Problem Teachers' Notes Hint Solution

This problem is a good example of a challenge which does not require high-level mathematics, but does need a systematic approach.

In order that children understand the requirements of this problem, it would be a good idea to begin by reversing the situation. Draw two bags on the board and label them $1$p and $3$p. Invite children to suggest the amounts you could pay using these bags if you are not allowed to open either of them [$1$p, $3$p and $4$p]. Introduce another bag to go with the first two which contains $5$p. This time, ask the children which amounts of money they wouldn't be able to make using these bags [$2$p, $7$p, anything of $10$p and above]. You could give them time in pairs to work on this, perhaps using a mini-whiteboard to make jottings.

Invite pairs to share their solutions and highlight those that have used a systematic approach, for example by starting with the smallest amount of money and working up in steps of a penny at a time. You can then introduce the problem as it stands. Give pairs chance to work together before once again sharing their ideas. Before they give the answer, ask children to explain how they went about finding the solution.

The whole class could then check the solution offered by trying to make all the different amounts from $1$p to $15$p.

How will you make $1$p?

How will you make $2$p? ...

How will you make sure you can pay for all the amounts from $1$p to $15$p?

You could challenge children to extend this problem by asking what amount they would need in a fifth bag to be able to make as many amounts as possible over $15$p.

Having real or fake coins available, along with some small bags, might help some children grasp this problem. Encouraging them to try ideas out is vital.