5806
/www/nrich/html/content/id/5806/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Well, here's a train route. The train starts at the top and makes a
number of visits to the stations. <br></br>
Now let's suppose that the train is going to make visits to three
stations (they do not have to be different stations - each station
can be visited several times!).<br></br>
<div style="text-align: center;"><mdo:image width="223" height="507" alt="Journeys" src="Journeys.jpg"></mdo:image></div>
<div style="text-align: left;">So the first station would be Dorby
- this will always be the case! (Why?)</div>
<div style="text-align: left;">Then the train can go on to Ender.
When at Ender it could return and visit Dorby again OR it could go
on to Floorin.</div>
<div style="text-align: left;">So two different journeys:</div>
<div style="text-align: left;">Dorby - Ender - Floorin</div>
<div style="text-align: left;">Dorby - Ender - Dorby</div>
<div style="text-align: left;"></div>
<div style="text-align: left;"></div>
<p style="text-align: left;">Your challenge is to find all the
different journeys for visiting four stations.</p>
<div style="text-align: left;"></div>
<div style="text-align: left;">You could then go on to find all the
different journeys for visiting more stations - try five.</div>
<div style="text-align: left;">How about six stations?</div>
<div style="text-align: left;"></div>
<p style="text-align: left;">Can you predict the number of
different journeys for visiting seven stations? Were you right?</p>
<div style="text-align: left;">How would you predict the number of
different journeys for visiting eight stations?</div>
<div style="text-align: left;"></div>
<div style="text-align: left;"></div>
<div style="text-align: left;"></div>
<p style="text-align: left;">You might like to then invent your own
routes that may go further than this one and then answer similar
questions that you can think up.</p>
<div style="text-align: left;"></div>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p><span class="editorial">We had a number of solutions come in
with answers from Connor, Isobel, Brian, Josh, Stephanie and
Ben.</span></p>
<p class="editorial">H.C. from SIS School sent us this very good
answer:</p>
I started calculating the number of ways through 4, 5 and 6
stations first.<br></br>
I did it in an orderly manner.You must start with Dorby and then
Ender and Dorby again so you will not miss any of the ways. Then
you can start with Dorby, Ender and Floorin and so on ... <br></br>
Here is the way I used from 4 stations to 6 stations: <br></br>
<br></br>
4 stations: <br></br>
1.Dorby-Ender-Dorby-Ender <br></br>
2.Dorby-Ender-Floorin-Ender <br></br>
3.dorby-Ender-Floorin-Gambolin <br></br>
<br></br>
5 stations: <br></br>
1.Dorby-Ender-Dorby-Ender-Dorby <br></br>
2.Dorby-Ender-Dorby-Ender-Floorin <br></br>
3.Dorby-Ender-Floorin-ender-Dorby <br></br>
4.Dorby-Ender-Floorin-Ender-Floorin <br></br>
5.Dorby-Ender-Floorin-Gambolin-Floorin <br></br>
<br></br>
6 stations: <br></br>
1.Dorby-Ender-Dorby-Ender-Dorby-Ender <br></br>
2.Dorby-Ender-Dorby-Ender-Floorin-Ender <br></br>
3.Dorby-Ender-Dorby-ender-Floorin-Gambolin <br></br>
4.Dorby-Ender-Floorin-Ender-Dorby-Ender <br></br>
5.Dorby-Ender-Floorin-Ender-Floorin-Ender <br></br>
6.Dorby-Ender-Floorin-Ender-Floorin-Gambolin <br></br>
7.Dorby-Ender-Floorin-Gambolin-Floorin-Ender <br></br>
8.Dorby-Ender-Floorin-Gambolin-Floorin-Gambolin <br></br>
<br></br>
3rd stat - 2 ways <br></br>
4th stat - 3 ways (2+1) <br></br>
5th stat - 5 ways (3+2) <br></br>
6th stat - 8 ways (5+3) <br></br>
7th stat - 12 ways (8+4) <br></br>
<br></br>
<p class="editorial">This is great - you have really gone about it
in a logical way.</p>
<p><span class="editorial">So, I wonder if anyone can predict how
many ways could be found for 8 stations? This is an example of the
Fibonacci sequence. You can find out more about this special
pattern of numbers in</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=2470&part=index" class="editorial">this article</a> <span class="editorial">.</span></p>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Simple Train Journeys</h2>
<br></br>
Well, here's a train route. The train starts at the top and makes a number of visits to the stations.<br></br>
Now let's suppose that the train is going to make visits to three stations (they do not have to be different stations - each station can be visited several times!).<br></br>
<div style="text-align: center;"><mdo:image alt="Journeys" height="507" src="Journeys.jpg" width="223"></mdo:image></div>
<div style="text-align: left;">So the first station would be Dorby - this will always be the case! (Why?)</div>
