In the Cups

Our department has a shared e-folder where we can share tasks we develop for use in specific courses. I've used it a lot this summer to support my planning for the intermediate algebra course I'm teaching. The activities there range from routine to inspiring--and both types have been useful!--which reminds of a quote I saw recently: Good teachers borrow, great teachers steal.

So I have been drawing heavily on the excellent activities there. Some tasks I use almost verbatim, and others I modify to suit my own purposes. One of the activities I found was a nice math modeling activity called "In the Cups". It is designed to support students with creating and using linear models to make predictions and solve problems.

The original document presents the following data to students:

 

Number of Cups	Height of stack (in cm)
1	7
2	7.6
3	8.2
4	8.7
5	9.2
7	10.1
8	10.6

The task proceeds to ask students to predict the height of 6-cup, 10-cup, and 25-cup stacks, and eventually invites students to find a model expressing the stack height as a function of the number of cups in the stack.

I noticed that the data in the table do not quite conform to a perfectly linear relationship--notice that the unit rate of change varies from 0.6cm to 0.5cm per cup--which got me thinking. I wondered if the author had introduced some random variation into the stack heights presented in the table in attempt to make the task seem more authentic. Or perhaps he or she had really measured some cups and the data were authentic approximations. Either way, I liked it.

I started measuring a stack of dixie cups I had in my office, then I realized that by doing the measuring myself, I was about to rob my students them of an opportunity to do some critical thinking. So I decided to replace the table of values with a photograph. 

That one picture (of 27 cups) contains all of the information needed to model the relationship, and it invites students to consider questions about the accuracy of the approximation. Replacing the table with the photograph felt like a more authentic way to present the task to my students.