# Strawberry Milk

*Keywords:*

This activity has a logic and reasoning focus.

AO elaboration and other teaching resources

AO elaboration and other teaching resources

rotate patterns through quarter and half turns

count possibilities

This problem is one of a series of 8 that builds up to some quite complicated maths based around the theme of no-three-in-a-line. That theme is not obvious here, though clearly it isn’t possible to put three bottles in a line in a 2 by 2 milk crate. However, the theme will develop as we move through the Levels. The other lessons in this sequence are Strawberry and Chocolate Milk, Level 1; __Three-In-A-Line__, Level 2; __No Three-In-A-Line__, Level 3; __More No-Three-In-A-Line__, Level 4; __No-Three-In-A-Line Again__, Level 5; No-More-In-A-Line; Level 6 and __No-Three-In-A-Line Game__, Level 6. The problems increase in difficulty through the sequence and the skills discovered in one problem are used in the next.

Now this present lesson has two key ideas. Consequently it might be best to split it into two lessons or, at least, work on it in two halves. The way that you deal with it will depend on the ability and experience of your class. We suggest one approach in the Teaching Sequence. So this problem is a bit unusual. The difficulty is that there are a lot of ideas floating around here.

There are two basic concepts behind this problem. The first is counting all possible arrangements and the second is noticing that some of these arrangements are ‘alike? and so might be considered to be the same.

The first part of this problem is about children trying to go through the following steps:

- find
answers to a problem;*some* - think about whether there are any
answers or not;*more* - try to explain why there are
answers.*no more*

We don’t necessarily expect children to find all of the answers by themselves. What we do expect though is that they will ** try** to find more answers than they have got and in the end have some systematic idea as to

**there are no more answers. This is because in the end these are three important skills that go throughout all mathematics (and maybe life itself); first being able to find**

*why***possibilities, then getting**

*some***possibilities and then**

*all***that there are no more. We work through this sequence in the Solution.**

*justifying*The second idea that this problem deals with is symmetry. In this case this involves noticing that turning some arrangements of the strawberry milk bottles through quarter turns, will get you to another arrangement. When we find two arrangements like this we say that they are ‘alike?. The aim then is to find such arrangements and put them into groups. In the end we want to see how many such groups there are. This is because then we know how many essentially different arrangements there are. This is just the number of groups that are not ‘alike?.

Although we have placed this problem in the Mathematical Processes? Strand you can see that it has elements of both Statistics and Geometry. On the Statistics side, we are trying to count all possibilities. This is a precursor to determining probabilities, which is an important part of Statistics. On the Geometry side, we shall need to talk about (rotational) symmetry in order to decide which arrangements of the bottles lead to different arrangements.

Once the class has completed these two lessons it should be ready for Strawberry and Chocolate Milk, another Level 1 problem.

There is a web site on the no-three-in-line problem. Its url is www.uni-bielefeld.de/~achim/no3in/readme.html. The material in this web site is too complicated for Level 1 students but it may be of interest to you. It shows a problem that mathematicians are working on right now and still can’t solve.

Mary the milk lady had a square milk crate that could hold four bottles. In how many ways can she put two strawberry milk bottles into the crate?

### Teaching sequence

- Talk about delivering milk. Ask

*What containers does milk come in?*

How is it delivered? What do the milk bottles travel in?

How big are milk crates?

How heavy are they? - Talk about Mary delivering milk.

*Why might she have a small crate?*

What shape is the crate? What do we know about that shape?

How many bottles can she get into her crate? - Tell the class Mary’s problem.

*How can you solve the problem?*

What might you need to help you? - After some discussion, let the class go into their groups or work alone.
- Help the children that need it.
- You may need to call them all together at some stage to see how many arrangements they have come up with. Get them to take turns in putting a picture of one of their arrangements on the board. Call each arrangement by the child’s name who found it (see Solution).
- Ask

*How do you know if there are any more?* - Try to get them to see the systematic approach that we used in the Solution to get all possible answers. As this is a new idea for the children, they may not all understand the concepts involved the first time round. In that case, follow up with the Extension problem. This will give them a chance to practice the ideas they have just seen.
- Let a few groups/children report back to the whole class. Try to choose groups that have used different approaches to the problem. Let the children put their pictures of the bottle arrangements on the wall.

(The lesson could be stopped here and continued the next day. If so, you will need to go over what you did the previous day and recall the crate arrangements with the children’s names attached. We have provided our own names in the Solution. We use those names in the rest of the Teaching Sequence.) - Ask the class

*What happens if you turn Charlie’s crate by a quarter turn?*

Does this give you someone else’s crate? Whose?

What happens to Hine’s crate if she turns it through a half turn? - Then try asking questions like

*If I have just turned the crate through a quarter turn and I’ve got Joe’s crate, whose crate did I start with?* - Quiz them about the various crates.

*Do any of these crates look the same?*

Do some of them have anything in common? - Call crate arrangements that can be rotated into each other ‘alike?.
- Ask them to go away and put alike crates together.

*Whose crates look alike?*

Whose crates are really different?

How many different answers are there?

Try to get them to see that rotating the crate around will move one arrangement to another (see the Solution). - Get them to report back on their progress.
- (This is an optional step that you may want to ignore or only ask of the more able students.) Look at the properties of the arrangements.

*Are there any crates that stay the same after a rotation of a quarter turn?*

Are there any crates that stay the same after a rotation of a half turn? - Suggest that they look at the Extension problem again to see if there are any answers that may be alike.
- Discuss their conclusions.