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/www/nrich/html/content/98/05/bbprob2/
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2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>This challenge is a bit different to my usual ones. I used it
many years ago when I wanted some quite young children to do lots
of adding in a more fun way.</p>
<table cellspacing="10">
<tbody>
<tr>
<td><mdo:image alt="" src="grid.gif"></mdo:image></td>
<td>You start with four numbers at the corners of a square. Then
you add up the numbers at the two ends of each side and put the
total in the middle of that side. So in my example 3 + 5 = 8, 5 + 4
= 9, 3 + 1 = 4, 1 + 4 = 5.</td>
</tr>
</tbody>
</table>
<table cellspacing="10">
<tbody>
<tr>
<td><mdo:image alt="" src="gridv.gif"></mdo:image></td>
<td>These four new answers give us the corners of a new square. The
corners are 8, 9, 5 and 4.</td>
</tr>
</tbody>
</table>
<table cellspacing="10">
<tbody>
<tr>
<td><mdo:image alt="" src="gridw.gif"></mdo:image></td>
<td>These four new numbers are added up and the answers put in the
centre of the edges of this new square. And so on and so on.</td>
</tr>
</tbody>
</table>
<p>The diagram gets more and more complicated, growing as shown
below:-</p>
<p style="text-align: center;"><mdo:image width="267" height="267" src="Picture%201.jpg" alt="pic1"></mdo:image></p>
<p style="text-align: center;"><mdo:image width="268" height="269" src="Picture%202.jpg" alt="pic2"></mdo:image></p>
<p style="text-align: center;"><mdo:image width="268" height="270" src="Picture%203.jpg" alt="pic3"></mdo:image></p>
<p style="text-align: center;"><mdo:image width="268" height="270" src="Picture%204.jpg" alt="pic4"></mdo:image></p>
<p style="text-align: center;"><mdo:image width="269" height="269" src="Picture%205.jpg" alt="pic5"></mdo:image></p>
<p style="text-align: center;"><mdo:image width="269" height="272" src="Picture%206.jpg" alt="pic6"></mdo:image></p>
<p></p>
<p></p>
<p>There is not much more to say, apart from have a go yourself.
Use any starting numbers at the corners. Can you estimate what the
size of the last four numbers will be?</p>
<p>What would happen if you used different shapes, for example
pentagons or hexagons?</p>
<p>What would happen if you used subtraction, always taking the
smaller from the bigger?</p>
<p>What would happen if you multiplied? Divided? What ...??</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">These two solutions came from West Flegg
Middle School. The first square uses addition and the second uses
subtraction. Do you see the patterns in the numbers? Did anyone
else end up with all the same numbers in the centre?</p>
<mdo:image alt="A square using addition" src="numsquare1.gif"></mdo:image>
<p><mdo:image alt="A square using subtraction" src="numsquare2.gif"></mdo:image></p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Number Squares</h2>
<br></br>
<p>This challenge is a bit different to my usual ones. I used it many years ago when I wanted some quite young children to do lots of adding in a more fun way.</p>
<table style="border-spacing: 10px;" border="1">
<tbody>
<tr>
<td><mdo:image alt="" src="grid.gif"></mdo:image></td>
<td>You start with four numbers at the corners of a square. Then you add up the numbers at the two ends of each side and put the total in the middle of that side. So in my example 3 + 5 = 8, 5 + 4 = 9, 3 + 1 = 4, 1 + 4 = 5.</td>
</tr>
</tbody>
</table>
<table style="border-spacing: 10px;" border="1">
<tbody>
<tr>
<td><mdo:image alt="" src="gridv.gif"></mdo:image></td>
<td>These four new answers give us the corners of a new square. The corners are 8, 9, 5 and 4.</td>
</tr>
</tbody>
</table>
<table style="border-spacing: 10px;" border="1">
<tbody>
<tr>
<td><mdo:image alt="" src="gridw.gif"></mdo:image></td>
<td>These four new numbers are added up and the answers put in the centre of the edges of this new square. And so on and so on.</td>
</tr>
</tbody>
</table>
<p>The diagram gets more and more complicated, growing as shown below:-</p>
<p style="text-align: center;"><mdo:image alt="pic1" height="267" src="Picture%201.jpg" width="267"></mdo:image></p>
<p style="text-align: center;"><mdo:image alt="pic2" height="269" src="Picture%202.jpg" width="268"></mdo:image></p>
<p style="text-align: center;"><mdo:image alt="pic3" height="270" src="Picture%203.jpg" width="268"></mdo:image></p>
