1148
/www/nrich/html/content/02/12/penta5/
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2011-02-01T00:00:01
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<td>Here are four cubes joined together:</td>
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<div class="c1"><mdo:image width="158" height="110" src="4cubes.gif" alt="three cubes joined horizontally with one joined at left end on top"></mdo:image></div>
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<td>We can draw this arrangement of cubes on dotty paper (isometric
paper) which gives us a way of drawing 3D objects more easily:</td>
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<div class="c1"><mdo:image width="293" height="257" src="isometricgrid.gif" alt="isometric grid showing arrangement of cubes above"></mdo:image></div>
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<td>How many other arrangements of four cubes can you find?</td>
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<td>Can you draw them on dotty paper? It's more difficult than it
looks!</td>
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<td>If you don't have any isometric paper, click <a href="http://nrich.maths.org/content/id/6676/DottedIsometricGrid_10mm.pdf">
here.</a></td>
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<td>Lots of you sent us excellent solutions for The Third
Dimension. Chris and Michael from Moorfield Junior School, and Lily
and Ruth from Brecknock Primary School in Camden managed to find
eight arrangements altogether, including the one which we drew in
the question. Lily explains how she systematically looked for them
all:</td>
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<blockquote>
<p>To do this problem I used unifix cubes to help me. First I
started with a long block, then I took one cube off and moved it to
different positions making sure I didn't do the same one twice. I
kept the cube in the middle and moved a second cube from the end to
make a square. After that I moved this brick into further
positions.</p>
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<td>Here is Ruth's drawing which shows these arrangements very
clearly:</td>
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<div class="c1"><mdo:image src="RuthCamden.gif" alt="eight arrangements of four cubes joined together" width="523" height="558"></mdo:image></div>
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<h2>The Third Dimension</h2>
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<td>Here are four cubes joined together:</td>
</tr>
<tr>
<td>
<div class="c1"><mdo:image alt="three cubes joined horizontally with one joined at left end on top" height="110" src="4cubes.gif" width="158"></mdo:image></div>
</td>
</tr>
<tr>
<td>We can draw this arrangement of cubes on dotty paper (isometric paper) which gives us a way of drawing 3D objects more easily:</td>
</tr>
<tr>
<td>
<div class="c1"><mdo:image alt="isometric grid showing arrangement of cubes above" height="257" src="isometricgrid.gif" width="293"></mdo:image></div>
</td>
</tr>
<tr>
<td>How many other arrangements of four cubes can you find?</td>
</tr>
<tr>
<td>Can you draw them on dotty paper? It's more difficult than it looks!</td>
</tr>
<tr>
<td>If you don't have any isometric paper, click <a href="http://nrich.maths.org/content/id/6676/DottedIsometricGrid_10mm.pdf">here.</a></td>
</tr>
</tbody>
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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=1148&part=index">This problem</a> helps learners to relate 3D solids to drawings of them. It requires a systematic approach.</div>
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<h3>Possible approach</h3>
<div>You might like to begin by drawing a shape on isometric dotty paper on your interactive whiteboard which uses, for example, three cubes joined together. Working in pairs, ask children to create the shape for themselves out of interlocking cubes. Ask one child in each pair to draw another shape on isometric paper and for their partner to make it, then swap over. You could repeat this as
necessary, depending on how much experience your class has of drawing in this way.</div>
<br></br>
<div>Introduce the problem itself by making the arrangement of four cubes shown in the picture. Leave pairs to try to make other arrangements for a while without saying much more at this stage, then bring the whole group together to discuss some issues. How are they checking that each shape they make is different from the shape/s they have already made? Has anyone got a good system for making the
shapes so that they are sure they'll be able to find them all? Draw attention to those who have developed a clear approach, for example by having three cubes the same and looking for all the positions that the fourth cube could go etc. Having talked together in this way, children should feel confident enough to find all the different arrangements.</div>
<br></br>
<div>Their models and drawings would make an interesting display, particularly if some pairs were encouraged to describe in words how they went about finding all the possibilities.</div>
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<h3>Key questions</h3>
<div>Is this arrangement different from this one? How do you know?</div>
<div>How will you know when you have found all the different arrangements?</div>
<br></br>
<h3>Possible extension</h3>
<div>Children could go on to make all the arrangements of five cubes - this is much more challenging and requires a very organised approach. Encourage them to use the four-cube arrangements as a basis for this extension.</div>
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<h3>Possible support</h3>
<div>If some learners are struggling with drawing the models, provide enough cubes for them to be able to keep each one as they make it.</div>
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<p>Use multilink cubes to make the arrangements. Keep each one so
you can make sure they're all different.</p>
<p>When you have drawn your arrangements on dotty paper, try giving
them to a friend. He or she can make the shapes as a check to see
whether you've drawn them correctly.</p>
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The Third Dimension
Here are four cubes joined together. How many other arrangements of
four cubes can you find? Can you draw them on dotty paper?
Mathematics Tools
Interlocking cubes
Using, Applying and Reasoning about Mathematics
Working systematically
3D Geometry, Shape and Space
Cubes
3D Geometry, Shape and Space
2D representations of 3D shapes
Admin
Upper primary mapping document