<div style="text-align: left;">Then the train can go on to Ender. When at Ender it could return and visit Dorby again OR it could go on to Floorin.</div>
<div style="text-align: left;">So two different journeys:</div>
<div style="text-align: left;">Dorby - Ender - Floorin</div>
<div style="text-align: left;">Dorby - Ender - Dorby</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;"> </div>
<p style="text-align: left;">Your challenge is to find all the different journeys for visiting four stations.</p>
<div style="text-align: left;"> </div>
<div style="text-align: left;">You could then go on to find all the different journeys for visiting more stations - try five.</div>
<div style="text-align: left;">How about six stations?</div>
<div style="text-align: left;"> </div>
<p style="text-align: left;">Can you predict the number of different journeys for visiting seven stations? Were you right?</p>
<div style="text-align: left;">How would you predict the number of different journeys for visiting eight stations?</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;"> </div>
<div style="text-align: left;"> </div>
<p style="text-align: left;">You might like to then invent your own routes that may go further than this one and then answer similar questions that you can think up.</p>
<div style="text-align: left;"> </div>
</div>
<br></br>
It might be helpful to try <a href="http://nrich.maths.org/public/viewer.php?obj_id=5807&part=index">Train Routes</a> before tackling this problem which will familiarise children with the way a train route might be pictured, and will help them develop systems for finding all possibilities.<br></br>
<br></br>
Initially, this problem could be introduced in a similar way as is suggested in the notes for <a href="http://nrich.maths.org/public/viewer.php?obj_id=5807&part=note">Train Routes</a> , but it would be good to focus more on looking for patterns and generalising in this case. You might like to work on the different routes for four stations as a whole class then ask small groups to look at five
and six stations so that you can pool results. Ask the children how they are recording the different routes - using initial capitals to stand for the stations is a great help, but share any good ways the pupils have found.<br></br>
<br></br>
In order to look for a pattern in the numbers of routes, it might be helpful to make a table, something like this:<br></br>
<br></br>
<table style="" border="1">
<tbody>
<tr>
<td style="text-align: center;">Number of station visits</td>
<td style="text-align: center;">Number of different journeys</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">2</td>
</tr>
<tr>
<td style="text-align: center;">4</td>
<td style="text-align: center;"> </td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;"> </td>
</tr>
</tbody>
</table>
<div>and so on ...</div>
<br></br>
<div>Encourage the class to look carefully at how the number of different journeys in each case is related to the number of different journeys for smaller numbers of station visits. Once they have identified the pattern, ask them to think about why the pattern occurs.</div>
<br></br>
<div>Making up their own rail networks and investigating them with similar questions would be a good next step. Alternatively, you could challenge them to devise networks with certain criteria.</div>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">How will you record the different journeys?<br></br>
Have you got a way of making sure you have got them all?<br></br>
Perhaps there is a way you could use the three-station journeys to
help you find all the four-station journeys?<br></br>
When you're looking for a pattern in the numbers, it might help to
go even simpler as well - how many journeys would there be for just
two stations?<br></br>
Can you see any way that the numbers of different journeys are
connected to each other? It might help to write them in a list or
table. <br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Abbreviating the stations to D, E, F & G<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="407" height="310" src="Ans2TrainsSimple.jpg" alt="Answers"></mdo:image></div>
<div style="text-align: center;"></div>
<br></br>
<br></br>
<br></br></mdoxml>
2
3
1
1
0
0
0
Simple Train Journeys
How many different journeys could you make if you were going to
visit four stations in this network? How about if there were five
stations? Can you predict the number of journeys for seven
stations?
Sequences, Functions and Graphs
Fibonacci sequence
Using, Applying and Reasoning about Mathematics
Working systematically
Decision Mathematics and Combinatorics
Route inspection problems
Using, Applying and Reasoning about Mathematics
Mathematical modelling
Using, Applying and Reasoning about Mathematics
Generalising