The Big Cheese

78 /www/nrich/html/content/00/05/bbprob1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p style="text-align: center;"><a href="/content/00/05/bbprob1/some%20cheese.jpg"><mdo:image src="some%20cheese.jpg"></mdo:image></a></p> <p>I met up with some friends yesterday for lunch. On the table was a good big block of cheese. It looked rather like a cube. As the meal went on we started cutting off slices, but these got smaller and smaller! It got me thinking ...</p> <p>What if the cheese cube was $5$ by $5$ by $5$ and each slice was always $1$ thick?</p> <p>It wouldn't be fair on everyone else's lunch if I cut up the real cheese so I made a model out of multilink cubes:</p> <div style="text-align: center;"><mdo:image alt="" src="5.gif"></mdo:image></div> <h5><em>You could of course, just make $5$ slices but I wanted to have a go at something else - keeping what is left as close to being a cube as possible.</em></h5> <p>You can see that it's a $5$ by $5$ by $5$ because of the individual cubes, so the slices will have to be $1$ cube thick.</p> <p>So let's take a slice off the right hand side, I've coloured it in so you can see which bit I'm talking about:</p> <div style="text-align: center;"><mdo:image alt="" src="5a.gif"></mdo:image> This now gets cut off and we have: <mdo:image alt="" src="5b.gif"></mdo:image><br></br> </div> <p>The next slice will be from the left hand side (shown in a different colour again):</p> <div style="text-align: center;"><mdo:image alt="" src="5c.gif"></mdo:image>Wel the knife cuts and we are left with: <mdo:image alt="" src="5d.gif"></mdo:image><br></br> </div> <h4 style="font-weight: bold;">Remember I'm setting myself the task of cutting so that I am left with a shape as close to a cube shape as possible each time.</h4> <p>So the next cut is from the top. Hard to cut this so I would have put it on its side!</p> <div style="text-align: center;"><mdo:image alt="" src="5e.gif"></mdo:image>I'll remove that and I'm left with the $4$ by $4$ by $4$ cube<mdo:image alt="" src="4.gif"></mdo:image></div> <p><br></br> I do three more cuts to get to the $3$ by $3$ by $3$ and these leave the block like this:</p> <div style="text-align: center;"><mdo:image alt="" src="4a.gif"></mdo:image> <mdo:image alt="" src="4b.gif"></mdo:image> <mdo:image alt="" src="3.gif"></mdo:image></div> I'm sure you've got the idea now so I don't need to talk as much about what I did: <div style="text-align: center;"><mdo:image alt="" src="3a.gif"></mdo:image> <mdo:image alt="" src="3b.gif"></mdo:image> <mdo:image alt="" src="2.gif"></mdo:image> and then onto:<mdo:image alt="" src="2a.gif"></mdo:image> <mdo:image alt="" src="2b.gif"></mdo:image> <mdo:image alt="" src="1.gif"></mdo:image></div> <div><br></br> That leaves you with two of the smallest size cube $1$ by $1$ by $1$.</div> <p>If we keep all the slices and the last little cube, we will have pieces that look like (seen from above):</p> <div style="text-align: center;"><mdo:image alt="" src="slices.gif"></mdo:image></div> <h4 style="text-align: center;">C H A L L E N G E</h4> Now we have thirteen objects to explore. <ul> <li>What about the areas of these as seen from above?</li> <li>What about the total surface areas of these?</li> <li>What about their volumes of the pieces?</li> </ul> <h4 style="text-align: center;">A L S O</h4> <p>Investigate sharing these thirteen pieces out so that everyone gets an equal share.</p> <br></br> What about ...? <p>I guess that once you've explored the pattern of numbers you'll be able to extend it as if you had started with a $10$ by $10$ by $10$ cube of cheese.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Chloe, Emily, Rebecca, Scarlett, Chloe, Alana, Isobel, Shauna and Joel from Griffithstown Primary School sent in various solutions that were all very interesting and showed hard work. Here is a summary taken from their suggestions.</p> <br></br> I found out that we can share the $13$ slices of the big cheese between $5$ people. This is how we did it:<br></br> The first person had a slice of cheese that was $25$cm$^2$.<br></br> The second person had $3$ slices of cheese that was $20$cm$^2$ + $4$cm$^2$ + $1$cm$^2$ = $25$cm$^2$.<br></br> The third person had $5$ slices of cheese that was $12$cm$^2$ + $6$cm$^2$ + $4$cm$^2$ + $2$cm$^2$ + $1$cm$^2$= $25$cm$^2$.<br></br> The fourth person had $2$ slices of cheese that was $16$cm$^2$ + $9$cm$^2$ = $25$cm$^2$.<br></br> The fifth person had $2$ slices of cheese that was $16$cm$^2$ + $9$cm$^2$ = $25$cm$^2$.<br></br> So everybody had an equal share of cheese but in different sizes.