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Themed maths · 14 February

Valentine’s Day maths activities for IGCSE and GCSE

A ready-to-teach lesson for 14 February: a 5-minute starter, a 35-minute main activity on a heart made from a square and two semicircles, a secret card swap and a bag of sweets, an extension and full worked answers.

Level
IGCSE and GCSE (Higher and Extended)
Time
40 minutes, plus a 10-minute extension
Topics
Area and perimeter of composite shapes with circles; Enlargement and area scale factor; Listing outcomes and probability; Probability without replacement
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 A12 minDesign a heart
Main: task B12 minSecret card swap
Main: task C11 minSweet hearts
Extension10 minFast finishers or homework

Starter (5 minutes)

No calculator.

  1. Find the area of a circle of radius 5 cm, in terms of π.
  2. Find the circumference of a circle of diameter 8 cm, in terms of π.
  3. A bag has 3 red and 5 white heart sweets. One is taken at random. Find P(red).
  4. Share 48 sweets in the ratio 3 : 5.

Main activity (35 minutes)

Task A: Design a heart (12 min)

A heart is made from a square of side 6 cm, turned to stand on a corner, with a semicircle on each of its two upper sides.

  1. Find the area of the heart, exactly and to 3 significant figures.
  2. Find the perimeter of the heart, to 3 significant figures.
  3. A bigger heart for a card is an enlargement with side 9 cm. Find its area, to 3 significant figures, using the area scale factor.

Task B: Secret card swap (12 min)

Friends each write their name on a card. The cards are shuffled and each friend takes one at random.

  1. There are 3 friends, A, B and C. List all the ways the cards can be handed out. Find the probability that nobody gets their own card.
  2. With 3 friends, find the probability that everybody gets their own card.
  3. With 4 friends there are 24 ways, and in 9 of them nobody gets their own card. Find this probability in its simplest form, and say whether it is more or less likely than with 3 friends.

Task C: Sweet hearts (11 min)

A bag holds 10 sweets: 4 red hearts and 6 pink hearts. Two sweets are taken without replacement.

  1. Draw a tree diagram. Find the probability that both sweets are red.
  2. Find the probability of one red and one pink.
  3. Find the probability of at least one red.

Extension (10 minutes)

For fast finishers, or as homework.

  1. Three people each pick a whole number from 1 to 6 at random. Find the probability that at least two of them pick the same number.
  2. A heart of the shape in task A must have an area of 100 cm2. Find the side of the square, to 3 significant figures.

For teachers

Teacher notes and full worked answers
  • Students can cut the heart from card: a square and two semicircles of diameter equal to its side.
  • In task A (c), a common slip is to multiply the area by 1.5 instead of 1.52.
  • Run task B for real with three or four volunteers and name cards before working it out.

Starter

  1. 25π cm2
    • π × 52 = 25π
  2. 8π cm
    • π × 8 = 8π
  3. 3/8
    • 3 out of 8
  4. 18 and 30
    • 8 parts: 48 ÷ 8 = 6. 3 × 6 = 18 and 5 × 6 = 30.

Task A: Design a heart

  1. 36 + 9π ≈ 64.3 cm2
    • Square: 6 × 6 = 36. Two semicircles of radius 3 make one circle: π × 32 = 9π.
    • 36 + 9π = 64.27…, which is 64.3 cm2 (3 s.f.).
  2. 30.8 cm
    • Two straight sides: 12 cm. Two semicircle arcs make one circle of diameter 6: 6π.
    • 12 + 6π = 30.84…
  3. 145 cm2
    • Length scale factor 9 ÷ 6 = 1.5, so area scale factor 1.52 = 2.25.
    • (36 + 9π) × 2.25 = 81 + 20.25π = 144.6…, which is 145 cm2.

Task B: Secret card swap

  1. 6 ways; 1/3
    • ABC, ACB, BAC, BCA, CAB, CBA (the card that A, B and C get, in order).
    • Nobody has their own card in BCA and CAB: 2 out of 6 = 1/3.
  2. 1/6
    • Only ABC: 1 out of 6.
  3. 3/8: slightly more likely than 1/3
    • 9/24 = 3/8 = 0.375, and 1/3 = 0.333…

Task C: Sweet hearts

  1. 2/15
    • 4/10 × 3/9 = 12/90 = 2/15
  2. 8/15
    • Red then pink: 4/10 × 6/9 = 24/90. Pink then red: 6/10 × 4/9 = 24/90.
    • 48/90 = 8/15
  3. 2/3
    • P(no red) = 6/10 × 5/9 = 30/90 = 1/3.
    • 1 − 1/3 = 2/3

Extension

  1. 4/9
    • P(all different) = 6/6 × 5/6 × 4/6 = 120/216 = 5/9.
    • 1 − 5/9 = 4/9
  2. 7.48 cm
    • Area = s2 + π(s/2)2 = s2(1 + π/4) = 1.785…s2.
    • s2 = 100 ÷ 1.785… = 56.0…, so s = 7.48.

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

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