Parallelogram 6 Year 8 17 Oct 2019The world’s greatest thief

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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 Sunday 27 Oct 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.

As most of you will have a half-term coming up, this Parallelogram is longer than usual and has a couple of extra questions. And your next parallelogram will be in two weeks time on 31st October, and it will be appropriately terrifying.

1. Just a few seconds

1 mark

1.1. When was a million seconds ago, roughly?

  • Yesterday
  • A week or so ago
  • A month or so ago
  • A year or so ago

1,000,000 / (60 × 60 × 24) = 11.6 days, or roughly a week ago.

1 mark

1.2. When was a billion seconds ago, roughly?

  • 3 years ago
  • 30 years ago
  • 300 years ago
  • 3,000 years ago

We can use our previous answer to give us a head start. We can jump to stating that:

1 billion seconds = 1,000 × 11.6 days = 11,600 days = 32 years ago.

1 mark

1.3. What was happening a trillion seconds ago, roughly?

  • Queen Victoria was on the throne
  • The Romans ruled Europe
  • The Bronze Age began
  • The Ice Age was at its coldest

We can use our previous answer to give us a head start. We can jump to stating that:

1 trillion seconds = 1,000 × 32 years = 32,000 years.

1 mark

1.4. The Earth’s history is divided into different Periods. What is the name of period that existed a quadrillion seconds ago?

  • Neogene Period
  • Paleogene Period
  • Cretaceous Period
  • Jurassic Period

We can use our previous answer to give us a head start. We can jump to stating that:

1 quadrillion seconds = 1,000 × 32,000 years = 32 million years.

The Paleogene Period lasted from 23 million to 66 million years ago.

2. The World’s Greatest Thief

Apollo Robbins is the world’s greatest thief, because he is the most incredible pickpocket that you will ever see. Fortunately, he steals only for entertainment.

When you watch this clip, bear in mind that Apollo Robbins has spent years studying and practicing pickpocketing in order to reach his level of immense skill. It is the same with anything, including mathematics – if you put in the effort, practice and concentration, then you can be a really good mathematician. True mastery of a discipline is never easy.

You will probably have to watch clip two or three times to appreciate all of Apollo Robbins’s clever manoeuvres.

2 marks

2.1 When the coin falls into his hand from above, where does it come from?

  • From out of his hand
  • From out of thin air
  • From a bird flying by
  • From the top of his head
  • From the volunteer

3. Junior Maths Challenge Problem (UKMT)

This is a tough Junior Maths Challenge problem, so you might need to rely on the hints. But, first, try it without the hints.

4 marks

3.1 Three congruent squares overlap as shown. The areas of the three overlapping sections are 2 cm2, 5 cm2 and 8 cm2 respectively.

The total area of the non-overlapping parts of the square is 117 cm2.

What is the side-length of each square?

  • 6 cm
  • 7 cm
  • 8 cm
  • 9 cm
  • 10 cm
Show Hint (–2 mark)
–2 mark

The total area of the three squares is the sum of the area of the non-overlapping parts and twice the areas of the overlapping sections, as each of these forms part of two squares. Why? If we look at, say, the 2 cm2 overlap, this appears in two overlapping squares, which is why it needs to be counted twice.

Show Hint (–1 mark)
–1 mark

Once you have the total area of all three squares, remember to divide the total area by three to find the area of each square. Then take the square root of one of the areas to find the length of one side of one of the squares.

The total area of the three squares is the sum of the area of the non-overlapping parts and twice the areas of the overlapping sections, as each of these forms part of two squares.

So the total area of the squares is, in cm2, 117 + 2 × (2 + 5 + 8) = 117 + 2 × 15 = 117 + 30 = 147.

Hence the area of one of the squares is one third of this, that is, 49 cm2.

Therefore the side-length of each square is 7 cm.

4. Relative speeds

This is an amazing video. Take a look.

It shows that two objects moving at the same speed and in the same direction behave towards each other as if nothing is moving. If the camera only showed the trampoline and the bouncing man, then you would have no idea that anything was moving.

It is pretty much the same situation when you consider that you are standing on the Earth. The Earth is spinning and you are moving at the same speed on its surface, so when you jump up in the air, you don’t land in a different place.

3 marks

4.1 How fast, approximately, do people move if they live on the Equator?

(The answers are given in miles per hour, because the answer works out quite nicely.)

  • 10 mph
  • 100 mph
  • 1,000 mph
  • 10,000 mph
  • 100,000 mph
Show Hint (–1 mark)
–1 mark

The Earth’s equator is roughly 25,000 miles round.

Show Hint (–1 mark)
–1 mark

How long does it take for someone at the equator to spin all the way around, along with the Earth, back to starting position?

The Earth is 25,000 miles round at the equator and it takes 24 hours to pin, so people at the equator move 25,000 miles in 24 hours, or at a speed of roughly 1,000 mph.

3 marks

4.2 Do people in the United States of America move faster or slower than people at the equator?

  • Faster
  • Slower

The United States of America is north of the equator, as you can see in the diagram on the left, so Americans (and Brits, and most of the people in the world) don’t travel so far each day, so they travel slower. The diagram on the right shows how the lines of latitude (the rings around the Earth) get shorter as you head north (or south) of the equator.

5. Line Rider

This is one of my favourite “games” at the moment – Line Rider. There is a billion tons of maths and physics involved in playing this game well, but there is a also a hell of a lot of fun. Take a look and see if you build a brilliant track. Visit Line Rider... but remember to hit the submit before you get too obsessed with Line Rider.

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 time.
  • 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.
  • It’s half-term, so the next Parallelogram is at 3pm on Thursday, 31st October.
  • 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 Parallelograms.

Cheerio, Simon.