I like big cups, I cannot lie.

We stacked cups in the first few days of school…

I’ve been stacking cups since…uh. I think my first NCTM Navigating Through… book was around 2002 or so. Its been a while. I have vivid memories of discussions in classes from room 125. Yep. It’s been while. Long before there were songs about Solo cups. My guess, a few of my sets of cups may be that old.

They’re a cheap resource. Find a buddy or two, each buy some different sizes, split them up and you’ve got some varied sets of cups. Hmmmm. What all can you do with cups?

I. This past week, I began by displaying a single cup and asking students to generate as many questions as they can about said cup. Set the timer.

II. Turn to your groups and share your questions. Then say whether it was mathematical in nature or not. Each group shares out 1 question with the whole class. Then if anyone had a question they wanted to share that had not been included.

Yes, we actually looked at the etymology of cup…wondering where the name originated.

III. a. I went with “Why am I stacking cups?” as my transition to the task. You guys are engineers today. Packaging designers, specifically. Design a box to ship a stack of 50 cups. They needed tools, so I gave each group 4 – 7 cups (did I mention some of these cups may actually be older than some students?), each group with a different size/brand of cup and a measuring device. Set the timer 5-7 minutes depending on class.

III b. As I monitor their work, I usually here a few moving in the wrong direction. I pause the timer and their discussions…attention at the board:

I need some help. One group has a stack of 5 cups measuring 14 cm, and their height for a stack of 50 cups would be 140 cm. Do you agree or disagree with their response? Turn to your group and discuss. Set the timer.

I have some varied responses usually. When I get to someone who disagrees, I ask how tall they think the box should be and they come to the board to explain their reasoning.

III. c. Yes, believe. You will sometimes have a class where no one disagrees with the 140 cm response. Have them to create a table of values to record their measures for 1 cup, 2 cups, 3 cups, etc. Set timer. Usually during this time you will hear the a-ha’s. Bring the class back together to discuss / share their thinking. Modeling how the cups would be stacked.

Okay, so moving on now.

IV. Once we feel fairly confident in our expressions. I ask them to find the height of a stack of ____ cups for their group.

V. Well, what if I had a box that was 80 cm tall, what is the largest amount of cups could I ship in that box?

VI. At that point, we share our expressions we’ve created for each type of cup. I put all cups on display and ask groups if they can match the cup with its expression for total height (cm).

This leads to some light bulb moments for a few students. They can now see how different parts of the expression represents different physical parts of the cup. I always thought it would be fun to list the expressions on cards and they have to match to the cups and play the Race Game from The Price is Right.

VII. For other practice, we use the expressions:

- simplify expression
- find the total height of 50 cups
- how many cups to make a stack of 80 cm?

VIII. Closer choices

- What’s one take-a-way from today’s task?
- Something I learned… realized… or was reminded of…
- How are the expressions alike? different?
- Which two expressions are most alike? Explain. Which two are most different? Explain.

IX. Systems

Next, have students compare their cup stack to another groups stack of cups. When will the two stacks be equal heights? Just using my groups’ expressions above, they get at least 6 practice problems. You can leave it as an open task – students can choose tables of values, creating equations to solve or even solve graphically. The key component is to ensure they interpret their solutions (x, y) = (cups, stack height) within the context of the scenario.