All change

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

Age

5 to 7

Challenge level

Primary

Number - Number and Place Value

Comparing Numbers

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

All Change printable sheet

Watch the video below (you do not need sound).

What do you notice?

What do you want to ask?

What are the 'rules' to this challenge, do you think?

How do you complete the challenge?

The rules are hidden below in case you would like to check them.

The aim of this challenge is to completely fill the grid with counters.

You will need a copy of the grid, 25 counters or buttons (or anything else that you have), a 1-6 dice, pencil and paper.

Throw the dice and place that number of counters anywhere on the grid.

Repeat this over and over again.

Each time you throw the dice, make a record (for example by keeping a tally) so that you know how many times you have thrown it so far.

Continue until you have completely filled the grid.

Make a note of the total number of throws that it took to fill the grid.

Now it is your turn! Have a go at the challenge for yourself (you may like to print off this sheet of the grid).

How many throws did it take to fill the grid completely?

Have some more goes to see if you can do it in fewer throws of the dice.

What is the smallest number of throws you did it in?

Do you think it would be possible to complete the grid in even fewer throws if you kept on trying? Why or why not?

Now take a look at two more videos, each one demonstrates a slightly different version of the challenge. (Once again you do not need sound.)

What is the same compared with the first version?

What is different?

Have lots of goes at these versions of the challenge.

How many throws did it take to complete the grid each time?

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

Teachers' Resources

All Change

All Change printable sheet

Watch the video below (you do not need sound).

What do you notice?

What do you want to ask?

What are the 'rules' to this challenge, do you think?

How do you complete the challenge?

The rules are hidden below in case you would like to check them.

The aim of this challenge is to completely fill the grid with counters.

You will need a copy of the grid, 25 counters or buttons (or anything else that you have), a 1-6 dice, pencil and paper.

Throw the dice and place that number of counters anywhere on the grid.

Repeat this over and over again.

Each time you throw the dice, make a record (for example by keeping a tally) so that you know how many times you have thrown it so far.

Continue until you have completely filled the grid.

Make a note of the total number of throws that it took to fill the grid.

Now it is your turn! Have a go at the challenge for yourself (you may like to print off this sheet of the grid).

How many throws did it take to fill the grid completely?

Have some more goes to see if you can do it in fewer throws of the dice.

What is the smallest number of throws you did it in?

Do you think it would be possible to complete the grid in even fewer throws if you kept on trying? Why or why not?

Now take a look at two more videos, each one demonstrates a slightly different version of the challenge. (Once again you do not need sound.)

What is the same compared with the first version?

What is different?

Have lots of goes at these versions of the challenge.

How many throws did it take to complete the grid each time?

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

This activity has been inspired by Doug Williams' Poly Plug resource. However, you do not need sets of Poly Plug to have a go at it.

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

Explain that you are going to show the class a video and you'd like everyone to watch it the first time without talking to each other. Ask learners to think about what they notice and what they want to ask as they watch.

Once the video has finished, invite comments. You could write up learners' observations and their questions on the board. Try not to pass judgement yourself, instead encourage other members of the group to respond to what is being said. You may wish to watch the video a few more times together so that you can clarify some points that have been raised.

Gradually, help the class to build up a sense of the 'rules' of this challenge and then give them the opportunity to have a go themselves. They could work collaboratively in pairs using a printed grid and counters. Remind them to keep track of the number of throws of the dice. You could suggest that once a pair has completed the challenge, they write up their total number of dice throws on the board, so that you will eventually have many examples.

When all pairs have completed the grid at least once, draw everyone's attention to the list on the board of the number of dice throws it has taken each time. What do they notice? If it does not come up naturally, ask what the smallest number of dice throws is so far. Do they think that it would be possible to complete the grid in even fewer dice throws if they kept on trying? Why or why not? Give pairs chance to discuss their thoughts and then invite comments to be shared. Encourage learners to articulate their reasoning clearly. Listen out for those who realise that throwing lots of higher numbers will fill the grid more quickly, in particular lots of sixes. How many sixes would they have to throw to fill the grid?

You could then introduce the second and third versions of the challenge in a similar way using the videos, discussing the 'rules' and giving learners time to have a go themselves. (It might be that the rules of the second and third versions have been discussed earlier in the lesson before those videos are shared with the class. This does not matter in the slightest!) Look out for the ways in which pairs deal with a six in the second and third games. This could 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 with 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

Where will you place those counters? 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 support

You could suggest that one child of the pair takes responsibility for recording the number of throws while the other actually throws the dice. They can then discuss where to place the counters together. This will hopefully mean they are less likely to forget to keep track of the total number of throws!

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?

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

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