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Themed maths · 22 July

Pi Approximation Day puzzles for A Level

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
A Level Maths (Year 12 and Year 13)
Time
35 minutes, plus a 10-minute extension
Topics
Approximation and error; Continued fractions and recurrence; Radians and small-angle approximations
Equipment
The starter is non-calculator. A calculator is useful 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 minContinued fractions as a recurrence
Main: task C10 minSmall angles and π
Extension10 minFast finishers or homework

Starter (5 minutes)

No calculator.

  1. Write 22/7 − 3 as a fraction.
  2. Work out 3 + 1/(7 + 1/15) as a single fraction.
  3. Convert 180° to radians and π/6 radians to degrees.
  4. Using sin θ ≈ θ, estimate sin 0.02.

Main activity (30 minutes)

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

Use your calculator’s value of π.

  1. Find the percentage error when 22/7 is used for π, to 3 significant figures.
  2. A satellite moves in a circle of radius 7000 km. By how many kilometres is one orbit overestimated if 22/7 is used for π? Give 3 significant figures.
  3. Which is the closer approximation to π: 22/7 or 355/113? By what factor, to the nearest whole number?

Task B: Continued fractions as a recurrence (10 min)

π = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + …)))). The approximations pn/qn follow pn = anpn−1 + pn−2 and qn = anqn−1 + qn−2, with a = 3, 7, 15, 1, 292, …

  1. Starting from 3/1 and 22/7, use the recurrence with a = 15 to find the next fraction.
  2. Use a = 1 to find the next fraction after that.
  3. Use a = 292 to find the next fraction.

Task C: Small angles and π (10 min)

Radians and small-angle approximations.

  1. Using sin θ ≈ θ, estimate 180 sin(1°) and compare it with π.
  2. Using cos θ ≈ 1 − θ2/2, estimate cos 0.1 and find the error in the approximation, to 2 significant figures.
  3. A wheel turns through 0.4 radians. Using π = 22/7, how many degrees is this, to the nearest degree?

Extension (10 minutes)

Archimedes’ bounds.

  1. Archimedes showed that 223/71 < π < 22/7. Find the width of the interval as a single fraction, and the midpoint to 5 decimal places.

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.
  • Task B uses the same kind of recurrence as the Fibonacci sequence: a nice link back to sequences.

Starter

  1. 1/7
    • 22/7 − 21/7 = 1/7
  2. 333/106
    • 7 + 1/15 = 106/15, so 3 + 15/106 = 333/106.
  3. π; 30°
    • 180° = π radians, so π/6 = 30°.
  4. 0.02
    • For small θ in radians, sin θ ≈ θ.

Task A: How good is 22/7?

  1. 0.0402%
    • (22/7 − π) ÷ π × 100 = 0.04024…
  2. 17.7 km
    • 2 × 7000 × (22/7 − π) = 17.70…
  3. 355/113, about 4740 times closer
    • |22/7 − π| = 0.0012644…, |355/113 − π| = 0.00000026676…
    • 0.0012644… ÷ 0.00000026676… = 4740.1…

Task B: Continued fractions as a recurrence

  1. 333/106
    • p = 15 × 22 + 3 = 333, q = 15 × 7 + 1 = 106
  2. 355/113
    • p = 1 × 333 + 22 = 355, q = 1 × 106 + 7 = 113
  3. 103993/33102
    • p = 292 × 355 + 333 = 103 993, q = 292 × 113 + 106 = 33 102

Task C: Small angles and π

  1. π; 180 sin 1° = 3.1414 (4 d.p.)
    • 1° = π/180 radians, so 180 sin(1°) ≈ 180 × π/180 = π.
    • The calculator gives 180 sin 1° = 3.14143…
  2. 0.995; error 0.0000042 (4.2 × 10−6)
    • 1 − 0.01/2 = 0.995
    • cos 0.1 = 0.995004165…, so the error is 0.0000041652…
  3. 23°
    • 0.4 × 180/π = 0.4 × 180 × 7/22 = 22.9…, so 23°.

Extension

  1. 1/497; 3.14185
    • 22/7 − 223/71 = (1562 − 1561)/497 = 1/497
    • Midpoint = (22/7 + 223/71) ÷ 2 = 3.141851…

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

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