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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?

Jumping Gerbils

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem:

This is certainly an exercise in 'visualisation' or rather an exercise in working through various visualisations until one is reached that does what is needed.

There are a number of ways to approach this problem but seeing it as a line being consumed from both ends simultaneously is a particularly helpful one.

Students may have to work quite hard just to find a first visualisation and then continue their effort until they find a visualisation that lets them 'see' how the problem may be solved.

Rates are a fundamental idea in mathematics and this problem offers a challenging context in which to encounter and consider these kinds of measure.

Possible approach :

For students who need a 'concrete' phase follow the idea in the 'Possible support' section below. For more able students the questions below should provide enough of a prompt and plenty of time should be allowed in which they can work towards a first visualisation and its subsequent improvements. Using the prompt questions in the 'Possible support' section too early will rob students of the important opportunity to arrive at their own visualisation.

For the very ablest students this problem provides a valuable context in which they may gain confidence and 'feel' for this type of problem.

Key questions :

  • What are we asked to find ?

  • What else might it be helpful to know so we can do that ?

  • Is there a way we might discover that ?

Possible extension :

Encourage abler students to see the connection between this context and the Swimmers problem.

Possible support :

Set up a line of 'tins' (plastic cubes maybe, the number doesn't matter but 20 maybe)
Have one student consuming as the gerbil and another student consuming as a labelling machine from the other end.
Ask students what information is needed so that the gerbil and the machine can eat their way along the line a second's worth at a time.

This should establish a visualisation and the next step is probably trial and improvement, adjusting the machine's rate and the number of 'tins' until the students can engage with the values given in the problem.
  • How long did each run last ?
  • How many more tins did the gerbil manage when it ran faster ?
  • How much less consuming did the machine manage on the gerbil's second run ?