7534
/www/nrich/html/content/id/7534/
1
2011-05-16T14:52:40
<mdoxml version="1.0"><br></br>
Here are nine shapes. You can download a set of these shapes to
print off <a href="/content/id/7534/AreaPerimeter.pdf">here</a>.<br></br>
<br></br>
<table border="0" align="center">
<tbody>
<tr>
<td>
<div> <mdo:image height="100" width="100" src="area1.jpg" alt=""></mdo:image></div>
</td>
<td> <mdo:image height="100" width="100" src="area2.jpg" alt=""></mdo:image></td>
<td> <mdo:image height="100" width="100" src="area3.jpg" alt=""></mdo:image></td>
</tr>
<tr>
<td> <mdo:image height="100" width="100" src="area4.jpg" alt=""></mdo:image></td>
<td> <mdo:image height="100" width="100" src="area5.jpg" alt=""></mdo:image></td>
<td> <mdo:image height="100" width="100" src="area6.jpg" alt=""></mdo:image></td>
</tr>
<tr>
<td> <mdo:image height="100" width="100" alt="" src="area7.jpg"></mdo:image></td>
<td> <mdo:image height="100" width="100" alt="" src="area8.jpg"></mdo:image></td>
<td> <mdo:image height="100" width="100" alt="" src="area9.jpg"></mdo:image></td>
</tr>
</tbody>
</table>
<br></br>
<br></br>
The challenge is to arrange the shapes in a 3 by 3 grid like
the one below:<br></br>
<br></br>
<div style="text-align: left;"><mdo:image height="269" width="353" src="areaperimetergrid.jpg" alt=""></mdo:image></div>
<div style="text-align: center;"> </div>
As you go from left to right, the area of the shapes must
increase.<br></br>
As you go from top to bottom, the perimeter of the shapes must
increase.<br></br>
All the shapes in the middle column must have the same area.<br></br>
All the shapes on the middle row must have the same
perimeter.<br></br>
<br></br>
What reasoning can you use to help you to decide where each card
must go?<br></br>
<br></br>
<br></br>
Here are the dimensions of nine rectangles (printable version
<a href="/content/id/7534/AreaPerimeter2.pdf">here</a>).<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td>
<div> </div>
<div style="text-align: center;">$2$ by $8$</div>
<div style="text-align: center;">rectangle</div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;">$4$ by $4$</div>
<div style="text-align: center;">square</div>
</td>
<td>
<div style="text-align: center;">$1$ by $15$</div>
<div style="text-align: center;">rectangle</div>
</td>
</tr>
<tr>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">$5$ by $5$</div>
<div style="text-align: center;">square</div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;">$3$ by $8$</div>
<div style="text-align: center;">rectangle</div>
</td>
<td>
<div style="text-align: center;">$2$ by $7$</div>
<div style="text-align: center;">rectangle </div>
</td>
</tr>
<tr>
<td>
<div style="text-align: center;"> </div>
<div style="text-align: center;">$1$ by $16$ </div>
<div style="text-align: center;">rectangle</div>
<div style="text-align: center;"> </div>
</td>
<td>
<div style="text-align: center;">$3$ by $6$ </div>
<div style="text-align: center;">rectangle</div>
</td>
<td>
<div style="text-align: center;">$1$ by $9$ </div>
<div style="text-align: center;">rectangle</div>
</td>
</tr>
</tbody>
</table>
<br></br>
<br></br>
Can you arrange them in the grid in the same way?<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=""></mdo:image><br></br>
The ticks represent the nine cards you've already placed.<br></br>
Can you create cards with dimensions for rectangles that could go
in the four blank spaces that satisfy the same criteria?<br></br>
Not all the spaces are possible to fill. Can you explain
why?<br></br>
<br></br>
Can you produce a set of cards that could be arranged in the same
way, if the card in the centre is a 1 by 5 rectangle?<br></br>
<br></br></mdoxml>
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<br></br>
<p><span class="editorial">Isabelle from South Wilts and Natalie from St.Andrews International School in Thailand both answered the first challenge correctly.</span></p>
<p>Here is Natalie's arrangement of the shapes:</p>
<p><mdo:image src="Natalie.png"></mdo:image></p>
<p> </p>
<p><span class="editorial">Here is how Isabelle described her strategy:</span></p>
<p>a) Write down the areas and perimeters of each shape.</p>
