2383
/www/nrich/html/content/id/2383/
1
2011-02-01T00:00:01
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<div>Find a cuboid (with edges of whole number lengths) that has a surface area of exactly $100$ square units.</div>
<div> </div>
<div>Is there more than one?</div>
<p>Can you find them all?</p>
<p>Can you provide a convincing argument that you have found them all?</p>
<p> <br></br>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=6097&part=">Click here for a poster of this problem</a>.</p>
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<div><span class="editorial">Steven and Ryan found that a 2x4x7
cuboid has a surface area of 100 square units.</span></div>
<br></br>
<mdo:image height="437" width="489" src="ryan.jpg" alt="image of a 2x4x7 cuboid"></mdo:image><br></br>
<p><span class="editorial">Justin from Mason Middle School found
that a 1x2x16 cuboid also satisfies the conditions:</span></p>
I back it up by saying that<br></br>
$1\times 2 = 2$, <br></br>
$2 \times16 = 32$ <br></br>
and $16 \times1 = 16$ <br></br>
and add them all up and you get $50$, and then multiply by $2$ and
you get a square area of $100$!<br></br>
<br></br>
<p class="editorial">Megan was able to show that these were the
only two possible solutions:</p>
Call the lengths of the 3 dimensions (height, depth, width) $x$,
$y$ and $z$.<br></br>
The surface area is $2xy+2xz+2yz$, as the area of each face is
calculated by multiplying its two sides together, and there are $2$
of each face. <br></br>
Hence, $2xy+2xz+2yz = 100$. <br></br>
Dividing by $2$, $xy+yz+xz = 50$. <br></br>
I will assume that $x$ is the shortest side and $z$ the longest to
avoid repeating solutions. Therefore I must find integer solutions
to the equation $xy+yz+xz = 50$ where $x < y < z$. <br></br>
Rearranging the equation, <br></br>
$yz+xz = 50-xy$<br></br>
$z(x+y) = 50-xy$<br></br>
$z = (50-xy)/(x+y)$. <br></br>
I used an excel spreadsheet with 3 columns, 1 for each of $x$, $y$
and $z$. <br></br>
In the $z$ column I write the rearranged formula. <br></br>
I then started from $x = 1$, $y = 1,2,3...$ looking for integer
values of $z$ until I reached a solution which had been repeated
(as here $y$ is bigger than $z$, so I would be repeating values
with $y$ and $z$ swapped around) or where $z$ became smaller than
$y$. <br></br>
For $x = 1$ I found only 1 solution, $(1, 2, 16)$. <br></br>
Checking with the original formula this does agree to a total of
$100$. <br></br>
I then continued repeating the procedure with $x = 2$, $y =
2$,$3$,$4$... and found a solution of $(2, 4, 7)$. <br></br>
With $x = 3$, $y = 3$,$4$,$5$... there are no solutions, as $z$
becomes smaller than $y$ where $y=5$ (and $z = 4.375$) and there
are no integer solutions before this. <br></br>
With $x = 4$, $y = 4,5,6...$ $z$ becomes smaller than $y$ when $y =
5$ $(z = 3.333)$ therefore there are no solutions where $x = 4$.
<br></br>
After $x = 4$, the intitial value of $z$ is always smaller than
$y$, therefore there are no further solutions. <br></br>
The only 2 solutions to the problem are: $(1, 2, 16)$ and $(2, 4,
7)$.<br></br>
<br></br>
<p class="editorial">Fred from Albion Heights School also offered
some non-integer solutions:</p>
$h = 1$, $w = 1$, $l = 24.5$<br></br>
<br></br>
$h = 2$, $w = 2$, $l = 11.5 $<br></br>
<br></br>
$h = 4$, $w = 4$, $l = 4.25$ <br></br>
<br></br>
$h = 2.5$, $w = 5$, $ l = 5$ <br></br>
<br></br>
$h = 1$, $w = 4$, $l = 9.2$ <br></br>
<br></br>
<p class="editorial">Well done to you all.</p>
<br></br>
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<h3>Why do this problem?</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=2383">This
problem</a> requires a lot of calculations of surface areas, within
a rich problem solving context.</div>
<h3>Possible approach</h3>
<div>Work with a specific cuboid, eg $2 \times 3 \times 5$, or a
breakfast cereal box, to establish how to calculate surface area of
cuboids. Students could practise working out surface area mentally
on some small cuboids made of multilink cubes.</div>
<br></br>
<div>Present the problem, ask students to keep a record of things
that they tried that didn't work (and what was wrong) as well as
things that did work. In this initial working session, try to
ensure that students are calculating surface area correctly.
<a href="/content/id/2383/examples%202383.xls">This spreadsheet</a>
may be useful (for teachers' eyes only!).</div>
<br></br>
It may be appropriate to draw a ladder on the board, with this on
the steps (starting from the bottom): <br></br>
- calculations going wrong <br></br>
- no solutions yet <br></br>
- one solution <br></br>
- some solutions <br></br>
- all solutions <br></br>
- why I am sure I have all the solutions <br></br>
- I'll change the question to... <br></br>
<br></br>
A short group discussion could suggest strategies to help students
move on up the ladder, before they continue with the problem.
<br></br>
<br></br>
This might be a good lesson in which to allocate fiveminutes at the
end to ask students to reflect on what they have achieved, which
methods and ideas were most useful, and what aspects of the problem
remain unanswered.<br></br>
<h3>Key questions</h3>
<ul>
<li>Have you found none/one/some or all of the solutions</li>
<li>Is there a cube that will work?</li>
<li>How might you organise a systematic search for the cuboids with
surface area $100$?</li>
</ul>
<h3>Possible extension</h3>
<div>The main extension activity could focus on the convincing
argument that all solutions have been found. Once this has been
answered, you might like to consider these extensions:</div>
<ul>
<li>Express the method for calculating surface area,
algebraically.</li>
<li>What surface area values will generate lots of cuboids and
which give none or just one?</li>
<li>Could you set up a spreadsheet to help with the
calculations?</li>
</ul>
<h3>Possible support</h3>
<div>In groups, or as a class, keep a record of all cuboids whose
surface areas have been calculated.</div>
<div>Award tenpoints for a bulls eye "$100$",
fivepoints for each $95-105$, and twopoints
for $90-110$.</div>
<div>Any miscalculated results could lose points, providing
motivation for peer checking, and helping each other.</div>
<br></br>
<div>A sheet showing a net of a cuboid, like <a href="/content/id/2383/SA%20net.doc">this</a> , may help students to
organise their working and ideas.</div>
<br></br>
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It may be simpler to focus on just three faces (the threedifferent
faces), rather than on all six.<br></br>
<br></br>
<a href="/content/id/2383/SA%20net.doc">This</a> net of a cuboid
may help.<br></br>
<br></br>
Try to be systematic:<br></br>
if the height is $1$, what are the possible combinations for the
width and depth?<br></br>
if the height is $2, 3, 4$... what are the possible combinations
for the width and depth?<br></br>
<br></br>
<br></br>
<br></br>
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Only two possible solutions:
</p>
<p>
1x2x16
</p>
<p>
2x4x7
</p>
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Cuboids
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
Admin
Workshop
Using, Applying and Reasoning about Mathematics
Working systematically
Measures and Mensuration
Surface and surface area
3D Geometry, Shape and Space
Cuboids
Using, Applying and Reasoning about Mathematics
Visualising
Numbers and the Number System
Factors and multiples
Information and Communications Technology
smartphone
Secondary Mapping Document
Area and volume LS