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Themed maths · Mid-October

Ada Lovelace Day maths for CBSE Class 10

Ada Lovelace is widely known for her notes on the Analytical Engine, which set out step-by-step methods for a machine to follow. In this lesson the steps build arithmetic progressions: a 5-minute starter, a 35-minute main activity, an extension and full worked answers. No calculator.

Level
CBSE Class 10 (Class 9 can try most of it)
Time
40 minutes, plus a 10-minute extension
Topics
Arithmetic progressions; Following an algorithm step by step; Sum of the first n terms of an AP; Number patterns
Equipment
No calculator. Every answer works out exactly.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A10 minFollow the algorithm
Main: task B12 minA loop that counts down
Main: task C13 minTwo loops race
Extension10 minFast finishers or homework

Starter (5 minutes)

Quick-fire mental maths.

  1. Find the 10th term of the AP 7, 12, 17, …
  2. Find the sum of the first 20 natural numbers.
  3. An AP has first term 4 and 8th term 39. Find the common difference.
  4. Is 101 a term of the AP 5, 9, 13, …? If so, which term?

Main activity (35 minutes)

Task A: Follow the algorithm (10 min)

Step 1: set n = 1 and S = 0. Step 2: add (2n − 1) to S. Step 3: if n = 15, write down S and stop; otherwise increase n by 1 and go back to step 2.

  1. Which numbers are added to S? What is written down at the end?
  2. What is written down if ‘n = 15’ in step 3 is changed to ‘n = 40’?
  3. Instead, the algorithm stops as soon as S is more than 500. What is written down?

Task B: A loop that counts down (12 min)

A program starts with x = 120. While x is positive it repeats: ‘write down x, then subtract 7 from x’.

  1. What is the 10th number written down?
  2. How many numbers are written down?
  3. Find the sum of all the numbers written down.

Task C: Two loops race (13 min)

Loop P starts at 3 and adds 8 each step. Loop Q starts at 63 and adds 5 each step. Both write down their starting number first.

  1. After how many steps do the two loops write down the same number? What is it?
  2. Find the sum of the first 20 numbers written down by loop P.
  3. If loop P never stopped, how many three-digit numbers would it write down?

Extension (10 minutes)

For fast finishers, or as homework.

  1. The sum 1 + 2 + 3 + … + n is a three-digit number with all three digits the same. Find n.
  2. Find the sum of all two-digit multiples of 7.

For teachers

Teacher notes and full worked answers
  • Ask students to trace the algorithm in a table with columns n and S before using any formula.
  • In task B (b), check that students include the last positive number, 1.
  • Ada Lovelace is widely known for her notes on the Analytical Engine. Keep the lesson about the maths.

Starter

  1. 52
    • a = 7, d = 5: a10 = 7 + 9 × 5 = 52
  2. 210
    • 20 × 21 ÷ 2 = 210
  3. 5
    • 4 + 7d = 39, so 7d = 35 and d = 5.
  4. Yes, the 25th term
    • an = 5 + 4(n − 1) = 4n + 1.
    • 4n + 1 = 101 gives n = 25, a whole number.

Task A: Follow the algorithm

  1. 1, 3, 5, …, 29; 225
    • The odd numbers 1, 3, 5, …, 29 are added: an AP with a = 1, d = 2 and 15 terms.
    • S15 = 15/2 × (1 + 29) = 225
  2. 1600
    • The sum of the first n odd numbers is n/2 × (1 + 2n − 1) = n2.
    • 402 = 1600
  3. 529
    • After n steps S = n2.
    • 222 = 484 is not more than 500, but 232 = 529 is.

Task B: A loop that counts down

  1. 57
    • An AP with a = 120 and d = −7: a10 = 120 − 9 × 7 = 57
  2. 18
    • 120 − 7(n − 1) > 0 gives n − 1 < 120/7 = 17 1/7, so n ≤ 18.
    • The 18th number is 120 − 119 = 1; the next would be −6.
  3. 1089
    • S18 = 18/2 × (120 + 1) = 9 × 121 = 1089

Task C: Two loops race

  1. After 20 steps; 163
    • After k steps, P has 3 + 8k and Q has 63 + 5k.
    • 3 + 8k = 63 + 5k gives 3k = 60, so k = 20 and the number is 163.
  2. 1580
    • a = 3, d = 8: a20 = 3 + 19 × 8 = 155.
    • S20 = 20/2 × (3 + 155) = 1580
  3. 112
    • P writes 3, 11, 19, …: the numbers that leave remainder 3 when divided by 8.
    • The first three-digit one is 107 and the last is 995.
    • (995 − 107) ÷ 8 + 1 = 111 + 1 = 112

Extension

  1. n = 36 (the sum is 666)
    • We need n(n + 1)/2 = 111a = 3 × 37 × a, so n or n + 1 is a multiple of 37.
    • n = 36: 36 × 37 ÷ 2 = 666. (n = 37 gives 703 and larger n give four digits.)
  2. 728
    • 14, 21, …, 98: an AP with 13 terms.
    • S = 13/2 × (14 + 98) = 13 × 56 = 728

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

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