1151
/www/nrich/html/content/03/01/penta3/
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2011-02-01T00:00:01
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<p>In the circle of numbers below each adjoining pair adds to make a square number:</p>
<br></br>
<div style="text-align: center;"><mdo:image alt="circle of numbers from top going clockwise: 14, 2, 7, 9, 16, 20, 29, 35." height="123" src="CycSq1.gif" width="125"></mdo:image></div>
<p>For example,</p>
<p>$14 + 2 = 16, 2 + 7 = 9, 7 + 9 = 16$</p>
<p>and so on.</p>
<p>Can you make a similar - but larger - cycle of pairs that each add to make a square number, using all the numbers in the box below, once and once only?</p>
<div style="text-align: center;"><mdo:image alt="Numbers given: 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 7, 19, 21, 28, 30, 34" height="74" src="CycSq2.gif" width="262"></mdo:image></div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">You might like to use this interactivity.</div>
<br></br>
<div style="text-align: left;"> </div>
<div style="text-align: left;"> </div>
<div style="text-align: left;"><a href="/content/03/01/penta3/CycSq.swf">Full screen version</a></div>
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<param name="movie" value="/content/03/01/penta3/CycSq.swf"></param>
<param name="flashplayerversion" value="8"></param>
</mdo:flash>
<div style="text-align: left;"><br></br>
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<p class="editorial">William from Tattingstone School said that he
tried four times before he came up with a solution to Cycling
Squares:</p>
<mdo:image height="344" width="209" src="willtatt.gif" alt="15-10-6-313-12-4-5-11-14-2-34-30-19-17-8-28-21"></mdo:image>
<p><span class="editorial">We're glad you didn't give up,
William. Beth and Pheobe from Exminster CP School and
Skye and Molly from Breckland Middle School also used trial and
error (or trial and improvement as I like to call it!).
Beth and Phoebe said it tested their skills of
perseverance.</span></p>
<p><span class="editorial">Matha, also from Tattingstone, said that
she started with the number $2$ and then added $14$ because it was
the only number she could have added to make a square number. She
goes on to say:</span></p>
<blockquote>I then added $11$ to $14$ to get $25$ which is another
square number and carried on like this.</blockquote>
<p>Martha sent in a drawing of her circle which is the same as
William's answer, just written in a different way:</p>
<mdo:image height="189" width="200" src="marthaTatt.gif" alt="circle of numbers going clockwise: 14, 11, 5, 4, 17, 13, 3, 6, 10, 15, 21, 28, 8, 17, 19, 30, 34, 2"></mdo:image>
<p>.</p>
<p class="editorial">Dominic from Stonehill took a
logical approach:</p>
<br></br>
<div>I listed the numbers and wrote down their pairs to make square
numbers.</div>
<div>Some of the numbers only had two possible combinations to make
square numbers, so I started with one of these. I put in $2$ first
and put $14$ and $34$ on either side of it.</div>
<div>From there, I put $11$ on the other side of the $14$ as these
were the only combinations, and so on.</div>
<div>Some numbers had four or even
five possible combinations but by trial and error I
moved these around until all the combinations made a square number.
</div>
<br></br>
<p class="editorial">Brandon, Antonia and Oliver from Mayhill
Junior used a similar method to Dominic. Emilie and
Bethany from Alverstoke Junior School said:</p>
<div>To solve the problem we first worked out the highest square
number was $64$.</div>
<div>We then worked out all the square numbers from $4$ to
$64$.</div>
<div>We then worked out which pairs of numbers made square numbers
and we used trial and error to solve the problem.
