7473
/www/nrich/html/content/id/7473/
1
2011-07-23T10:10:45
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<p><em>This activity has been particularly created for the most able. (The pupils that you come across in many classrooms just once every few years.)</em></p>
<p>This activity also follow on for those who have successfully worked at <a href="http://nrich.maths.org/66&part=">Doplication.</a> <br></br>
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Let me help you visualise this representation of a 3D situation.<br></br>
It's kind of two set of three adjacent cubes with balls at each vertex.<br></br>
Towards the bottom left of the picture are the eight large balls, then going backwards there are eight medium red balls and further back are eight small ones.<br></br>
In the centre of the front three cubes are three orange balls and at the centre of the rear cubes are three brown ones.</p>
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<div style="text-align: center;"> <mdo:image width="401" height="163" src="new6cubes.jpg" alt="new6"></mdo:image></div>
<div style="text-align: left;">In Doplication you would have had;</div>
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<div style="text-align: center;"> <mdo:image width="249" height="175" src="Axitate%20version%204%3F3.jpg" alt="axitate version"></mdo:image></div>
<div style="text-align: left;">which we called $4$ ? $3$.</div>
<div style="text-align: left;">In this activity it would be called something different involving $4, 3, 2$. Just for now I am going to use @, so the diagram above represents $4$@$3$@$2$.</div>
<div style="text-align: left;">I invite you to explore many such arrangements and find a way of recording your results.</div>
<div style="text-align: left;">You might like to just look at the arrangements that are like cubes - $3$@$3$@$3$ , $4$@$4$@$4$, $5$@$5$@$5$, etc.</div>
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<div style="text-align: left;">As with most mathematics it's good to compare so maybe compare the results for squares in Doplication and cubes in 3D Stacks.</div>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<div>This <a href="http://nrich.maths.org/7473&part=">activity</a> is
specially designed for the highest-attaining pupils that you ever
come across. It acts as a further extension to <a href="http://nrich.maths.org/66&part=">Doplication</a>. It's an
activity that is intended to give opportunities for those pupils to
explore deeply using their intuition and flair in the areas of both
spatial awareness and number replationships and
patterns. </div>
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<h3>Possible approach</h3>
<div>As this is designed for the highest attaining, it might be
presented as on the website or in a one-to-one situation,
encouraging discussion between adult and pupil. The pupils
may need access to a spreadsheet once many number results are being
acquired.</div>
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<h3>Key questions</h3>
<div>Tell me about what you have found.</div>
<div>Can you describe the ways that you arrived at these
numbers?</div>
<div>How did you construct this on the spreadsheet you used?</div>
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If its useful for the teacher here are some little bits.<br></br>
<mdo:image width="595" height="52" src="some%20cube%20results.jpg" alt="cube results"></mdo:image><br></br>
Some more for the cube ones.<br></br>
<mdo:image width="711" height="347" src="extra%20resulst.jpg" alt="extras"></mdo:image><br></br>
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For results from $3$ to $7$ some layers to show the answers<br></br>
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<mdo:image width="542" height="407" alt="general" src="some%20general%20results.jpg"></mdo:image><br></br>
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5
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1
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3D Stacks
Can you find a way of representing these arrangements of balls?
Using, Applying and Reasoning about Mathematics
Representing
Using, Applying and Reasoning about Mathematics
Visualising