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On the Road

Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at 17.00 then it overtook the bike at 18.00. At what time did the bike and the scooter meet?

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There and Back

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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Escalator

At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?

Swimmers

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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 crossing similarly.

Next involve some numbers and some calculation to connect some arithmetic with the context that students are starting to visualise.

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 problem solving.

Key questions :

  • Why wasn't the second crossing in the same place as the first?

  • If the swimmers can keep this going indefinitely will they have eventually had a crossing point everywhere along the length of the pool?

  • 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.

An Unhappy End

Circuit Training