How many metres shorter is the shortcut? Round to the nearest metre. → 34Prompt: Find the diagonal with Pythagoras.
  2. A box is 4 cm by 6 cm by 12 cm. What is the length of the longest straight rod that fits inside, from one corner to the opposite corner, in cm? → 14Prompt: First find the diagonal of the base with Pythagoras.

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. A box is 6 cm by 6 cm by 7 cm. What is the length of the longest straight rod that fits inside, from one corner to the opposite corner, in cm? → 11Prompt: First find the diagonal of the base with Pythagoras.
  2. A rectangular park is 63 m long and 56 m wide. Instead of walking along two sides, you cut across the diagonal. How many metres shorter is the shortcut? Round to the nearest metre. → 35Prompt: Find the diagonal with Pythagoras.

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: Use Pythagoras to find a shorter side.
  • 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

  • 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-8.

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

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Pythagoras in real life: ladders, screens, shortcuts and boxes

Name

Date

Score/ 8

  1. Question 1

    A box is 2 cm by 10 cm by 11 cm. What is the length of the longest straight rod that fits inside, from one corner to the opposite corner, in cm?

    Answer

  2. Question 2

    A 5 m ladder leans against a wall. Its foot is 3 m from the wall. How high up the wall does it reach, in m?

    3 m?5 m

    Answer

  3. Question 3

    A rectangular park is 100 m long and 63 m wide. Instead of walking along two sides, you cut across the diagonal. How many metres shorter is the shortcut? Round to the nearest metre.

    Answer

  4. Question 4

    A box is 4 cm by 12 cm by 18 cm. What is the length of the longest straight rod that fits inside, from one corner to the opposite corner, in cm?

    Answer

  5. Question 5

    A 10 m ladder leans against a wall. Its foot is 6 m from the wall. How high up the wall does it reach, in m?

    6 m?10 m

    Answer

  6. Question 6

    A rectangular park is 84 m long and 36 m wide. Instead of walking along two sides, you cut across the diagonal. How many metres shorter is the shortcut? Round to the nearest metre.

    Answer

  7. Question 7

    A box is 1 cm by 4 cm by 8 cm. What is the length of the longest straight rod that fits inside, from one corner to the opposite corner, in cm?

    Answer

  8. Question 8

    A 13 m ladder leans against a wall. Its foot is 5 m from the wall. How high up the wall does it reach, in m?

    5 m?13 m

    Answer

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Answer key: Pythagoras in real life: ladders, screens, shortcuts and boxes

  1. 1. 15

  2. 2. 4

  3. 3. 45

  4. 4. 22

  5. 5. 8

  6. 6. 29

  7. 7. 9

  8. 8. 12