Square Milk Bottle Crates

Qi-xiao has a problem. He works with a company that uses square-bottomed milk crates. These can be divided up to contain a square number of bottles fitting the crates with dividers. Using 2 dividers Qi-xiao can fit 4 bottles into a crate. Using 4 dividers he can fit 9 bottles another sized crate. Both of these situations are shown in the picture.

On Friday Qi-xiao’s boss wants to know how many bottles can be fitted into a square crate if 30 dividers are used.
On Tuesday Qi-xiao’s boss wants to know how many dividers would be needed for a square crate that could hold 100 milk bottles.

Teaching sequence

1. Show the students the picture of a carton with 4 dividers and ask:
   What mathematical questions could we pose about this carton?
2. It is possible that the students will come up with a similar problem – at that point pose the problem for the students to solve.
3. Ask for their initial ideas for getting started on the problem.
4. As the students work on the problem ask questions that encourage them to think about their problem solving strategies:
   What strategy are you using to solve the problem? Are you making progress with it?
   How are you keeping track of what you do?
   Have you seen similar problems before?
5. Also ask questions that encourage the students to look for and explain patterns in the number of dividers.
   What have you found out about the dividers?
   Does that help you solve the problem? How?
   What patterns can you see in the problem?
   What helps you find patterns in problems like these? (tables)
6. Share answers and justifications.

Extension to the problem

Take the square crates again. Qi-xiao wants to stay a step ahead of his boss. So he decides to work out some formulae.

If Qi-xiao’s boss gives him 2d dividers, what is the maximum number of bottles that he can put in the crate?

If Qi-xiao boss wants to fit b² bottles in a square crate, how many dividers would he needed?

There are a number of possible solutions to these problems. They include solving a simpler problem using a table and using algebra. We give them both below.

Option 1: The 30 dividers problem.

Using a table and considering simpler cases first. It soon becomes clear that each bottle must fit into a square hole formed by the dividers. That means that an equal number of dividers have to be used in each direction. What’s more there is always one more square per side than the number of dividers perpendicular to that side.

The table can be continued up to 30 if the pattern is not obvious to students from the start.

The 100 milk bottles problem.

One way to do this is to work backwards in the table. Since 100 = 10², and 10 – 1 = 9, then 2 x 9 is the number of dividers that have to be used. So Qi-xiao’s boss will have to find 18 dividers.

Solution to the extension

The square numbers become evident and need to be linked algebraically with the number of dividers. The number of dividers occurs in pairs. If each pair is considered as a single case then the table becomes.

Focusing on the number of dividers 2d we can generate a quadratic formula for the number of milk bottles. So

the number of bottles = (d + 1)².

If we have b² bottles, then we can work from the right to the left in the table.
Corresponding to b² we have 2(b – 1) in the middle column. So

the number of dividers = 2(b – 1).

Option 2: The 30 dividers problem.

By drawing a diagram it can be seen that half the number of dividers are used widthways and half lengthways. So 15 dividers will be used on one side of the crate.

Hence 15 divides on one side will give 16 bottles spaces. Thus the total number of bottles spaces in the square crate is 16 x 16 = 256.

The 100 milk bottles problem.

100 bottles mean 10 bottles on each side so there are 9 dividers on each side. Hence there are 18 dividers altogether.