Parallelogram 16 • Level 1 • 21 Dec 2023Wobbly circles

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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.
Finish by midnight on Sunday 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.

1. Keep on rolling!

Circles, famously, roll pretty well.

Watch this video where Matt Parker joins two circles to make a wobbly shape, which still rolls surprisingly well.

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

1 mark

1.1 When rolling a circle, what happens to its center of mass?

It stays at a constant height
It moves up and down
It remains in contact with the surface its rolling on
(Not answered)

2 marks

1.2 When rolling a square, what happens to its center of mass?

It stays at a constant height
It moves up and down
It remains in contact with the surface its rolling on
(Not answered)

2 marks

1.3 In the wobbly circles, where is the center of mass?

It changes from one center to the other
The midpoint between the centers
It has two centers of mass
(Not answered)

2 marks

1.4 On a penny farthing bicycle the big wheel has a circumference of 3m and the small wheel has a circumference one quarter the size of the big wheel.

After the bike has traveled 9m, what is the total number of revolutions of both wheels combined?

9
4
15
0
(Not answered)

Wheels turn along their circumference, so after 9m the big wheel has completed 93=3 revolutions.

The small wheel, however, has a circumference one quarter the size, so 34m or 75cm.

900cm / 75cm = 12 revolutions.

So in total the wheels have turned 3 + 12 = 15 revolutions.

2. Going round in circles.

2 marks

2.1 In the diagram, the smaller circle touches the larger circle and also passes through its centre.

What fraction of the area of the larger circle is the smaller circle?

The small circle is half the area of the large
The small circle is one third the area of the large
The small circle is one quarter the area of the large
The small circle is one sixth the are of the large
(Not answered)

First think of a square: if we half the length, what happens to the area?

A 6cm square has an area of 36cm², if we half that length we get a 3cm square with area 9cm².

So the area has been quartered!

We can apply this to the circle too, if we half the diameter we will quareter the area (because it is both half the width AND the height of the original circle).

You might also be able to look at this dagram and convince yourself (correctly) that the shaded area is equivalent to the area of the smaller circles.

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 week.
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.
The next Parallelogram is next week, at 3pm on Thursday.
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 and Ayliean.