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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Coke Machine

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem challenges learners to use their visualisation skills to gain an understanding of what happens when a coin rolls inside a rectangle, while providing a context for practising methods of calculation of circumferences and arcs.

Possible approach

Start by asking the learners to consider the question "How far forward would a bicycle travel if its wheels turned through one complete revolution?" Then show the first part of the problem, and ask what the difference is between the bicycle problem and the coin in the box problem - this should make it clear that the corners are key.

Now show the second diagram. Intuition may suggest that if the coin is travelling on two sides, each side would not need to be as long in order to get the whole circumference to touch, but having a corner where part of the circumference doesn't touch makes things interesting! Learners could draw corners on paper and roll a cardboard circle along them, highlighting on their circle the parts that touch and the parts that don't.

Now look at what happens when the coin rolls around the inside of a tray. Ask the learners to discuss in pairs whether all of the circumference of the coin will touch on one circuit of a $4$ by $3$ tray, and then share ideas before trying it with the interactivity.

One way of recording what happens is to draw a line $14$ units long (perimeter of tray) and mark all the key sections such as corners, and the points where the coin has made a complete revolution.

Key questions

How far does the centre of the coin travel as it makes one revolution?
What happens at the corners?

Possible extension

Try the problem Five Circuits, Seven Spins.
There are more ideas, explanations and problems to work on in the article A Rolling Disc.
How many times does the disc rotate about its own centre when it makes one revolution around the tray?
What happens if corners are not $90^\circ$?

Possible support

Physically manipulating a circular object inside a frame can make it clearer what's happening at the corners.
The problem Roundabout may be a good starting point.