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/www/nrich/html/content/01/10/bbprob2/
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2011-02-01T00:00:01
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<mdoxml version="1.0"><br></br>
<p>Ana and Ross looked in a trunk in the attic. They found old
cloaks and gowns, hats and masks.</p>
<p>"We could make costumes up for <mdo:image src="pic1.gif" alt="Hallowe'en."></mdo:image> ", said
Ana.</p>
<p>"I could wear a black cloak, green mask and pointy hat", said
Ana.</p>
<table cellspacing="0" cellpadding="5">
<tbody>
<tr>
<td><mdo:image src="pic2.gif" alt="Ross in a black cloak and Dracula mask. "></mdo:image></td>
<td>"Look at me in the black cloak and Dracula mask. Put the crown
on me please", said Ross.</td>
<td><mdo:image src="pic3.gif" alt="Witch."></mdo:image></td>
</tr>
</tbody>
</table>
<mdo:image src="pic4.gif" alt=""></mdo:image> <br></br>
<p>How many <strong>possible</strong> costumes could Ross have made
if he kept the black cloak and Dracula mask but wore one of the
different hats?</p>
<br></br>
<p>Here are two other costumes that Ana and Ross made up.</p>
<mdo:image src="pic5.gif" alt="Two other costumes."></mdo:image> <br></br>
<p>List or draw all of the different cloaks and gowns, masks and
hats that they have used.</p>
<p>If they changed around their clothes, hats and masks, how many
different costumes do you <strong>think</strong> Ana and Ross could
make?</p>
<p>Find out if your guess was correct.</p>
<br></br>
<table>
<tbody>
<tr>
<td>Ross found extra masks in the trunk.</td>
<td><mdo:image src="pic6.gif" alt="Mask"></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="pic7.gif" alt="Another mask."></mdo:image></td>
<td>"I think I'll change my face!" laughed Ana.</td>
</tr>
</tbody>
</table>
<p>I wonder how many other costumes Ana and Ross could put together
if they put one of these other masks on.</p>
<p>Can you find out?</p>
<br></br>
<table>
<tbody>
<tr>
<td>Ana looked deep into the trunk. "I've found a different cape",
she cried. "Now we can make even more costumes!"
<p>How many different <mdo:image src="pic1.gif" alt="Hallowe'en."></mdo:image> costumes could Ana and Ross make
altogether?</p>
</td>
<td><mdo:image src="pic8.gif" alt="Cape."></mdo:image></td>
</tr>
</tbody>
</table>
<br></br>
<p>How can you be sure that you have found all of the
<strong>possible</strong> costumes?</p>
<br></br>
<p>I wonder how many more different costumes they could
<strong>possibly</strong> make if they had found this bag of old
shoes....</p>
<mdo:image src="pic9.gif" alt="Shoes."></mdo:image> <br></br></mdoxml>
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<span class="editorial">Ali worked hard at this problem and sent us a <a href="/content/01/10/bbprob2/halloween.xlsx">spreadsheet</a> of the solution. Ali says:</span><br></br>
<br></br>
The way we solved the problem is as follows:<br></br>
1) First, we put all the data into a table.<br></br>
2) We worked out the combinations for each possibility, then we started to see patterns forming in our solutions table. For example, we saw that in the "masks" column, there is an alternating pattern of "green" and "Dracula".<br></br>
3) Then, we recognised the same logic in the "Hats" column and the "Cloak" column. There is no variation in the "Gown" column since there is only one type of gown.<br></br>
4) The answer to the first question is $24$ combinations.<br></br>
<br></br>
<span class="editorial">If we look at Ali's spreadsheet, we can see that there are $24$ combinations using:<br></br>
a black cloak or a wizard's cloak<br></br>
the grey gown<br></br>
six different hats (what Ali has called pointy, wizard, crown, Santa, school and muslim)<br></br>
a green mask or a Dracula mask</span><br></br>
<br></br>
<br></br>
5) We used the same method for the next question <span class="editorial">(using the two new masks, what Ali has called 'old' and 'clown')</span>, but in the spreadsheet we used red to distinguish this part of the question from the others.<br></br>
6) The answer to the second question is another $24$ combinations, making the total of $48$ combinations.<br></br>
<br></br>
7) We used the same method for the third question, using yellow to denote the new information on the yellow cloak, as well as the combinations found using this new information.<br></br>
8) The answer to the third question is another $24$ combinations, making the total of $72$ combinations.<br></br>
<br></br>
9) For the last part (with addition of the shoes), we added the new information in green in our table.<br></br>
10) Then we worked through the combinations as before, noticing the repeating patterns.<br></br>
11) The answer to the fourth question is a grand total of $360$ combinations.<br></br>
<br></br>
<span class="editorial">Wow - you have worked very systematically, Ali - well done!</span></mdoxml>
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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=95&part=">This
activity</a> is one of the many that tackle the mathematical topic
of combinations.</div>
<br></br>
<h3>Possible approach</h3>
<div>text</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about the way you have worked out your
solutions.</div>
<br></br>
<h3>Possible extension</h3>
<div>text</div>
<br></br>
<h3>Possible support</h3>
<div>text</div>
<br></br>
<br></br></mdoxml>
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Combinations, often thought of in terms of possibilites, is an
important mathematical topic for young children. They are called
upon to think logically, to develop strategies, to organise results
and to communicate clearly. Each of these processes can be
approached and tackled in a variety of ways. This allows children
to enter and exit the problem and to engage with it in appropriate
ways and at an appropriate level. Teacher's questions around
results will scaffold children's thinking and move them along in
their efforts and strategies. Subsequently, children can be asked
to use higher order, or critical, thinking skills to assess and
evaluate the effectiveness and efficiency of the different
strategies. <br></br>
<br></br>
This Let's Investigate problem is topical for the season of
Halloween. Even if Halloween is not part of the local culture,
setting problems within such a context is an opportunity to talk
about the celebrations and traditions of other cultures. <br></br>
<br></br>
Each step of the problem introduces another level of complexity, as
the number of possibilities grow. This establishes the need for
results to be ordered to avoid repetitions or omissions. Some
children will need to draw and cut out the costume pieces and then
put them together. Checklists can be used while others, operating
more abstractly, can be guided toward making a table or matrix to
listing and combine the items or find the permutations.<br></br></mdoxml>
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Halloween Investigation
Ana and Ross looked in a trunk in the attic. They found old cloaks
and gowns, hats and masks. How many possible costumes could they
make?
Using, Applying and Reasoning about Mathematics
Investigations
Using, Applying and Reasoning about Mathematics
Working systematically
Decision Mathematics and Combinatorics
Combinations