If you’re interested in implementing 3 Act Math or other open-ended problems, check out 5 Practices for Orchestrating Productive Mathematics Discussions. It’s a great resource for those, like me, who struggle when trying something new. This is the 3rd post detailing my experience with the strategies discussed in the book. Here are the lessons I’ve written about so far:

- Filling it Up! (Solving Linear Equations)
- Nail Polish (Solving Linear Inequalities)
- Elevator or Stairs? (Systems of Equations)

# Learning Goal

“To ensure that a discussion will be productive, teachers need to have clear learning goals for what they are trying to accomplish in the lesson, and they must select a task that has the potential to help students achieve those goals…The key is to specify a goal that clearly identifies what students are to know and understand about mathematics as a result of their engagement in a particular lesson.” (Smith & Stein, 2011)

My learning goal for this lesson was to **understand the meaning and purpose of finding the point of intersection in a system of equations and make connections between representations**. I chose Elevator or Stairs because it’s a dilemma I’ve been intrigued by in the past, it provides a context for systems of equations that everyone can relate to, and selfishly, I spent a lot of time making the task so why not use it?

By the way, check out Dr. Clayton Edward’s post about his implementation of this same lesson. He used it in a 6th grade class and provides amazing insight into student and teacher thinking. I also really like that it shows open-ended problems aren’t one size fits all. My learning goal was related to systems of equations, but others could use the task for many other concepts as well.

# Anticipating

“Anticipating students’ responses involves developing considered expectations about how students might mathematically interpret a problem, the array of strategies—both correct and incorrect—that they might use to tackle it, and how those strategies and interpretations might relate to the mathematical concepts, representations, procedures, and practices that the teacher would like his or her students to learn.” (Smith & Stein, 2011)

For the anticipation process, I had to anticipate student responses and mistakes for two different phases of the problem. Although I used this to introduce systems of equations, it does not immediately line up with that concept. Instead, the initial question, “Will the elevator or stairs be faster?” will require ratios and not necessarily force students into a systems situation. However, the sequel question, “At what floor would both the elevator and stairs take the same amount of time?” is perfect for systems, and it is the question that will drive the rest of my lesson (and much of the systems unit).

Therefore, I started by creating the following chart for the first question and worked the problem out in ways that I thought the students might approach it.

After anticipating solutions and mistakes for the first question, “will the elevator or stairs be faster (for 3 floors)?” I used the same process to predict responses for the sequel question, “at what floor will both take the same amount of time?”

# Monitoring

“One way to facilitate the monitoring process is for the teacher, before beginning the lesson, to create a list of solutions that he or she anticipates that students will produce and that will help in accomplishing his or her mathematical goals for the lesson.” (Smith & Stein, 2011)

After creating the list of possible solutions in the anticipation stage, I actively looked for these strategies during the group work session in order to take note of which groups were using what strategies. Since systems of equations was my focus for the overall lesson, I narrowed my focus specifically toward the strategies used to solve the sequel question. Here’s the monitoring chart.

As the groups worked, I noticed that most were using tables to try to figure out where the point of intersection was. In addition, a few students wrote equations and used a calculator to see what the graph looked like. Finally, a couple students used what I thought was the coolest method of the day (which I didn’t anticipate). They set the equations equal to each other and solved for x.

I was really excited about this because they unknowingly used a method that we would be learning a few days later.

# Selecting

“The selection of particular students and their solutions is guided by the mathematical goal for the lesson and the teacher’s assessment of how each contribution will contribute to that goal. Thus, the teacher selects certain students to present because of the mathematics in their responses.” (Smith & Stein, 2011)

Since most of the groups used the table method, I had one group present this method from each class. I also selected a group that used equations. If there were multiple equation groups, then I selected the group who set the equations equal to each other because I really wanted to show off that strategy.

Also, I chose speakers based on who has and hasn’t presented in the past. I want to make sure as many students as possible get a chance to be the center of attention throughout the year.

# Sequencing

“Having selected particular students to present, the teacher can then make decisions regarding how to sequence the student presentations. By making purposeful choices about the order in which students’ work is shared, teachers can maximize the chances of achieving their mathematical goals for the discussion.” (Smith & Stein, 2011)

For the sequencing process, I decided to have a group present the most commonly used strategy first. This is a recommendation from the book that I like because it allows the highest percentage of students to have investment in the presentation process. Since most of the groups used the table method, they would be immediately curious to see how the first presentation compared to their work.

After a group presented the table method, I had a group that used equations present next. Specifically, I wanted to highlight the few groups who set the equations equal to each other in order to arrive at a solution.

# Connecting

“Rather than having mathematical discussions consist of separate presentations of different ways to solve a particular problem, the goal is to have student presentations build on one another to develop powerful mathematical ideas.” (Smith & Stein, 2011)

After the presentations were complete, I led the students through a connection handout. These handouts are my preferred way to ensure that the learning goal is met. The handout starts by highlighting key methods at the top. From there, leading questions are asked and big ideas for the unit (I stole them from New Visions for Public Schools) are mentioned in order to make connections and prepare students for developing procedural fluency in the coming days.

# Materials:

Here are the links to all of the materials mentioned in this post and used during the lesson.

Smith, M., & Stein, M. (2011). *5 Practices for Orchestrating Productive Mathematics Discussions*. Reston, VA: National Council of Teachers of Mathematics

A mathematical thought:

We could slow down the stairs by introducing one more stair.

In that case, we have:

5 stairs / 1.5 seconds = 65 stairs / x seconds.

Multiplying the left hand side by 13/13 gives:

65 stairs / 19.5 seconds.

[The denominator calculated as:

1.5×13 = (1 + 0.5)13 = 13 + 6.5 = 19.5.]

On the other hand, we could speed up the elevator.

Say, 4.5 ft / 3 seconds, which is 1.5 ft / 1 second.

Multiplying the latter rate by 20/20 gives:

30 ft / 20 seconds.

Summarizing:

Slowing down the stair-choice gave 19.5 seconds, and speeding up the elevator-choice gave 20 seconds. If the slower-stairs are quicker than the faster-elevator, then the true stairs are also quicker than the true elevator.

Perhaps not a method of solution likely to arise with young students, but, I think, not a bad idea [around estimation] to discuss with teachers.

This is awesome, Maya! Another reason to really like open-ended tasks. This is a strategy I would never have thought of. Really cool! Thank you for sharing!