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.