2927
/www/nrich/html/content/id/2927/
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2011-02-01T00:00:01
<mdoxml version="1.0"><br></br>
<p>For this problem, you will need to print off a set of <a href="/content/id/2927/quadrilateral%20cards.doc">quadrilateral cards</a> . You'll also need a copy of <a href="/content/id/2927/Property%20Chart.doc">the game grid</a><br></br>
<br></br>
Shuffle eight cards, and lay them on the grid in the spaces marked "property card". Your challenge is to draw a quadrilateral in each square, so that the quadrilateral has both the properties at the top of the column and at the start of the row. There might be some that aren't possible! If you like, you could play this with a friend, taking turns to draw shapes. If you can't draw a shape, pass
and see whether your partner can, the winner is the one who draws the last shape.<br></br>
<br></br>
You might find it helpful to draw the shapes on spotty paper; there are squares <a href="http://www.curriculum.edu.au/maths300/m300bits/000dotsq.htm">here</a> and isometric paper <a href="http://www.curriculum.edu.au/maths300/m300bits/000dotis.htm">here</a> (this might help you find areas and angles).<br></br>
</p>
<div style="text-align: center;"><mdo:image alt="Picture of board and some cards" height="403" src="Property%20Chart.gif" width="328"></mdo:image></div>
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<div style="text-align: left;">You could play the game using <a href="/content/id/2927/triangle%20cards.doc">triangle cards</a> instead.</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">
<div style="clear: both;"> </div>
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Now you've had a go at the challenge, here are some questions you could think about. Use the quadrilateral cards for these.</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;"> </div>
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<p>Can you select 8 cards and arrange them so that you can fill in all of the squares? What cards did you use? What about none of the squares?<br></br>
<br></br>
What's the smallest number of different shapes you need to fill in the grid? What shapes are these, and what cards did you use?<br></br>
</p>
<div style="text-align: center;"><mdo:image alt="Quadilateral cards" height="400" src="Prop%20Chart.gif" width="400"></mdo:image></div>
<div class="framework"><span style="font-weight: bold;">This problem is based on the game Nine Pin Shape Draw from "Geometry Games", a photocopiable resource produced by Gillian Hatch and available from the</span> <a href="http://www.atm.org.uk//buyonline/products/act061.html" style="font-weight: bold;">Association of Teachers of Mathematics</a></div></mdoxml>
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<span class="editorial">Samantha from Hamlin sent us her work on
this problem. She found that it</span> <span class="editorial" style="font-style: italic;">is</span> <span class="editorial">possible to choose cards so that all of the boxes
can be filled in. In fact, she chose cards so that she could fill
in all of the boxes using a 1x1 square! Here is her example</span>
:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td> </td>
<td>Has all equal angles</td>
<td>Has rotational symmetry</td>
<td>Has more than 1 axis of symmetry</td>
<td>Has area of 1 unit</td>
</tr>
<tr>
<td>Has more than 2 equal angles</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>Has more than 1 right angle</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>Has more than 2 equal sides</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>Has 2 pairs of parallel sides</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br></br>
<span class="editorial">Well done to Christine, Peter, Rebecca and
Josh from Ely St John's School who found two more ways to choose
cards so that all the boxes could be filled in. They decided
to use rectangles as well as squares. Here is one of their
solutions.</span> <br></br>
<table border="1">
<tr>
<td> </td>
<td>Has more than 1 axis of symmetry</td>
<td>Has all equal angles</td>
<td>Has more than 2 equal angles</td>
<td>Has 2 pairs or parallel sides</td>
</tr>
<tr>
<td>Has more than 1 right angle</td>
<td style="text-align: center; font-weight: bold;">square</td>
<td style="text-align: center; font-weight: bold;">
square </td>
<td style="text-align: center; font-weight: bold;">square</td>
<td style="text-align: center; font-weight: bold;">square</td>
</tr>
<tr>
<td style="text-align: left;">Has more than 2 equal angles</td>
<td style="text-align: center; font-weight: bold;">square</td>
<td style="text-align: center; font-weight: bold;">square</td>
<td style="text-align: center; font-weight: bold;">square</td>
<td style="text-align: center; font-weight: bold;">square</td>
</tr>
<tr>
<td style="text-align: center;">Has area of 1 unit
<div> </div>
</td>
<td style="text-align: center; font-weight: bold;">square</td>
<td style="text-align: center; font-weight: bold;">square</td>
<td style="text-align: center; font-weight: bold;">square</td>
<td style="text-align: center; font-weight: bold;">square</td>
</tr>
<tr>
<td style="text-align: center;">
<div>Has area of 2 units</div>
<div> </div>
</td>
<td style="text-align: center; font-weight: bold;">rectangle</td>
<td style="text-align: center; font-weight: bold;">rectangle</td>
<td style="text-align: center; font-weight: bold;">rectangle</td>
<td style="text-align: center; font-weight: bold;">rectangle</td>
</tr>
</table>
<br></br>
<span class="editorial">George found that it is possible to choose
