Something in Common: Train length
The ‘Something in Common’ collection from John Burke contains problems that draw together different areas of mathematics. The materials are ideal to use in enhancing problem solving skills and in teaching for understanding.
Each investigation comes with several different versions of a problem that have ‘something in common’. Having discovered that the problems are linked by ‘something in common’, an extension task is to prove that the links will always work. The ‘Train Length’ resource brings together work on forming and solving equations, working with rates such as speed and manipulating fractions.
Here is an example:

The full length of the train takes 10 seconds to pass a signal

The full length of the train takes 30 seconds to completely pass through a tunnel

The tunnel is 560 metres long

The train travels at a constant speed
How long is the train?
We begin by letting the length of the train be L metres. Given that the width of the signal is negligible in relation to the train length, and the trains takes 10 seconds to pass the signal, we conclude:
The tunnel is 560 metres long. To this must be added the new the length of the train L.
Hence
We now have two equations for the speed in terms of L. Equating the two equations gives
Hence, the length of the train is 280 metres.
For each of the problems in the resource, the length of the train is half the length of the tunnel. How do we choose periods of time to guarantee this relationship?
Let

T_{S} be the time to pass the signal.

T_{T} be the time to pass the tunnel.

D be the length of the tunnel.
We require that L = 0.5 D
We now have
The time for the train to pass the signal must be one third of the time for the train to pass completely through the tunnel.
We have recently added the complete collection of resources from John Burke’s ‘Something in Common’ materials, which comprise; investigations, puzzles and other enrichment activities.
You can also join us at the STEM Centre to experience ideas for teaching through understanding on Enthuse funded courses, details of which are here.