6398
/www/nrich/html/content/id/6398/
1
2011-05-27T14:31:00
<mdoxml version="1.0"><p>Charlie has been drawing rectangles:<br></br>
<br></br>
<mdo:image src="equal1.png"></mdo:image> <mdo:image src="equal2.png"></mdo:image><br></br>
<br></br>
The first rectangle has a perimeter of 30 units and an area of 50 square units.<br></br>
The second rectangle has a perimeter of 24 units and an area of 20 square units.<br></br>
<br></br>
Charlie wondered if he could find a rectangle whose perimeter and area have the same numerical value. <br></br>
<br></br>
Can you find a rectangle that satisfies this condition?<br></br>
<br></br>
Alison says "There must be lots of rectangles whose perimeter and area have the same numerical value."<br></br>
Charlie is not so sure.<br></br>
Can you find more examples of such rectangles?<br></br>
<br></br>
Can you come up with a convincing argument to help Charlie and Alison decide who is right?<br></br>
<br></br>
<a href="http://nrich.maths.org/7996">Click here for a poster of this problem</a>.</p></mdoxml>
<mdoxml version="1.0">
<br></br>
<p><span class="editorial">We had lots of solutions to this problem, so well done to everyone who submitted an answer!</span></p>
<p><span class="editorial">David, Noah, Felix, Tom, Amy and Laura from Bristol Grammar School worked on this problem together.</span> <span class="editorial">David says:</span></p>
<p>We found that the 6 by 3 rectangle works, because 6+3+6+3=18 and 6x3=18, so this has equal area and perimeter.<br></br>
<br></br>
<span class="editorial">Laura and Amy say:</span></p>
<p>We have been working systematically to list all the possible rectangles, e.g. 1x1 1x2 1x3 1x4 ... and deciding whether there are cases with 1, 2 or 3</p>
<div style="margin-left: 40px;">2x1 2x2 2x3 2x4...<br></br>
3x1 3x2 3x3 3x4...</div>
<p>We noticed a diagonal pattern for where the perimeter becomes less than the area.<br></br>
We are still working on a solution!</p>
<p><span class="editorial">Well done to Gourav from India, Radha from Stanhope, Kirstey from da Vinci College and Caitlin from Marshfield Primary who all correctly found rectangles with the same area and perimeter, such as 4x4 and 3x6.</span></p>
<p><span class="editorial">Hannah from Leicester Girls High made some good notes on even and odd numbers:</span></p>
<p>I realised that at least one of the length or the width of the rectangle has to be even. The perimeter will always be even, because the length is multiplied by 2, making it even, and is added to the width which has been multiplied by 2, also making it even. But if both the length and the width are odd, then the area will be odd, meaning that it is impossible for the perimeter to be the same as
the area.</p>
<p><span class="editorial">Click <a href="/content/id/6398/DexterExplanation.xlsx">here</a> to see how Dexter from Wilson's School used diagrams to understand different rectangles and how he also used some algebra to find examples of rectangles where the perimeter is equal to the area.</span></p>
<p> </p>
<p><span class="editorial">Lucy from Belgium noticed visually what was happening to rectangles with whole number side lengths:</span></p>
<p>Each square along the edges accounts for one unit of perimeter, except for the four corner squares. They account for two units of perimeter but only one unit of area. This means the perimeter is 4 more than the area and there will have to be some squares in the middle that will only be counted for the area and not the perimeter.</p>
<p><span class="editorial">Bhavik from Queen Elizabeth's School for Boys also considered rectangles with whole number side lengths, and came to the same conclusion. Click <a href="/content/id/6398/BhavikExplanation.xls">here</a> to see his very clear explanation of why there can only be two such rectangles where the area is equal to the perimeter.</span></p>
<p> </p>
<p><span class="editorial">Nathan from Rushmore Primary made an attempt at using algebra, which was continued by Vicki from Farnborough Hill, Eliza and Jacqueline from Chevalier College, Australia and Tom. Vicki noted that for a rectangle <em>x</em> by <em>y</em>, the area is equal to the perimeter if:</span> $$ \begin{align*} xy &= 2x+2y \\ xy-2y &= 2x \\ y(x-2) &= 2x\\ y &=
\frac {2x}{x-2} \end{align*} $$</p>
<p> </p>
<p><span class="editorial">Niharika then looked at the possible values of <em>x</em> and <em>y</em> from this equation:</span> $$ \begin{align*} y &= \frac {2x}{x-2} \text{ and } x ,y> 0 \\ \Rightarrow x-2 &> 0 \\ x&> 2 \end{align*} $$ There are an infinite number of these rectangles.</p>
<p> </p>
