Stacking Cups Activity (Systems of Equations)

This winter I observed a lesson in an 8th grade math class that used a cup stacking activity to introduce systems of equations. I really like the activity and I thought it provided the students with some rich mathematical exploration. This activity was something that I was very interested in using in my own classroom someday. I made a few changes and decided I would share!

I started planning this lesson with one thought: what do I want my students to be able to do once this activity is done. With that in mind I went to search for the CCSS and Mathematical Practices that aligned with this activity.

Common Core State Standards:
8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to the points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Mathematical Practices:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
8. Look for and express regularity in repeated reasoning

I also had some questions that motivated me while working on this activity. Here they are:

• What does understanding look like for this?
• How does this activity help with understanding?
• What will students do?
• How will I check for understanding?
• How will I have students consolidate?

This activity is meant to be an introduction to systems of equations, or an activity to do right after the topic is introduced. So I decided that if a student had understanding of these topics they would be able to write equations that represent real world scenarios, they would be able to graph these equations and find their solution, and they would be able to explain to me what this solution represents. This activity will help with their understanding because it will walk them through writing equations, graphing them, and interpreting their solutions. As far as checking for understanding, I have created a worksheet for the students to do after this activity that I might collect and check. However, this would all depend on my classroom structure at the time. To have my students consolidate I have given room at the end of the activity for a reflection.

Now for the lesson. I have broken it up into four parts: the warm-up, the introduction, the investigation, and the reflection. The warm-up consists of two questions that will brush students up on some skills that will be needed for the investigation. The introduction includes a 101qs video and a group discussion. The investigation is the Cup Stacking Activity. Finally the reflection includes another group discussion and a written reflection.

Warm-Up

For this warm-up I would either put these problems on a projector and have students write their solutions down or else I would pass out this warm-up on a paper. After it seems like most students are done I will pull the class together and we will discuss the warm-up.

Graphing and Writing Equations Warm-Up

Introduction

The purpose of this introduction to to get the students ready to complete the activity. First I would have the students watch Andrew Stadel's 101 questions video called Stacking Cups. The purpose of this is to get the students interested in finding a solution to this activity. 

I would have the students watch this video two times and then I would have a group discussion. I would ask my students what they Notice and what they Wonder about the video. I would record these answers on the board. Then I would have the students watch the video one more time. I would tell them that their task is "How many cups does it take for the stacks to be equal in height?". At this point I would introduce our activity, distribute materials (yardstick, two stacks of different cups, and the activity). 

Before I started I also wanted to anticipate some student strategies. The first thing that I anticipated was that students would assume that the height of two cups is twice the height of only one cup. If I saw any students using this strategy I would remind them to remeasure their stacks of cups and look and see if the height is really doubling. I also anticipated students not knowing where this change in height was coming from, that is why in the first problem of the activity I have the students label the different measurements of the cups. Hopefully by doing this the students can see that the height is changing each time by the height of the lip of the cup. These were some things specifically that I would be looking for throughout the activity. 

Investigation

The students should complete this activity in groups of 2-4 students. I will be walking around the room while the students are working so I can help them if they are struggling. I want a healthy bit of frustration to happen while the students are working. I think that this frustration is where awesome learning happens. 

Cup Stacking Activity 

Reflection

The reflection is the most important part of this lesson. With 15 minutes left in the hour (or once everyone is done with the activity) I would bring the whole class back together and we would discuss the activity. I would reference the Notice and Wonder that we did earlier, and see how many of the "wonders" we answered. I would also share some of the students work and we would share our answers. At this point I would show the video Stacking Cups - Act 3. Finally I would have my students complete the reflection at the bottom of their activity if they haven't done so already. I may or may not collect these depending on how well I think the activity went. 

I also created a follow up worksheet for this activity. This activity uses a lot of the same concepts used in the Cup Stacking Activity. I might have the students do this activity after they have been working on systems for a day or two. 

Follow Up Worksheet 

Variations of the Activity 

Noelani Davis- This is a variation on the cup stacking activity that uses the same 101 questions video. Noelani has a cool idea using Hint Cards with her students too! If her students are struggling she gives them a Hint Card to help them. I also adapted my follow up worksheet from Noelani's. 

Tara Maynard- Tara's class was where I first saw this lesson, her student teacher did this activity with her 8th grade students. Tara's activity has a really cool Part 2 that you should check out!

Dan Meyer- Dan has a variation that has students estimate their teachers height in cups!

Andrew Stadel- Here is the link again for the Stacking Cups video this activity is based on.