Strawberry and Chocolate Milk
AO elaboration and other teaching resources
AO elaboration and other teaching resources
This activity has a logic and reasoning focus.
rotate patterns through quarter and half turns
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 are Strawberry 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.
We suggest that you do Strawberry Milk before you do the present lesson as this lesson is meant to recall and reinforce the ideas that the children met in that lesson. These two main ideas were counting all possible arrangements and 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 students trying to go through the following steps:
- find some answers to a problem;
- think about whether there are any more answers or not;
- try to explain why there are no more answers.
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 why 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 some possibilities, then getting all possibilities and then justifying that there are no more. We work through this sequence in the Solution.
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.
There is a web site on the no-three-in-line problem. Its url is www.uni-bielefeld.de/~achim/no3in/readme.html. This is still an open problem in mathematics and has an interesting number of sub problems relating to symmetry.
Mary the milk lady had a square milk crate that would hold four bottles. In how many ways can she fill it with strawberry and chocolate milk bottles?
- 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 will probably need to call them all together at some stage to see how many arrangements they have come up with. Put children’s names to the different arrangements. Let them put their pictures of the bottle arrangements on the wall so that the class can refer to them. These can be added to if another one is discovered.
How do you know if there are any more?
Can two arrangements actually be the same?
- Talk about symmetry and rotating the square so that one arrangement goes to another. But let them decide when two things are ‘alike? or the ‘same?.
- Any children who are able to justify their answer could try the Extension.
- Let a few groups/children report back to the whole class. Try to choose groups that have used different approaches to the problem.
Suppose that Mary had a 3 by 3 crate. In how many ways could she fill the crate with two bottles of strawberry milk and seven bottles of chocolate milk?
We imagine that, no matter how the children try to do this, they will first of all do it quite unsystematically. This is what you would expect at step (i) of the process of finding all answers. So they will probably come up with several ways of putting the bottles into the crate but not be sure that two of them could be considered to be the same and not be able to see why there are no more.
Let’s talk about what crate arrangemnets are ‘alike? for a moment (see Strawberry Milk, Level 1). We could think of the two filled crates below as being alike because it is possible to rotate one around through a quarter turn, until it looks exactly the same as the other.
This should provoke a discussion. Are there other ways of filling the crate that look the same as the one above? You might list these in a column one under the other. There are four possible arrangements that could be in this column.
So, if we decide that these four are all the same then there are only 6 ways of filling Mary’s crate. These are shown below.
Perhaps the best way to see that there are no more is to be systematic (this will complete steps (ii) and (iii)). So, in the first case suppose that there are no chocolate bottles. This gives only one case.
Then suppose that there is one chocolate bottle. There is only one case here too because the four possible contenders are alike. They can be rotated into each other.
With two chocolate bottles, there are two possibilities: one where the chocolate bottles are next to each other and the other where they are not. (This is essentially the situation of Strawberry Milk, Level 1). Again, other arrangements look possible but each one is just a rotation of the situation in the picture above.
With three chocolate bottles there is only one place for the strawberry bottle. With four chocolate bottles there is again only one possibility.
Being systematic we get the eight possibilities below.
The system here is achieved by first putting a strawberry bottle in one corner and seeing where the other one can go without repeating a situation that has occurred before.
When that case has been exhausted, put one strawberry bottle in the middle square along an edge and then move the other one around.
Be symmetry, the only position left for the first strawberry bottle is the centre position. However, there is then no square to put the other bottle in that hasn’t been used before. Hence there are only eight possibilities here.