## Ice Cream

Let's all go down to the favourite cafe that sells ice cream which you choose from different tubs.

Suppose that there's Apricot, Banana and Citrus.

There is just one rule about what you can choose, and here it is:

YOU CANNOT CHOOSE A SELECTION OF ICE CREAM FLAVOURS THAT INCLUDES TOTALLY WHAT SOMEONE HAS ALREADY CHOSEN!

This means that if someone has chosen Banana and Citrus I cannot then go and choose all three but I could choose to have Apricot on its own.

So perhaps something like this happens:-

Sarah, the first child, chooses Apricot and Citrus.

Tim, the second child chooses Banana and Citrus [this obeys the rule because Sarah's choice was not Banana on its own nor was it Citrus on its own].

Raj, the third child, chooses Citrus.

Zoe, the fourth child, chooses Banana.

Matt, has to be the last child because he can only choose Apricot and then there are no other choices left.

In this example, with these children making these choices, only five children can have ice cream [using our rule].

But suppose more children wanted ice cream and so they got together to work out how this could be done.

They might come up with an idea like this:-

[I'm using the short-hand this time of A B C where A is Apricot, B is Banana and C is Citrus.]

1st. choice > A B C
2nd. choice > A B
3rd. choice > A C
4th. choice > B C
5th. choice > A
6th. choice > B
7th. choice > C

So seven children altogether. I think that the children can have different sized scoops so that even if they only have one flavour they have as much ice cream as someone choosing three flavours!

If these children weren't very good at working things out they could come up with the worst way; that would be like this:-

1st. choice > A
2nd. choice > B
3rd. choice > C

AND THAT'S ALL!

Well that's what it's like when there are three flavours. At the most, seven children can go in that order and get those choices of ice cream. At worst, only three children can go and get ice cream.

Mind you I think that there are other ways of getting seven.

Have a go and find all the different ways of having seven children getting ice cream. Remember that when someone goes up and makes their choice they have to obey that rule:

YOU CANNOT CHOOSE A SELECTION OF ICE CREAM FLAVOURS THAT INCLUDES TOTALLY WHAT SOMEONE HAS ALREADY CHOSEN!

And FINALLY ...

I wonder what would happen if ...?

Please send in any results that you get along the way. I have to go - my mouth is watering for some ice cream!

### Why do this problem?

This activity , when done practically, usually gives great excitement and leads to an extensive discussion as learners try to explain their reasoning to one another.

### Why do this problem?

This investigation offers an opportunity for children to work creatively, as there is no obvious place to start.  However, it may well give you chance to discuss systematic approaches once they have been working on it for a while.

### Possible approach

You'll probably get the best discussion and thoughts from the pupils if you 'act' out the situation. You are the ice cream vendor with a selection of cards, counters, cubes etc. to represent the ice cream. Make sure that the class can see the choices that the previous pupil(s) made and encourage them to check if they've kept to the rule, helping as little as possible.

Once they have got the idea, invite them to work on the problem, perhaps in pairs. Allow them to make their own decisions as to the equipment they use and the way they record - sharing these can be just as valuable as sharing solutions.

This activity would lend itself to a display and so you may want children to work on large sheets of paper which could be the stimulus for a plenary and could then be put straight up on the classroom walls. You could invite some children to explain how they have found different ways of seven children having ice cream. Draw attention to those who have found a systematic way of finding solutions.

### Key questions

Why did you choose that flavour?
Is there a way of choosing to let more pupils have a choice?

### Possible extension

As the problem suggests, the thinking can be extended by looking at larger numbers of flavours.

### For the exceptionally mathematically able

From the "Possible Extension" the challenge would be to find a general formulae that would take into account the number of flavours and how many scoops were alloed each time to produce the largest numbers of hungry ice cream addicts who could have their fill. Then there would be the need to prrove it to be correct.

### Possible support

This problem is a good context in which to try to stand back and encourage pupils to support each other and explain ideas themselves. Working in pairs might be an appropriate way to support this.