Pairs of Numbers
Pairs of Numbers printable sheet
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?
Say how you got your answer.
Now put the counters into pairs to make $12$.
- Can you use them all?
- Say how you got your answer.
Now put the counters into pairs to make $13$.
- Can you use them all?
- Say how you got your answer.
Now put the counters into pairs to make $11$.
- Can you use them all?
- Say how you got your answer.
It might help to have counters numbered from $1$ - $10$ to do this problem.
Thank you to everybody who sent in their solutions to this challenge. We were sent three good solutions that explained the answers very clearly. Well done to all of you!
Jacob and Michael from Cloverdale Catholic School in Canada wrote:
Because there are ten counters there are only four matches. (1+9, 6+4, 3+7, 2+8).
Apparently, there is only one five so 5+5 is not possible.
There is no zero so 0+10 is not possible.
If there are eleven counters, there are five possible matches. (1+10, 2+9, 3+8, 4+7, 5+6).
Benjamin from West Oxford Primary School wrote:
Pairs that make 10 are 1+9, 2+8, 3+7, 4+6
Eight counters were used. Counters 10 and 5 were not used.
Pairs that make 12 are 2+10, 3+9, 4+8, 5+7
Eight counters were used. Counters 1 and 6 were not used.
Pairs that make 13 are 3+10, 4+ 9, 5+8, 6+7
Eight counters were used. Counters 1 and 2 were not used.
Pairs that make 11 are 1+10, 2+9, 3+8, 4+7, 5+6
Ten counters were used. All counters were used.
Casper from Torbay Primary School in New Zealand sent in the following:
1. The pairs 9-1, 8-2, 7-3, 6-4 work
That leaves 10 and 5, 10 because 10+0 is ten but there is no zero and 5 because double 5 is ten but there is only one 5.
2. The pairs 10-2, 9-3, 8-4, 7-5 work
That leaves 1 and 6, 6 because double six is 12 but there is only one six and 1 because 1+11 is twelve but there is no eleven.
3. The pairs 10-3, 9-4, 8-5, 7-6 work
That leaves 1 and 2, 1 because 1+12 is thirteen but there is no twelve and 2 because 2+11 is thirteen but there is no eleven.
4. The pairs 10-1, 9-2, 8-3, 7-4, 6-5 work
That leaves none so they all can be paired.
Pairs of Numbers
Pairs of Numbers printable sheet
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?
Say how you got your answer.
Now put the counters into pairs to make $12$.
- Can you use them all?
- Say how you got your answer.
Now put the counters into pairs to make $13$.
- Can you use them all?
- Say how you got your answer.
Now put the counters into pairs to make $11$.
- Can you use them all?
- Say how you got your answer.
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.
Key questions
What goes with this number to make $10$/$11$ etc?
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.