<p style="text-align: center;"><mdo:image alt="pic4" height="270" src="Picture%204.jpg" width="268"></mdo:image></p>
<p style="text-align: center;"><mdo:image alt="pic5" height="269" src="Picture%205.jpg" width="269"></mdo:image></p>
<p style="text-align: center;"><mdo:image alt="pic6" height="272" src="Picture%206.jpg" width="269"></mdo:image></p>
<p> </p>
<p> </p>
<p>There is not much more to say, apart from have a go yourself. Use any starting numbers at the corners. Can you estimate what the size of the last four numbers will be?</p>
<p>What would happen if you used different shapes, for example pentagons or hexagons?</p>
<p>What would happen if you used subtraction, always taking the smaller from the bigger?</p>
<p>What would happen if you multiplied? Divided? What ...??</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This is an engaging <a href="http://nrich.maths.org/public/viewer.php?obj_id=43&part=">activity</a> that involves both computational skills and organisational skills. I have found that quite young children (6-8 year olds) really enjoy it and do a great deal of work in performing many calculations without really realising it. Quite a few of the children use pentagons and hexagons to vary
it a bit.</div>
<br></br>
<h3>Possible approach</h3>
<div>Doing a "demonstration" square with the class, making suggestions along the way, works well.</div>
<br></br>
<div>Having some pre-printed sheets available is helpful for some pupils, although just seeing the final shape on inner squares can be very confusing for some.</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about anything you have noticed.</div>
<div>What numbers did you start with?</div>
<div>Do you have any ideas about the number you might end up with in the middle?</div>
<br></br>
<h3>Possible extension</h3>
<div>You can reverse the process and see if children can come up with what the starting numbers could be to produce a certain specified result in the middle. Asking them how they would work this out leads to some interesting insights into children's methods of thinking through addition and subtraction. Older children (10-11 year olds) could try the reverse when it is addition taking place. The
other thing is to try to predict the result in the middle when you have certain prescribed starting numbers.</div>
<br></br>
<h3>Possible support</h3>
<div>With some pupils it is useful to have an adult (more mature thinker) who can keep two fingers at the two places holding the numbers that the pupil is considering at that time.</div>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
You could try altering just one of your starting numbers to see
what effect this has. Can you predict what will happen before you
try it?<br></br>
You could set yourself some targets for the middle numbers and see
how close you can get.<br></br>
What are the similarities and differences betweeen the different
squares you produce?<br></br>
Can you explain these similarities and differences?<br></br>
You might find a calculator handy!<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">This activity has been seen around for some time in different
forms, I really do not know if I was the first to use it, I guess
not, but I started using it when I was involved in the C.A.N.
[Calculator Aware Number Curriculum] project. I found that year 2
children really enjoyed it and did a great deal of work in doing
many calculations without really realising it. Quite a few of the
children used pentagons and hexagons to vary it a bit. With years 3
and 4 I find that using ``differences'' instead of adding leads to
quite a lot of interest when it comes to predicting what the final
result will be. With the brighter children of this age you can
reverse it and see if they can come up with what the starting
numbers could be to produce a certain specified result in the
middle. Asking them how they would work this out leads to some
interesting insights into children's methods of thinking through
addition and subtraction. At the top end of Key Stage 2 the
children could try the reverse when it is addition taking place.
The other thing is to try to predict the result in the middle when
you have certain prescribed starting numbers.<br></br></mdoxml>
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Number Squares
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
Calculations and Numerical Methods
Addition & subtraction
Using, Applying and Reasoning about Mathematics
Investigations