1788
/www/nrich/html/content/03/07/cupboardlove3/
1
2011-02-01T00:00:01
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These images show squares split in half:<br></br>
<div align="center">
<p><mdo:image width="400" height="139" src="halving1.gif" alt="squares shaded in half in different ways"></mdo:image></p>
<p><mdo:image width="397" height="218" src="halving2.gif" alt="squares with geometric pattern with them shaded in half in different ways"></mdo:image>
</p>
</div>
<div>How might you check that each was correct?</div>
<div>Can you think of more ways to split a square into two
halves?</div>
<div>You might like to use squared paper to draw your ideas on or
to print out the uncoloured squares above.</div>
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<span class="editorial">These ideas were sent in by Donna of Zion
Heights. Can you think of any more?</span><br></br>
<br></br>
To check that each way is correct, you can measure the area of each
red/white piece and see if the combined total area of the white
pieces is equal to the total area of the red pieces. You can also
cut out the white and red pieces and see if you can rearrange them
so that every white piece has a matching, congruent red
piece.<br></br>
<br></br>
Here are two more possible ways to cut a square in half:<br></br>
<mdo:image height="151" width="273" alt="Halved squares" src="szZT4y-squares.GIF"></mdo:image><br></br>
Krystof also sent in another way to cut a square in half:<br></br>
<br></br>
<mdo:image height="208" width="200" src="sol2.gif" alt=""></mdo:image> <br></br>
<p><span class="editorial">Thank you to both of you.</span></p>
<p class="editorial">Please don't worry that your solution is not
"complete" - we'd like to hear about anything you have tried.
Teachers - you might like to send in a summary of your children's
work.</p>
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<h2>Halving</h2>
<br></br>
These images show squares split in half:<br></br>
<div align="center">
<p><mdo:image alt="squares shaded in half in different ways" height="139" src="halving1.gif" width="400"></mdo:image></p>
<p><mdo:image alt="squares with geometric pattern with them shaded in half in different ways" height="218" src="halving2.gif" width="397"></mdo:image></p>
</div>
<div>How might you check that each was correct?</div>
<div>Can you think of more ways to split a square into two halves?</div>
<div>You might like to use squared paper to draw your ideas on or to print out the uncoloured squares above.</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=1788&part=index">This exploratory problem</a> is a fantastic way to consolidate children's understanding of halving and halves. It also gives learners experience of mathematical proof.</div>
<br></br>
<h3>Possible approach</h3>
<div>To introduce the problem, you could use <a href="/content/03/07/cupboardlove3/FractionNew.ppt">this PowerPoint</a> presentation. It shows squares being divided in different ways and would provoke interesting discussion. (It loops so starts again when it reaches the end.) You may want to show the images in turn and after allowing some time for them to talk in pairs, ask children to
raise their hands if they think the squares are divided in half. This allows children to clarify their understanding of halving before going on to the main task.</div>
<br></br>
<div>It would be a good idea to give pairs of children a print-out of <a href="/content/03/07/cupboardlove3/Squares.doc">these sheets</a> , the first of which contains the smaller squares, and the second the larger squares in the problem. Pupils will have many ideas of their own when it comes to checking or proving that the squares in the problem really are split in half. Some will be able to
explain orally, in terms of looking for red and white parts that are exactly the same shape and size. Others will prefer to draw them on squared paper and count squares. Some may want to cut out the squares and physically lay the halves on top of each other. Alternatively, it might be possible in some instances to cut out the square and fold it to show that it has been split in half. Learners can
use the uncoloured squares on the sheet to investigate different ways of shading halves.</div>
<br></br>
<div>In the plenary, as well as giving the class a chance to share their solutions and ways of working, you might like to distinguish between the squares that have been divided into two pieces (one red, one white) from those where there are more than two pieces. In all cases, we are comparing the total amount of red with the total amount of white.</div>
<br></br>
<h3>Key questions</h3>
<div>Have you thought about drawing the squares on squared paper and then using the small squares to help you check that they are split in half?</div>
<div>How about cutting out your square? How would this help to check that you had divided it in half?</div>
<br></br>
<h3>Possible extension</h3>
Some children might enjoy investigating how to split other shapes in half.<br></br>
<br></br>
<h3>Possible support</h3>
<div><a href="/content/03/07/cupboardlove3/SquaresonGrid.doc">This sheet</a> has the larger squares drawn on a grid of squares which may be useful for many children.</div>
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<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">Have you thought about using the squares on your squared paper to
help you check your drawings?<br></br>
How about cutting out your square? How
would this help to check that you had divided it in half?<br></br></mdoxml>
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Halving
These pictures show squares split into halves. Can you find other
ways?
Using, Applying and Reasoning about Mathematics
Investigations
2D Geometry, Shape and Space
Squares
Using, Applying and Reasoning about Mathematics
Practical Activity
Fractions, Decimals, Percentages, Ratio and Proportion
Fractions
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