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<p>On each of these grids the counters lie at the four corners of a
square:</p>
<p><mdo:image width="504" height="144" alt="Three 4 by 4 grids. Each has 4 counters positioned at the corners of a square" src="demosquares.gif"></mdo:image></p>
<p>What is the greatest number of counters you can place on the
grid below without four of them lying at the corners of a
square?</p>
<p><mdo:image width="288" height="288" alt="Empty 4 by 4 grid" src="largesquare.gif"></mdo:image></p>
<p style="font-style: italic;">This problem is taken from
'Mathematical Challenges for Able Pupils Key Stages 1 and 2',
published by DfES. You can find out more about this book, including
how to order it, on the Standards website <a href="http://www.standards.dfes.gov.uk/primary/publications/mathematics/able_pupils_challenges/">
here</a>.</p>
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<table width="100%" cellpadding="5" border="0">
<tbody>
<tr>
<td colspan="2" class="editorial">Thank you to all those who
answered this problem, which is not as straightforward as it may
seem.</td>
</tr>
<tr>
<td colspan="2" class="editorial">John and Chadwick at Coleridge
Community College sent in these solutions:</td>
</tr>
<tr>
<td valign="middle" align="center">
<p style="text-align: left;"><mdo:image width="144" height="144" alt="" src="John.gif"></mdo:image></p>
<p style="text-align: left;">John</p>
</td>
<td>
<div class="c1">
<p><mdo:image width="144" height="144" alt="" src="Chadwick.gif"></mdo:image></p>
<p>Chadwick</p>
</div>
</td>
</tr>
<tr>
<td colspan="2" class="editorial">Kenny and Bon from Our Lady's
School in Welwyn Garden City found these arrangements:</td>
</tr>
<tr>
<td>
<div class="c1">
<p><mdo:image width="144" height="144" alt="" src="Kenny.gif"></mdo:image></p>
<p>Kenny</p>
</div>
</td>
<td>
<div class="c1">
<p><mdo:image width="144" height="144" alt="" src="Bon.gif"></mdo:image></p>
<p>Bon</p>
</div>
</td>
</tr>
<tr>
<td colspan="2" class="editorial">Children from the Challenge Club
at Lodgefields Primary School, Crewe have found more
arrangements:</td>
</tr>
<tr>
<td>
<div class="c1">
<p><mdo:image width="144" height="144" alt="" src="Emily.gif"></mdo:image></p>
<p>Emily</p>
</div>
</td>
<td>
<div class="c1">
<p><mdo:image width="144" height="144" alt="" src="Ian.gif"></mdo:image></p>
<p>Ian</p>
</div>
</td>
</tr>
<tr>
<td>
<div class="c1">
<p><mdo:image width="144" height="144" alt="" src="Christopher.gif"></mdo:image></p>
<p>Christopher</p>
</div>
</td>
<td>
<div class="c1">
<p><mdo:image width="144" height="144" alt="" src="David_Daniel.gif"></mdo:image></p>
<p>David and Daniel</p>
</div>
</td>
</tr>
<tr>
<td colspan="2" class="editorial">Maybe there are other
possibilities?</td>
</tr>
</tbody>
</table>
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<h2>Square Corners</h2>
<br></br>
<p>On each of these grids the counters lie at the four corners of a square:</p>
<p><mdo:image alt="Three 4 by 4 grids. Each has 4 counters positioned at the corners of a square" height="144" src="demosquares.gif" width="504"></mdo:image></p>
<p>What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?</p>
<p><mdo:image alt="Empty 4 by 4 grid" height="288" src="largesquare.gif" width="288"></mdo:image></p>
<p style="font-style: italic;">This problem is taken from 'Mathematical Challenges for Able Pupils Key Stages 1 and 2', published by DfES. You can find out more about this book, including how to order it, on the Standards website <a href="http://www.standards.dfes.gov.uk/primary/publications/mathematics/able_pupils_challenges/">here</a>.</p>
</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=1142&part=index">This problem</a> challenges pupils' understanding of the properties of squares in the sense that squares do not necessarily have to be orientated so that their sides are horizontal and vertical. It is a good context in which to encourage children to find a systematic approach.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could begin by playing the game <a href="http://nrich.maths.org/public/viewer.php?obj_id=2526&part=index">Square It</a> a few times as a class. This will provoke discussion amongst pupils about what makes a square a square, and you may wish to address the misconception that tilted squares are called 'diamonds'.</div>
<br></br>
<div>Introduce the problem, either using <a href="/content/02/11/penta4/SquareCornersLargeGrid.doc">the grid</a> on an OHP with counters or on the interactive whiteboard and ask children to begin to work on it in pairs. They could use the <a href="/content/02/11/penta4/SquareCornersLargeGrid.doc">large grid</a> with counters and/or <a href="/content/02/11/penta4/SquareCornersGrids.doc">this
sheet</a> of smaller grids. Remind them to check for squares! After a suitable length of time, share results so far amongst the whole group. What is the largest number of counters so far? Ask a pair or pairs of learners to come up and recreate their arrangement of counters on the screen so that everyone can check there aren't any squares.</div>
<br></br>
<div>At this point, challenge the class to come up with a way or working that will ensure the largest number of counters is definitely found. How will they know that all arrangements have been tested? Some children may suggest a system based on, for example, placing counters on the grid from top left to bottom right and each time checking that a square has not been made. You could model the
beginnings of a strategy and then give the class more time to investigate the problem. It may be that you split the group up to investigate different 'families' of arrangements.</div>
<br></br>
<div>In the plenary, you could discuss the solutions and what makes one arrangement of counters different to another. Will rotations and reflections be considered different or the same?</div>
<br></br>
<h3>Key questions</h3>
<div>How will you know that you have definitely found the largest number of counters?</div>
<div>Are you sure there aren't any squares on your grid?</div>
<br></br>
<h3>Possible extension</h3>
<div>Children could investigate larger grids and see whether there is a pattern to the number of counters by looking at smaller grids too. Is it possible to predict the largest number of counters in any size grid?</div>
<br></br>
<h3>Possible support</h3>
<div>Some pupils could start with a three by three grid.</div>
<br></br>
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<p>One way to start would be to find all the possible places for
the four counters to go that would actually make a square.</p>
<p>Or you could fill in the grid with counters one by one, checking
to make sure you haven't made a square.</p>
<p>Being systematic (i.e.doing things in a logical order) is the
key!</p>
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Square Corners
What is the greatest number of counters you can place on the grid
below without four of them lying at the corners of a square?
Using, Applying and Reasoning about Mathematics
Visualising
Using, Applying and Reasoning about Mathematics
Practical Activity
Using, Applying and Reasoning about Mathematics
Working systematically
2D Geometry, Shape and Space
Squares
Mathematics Tools
Counters
Admin
Upper primary mapping document