Parallelogram 18 Year 10 6 Feb 2020Smilla’s Sense of Snow

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau 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. Smilla’s Sense of Snow

First a book, then a movie, this is the story of Smilla, who grew up in Greenland and who has an incredible knowledge and feeling for snow and all its properties and secrets. Moving from Greenland to Denmark, she finds herself using her scientific and snowy background to explain a mysterious death. In this clip, she talks about the two things that are most important to her, snow and...

(If you have problems watching the video, right click to open it in a new window)

2 marks

1.1 Which type of number does Smilla hint at, but NOT mention?

  • Natural numbers
  • Negative numbers
  • Fractions
  • Irrationals

2. Swearing on Mumsnet

This graph made me smile. It shows the amount of swearing for different swear words appearing on the Brexit discussion pages on the social media site Mumsnet. I have removed the swear word labels, but you can probably guess some of them.

You might wonder why there are certain peaks? They are probably linked to the initial referendum, the election of Mrs May, submitting Article 50, etc.

The point of the this section is to look at the surprising things that people can plot.

2 marks

2.1 In 2019, there is a curious double spike for most of the swear words. Roughly, how many days apart are the two peaks?

  • 30 days
  • 60 days
  • 90 days
  • 100 days
  • 120 days

Each vertical line marks 6 months, or roughly 180 days. The space between the double peaks is about 1/3 of the distance between vertical lines, so we are looking at 1/3 x 180 days = roughly 60 days.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 The diagram shows three rectangles.

What is the value of x?

  • 108
  • 104
  • 100
  • 96
  • 92
Show Hint (–2 mark)
–2 mark

We let the angles in the triangle that is formed by the rectangles be p°, q° and r°, as shown in the figure.

The angles of a rectangle are 90°, as shown. The angles on the line at the points where the bottom rectangle meets the other two are each 180°, as shown.

The total of the angles at a point is 360°. Therefore, from the angles at the points where the rectangles meet, we have the following three equations.

p+90+90+x=360,
q+180+90+43=360, and
r+180+90+29=360.

It follows that

p=180x,
q=47, and
r=61.

Therefore, because the angles in a triangle total 180°,

180x+47+61=180,

from which it follows that

180x+108=180,

and hence x=108.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

4.1 In the diagram shown, PQ=SQ=QR and SPQ=2×RSQ.

What is the size of angle QRS?

  • 20°
  • 25°
  • 30°
  • 35°
  • 40°
Show Hint (–4 mark)
–4 mark

We let QRS=x°. Because SQ=QR, the triangle SQR is isosceles. Therefore RSQ=QRS=x°.

Because SPQ=2×RSQ, we deduce that SPQ=2x°. Since PQ=SQ, the triangle SPQ is isosceles. Therefore PSQ=SPQ=2x°.

It follows that RSP=2x°+x°=3x°.

We now apply the fact that the angles of a triangle have sum 180° to the triangle PRS. This gives x+2x+3x=180. Hence 6x=180. Therefore x=30. It follows that QRS is 30°.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The diagram shows seven circular arcs and a heptagon with equal sides but unequal angles.

The sides of the heptagon have length 4.

The centre of each arc is a vertex of the heptagon, and the ends of the arcs are the midpoints of the two adjacent sides.

What is the total shaded area?

  • 12π
  • 14π
  • 16π
  • 18π
  • 20π

The shaded area is made up of seven sectors of circles each of radius 2.

The area of a circle with radius 2 is π×22=4π.

The area of a sector of a circle is directly proportional to the angle subtended by the sector at the centre of the circle. A full circle subtends an angle of 360° at its centre. Therefore, if the total of the angles subtended by the shaded sectors shown in the figure is x×360°, the total shaded area is x×4π.

The internal angles of a polygon with n sides add up to n2×180°. Therefore, the internal angles of a heptagon add up to 5×180°.

The total of the angles at a point is 360°. Therefore the total of the angles at the seven vertices of the heptagon is 7×360°. Since the internal angles have total 5×180°, the total of the external angles is 7×360°5×180°=7×360°52×360°=92×360°.

Therefore the total shaded area is 92×4π, that is, 18π.

6. Rob Eastaway’s puzzle

Rob Eastaway is the author of several great books about maths, and his most recent publication is “Maths on the Back of an Envelope: Clever ways to (roughly) calculate anything”.

3 marks

6.1 Here is a puzzle that he recently published online.

The numbers 1 to 9 have been written on cards and placed into two columns, like this:

Here's the puzzle: move just one card so that the two columns add to the same total.

Rules:

  1. You cannot move one card so that it covers another card.
  2. You cannot turn a card over, so that the number is no longer visible.
  3. You cannot move a number to the top corner of another number, so that it becomes an exponential.

Which card do you move?

Correct Solution: 9

Show Hint (–1 mark)
–1 mark

Moving a card can mean sliding it, but it can also mean rotating it?

Show Hint (–1 mark)
–1 mark

If you find the correct solution, both columns will add to 21.

The right column sums to 21. If you rotate the 9 in left column, so it becomes a 6, it also sums to 21.

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.