<br></br> <br></br> <p><span class="editorial">Cherian from Quarry Bay School sent in the following;</span></p> <br></br> Solution for the third investigation for The Big Cheese. My solution:<br></br> First you imagine the two $1$ by $1$ by $1$ cubes put on top of each other.<br></br> Then you put the $1$ by $1$ by $2$ layer beside the other one.<br></br> After that, put another $1$ by $1$ by $2$ layer in front of it, so that you will have a $2$ by $2$ by $2$ cube.<br></br> Now you have to put $2$ by $2$ by $1$ layer on top of it.<br></br> Then you put $2$ by $3$ by $1$ layer on the left side.<br></br> After that you put a $3$ by $3$ by $1$ layer behind it.<br></br> Then you put another one on your cube.<br></br> Now you put a $3$ by $4$ by $1$ layer beside it.<br></br> Then put a $4$ by $4$ by $1$ layer behind it.<br></br> Now you have a $4$ by $4$ by $4$ cube.<br></br> After that you put a $4$ by $4$ by $4$ layer on top of it.<br></br> Then put a $4$ by $5$ by $1$ layer beside it.<br></br> Now put a $5$ by $5$ by $5$ layer behind it, so now you have a $5$ by $5$ by $5$ cube!!<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><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=78&part=">This activity</a> is a rich environment in which children can explore numbers, shapes and/or measures. It therefore has the potential to involve a lot of visualisation and calculation, and to stimulate lively discussion.</div> <br></br> <h3>Possible approach</h3> <div>It would be helpful to introduce this activity using a model of the cheese, for example made out of multilink cubes. As you tell the story and explain how you're going to cut the cheese, ask learners to picture what the cheese will look like once the cut has been made. Invite some of them to describe the resulting block, before demonstrating what it looks like with the model itself. You might like to ask the group to try the last few cuts for themselves in pairs rather than going through them all as a whole class.</div> <br></br> <div>Once all the cuts have been made, you could have models of the pieces from multilink, or you could have images of the pieces on the board. Before saying much else, ask children what they notice. This will bring up many interesting observations, some of which the children may wish to pursue, or you can make suggestions based on the text of the problem itself.</div> <br></br> <div>Allow children to choose the materials they need to work on their investigation. Some may be happy to visualise the cutting, others may need to use suitable materials like multilink or plasticene. All of them are likely to want to write and/or draw at some stage.</div> <br></br> <div>This activity lends itself to a 'messy maths wall' - a place where learners can contribute findings over an extended period. You can then plan to look together at everyone's work and talk about what they have explored.</div> <br></br> <div>If you are handy with materials you could have made all the thirteen cuboids and stuck them together with some tacky material beforehand.</div> <br></br> <br></br> <h3>Key questions</h3> <div>Tell me about the shapes that you've got.</div> <div>Tell me about what you're doing.</div> <div>Which is the biggest piece? How do you know?</div> <br></br> <h3>Possible extension</h3> <div>Some learners could investigate what happens when the starting piece is a larger size. Encourage them to ask questions of their own: "I wonder what would happen if ...?"</div> <div> </div> <h3>For the exceptionally able</h3> This pupil could be presented with the same idea as using a $5$ by $5$ by $5$ piece of cheese but consider using a knife to carefully cut cubes from it. I would set the challenge to find all the different ways of cutting cubes (with as little waste as possible) from it. For example you can obviously cut $125$ cubes of size $1$ - so that's one way. Can you cut it to produce some other size as well as $1$'s? So, how many ways altogether?<br></br> <h3>Possible support</h3> <div>Having cubes available will help pupils to make a start on this challenge, although some may need help from adults or peers when dismantling the cubes as they make the 'cuts'.