6529
/www/nrich/html/content/id/6529/
1
2011-02-01T00:00:01
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<div style="text-align: center;"><mdo:image height="349" width="256" src="1stHolePic.jpg" alt="1st pic"></mdo:image></div>
<br></br>
<div style="text-align: left;">Here we have three solid cubes and
three cubes that have holes.</div>
They're just the first three in a series that could go on and
on.<br></br>
<div style="text-align: left;"> </div>
<div style="text-align: left;">I was wondering about the number of
cubes used in each ...</div>
<br></br>
<div style="text-align: left;">Then I thought about the difference
between those numbers.</div>
<br></br>
So, for example, I found that the first cube, $3$ by $3$ by $3$,
used $27$ cubes.<br></br>
<br></br>
The same cube with holes used $20$ cubes, a difference of
$7$.<br></br>
<br></br>
<div style="text-align: left;">You can call these types of cubes
what you like but I called them "Solid" and "Frame".</div>
<br></br>
How about exploring the numbers for the next few Solid and Frame
cubes?
<div style="text-align: left;"></div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">Do you notice any patterns?</div>
<div style="text-align: left;">Can you explain any of the
patterns?</div>
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<p class="editorial">We had a few solutions sent in for this
challenge. I think that maybe there were other pupils who got so
far, on their own or with the teacher, and decided it was not worth
sending in. We would like to reassure you that all journeys in the
solving of challenges - however far you get - we welcome!</p>
<br></br>
<p class="editorial">Here is the excellent result that we received
from class $5$A at St. Christopher's School in Penang,
Malaysia.</p>
<mdo:image height="539" width="629" alt="new Sol" src="NewCubeSolution.jpg"></mdo:image><br></br>
<p><span class="editorial">Well Done !</span></p>
<br></br>
<p><span class="editorial">From Jenthghe and Olivia we
received</span>.</p>
<br></br>
<br></br>
For this problem we used a trial and error method!<br></br>
For solid cube number $1$ we did, $3$x$3$x$3$=$27$<br></br>
For frame cube number $1$ we did $3$x$3$x$3$=$27-7=20$ we took away
$7$ because there was seven holes in the cube.<br></br>
We did the same for solid and frame cube number $2$ but this time
we found that we had to do $4$x$4$x$4$=$64$ (for the solid cube),
then find what the frame cube had which was $4$x$4$x$4$=$64-24=40$
(for the frame) we took $24$ away because that was how many holes
were in the cube.<br></br>
For solid and frame number $3$ we did $5$x$5$x$5$=$125$ (for the
solid cube) and for the frame we did $5$x$5$x$5$=$125-65=60$. we
took away $65$ because there were $65$ cubes missing.<br></br>
<br></br>
<p class="editorial">This was a good start to this challenge but a
closer look at the frame cubes would be necessary. Then you might
see that the $24$ you take away from $64$ cubes for the second cube
only takes account of the $4$ cubes on each face that were removed.
There was also some cubes removed from the centre of the solid cube
and they would form a $2$ by $2$ by $2$ cube. This would mean a
total of $32$ cubes altogether.</p>
<br></br>
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<h2>Holes</h2>
<br></br>
<div style="text-align: center;"><mdo:image alt="1st pic" height="349" src="1stHolePic.jpg" width="256"></mdo:image></div>
<br></br>
<div style="text-align: left;">Here we have three solid cubes and three cubes that have holes.</div>
They're just the first three in a series that could go on and on.<br></br>
<div style="text-align: left;"> </div>
<div style="text-align: left;">I was wondering about the number of cubes used in each ...</div>
<br></br>
<div style="text-align: left;">Then I thought about the difference between those numbers.</div>
<br></br>
So, for example, I found that the first cube, $3$ by $3$ by $3$, used $27$ cubes.<br></br>
<br></br>
The same cube with holes used $20$ cubes, a difference of $7$.<br></br>
<br></br>
<div style="text-align: left;">You can call these types of cubes what you like but I called them "Solid" and "Frame".</div>
<br></br>
How about exploring the numbers for the next few Solid and Frame cubes?
<div style="text-align: left;"> </div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">Do you notice any patterns?</div>
<div style="text-align: left;">Can you explain any of the patterns?</div>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=6529&part=">problem</a> gives opportunities for pupils to explore, to discover, to analyse and communicate. It allows pupils to approach it in whatever way they find most helpful. It also provides opportunities for using and extending visualising skills.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could show the group models of the first solid cube and first frame cube. Invite them to talk to a partner about them and to share their observations with the whole class. You could then ask pairs to suggest what the 'next' two cubes would look like. You could also ask them to explain why they think you started with a $3$ by $3$ by $3$ cube rather than, say, a $2$ by $2$ by $2$
one.</div>
<br></br>
<div>You may like to challenge learners to find the number of small cubes each is made up of and they could share their methods for doing so. Some may need to make their own model to help them, but others will be able to calculate the number of cubes using certain facts they know. By sharing different methods, some children may take on a new method because they find it works better for them
compared with the original way they chose. Pupils can then work on the challenge itself in pairs or small groups.</div>
<br></br>
<div>Give the class the freedom to choose their own way of recording their findings and share these in the plenary, as well as sharing results. Encourage explanations of the number patterns, rather than just 'pattern spotting'.</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about about what you've found.</div>
<div>Are there other ways of showing what you've found?</div>
<br></br>
<h3>Possible extension</h3>
<div>Learners could explore what happens when they count the square surfaces that are visible on each small cube.</div>
<br></br>
<h3>Possible support</h3>
<div>Having plenty of interlocking cubes available will aid some children, whereas others may wish to use computer drawing programs and/or spreadsheet software.</div>
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<mdoxml version="1.0">How will you record what you find?<br></br>
You might find interlocking cubes useful, or perhaps computer
drawing programs, or even spreadsheet software?<br></br></mdoxml>
2
3
1
1
0
0
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Holes
I've made some cubes and some cubes with holes in. This challenge
invites you to explore the difference in the number of small cubes
I've used. Can you see any patterns?
3D Geometry, Shape and Space
Cubes
Using, Applying and Reasoning about Mathematics
Visualising
Sequences, Functions and Graphs
Sequences
Mathematics Tools
Interlocking cubes