# Buying a Balloon

## Problem

*Buying a Balloon printable sheet*

Lola bought a balloon at the circus. She paid for it using six coins.

How much might the balloon have cost?

What is the largest amount Lola could have paid?

What is the smallest amount Lola could have paid?

Imagine that Lola has two different types of coin.

How much might the balloon cost now?

Can you find all the possible prices? How do you know you have found them all?

Which of your answers seems a reasonable amount to pay for a balloon?

## Getting Started

What is the largest amount of money we could make?

What is the smallest amount we could make?

How will we know when we have all the possibilities?

You could try all the combinations with just 1ps and 2ps first.

## Student Solutions

Mrs Fother's class sent us in their answers to this problem:

First we made a list of all the possible coins Lola might have used to pay the clown. These were 1p, 2p, 5p, 10p, 20p, 50p, £1 and £2.

If Lolla paid using only 1p pieces, she would have paid 6p, which we thought was too little for a balloon.

If she paid using only £2 coins, she would have paid £12, which we thought was far too much for a balloon.

Then we looked at how many different prices Lola could have paid using exactly two different types of coin.

With the 1p and the 2p she could have paid 7p, 8p, 9p, 10p or 11p.

With the 1p and the 5p she could have paid 10p, 14p, 18p, 22p or 26p.

Here we thought we saw a pattern. We started off with five 1p coins and one of the other type of coin, and then to get the next largest amount we took away one 1p coin and added another of the other type of coin. So, as we did above, to go from 10p (five 1p coins and one 5p coin) to 14p (four 1p coins and two 5p coins) we took away 1p and added 5p. This is the same as adding 4p.

With the 1p and the 10p the smallest amount she could have paid was five 1p coins and one 10p coin, which makes 15p. 10p - 1p = 9p so we need to add 9p to 15p to get the next smallest amount of money - this is 24p, which is four 1p coins and two 10p coins.

Having found this pattern, we then split into groups to look at how many different prices Lola could have paid using exactly three different types of coin.

Here are some of our results:

With the 1p, the 2p and the 5p, she could have paid 11p, 12p, 13p, 14p, 15p, 16p, 17p, 19p, 20p or 23p.

With the 5p, the 10p and the 50p, she could have paid 80p, 85p, 90p, 95p, £1.25, £1.30, £1.35, £1.70, £1.75 or £2.15.

Thank you very much, Mrs Fother's class!

## Teachers' Resources

**Why do this problem?**

This problem offers an opportunity for learners to use numerical operations (addition, subtraction and possibly multiplication) and can be used to highlight ways of working systematically.

### Possible approach

The problem could be introduced through story and a real balloon can also be used to engage the children. Children can be asked if they have had balloons at home, the types of occasions when balloons are used as decorations, where they can be purchased and how much they might cost.

Give children time to work on the problem for a few minutes with large sheets of paper available for them to record any solutions. Then invite some children to suggest some different amounts, checking that they can be made with exactly six coins. You could ask what the largest amount Lola could have paid was, and the smallest amount. It might be appropriate for you to narrow down the problem at this stage so that you are able to emphasise ways of working systematically, so challenge the class to find ALL the different amounts which could be made with two types of coin. You could suggest that everyone tries using 1p and 2p coins first. Invite learners to record their ways on strips of paper (each way on a separate strip) as this will make it easier later.

Having given the group time to work on this, draw them together to find out the different amounts they have made. Ask children to come and stick a strip on the board so you begin to collate some different combinations. Once you have quite a few (there are seven altogether), ask the children how they know whether or not they have all the possible solutions. At this stage, you may be able to highlight some methods that you noticed while the children were working, and you can ask learners for their suggestions. Take up one of these (for example starting with all lowest value, then swapping one of those for the next value up, then swapping another lowest for another higher value etc.) and order the strips of paper to reflect this on the board. In this way, pupils will notice any gaps and having this modelled will help on future occasions.

You could then challenge different groups of children to work on a different pair of coins so that the task is shared between the class. Can they tell you how many different solutions there will be for each pair of coins?

### Key questions

### Possible extension

Children could go on to find all the possible combinations of six coins in a similar way if, for example, three different coins can be used.

### Possible support

Having plastic (or real) coins available will help the children identify, name and sort to find possible answers.