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Secondary Magazine - Issue 80: Focus on


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 14 February 2011 by ncetm_administrator
Updated on 01 March 2011 by ncetm_administrator

 

Secondary Magazine Issue 80Freight train photograph by Roger Geach used under the Creative Commons Attribution-ShareAlike 2.0 Generic (CC BY-SA 2.0) licence
 

Focus on...shunting

Shunting train photograph by Martin Addison used under the Creative Commons Attribution-ShareAlike 2.0 Generic (CC BY-SA 2.0) licence


Shunting photograph by Martin Addison

Shunting problems, in which the challenge is to rearrange railway trucks with as few shunts as possible, provide situations in which a teacher can focus on some particular ways of acting mathematically.

The Shunting of Railway Wagons is an article published in 1994 in the IMA journal, Teaching Mathematics and its Applications. In it, Alan Jasper, a secondary school mathematics teacher, describes how he became fascinated by shunting problems when he first came across them in Hugh Burkhardt’s book, The Real World and Mathematics - which is about the place of real problem solving in the mathematics curriculum.
 

It [the shunting of railway wagons] was used as an example to show how binary notation could be used for automatic decision-making. The article gave a worked example, outlining the method involved—and it worked. I couldn't see why (the method worked) but I wanted to try it out on some examples of my own. I was now hooked! I wanted to know more and more about it and also to give some of my pupils the opportunity of having a go!

Alan Jasper begins his quest to try to find some ‘principles’ of efficient shunting by deciding to explore first a particular, relatively simple, shunting situation, about which he asks himself some questions:

  • Can I rearrange the trucks in the order that I want by shunting them about on this track layout?
  • If I achieve my aim, how can I know whether or not I have done it in the least number of shunts?
  • What tactics should I use in this situation?

To help himself address these questions, Alan adopts a convention (about always moving the trucks in one siding before moving the trucks in another siding) in the hope that by keeping the process as orderly as possible some effective tactics will become apparent. He also develops a notation, which he uses to show actions and results in a way that he hopes will facilitate his thinking about them.

In doing these things, Alan is already acting mathematically by:

  • first exploring a simple case
  • setting himself a specific challenge, and asking questions about it
  • adopting a convention which he makes explicit
  • developing a notation in order to reveal structure.
orange circles

Getting started
Alan Jasper describes how he introduced shunting to students in a Year 7 mixed ability class and in a Year 11 ‘Higher Level’ class so that the students in both classes became fruitfully engaged in exploring shunting ‘tactics’.

I decided to give my Year 7 (mixed ability) class the first opportunity to have a go at Shunting. I was concerned as to how long it would take them to get to grips with it but I was surprised at how quickly they picked it up. I think this was partly due to the fact that I asked several pupils to act as wagons in order to demonstrate the shunting process and then allowed the class to solve simple problems—by moving pupils (who were carrying lettered cards). This was later reinforced by moving some magnetic letters about on the white board and using this to develop the notation I wanted them to use.

A simple introductory problem for students to tackle first – possibly as Alan describes in his article – might be, for example:

Shunt the trucks from...                        ...to...
 
railway shunting diagram

The challenge is to shunt the trucks from their original order (blue, green, red – or ABC) on the inward line, so that eventually they are on the outward line in a new order (green, blue, red – or BAC).

It could be done in the following way.

Engine comes in on the outward line, and shunts (pulls) blue truck and green truck into siding:

railway shunting diagram

Points are changed, and engine shunts (pushes) blue truck and green truck onto the outward line:

railway shunting diagram

Engine goes back to get red truck:

railway shunting diagram

Engine shunts (pulls) red truck into siding:

railway shunting diagram

Points are changed, and engine shunts (pushes) red truck onto the outward line:

railway shunting diagram

Engine goes off (down the inward line) leaving the three trucks on the outward line in the new order:

railway shunting diagram

Ten engine-movements, five point-changes, one un-coupling of trucks and one coupling of trucks are required to do this simple bit of shunting!

In Starting Points (Banwell, Saunders and Tahta, 1986) we are reminded that:

Discussion may reveal that there are two basic operations – bringing a truck (or pupil) into the siding and taking one out of the siding. These may be coded by two letters or symbols. One class chose ‘I’ and ‘O’ for ‘in’ and ‘out’ and then gradually changed to numerals 1 and 0.

If we adopt this notation, then shunting from order ABC in direction of intended travel to BAC is represented by 110010, as was shown above.

