Problem Teachers' Notes Hint Solution

All Change

Here are three challenges for you to try.

In each case, the idea is to complete the game in as few throws of the die as possible.

Let us know how you get on! Do you think it would be possible to do it in any fewer? Why?

Which game needed the fewest number of moves? Will that always be the case, do you think?

Why?

Challenge 1

Throw the die and click on that number of plugs.

How many throws do you need to change the colour of all the plugs?

Challenge 2

This time when you throw the die you must click on that number of plugs but they must all be in the same row.

How many throws do you need to change the colour of all the plugs?

Challenge 3

Now when you throw the die, you click on that number of plugs all in the same row and you must finish a row before starting a new one.

How many throws do you need to change the colour of all the plugs?

This activity has been inspired by Doug Williams' Poly Plug resource. You can find out more details, including how to order sets of Poly Plug, on the Mathematics Task Centre website. However, you do not need sets of Poly Plug to have a go at this activity .

Why do this problem?

At the basic level, these challenges offer chances for children to practise number recognition, one-to-one correspondence and counting. However, some will begin to analyse and compare the three versions, explaining their findings and possibly drawing on ideas associated with probability.

Possible approach

All of the challenges could be adapted to be played as two-player games. Each child would need his/her own Poly Plug board, or his/her own large grid with $25$ counters, and the pair would need a $1$ to $6$ die. The idea then would be to fill your own board/grid completely before your partner fills his/hers.

You could introduce the first challenge using the interactivity, inviting different children to 'throw' the die and click the plugs. Once all the plugs have changed colour, draw the group's attention to the total number of die throws that were needed. Do they think it could be done in fewer throws? You could have one more go all together using the interactivity and compare the number of throws needed this time. Ask the children why more/fewer throws were needed the second time round and listen out for responses which show some understanding that if higher numbers are thrown, fewer throws might be needed in total.

Invite the group to have a go in pairs (either working together on one grid/board or in competition with each other). Who will manage it in the fewest number of throws? Ask the children how they will remember the number of throws they've had and take on board suggestions about recording in some way. You may also wish to encourage them to record the actual numbers they have thrown.

Bring everyone together and share the number of throws that each pair had needed. You might like to look in more detail at, for example, the lowest three or four. Would it be possible to do it in any fewer? Listen out for children who are able to explain that the minimum number of throws would be five. This might be four sixes and a one, or five fives.

You could introduce the other versions of the challenge in similar ways. Observe how pairs deal with a six in the second and third games. This would be a good discussion point. Encourage learners to consider the fewest possible throws for each version but also to compare the versions with each other. In the second version, how can you "keep your options open" so that you are more likely to be able to do something on your next throw? How does this compare to the third version? It could be interesting to have half the class playing the second version at the same time as the other half plays the third version and then to compare the number of throws needed.

Key questions

Which counters/plugs will you choose? Why?

Could it be done in fewer throws? How do you know?

How might you play differently next time?

How can you "keep your options open" in the second game?

Possible extension

Some children might enjoy making up their own version/s. You could set a particular challenge, for example, can they create a version which they know will be hard to complete in a small number of throws?

Possible support

You may wish to help some children by recording their throws for them, at least to begin with, so they don't have to worry about that aspect.