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Themed maths · 22 July

Pi Approximation Day puzzles for IGCSE and GCSE

22 July is Pi Approximation Day: written day first, 22/7 is the famous fraction for π. A short summer puzzle set: a 5-minute starter, three 10-minute puzzles, an extension and full worked answers.

Level
IGCSE and GCSE (Higher and Extended)
Time
35 minutes, plus a 10-minute extension
Topics
Fractions, decimals and rounding; Percentage error; Circles: circumference and area
Equipment
The starter is non-calculator. A calculator is allowed in the main activity.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A10 minHow good is 22/7?
Main: task B10 minThe best fraction
Main: task C10 minCircle puzzles
Extension10 minFast finishers or homework

Starter (5 minutes)

No calculator.

  1. Write 22/7 as a mixed number.
  2. Using π = 22/7, find the circumference of a circle of diameter 21 cm.
  3. Using π = 22/7, find the area of a circle of radius 7 cm.
  4. Round 3.142857 to 3 decimal places and to 3 significant figures.

Main activity (30 minutes)

Task A: How good is 22/7? (10 min)

Use the π key on your calculator.

  1. Write 22/7 and π as decimals to 5 decimal places. Which is bigger?
  2. Find the percentage error when 22/7 is used for π, to 2 significant figures.
  3. A round pond has radius 14 m. Find its area using 22/7 and using the π key, to 1 decimal place. What is the difference?

Task B: The best fraction (10 min)

Which fraction with a small denominator is closest to π?

  1. For denominators 1 to 8, the closest fractions to π are 3/1, 6/2, 9/3, 13/4, 16/5, 19/6, 22/7 and 25/8. Which is closest?
  2. Write 355/113 as a decimal to 7 decimal places. How many decimal places of π does it match?
  3. Find 355/113 − 22/7 as a single fraction.

Task C: Circle puzzles (10 min)

Use π = 22/7.

  1. A wheel has diameter 70 cm. How many complete turns does it make in 1 km?
  2. A semicircle has diameter 28 cm. Find its perimeter.
  3. A circle has circumference 132 cm. Find its radius and its area.

Extension (10 minutes)

For fast finishers.

  1. Archimedes showed that π is between 223/71 and 22/7. Write both as decimals to 4 decimal places and find the difference between them as a fraction.

For teachers

Teacher notes and full worked answers
  • Pi Approximation Day is 22 July because 22/7, written day first, is the famous fraction for π.
  • This is a short set for the last days of term or summer school: each puzzle stands alone.
  • In task C (a), insist on complete turns: 454, not 455.

Starter

  1. 3 1/7
    • 22 ÷ 7 = 3 remainder 1
  2. 66 cm
    • 22/7 × 21 = 66
  3. 154 cm2
    • 22/7 × 49 = 154
  4. 3.143 and 3.14
    • 3 d.p.: 3.143; 3 s.f.: 3.14

Task A: How good is 22/7?

  1. 3.14286 and 3.14159: 22/7 is bigger
    • 22 ÷ 7 = 3.142857…
    • π = 3.141592…
  2. 0.040%
    • (22/7 − π) ÷ π × 100 = 0.0402…
  3. 616 m2 and 615.8 m2; 0.2 m2
    • 22/7 × 196 = 616
    • π × 196 = 615.75…, which is 615.8
    • 616 − 615.8 = 0.2

Task B: The best fraction

  1. 22/7
    • Find how far each is from 3.14159: 22/7 is 0.00126 away; the next best, 16/5, is 0.0584 away.
  2. 3.1415929; 6
    • 355 ÷ 113 = 3.14159292…
    • π = 3.14159265…: the first 6 decimal places match.
  3. −1/791
    • (355 × 7 − 22 × 113) ÷ (113 × 7) = (2485 − 2486)/791 = −1/791

Task C: Circle puzzles

  1. 454 complete turns
    • One turn: 22/7 × 70 = 220 cm.
    • 100 000 ÷ 220 = 454.5…, so 454 complete turns.
  2. 72 cm
    • Arc: ½ × 22/7 × 28 = 44. Add the diameter: 44 + 28 = 72.
  3. 21 cm; 1386 cm2
    • 2 × 22/7 × r = 132, so r = 21.
    • 22/7 × 212 = 1386

Extension

  1. 3.1408 and 3.1429; 1/497
    • 223 ÷ 71 = 3.14084…, 22 ÷ 7 = 3.14285…
    • 22/7 − 223/71 = (1562 − 1561)/497 = 1/497

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

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