You may wish to engage students in a short conversation about the relevance of this activity to real-world problems (e.g. reducing waste in landfills, reducing costs of production, shipping products and stocking store shelves more efficiently, etc.)
The primary focus of this activity is to provide students with opportunities to explore linear functions, surface area, and volume in real-world contexts. During the first stage of the activity, as students build linear models for the height of cups, it is powerful to share the following graphics (see Figure 1, Figure 2) with students (better yet, construct the model in front of class, using a meter stick to represent the "best fit" line.
The stacked cups provide a concrete example of slope and y-intercept that may help students grasp these ideas more conceptually.
Numerous configurations of cups exist. Because stacks must contain the same number of cups, listing the divisors of 100 (1, 2, 4, 5, 10, 20, 25, 50, 100) helps students generate configurations that may otherwise be overlooked. The task of finding the dimensions of a cardboard container that will minimize cost and resources should be considered open-ended, with multiple correct answers possible. For instance, while cylindrical containers may have smaller surface area than box-shaped containers (i.e., prisms), boxes may be packed more tightly when stored for shipping.
In the partial list of solutions provided, the cylindrical container holding 4 stacks of 25 cups each was the least expensive in terms of cost for cardboard (i.e., $0.36). However, square prism containers holding 4 stacks of 25 cups can be stacked more easily for shipping in bulk. The slightly higher cost for materials (i.e., $0.37) will likely be offset by savings in time loading and unloading shipments. Furthermore, space will necessarily exist between cylindrical containers (rather than in the container itself). A nice extension of the problem is to see whether there is a difference between the number of circular versus the number of square containers of 4 stacks of 25 cups that will fit into a shipping crate.
Determining a method of packaging cups that will be as inexpensive as possible, while using the least possible resources, is a complex task. Students should be encouraged to appreciate the richness and complexity of the problem without becoming overly preoccupied with finding the best solution. Considerable time should be provided to students to discuss the advantages and disadvantages of various containers.
Here is a suggested schedule for engaging the problem:
Day 1: Construct a linear model describing the height of n cups; Brainstorm 3 or more configurations of 100 cups.
Homework for Day 2: Calculate the surface area, volume, and cost for one configuration.
Days 2 and 3: Calculate surface area, volume, and cost for additional configurations; discuss/share results. Discuss strategies for minimizing surface area while also minimizing other resources used.
See the attached solution sheet for possible solutions. Note that the solutions on the worksheet have been calculated using coffee cups with the following dimensions: height = 16.2 cm and radius of the mouth of the cup = 4.5 cm. (Note: this conforms with dimensions of the large cup of several popular coffee retailers).