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Themed maths · 22 July

Pi Approximation Day puzzles for IB Maths

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 on errors and continued fractions, an extension and full worked answers.

Level
IB Maths SL and HL (AA and AI)
Time
35 minutes, plus a 10-minute extension
Topics
Approximation, absolute and percentage error; Continued fractions and sequences; Arc length and radians
Equipment
The starter is non-calculator. A GDC 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
Main: task C10 minThe best fraction
Extension10 minFast finishers or homework

Starter (5 minutes)

No GDC.

  1. Write 22/7 as a mixed number.
  2. Work out 355/113 − 3 as a fraction.
  3. Using π ≈ 22/7, find the circumference of a circle of diameter 14.
  4. Find the arc length of a sector of radius 7 and angle 2 radians.

Main activity (30 minutes)

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

Use your GDC’s value of π.

  1. Find the absolute error when 22/7 is used for π, to 3 significant figures.
  2. Find the percentage error, to 3 significant figures.
  3. A circular running track has radius 1 km. By how many metres is its length overestimated if 22/7 is used for π? Give 3 significant figures.

Task B: Continued fractions (10 min)

π can be written as 3 + 1/(7 + 1/(15 + 1/(1 + …))). Stopping early gives fractions close to π.

  1. Work out 3 + 1/7.
  2. Work out 3 + 1/(7 + 1/15) as a single fraction.
  3. Work out 3 + 1/(7 + 1/(15 + 1/1)) as a single fraction. How many decimal places of π does it get right?

Task C: The best fraction (10 min)

A fraction p/q is a good approximation when it is close to π for the size of q.

  1. For each denominator from 1 to 10, find the fraction closest to π. Which one is closest overall?
  2. Find the absolute error of 355/113, to 3 significant figures.
  3. How many times smaller is the error of 355/113 than the error of 22/7? Give the nearest whole number.

Extension (10 minutes)

Archimedes’ bounds.

  1. Archimedes showed that 223/71 < π < 22/7. Find the width of this interval as a single 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.
  • Task B is a nice first look at continued fractions; HL students can explore why the large term 15 makes 22/7 so good.

Starter

  1. 3 1/7
    • 22 = 3 × 7 + 1
  2. 16/113
    • 355 − 339 = 16
  3. 44
    • 22/7 × 14 = 44
  4. 14
    • rθ = 7 × 2 = 14

Task A: How good is 22/7?

  1. 0.00126
    • 22/7 − π = 3.142857… − 3.141592… = 0.0012644…
  2. 0.0402%
    • 0.0012644… ÷ π × 100 = 0.04024…
  3. 2.53 m
    • 2 × 1000 × (22/7 − π) = 2.528…

Task B: Continued fractions

  1. 22/7
    • 3 + 1/7 = 22/7
  2. 333/106
    • 7 + 1/15 = 106/15, so 1/(106/15) = 15/106.
    • 3 + 15/106 = 333/106
  3. 355/113; 6 decimal places
    • 15 + 1 = 16; 7 + 1/16 = 113/16; 3 + 16/113 = 355/113.
    • 355/113 = 3.1415929…; π = 3.1415926…: they agree to 6 decimal places.

Task C: The best fraction

  1. 22/7
    • The best for each q is the nearest whole number to qπ, over q: 3/1, 6/2, 9/3, 13/4, 16/5, 19/6, 22/7, 25/8, 28/9, 31/10.
    • 22/7 is the closest, about 0.0013 away.
  2. 0.000000267 (2.67 × 10−7)
    • 355/113 − π = 0.000000266764…
  3. 4740
    • 0.0012644… ÷ 0.00000026676… = 4740.1…

Extension

  1. 1/497
    • 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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