</div>
<br></br>
<p class="editorial">Very well done to you all.</p>
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<h2>Cycling Squares</h2>
<br></br>
<p>In the circle of numbers below each adjoining pair adds to make a square number:</p>
<br></br>
<div style="text-align: center;"><mdo:image alt="circle of numbers from top going clockwise: 14, 2, 7, 9, 16, 20, 29, 35." height="123" src="CycSq1.gif" width="125"></mdo:image></div>
<p>For example,</p>
<p>$14 + 2 = 16, 2 + 7 = 9, 7 + 9 = 16$</p>
<p>and so on.</p>
<p>Can you make a similar - but larger - cycle of pairs that each add to make a square number, using all the numbers in the box below, once and once only?</p>
<div style="text-align: center;"><mdo:image alt="Numbers given: 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 7, 19, 21, 28, 30, 34" height="74" src="CycSq2.gif" width="262"></mdo:image></div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">You might like to use this interactivity.</div>
<br></br>
<div style="text-align: left;"> </div>
<div style="text-align: left;"> </div>
<div style="text-align: left;"><a href="/content/03/01/penta3/CycSq.swf">Full screen version</a></div>
<mdo:flash height="400" id="/content/03/01/penta3/CycSq.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="/content/03/01/penta3/CycSq.swf"></param>
<param name="flashplayerversion" value="8"></param>
</mdo:flash>
<div style="text-align: left;"><br></br>
</div>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=1151&part=index">This problem</a> is a challenging one using the learner's knowledge of square numbers and giving practice in addition in an unusual context. It is a useful problem through which to discuss different methods of approach.<br></br>
<h3>Possible approach</h3>
<div>You could start by reminding the group about square numbers, perhaps by playing a game of 'I like ...' using <a href="/content/03/01/penta3/like100.swf">this interactivity</a> where you drag numbers you 'like' (i.e. are part of a set) to one side and numbers you 'don't like' (i.e. are not in your set) to the other. The children then have to ask questions with yes/no answers to determine the
name of your set.</div>
<br></br>
<div>You could then use the small cycle given at the beginning of the problem to introduce the idea of a cycle of squares, encouraging learners to check that each pair of numbers does indeed add to a square number.</div>
<div>(The cycle goes like this: $14 - 2 - 7 - 9 - 16 - 20 - 29 - 35 -$.)</div>
<br></br>
<div>Then you could go on to the actual problem, possibly using the <a href="/content/03/01/penta3/CycSq.swf">interactivity</a>. Learners could work on this in pairs either from a printed sheet or on a computer using the interactivity. Learners who do not have access to the interactivity might find it useful to have numbered counters that can be moved round experimentally.</div>
<br></br>
<div>At the end of the lesson, all learners could be involved in a discussion of how they tackled the problem, building up the cycle on the board or interactive whiteboard, number by number. You could invite them to comment about the advantages/disadvantages of the different methods used and they may decide that one method is particularly efficient.</div>
<br></br>
<h3>Key questions</h3>
<div>Can you think of a good way to start?</div>
<div>Have you made a list of square numbers up to $100$?</div>
<div>Which of the numbers given can be added to that one to make a square number?</div>
<div>If you cannot go on from there, what could you try instead?</div>
<br></br>
<h3>Possible extension</h3>
Learners could make their own cycle of squares using different numbers, or invent cycles using a different number property.<br></br>
<h3>Possible support</h3>
Making a list of the square numbers up to $100$ will help some children get started on this problem. Encourage the learner to choose one number and see which of the numbers given can be added to it to make a square number. Numbered counters to work with could be very helpful and a calculator might free-up some children who find calculation difficult.<br></br>
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<p>Make a list of the square numbers up to $8\times 8$.</p>
<p>Using the interactivity might help you get started.
Don't forget you can always change your mind and alter your
solution as you go along!</p>
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Cycling Squares
Can you make a cycle of pairs that add to make a square number
using all the numbers in the box below, once and once only?
Using, Applying and Reasoning about Mathematics
Trial and improvement
Calculations and Numerical Methods
Addition & subtraction
Numbers and the Number System
Square numbers
Information and Communications Technology
Interactivities
Admin
Upper primary mapping document