7535
/www/nrich/html/content/id/7535/
1
2011-06-06T14:05:09
<mdoxml version="1.0"><span style="font-style: italic;">This problem follows on
from</span> <a href="http://nrich.maths.org/7534&part=" style="font-style: italic;">Changing Areas, Changing
Perimeters</a>.<br></br>
<br></br>
<br></br>
Here are the dimensions of nine cuboids. You can download a set of
cards <a href="/content/id/7535/AreaVolume.pdf">here</a>.<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">1 by 2 by 28</div>
<div style="text-align: center;">cuboid </div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">4 by 4 by 4</div>
<div style="text-align: center;">cube</div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">2 by 4 by 7</div>
<div style="text-align: center;">cuboid </div>
<div style="text-align: center;"> </div>
</td>
</tr>
<tr>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">1 by 2 by 26</div>
<div style="text-align: center;">cuboid </div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">2 by 4 by 6</div>
<div style="text-align: center;">cuboid </div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">4 by 5 by 6</div>
<div style="text-align: center;">cuboid </div>
<div style="text-align: center;"> </div>
</td>
</tr>
<tr>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">4 by 5 by 7</div>
<div style="text-align: center;">cuboid </div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">1 by 2 by 24</div>
<div style="text-align: center;">cuboid </div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">1 by 4 by 14</div>
<div style="text-align: center;">cuboid </div>
<div> </div>
</td>
</tr>
</tbody>
</table>
<br></br>
<br></br>
The challenge is to arrange them in a 3 by 3 grid like the one
below:<br></br>
<mdo:image height="278" width="386" src="volumegrid.jpg" alt="area and volume grid"></mdo:image><br></br>
<br></br>
<br></br>
As you go from left to right, the surface area of the shapes must
increase.<br></br>
As you go from top to bottom, the volume of the shapes must
increase.<br></br>
All the cuboids in the middle column must have the same surface
area.<br></br>
All the cuboids on the middle row must have the same volume.<br></br>
<br></br>
<span style="font-weight: bold;">What reasoning can you use to help
you to decide where each cuboid must go?</span><br></br>
<br></br>
<br></br>
Once you've placed the nine cards, take a look at the extended grid
below:<br></br>
<mdo:image height="300" width="300" src="grid2.jpg" alt="extended grid"></mdo:image><br></br>
The ticks represent the nine cards you've already placed. Can you
create cards with dimensions for cuboids that could go in the four
blank spaces that satisfy the same criteria? <br></br>
<br></br>
<br></br>
Can you design a set of cards of your own with a different cuboid
in the centre?<br></br></mdoxml>
<mdoxml version="1.0">
<p><span class="editorial">Well done to Philip from Wilson's School who submitted <a href="/content/id/7535/philip%20solution.pdf">this solution</a>.</span></p>
<p> </p>
</mdoxml>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
Working on this problem will give students a deeper understanding
of the relationship between volume and surface area, and how they
change as the dimensions of a cuboid are altered.<br></br>
<h3>Possible approach</h3>
<div>This problem follows on from <a href="http://nrich.maths.org/7534&part=note">Changing Areas,
Changing Perimeters</a>. We suggest students start with the
rectangles task from that problem to introduce them to the
structure of the grid they will be using here.</div>
<div> </div>
<div>Once students are familiar with the grid structure, hand out
<a href="/content/id/7535/AreaVolume.pdf">these cards</a> and
invite students to work in pairs to arrange the cards in a grid
like this:</div>
<div> <mdo:image height="278" width="386" alt="" src="volumegrid.jpg"></mdo:image></div>
<div>Students could use multilink cubes or draw each cuboid on <a href="http://nrich.maths.org/content/id/6676/DottedIsometricGrid_10mm.pdf">
isometric paper</a> to support them as they work on the task.</div>
<br></br>
For those who finish quickly, ask them the question from the
problem about extending the grid like this:<br></br>
<mdo:image height="300" width="300" alt="" src="grid2.jpg"></mdo:image><br></br>
<br></br>
Towards the end of the lesson, bring the class together to share
any efficient strategies they used to complete the task.<br></br>
<br></br>
Pose the question <span style="font-weight: bold;">"If I know two
cuboids have the same volume, how can I decide, just by looking at
their dimensions, which has the greater surface area?"</span><br></br>
<br></br>
Draw out students' ideas about the properties of long and thin
cuboids as opposed to those that are almost cubes.<br></br>
(This is the three dimensional analogue of short and fat rectangles
having a smaller perimeter than long thin ones, when their areas
are equal.)<br></br>
<br></br>
Finally, discuss the possible content of the four extra spaces in
the extended grid and strategies they used to generate
possibilities.<br></br>
<h3>Possible extension</h3>
Challenge students to design their own set of nine cards that can
be arranged in this way. If students are restricted to whole
numbers it is quite challenging to create cuboids with equal
surface areas.<br></br>
<br></br>
<h3>Possible support</h3>
<a href="http://nrich.maths.org/7280&part="></a><a href="http://nrich.maths.org/7534&part=">Changing Areas,
Changing Perimeters</a> provides a two-dimensional version of this
three-dimensional problem.<br></br>
<br></br>
<br></br></mdoxml>
<mdoxml version="1.0">Try working on <a href="http://nrich.maths.org/7534&part=">Changing Areas,
Changing Perimeters</a> first, as it is a two-dimensional version
of this three-dimensional problem.<br></br>
<br></br>
If I know two rectangles have the same area, how can I decide, just
by looking at their dimensions, which has the greater
perimeter?<br></br>
If I know two cuboids have the same volume, how can I decide, just
by looking at their dimensions, which has the greater surface
area?<br></br>
<br></br>
<br></br></mdoxml>
<mdoxml version="1.0"><table border="1">
<tr>
<td>2 by 4 by 6 cuboid </td>
<td>1 by 2 by 24 cuboid </td>
<td>1 by 2 by 26 cuboid </td>
</tr>
<tr>
<td>2 by 4 by 7 cuboid </td>
<td>1 by 4 by 14 cuboid </td>
<td>1 by 2 by 28 cuboid </td>
</tr>
<tr>
<td>4 by 4 by 4 cube</td>
<td>4 by 5 by 6 cuboid </td>
<td>4 by 5 by 7 cuboid </td>
</tr>
</table>
<div> </div>
<div> </div>
<table border="1">
<tr>
<td> </td>
<td> </td>
<td>1 by 1 by 36.5</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>4 by 4 by 3.5</td>
<td> </td>
<td> </td>
<td> </td>
<td>1 by 1 by 56</td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>4.9 by 5 by 5</td>
<td> </td>
<td> </td>
</tr>
</table>
<div> </div>
<div> </div>
<br></br></mdoxml>
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Changing areas, changing volumes
How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?
Measures and Mensuration
Surface and surface area
Measures and Mensuration
Volume and capacity
3D Geometry, Shape and Space
Cubes
3D Geometry, Shape and Space
Cuboids