This problem solving activity has a logic and reasoning focus.
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?
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:
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 are three important skills that are fundamental to all of mathematics; being able to find some possibilities, getting all possibilities, and justifying that there are no more possibilities. 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 alike arrangements, put them into groups, and calculate how many alike groups there are. This will confirm the number of different arrangements, or the number of groups that are not ‘alike'. To do this, students should have some knowledge of what is meant by a quarter turn.
The basis for two arrangements being alike is discussed in the Level 1 logic and reasoning 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?
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, we need to look at where the end 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. However, 1 and 3 are alike because a quarter turn of the crate will take 1 to 3. Therefore, we have two non-alike arrangements (see the Solution to the Level 1 Strawberry Milk problem.)
We also now know that we don’t have to think of any other arrangement that starts and ends with a corner square. This is because any such possibility would be alike with arrangements 1 and 2. Use the quarter turn test to demonstrate this.
This now only leaves a line of three down the middle of the grid. Anything else can be turned into this, or one of the previous solutions, 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. Is the answer 4?
Do you get the same answer for 6 cartons in a line in a 6 by 6 crate? What is the general pattern?
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/three-line at 8:56pm on the 26th February 2024