7233
/www/nrich/html/content/id/7233/
1
2011-02-01T00:00:01
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<p style="text-align: center;"><mdo:image alt="1-10" height="45" src="7233.png" width="516"></mdo:image></p>
<p>If you have ten counters numbered $1$ to $10$, how many can you put into pairs that add to $10$?<br></br>
<br></br>
Can you use them all?<br></br>
Say how you got your answer.<br></br>
<br></br>
</p>
Now put the counters into pairs to make $12$.
<ul>
<li>Can you use them all?</li>
<li>Say how you got your answer.</li>
</ul>
Now put the counters into pairs to make $13$.
<ul>
<li>Can you use them all?</li>
<li>Say how you got your answer.</li>
</ul>
<br></br>
Now put the counters into pairs to make $11$.
<ul>
<li>Can you use them all?</li>
<li>Say how you got your answer.</li>
</ul></mdoxml>
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<p class="editorial">James from Evenlode sent a very well-explained
solution this problem. He said:</p>
To make pairs to $10$ you need to leave out $5$ and $10$, because
you would need a double for the $5$ and a $0$ for the $10$.<br></br>
To make pairs to $12$ you need to leave out $1$ and $6$, because
you need $11$ for the $1$ and double for the $6$.<br></br>
To make pairs to $13$ you need to leave out $1$ and $2$, because
you need numbers higher than $10$ ($11$ and $12$) .<br></br>
To make pairs to $11$ ... you don't need to leave ANYTHING out,
because $11$ is an odd number so you don't need any doubles AND you
don't need partners bigger than $10$. <br></br>
<br></br>
<p class="editorial">Thank you also to Johannes, Fred and Lok from
St Barnabas, Henry from Moulsford, children from Jebel Ali
Primary School, Alex from Templars Primary, Alex from Maidstone and
Beth from Torpoint who all sent clear solutions. </p>
<br></br></mdoxml>
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<mdoxml version="1.0"><div class="embed">
<h2>Pairs of Numbers</h2>
<br></br>
<p style="text-align: center;"><mdo:image alt="1-10" height="45" src="7233.png" width="516"></mdo:image></p>
<p>If you have ten counters numbered $1$ to $10$, how many can you put into pairs that add to $10$?<br></br>
<br></br>
Can you use them all?<br></br>
Say how you got your answer.<br></br>
<br></br>
</p>
Now put the counters into pairs to make $12$.
<ul>
<li>Can you use them all?</li>
<li>Say how you got your answer.</li>
</ul>
Now put the counters into pairs to make $13$.
<ul>
<li>Can you use them all?</li>
<li>Say how you got your answer.</li>
</ul>
<br></br>
Now put the counters into pairs to make $11$.
<ul>
<li>Can you use them all?</li>
<li>Say how you got your answer.</li>
</ul>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/7233&part=">This problem</a> looks simple to start with, but it has a certain complexity. It is a great opportunity to encourage children to justify their thinking, which they may find quite difficult at first. <br></br>
<h3>Possible approach</h3>
<div>All children will need access to ten counters or number cards numbered from $1$ - $10$. Having counters to move around will help free up their thinking and means they can try out lots of ways without the fear of having something committed to paper which might be wrong. Some children may also need some unnumbered counters or Multilink cubes to help them with the
calculations.</div>
<div> </div>
<div>You could start with the whole group by either giving the question aurally or using the <a href="/content/id/7233/PairsNos.swf">interactivity</a>. Alternatively, you could put the children in pairs straight away and give them <a href="/content/id/7233/Sheet7233A.pdf">this sheet</a> to work on with numbered counters. If you print it onto thin card and laminate it, it can be
re-used many times. You may wish to encourage pairs to record their solutions, perhaps on mini-whiteboards or paper or even in the form of photograps. </div>
<div> </div>
<div>After a suitable length of time, you could bring everyone together to use the interactivity to share their solutions. At this point, having recordings might be very helpful so that each pair can compare their own solution with that on the board. You could use this opportunity to ask whether everyone has the same answer each time and if so, why.</div>
<div> </div>
<div>Allow time for children to explain why specific counters are left over each time for totals $10$, $12$ and $13$, but not for $11$. Could they suggest other totals which would leave some counters left over? Are there any other totals which would use up all the counters? </div>
<br></br>
<h3>Key questions</h3>
<div>What goes with this number to make $10$/$11$ etc?</div>
<div>Which numbers can't you use this time? Why?</div>
<div>What is different when you are making $11$? Why is it different?</div>
<div>Can you see any difference between using odd and even numbers?</div>
<br></br>
<h3>Possible extension</h3>
Children could try to find other numbers of which can be made from pairs of the numbers $1$ - $10$. Are there any number which can't be used?<br></br>
What can they do if they use the numbers from $1$ - $12$ instead of $1$ - $10$?<br></br>
<h3>Possible support</h3>
Some learners may need support with the calculations, so having number lines, blank counters or other equipment available will be useful. This task offers children the chance to practice adding numbers in a meaningful context.<br></br>
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It might help to have counters numbered from $1$ - $10$ to do this
problem.<br></br></mdoxml>
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Pairs of Numbers
If you have ten counters numbered 1 to 10, how many can you put
into pairs that add to 10? Which ones do you have to leave out?
Why?
Calculations and Numerical Methods
Addition & subtraction
Numbers and the Number System
Properties of numbers
Numbers and the Number System
Counting
Numbers and the Number System
Odd and even numbers
Using, Applying and Reasoning about Mathematics
Practical Activity
Admin
Lower primary mapping document