1008
/www/nrich/html/content/00/06/penta5/
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2011-02-01T00:00:01
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<p>When intergalactic Wag Worms are born they look just like a cube.<br></br>
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<mdo:image src="fig1.gif"></mdo:image><br></br>
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Each year they grow another cube in any direction (except on their faces of course). So a Wag Worm in its second year might look like any of these.</p>
<mdo:image alt="Cube" src="fig2.gif"></mdo:image><br></br>
<p>The next year it grows another cube onto its second cube (not its face cube). So one shape a Worm in its third year can be is this:</p>
<mdo:image alt="Cube" src="fig3.gif"></mdo:image><br></br>
<p>Find all the shapes that a Wag Worm in its third year can be.</p>
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<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6815&part=">Click here for a poster of this problem</a>.</div>
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<p><span class="editorial">Many people produced good results and it
was interesting to see the different ways the problem was
interpreted.</span></p>
<p><span class="editorial">Did you decide that you can add another
cube onto one that you have already added to or not? So, in the
following, can the latest blue cube go in the places pictured in
the 2nd and 3rd diagrams?</span></p>
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<div style="text-align: center;"><mdo:image height="149" width="336" alt="No 2" src="no2Wagworms.jpg"></mdo:image></div>
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<p><span class="editorial">Michael wrote in to say:</span></p>
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There were five places where a cube could be added on to each of
the five Wag Worms, and $5\times5 = 25$. So, there are $25$
different types of Wag Worm in its third year.<br></br>
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<p><span class="editorial">The Gateway School sent in two solutions
of $25$ as well.</span></p>
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<p class="editorial">Rowena wrote the following good
explanation</p>
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I know that a cube has $6$ faces. I looked at the Wag Worm in its
second year, you can add a cube to all of the $6$ faces except the
one with eyes, so there are $5$ possible Wag Worms.<br></br>
For the Wag Worms in their third year, I looked at adding a cube to
the 'non-eye cube' and the 'eye cube'.<br></br>
For the 'non-eye cube' there are $5$ visible faces (one is attached
to the 'eye cube') so I could make $5$ different Wag Worms.<br></br>
For the 'eye cube' again there are $5$ visible faces, but one has
eyes, so I could have $4$ places to put the cube.<br></br>
This is $5+4=9$ possible Wag Worms for each of the 5 Worms in their
second year. So in total there would be $5\times9 = 45$ different
Wag Worms in their third year.<br></br>
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<p><span class="editorial">Emma and Oscar went further and tried
the fifth year of growth. Here are some of their
results:</span></p>
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<div style="text-align: center;"><mdo:image height="224" width="449" src="wagworm5.jpg" alt="wagworm5"></mdo:image></div>
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<h2>Wag Worms</h2>
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<p>When intergalactic Wag Worms are born they look just like a cube.<br></br>
<br></br>
<mdo:image src="fig1.gif"></mdo:image><br></br>
<br></br>
Each year they grow another cube in any direction (except on their faces of course). So a Wag Worm in its second year might look like any of these.</p>
<mdo:image alt="Cube" src="fig2.gif"></mdo:image><br></br>
<p>The next year it grows another cube onto its second cube (not its face cube). So one shape a Worm in its third year can be is this:</p>
<mdo:image alt="Cube" src="fig3.gif"></mdo:image><br></br>
<p>Find all the shapes that a Wag Worm in its third year can be.</p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1008&part=index">This activity</a> gives you a good opportunity to encourage pupils to make sense of, and interpret, situations. It also gives them a chance to share ways of convincing you they have found all the possibilities, which is quite a challenge in this problem.</div>
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<h3>Possible approach</h3>
<div>It would be good to introduce the problem using some interlocking cubes. Draw a face on the face of one cube, or stick a sticker on a face, and explain to the group that this is what the Wag Worms look like when they are born. Continue by telling the children that each year, they grow one more cube. You could show one example and then ask some children to make other two-year old Wag Worms so
that you have all the possibilities. Encourage children to explain how they know they have got them all - each Worm has the second cube on a different face and there are five faces that this second cube could go on so there are five possibilities.</div>
