This activity has a logic and reasoning focus
This problem develops two concepts. 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 has students following these 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.
The students should be encouraged to try to find a number of answers and to reach a point where they have some systematic idea as to why there are no more answers. There three important skills that are fundamental to all of mathematics (and maybe life itself); first being able to find some possibilities, then gettingall possibilities and then justifying that there are no more. We work through this sequence in the Solution.
The second idea in this problem is symmetry. This involves noticing that turning some arrangements of the milk cartons through quarter turns, will give another arrangement. The two arrangements are said to be ‘alike'. The aim is to find such arrangements, put them into groups and find how many such groups there are. This will confirm the number of different arrangements, or the number of groups that are not ‘alike'.
Note that the basis for two arrangements being alike is discussed in Level 1 problem Strawberry Milk
Hannah has a square crate that can hold nine cartons of milk. In how many ways can she put three cartons of strawberry milk in the crate so that they form a line?
- Tell the class Hannah’s problem. Ask:
What might you need to help you solve the problem? How will you record what you find out?
- As students work on the problem, ask questions that support them to explore all possibilities.
- As some solutions emerge, have several students present a picture of one of their arrangements. Give the student's name to each.
- Ask all the students:
How do you know if there are any more?
How do you know if two answers are alike?
- Encourage them to use a systematic approach to find all the possible arrangements. (See the Solution).
- As appropriate, have students explore the Extension.
Hannah has another square milk crate. It can hold 16 bottles. In how many ways can she put four cartons of milk in her crate so that they form a line?
To be systematic, look at where the end carton of the three in a line can be. Work round the crate starting at the top left-hand corner of the crate. This appears to give 3 answers but 1 and 3 are alike because a quarter turn of the crate will take 1 to 3. So we have two non-alike arrangements here. (See the Solution to the Strawberry Milk problem.)
We also now know that we don’t have to think of any other end carton as being in a corner square. This is because any such possibility and 1 or 2 would be alike. Try the quarter turn test.
This now only leaves an end carton of a line of three being in the middle square on a side. But this gives only one possibility. Anything else can be turned into this by a (series of) quarter turn(s).
So there are three different solutions, 1, 2, and 4.
By going through the methods of Strawberry Milk and the Solution above, we get the following three possibilities.
You might like to think about what happens with 5 cartons in a line in a 5 by 5 crate. We think that the answer is 4.
Do you get the same answer for 6 cartons in a line in a 6 by 6 crate? What is the general pattern?