In this lesson, students will investigate two related problems. The first involves arranging soda cans on a shelf; students will explore this scenario by arranging pennies in a rectangular region. The second problem is more complex—students will attempt to determine the best possible package to carry more than one can of soda; that is, they will attempt to improve the designs currently used for six-packs and twelve-packs.
To stimulate interest in the lesson, allow students to do some research about soda by visiting the American Beverage Association web site. Students can share with the class some of the facts that they learn. Alternatively, you can find some information about soda on the web and present it to the class.
Then, give students the following problem to consider:
The Soda Warehouse arranges soda cans on refrigerated shelves. The shelves are rectangular. What arrangement will allow the maximum number of cans to be placed on a shelf?
To investigate this problem, give each student a copy of the Soda Cans
activity sheet and about 50 pennies. (Other congruent circular shapes, such as bingo chips, tokens, or mock coins, could also be used.) Explain to students that the rectangle on the activity sheet represents the shelf, and the pennies represent soda cans. Students may point out that this model does not account for the volume of the can, only the area of the cross section. If that happens, explain that the shelves are arranged one on top of another with only enough room for the height of one can between them. Consequently, this model is sufficient to investigate the problem.
For this part of the lesson, students should work individually. Note that the purpose of this activity is for students to consider various arrangements. For the moment, it is not necessary for students to determine the exact area that will be covered by cans, to prove that the cans will fit, or to verify that they have found the maximum number. Such calculations will, however, be required later when students investigate the second problem in this lesson.
As students explore, circulate through the room and note the results. To motivate students to find better arrangements, you might consider offering a small prize for the student who is able to fit the most cans on the shelf.
Once students have considered various arrangements and understand how the cans fit together, proceed to the second problem of this lesson:
Consider the packaging that is used to hold soda cans. How might this packaging be made more efficient? That is, can you place the soda cans in some other arrangement to reduce the amount of packaging that is needed?
Explain that soda companies often package cans in twelve-packs. Some packages are arranged in a 3 by 4 pattern, and other packages are arranged in a 2 by 6 pattern (known as a fridge pack). On the overhead or board, display the two arrangements shown below. Ask students to consider the question, "Which of these arrangements is more efficient? That is, which one has less unused space in the package?"
Allow students to think about this question for a minute; then, discuss it with a partner. After partners have shared, ask the class for their opinions. [Students will likely notice that neither arrangement is more efficient. Each package can be divided into twelve congruent squares, one square around each can, and there is the same amount of unused space in each square.]
Ask students to consider if there is another rectangular twelve-pack arrangement of m × n cans that would be more efficient. [No. All rectangular arrangements would have the same amount of unused space.]
Given this lead-in, allow students to work in pairs or small groups to devise an arrangement that would be more efficient. Be sure to point out that they can design a package for more or fewer than twelve cans, but explain that the minimum is three cans. You can also impose a maximum, but it is usually not necessary. However, students should consider that if there are too many cans in a package, customers will not be able to lift the package and carry it home.
To guide this investigation, you may want to assist students in determining a measurement for gauging the efficiency of a package. The simplest measurement is to consider the ratio of filled space within a package to total space. That is,
efficiency = area covered by cans ÷ area of entire package
At this point, you may want to have students calculate the efficiency of the twelve-packs considered earlier. [The radius of a soda can is approximately 3.2 centimeters, so the area covered by one can is approximately 32.15 square centimeters. The area of the square that surrounds a can is roughly 40.96 square centimeters. Therefore, the efficiency of the packages above is approximately 32.15 ÷ 40.96 = 78.49%. Note, however, that the actual dimensions of the soda can are irrelevant. If twelve circles of radius, r, are arranged as shown above, the efficiency is given by the equation:
which shows that for any radius, the efficiency will remain constant with this arrangement.]
Allow students to work on the problem for the remainder of class. As they work, circulate and talk to the groups. At the end of class—or at the beginning of the following class, if you want students to continue to work on the project at home—have each group of students present their best arrangement.
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