9772
/www/nrich/html/content/id/9772/
1
2013-01-05T11:45:43
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<mdoxml version="1.0">I've been playing around with cuisenaire rods, this is what I came to.<br></br>
There are five different ways for us to make the pink rod using just red and white rods.
<div style="text-align: center;"><mdo:image src="KS3%20Q2.gif"></mdo:image></div>
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<div style="text-align: center;"> </div>
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We count white, white, red as different from the white, red, white even though they both use two white rods and one red rod.<br></br>
<div style="text-align: center;"><mdo:image src="2egs.jpg"></mdo:image></div>
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Using the interactivity below, can you work out how many different ways there are, using only the red and white rods, to make up:<br></br>
<br></br>
<ul>
<li>The light green rod?</li>
<li>The yellow rod?</li>
<li>The dark green rod?</li>
<li>The black rod?</li>
</ul>
<div><a href="/content/id/4338/cuisenaire.swf">Full Screen Version</a></div>
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Without using the interactivity, how many different ways are there to make up the orange rod (equivalent to 10 white rods)?<br></br>
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Can you explain the pattern?<br></br>
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<mdoxml version="1.0"><br></br>
<p class="editorial">Well done to Hannah who sent in this
generalisation:</p>
Using the red and white blocks only, you add together the amounts
of ways to form the two previous blocks:<br></br>
2 ways to form the red block + 3 ways to form the light green
block<br></br>
= 5 ways to form the pink block<br></br>
<br></br>
Using the red, white and light green blocks only, you add together
the amounts of ways to form the three previous blocks:<br></br>
2 ways to form the red block + 4 ways to form the light green block
+ 7 ways to form the pink block<br></br>
= 13 ways to form the yellow block<br></br>
<br></br>
<p class="editorial">It is interesting to think about why this
happens.</p>
<div><span class="editorial">Imagine that you are just using the
white and red blocks only and that you have found that there are 2
ways to form the red block and 3 ways to form the light green
block.</span></div>
<div><span class="editorial">In order to form the pink rod you will
need to either add a red block to the 2 ways you formed the red
block or add a white block to the 3 ways you formed the light green
block.</span></div>
<div><span class="editorial">This will give you the 5 ways of
forming the pink rod.</span></div>
<p class="editorial">You can use similar reasoning to explain
Hannah's other solution regarding the use of red, white and light
green blocks.</p>
<br></br></mdoxml>
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<mdoxml version="1.0"><div class="embed">
<h2>Rods and Rods</h2>
I've been playing around with cuisenaire rods, this is what I came to.<br></br>
There are five different ways for us to make the pink rod using just red and white rods.
<div style="text-align: center;"><mdo:image src="KS3%20Q2.gif"></mdo:image></div>
<br></br>
<div style="text-align: center;"> </div>
<br></br>
<br></br>
We count white, white, red as different from the white, red, white even though they both use two white rods and one red rod.<br></br>
<div style="text-align: center;"><mdo:image src="2egs.jpg"></mdo:image></div>
<br></br>
<div style="text-align: center;"> </div>
<br></br>
Using the interactivity below, can you work out how many different ways there are, using only the red and white rods, to make up:<br></br>
<br></br>
<ul>
<li>The light green rod?</li>
<li>The yellow rod?</li>
<li>The dark green rod?</li>
<li>The black rod?</li>
</ul>
<div><a href="/content/id/4338/cuisenaire.swf">Full Screen Version</a></div>
<mdo:flash height="400" id="/content/id/9772/cuisenaire.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="/content/id/9772/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>
<br></br>
<br></br>
Without using the interactivity, how many different ways are there to make up the orange rod (equivalent to 10 white rods)?<br></br>
<br></br>
Can you explain the pattern?<br></br>
<h3><br></br>
</h3>
</div>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org.uk/9772">This problem</a> can help pupils extend their spatial understanding related to number sense. It can be used to acquaint pupils with the attributes of the cuisenaire rods.<br></br>
<br></br>
<h3>Possible approach</h3>
<div>A time to play with the rods if pupils are not used to using them would be essential. If you do not have access to the rods then pupils could have some time with the general cuisenaire environment to be found <a href="http://nrich.maths.org.uk/4348">here.</a> The challenge could begin my working on the pink rod ideas altogether and having some clear discussion as to why the two examples
shown lower down although using the same rods are counted as different.<br></br>
The pupils can then work indiviually or in groups to tackle the other questions.<br></br>
<h3>Key questions</h3>
Do you think there are any more to find?<br></br>
Are any of yours the same? (Good to ask both when there <em>is</em> and <em>is not</em> a slip-up in their examples)<br></br>
Tell me about how you found these.<br></br>
<h3>Possible extension</h3>
Suggest other different coloured pairs of rods that could be tested to see if they can be put together to equal the largest of the rods.<br></br>
Two bigger rods can be put together to for a much longer length for the pupils to try to work on using pairs of different rods (see here below)
<div style="text-align: center;"><mdo:image src="ext.jpg"></mdo:image></div>
<h3>Possible support</h3>
Some pupils who are using the rods and have problems with fine motor skills made need to have someone to arrange the rods as they require.</div></mdoxml>
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<mdoxml version="1.0"><br></br>
Pick any colour of rod. Consider how many different ways there are to make it with a white rod on the right-hand end, and how many ways there are with a red rod on the right-hand end.<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">w: 1 (w)
<br></br><br></br>r: 2 (ww, r)
<br></br><br></br>lg: 3 (www, wr, rw)<br></br><br></br>
(p: 5)
<br></br><br></br>y: 8 (wwwww, wwwr, wwrw, wrww, wrr, rwww, rwr, rrw)
<br></br><br></br>dg: 13 (wwwwww, wwwwr, wwwrw, wwrww, wwrr, wrwww, wrwr, wrrw, rwwww, rwwr, rwrw, rrww, rrr)
<br></br><br></br>b: 21
<br></br><br></br>Fibonacci sequence: To make a rod of length x, we can add a white rod to all the different rods of length x-1 or we can add a red rod to all the rods of length x-2. Let w(x) = number of ways of making rod length x, then we have w(x) = w(x-1) + w(x-2).
<br></br><br></br>For white, red and light green rods, we have w(1) = 1, w(2) = 2, w(3) = 4 and for x > 3, w(x) = w(x-1) + w(x-2) + w(x-3). <br></br><br></br>In general, using rods of length n, m, ..., p, once we've established answers up to w(p), we have for x > p, w(x) = w(x-n) + w(x-m) + ... + w(x-p).<br></br></mdoxml>
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Rods and Rods
Using only the red and white rods, how many different ways are there to make up the other colours of rod?
Numbers and the Number System
Patterned numbers