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/www/nrich/html/content/99/12/bbprob1/
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2011-02-01T00:00:01
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<p>I was on a train the other day and I was looking out of the
window as we went through a station and I saw lots and lots of
carriages waiting in another line. They were going to be linked
together to make a new train going on a long journey - so I thought
!!</p>
<p>It was then that my train went round a curve in the track and I
looked out of the window and saw the front of my train and quickly
turned my head and saw the back of the train.</p>
<p>After a few more miles I had seen that my train had ten
carriages to make up the whole train.</p>
<p style="text-align: left;"><mdo:image height="40" width="511" alt="pic1" src="Picture%201.jpg"></mdo:image></p>
<br></br>
<p>I thought back about the carriages I had seen at the station and
wondered about making them into several trains.</p>
<p style="text-align: left;"><mdo:image height="131" width="210" src="Picture%202.jpg" alt=""></mdo:image></p>
<br></br>
<p>I thought my train was rather special having ten carriages so I
want to put this challenge to you all.</p>
<ol>
<li>Suppose there are $24$ carriages.</li>
<li>They're going to be put together to make up some trains.</li>
<li>The smallest train you are allowed is one with
two carriages.</li>
<li>You must include at least one "ten-carriage train".</li>
</ol>
<p>That's all really.</p>
<p>Here are some to start you off.</p>
<p style="text-align: left;"><mdo:image height="224" width="509" alt="pic3" src="Picture%203.jpg"></mdo:image></p>
<br></br>
<p>You could draw these.<br></br>
You could use cubes/ beads/ boxes or whatever to stand for the
carriages. So it might also look like:-</p>
<mdo:image src="fig1.gif" alt=""></mdo:image><br></br>
<p>Next:-</p>
<p><mdo:image height="177" width="512" alt="pic4" src="Picture%204.jpg"></mdo:image></p>
<br></br>
<p>and if you used squares or cubes it might look like:-</p>
<mdo:image src="fig2.gif" alt=""></mdo:image><br></br>
<p>So now it's your turn. See what different train arrangements you
can make.</p>
<p>Remember the four "Rules" above.</p>
<p>
<comment> #set var="roll-text" value="You can scan, photocopy, draw - then cut, rearrange and paste. Messy!" </comment>
<comment> #set var="roll-text" value="" </comment></p>
<p>It would be good to think if you have been using some kind of
special way of getting new ones. Maybe you've found a pattern that
helps you get lots of answers?</p>
<p>Please write to us and tell us about your ways of doing
this.</p>
<p><comment> #set var="roll-text" value=
"You can scan, photocopy, draw - then cut, rearrange and paste. Messy!" </comment>
<comment> #set var="roll-text" value="" </comment></p>
<p>FINALLY</p>
<p>You simply have to ask "I WONDER WHAT WOULD HAPPEN IF ...?"</p>
<p>Here are some to start you off.</p>
<p>"I wonder what would happen if I had to only make
three trains?"<br></br>
"I wonder what would happen if I had to make only
four trains?"<br></br>
"I wonder what would happen if I had to have all the trains
different lengths?"<br></br>
and so on<br></br>
and so on<br></br>
and so on.</p>
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<p><span class="editorial">There are lots of answers to this
problem, depending on what questions you choose to ask.</span></p>
<br></br>
<br></br>
<p class="editorial">Have a go yourself, and if you discover
anything interesting,<a href="mailto:%20nrich@damtp.cam.ac.uk">e-mail</a> us to tell us what
you've done! Please don't worry that your solution is not
"complete" - we'd like to hear about anything you have tried.
Teachers - you might like to send in a summary of your children's
work.</p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=73&part=">activity</a> appeals to many pupils much more than being presented with "sums" to do. It may make use of number bonds and facts that the pupils already know.</div>
<br></br>
<h3>Possible approach</h3>
<div>Use something fairly large to represent the $24$ carriages - even carriages from a toy train set would be great! You could create an IWB file that allowed you to create multiple copies of the carriage and to move them around the screen.</div>
<br></br>
<div>Encourage children to suggest some ways of making the trains to start with and display them using whatever you have chosen. If possible, keep these to be referred to later. Give children time to work in pairs on the challenge. You may want them to put each solution on a separate strip of paper, because then you could use these in the plenary to order the solutions in some way and this will
help the group work out if they have missed any out.</div>
<br></br>
<h3>Key questions</h3>
<div>How many carriages here?</div>
<div>Which train has most carriages?</div>
<div>How many carriages have you used?</div>
<br></br>
<h3>Possible extension</h3>
<div>Explore the results for $20, 21, 22$ and $23$ carriages, and compare them.</div>
<div>Encourage children to ask "I wonder what would happen if ...?".</div>
<br></br>
<h3>Possible support</h3>
<div>Some of the youngest pupils may need help in counting accurately and not counting the same carriages twice.</div>
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<mdoxml version="1.0">This is a rather useful activity for pupils as young as year 1/2 -
more able. The activity came about in order to achieve something
which would be interesting for the pupils and take the children on
in their understanding of place value. On the whole I find that
youngsters can work very well with numbers up to about 36 without
having to start grouping in tens. <br></br>
<br></br>
But when the need arises to start place value considerations I
believe it is good to use numbers that they have already dealt
with, without them considering the grouping of the tens. The
results from children have shown that this activity allows for a
lot of discussion and the beginnings of certain patterning ideas.
<br></br>
<br></br>
It has also been useful to help pupils start creating their own
systems for doing these kinds of investigations. I shall be very
pleased to hear how very, very able year 1/2 children do and how
far they extend it.<br></br></mdoxml>
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Train Carriages
Suppose there is a train with 24 carriages which are going to be
put together to make up some new trains. Can you find all the ways
that this can be done?
Using, Applying and Reasoning about Mathematics
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