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Themed maths · 22 July

Pi Approximation Day puzzles for CBSE Class 9 and 10

22 July is Pi Approximation Day: written day first, 22/7 is the famous fraction for π. A short no-calculator puzzle set: a 5-minute starter, three 10-minute puzzles, an extension and full worked answers.

Level
CBSE Class 9 and 10
Time
35 minutes, plus a 10-minute extension
Topics
Number system: decimal expansions of rational numbers; Real numbers; Areas related to circles
Equipment
No calculator. Use π = 22/7 where a question says so.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A10 minThe decimal of 22/7
Main: task B10 minRational numbers near π
Main: task C10 minCircle puzzles
Extension10 minFast finishers or homework

Starter (5 minutes)

Quick-fire mental maths. Use π = 22/7.

  1. Write 22/7 as a mixed number.
  2. Find the circumference of a circle of diameter 14 cm. (Use π = 22/7.)
  3. Find the area of a circle of radius 7 cm. (Use π = 22/7.)
  4. Is 22/7 rational or irrational? Is π?

Main activity (30 minutes)

Task A: The decimal of 22/7 (10 min)

Rational numbers have decimal expansions that end or repeat.

  1. Divide 22 by 7 by long division until the remainders repeat. What is the repeating block?
  2. What is the 100th digit after the decimal point of 22/7?
  3. Without dividing, explain whether 355/113 has a terminating decimal expansion.

Task B: Rational numbers near π (10 min)

22/7 and 355/113 are both close to π.

  1. Find 22/7 − 355/113 as a single fraction. Which of the two fractions is bigger?
  2. Find a rational number between 355/113 and 22/7 by adding the numerators and adding the denominators.
  3. Archimedes showed that π is between 223/71 and 22/7. Find the difference between these two fractions.

Task C: Circle puzzles (10 min)

Use π = 22/7.

  1. A circle has circumference 88 cm. Find its radius and area.
  2. A wheel of diameter 63 cm makes 500 turns. How far does it travel, in metres?
  3. Find the area of a sector of radius 21 cm and angle 60°.

Extension (10 minutes)

For fast finishers.

  1. A square and a circle have the same perimeter, 88 cm. Which has the bigger area, and by how much? (Use π = 22/7.)

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.
  • A short set for any lesson near 22 July: each puzzle stands alone.
  • Task A links to the Number system and Real numbers chapters: terminating and repeating decimals.

Starter

  1. 3 1/7
    • 22 = 3 × 7 + 1
  2. 44 cm
    • 22/7 × 14 = 44
  3. 154 cm2
    • 22/7 × 49 = 154
  4. 22/7 is rational; π is irrational
    • 22/7 is a ratio of integers. π cannot be written as one.

Task A: The decimal of 22/7

  1. 142857
    • Remainders 1, 3, 2, 6, 4, 5, then 1 again, giving digits 1, 4, 2, 8, 5, 7.
    • 22/7 = 3.142857142857…
  2. 8
    • The block 142857 has 6 digits. 100 = 6 × 16 + 4.
    • The 4th digit of the block is 8.
  3. No: it is non-terminating and repeating
    • 113 is prime and is not 2 or 5, so the denominator is not of the form 2m5n.

Task B: Rational numbers near π

  1. 1/791; 22/7 is bigger
    • (22 × 113 − 355 × 7)/(7 × 113) = (2486 − 2485)/791 = 1/791
  2. 377/120
    • (355 + 22)/(113 + 7) = 377/120
    • 377 × 113 − 355 × 120 = 42 601 − 42 600 = 1 > 0, so 377/120 > 355/113; and 22 × 120 − 377 × 7 = 2640 − 2639 = 1 > 0, so 377/120 < 22/7.
  3. 1/497
    • 22/7 − 223/71 = (1562 − 1561)/497 = 1/497

Task C: Circle puzzles

  1. 14 cm; 616 cm2
    • 2 × 22/7 × r = 88, so r = 14.
    • 22/7 × 14 × 14 = 616
  2. 990 m
    • One turn: 22/7 × 63 = 198 cm.
    • 500 × 198 = 99 000 cm = 990 m
  3. 231 cm2
    • 60/360 × 22/7 × 21 × 21 = 1/6 × 1386 = 231

Extension

  1. The circle, by 132 cm2
    • Square: side 22, area 484.
    • Circle: radius 14, area 616.
    • 616 − 484 = 132

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

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