<p>b) The only three shapes that share an area are G, A & C therefore they must occupy the middle column.</p>
<p>c) B, D & I have areas less than 14 so must occupy the left column and E, F & H have areas greater than 14 so must occupy the right column.</p>
<p>d) The shapes with perimeter 20 (B, C & H) must go in the bottom row, those with perimeter 18 must go in the middle row and those with perimeter 16 must go in the top row.</p>
<p>There is only one way this can be achieved so by a process of elimination the solution is as above.</p>
<p> </p>
<p> </p>
<p>Isabelle also answered the second challenge correctly:</p>
<p> </p>
<div> </div>
<div>
<table border="0" cellpadding="1" cellspacing="1" style="width: 500px;">
<thead>
<tr>
<th scope="row"> </th>
<th scope="col"> </th>
<th scope="col"> </th>
<th scope="col">AREA</th>
<th scope="col"> </th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row"> </th>
<td> </td>
<td style="text-align: center;">-</td>
<td style="text-align: center;">=</td>
<td style="text-align: center;">+</td>
</tr>
<tr>
<th scope="row"> </th>
<td style="text-align: center;">-</td>
<td style="text-align: center;">2 by 7</td>
<td style="text-align: center;">4 by 4</td>
<td style="text-align: center;">3 by 6</td>
</tr>
<tr>
<th scope="row">PERIMETER</th>
<td style="text-align: center;">=</td>
<td style="text-align: center;">1 by 9</td>
<td style="text-align: center;">2 by 8</td>
<td style="text-align: center;">5 by 5</td>
</tr>
<tr>
<th scope="row"> </th>
<td style="text-align: center;">+</td>
<td style="text-align: center;">1 by 15</td>
<td style="text-align: center;">1 by 16</td>
<td style="text-align: center;">
<div>3 by 8</div>
</td>
</tr>
</tbody>
</table>
</div>
<div> </div>
<div><span class="editorial">Krystof from Uhelny Trh, Prague, filled in one of the squares of the extended grid:</span></div>
<div> </div>
<div><mdo:image src="krystof.png"></mdo:image></div>
<div><br></br>
<span class="editorial">It's possible to fill in the box on the left too. I wonder if you can think of a way, and convince yourself that it's impossible to fill in the top and right boxes.</span></div>
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<h3>Why do this problem?</h3>
Working on this problem will give students a deeper understanding
of area and perimeter, and how they change as a shape is altered.
The problem will address some students' misconception that as area
increases, perimeter must necessarily increase too. <br></br>
<h3>Possible approach</h3>
<div>Display a simple shape made out of squares on a small square
grid:</div>
<div> </div>
<div style="text-align: center;"> <mdo:image height="261" width="268" alt="" src="charlieshape.jpg"></mdo:image></div>
<div> </div>
<div>"On your squared paper, shade in squares to make a different
shape with the same area as mine."</div>
<div>"My shape has a perimeter of 14. Does anyone else have a shape
with a perimeter of 14?"</div>
<div>Collect any examples and display them.</div>
<div>"Does anyone have a shape with a perimeter less than
14?"</div>
<div>Again, display any examples.</div>
<div>"Does anyone have a shape with a perimeter greater than
14?"</div>
<div>Once more, display any examples.</div>
<div>If there are no examples for any of the categories, challenge
them to find suitable shapes.</div>
<div> </div>
<div>"Area and perimeter are two attributes of these shapes. On <a href="/content/id/7534/OrderedRobots.pdf">this picture</a> the
robots have been arranged according to two of their attributes. Can
you work out how they have been arranged?"</div>
<div> </div>
<div>Draw out the key ideas:</div>
<div>As you go from left to right, the width of the
robots increases.</div>
<div>As you go from top to bottom, the height of the robots
increases.</div>
<div>All the robots in the middle column have the same width.</div>
<div>All the robots on the middle row have the same
height. </div>
<div> </div>
<div>Now display <a href="/content/id/7534/OrderedRobots2.pdf">this
picture</a> to summarise what they have (hopefully) noticed
and to introduce the type of grid the students will be using for
the rest of the problem.</div>
<div> </div>
