2724
/www/nrich/html/content/id/2724/
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2011-02-01T00:00:01
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I have some rods that are different colours and different
lengths.<br></br>
<br></br>
I could make a yellow rod using white and red rods in many
different ways. Here are some of them:<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image height="195" width="200" src="yellowredwhite.png" alt=""></mdo:image></div>
<p>Can you find all the ways to match the yellow rod using reds and
whites?</p>
<p>How will you know that you haven't missed any out?</p>
<p>You could use the interactivity below to try out your ideas, or
you could use our <a href="http://nrich.maths.org/4348&part=">Cuisenaire
Environment</a>.</p>
<div> </div>
<div><a href="/content/id/2724/Cuisenaireywr.swf">Full screen
version</a></div>
<mdo:flash height="400" width="550"><param value="/content/id/2724/Cuisenaireywr.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>
<br></br>
The dark green rod is longer than the yellow rod.<br></br>
Can you find all the ways of matching the dark green rod using
white and red rods?<br></br>
How will you know that you have found them all?<br></br>
<br></br>
<p>Why not use the interactivity below to find all the ways of
matching the dark green rod, or again, you could use our <a href="http://nrich.maths.org/4348&part=">Cuisenaire
Environment</a>.</p>
<br></br>
<a href="/content/id/2724/Cuisenairegwr.swf">Full screen
version</a><br></br>
<br></br>
<mdo:flash height="450" width="550"><param value="/content/id/2724/Cuisenairegwr.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param><param value="450" name="height" ></param><param value="550" name="width" ></param></mdo:flash><br></br>
<br></br>
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<p class="editorial">Jack, Alice, Phoebe, Eva and Holly from
Georgeham C of E School explained their findings:</p>
<br></br>
We did all the 1s.<br></br>
<br></br>
Then we did one 2 block on the right filling in the spaces with
1s.<br></br>
We moved the 2 block over a square, filled that with 1s.<br></br>
Then we moved the 2 block over again and filled the space with
1s.<br></br>
We moved the 2 block for the last time with 1s in the spaces.<br></br>
<br></br>
The pattern for two 2 blocks is... there's a single white square on
the left with two 2 blocks on the right.<br></br>
There's another with the single white square in between the two 2
blocks.<br></br>
The third is the single white square on the right with the two 2
blocks on the left.<br></br>
<br></br>
<p class="editorial">So, I think that makes eight ways in total.
I love the way you have done this in a careful order.
As Chelsea from Templars Primary said:</p>
... the trick is to work systematically ...<br></br>
<p class="editorial">Laura sent in her solution as pictures copied
from the interactivity, which is very helpful - thank you,
Laura.</p>
<p class="editorial">Here is her solution to the first challenge
which we can compare with the solution from Georgeham:</p>
<br></br>
<mdo:image width="194" height="300" alt="partitioning 5" src="cuissol1.gif"></mdo:image><br></br>
<br></br>
<p class="editorial">Laura's pictures are in a slightly different
order to the children from Georgeham's. I wonder which you
think is more systematic? </p>
<p class="editorial">Here Laura shows the thirteen different ways
to make the green rod using reds and whites:</p>
<br></br>
<mdo:image width="190" height="391" alt="partitioning 6" src="cuissol2.gif"></mdo:image><br></br>
<br></br>
<p class="editorial">Would you have done them in the same
order? Why or why not?</p>
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<h2>Cuisenaire Counting</h2>
<br></br>
I have some rods that are different colours and different lengths.<br></br>
<br></br>
I could make a yellow rod using white and red rods in many different ways. Here are some of them:<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image alt="" height="195" src="yellowredwhite.png" width="200"></mdo:image></div>
<p>Can you find all the ways to match the yellow rod using reds and whites?</p>
<p>How will you know that you haven't missed any out?</p>
<p>You could use the interactivity below to try out your ideas, or you could use our <a href="http://nrich.maths.org/4348&part=">Cuisenaire Environment</a>.</p>
<div> </div>
<div><a href="/content/id/2724/Cuisenaireywr.swf">Full screen version</a></div>
<mdo:flash height="400" id="/content/id/2724/Cuisenaireywr.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/2724/Cuisenaireywr.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>
The dark green rod is longer than the yellow rod.<br></br>
Can you find all the ways of matching the dark green rod using white and red rods?<br></br>
How will you know that you have found them all?<br></br>
<br></br>
<p>Why not use the interactivity below to find all the ways of matching the dark green rod, or again, you could use our <a href="http://nrich.maths.org/4348&part=">Cuisenaire Environment</a>.</p>
<br></br>
<a href="/content/id/2724/Cuisenairegwr.swf">Full screen version</a><br></br>
<br></br>
<mdo:flash height="450" id="/content/id/2724/Cuisenairegwr.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/2724/Cuisenairegwr.swf"></param>
<param name="flashplayerversion" value="7"></param>
<param name="height" value="450"></param>
<param name="width" value="550"></param>
</mdo:flash><br></br>
