## Pairs of Numbers

If you have ten counters numbered $1$ to $10$, how many can you put into pairs that add to $10$?

Can you use them all?

Now put the counters into pairs to make $12$.
• Can you use them all?
Now put the counters into pairs to make $13$.
• Can you use them all?

Now put the counters into pairs to make $11$.
• Can you use them all?

### Why do this problem?

This problem 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.

### Possible approach

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.

You could start with the whole group by either giving the question aurally or using the interactivity.  Alternatively, you could put the children in pairs straight away and give them this sheet 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.

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.

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?

### Key questions

What goes with this number to make $10$/$11$ etc?
Which numbers can't you use this time?  Why?
What is different when you are making $11$? Why is it different?
Can you see any difference between using odd and even numbers?

### Possible extension

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?
What can they do if they use the numbers from $1$ - $12$ instead of $1$ - $10$?

### Possible support

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.