Students could explore the different orders into which it is possible, on this particular track layout with one siding, to shunt three trucks, then four trucks, then five trucks…  (They will need to make assumptions about the maximum number of trucks that the siding can hold at any one time).

Railway points  photograph by S. Terfloth used under the Creative Commons Attribution-ShareAlike 2.5 Generic (CC BY-SA 2.5)  licence
Railway points photograph by S. Terfloth

They might also try to find conditions that a string of 0s and 1s necessarily satisfies if it represents the shunting of a train of trucks from one order to another.

green circles 

More complex situations
A natural next step might be to investigate track situations with more than one siding, extending, developing or re-inventing our notation in order to record more complex shunts.

For example, Alan Jasper in The Shunting of Railway Wagons challenges himself to rearrange five trucks from the order DCEAB on a mainline to the order ABCDE, by shunting the trucks in and out of two sidings. He labels the sidings, S0 and S1, and solves the puzzle in two stages, each of which consists of several pushes and pulls, and which he denotes as Shunt 1 and Shunt 2. (Students will realise that they need to know precisely to what they and others are referring when they use the word ‘shunt’. Here, Alan Jasper is using ‘complete shunt’ to mean putting each truck from a queue into either S1 or S2, and then shunting them back to produce a new sequence of trucks.) The following diagrams show the positions of the trucks at crucial times during the whole process, together with Alan Jasper’s notation.

Shunt 1 changes the order from DCEAB to DEABC

railway shunting diagram

railway shunting diagram

S0 : DEAB
S1 : C

railway shunting diagram

Shunt back S1 : C
Shunt back S0: DEABC

Shunt 2 changes the order from DEABC to ABCDE

railway shunting diagram

S0 : ABC
S1 : DE

railway shunting diagram

Shunt back S1 : DE
Shunt back S0 : ABCDE

Alan Jasper’s Shunt 1 and Shunt 2, shown above, would both in reality involve a sequence of several actions.
 
It is possibly because, in reality, even relatively simple shunts require many pushes, pulls, point changes, un-couplings and couplings that shunting can appear complex, and the ‘structure’ of successive arrangements of trucks in different locations can therefore be obscured.

Shunting yard photograph by Rajithmohan used under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) licence
Shunting yard photograph by Rajithmohan

You may like to compare your students’ effective approaches, and how they choose to record their findings, with the ways of displaying and recording combinations of shunts developed by Alan Jasper and his students. If you are feeling brave you can try to follow how ‘a meaningful use of number bases … arose naturally’.

purple circles 

Classic model railways shunting puzzles

Model railway photograph by Benkid77 used under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)  licence
Model railway photograph by Benkid77

Model railway enthusiasts enjoy constructing various track layouts, and then shunting trucks on them! Two well-known layouts for model railway shunting puzzles are the classic British Inglenook Sidings and the American Timesaver.

Students can set up many different particular Inglenook Sidings puzzles. The challenge is to use an engine that is originally on an inward mainline, A, to assemble on an outward line, D, a train that is composed of five trucks which you have pre-selected from eight trucks that are originally in two sidings, B and C. Therefore, if you start with this ….

Inglenook Sidings style puzzle

… you can choose to try to achieve many different truck arrangements on the outward line.

On the more complex Timesaver layout, the challenge is to move trucks from starting positions that you set up yourself to selected destinations that you also set up yourself. Therefore, again, it is possible to explore very many different particular problems.

Students may enjoy shunting trucks virtually on interactive computer representations of the two model-railway shunting layouts just mentioned. They can explore a Riverside Yard puzzle, which is on the Inglenook Sidings layout, and the Skowhegan & Athens puzzle, which is posed in the Timesaver situation. The Dawson Station Railroad Shunting Puzzle is another interactive Inglenook Sidings puzzle.

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Some other starting points
A problem which is simple to pose, but not so simple to solve, is the classic sequential movement railway shunting puzzle:

How do you get two trains travelling in opposite directions on a single track to pass each other using just one siding that can hold only one truck?

trains travelling in opposite directions puzzle

Students may be interested in this discussion of a real shunting situation in Sweden.

At NRICH students will find a truck Shunting Puzzle and another Shunting Puzzle, in which they move, or visualise moving, counters on a board.

Image Credits
Page header - Shunting train photograph by Roger Geach some rights reserved
Shunting train photograph by Martin Addison some rights reserved
Railway points photograph by S. Terfloth some rights reserved
Shunting yard photograph by Rajithmohan some rights reserved
Model railway photograph by Benkid77 some rights reserved

 
 
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