<br></br>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>In <a href="http://nrich.maths.org/2724&part=">this activity</a>, children have an environment in which to explore partitioning of numbers. Of course, it is essential to have cuisenaire rods available for them to solve this problem practically too. These notes are written to suggest that the activity could be used to focus on sharing good ways to find all possibilities
which can then be applied to further challenges. </div>
<br></br>
<h3>Possible approach</h3>
<div>If your children are not already familiar with Cuisenaire rods, it would be good to give them time to 'play' with the rods before having a go at this activity.</div>
<div> </div>
<div>If you are working with a small group, it might be appropriate to introduce the first challenge (making a yellow with reds and whites) using the rods themselves. If you are working with the whole class, then it might be better to use the <a href="/content/id/2724/Cuisenairegwr.swf">interactivity</a> or <a href="http://nrich.maths.org/4348&part=">Cuisenaire Environment</a> on the
interactive whiteboard. Challenge the children to find a way of making the yellow from red and/or white rods, and ask a pupil to make an arrangement. Ask for another, different, arrangement and invite someone else to make that next to the first. Then set them off to see whether they can find all the different ways, perhaps working in pairs. </div>
<div> </div>
<div>After a suitable length of time, draw everyone together again to talk about what they have done so far. At this stage, you may need to have a discussion about what 'different' means. Ask the group how they are making sure they don't leave out any possibilities. It could be that some learners have developed a system for creating or ordering the different ways, for example by
starting with all whites, then putting in one red etc. If this isn't the case, ask pairs to draw one arrangement on a strip of paper and pin as many different arrangements as they have found on the board. You can then ask the children to order the arrangements by moving the strips. This way, it will be easier to see if they have missed any out. You could complete the
solution of this part of the problem all together and then invite pupils to comment on what they might do to solve the problem if they are given a differently coloured rod (e.g. dark green) to make from reds and whites. Do they have a sense of the need for a system?</div>
<div> </div>
<div>As they work on the second part of the challenge in their pairs, you will be able to listen and observe. Have they got the idea of creating possibilities in a particular order? Are they using the same order that the whole group used, or have they developed their own? </div>
<div> </div>
<div>Allow time at the end of the session for children to look at what other pairs have done. It might be that this activity (and any extensions you suggest - see below) form a 'simmering' task over a period of a few days or more. Allocate a space on the wall for children to add to the possible arrangements and then come back to their contributions at a later date. </div>
<br></br>
<h3>Key questions</h3>
<div>Can you use just white rods?</div>
<div>Can you use just red rods?</div>
<div>Can you find another way to make the yellow rod?</div>
<div>How could you put those rods in a different order to make the yellow?</div>
<div>How will you know that you have found all the possiblities?</div>
<div>How could you use what you've done to help find all the possible ways for making the dark green rod?</div>
<br></br>
<h3>Possible extension</h3>
<div>Some children may like to continue to work in a systematic way to find all possible ways of making a black, then brown, then blue, then orange rod from reds and whites. Alternatively, they may like to apply their method to finding all the ways of making different rods using light green in addition to red and white rods. <a href="/content/id/2724/Cuisenaireywrg.swf">This
interactivity</a> could be useful for the latter. </div>
<br></br>
<h3>Possible support</h3>
<div>Children might find it helpful to use cm squared paper with the cuisenaire rods for recording purposes. </div>
<br></br>
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<mdoxml version="1.0">Can you use just white rods? <br></br>
Can you use just red rods?<br></br>
Can you use a mixture of red and white rods?<br></br>
Could you put the rods in a different order?<br></br></mdoxml>
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<mdoxml version="1.0">Six:<br></br>1, 1, 1, 1, 1, 1<br></br>2, 1, 1, 1, 1<br></br>1, 2, 1, 1, 1<br></br>1, 1, 2, 1, 1<br></br>1, 1, 1, 2, 1<br></br>1, 1, 1, 1, 2<br></br>2, 2, 1, 1<br></br>2, 1, 2, 1<br></br>2, 1, 1, 2<br></br>1, 2, 2, 1<br></br>1, 2, 1, 2<br></br>1, 1, 2, 2<br></br>2, 2, 2<br></br><br></br>Five:<br></br>1, 1, 1, 1, 1<br></br>2, 1, 1, 1<br></br>1, 2, 1, 1<br></br>1, 1, 2, 1<br></br>1, 1, 1, 2<br></br>2, 2, 1<br></br>2, 1, 2<br></br>1, 2, 2<br></br>3, 1, 1<br></br>1, 3, 1<br></br>1, 1, 3<br></br>2, 3<br></br>3, 2<br></br></mdoxml>
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Cuisenaire Counting
Here are some rods that are different colours. How could I make a
dark green rod using yellow and white rods?
Using, Applying and Reasoning about Mathematics
Working systematically
Mathematics Tools
Cuisenaire rods
Decision Mathematics and Combinatorics
Combinations
Information and Communications Technology
Interactivities
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