Ohio Resource Center

# Cup Packing

Estimated Time
Two or three 50-minute class sessions
Prerequisites
• Some familiarity with linear functions
• Volume and surface area of various 3-dimensional solids
Materials Needed
• Class set of centimeter rulers
• Identical hot beverage cups (enough for 5 per student)
• Internet access (not required)
Ohio Standards Alignment

### Topics

Equation of a line, volume and surface area of prisms and cylinders

### Overview

Students are provided with 5 hot beverage cups and are asked to determine the dimensions of the least expensive cardboard container that will hold 100 cups. This activity requires students to construct a linear function modeling the height of a stack of cups. An appealing feature of this modeling is the concrete representation of slope (the height of the rim of one cup) and y-intercept (the height of the first cup including the rim).  Students also investigate the surface area and volume of various 3-dimensional solids as they look for packages that are as inexpensive as possible to produce. Students are encouraged to use technology to calculate areas and perimeters of various geometric shapes and to explore relationships between surface area and volume.

### The "Hook"

You have been hired by a company that makes disposable drinking cups for a popular coffee retailer.  The company typically ships cups in boxes of 100 cups.  They need to determine a method of packaging cups that will be as inexpensive as possible.

Critical Question
What is the least expensive package that will hold 100 cups?

### The Investigation

Distribute the "Cup Packing" activity packet to each student.

Read the following directions to students (from the activity packet):

You have been hired by a company that makes disposable drinking cups for a popular coffee retailer.  The company typically ships cups in boxes of 100 cups.  They need to determine a method of packaging cups that will be as inexpensive as possible, while using resources as efficiently as possible.  What is the least expensive box that will hold 100 cups?

In solving this problem, you should make the following assumptions:

• If cups are split into several stacks, each stack must contain the same number of cups (this introduces some work with divisibility and factors into the problem).
• The cost of the cardboard packing material (cardboard) is \$0.0001 per square centimeter.
• For simplicity's sake, assume that the packaging does not overlap (i.e., no glued tabs).

Critical Question
What do we need to do to solve this problem?

You may work with a partner.  I'll provide you with 5 cups.  Help students identify the following tasks:

• Make a table and record in it the measurement data (number of cups and height of a stack of cups).
• Make a coordinate graph of the data set.
• Using your graph and/or table, determine a formula that describes how tall a stack of n cups would be.
• Experiment with different configurations of 100 cups.
• For each configuration, determine the amount of cardboard needed to contain the cups, the total cost of the cardboard, and the volume of the container.
• Based on your cost and volume calculations, recommend the inside dimensions of a carton that will be as inexpensive as possible and/or use the least resources.

Pair up students, then distribute 5 identical hot beverage cups and a centimeter ruler to each pair. Each pair should have the same kind of cups, since it is possible that pairs will combine cups into larger stacks in an effort to find dimensions.

### Teaching Tips

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).

### Supporting Resources

Internet access may prove useful for students, though such access is not required.  In particular, websites that discuss the mathematics of "packing" are helpful as students attempt to find packaging techniques that minimize surface area and/or volume.  For instance, Erich's Packing Center  is a particularly good resource.

Students may also find it helpful to research formulas for perimeters and areas of regular polygons.  Sites such as the Mainland High School Algebra LAB  and Dummies.com are helpful in this regard.

### Citation

Adapted by Michael Todd Edwards from Friel, S., Rachlin, S., & Doyle, D. (2001).  Stacking cups. In Navigating Through Algebra in Grades 6-8 (pp. 41-43).  Reston, VA: National Council of Teachers of Mathematics.