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. 

Interview Reflection

This semester I have been observing in an 8th grade math class at Creekside Middle School in Zeeland Michigan. During my time observing I also conducted three student interviews. During these interviews I gave a student a math problem to work on, and my goal was to try and understand their thought process. While the students were working I would ask them questions to try and get them to explain their thinking. The purpose of this post is to reflect on the interview process and talk about what I would do different next time.

Before this assignment I had never done a student interview, so my first goal was to find a couple resources that would help me. I used NRICH to search for challenging problems to give the students I would be interviewing. One of my favorite problems that I found was called What's it Worth. Even though I did not end up using this problem for my interviews it was one that I seriously considered. Then I found a Student Interview article that gave a lot of great guidance on interviews including questions to ask the students, what to look for, and math problems to use.

I ended up giving all of my students the same interview question. Initially I was going to give them a choice of a couple problems but then I realized that I really wanted to be able to compare the thinking of the students I interviewed. I thought the best way to do this was by giving them the same problem. One of the problems that I found in the Student Interview article aligned with what the students had been doing in class. This is the problem that I ended up choosing:

A restaurant has square tables. Four people can sit at 1 table. Six people can sit at 2 tables. Eight people can sit at 3 tables. How many people can sit at 37 tables?

Before I started my interview I wanted to decide what I wanted to look for while the students were working on their problems. I made a list of things I would be happy to see in the students work.

My List of Awesome Mathematical Modeling:

-Table of Values

-Equation

-Picture/Diagram

-Recognizing a Pattern

I am not going to lie, I was very nervous to do these interviews at first. I was worried that I would not know what to ask the students while they were working on the problem. So again I went back to the Student Interview article and I found an AWESOME list of non-leading questions that I could ask during my interviews. 

My List of Awesome Non-Leading Interview Questions:

What do you predict will happen?

Can you solve it in a different way?

How did you figure it out?

Why did you______?

You wrote ______. How did that help you?

I noticed that you stopped what you were doing just now. What were you thinking?

I don't know what you mean by that, will you show me?

Will you draw a picture of that?

You started with _____ and then went to ____. Tell me your thinking.

Can you tell me what ______ means?

Are you right? How do you know?

Is there another way to show me? What is it?

Are there things that you already know that would help you solve this problem?

I was overall very pleased with the way that my interviews went. Two of the three students I interviewed used a table, a picture, and an equation to represent their work. All three of the students identified the pattern (increases by two) in the tables. While only one student had a correct final answer, one other student just made a mental math error that resulted in an incorrect solution. 

The most interesting interview was with my second student. This student did not use tables or an equation to model their thinking. They thought about the problem as adding "sets" of tables. During the interview I was confused why they took this approach, because I knew they had done problems similar to this in class. So then once the interview was almost finished and the student had given me their final solution, I asked them to draw what they thought the tables in this problem looked like. More specifically I asked what they thought "8 people can sit at 3 tables" meant. The student then drew 3 separate tables all with 8 people sitting around the table. Suddenly I realized why I was so confused! I had not stated in the problem that the tables were end to end, so this student thought that 3 tables meant 3 separate tables. If I would have asked this question sooner, I would have explained to the student that I meant for the tables to be end to end. Maybe by doing this the student would have solved the problem using a different method.  Here is the work of the three students that I interviewed.  

I learned a lot from conducting these interviews. I now have a couple resources that have great problems that I can use in my classroom someday, and a list of great  questions to ask students while they are working on these problems. The biggest take away I have from this assignment is learning the signs of when a student is struggling and when a student in confident in their work. I noticed that when a student did not feel comfortable with what they were doing they normally slowed down their work, looked to me for validation, or talked quietly to themselves. When one of the students was showing me one of these signs I tried to find a question on my list that could guide the student. Interviews like this could be very valuable to the teacher because by the time the interview is over the teacher will know exactly what their student knows and what they don't. The biggest drawback? Finding time in your plans to do interviews like this.