Parallelogram 10 Year 10 21 Nov 2019The Other Hamilton

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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.
  • 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. Radical mathematics

Khan Academy has created dozens of great online videos about mathematics, and hundreds more about a ton of other subjects, ranging from history to science. Take a look at this one, which is all about “Comparing irrational numbers with radicals”.

Did that make sense? If not, then re-watch the video before trying the questions below.

2 marks

1.1. Which is bigger: 42 or 33?

  • 42 is bigger
  • 33 is bigger
  • Neither, they are the same

If we square both numbers 422=16×2 and 332=9×3, so the former is bigger.

2 marks

1.2. Which is bigger: 23 or 5?

  • 23 is bigger
  • 5 is bigger
  • Neither, they are the same

If we square both numbers 232=4×3 and 52=5, so the former is bigger.

2 marks

1.3. Which is bigger: 34 or 36?

  • 34 is bigger
  • 36 is bigger
  • Neither, they are the same

If we square both numbers 342=9×4 and 362=36, so they are the same.

Alternatively, 34=3×2=6, while 36=6.

2. William Rowan Hamilton

The biggest theatre show in recent years to hit New York is a musical called “Hamilton”. It tells the story of Alexander Hamilton, one of the founding fathers of the United States of America, and the man responsible for setting up the American banking system. Many of the songs involve rap.

It is incredible that someone (the brilliant Lin-Manuel Miranda) was able to create a hit show based on the life of a long-dead politician, whose life story was unknown to many Americans.

Anyway, what has this got to do with maths? Tim Blais has a created a song and video based on the opening song from “Hamilton” – it is not about American politician Alexander Hamilton, but rather it is about the Irish mathematician William Rowan Hamilton. Watch the video and answer the three questions below.

1 mark

2.1 How cool was that?

  • Very cool
2 marks

2.2 William Rowan Hamilton invented (or discovered) a new type of number, which you will not cover during your time in school, even in sixth form. These numbers are called...

  • imaginary
  • complex
  • transcendental
  • quaternions
  • irrational
2 marks

2.3 The song describes one of the key properties of Hamilton’s newly discovered number system. Which number is equivalent to all of these? i2 = j2 = k2 = ijk

  • -1
  • 0
  • 1
  • x
  • e

3. Intermediate Maths Challenge Problem (UKMT)

3 marks

3.1 Each of the diagrams below shows a circle and four small squares. In each case, the centre of the circle is the point where all four squares meet.

In one of the diagrams, exactly one third of the circle is shaded. Which one?

Show Hint (–1 mark)
–1 mark

The straight edges of the shaded sector which occupies one third of the area of the circle will make an angle equal to one third of a complete revolution, that is, an angle of 13×360°=120°.

In diagram A the straight edges of the shaded sector make an angle of 90°. In diagram C these edges make an angle equal to 90° + 45° = 135°. The angles in diagrams D and E are even larger. So in none of these cases is exactly one third of the circle shaded.

This leaves diagram B where the angle is between 90° and 135°. Therefore, this is the diagram where the angle is 120°. Hence it is the diagram where the shaded area occupies one third of the circle.

4. Intermediate Maths Challenge Problem (UKMT)

5 marks

4.1 A list of positive integers has a median of 8, a mode of 9 and a mean of 10.

What is the smallest possible number of integers in the list?

  • 5
  • 6
  • 7
  • 8
  • 9
Show Hint (–2 mark)
–2 mark

Because the list has a mode of 9, the integer 9 must occur at least twice in the list. Therefore, as the mean is 10, the list must include at least one integer which is greater than 10.

Because the list has a mode of 9, the integer 9 must occur at least twice in the list. Therefore, as the mean is 10, the list must include at least one integer which is greater than 10.

So the list includes at least three integers that are greater than 8. Because the median is 8, the list must also include at least three integers that are less than 8. It follows that there are at least six integers in the list.

We now try and find a list of six integers with the required properties. We can suppose from what we already know that the list, in increasing order, is

p,q,r,9,9,u

where r<8 and u>10.

For this list to have median 8, we need to have 12r+9=8, and therefore r=7. So the list, in increasing order is

p,q,7,9,9,u

with u>10.

For this list to have mode 9, no other integer can occur more than once in the list. Therefore pq and q7. So p<q<7.

For this list to have mean 10, we need to have p+q+7+9+9+u=6×10=60, that is, p+q+u+25=60.

Therefore we require that p+q+u=35, with p<q<7, and u>10.

There is more than one way to choose p, q and u so as to meet these conditions. For example, the list

5, 6, 7, 9, 9, 24

has a median of 8, a mode of 9 and a mean of 10.

We have found a list of 6 positive integers with the given properties. We have shown that no list of fewer than 6 positive integers has these properties. We conclude that the smallest possible number of integers in the list is 6.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 Every day, Aimee goes up an escalator on her journey to work. If she stands still, it takes her 60 seconds to travel from the bottom to the top. One day the escalator was broken so she had to walk up it. This took her 90 seconds.

How many seconds would it take her to travel up the escalator if she walked up at the same speed as before while it was working?

  • 30
  • 32
  • 36
  • 45
  • 75
Show Hint (–2 mark)
–2 mark

We suppose that the distance from the bottom to the top of the escalator is d metres.

Since it takes Aimee 60 seconds to travel this distance when she stands still, the escalator moves upwards at a speed of d60 metres per second.

Since it takes Aimee 90 seconds to walk from the bottom to the top of the escalator when it is not moving, Aimee walks up the escalator at a speed of d90 metres per second.

Show Hint (–1 mark)
–1 mark

It follows that when Aimee is walking at the same speed up the moving escalator she is moving at a speed of d60+d90 metres per second.

We suppose that the distance from the bottom to the top of the escalator is d metres.

Since it takes Aimee 60 seconds to travel this distance when she stands still, the escalator moves upwards at a speed of d60 metres per second.

Since it takes Aimee 90 seconds to walk from the bottom to the top of the escalator when it is not moving, Aimee walks up the escalator at a speed of d90 metres per second.

It follows that when Aimee is walking at the same speed up the moving escalator she is moving at a speed of d60+d90 metres per second. Now

(d/60 + d/90) = d(1/60 + 1/90) = d ((3 + 2)/180) = d (5/180) = d/36.

So Aimee moves up the escalator at a speed of d36 metres per second. At this speed it takes Aimee 36 seconds to travel the d metres up the escalator.

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.