Week 5Breakfast curiosity – Part 2

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Fermi questions

Last week, I asked you a question about pouring the UK’s cereal bowls into Wembley stadium. This sort of question is known as a Fermi question, because the great physicist Enrico Fermi enjoyed asking his students these questions. They generally involve some guesswork, some general knowledge and some arithmetic. Here are two more Fermi questions for you to think about.

2 marks

1.1. How many times could you say the alphabet in 24 hours? Assume that you will need to take appropriate breaks and will speak at a normal pace.

  • 100
  • 500
  • 5,000
  • 75,000
  • 750,000

The alphabet contains 26 letters, so you have say 26 words (ay, bee, see, dee, eee) and that takes about 10 seconds at normal speed. So, that’s 6 alphabets in a minute, 360 in an hour, but let’s say 300/hour allowing for breaks. During a day, you might be able to do this for 15 hours (to allow for lunch and sleep and toilet breaks). So I reckon you could say the alphabet 4,500 times in a day.

The idea for this question came from Teacher Tool Kit, where you will find lots of other Fermi questions.

4 marks

1.2. If the whole world moved to Wales, and spread themselves out evenly, how much space would each person have on average?

  • 0.3 m2 per person
  • 3 m2 per person
  • 30 m2 per person
  • 300 m2 per person
  • 3000 m2 per person

First, we need to know the area of Wales. You could look it up and find that the answer is 20,735 km2. However, in Fermi questions you should really try and work it from sensible guesses. My vision of Wales is a largely rectangular block, 100 km wide by 200 km north to south, which is an area of 20,000 km2 (which is actually quite close to the true answer).

I could do this now or later, but I am going to convert this into (m2), as it will be easier to work with. There are 1 million (1,000 × 1,000) m2 in a km2, so that is 20,000,000,000 m2 or 20 billion m2.

The world population is 7 billion. This is something you should probably already know.

That means the area per person = area ÷ population = 20 billion ÷ 7 billion ≈ 3 m2 per person.

2. Curious Cheerios

Last week, we looked at the magnetic properties of cereal, and this week we’ll look at the weird properties associated with cheerio buoyancy. Watch carefully, have your mind blown, rebuild your brain and then answer the question that follows.

1 mark

2.1. Which creatures benefit from the Cheerio’s effect?

  • Birds
  • Fish
  • Amphibians
  • Worms
  • Insects

3. Intermediate Maths Challenge Problem

3 marks

3.1 The diagram shows a rectangle placed on a grid of 1 cm × 1 cm squares.

What is the area of the rectangle in cm2?

  • 15
  • 22.5
  • 30
  • 36
  • 45
Show Hint (–1 mark)
–1 mark

Try working out the area of everything except the rectangle you are most interested in.

Show Hint (–1 mark)
–1 mark

The area of the big square is 8 × 8 = 64 cm2. Each pair of triangles can be joined to form a square of 3 × 3 and 5 × 5 cm2.

The total area of the 8 × 8 grid in cm2 is 82 = 64.

The two larger triangles in the top left and bottom right corners of the grid make up a 5 × 5 square with area 25 cm2.

The two smaller triangles in the other corners of the grid make up a 3 × 3 square with area 9 cm2.

Hence the area of the rectangle, in cm2, 64 - 25 - 9 = 30.

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelogams.

Cheerio, Simon.