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<div>Ask them to guess what three-year-old Wag Worms look like and after clarifying the 'rules', set them off to try to find all the worms in pairs. Having a large supply of cubes will help and if you have enough, you'll be able to let the children keep a model of each worm rather than asking them to record their work, which might be quite tricky. If cubes are in short supply, you could make a
class set of all the Worms, so designate a particular table to collecting them. Of course you could take photos as well.</div>
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<div>Bring the children together to talk about how they have gone about finding the Worms. Invite those children who had some sort of system to explain what it was so everyone could replicate that particular way. For example, one way would be to start with the first two-year old Worm and look at all the different possible places to put a third cube, then to look at a second two-year-old Worm and
do the same thing etc. If the children have struggled to use a system at all, you could ask them to look at the Worms they've made and sort them into groups. This might help them notice ones they have missed out.</div>
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<h3>Key questions</h3>
<div>How did you find these?</div>
<div>Do you have a system (method, idea, recipe) for getting more solutions?</div>
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<h3>Possible extension</h3>
<div>Encourage the pupils to ask, "I wonder what would happen if I ...?". For example, they could go on to look at four cubes. In this case, they may need to consider what to do when they want to add a further cube but this cube would cover the Worm's face even though it is not connected to it:</div>
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<div><mdo:image alt="" height="103" src="Picture%204.jpg" width="147"></mdo:image></div>
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<div>Would it be ok to add another cube on the underneath side of the green?</div>
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<h3>Possible support</h3>
<div>You could suggest that children start with one of the two-year-old Worms and find all the possible ways of adding one cube.</div>
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Do you have a system or way of getting more solutions?<br></br>
How will you know you have found them all?<br></br>
How would it help to start with the two-year-old wag worms?<br></br>
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Max found 25 possible Wag Worms;well done! <br></br>
<mdo:image width="100" height="108" src="worm1.gif" alt="1"></mdo:image><mdo:image width="100" height="94" src="worm2.gif" alt="2"></mdo:image><mdo:image width="100" height="103" src="worm3.gif" alt="3"></mdo:image><mdo:image width="100" height="63" src="worm4.gif" alt="4"></mdo:image><mdo:image width="100" height="74" src="worm5.gif" alt="5"></mdo:image><mdo:image width="100" height="95" src="worm6.gif" alt="6"></mdo:image><mdo:image width="100" height="65" src="worm7.gif" alt="7"></mdo:image><mdo:image width="100" height="107" src="worm8.gif" alt="8"></mdo:image><mdo:image width="100" height="102" src="worm9.gif" alt="9"></mdo:image><mdo:image width="100" height="74" src="worm10.gif" alt="10"></mdo:image><mdo:image width="100" height="95" src="worm11.gif" alt="11"></mdo:image><mdo:image width="56" height="100" src="worm12.gif" alt="12"></mdo:image><mdo:image width="100" height="111" src="worm13.gif" alt="13"></mdo:image><mdo:image width="100" height="110" src="worm14.gif" alt="14"></mdo:image><mdo:image width="100" height="97" src="worm15.gif" alt="15"></mdo:image><br></br>
<mdo:image width="57" height="100" src="worm16.gif" alt="16"></mdo:image><mdo:image width="100" height="110" src="worm17.gif" alt="17"></mdo:image><mdo:image width="100" height="98" src="worm18.gif" alt="18"></mdo:image><mdo:image width="100" height="108" src="worm19.gif" alt="19"></mdo:image><mdo:image width="100" height="101" src="worm20.gif" alt="20"></mdo:image><mdo:image width="100" height="74" src="worm21.gif" alt="21"></mdo:image><mdo:image width="100" height="66" src="worm22.gif" alt="22"></mdo:image><mdo:image width="100" height="100" src="worm23.gif" alt="23"></mdo:image><mdo:image width="100" height="96" src="worm24.gif" alt="24"></mdo:image><mdo:image width="100" height="110" src="worm25.gif" alt="25"></mdo:image><br></br>
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Well done Olivia (Stamford High School, Lincolnshire) for sending
in lots of shapes for 5 year old Wag Worms. She suspects there
could be as many as 95 - BUT if you make the rule that each new
cube can only grow on the previous year's cube, then there would be
fewer possibilities. That way, the Wag Worm stays looking like a
worm, and doesn't look like it's growing arms.<br></br>
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Wag Worms
When intergalactic Wag Worms are born they look just like a cube.
Each year they grow another cube in any direction. Find all the
shapes that five-year-old Wag Worms can be.
Using, Applying and Reasoning about Mathematics
Working systematically
Mathematics Tools
Interlocking cubes
Decision Mathematics and Combinatorics
Combinations