Why do this problem:
Non-standard problems can stimulate deeper thinking.
At first glance it may seem surprising that the two given
distances fix the length of the pool. This context will also
stimulate visualisation, as students 'picture' the motion of each
swimmer and then watch for the moments when they cross.
Possible approach :
It is important that the context is well understood so that
visualisation can be productive. To help this, bring two volunteer
walkers to the front of the classroom. Place them on opposite sides
and direct one walker to move a little more briskly than normal and
the other to walk a little more slowly.
Set them off and at their first crossing pause them and mark the
position. Set them off again and catch the position of the second
Next involve some numbers and some calculation to connect some
arithmetic with the context that students are starting to
Invent a pool length (say 100m) and two swimming speeds (2m/s and
2.5 m/s are good). Invite students to calculate the crossings that
happen with these values.
The values can then be altered, using calculators when required,
until students have a good feel for this problem context.
Use the questions and additional activity below to draw students
into this simple but potentially rich context of enquiry and
Key questions :
Why wasn't the second crossing in the same place as the
If the swimmers can keep this going indefinitely will they have
eventually had a crossing point everywhere along the length of the
What happens if they both swim twice as fast?
What would happen if the pool was twice as long?
Possible extension :
- What, in general, is the connection between the two given
distances and the length of the pool?
- What would be the effect, if any, of having both distances
given from the same end?
- Do any patterns emerge as you consider the sequence of crossing
places or crossing times? And if so can you account for these?
- Think about how you have visualised these swimmers going up and
down the pool, what have you pictured in your mind to help you
solve this problem? Can you connect any of your results to the
questions above into that visualisation?
- Do any of the results prompt you to look for new ways to
visualise this general situation?
Possible support :
These two problems may be easier introductions to this kind of
context for some students.