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Themed maths · 22 December

National Mathematics Day activities for IB Maths

22 December is National Mathematics Day in India, on the birthday of Srinivasa Ramanujan. This lesson pack has a 5-minute starter, a 35-minute main activity, an HL extension and full worked answers.

Level
IB Maths SL and HL (AA and AI)
Time
40 minutes, plus a 10-minute extension
Topics
Sequences, series and sigma notation; Proof by induction (HL); Counting: partitions and compositions; Systems of linear equations; Number: factors and divisors
Equipment
The starter is non-calculator. A GDC is optional.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A9 minTwo ways to make 1729
Main: task B9 minSums of cubes
Main: task C9 minPartitions and compositions
Main: task D8 minThe birthday magic square as a linear system
Extension10 minFast finishers or homework

Starter (5 minutes)

No GDC.

  1. Work out 13 + 123.
  2. Find Σr=14 r3.
  3. In how many ways can 4 be written as a sum of positive integers if order does not matter?
  4. Solve x + y = 19, x − y = 1. Then find x3 + y3.

Main activity (35 minutes)

Task A: Two ways to make 1729 (9 min)

1729 is the smallest positive integer that is a sum of two positive cubes in two different ways.

  1. Suppose a3 + b3 = 1729 and a + b = 13. Use a3 + b3 = (a + b)((a + b)2 − 3ab) to find ab, and hence a and b.
  2. Repeat with a + b = 19.
  3. Explain why a + b must be a factor of 1729, and list the factors of 1729 = 7 × 13 × 19.

Task B: Sums of cubes (9 min)

Σr=1n r3 = (n(n + 1)/2)2.

  1. Find Σr=120 r3.
  2. Find Σr=1120 r3.
  3. (HL) Prove the formula by induction.

Task C: Partitions and compositions (9 min)

A partition writes a number as a sum of positive integers where order does not matter. A composition is the same but order matters (so 2 + 1 and 1 + 2 are different).

  1. How many partitions does 5 have?
  2. How many partitions of 7 have all parts different? How many have all parts odd?
  3. Explain why 7 has 26 = 64 compositions.

Task D: The birthday magic square as a linear system (8 min)

Ramanujan was born on 22-12-1887. In this 4 × 4 magic square the top row is 22 | 12 | 18 | 87, and every row, column and diagonal has the same total.

22121887
ab925
1024c16
19d2311

  1. Write down four linear equations in a, b, c and d (use row 2, row 3, column 2 and the leading diagonal), and solve them.

Extension (10 minutes)

Number puzzles.

  1. How many positive divisors does 1728 = 26 × 33 have?
  2. Show that 4104 = 23 + 163 = 93 + 153. Use the method of task A to find the pair for a + b = 24.

For teachers

Teacher notes and full worked answers
  • Facts in this pack: 22 December is National Mathematics Day in India, and Ramanujan was born on 22 December 1887 (source: Press Information Bureau, Government of India (PDF)).
  • Task A is a neat use of the symmetric functions a + b and ab: the same idea as the sum and product of the roots of a quadratic.
  • For task C, Euler’s result (distinct parts = odd parts) is a lovely thing to state; HL students can look up the bijection as an exploration idea.

Starter

  1. 1729
    • 1 + 1728 = 1729
  2. 100
    • 1 + 8 + 27 + 64 = 100
  3. 5
    • 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
  4. x = 10, y = 9; 1729
    • Add: 2x = 20, so x = 10 and y = 9.
    • 1000 + 729 = 1729

Task A: Two ways to make 1729

  1. ab = 12; 1 and 12
    • 1729 = 13(169 − 3ab), so 169 − 3ab = 133 and ab = 12.
    • a and b are the roots of t2 − 13t + 12 = 0: 1 and 12.
  2. ab = 90; 9 and 10
    • 1729 = 19(361 − 3ab), so 361 − 3ab = 91 and ab = 90.
    • t2 − 19t + 90 = 0 gives 9 and 10.
  3. 1, 7, 13, 19, 91, 133, 247, 1729
    • 1729 = (a + b)(a2 − ab + b2) and both brackets are positive integers.
    • The 8 factors are 1, 7, 13, 19, 91, 133, 247 and 1729.

Task B: Sums of cubes

  1. 44 100
    • (20 × 21/2)2 = 2102 = 44 100
  2. 41 075
    • 44 100 − (10 × 11/2)2 = 44 100 − 3025 = 41 075
  3. See the steps
    • n = 1: 1 = (1 × 2/2)2. True.
    • Assume true for n = k. Then Σ1k+1 r3 = k2(k + 1)2/4 + (k + 1)3 = (k + 1)2(k2 + 4k + 4)/4 = ((k + 1)(k + 2)/2)2.
    • So true for n = k + 1; true for n = 1; so true for all positive integers n.

Task C: Partitions and compositions

  1. 7
    • 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1
  2. 5 and 5
    • Different parts: 7, 6+1, 5+2, 4+3, 4+2+1.
    • Odd parts: 7, 5+1+1, 3+3+1, 3+1+1+1+1, 1+1+1+1+1+1+1.
    • Both counts are 5. The two counts are always equal (Euler).
  3. 64
    • Write 7 as 1 + 1 + 1 + 1 + 1 + 1 + 1. There are 6 plus signs.
    • Each plus sign is either kept or merged, independently: 26 = 64 compositions.

Task D: The birthday magic square as a linear system

  1. a = 88, b = 17, c = 89, d = 86
    • The total is 22 + 12 + 18 + 87 = 139.
    • Row 2: a + b = 105. Row 3: c = 89. Column 2: b + d = 103. Diagonal: b + c = 106.
    • So b = 17, a = 88, d = 86.

Extension

  1. 28
    • A divisor is 2i3j with 0 ≤ i ≤ 6 and 0 ≤ j ≤ 3: 7 × 4 = 28.
  2. ab = 135; 9 and 15
    • 8 + 4096 = 729 + 3375 = 4104.
    • 4104 = 24(576 − 3ab), so 576 − 3ab = 171 and ab = 135.
    • t2 − 24t + 135 = 0 gives 9 and 15.

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

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