Stacks Of Cups

Stacks of Cups Target audience: Algebra I, Algebra II Complete the activity in groups of four students.

Objective: To measure stacks of cups and determine a linear relationship between the number of cups and the height of the stack. Materials: Several different types of cups in stacks of 5 each. (Most restaurant managers will give you 10 free plastic kid cups.) Rulers that measure in cm with mm marks. Teacher Note: Introduce the activity by having a student read the problem scenario at the top of the page. When students get to number 4, they will call you over for “quality control”. Grab the ruler and a stack of 3 cups. Ask them to use their equation to tell you the height. If they are within a millimeter they pass inspection. If not, they try again until they are. When students get to numbers 7 and 8, they will need to be able to deal with greatest integer function. This can be done informally by talking about truncating the decimal off the number they get when they plug in the height of the carton. This activity can also be used when discussing arithmetic sequences.

NAME__________________________________________________________________ DATE_______________________ Stacks of Cups

You have been hired to be the package design director for a company that designs all kinds of cups – Styrofoam hot cups, plastic cold cups, and more – of different sizes. Your design team has been given the assignment of designing a cardboard carton that could be used to package the cups for sale. Your supervisor has given you the following requirements: • • • •

All the faces of the carton are to be rectangular. The square base should be just large enough to accommodate a single stack of cups. The height of your carton should be given as a function of the number of cups it will hold. All measurements should be given in centimeters.

Since this is a new product for the team, you decide to study actual stacks of cups to help you discover which features of the cup affect the height of a stack. To do this, the team uses two types of cups found around the office. Complete the study and summarize your findings in a report to your supervisor. 1. Measure a stack of one, two, three, four, and five cups and record the data in the tables. In addition, measure the diameter of the top and bottom of the cups. Use the data to look for patterns that might help you determine a relationship between the height of the stack and the number of cups in the stack. Choose one of these types of cups for further analysis. Cup #1 # of Cups Height of Stack 1 2 3 4 5 Diameter of Top _______ Bottom _______

Cup #2 # of Cups Height of Stack 1 2 3 4 5 Diameter of Top _______ Bottom _______

2. Make a plot of the data from the stack your team has chosen to analyze. Don’t forget to include your scale and labels.

3. Predict, without measuring, the height of a stack of 16 cups _______, 50 cups ______ . 4. Define a function, h , such that h ( n ) gives the height of a stack, h , in terms of the number of cups in that stack, n . When your team has this result, call in your quality control inspector (this is your teacher) to verify your results.

5. As you are planning your report to your supervisor, your team realizes that it has not dealt with the dimensions of the base of the carton. What would the dimensions of the base be for the cup you are analyzing? ____________ What would the dimensions of the base be if the cup selected had as its largest base a circle of radius R? ____________ Several days after submitting your report to the supervisor, she sends word that it was extremely helpful. She also reports that the style and size of cup have been selected. The cup is of similar style to the one you described in your report, but the lower base will be a circle of diameter 7 cm, the upper base will be a circle with diameter 10 cm, and the height of one cup will be 13 cm. Based on the report your team wrote, she has determined the formula for the height of the carton will be h ( n ) = 0.5n + 12.5 . 6. If you decide to increase an existing stack by 2 cups, the height of the stack increases by a constant. What is the constant for 2 additional cups? What if you were to increase the stack by 20 cups? What about an increase of k cups?

2 cups ________

20 cups ________

k cups ________

Adapted from Navigations Series: Navigating through Algebra in Grades 6 – 8 (NCTM – 2004) and Pacesetter Mathematics with Meaning (ETS - 1993)

NAME________________________________________________________________ DATE________________________ Stacks of Cups Report

In the statement of the original problem, you were told that as package design director you needed to summarize your findings in a report to your supervisor. Usually you and she would meet to discuss the team’s findings, however, she is out of town at a meeting and you have to fax the report to her. You must describe your findings in more detail than usual since you will not be able to meet with her before she meets with the marketing director to discuss the project. Specifically, your report should contain the following information: •

The function your team proposes to use in determining the height of a carton that could be used to package the cups if they are of the same size and shape as one of the types of cups found around the office. (Be sure to include the meaning of any variables you use.)

•

A brief description of what the team did to arrive at the function above.

•

A more detailed explanation of how the specific numbers used in your function relate to the physical attributes of the cups. To help with this explanation, you should show her the function you would use for the other type of cup found around the office.

•

The minimum number of cups needed to come up with the function for any type of cup and why she would not need more than this number.

•

Any other information she might need to find the dimensions of the carton.

NAME________________________________________________________________ DATE________________________ Stacks of Cups Assessment (10 points)

Directions: Show all work and give clear explanations in order to receive full credit.

Type of Cup __________________________________________ 1.

2.

You have been given two cups of the same type.

a.

Find a function h that gives the height in cm of a stack of n of these cups. Explain how you derive your function. (4 points)

b.

Use your function to find the height of a stack of 17 of this type of cup. (2 points)

Adolfo says, “I have a stack of 11 cups of the same kind that is 21 cm high.” Bailey says, “I also have a stack of 11 cups of the same kind that is 21 cm high.” Carla says, “Aha! That means the two of you have the same kind of cup!” Is Carla right? Explain carefully why you think Carla is or is not right. Use examples and mathematical models (graphs and equations) in your explanation. (4 points)