<div>"We could arrange shapes in a 3 by 3 grid in the same way,
sorting them by their area and perimeter instead of the height and
width."</div>
<div> </div>
<div>Give each pair of students the <a href="/content/id/7534/AreaPerimeter.pdf">cards</a> for the first
part of the problem, display the grid on the board and ensure that
students understand what they have to do:</div>
<div> <mdo:image height="269" width="353" alt="" src="areaperimetergrid.jpg"></mdo:image></div>
<br></br>
As each pair finishes, they can be given the <a href="/content/id/7534/AreaPerimeter2.pdf">second set of cards</a>
to work on in the same way. 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 ways they found to compare areas and perimeters
without having to work them all out.<br></br>
<br></br>
In sharing feedback on the first activity, ask students what they
notice about the shapes on the top row of the grid. <br></br>
<br></br>
To explain why all shapes drawn by cutting corners out of a 4 by 4
square have a perimeter of 16, these images might be useful:<br></br>
<mdo:image height="143" width="500" alt="" src="bittensquare.jpg"></mdo:image><br></br>
"How much perimeter has been lost by cutting out the pink
rectangle? How much has been gained?"<br></br>
Shapes on the second and third row can be compared in the same
way.<br></br>
<br></br>
For the second activity, we want students to recognise:<br></br>
"Rectangles that are closer to squares have smaller perimeters than
long thin rectangles with the same area"<br></br>
<br></br>
One prompt that could draw out this thinking might be:<br></br>
"If two rectangles have the same area but different perimeters, how
can I decide which has the greater perimeter?"<br></br>
<br></br>
Finally, discuss the possible content of the four extra spaces in
the extended grid, focussing in particular on why some of the
spaces are impossible to fill in.<br></br>
<h3>Possible extension</h3>
<div>Finally, students could be challenged to create their own set
of cards with a $1$ by $5$ rectangle as the central card. This
forces them to consider rectangles whose side lengths are not whole
numbers.</div>
<div> </div>
<div><a href="http://nrich.maths.org/7535&part=">Changing
Areas, Changing Volumes</a> explores in a similar way the
relationship between volume and surface area of different
cuboids.</div>
<br></br>
<h3>Possible support</h3>
<a href="http://nrich.maths.org/7280&part=">Area and
Perimeter</a> provides a good starting point for the thinking
expected of students in this problem. <br></br>
<br></br>
<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
Working on <a href="http://nrich.maths.org/7280&part=">Area and
Perimeter</a> might give you some useful insights to help you to
solve this problem. <br></br>
<br></br>
If you are having trouble making sense of the grid, have a look at
<a href="/content/id/7534/OrderedRobots2.pdf">this image</a>. The
robots have been arranged according to their width and height, in
the same way that the cards need to be arranged according to their
area and perimeter:<br></br>
<br></br>
As you go from left to right, the width of the robots increases.
<br></br>
As you go from top to bottom, the height of the robots increases.
<br></br>
All the robots in the middle column have the same width. <br></br>
All the robots on the middle row have the same height. <br></br></mdoxml>
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<table border="1">
<tbody>
<tr>
<td>2 by 7</td>
<td>4 by 4</td>
<td>3 by 6</td>
</tr>
<tr>
<td>1 by 9</td>
<td>2 by 8</td>
<td>5 by 5</td>
</tr>
<tr>
<td>1 by 15</td>
<td>1 by 16</td>
<td>3 by 8</td>
</tr>
</tbody>
</table>
<div> </div>
<table border="1">
<tr>
<td> </td>
<td> </td>
<td>impossible</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>e.g. 0.5 by 9.5</td>
<td> </td>
<td> </td>
<td> </td>
<td>impossible</td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>e.g. 0.5 by 32</td>
<td> </td>
<td> </td>
</tr>
</table>
<div> </div>
<br></br></mdoxml>
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Changing areas, changing perimeters
How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?
Measures and Mensuration
Area
Measures and Mensuration
Perimeters