Jumping Gerbils
Problem
A strange scene : imagine a conveyer belt, with tins placed at regular intervals onto the belt at one end, moving along at a steady rate towards a labelling machine at the far end. A gerbil starts from a tin at the beginning of the belt and steadily jumps from tin to tin, finally jumping off immediately before the labelling machine.
When the gerbil jumps at a rate of one jump per second he arrives at the labelling machine in 20 jumps but if he speeds up and manages two jumps per second he can fit in 32 jumps before the end of the line.How many tins are there on the line ?
Getting Started
- 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 ?
Student Solutions
James from Wellingborough had the best explanation with his answer
The gerbil jumps at a rate of one jump each second, first time, and two per second, the second time, and if the labelling machine consumes tins at a constant m 'jumps' per sec and if there are j 'jumps' taken between the gerbil and the machine together then j is the rate times the time taken :
j = 20( m + 1 )
and
j = 16( m + 2 ) . . . time is 16 seconds because the gerbil makes 32 jumps at 2 jumps per second
Therefore 80 jumps, which means 81 tins if the machine is just about to take a tin when the gerbil starts or 80 tins if it just has taken one.
Teachers' Resources
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 ?