Parallelogram 4 Year 10 3 Oct 2019Little Big League

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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.

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are doing at the moment in your classroom. Be prepared to encounter all sorts of weird ideas, including a few questions that appear to have nothing to do with mathematics at all.

  • Tackle each Parallelogram in one go. Don’t get distracted.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Little Big League

In the 1994 film “Little Big League”, a 12-year-old boy called Billy suddenly becomes the manager of the Minnesota Twins baseball team. In this clip, Billy has to solve a maths problem for his homework before he can focus on the rapidly approaching baseball game.

A player comes up with a correct solution, but it is not very well explained, so let’s try a similar problem.

Suppose Mr Smith can paint the entire house in 12 hours, and Mrs Smith takes just 8 hours to paint the same house. How long would it take the Mr and Mrs Smith working together to paint the house?

1 mark

1.1. First, which of the following answers is correct?

  • Working together, they will take even more time than the slower painter.
  • Working together, they will take even less time than the faster painter.
  • Working together, they will take less time than the slower painter, but more time than the quicker painter.
4 marks

1.2. How long will it take Mr and Mrs Smith to paint the house if they work together?

Give your answer in minutes (not hours and minutes).

Correct Solution: 288

Show Hint (–1 mark)
–1 mark

How much of the house can Mr Smith paint in 1 hour, and how much of the house can Mrs Smith paint in 1 hour? So, working together, how much of the house can they paint in 1 hour?

If it takes Mr Smith 12 hours to paint a house, then he paints 112 of a house per hour.

If it takes Mrs Smith 8 hours to paint a house, then she paints 18 of a house per hour.

Together they paint 112+18=524 of a house per hour.

This means they will take 245 hours to paint the house.

(This previous step sometimes makes me scratch my head. Imagine they can paint 13 of a house in 1 hour, then it will take them 31 (or 3) hours to paint the whole house.)

245 hours = 4.8 hours = 240 minutes + 48 minutes = 288 minutes.

2. Proving √2 is irrational

Two weeks ago, I mentioned that a follower of Pythagoras was executed for proving that 2 is an irrational number. But, you might be wondering, how does someone prove that a number is a irrational. How can you prove that a number cannot be written down as a fraction or a ratio?

Here is an outline of such a proof, but...

WARNING – this proof is not simple. You will have to think carefully about each and every step.

This type of proof is known as a ‘proof by contradiction’, which means we are going to start by assuming the opposite of what we want to prove. So, we are going to “try” to prove that 2 is rational! By trying to prove this, we should discover something that would be impossible, which means that our starting point (2 is rational) must have been impossible.

Let’s get going.

Let's imagine that it is possible to come up with such a ratio to produce 2.

In fact, make this the simplest ratio you can have - cancelling out any common factors. Remember this. No common factors.

Let's call it AB.

So, AB = 2

We know that A and B could not both be even, because they would then have a common factor of 2, which we would have cancelled out already.

Now, let's multiply both sides of the equation by itself, so

A2B2=2

Next multiply both sides by B2, ending up with

A2=2B2

This means that A2 must be an even number - because 2 times anything is even.

And that makes A an even number too – because it cannot be an odd number, bearing in mind that an odd number multiplied by itself is always odd.

So, let’s say A=2a, and let’s plug it into the equation and simplify.

2a2=2B2

4a2=2B2

2a2=B2

This means that B2 must be an even number - because 2 times anything is even.

And that makes B an even number too – because it cannot be an odd number, bearing in mind that an odd number multiplied by itself is always odd.

So, let’s say B=2b, and let’s plug it into the equation and simplify.

2a2=B2

2a2=2b2

2a2=4b2

Let’s re-arrange a bit more…

a2=2b2

a2b2=2

ab = 2.

Wow! We started out by saying AB was the simplest way to represent 2, but now we know that ab = 2, and we know that a and b are only half of A and B. So we have a simpler ratio for 2… but we started off saying we were already at the simplest ratio.

We have contradicted ourselves, so our initial attempt to represent 2 as a fraction must be impossible.

CONCLUSION – this proof is tricky, and you need to follow it carefully, but hopefully get a sense of how mathematicians know that 2 is irrational.

2 marks

2.1 Did you follow some or all of that?

  • Yes
  • No
2 marks

2.2 Did you find it at least a bit interesting or boring?

  • A bit interesting
  • Boring

3. Intermediate Maths Challenge Problem (UKMT)

3 marks

3.1 Amrita is baking a cake on a Thursday. She bakes a cake every fifth day.

How many days will it be before she next bakes a cake on a Thursday?

  • 5
  • 7
  • 14
  • 25
  • 35

Amrita bakes a cake again after 5, 10, 15, 20, 25, 30, 35 days and so on. Thursdays come after 7, 14, 21, 28, 35 days and so on. The first number which appears in both of these lists is 35. So Amrita next bakes a cake on a Thursday after 35 days.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 The net shown consists of squares and equilateral triangles. The net is folded to form a rhombicuboctahedron, as shown.

When the face marked P is placed face down on a table, which face will be facing up?

  • A
  • B
  • C
  • D
  • E

The diagram shows a vertical cross-section through the centre of the rhombicuboctahedron when it is placed so that the face marked P is face down.

The cross-section cuts through a ring of eight square faces. These are the eight square faces that form the central band of the net. They are labelled as shown. We see from the diagram that the face D will be facing up.

There will be more next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.