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> You could make a note of the area of each piece as seen from above, and group them according to the cube that you started with i.e. in threes. This might help you notice some patterns to investigate. What happens if you add in these groups? <br></br> You could look at the difference in size of successive pieces. <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <br></br> <mdo:image height="159" width="478" alt="3 -B" src="3slicesB.gif"></mdo:image><br></br> [3 slices B]<br></br> We can take the 3 by 4 and split it into two triangular pieces:<br></br> <mdo:image height="120" width="376" alt="sq 2 tri A" src="Sq2TriA.gif"></mdo:image><br></br> [Sq2TriA]<br></br> Each of these triangular pieces can be placed on two adjacent side of the 3 by 3 square:<br></br> <br></br> <mdo:image height="263" width="279" alt="" src="Sq2TriB.gif"></mdo:image><br></br> [Sq2TriB]<br></br> When put together we have a large triangular shape using 21 little squares, and of course 21 is a triangular number.<br></br> <br></br> <mdo:image height="252" width="256" alt="sq2triC" src="sq2TriC.gif"></mdo:image><br></br> [Sq2TriC]<br></br> <br></br> For the more able or older pupils there is much to explore - particularly if you are able to extend the cheese to a 20 x 20 x 20 block - the following could be good doors to open:<br></br> <div style="margin-left: 40px;">1/ Looking at the total sizes of slices that will follow (eg for the 5 x 5 x 5 it will be adding the 1, 1, 2, 4, 4, . . . . 20, 25 and getting 125), but considering consecutive sizes of starting blocks.</div> <div style="margin-left: 40px;">2/ Consider the surface areas of the slices that are cut off.</div> <div style="margin-left: 40px;">3/ Examine the total surface areas of the slices that will come from a particular block and then considering consecutive sized blocks.</div> <div style="margin-left: 40px;">4/ Think of the block that is left after a slice has been taken and look at its surface area.</div> <div style="margin-left: 40px;">5/ With all of these "patterns" it is good (in fact rather exciting!) to look at the Digital Roots!</div> <div style="margin-left: 40px;">6/ Each time you look at a new sized block (e.g. the 8 x 8 x 8) looking at how, when all the slices have been cut, you can share out the slices to give an equal amount to the (8)people.</div> <div>Here's a spreadsheet and a bit of explanation of the columns.</div> <div>A is the size of the starting cube of cheese B,</div> <div>C is the size of successive slices to be removed - as B by C being the "cut" area</div> <div>D is the area that was cut through E is the digital root of column D</div> <div>F is the volume that we'll get when all the slices have been cut, starting from here</div> <div>G is the digital root of column F</div> <div>H is the surface area of the slice taken</div> <div>I is the total surface area of all the slices to come starting from here</div> <div>J, L, N are parts of I written out separately with</div> <div>K, M and O being the digital roots</div> <div>P is the surface area of the block that is left when the latest slice has been cut</div> <div>Q, R, S are the digital roots of P split up.</div> <br></br> <br></br> [Spread A]<br></br> <br></br> [Spread B]<br></br> REMOVED FROM ORIGINAL PROBLEM AFTER ALL THE CUTS WERE MADE;<br></br> <br></br> <br></br> So we had cut off in order:- a 5 by 5; 4 by 5; 4 by 4; 4 by 4; 3 by 4; 3 by 3; 3 by 3; 2 by 3; 2 by 2; 2 by 2; 2 by 1; 1 by 1 and you're left with a 1 by 1. When these slices are laid down their top surface measures :- 25, 20, 16, 16, 12, 9, 9, 6, 4, 4, 2, 1, and 1 left at the end. I felt that this series of numbers could well be worth exploring. So why don't you have a go too! You might find it useful to write them in the opposite order going up instead of going down. 1, 1, 2, 4, 4, 6, 9, 9, 12, 16, 16, 20, 25 . . . . . As well as looking at the numbers, you might like to use cubes, like I did and then make the 13 cuboids. Some ideas to try:- Investigate trying to share these 13 pieces out so that everyone gets an equal share. Try using the 13 cuboids to re-construct the 5 by 5 by 5 cube and explore different ways of doing that!! What about the surface area of each of the 13 slices? What about the surface area of the big block after each slice has been cut off. What about . . . . . . . . ? I guess that once you've explored the pattern of numbers you'll be able to extend it as if you had started with a 10 by 10 by 10 cube of cheese.<br></br> <br></br> There are lots of answers to this problem, depending on what questions you choose to ask. Have a go yourself, and if you discover anything interesting, let us know what you've done! 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. <br></br></mdoxml> 2 3 0 1 0 0 0 The Big Cheese Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with? Measures and Mensuration Area Using, Applying and Reasoning about Mathematics Investigations Mathematics Tools Interlocking cubes Measures and Mensuration Surface and surface area