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Themed maths · 14 February

Valentine’s Day maths for IB Maths SL and HL

A ready-to-teach lesson for 14 February: a 5-minute starter, a 35-minute main activity on a heart made from simple shapes, the probability that nobody draws their own name in a secret card swap and the area of a heart-shaped curve, an extension and full worked answers.

Level
IB Maths SL and HL (AA and AI)
Time
40 minutes, plus a 10-minute extension
Topics
Area and perimeter of composite shapes; Counting and probability; Graphs of implicitly defined curves; Definite integrals and area; Optimisation with a GDC
Equipment
The starter is non-calculator. A GDC is needed for the main activity.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A10 minA heart from simple shapes
Main: task B13 minSecret card swap
Main: task C12 minA heart-shaped curve
Extension10 minFast finishers or homework

Starter (5 minutes)

No GDC.

  1. Find the area of a circle of radius 3, in terms of π.
  2. Work out 4!.
  3. Two fair coins are tossed. Find the probability of two heads.
  4. Solve |x|2/3 = 4.

Main activity (35 minutes)

Task A: A heart from simple shapes (10 min)

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

  1. Take s = 10. 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 card designer wants a heart of this shape with area 500 cm2. Find s, to 3 significant figures.

Task B: Secret card swap (13 min)

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

  1. In how many different ways can the four cards be handed out?
  2. List the ways in which nobody gets their own card. Find the probability that nobody gets their own card.
  3. Find the probability that exactly one friend gets their own card.
  4. (HL) For a class of 30, the probability that nobody gets their own card is very close to 1/e. Find 1/e to 3 significant figures.

Task C: A heart-shaped curve (12 min)

The curve x2 + (y − |x|2/3)2 = 1 is heart-shaped. Plot it on your GDC as two functions.

  1. Show that the curve is y = |x|2/3 ± √(1 − x2) for −1 ≤ x ≤ 1. Find where it crosses the y-axis.
  2. Show that the vertical distance between the two halves of the curve at x is 2√(1 − x2). Hence explain why the area inside the heart equals the area of a circle of radius 1, and write it down.
  3. Use your GDC to find the highest points of the heart, to 3 significant figures.

Extension (10 minutes)

A matching problem.

  1. People each choose a whole number from 1 to 20 at random. Find the probability that two people choose the same number. How many people are needed for the probability that at least two of them match to be more than ½?

For teachers

Teacher notes and full worked answers
  • Run task B for real: shuffle four name cards with four volunteers a few times before working it out.
  • In task B (c), many students think ‘exactly three right’ is possible: ask why it is not.
  • Task C is a good GDC-skills task: students need both halves of the curve and must use |x| correctly.

Starter

  1. 9π
    • π × 32 = 9π
  2. 24
    • 4 × 3 × 2 × 1 = 24
  3. 1/4
    • ½ × ½ = ¼
  4. x = 8 or −8
    • |x| = 43/2 = 8, so x = ±8.

Task A: A heart from simple shapes

  1. 100 + 25π ≈ 179 cm2
    • Square: 102 = 100. Two semicircles of radius 5 make one circle: π × 52 = 25π.
    • 100 + 25π = 178.5…, which is 179 cm2 (3 s.f.).
  2. 51.4 cm
    • Two straight sides: 20 cm. Two semicircular arcs make one circle of diameter 10: 10π.
    • 20 + 10π = 51.41…
  3. 16.7 cm
    • Area = s2 + π(s/2)2 = s2(1 + π/4).
    • s2 = 500 ÷ 1.7853… = 280.0…, so s = 16.73…

Task B: Secret card swap

  1. 24
    • 4! = 24
  2. 9 ways; 3/8
    • Call the friends A, B, C, D. The orders with nobody in their own place are BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBA: 9 of them.
    • 9/24 = 3/8
  3. 1/3
    • Choose the lucky friend: 4 ways. The other three must all miss: 2 ways (for example BCA or CAB).
    • 4 × 2 = 8, and 8/24 = 1/3.
  4. 0.368
    • 1/e = 0.36787…
    • The exact probability for 30 people agrees with 1/e to many decimal places.

Task C: A heart-shaped curve

  1. (0, 1) and (0, −1)
    • (y − |x|2/3)2 = 1 − x2, so y − |x|2/3 = ±√(1 − x2).
    • At x = 0: y = ±1.
  2. π
    • Top minus bottom: (|x|2/3 + √(1 − x2)) − (|x|2/3 − √(1 − x2)) = 2√(1 − x2).
    • That is the same as for the circle x2 + y2 = 1, so the area is the same: ∫−11 2√(1 − x2) dx = π.
  3. (±0.617, 1.51)
    • Maximise y = x2/3 + √(1 − x2) for 0 < x < 1: the maximum is at x = 0.6166…, y = 1.5117…
    • By symmetry the other top is at x = −0.617.

Extension

  1. 1/20; 6 people
    • Two people: the second matches the first with probability 1/20.
    • With k people, P(no match) = (20 × 19 × … × (21 − k)) ÷ 20k.
    • k = 5: P(no match) = 0.581…, so P(match) = 0.419. k = 6: P(no match) = 0.436…, so P(match) = 0.564 > ½.

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

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