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Themed maths · Mid-October

Ada Lovelace Day maths activities for IGCSE and GCSE

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 students follow and write their own: a 5-minute starter, a 35-minute main activity on flowcharts, loops and sequences, an extension and full worked answers.

Level
IGCSE and GCSE (Higher and Extended)
Time
40 minutes, plus a 10-minute extension
Topics
Following algorithms and flowcharts; Linear sequences and the nth term; Geometric sequences; Quadratic sequences; Triangular numbers
Equipment
The starter is non-calculator. A calculator is allowed in the main activity.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A10 minFollow the flowchart
Main: task B12 minLoops that print sequences
Main: task C13 minPixel patterns
Extension10 minFast finishers or homework

Starter (5 minutes)

No calculator.

  1. Write down the next two terms: 4, 11, 18, 25, …
  2. Find the nth term of 4, 11, 18, 25, …
  3. Input 6 into this machine: square it, subtract 10, then halve the result. What comes out?
  4. Write down the first three terms of the sequence with nth term n2 + 1.

Main activity (35 minutes)

Task A: Follow the flowchart (10 min)

Step 1: start with n = 1. Step 2: work out T = n(n + 1) ÷ 2 and write it down. Step 3: if T is more than 50, stop; otherwise add 1 to n and go back to step 2.

  1. Write down the first five numbers the flowchart produces.
  2. What is the last number written down?
  3. How many numbers are written down altogether?

Task B: Loops that print sequences (12 min)

Program P starts with x = 7 and repeats: ‘print x, then add 6 to x’. Program Q starts with x = 2 and repeats: ‘print x, then multiply x by 3’.

  1. Find an expression for the nth number printed by program P.
  2. Find the 30th number printed by program P.
  3. Does program P ever print 400? Explain.
  4. Find the 7th number printed by program Q.

Task C: Pixel patterns (13 min)

A computer draws patterns of pixels. Patterns 1 to 5 use 2, 7, 14, 23 and 34 pixels.

  1. How many pixels does pattern 6 use?
  2. Find the nth term.
  3. Which pattern uses 287 pixels?

Extension (10 minutes)

For fast finishers, or as homework.

  1. A loop starts with the number 1 and repeats: ‘double the number, then add 1’. How many times does the loop run before the number is first more than 1000? What is the number then?
  2. Find the 100th number the flowchart in task A would produce if it never stopped.

For teachers

Teacher notes and full worked answers
  • Ask students to write their own three-step flowchart for a friend to follow: writing exact instructions is the point of the lesson.
  • In task B (c), students should use the nth term rather than keep adding 6.
  • Ada Lovelace is widely known for her notes on the Analytical Engine. Keep the lesson about the maths.

Starter

  1. 32, 39
    • The terms go up by 7 each time.
  2. 7n − 3
    • The difference is 7, so start with 7n. The first term is 4 = 7 − 3, so the nth term is 7n − 3.
  3. 13
    • 62 = 36, 36 − 10 = 26, 26 ÷ 2 = 13
  4. 2, 5, 10
    • 1 + 1 = 2, 4 + 1 = 5, 9 + 1 = 10

Task A: Follow the flowchart

  1. 1, 3, 6, 10, 15
    • n = 1, 2, 3, 4, 5 give 1, 3, 6, 10, 15: the triangular numbers.
  2. 55
    • n = 9 gives 45, which is not more than 50, so carry on.
    • n = 10 gives 55, which is more than 50, so stop.
  3. 10
    • One for each of n = 1 to 10.

Task B: Loops that print sequences

  1. 6n + 1
    • P prints 7, 13, 19, …: difference 6, and 7 = 6 + 1.
  2. 181
    • 6 × 30 + 1 = 181
  3. No
    • 6n + 1 = 400 gives n = 66.5, which is not a whole number.
  4. 1458
    • Q prints 2, 6, 18, 54, …: each number is 3 times the one before.
    • The 7th is 2 × 36 = 2 × 729 = 1458.

Task C: Pixel patterns

  1. 47
    • The differences are 5, 7, 9, 11, so the next difference is 13.
    • 34 + 13 = 47
  2. n2 + 2n − 1
    • The second difference is 2, so start with n2.
    • Subtract n2 from each term: 1, 3, 5, 7, 9, which is 2n − 1.
    • So the nth term is n2 + 2n − 1.
  3. Pattern 16
    • n2 + 2n − 1 = 287 gives n2 + 2n − 288 = 0.
    • (n + 18)(n − 16) = 0, so n = 16.

Extension

  1. 9 times; 1023
    • 1 → 3 → 7 → 15 → 31 → 63 → 127 → 255 → 511 → 1023
    • That is 9 runs of the loop. (After k runs the number is 2k+1 − 1.)
  2. 5050
    • 100 × 101 ÷ 2 = 5050

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

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