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Beads and Bags




This challenge involves three beads and lots of bags.
There are as many bags as you need.

Find a way to put the beads into some of the bags.
Find another different way to put the beads into some of the bags.
How many different ways can you make?
Have you tried putting a bag into another bag?

Can you record your ways?
How do you know you've got them all?

Can you give your recording to a friend to see if they can re-make your way?

Can you record your ways without pictures?
Can you record using numbers?

Does your partner always make the way you expect?
Can you refine your recording so that your partner always gets it right?
 
The photograph of beads was taken by Danielle Keller http://commons.wikimedia.org/wiki/File:Wooden_beads.JPG
 

Why do this problem?

This task provides an opportunity for children to develop and refine their own recording system with a clear purpose in mind. The activity allows learners to think creatively in the way they approach the problem and in the recording, and they will need to work systematically if they are to find all the possibilities. 

Possible approach

Have three beads/marbles/counters and some bags available. Ideally, transparent bags would be useful as this will help identify the number of objects contained in each. Invite a child to put the three beads into some bags in any way they choose. Ask another child to put the three beads into bags in a different way. You could ask a third child to do it another way too.
 
Explain to the children that their first task is to find all the different ways that we can put three beads into bags. Set them off on the challenge, perhaps working in pairs, and make sure that beads and bags are available for everyone if they would like them. After a short time, draw the whole group together and ask how the children are remembering the ways they have already found and how they are making sure that they don't repeat any. Tease out the need to record in some way and perhaps share some suggestions for ways to do this. (At this stage, encourage the pupils to record in whatever way they find works well for them.)
 
Allow more time for pairs to continue working and then bring everyone together again. You may focus to start with on how they know they have got all the different ways. Invite a pair to record their ways on the board. Do any other pairs have more ways? Continue to ask the whole group until the children haven't got any new arrangements. You may need to talk about what they are 'allowing'. For example, can you have an empty bag? Can you put a bag inside another bag? If they haven't done so already, encourage them to think about the possibilities that include putting one or more bags inside another one. (At this point you may need to give them more time in their pairs.) Invite a pair to offer a way to check we haven't left any arrangements out. This will require a system, for example having all the beads in one bag, then splitting them into two bags, then three bags, and then considering bags inside bags.
 
Subsequently, (or alternatively) you can focus on the recording. Explain to the class that you're keen to find out whether you could look at someone's work and be able to put the beads in bags to match what they had written or drawn. Ask each pair to record one way of placing the beans on their mini whiteboard. Encourage two pairs to team up and swap whiteboards. Was each pair able to recreate the arrangement of beads?
 
You can then challenge pairs or groups of four to come up with better ways of recording so that another pair or group would be able to get the arrangement right.
 
Share some of their ideas and then explain that mathematicians like to find ways of recording that are quick and efficient, in other words ways that don't take long to do but are easy to understand. The final challenge therefore is for groups/pairs to develop a way of recording that meets these criteria.  

Key questions

Tell me about the ways you've found so far.
How will you remember them?
How will you know when you've got them all?
How could you check that you've got them all?
How could you record that arrangement?
How could you make your recording clearer/easier to understand/quicker to do?  

Possible extension

This could be an ongoing 'simmering' activity where children could continue to think about improving their recording methods over a period of a few days or weeks. You could devote some wall space to the activity so that children could post up their suggestions.
 
Some children may want to investigate the number of ways of placing four beads in bags.  

Possible support

For some children it will be enough to either focus on ways of recording or on finding all possibilities. If the latter, try to provide enough bags and beads so that they can physically make up all the different ways without having to re-use beads and bags.