<p><span class="editorial">Substituting in different values of <em>x</em> and <em>y</em> and checking the answers are correct is a good problem solving skill - well done to Krystof and Mimas who did this. Also Shashank from India drew a graph of the possible <em>x</em> and <em>y</em> values:</span></p>
<p style="text-align: center;"><mdo:image style="width: 298px; height: 292px;" src="ShashankExplanation.jpg" alt=""></mdo:image></p>
<p><br></br>
<span class="editorial">Esther commented that:</span><br></br>
<br></br>
Although you can always put a number into the formula I used, $2x+2y=xy$, the end result is not always an integer.<br></br>
In fact, the only rectangles like this with side lengths integers are a 3x6 rectangle and a 4x4 square.<br></br>
Any other numbers substituted into the formula as <em>x</em> will give a decimal output for <em>y</em>.<br></br>
For example, if you put in 5 as <em>x</em>, your value for <em>y</em> would be 10/3.<br></br>
</p>
<p><span class="editorial">Well done to Miss Gerrard's 2BE Class from Perth High School, who noticed some interesting connections between area and perimeter:</span></p>
<p>We all found 2 rectangles that worked fairly quickly. They were the square with sides of 4 (A=P=16) and the rectangle with sides 3 and 6 (A=P=18).</p>
<p> </p>
<div>Then we decided that we would try using decimals to see if we could find any more as we were getting stuck. One of us found out that a rectangle of sides 10 and 2.5 worked as it gave us an area and perimeter of 25.</div>
<div> </div>
<div>After we found this rectangle someone else in the class managed to spot a pattern linking the numbers. Start at the square with sides 4 by 4.</div>
<div>Then look at the rectangle with side length of 3 and width of 6.</div>
<div>The difference between the lengths 3 and 4 is 1, the difference between the widths 4 and 6 is 2.</div>
<div>To get from 3 to the next length you halve the difference between 4 and 3 and subtract this from 3, to get 2.5.</div>
<div>To get from 6 to the next width you double the difference between 4 and 6 and add this to 6, to get 10.</div>
<div>We found out that if you continue halving the difference in lengths and subtracting this and doubling the width and adding it on you can find many more rectangles with equal perimeter and area:</div>
<div style="margin-left: 80px;">$4$ by $4$</div>
<div style="margin-left: 80px;">$3$ by $6$</div>
<div style="margin-left: 80px;">$2\frac{1}{2}$ by $10$</div>
<div style="margin-left: 80px;">$2\frac{1}{4}$ by $18$</div>
<div style="margin-left: 80px;">$2\frac{1}{8}$ by $34$</div>
<div style="margin-left: 80px;">...</div>
<p><span class="editorial">Well done to you all.</span></p>
<br></br>
</mdoxml>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
Sometimes area and perimeter of rectangles are taught separately,
and are often confused. In this problem students consider the
relationship between them and are being challenged to engage in
some sophisticated mathematical thinking.<br></br>
<h3>Possible approach</h3>
<div>Show the students <a href="/content/id/6398/charlie1.jpg">this
image</a> and ask them to work out the area and perimeter of each
rectangle. </div>
<div>Collect answers.</div>
<div>"That's interesting, the first rectangle has an area that is
numerically greater than the perimeter, but the second one has an
area that is numerically less than the perimeter. I wonder if you
could find a rectangle whose area and perimeter are numerically the
same?"</div>
<div>Set students to work on this challenge, perhaps encouraging
them to work in pairs so they can share ideas on how to
proceed.</div>
<div>"If you manage to find a rectangle that satisfies my
conditions, see if you can find a few more."</div>
<div> </div>
<div>Circulate and observe the methods and reasoning students are
using. Look out for students who:</div>
<ul>
<li>fix one attribute (side length, area, perimeter) and vary
the others using trial and improvement</li>
<li>fix one attribute and use algebra to solve for the other
attributes</li>
<li>write an algebraic expression for area and perimeter, equate
them, and substitute values into the resulting equation</li>
</ul>
<div>For students who are struggling to get started:</div>
<div>"What is the same about the two rectangles we started
with?"</div>
<div>"What could you change?"</div>
<div>"How does the area and perimeter change as you change the
height of the rectangle?"</div>
<div> </div>
<div> </div>
<div>Once everyone has had a chance to find a few rectangles that
satisfy the condition, collect together the dimensions on the
board.</div>
<div>Invite students to share any different strategies you observed
them using as they were working.</div>
<div>"I'd like you to have a go at finding a few more rectangles,
using several different strategies." </div>
