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Maths about me: a first-lesson icebreaker for A Level Maths

A first lesson of Year 12 or Year 13 that gets everyone doing maths straight away: number facts built from a birthday, ‘Who am I?’ riddles from GCSE and Year 12, the four 4s challenge and two short proofs. Everything is on paper and nothing personal is collected.

Level
A Level Maths (Year 12 and Year 13)
Time
40 minutes, plus a 10-minute extension
Topics
Proof; Sequences and series; Algebra and functions; Exponentials and logarithms
Equipment
The starter is non-calculator. A calculator is useful for task A.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A12 minMaths about me
Main: task B10 min‘Who am I?’ in pairs
Main: task C13 minFour 4s
Extension10 minFast finishers or homework

Starter (5 minutes)

‘Who am I?’ riddles on the board as students arrive.

  1. I am a positive integer and my square is 6 more than me. Who am I?
  2. I am 1 + 2 + 3 + … + 20. Who am I?
  3. I am log2 32 − log3 9. Who am I?
  4. I am the coefficient of x2 in (1 + x)6. Who am I?

Main activity (35 minutes)

Task A: Maths about me (12 min)

Work through Sam’s made-up numbers first. Then write the same facts about your own numbers on paper.

  1. Sam was born on day 18 of month 12 in 2008. Write 12 and 2008 as products of prime factors, and find their HCF and LCM.
  2. Write 2008 in binary.
  3. How many terms of the arithmetic sequence 12, 18, 24, … are less than 2008, and what is their sum?

Task B: ‘Who am I?’ in pairs (10 min)

Solve these with a partner. Then each write a riddle of your own, on a topic from last year, with exactly one answer and swap.

  1. I am the only positive integer n with 2n = n2 + 7. Who am I?
  2. Find every two-digit number that equals the sum of its digits plus the product of its digits.
  3. I am the sum to infinity of a geometric series with first term 6 whose first two terms add to 10. Who am I?

Task C: Four 4s (13 min)

Use exactly four 4s to make each target. Allowed: +, −, ×, ÷, brackets, powers, √ and ! (4! = 24). You may join two 4s to make 44. Other answers are possible.

  1. Make 0 and 24.
  2. Make 19.
  3. Make 1024.
  4. Make 11 using √.

Extension (10 minutes)

For fast finishers.

  1. Prove that the difference between a two-digit number and the number with its digits reversed is always a multiple of 9.
  2. Prove that n3 − n is a multiple of 6 for every integer n.

For teachers

Teacher notes and full worked answers
  • Nothing personal is collected: students work on paper, and nothing is typed into the site or stored. If a student would rather not use a real birthday, any date works.
  • Pairs for task B: each student writes one riddle with exactly one answer and swaps with a partner, who checks that only one number fits.
  • Mindset prompts for the first lesson (for you to read out or put on the board; nothing is collected or stored): ‘A time maths felt hard, and what helped’; ‘One topic from last year I could teach a friend’; ‘What I do when I am stuck’; ‘How I want this class to feel’.
  • The extension proofs are direct proofs of the kind in the Proof topic: ask for a concluding sentence each time.

Starter

  1. 3
    • x2 − x − 6 = (x − 3)(x + 2) = 0
    • x = 3
  2. 210
    • 20 × 21 / 2 = 210
  3. 3
    • 5 − 2 = 3
  4. 15
    • 6C2 = 15

Task A: Maths about me

  1. 12 = 22 × 3, 2008 = 23 × 251; HCF = 4, LCM = 6024
    • 2008 = 8 × 251, and 251 is prime (not divisible by 2, 3, 5, 7, 11 or 13, and 172 > 251)
    • HCF = 22 = 4; LCM = 23 × 3 × 251 = 6024
  2. 11111011000
    • 2008 = 1024 + 512 + 256 + 128 + 64 + 16 + 8
  3. 333 terms; sum 335 664
    • 12 + 6(n − 1) < 2008 gives n − 1 < 332.67, so n = 333 (the last term is 2004).
    • S = 333/2 × (12 + 2004) = 335 664

Task B: ‘Who am I?’ in pairs

  1. 5
    • Try n = 1, 2, 3, 4, 5: 2n − n2 = 1, 0, −1, 0, 7.
    • For n ≥ 5 the gap keeps growing, so 5 is the only answer.
  2. 19, 29, 39, 49, 59, 69, 79, 89 and 99
    • 10a + b = a + b + ab gives 9a = ab, so b = 9.
  3. 18
    • 6 + 6r = 10 gives r = 2/3
    • S∞ = 6 / (1 − 2/3) = 18

Task C: Four 4s

  1. 44 − 44 = 0 and 4 × 4 + 4 + 4 = 24
    • 44 − 44 = 0
    • 16 + 8 = 24
  2. 4! − 4 − 4/4 = 19
    • 24 − 4 − 1 = 19
  3. 44 + 4/4 = 1024
    • 45 = 1024
  4. 44/(√4 × √4) = 11
    • √4 × √4 = 4, and 44 ÷ 4 = 11

Extension

  1. (10a + b) − (10b + a) = 9(a − b)
    • The number is 10a + b; its reverse is 10b + a.
    • The difference is 9a − 9b = 9(a − b), a multiple of 9.
  2. n3 − n = (n − 1)n(n + 1)
    • It is the product of three consecutive integers.
    • One of any two consecutive integers is even, and one of any three is a multiple of 3, so the product is a multiple of 2 × 3 = 6.

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

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