5808
/www/nrich/html/content/id/5808/
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2011-02-01T00:00:01
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<div style="text-align: center;">This activity is all about
imagining, like you might when you listen to a story or poem.</div>
<br></br>
<div style="text-align: center;">There is no need to write or draw
anything</div>
<br></br>
<div style="text-align: center;">BUT</div>
<div style="text-align: center;">talking about it could be
good!</div>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="150" height="150" alt="imagine!" src="imagine2.png"></mdo:image></div>
<br></br>
<br></br>
<br></br>
Imagine a 4 by 4 by 4 cube hanging in front of you with just the
front face facing you. <br></br>
The cube is made up of 4x4x4, 64 cubes. <br></br>
You drill a hole through the four corner cubes, that are facing
you, all the way through to the back. <br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="192" height="141" src="Drilling.jpg" alt="Drilling"></mdo:image></div>
<br></br>
<br></br>
<br></br>
A friend looks down on the cube, from above, and they also drill
four holes through their four corner cubes all the way through to
the bottom. <br></br>
You and your friend examine all the 64 small cubes. <br></br>
<br></br>
How many will not have holes drilled through them? <br></br>
<br></br>
What if the same kind of activity happened with a new 4 by 4 by 4
cube but, this time, you drill holes through the four centre cubes
that are facing you and your friend drills holes through the four
centre cubes that are facing them. <br></br>
You and your friend examine all the 64 small cubes. <br></br>
How many will not have holes drilled through them?<br></br>
<br></br>
Please do send in your solutions, describing how you worked them
out.<br></br>
<br></br>
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<p><span class="editorial">Well done to all of you who sent in
solutions to this problem - there are too many to mention you all,
I'm afraid. Not so many of you answered both parts of the question,
though! Mrs Alford's Maths Challenge Group at West Lodge Middle
School, Mathew and Sam from Woodfall Primary and Linden from
Comberton Village College explained their thinking clearly. West
Lodge pupils say:</span></p>
40 cubes were not drilled.<br></br>
<div style="clear: both;">By drilling a hole in each corner of the
face that is facing you, you get 16 cubes that have been drilled
through.<br></br>
<div style="clear: both;">When the friend drills through, 2 cubes
on each edge have already been drilled so they don't count.<br></br>
<div style="clear: both;">A total of 8 extra cubes have been
drilled.<br></br>
<div style="clear: both;">So 8 + 16 = 24 (cubes drilled.)<br></br>
<div style="clear: both;">So 64 - 24 = 40 not drilled.</div>
<div style="clear: both;"></div>
<div style="clear: both;"></div>
<p class="editorial" style="clear: both;">Sakib from Swanlea School
sent us a picture which really helps the visualisation:</p>
<div style="clear: both;"></div>
<div style="clear: both;"><mdo:image width="500" height="321" alt="picture of cube being drilled" src="sol.gif"></mdo:image></div>
</div>
</div>
</div>
</div>
<br></br>
<br></br>
<p class="editorial">William from St Patrick's School found a
solution to the second part of the problem too:</p>
If you were to drill through the centre 4 cubes (that are also 4
cubes deep) then 4 cubes x 4 cubes = 16.<br></br>
Than if you drilled down from the top, only 2 cubes would have new
holes <br></br>
4 x 2 = 8, 8 + 16 = 24, 64 - 24 = 40.<br></br>
<br></br>
<p class="editorial">Another William, from Dane Court Grammar,
agreed with this.</p>
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<h2>Cube Drilling</h2>
<br></br>
<div style="text-align: center;">This activity is all about imagining, like you might when you listen to a story or poem.</div>
<br></br>
<div style="text-align: center;">There is no need to write or draw anything</div>
<br></br>
<div style="text-align: center;">BUT</div>
<div style="text-align: center;">talking about it could be good!</div>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="imagine!" height="150" src="imagine2.png" width="150"></mdo:image></div>
<br></br>
<br></br>
<br></br>
Imagine a 4 by 4 by 4 cube hanging in front of you with just the front face facing you.<br></br>
The cube is made up of 4x4x4, 64 cubes.<br></br>
You drill a hole through the four corner cubes, that are facing you, all the way through to the back.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="Drilling" height="141" src="Drilling.jpg" width="192"></mdo:image></div>
<br></br>
<br></br>
<br></br>
A friend looks down on the cube, from above, and they also drill four holes through their four corner cubes all the way through to the bottom.<br></br>
You and your friend examine all the 64 small cubes.<br></br>
<br></br>
How many will not have holes drilled through them?<br></br>
<br></br>
What if the same kind of activity happened with a new 4 by 4 by 4 cube but, this time, you drill holes through the four centre cubes that are facing you and your friend drills holes through the four centre cubes that are facing them.<br></br>
You and your friend examine all the 64 small cubes.<br></br>
How many will not have holes drilled through them?<br></br>
<br></br>
Please do send in your solutions, describing how you worked them out.<br></br>
<br></br>
</div>
<h3>Why do this problem?</h3>
The whole idea of this problem is to invite children to picture something in their mind. In this instance, pupils will need to be familiar with properties of a cube. Visualising, in this instance picturing something in your head and "doing something to it", is a useful tool to have when solving problems. This problem focuses entirely on visualisation, encouraging pupils to have a go at
imagining. The more opportunities we give for children to practise visualisation, the more comfortable they will become with using it of their own accord.<br></br>
<h3><br></br>
Possible approach</h3>
Ideally, it would be good to encourage your class to tackle this challenge purely by trying to imagine what is happening.<br></br>
<br></br>
To convince you and each other of their solutions, they will need to explain particularly carefully what they are picturing, which can be quite tricky, and you may find that they gesticulate rather a lot! In order to reach a joint conclusion, you might find it helpful to make a model of the cube from interlocking cubes.<br></br>
<h3><br></br>
Key questions</h3>
When you drill your holes, can you picture which small cubes have got holes in them?<br></br>
How many do you drill holes in?<br></br>
Now think about your friend. Can you picture the small cubes that he or she drills holes in?<br></br>
Are any of the cubes your friend drills ones that you've already drilled?<br></br>
Are some of these cubes ones that you haven't already drilled?<br></br>
<h3><br></br>
Possible extension</h3>
Children could investigate different ways of drilling holes.<br></br>
<h3><br></br>
Possible support</h3>
It would be a good idea to try <a href="/5821">Start Cube Drilling</a> before tackling this problem as that would make a nice introduction.<br></br></mdoxml>
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When you drill your holes, can you picture which small cubes have
got holes in them?<br></br>
How many do you drill holes in?<br></br>
Now think about your friend. Can you picture the small cubes that
he or she drills holes in?<br></br>
Are any of the cubes your friend drills ones that you've already
drilled?<br></br>
Are some of these cubes ones that you haven't already
drilled?<br></br></mdoxml>
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<div style="text-align: left;">Here are two cubes helping you to
see the answer. The Blue cubes are the ones without holes in.</div>
<div style="text-align: center;"></div>
<div style="text-align: center;"><mdo:image width="227" height="244" alt="Ans" src="DrilledAns.jpg"></mdo:image></div>
<div style="text-align: center;"><mdo:image width="249" height="243" alt="Ans3" src="Ans3.jpg"></mdo:image></div>
<div style="text-align: center;"></div>
<div style="text-align: center;"></div>
So its 40 in the first case and 40 in the second case.<br></br>
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Cube Drilling
Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?
3D Geometry, Shape and Space
Cubes
Using, Applying and Reasoning about Mathematics
Visualising