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Themed maths · 22 December

National Mathematics Day activities for Class 9 and Class 10

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 extension and full worked answers, all without a calculator.

Level
CBSE Class 9 and Class 10
Time
40 minutes, plus a 10-minute extension
Topics
Real numbers: prime factorisation, HCF and LCM; Algebraic identities: sum of cubes; Patterns, sequences and quadratic equations; Counting: partitions; Magic squares and number sense
Equipment
No calculator. Every answer is a whole number or an exact fraction.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A10 minThe number 1729
Main: task B9 minPartitions
Main: task C8 minA birthday magic square
Main: task D8 minSums of cubes
Extension10 minFast finishers or homework

Starter (5 minutes)

Quick-fire mental maths.

  1. Work out 123.
  2. Work out 103 + 93.
  3. Write 1729 as a product of prime factors. (Hint: try 7.)
  4. In how many ways can 4 be written as a sum of positive whole numbers, if the order does not matter? (4 itself counts.)

Main activity (35 minutes)

Task A: The number 1729 (10 min)

1729 is the smallest positive whole number that can be written as the sum of two positive cubes in two different ways.

  1. Show that 1729 = 13 + 123 = 93 + 103.
  2. Use the identity a3 + b3 = (a + b)(a2 − ab + b2) to write 93 + 103 as a product of two whole numbers.
  3. 4104 = 23 × 33 × 19. Find the HCF and LCM of 1729 and 4104.

Task B: Partitions (9 min)

A partition of a number is a way of writing it as a sum of positive whole numbers, where the order does not matter. For example, 3 has three partitions: 3, 2 + 1 and 1 + 1 + 1.

  1. List all the partitions of 5. How many are there?
  2. How many partitions does 6 have?
  3. How many partitions of 6 use only different numbers? How many use only odd numbers? What do you notice?

Task C: A birthday magic square (8 min)

In a magic square every row, every column and both diagonals add up to the same total. Ramanujan was born on 22-12-1887. This 4 × 4 magic square has the date 22 | 12 | 18 | 87 as its top row. Four numbers are missing.

22121887
?17925
1024?16
19?23?

  1. Find the magic total.
  2. Find the four missing numbers.
  3. Check that the four corners, and the four numbers in the middle, also add up to the magic total.

Task D: Sums of cubes (8 min)

13 = 1, 13 + 23 = 9, 13 + 23 + 33 = 36, …

  1. Find 13 + 23 + 33 + 43. What kind of numbers are 1, 9, 36 and your answer?
  2. The pattern is 13 + 23 + … + n3 = (n(n + 1)/2)2. Use it to find 13 + 23 + … + 123.
  3. For which n is the sum 44 100?

Extension (10 minutes)

For fast finishers, or as homework.

  1. 1729 = 7 × 13 × 19, and 7, 13, 19 is an AP with common difference 6. Find the smallest number that is the product of three primes in an AP with common difference 6.
  2. Show that 4104 is also a sum of two cubes in two different ways: 4104 = 23 + 163 = 93 + 153.
  3. Make your own magic square from your birthday, as in task C. Start with your date as the top row and find a way to complete it. (There is no single answer.)

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. Both are on our National Mathematics Day puzzles page with official sources.
  • For the partitions task, ask students to list systematically (largest part first) so that none are missed.
  • The magic square is a good place to talk about checking: one missing number found two ways is a strong check.

Starter

  1. 1728
    • 12 × 12 = 144, and 144 × 12 = 1728.
  2. 1729
    • 1000 + 729 = 1729
  3. 7 × 13 × 19
    • 1729 ÷ 7 = 247, and 247 = 13 × 19.
  4. 5
    • 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1

Task A: The number 1729

  1. 1 + 1728 = 729 + 1000 = 1729
    • 1 + 1728 = 1729 and 729 + 1000 = 1729.
  2. 19 × 91
    • a + b = 19 and a2 − ab + b2 = 81 − 90 + 100 = 91.
    • So 1729 = 19 × 91.
  3. HCF 19; LCM 373 464
    • 1729 = 7 × 13 × 19 and 4104 = 23 × 33 × 19, so HCF = 19.
    • LCM = 23 × 33 × 7 × 13 × 19 = 216 × 1729 = 373 464.

Task B: Partitions

  1. 7
    • 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1
  2. 11
    • 6; 5+1; 4+2; 4+1+1; 3+3; 3+2+1; 3+1+1+1; 2+2+2; 2+2+1+1; 2+1+1+1+1; 1+1+1+1+1+1
  3. 4 and 4: the same
    • Different numbers: 6, 5 + 1, 4 + 2, 3 + 2 + 1.
    • Only odd numbers: 5 + 1, 3 + 3, 3 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1.
    • Both counts are 4. (This is true for every number, not just 6.)

Task C: A birthday magic square

  1. 139
    • Add the top row: 22 + 12 + 18 + 87 = 139.
  2. 88, 89, 86 and 11
    • Row 2: 139 − (17 + 9 + 25) = 88.
    • Row 3: 139 − (10 + 24 + 16) = 89.
    • Column 2: 139 − (12 + 17 + 24) = 86.
    • Row 4: 139 − (19 + 86 + 23) = 11.
  3. Both give 139
    • Corners: 22 + 87 + 19 + 11 = 139.
    • Middle: 17 + 9 + 24 + 89 = 139.

Task D: Sums of cubes

  1. 100; they are all square numbers
    • 1 + 8 + 27 + 64 = 100.
    • 1 = 12, 9 = 32, 36 = 62, 100 = 102, and 1, 3, 6, 10 are the triangular numbers.
  2. 6084
    • n(n + 1)/2 = 12 × 13/2 = 78, and 782 = 6084.
  3. n = 20
    • 44 100 = 2102, so n(n + 1)/2 = 210.
    • n2 + n − 420 = 0, so (n + 21)(n − 20) = 0 and n = 20.

Extension

  1. 935 = 5 × 11 × 17
    • Try the smallest first terms: 2, 8, 14 and 3, 9, 15 are not all prime.
    • 5, 11, 17 are all prime, so the answer is 5 × 11 × 17 = 935.
  2. 8 + 4096 = 729 + 3375 = 4104
    • 23 + 163 = 8 + 4096 = 4104.
    • 93 + 153 = 729 + 3375 = 4104.
  3. Answers vary
    • Share and check each other’s squares: every row, column and diagonal must give the same total.

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

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