Parallelogram 21 Year 10 5 Mar 2020Pieces of Eight

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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. Pieces of 8

2 marks

1.1 What is interesting about this equation?

61 - (8 + 8 + 8 + 8 + 8)

  • The answer is the same if it is reflected left/right
  • The answer is the same if it is reflected up/down
  • The answer is the same if it is rotated by 90°
  • The answer is the same if it is rotated by 180°
  • The answer is the same if it is rotated by 270°

The answer is the same if it is rotated by 180°

61 - (8 + 8 + 8 + 8 + 8)

=

(8 + 8 + 8 + 8 + 8) - 19

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1. Four of these tiles may be put side by side so that they simultaneously spell two imperial units of length. Which tile is left out?

  • A
  • B
  • C
  • D
  • E
Show Hint (–1 mark)
–1 mark

Think in terms of feet and inches, but longer.

The words are MILE and YARD.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 The time shown on a digital clock is 5.55. How many minutes will pass before the clock next shows a time for which all the digits are the same?

  • 71
  • 255
  • 316
  • 377
  • 436

The digits will next be all the same at 11.11, i.e. in 5 hours and 16 minutes time.

4. Intermediate Maths Challenge Problem (UKMT)

5 marks

4.1. The diagram shows a regular dodecagon (a polygon with twelve equal sides and equal angles).

What is the size of the marked angle?

  • 67.5°
  • 72°
  • 75°
  • 82.5°
  • 85°
Show Hint (–1 mark)
–1 mark
Show Hint (–1 mark)
–1 mark

Each side of the dodecagon subtends an angle of 30° at the centre of the circumcircle of the figure (the circle which passes through all 12 of its vertices). Thus AOB=150°

Show Hint (–1 mark)
–1 mark

The angle subtended by an arc at the centre of a circle is twice the angle subtended by that arc at a point of the circumference, APB

Each side of the dodecagon subtends an angle of 30° at the centre of the circumcircle of the figure (the circle which passes through all 12 of its vertices).

Thus AOB=150° and, as the angle subtended by an arc at the centre of a circle is twice the angle subtended by that arc at a point of the circumference, APB=75°.

5. Factoring quadratic equations

Some of you will have learned about factoring quadratic equations. This video by Po_Shen Loh gives a new spin on an ancient (and largely forgotten) approach to factoring quadratics, which you might find helpful.

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

Using the approach in the video, or any other technique, factorise the quadratic below and find the possible values of x.

x220x+64=0

2 marks

5.1 What is the larger value of x?

Correct Solution: 16

2 marks

5.2 What is the smaller value of x?

Correct Solution: 4

x220x+64=x4x16=0

so x = 16 or 4.

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.