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'Buying a Balloon' printed from http://nrich.maths.org/
Why do 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.
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 might they 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 Lolla 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 six of each of the two lowest value coins only. Invite them 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.
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?
Children could go on to find all the possible combinations of six coins in a similar way if $1$ps, $2$ps and $5$ps are available. Some may be able to use the solution for just $1$ps and $2$ps to help.
Having plastic (or real) coins available will help the children identify, name and sort to find possible answers.