1873
/www/nrich/html/content/03/07/penta4/
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2011-02-01T00:00:01
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You will need a mirror for this activity.<br></br>
<br></br>
Here is a square:
<div align="center">
<p><mdo:image width="159" height="159" src="square.gif" alt="square"></mdo:image></p>
<p align="left">Where can you put the mirror across the square so
that you can still 'see' the whole square?<br></br>
How many different positions are possible?<br></br>
How many lines of symmetry does a square have?</p>
<p align="left">Can you reflect part of the square so that you can
see a smaller square?<br></br>
A rectangle? A kite? A hexagon? An octagon?<br></br>
What do all the shapes have in common?</p>
<p align="left" class="acknowledgement"><span style="font-style: italic;">This problem is taken from 'Starting from
Mirrors' by David Fielker, published by BEAM Education. It can be
purchased from the</span> <a href="http://www.beam.co.uk" style="font-style: italic;">BEAM website</a><span style="font-style: italic;">.</span></p>
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<p class="editorial">Ruth has sent in some ideas for the first part
of the question. She says you can put a mirror across the middle or
diagonally from corner to corner to see the whole square. She sent
these diagrams:</p>
<table width="100%" cellpadding="5" border="0">
<tbody>
<tr>
<td>
<div align="center"><mdo:image width="67" height="65" alt="square with horizontal line across the middle" src="image002.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="68" height="66" alt="square with diagonal line from bottom left corner to top right corner" src="image001.gif"></mdo:image></div>
</td>
</tr>
</tbody>
</table>
<div class="editorial">Gemma from Greenleas Primary School managed
to find a few more ways of seeing the whole square, as well as some
other shapes:</div>
<br></br>
<div><mdo:image width="350" height="553" src="Mirror%20square%20solution%201.gif" alt="Mirror square"></mdo:image></div>
<div class="editorial">Siobhan from Daubeny Middle School also sent
in a very nice solution. She says:</div>
<div class="editorial"></div>
<div class="editorial"></div>
<br></br>
<div>There are four positions where the mirror can be put and you
can still 'see' the whole square: vertically in the centre,
horizontally in the centre, and on the two diagonals. So the square
has four lines of symmetry. To reflect part of the square to make
another shape, half the number of the sides of the shape you want
to make would need to be reflected into the mirror. The shape you
make will have at least one line of symmetry. The diagrams shows
the positions of the mirror to make more shapes.</div>
<br></br>
<div><mdo:image width="600" height="449" src="siobhan.gif" alt=""></mdo:image></div>
<br></br>
<span class="editorial">Thank you for writing to us Siobhan, Ruth
and Gemma. Well done!</span>
<div class="editorial"></div>
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<h2>Let's Reflect</h2>
<br></br>
You will need a mirror for this activity.<br></br>
<br></br>
Here is a square:
<div align="center">
<p><mdo:image alt="square" height="159" src="square.gif" width="159"></mdo:image></p>
<p align="left">Where can you put the mirror across the square so that you can still 'see' the whole square?<br></br>
How many different positions are possible?<br></br>
How many lines of symmetry does a square have?</p>
<p align="left">Can you reflect part of the square so that you can see a smaller square?<br></br>
A rectangle? A kite? A hexagon? An octagon?<br></br>
What do all the shapes have in common?</p>
<p align="left" class="acknowledgement"><span style="font-style: italic;">This problem is taken from 'Starting from Mirrors' by David Fielker, published by BEAM Education. It can be purchased from the</span> <a href="http://www.beam.co.uk" style="font-style: italic;">BEAM website</a><span style="font-style: italic;">.</span></p>
</div>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1873&part=index">This problem</a> enables children to explore the line symmetry of 2D shapes. The practical nature of the activity means that experimentation is possible and therefore it is readily accessible.</div>
<br></br>
<h3>Possible approach</h3>
<div>Ideally, children should each have a mirror for this activity. Being able to move the mirror yourself and seeing the effect is much more powerful than watching someone else doing it.</div>
<br></br>
<div>You could introduce the first part of the problem orally, giving children a chance to experiment with their mirrors. Printing off <a href="/content/03/07/penta4/Squares.doc">sheets of squares</a> may be useful. Once one way is shared between the whole group, challenge pairs of children to come up with all the other ways of seeing the whole square. Record them on the board as lines on the
square and ask the children what they notice (the lines all correspond with the square's lines of symmetry). The rest of the problem can be tackled by the children at their own pace.</div>
<br></br>
<div>You could have some square templates on the board, one labelled rectangle, one kite etc and learners could come to the front to mark on a line which works. Encourage them to make generalisations about where the mirror must be placed in each case.</div>
<br></br>
<h3>Key questions</h3>
<div>Where are the lines of symmetry on a rectangle/kite/hexagon/octagon?</div>
<div>How does this help you think about where to place the mirror on the square?</div>
<br></br>
<h3>Possible extension</h3>
<div>Learners could investigate which other 2D shapes they can 'see' using their mirror. Which are not possible and why? What would happen if the shape used was different? For example, if it was a triangle rather than a square?</div>
<br></br>
<h3>Possible support</h3>
<div>Having plastic/wooden shapes available to remind children of their properties might be helpful. Other learners might appreciate having some rough paper or a mini-whiteboard to sketch on as they tackle this problem.</div>
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<p>You might find it useful to print off <a href="/content/03/07/penta4/Squares.doc">this sheet of
squares</a>.<br></br>
Try experimenting by placing the mirror across the middle of the
square to start with then move it gradually in one direction to see
what happens.<br></br>
Try a different starting position now - perhaps still across the
middle, but at a different angle.</p>
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Let's Reflect
Where can you put the mirror across the square so that you can
still "see" the whole square? How many different positions are
possible?
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