Why do this problem?
offers a chance to think about the links between
distance and speed. By solving parts of the problem mentally, there
is an opportunity for students to use their visualisation skills
and then discuss the different ways of thinking about the
Explain the scenario to the class - the boats start from
opposite ends and always turn when they get to the other end
without slowing down. Ask them to consider whereabouts on the lake
the boats will meet for the first time if they travel at the same
speed, and where they will meet for the second time, third time and
so on. Encourage them to picture it without writing anything down
if they can. (It is important to highlight that we're interested in
the boats being in the same position on the lake, regardless of
which way they are travelling, so a meeting point could be when
they cross each other or when one overtakes the other.)
Allow students to share in pairs or small groups their answer to
this first question and their different ways of thinking about it.
Then set them the task of investigating how the meeting points
change if the ratios of the speeds of the boats are changed.
Encourage groups to share their ways of approaching the problem,
and give them a chance to describe their way of seeing it. Sharing
different strategies for recording can also be useful. Then present
the problem of finding the ratio of speeds and length of the lake
from the three statements given in the problem. The natural
extension to this is to experiment with different distances and see
how it affects the ratio of speeds.
How do you picture what happens for different ratios of
How can you record what you are picturing?
Around and Back
is a harder problem about ratio of speeds which
can be solved using some of the same techniques.
might be a simpler context to think about as
an introduction to this problem.
It is well worth looking at the problem
and its solutions to see a variety of approaches to
this sort of problem.