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Themed maths · 31 October

Halloween maths activities for IGCSE and GCSE

A ready-to-teach Halloween lesson for secondary classes: a 5-minute starter, a 35-minute main activity in four short tasks, an extension and full worked answers. It practises growth, spheres, tree diagrams and straight-line graphs.

Level
IGCSE and GCSE (Higher and Extended)
Time
40 minutes, plus a 10-minute extension
Topics
Percentage and exponential growth; Volume and surface area of a sphere; Similar shapes; Tree diagrams; Coordinate geometry: gradient, midpoint, distance, perpendicular lines
Equipment
The starter is non-calculator. A calculator is needed for the main activity.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A10 minZombie outbreak
Main: task B10 minPumpkin geometry
Main: task C8 minTrick or treat
Main: task D7 minHaunted house maze
Extension10 minFast finishers or homework

Starter (5 minutes)

Quick-fire, no calculator. Show one at a time or all four on the board.

  1. A pumpkin costs £2.40. In a Halloween sale it is reduced by 25%. What is the sale price?
  2. A bag holds 5 toffees and 3 chocolate eyeballs. One sweet is taken at random. What is the probability it is a chocolate eyeball?
  3. Find the midpoint of the points (−2, 7) and (6, 1).
  4. Write 23 × 24 as a single power of 2, then work out its value.

Main activity (35 minutes)

Task A: Zombie outbreak (10 min)

At midnight on 31 October there are 50 zombies in a town. The number of zombies increases by 40% every hour.

  1. How many zombies are there after 1 hour?
  2. Explain why the number of zombies after t hours is 50 × 1.4t. Use it to find the number after 6 hours, to the nearest whole number.
  3. The town has 12 000 people. After how many whole hours does the number of zombies first go above 12 000?

Task B: Pumpkin geometry (10 min)

A pumpkin is modelled as a sphere with diameter 24 cm. Volume of a sphere = 4⁄3πr3; surface area = 4πr2.

  1. Find the volume of the pumpkin. Give your answer to 3 significant figures.
  2. Find the surface area of the pumpkin, to 3 significant figures.
  3. The pumpkin is hollowed out to make a lantern. The hollow is a sphere of radius 10 cm. What percentage of the pumpkin’s volume is removed? Give your answer to 1 decimal place.
  4. A smaller pumpkin is mathematically similar, with diameter 18 cm. Find its volume to 3 significant figures.

Task C: Trick or treat (8 min)

At each house, the probability of getting a treat is 0.7. Otherwise you get a trick. Houses are independent.

  1. Draw a tree diagram for two houses. Find the probability of two treats.
  2. Find the probability of exactly one treat.
  3. A bowl holds 6 toffees and 4 lollipops. Two sweets are taken without replacement. Find the probability that both are the same type.

Task D: Haunted house maze (7 min)

On a plan of a haunted house, a ghost glides in a straight line from the door D(1, 2) to the treasure chest C(9, 8).

  1. Find the length of DC.
  2. Find the equation of the line DC in the form y = mx + c.
  3. A secret wall runs through the midpoint of DC, at right angles to DC. Find its equation in the form ax + by = c.
  4. Is the trapdoor T(8, 1) on the secret wall? Show how you know.

Extension (10 minutes)

For fast finishers, or as homework.

  1. Scientists find a cure. Each hour the zombies first increase by 40%, then 30 of them are cured. Show that if there are exactly 75 zombies, the number never changes. What is the smallest whole number of zombies for which the outbreak still grows?
  2. Find the shortest distance from the trapdoor T(8, 1) to the ghost’s path DC.

For teachers

Teacher notes and full worked answers
  • Common slip in the zombie task: adding 40% of the original 50 every hour (linear growth) instead of multiplying by 1.4.
  • In the pumpkin task, students often use the diameter as the radius. Ask them to say the radius out loud first.
  • The percentage in part (c) needs no π at all: let students spot that the 4⁄3π cancels.
  • Extension 1 is a first look at a fixed point: a good bridge to iteration.

Starter

  1. £1.80
    • 25% of £2.40 is £0.60.
    • £2.40 − £0.60 = £1.80
  2. 3/8
    • There are 5 + 3 = 8 sweets, and 3 are eyeballs, so the probability is 3/8.
  3. (2, 4)
    • Average the x-coordinates: (−2 + 6) ÷ 2 = 2.
    • Average the y-coordinates: (7 + 1) ÷ 2 = 4.
  4. 27 = 128
    • Add the powers: 3 + 4 = 7, so 23 × 24 = 27.
    • 27 = 128

Task A: Zombie outbreak

  1. 70
    • An increase of 40% means multiplying by 1.4.
    • 50 × 1.4 = 70
  2. 376 zombies
    • Each hour the number is multiplied by 1.4, so after t hours it has been multiplied by 1.4 a total of t times.
    • 50 × 1.46 = 50 × 7.529536 = 376.4768
    • So about 376 zombies.
  3. 17 hours
    • Try values: 50 × 1.416 ≈ 10 889, which is less than 12 000.
    • 50 × 1.417 ≈ 15 245, which is more than 12 000.
    • So it first goes above 12 000 after 17 hours.

Task B: Pumpkin geometry

  1. 7240 cm3
    • The radius is 24 ÷ 2 = 12 cm.
    • 4⁄3 × π × 123 = 2304π = 7238.2…
    • 7240 cm3 (3 s.f.)
  2. 1810 cm2
    • 4 × π × 122 = 576π = 1809.5…
    • 1810 cm2 (3 s.f.)
  3. 57.9%
    • Volume removed = 4⁄3π × 103; whole pumpkin = 4⁄3π × 123.
    • The 4⁄3π cancels: 1000 ÷ 1728 = 0.5787…
    • 57.9%
  4. 3050 cm3
    • Length scale factor = 18 ÷ 24 = 3/4, so the volume scale factor is (3/4)3 = 27/64.
    • 2304π × 27/64 = 972π = 3053.6…
    • 3050 cm3 (3 s.f.)

Task C: Trick or treat

  1. 0.49
    • 0.7 × 0.7 = 0.49
  2. 0.42
    • Treat then trick: 0.7 × 0.3 = 0.21. Trick then treat: 0.3 × 0.7 = 0.21.
    • 0.21 + 0.21 = 0.42
  3. 7/15
    • Both toffees: 6/10 × 5/9 = 30/90.
    • Both lollipops: 4/10 × 3/9 = 12/90.
    • 30/90 + 12/90 = 42/90 = 7/15

Task D: Haunted house maze

  1. 10 units
    • Across 8, up 6.
    • √(82 + 62) = √100 = 10
  2. y = 3/4x + 5/4
    • Gradient = 6 ÷ 8 = 3/4.
    • Through (1, 2): 2 = 3/4 × 1 + c, so c = 5/4.
  3. 4x + 3y = 35
    • Midpoint M = (5, 5).
    • Perpendicular gradient = −4/3.
    • y − 5 = −4/3(x − 5) gives 3y − 15 = −4x + 20, so 4x + 3y = 35.
  4. Yes
    • 4 × 8 + 3 × 1 = 32 + 3 = 35, so T satisfies the equation and lies on the wall.

Extension

  1. 76 zombies
    • 75 × 1.4 − 30 = 105 − 30 = 75, so 75 stays at 75.
    • With Z zombies the next hour has 1.4Z − 30. This is bigger than Z when 0.4Z > 30, that is when Z > 75.
    • So the smallest whole number is 76.
  2. 5 units
    • T lies on the secret wall, which meets DC at right angles at M(5, 5).
    • So the shortest distance is TM = √(32 + 42) = 5.

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

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