<div>"While you are working, think about how many different
rectangles we could possibly find."</div>
<div> </div>
<div>Finish off by asking students to share their ideas about how
many different rectangles satisfy the criteria, together with
convincing arguments about why there are infinitely many.</div>
<br></br>
<h3>Possible extension</h3>
<div>Ask students to consider other polygons with numerically equal
areas and perimeters - those who have met Pythagoras' theorem could
investigate right-angled and isosceles triangles, and those who
have met trigonometry could work on regular polygons.</div>
<div> </div>
<div>Students could be invited to consider cuboids whose surface
area is numerically equal to their volume.</div>
<br></br>
<h3>Possible support</h3>
<div>A more scaffolded introduction to the problem:</div>
<div> </div>
<div>Tell the students you are thinking of a rectangle. Ask them to
work out its dimensions if:</div>
<div>the area is 24 and the perimeter is 20</div>
<div>the area is 24 and the perimeter is 22</div>
<div>the area is 24 and the perimeter is 28</div>
<div>the area is 24 and the perimeter is 50</div>
<div> </div>
<div>Record the solutions on the board. Ask the students to comment
on anything they notice. (This might be to do with the shape of the
rectangles, or perhaps the evenness of the perimeters.) </div>
<div> </div>
<div>Repeat the process keeping the perimeter fixed this time, to
20.</div>
<div>Can they find the dimensions of rectangles with areas of 9,
16, 21, 24, 25? </div>
<div> </div>
<div> </div>
<div>Another activity to help students to become fluent in working
out the different attributes of rectangles:</div>
<div>Students could make up their own card matching game where each
set contains three cards about a specific rectangle, one with area,
one with perimeter and one with the dimensions. Students have to
find all three in a set. Each student produces 8 sets, shuffles
them and hands them on to their neighbour to sort.</div>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
Find the dimensions of the following rectangles:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td style="text-align: center;">Dimensions</td>
<td style="text-align: center;">Area</td>
<td style="text-align: center;">Perimeter</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">9</td>
<td style="text-align: center;">20</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">16</td>
<td style="text-align: center;">20</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">21</td>
<td style="text-align: center;">20</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">24</td>
<td style="text-align: center;">20</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">25</td>
<td style="text-align: center;">20</td>
</tr>
</tbody>
</table>
<br></br>
Draw the rectangles.<br></br>
What do you notice about the shapes of rectangles with a
fixed perimeter as their areas
increase?<br></br>
<br></br>
Find the dimensions of the following rectangles:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td style="text-align: center;">Dimensions</td>
<td style="text-align: center;">Area</td>
<td style="text-align: center;">Perimeter</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">24</td>
<td style="text-align: center;">20</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">24</td>
<td style="text-align: center;">22</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">24</td>
<td style="text-align: center;">28</td>
</tr>
<tr>
<td style="text-align: center;"></td>
<td style="text-align: center;">24</td>
<td style="text-align: center;">50</td>
</tr>
</tbody>
</table>
<br></br>
Draw the rectangles.<br></br>
What do you notice about the shapes of rectangles with a fixed
area as their perimeters increase?<br></br>
<br></br>
<br></br></mdoxml>
<mdoxml version="1.0">For a rectangle of sides a and b, Area = Perimeter when<br></br>
<br></br>
ab = 2a + 2b<br></br>
<br></br>
so b = 2a / (a - 2)<br></br>
<br></br>
Here are some possible solutions:<br></br>
<table border="1">
<tr>
<td>a</td>
<td>b</td>
</tr>
<tr>
<td>3</td>
<td>6</td>
</tr>
<tr>
<td>4</td>
<td>4</td>
</tr>
<tr>
<td>5</td>
<td>3 1/3</td>
</tr>
<tr>
<td>6</td>
<td>3</td>
</tr>
<tr>
<td>8</td>
<td>2 2/3</td>
</tr>
<tr>
<td>10</td>
<td>2.5</td>
</tr>
<tr>
<td>12</td>
<td>2.4</td>
</tr>
</table>
<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
2
3
0
0
1
0
0
Can they be equal?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Measures and Mensuration
Area
Measures and Mensuration
Perimeters
2D Geometry, Shape and Space
Rectangles
Using, Applying and Reasoning about Mathematics
Making and testing hypotheses
Information and Communications Technology
smartphone