4332
/www/nrich/html/content/id/4332/
1
2011-02-01T00:00:01
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Matt made a train with the Cuisenaire rods. It looked like
this:<br></br>
<br></br>
<mdo:image width="569" height="29" src="rods1.gif" alt="light green rod, pink rod, black rod, dark green rod"></mdo:image><br></br>
<br></br>
Katie made one too, exactly the same length, with just white rods.
Hers looked like this:<br></br>
<br></br>
<mdo:image width="568" height="29" src="rods2.gif" alt="20 white rods"></mdo:image><br></br>
<br></br>
Can you make some trains the same length as Matt's train, with rods
of just one colour? <br></br>
How many different ones can you make? <br></br>
<br></br>
You may like to use the interactivity below to help you. Click on
'Rods' to choose your rods. <br></br>
<br></br>
<a href="/content/id/4332/cuisenaire.swf">Full Screen Version</a>
<br></br>
<mdo:flash height="400" width="550"><param value="/content/id/4332/cuisenaire.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param><param value="400" name="height" ></param><param value="550" name="width" ></param></mdo:flash><br></br>
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<p class="editorial">Thank you for your responses to this problem.
Particularly good explanations were sent from Danny who goes to
High Ash School, Shubha from Northveil Elementary and Abhijit from
Bhavans School. Danny said:</p>
<div>First, we counted how many squares made up the train.
$20$.</div>
<br></br>
<div style="clear: both;">Then we worked out that we could use $10$
red rods, each of length $2$, to make a train of length $20$.<br></br>
<div style="clear: both;">Then we used $5$ pink rods each of length
$4$.<br></br>
<div style="clear: both;">Then $4$ yellow rods each of length
$5$.<br></br>
<div style="clear: both;">Then $2$ orange rods each of length $10$.</div>
<div style="clear: both;"></div>
<p class="editorial" style="clear: both;">Shubha gave a little bit
more detail:</p>
<div style="clear: both;"></div>
<div style="clear: both;">You have to look at the multiplication
tables for the numbers $1$ through $10$.
<br></br>
<div style="clear: both;">Find all the combinations that have an
answer of $20$. They are $1 \times 20$ = $20$;
$2 \times 10$ = $20$;
$4 \times 5$ = $20$;
$5 \times 4$ = $20$;
$10 \times 2$ = $20$
<br></br>
<div style="clear: both;">This means that by using the following
combinations we can make the train length to be $20$ blocks:<br></br>
<div style="clear: both;">$20$ rods that have a length of $1$
block<br></br>
<div style="clear: both;">$10$ rods that have a length of $2$
blocks<br></br>
<div style="clear: both;">$5$ rods that have a length of $4$
blocks<br></br>
<div style="clear: both;">$4$ rods that have a length of $5$
blocks<br></br>
<div style="clear: both;">$2$ rods that have a length of $10$
blocks</div>
<div style="clear: both;">So it means that you can only make $5$
different trains of the same length as Matt's train.</div>
<div style="clear: both;"></div>
<p class="editorial" style="clear: both;">(The first train you
mention, Shubha, has already been made by Katie in the question, so
there are four others as Danny concluded.) Well done to you
all.</p>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
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<h2>Same Length Trains</h2>
<br></br>
Matt made a train with the Cuisenaire rods. It looked like this:<br></br>
<br></br>
<mdo:image alt="light green rod, pink rod, black rod, dark green rod" height="29" src="rods1.gif" width="569"></mdo:image><br></br>
<br></br>
Katie made one too, exactly the same length, with just white rods. Hers looked like this:<br></br>
<br></br>
<mdo:image alt="20 white rods" height="29" src="rods2.gif" width="568"></mdo:image><br></br>
<br></br>
Can you make some trains the same length as Matt's train, with rods of just one colour?<br></br>
How many different ones can you make?<br></br>
<br></br>
You may like to use the interactivity below to help you. Click on 'Rods' to choose your rods.<br></br>
<br></br>
<a href="/content/id/4332/cuisenaire.swf">Full Screen Version</a><br></br>
<mdo:flash height="400" id="/content/id/4332/cuisenaire.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/4332/cuisenaire.swf"></param>
<param name="flashplayerversion" value="7"></param>
<param name="height" value="400"></param>
<param name="width" value="550"></param>
</mdo:flash><br></br>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=4332&part=index">This problem</a> is a good way for children to gain familiarity with factors and multiples in a non-threatening environment.</div>
<br></br>
<div>You could also make this an opportunity to encourage the children to have a system for making sure that they find all the possible solutions.</div>
<br></br>
<h3>Possible approach</h3>
<div>Using the interactivity on an interactive whiteboard or via a projector would be a good way to introduce the problem.</div>
<br></br>
<div>Ideally, it would be good for the pupils to then work in pairs with "real" Cuisenaire rods and talk about how they are solving the problem.</div>
<br></br>
<div>Returning again to the whiteboard and interactivity will allow the whole group to share their solutions.</div>
<br></br>
<div>You could model starting with the red rod and working up to the green, then pink, then yellow etc if the children themselves do not find some good ways.</div>
<br></br>
<h3>Key questions</h3>
<div>How many white rods did Katie use?</div>
<div>How many red rods did you need to make the same length?</div>
<div>Which colour rods fit in exactly?</div>
<div>Which colour rods cannot be fitted in exactly?</div>
<div>How will you know that you have found them all?</div>
<div>How can you record what you have done?</div>
<br></br>
<h3>Possible extension</h3>
Learners could try using different numbers of white rods to make "same length trains" with rods of just one colour. Using $21$, $22$, $23$ and $24$ could prove interesting.<br></br>
<h3>Possible support</h3>
Try to use real Cuisenaire rods if at all possible, otherwise use the interactivity and work through the different lengths. You could suggest recording on squared paper.<br></br>
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You could try with the next shortest rod first.<br></br>
Have you counted how many white rods Katie used? How might that help? <br></br>How will you know that you have found them all?<br></br></mdoxml>
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10 x 2<br></br>
5 x 4<br></br>
4 x 5<br></br></mdoxml>
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Same Length Trains
How many trains can you make which are the same length as Matt's,
using rods that are identical?
Mathematics Tools
Cuisenaire rods
Information and Communications Technology
Interactivities
Numbers and the Number System
Factors and multiples
Using, Applying and Reasoning about Mathematics
Working systematically