Fair Exchange
In your bank, you have three types of coins. The number of spots shows how much they are worth.
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1  2  5 
Can you choose coins to exchange with the groups below to make the same total?
Can you find another way to do each one?
What is the total you've got to make?
How many more do you need?
Can you do it in a different way?
What is the largest coin you could use?
Could you make that amount with just twos?
Jamie from Waddington Redwood Primary told us:
I used a 1p and two 2p coins for 5p.
I used four 2p coins for 8p.
I used five 2p coins and a 1p for 11p.
Pranav from Vardhana School looked at how many different ways you can make each total:
First, we must find the individual values of each number:The 5 coin can be made with:
This is really helpful, Pranav. Pranav then went on to make a table of the number of ways the values from 1 up to 14 could be made with the coins. However, it's very difficult to make sure that we don't count some ways which are the same twice.
Meg used Pranav's method to find all the ways of making 8 and 11 with these coins, and wrote a list.
To make sure I count all the ways to make 8 and don't count any twice, I'll first list all the ways which use 5p, then list the ways that don't use 5p. I found that there are seven possible ways to do this. Here is the list:5 + 2 + 1
5 + 1 + 1 + 1
2 + 2 + 2 + 2
2 + 2 + 2 + 1 + 1
2 + 2 + 1 + 1 + 1 + 1
2 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
Then I did a similar thing for 11p. First I listed the ways using two 5ps (one way only), then the ways which use one 5p, then the ways that don't use any 5ps at all. I found that there are eleven possible ways to do this in total. Here is the list:
5 + 5 + 1
5 + 2 + 2 + 2
5 + 2 + 2 + 1 + 1
5 + 2 + 1 + 1 + 1 + 1
5 + 1 + 1 + 1 + 1 + 1 + 1
2 + 2 + 2 + 2 + 2 + 1
2 + 2 + 2 + 2 + 1 + 1 + 1
2 + 2 + 2 + 1 + 1 + 1 + 1 + 1
2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1
2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
Well done everyone for these solutions! If you have had a go at this puzzle, why not challenge yourself with some different totals, or even some different coins?
Why do this problem?
This problem gives opportunities for children to practise numbers bonds in the context of a game. Children can try out different options to find sets with equal numbers of spots in them.
You could focus on encouraging learners to work systematically to find all possibilities.
Possible Approach
Introduce the activity to the class on an interactive whiteboard and ask them to choose a number to match from the three choices 5, 8 and 11. The 'coins' in the game have the same number of spots as the number they represent which makes them easier for children to work with than either toy coins or real money. It would be possible to use real coins or toy money as an alternative.
Ask the children what each of the target numbers is in turn: 5, 8 and 11. Then see if they can suggest different sets of coins that have the same value and try them out using the interactivity.
The children could then go on to creating their own equivalent sets of coins either using coins cut out from this Word document and pdf card showing coins worth 1, 2 and 5.
Key questions
Possible extension
Children can choose their own target numbers and see how many different equivalent sets they can make using coins worth 1, 2 and 5. They could use real coins instead of the printed version and even move on to higher deniminations such as 10p or 20p.
Possible support
Plenty of practice with exchanging small collections of coins may be needed by some children. Understanding that five penny pieces are worth the same as one 5p piece is tricky and may take time to establish with young children.