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2011-02-01T00:00:01
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I found a number of small bracelets and rings on a table some time
ago and I noticed how some were on their own, others were touching
at the edges, others were overlapping each other and some small
ones had found themselves inside larger ones.
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<td>I took two of these, one ring and one bracelet, and explored
what possibilities there were.</td>
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<p>I thought that this would be the next challenge for you all. To
look at the situation when you have three rings, circles, bracelets
. . . . it doesn't matter what they are really or what size they
are. They could even expand and get bigger or get smaller if you
liked. But, thinking of the four things I noticed at the
start:-</p>
<p>1) TOUCHING</p>
<p>2) OVERLAPPING</p>
<p>3) SEPARATE</p>
<p>4) IN/OUT SIDE</p>
<p>I wonder what would be the number of ways in which 3 such
circles could be?</p>
<p>Here are some ways, remember I said they could be different
sizes each time, but I've coloured them so that it is easy to know
which one we are talking about.</p>
<mdo:image alt="" src="diagc.gif"></mdo:image> <br></br>
<p>Well I feel you could carry on at this point, just a few points
to remember:-</p>
<p>When writing you must say something about each of the three
circle/rings/bracelets.</p>
<p>Three separate ones could be anywhere yet separate and they
would all count as one arrangement, and the same kind of things
goes for any other arrangement, if the words are the same then, for
this challenge the arrangement is the same.</p>
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You could now ask "I wonder what would happen if.....?" <br></br></mdoxml>
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<p><span class="editorial">We haven't had any solutions to this
investigation yet and you might have gone down many different
routes. However, if you discover anything interesting, please
do</span> <a href="mailto:nrich@damtp.cam.ac.uk" class="editorial">let us know</a> <span class="editorial">. Please don't
worry that your solution is not "complete" - we'd like to hear
about anything you have tried. Teachers - you might like to send in
a summary of your children's work.</span></p>
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<h2>3 Rings</h2>
<br></br>
I found a number of small bracelets and rings on a table some time ago and I noticed how some were on their own, others were touching at the edges, others were overlapping each other and some small ones had found themselves inside larger ones.
<table border="1">
<tbody>
<tr>
<td><mdo:image alt="" src="diaga.gif"></mdo:image></td>
</tr>
</tbody>
</table>
<table>
<tbody>
<tr>
<td><mdo:image alt="" src="diagb.gif"></mdo:image></td>
<td>I took two of these, one ring and one bracelet, and explored what possibilities there were.</td>
</tr>
</tbody>
</table>
<p>I thought that this would be the next challenge for you all. To look at the situation when you have three rings, circles, bracelets . . . . it doesn't matter what they are really or what size they are. They could even expand and get bigger or get smaller if you liked. But, thinking of the four things I noticed at the start:-</p>
<p>1) TOUCHING</p>
<p>2) OVERLAPPING</p>
<p>3) SEPARATE</p>
<p>4) IN/OUT SIDE</p>
<p>I wonder what would be the number of ways in which 3 such circles could be?</p>
<p>Here are some ways, remember I said they could be different sizes each time, but I've coloured them so that it is easy to know which one we are talking about.</p>
<mdo:image alt="" src="diagc.gif"></mdo:image><br></br>
<p>Well I feel you could carry on at this point, just a few points to remember:-</p>
<p>When writing you must say something about each of the three circle/rings/bracelets.</p>
<p>Three separate ones could be anywhere yet separate and they would all count as one arrangement, and the same kind of things goes for any other arrangement, if the words are the same then, for this challenge the arrangement is the same.</p>
<br></br>
You could now ask "I wonder what would happen if.....?"</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=38&part=">This activity</a> is a very simple investigation which allows for many levels of participation. There are those youngsters who can use three different sized rings and just "play around" with them. This provides a wonderful basis for most useful discussion as to the positioning of each of the three rings and their physical
relationship to each other. Children who go about it more systematically seem to have great enjoyment in the systems that they use. These two groups have to deal with the fact that some arrangements may 'look' very different but in the context of this challenge are equivalent. This again leads to very worthwhile discussion.</div>
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<h3>Possible approach</h3>
<div>I usually provide the children with just two different sized rings and have the third one as an imaginary one that they can alter the size of for their own use, each time.</div>
<br></br>
<h3>Key questions</h3>
<div>How would you talk about this ring?</div>
<div>What is this ring doing?</div>
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<h3>Possible extension</h3>
<div>The more advanced youngsters who have no problems with the "look" and can "see" equivalent situations are usually able to go through the activity fairly quickly and are soon asking "I wonder what would happen if ... they were squares . . . the rings were all the same size. . . ?"</div>
<h3> </h3>
<h3>Possible support</h3>
<div>Having an adult working with a small group so as to help the language to flow can sometimes be very effective.</div>
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<mdoxml version="1.0">You could cut out two rings from card if you haven't got any
jewellery to use to try things out.<br></br></mdoxml>
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3 Rings
If you have three circular objects, you could arrange them so that
they are separate, touching, overlapping or inside each other. Can
you investigate all the different possibilities?
Using, Applying and Reasoning about Mathematics
Investigations
Using, Applying and Reasoning about Mathematics
Working systematically
Decision Mathematics and Combinatorics
Combinations