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    PS Home| Information|Level 1|Level 2| Level 3| Level 4| Level 5| Level 6

    Strawberry and Chocolate Milk

    Logic and Reasoning, Level 1

    The Problem

    Mary the milk lady had a square milk crate that would hold four bottles. In how many ways can she fill it withstrawberry and chocolate milk bottles?

    What is the problem about?

    This problem is one of a series of 8 that builds up to some quite complicated maths based around the theme ofno-three-in-a-line. That theme is not obvious here, though clearly it isnt possible to put three bottles in a line in a2 by 2 milk crate. However, the theme will develop as we move through the Levels. The other lessons areStrawberry Milk, Level 1; Three-In-A-Line, Level 2; No Three-In-A-Line, Level 3; More No-Three-In-A-Line, Level4; No-Three-In-A-Line Again, Level 5; No-More-In-A-LineLevel 6; andNo-Three-In-A-Line Game, Level 6.

    We suggest that you do Strawberry Milkbefore you do the present lesson as this lesson is meant to recall andreinforce 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 thesame.

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

    I. find someanswers to a problem;II. think about whether there are any moreanswers or not;III. try to explain why there are no moreanswers.

    We dont necessarily expect children to find all of the answers by themselves. What we do expect though is thatthey will tryto find more answers than they have got and in the end have some systematic idea as to whythereare no more answers. This is because in the end these are three important skills that go throughout allmathematics (and maybe life itself); first being able to find somepossibilities, then getting allpossibilities andthenjustifyingthat 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 somearrangements of the strawberry milk bottles through quarter turns, will get you to another arrangement. When wefind two arrangements like this we say that they are alike?. The aim then is to find such arrangements and putthem into groups. In the end we want to see how many such groups there are. This is because then we knowhow 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 elementsof both Statistics and Geometry. On the Statistics side, we are trying to count all possibilities. This is a precursor

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    to determining probabilities, which is an important part of Statistics. On the Geometry side, we shall need to talkabout (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 iswww.uni-bielefeld.de/~achim/no3in/readme.html.This is still an open problem in mathematics and has an interesting number of sub problems relating tosymmetry.

    Achievement Objectives

    Logic and Reasoning

    o classify objects;o interpret information and results in context;o use words and symbols to continue patterns.

    Geometry, Level 1

    o create and talk about symmetrical and repeating patterns.

    Statistics, Level 1

    o collect everyday objects, sort them into categories, count the number of objects in each category, anddisplay and discuss the results.

    Problem Solving

    o devise and use problem solving strategies to explore situations mathematically (guess and check, besystematic, use equipment, act it out, draw a diagram).

    Resources

    o Copymaster of the problem (English)o Copymaster of the problem (Mori).

    o Coloured pens and paper.

    o Copymaster of 2 by 2and3 by 3crates.

    Specific learning outcomes

    The children will be able to:

    o explore coloured arrangements;o recognise similar situations.

    Teaching sequence

    1. Talk about delivering milk. AskWhat 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?

    2. Talk about Mary delivering milk.Why might she have a small crate?

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    What shape is the crate? What do we know about that shape?How many bottles can she get into her crate?

    3. Tell the class Marys problem.How can you solve the problem?What might you need to help you?

    4. After some discussion, let the class go into their groups or work alone.5. Help the children that need it.

    6. You will probably need to call them all together at some stage to see how many arrangements theyhave come up with. Put childrens names to the different arrangements. Let them put their pictures ofthe bottle arrangements on the wall so that the class can refer to them. These can be added to ifanother one is discovered.

    7. AskHow do you know if there are any more?Can two arrangements actually be the same?

    8. Talk about symmetry and rotating the square so that one arrangement goes to another. But let themdecide when two things are alike? or the same?.

    9. Any children who are able to justify their answer could try the Extension.10. Let a few groups/children report back to the whole class. Try to choose groups that have used different

    approaches to the problem.

    Extension

    Suppose that Mary had a 3 by 3 crate. In how many ways could she fill the crate with two bottles of strawberrymilk and seven bottles of chocolate milk?

    Solution

    We imagine that, no matter how the children try to do this, they will first of all do it quite unsystematically. This iswhat you would expect at step (i) of the process of finding all answers (see What is the problem about?). So theywill probably come up with several ways of putting the bottles into the crate but not be sure that two of themcould be considered to be the same and not be able to see why there are no more.

    Lets talk about what crate arrangemnets are alike? for a moment (see Strawberry Milk, Level 1). We could thinkof 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 thiscolumn.

    So, if we decide that these four are all the same then there are only 6 ways of filling Marys crate. These areshown 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 possiblecontenders are alike. They can be rotated into each other.

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    With two chocolate bottles, there are two possibilities: one where the chocolate bottles are next to each otherand the other where they are not. (This is essentially the situation of Strawberry Milk, Level 1). Again, otherarrangements 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 isagain only one possibility.

    Extension

    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 onecan 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 thenmove the other one around.

    Be symmetry, the only position left for the first strawberry bottle is the centre position. However, there is then nosquare to put the other bottle in that hasnt been used before. Hence there are only eight possibilities here.

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