Rollin' Rollin' Rollin'
Why do this problem?
This
problem is interesting because the answer is not at all obvious
and will challenge students' perceptions. It is a great problem in
visualisation and translating a visual concept into numbers or
algebra.
Possible approach
Start with the problem of two disks of the same size, one
rolling around the other.
Discuss the problem and canvass the class for their
instinctive response: how many revolutions will the first disk
make? Ask them to go off and justify this response. Those who
incorrectly assume that the disk turns one time will either become
aware of their error or produce a faulty justification of their
result.
The class could then discuss their result and give their
explanations (faulty or otherwise). Choose two students with
different results to present their ideas. Who convinces the rest of
the class? [note: if the whole class is correct, then who can give
the clearest explanation? If the whole class is incorrect then
watch the animation and ask at what point the rolling disk is in
the same orientation as at the start]
Even when the correct answer has been worked out, students are
likely to want to demonstrate this with physical objects such as a
pair of coins.
The activity contains many extensions which are likely to
become accessible once the initial problem is grasped.
Key questions
- How far does the centre of the rolling disk travel?
- Can you visualise the locus of a point on the edge of the
rolling disk?
Possible extension
Can students find a rule for the number of revolutions when a disk
of radius $r$ rolls around a disk of radius $m$? Rolling around a
polygon could also usefully be investigated.
Possible support
If you have access to 'Spirograph'-type resources the demonstration
of a disk rolling around another becomes easier due to the teeth on
the wheels. Playing with the interactivity for various settings
also will help to develop intuition.