cards so that none of the boxes can be filled in. Here's the
example he sent us:</span><br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td> </td>
<td>More than one axis of symmetry</td>
<td>Just two pairs of parallel sides</td>
<td>Rotational symmetry</td>
<td>All angles equal</td>
</tr>
<tr>
<td>Just one axis of symmetry</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>Just one pair of parallel sides</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>Just two equal angles</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>One right angle</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=2927&part=">This
game</a> provides an interesting context in which to consider the
properties of quadrilaterals (or triangles), and has a particular
focus on the combinations of properties that are possible.<br></br>
<h3>Possible approach</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=2924&part=">Quadrilaterals
Game</a> could be used either at the beginning or at the end of
this problem.</div>
<br></br>
<div>Use the instruction on the <a href="http://nrich.maths.org/public/viewer.php?obj_id=2927">problem
page</a> to set up and play the game for about half a lesson, then
move the group on to the questions at the end of the problem.</div>
<br></br>
<h3>Key questions</h3>
<ul>
<li>Which shapes are most useful in this game?</li>
<li>Which property cards are 'good' and 'bad' and why?</li>
<li>Tell me two cards where there is no shape that works for
both.</li>
</ul>
<h3>Possible extension</h3>
<div>If only the quadrilaterals are visible on the board can you
identify the property cards in each position? In what other ways
can you adapt/invert/develop this game to make new and possibly
harder challenges?</div>
<br></br>
<div>Suggest students have a go at <a href="http://nrich.maths.org/public/viewer.php?obj_id=2925&part=">Shapely
Pairs</a></div>
<h3>Possible support</h3>
<div>The game could be played as a whole class - shuffle and
arrange the property cards on the board so that everyone has the
same question. Groups of 3 or 4 then work together filling in the
grid and checking each others work. A correct shape or gap will
earn 10 points, but each incorrect shape or gap will lose 10
points. After a set time, all the grids are displayed, and students
try to find errors in the other groups' work, in order to establish
the scores and the winners. They may be ready to try the problem as
stated after this!</div>
<div>Another way in to the problem could be to produce some partly
completed grids and ask students to finish them, or produce some
completed grids with a few deliberate errors for students to find
and correct.</div>
<h3><span style="font-weight: bold;">Additional comment</span></h3>
Teachers may be interested in Gillian Hatch's article <a href="http://nrich.maths.org/public/viewer.php?obj_id=2928&part=index">
Using Games in the Classroom</a> in which she analyses what goes on
when mathematical games are used as a pedagogic device.<br></br></mdoxml>
<mdoxml version="1.0">You'll have to use some experimentation here, to get a feel for what does and doesn't work.<br></br><br></br>Do you think you could fill the whole grid using just one shape, if you chose the right 8 cards? What sort of quadrilateral might have lots of nice properties?<br></br><br></br>It's pretty obvious that you can't come up with a shape that has just 1 axis of symmetry and more than 1 axis of symmetry, so if you put one of these at the start of a row and the other at a start of a column, you know that you won't be able to fill in the square where they intersect. Can you think of any other combinations like this? Is it possible to choose them so that you can't fill in <span style="font-style: italic;">any</span> of the squares?<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<div>The answers given below are using the quadrilateral cards.</div>
<div>
<div style="clear: both;"></div>
<br></br><br></br>
</div>
<div></div>
<div>It is possible to fill in the whole grid, in fact using just one shape. For example, a 1x1 square has all 8 of the following properties:</div>
<div></div>
<div>More than two equal sides</div>
<div>More than one axis of symmetry</div>
<div>Rotational symmetry</div>
<div>All angles equal</div>
<div>Two pairs of parallel sides</div>
<div>Area of 1 unit</div>
<div>More than two equal angles</div>
<div>More than one right angle</div>
<div>
<div style="clear: both;"></div>
<br></br><br></br>
</div>
<div></div>
<div>It is also possible to select 8 cards so that no squares can be filled in. For example, we could have the following arrangement:</div>
<div></div>
<table border="1"><tbody><tr><td></td>
<td>More than one axis of symmetry</td>
<td>Just two pairs of parallel sides</td>
<td>Rotational symmetry</td>
<td>All angles equal</td>
</tr>
<tr><td>Just one axis of symmetry</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr><td>Just one pair of parallel sides</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr><td>Just two equal angles</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr><td>One right angle</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<div></div>
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Property chart
A game in which players take it in turns to try to draw
quadrilaterals (or triangles) with particular properties. Is it
possible to fill the game grid?
Using, Applying and Reasoning about Mathematics
Games
2D Geometry, Shape and Space
Quadrilaterals
2D Geometry, Shape and Space
Mixed triangles