Speeding Boats

Why do this problem?

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 problem.

Possible approach

This printable worksheet may be useful: Speeding Boats

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.

Key questions

How do you picture what happens for different ratios of speeds?

Possible extension

Around and Back is a harder problem about ratio of speeds which can be solved using some of the same techniques.