Why do this
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.
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
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
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
How far does the centre of the coin travel as it makes one
What happens at the corners?
Try the problem
Five Circuits, Seven Spins.
There are more ideas, explanations and problems to work on in the
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 $90^\circ$?
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.