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If you’re interested in implementing 3 Act Math or other open-ended problems, check out 5 Practices for Orchestrating Productive Mathematics Discussions. This is the 5th post detailing my experience with the strategies discussed in the book. Here are the lessons I’ve written about so far:

1. Domino Skyscraper (Exponential Functions)

“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)

I had three learning goals for the lesson. (1) Create a need for the exponential functions formula. (2)Understand that exponential functions have a multiplying rate of change. (3)Provide a context for the meaning of a and b. I chose Domino Skyscraper because it sets up really well for the learning goals, and the kids enjoy seeing the guy jump when the last domino falls to the ground. Bonus points right there.

However, I’ve done this lesson in the past, and it was difficult for my Algebra 1 kids because they don’t have a lot of tools to solve the problem. They mainly just know how to create a table and continuously multiply by the scale factor until the answer is achieved. Despite this, I decided to use the lesson because it creates “a headache” like Dan Meyer mentions, and the usefulness for the exponential functions formula is seen vividly.

“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 tried to predict different strategies and mistakes by working the problem out in ways I thought the students may choose. I made the following charts for organization.

As mentioned, the students have not learned other methods that can apply to this problem yet. Therefore, my anticipation phase centered not so much around the correct solution methods but instead the possible misconceptions.

“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 and mistakes in the anticipation stage, I actively looked for these during the group work session in order to take note of which groups were doing what. Here’s the blank monitoring chart I used.

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Also, here’s a sample page of monitoring notes from class.

The kids quickly realized that the problem was pretty frustrating because they only had a table as a tool to solve it. Also, initially several groups tried to create a linear equation based on the given information (y = 1.5x + 5). Other groups tried to divide the height of the skyscraper by 1.5. Overall, only one group figured out how to write a correct exponential function to match the situation.

Since there was a lot of struggle (most of it unproductive), I decided to stop the group work portion of the lesson early. It was now time to teach the formula and let the kids apply it to the problem. However, my main goal of creating a headache was achieved because the students truly saw a need for a more efficient method. Also, the mistakes were great because they can now be used to discuss why this situation is not linear (see the connection handout). This will hopefully solidify some conceptual understanding of what an exponential function is.

“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)

For the most part (except one class), I only chose one group to present the table method. In most classes, only one or two groups stuck with the method long enough to solve the problem so the selection process didn’t require much thought. Therefore, I just chose a speaker based on who hasn’t presented in the past.

In the one class that had a group figure out the equation, I had them present as well.

“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)

Not much to report here. One group presented the table method per class. In the class that had an equation group, I had the table presented first and equation second. I decided to do this based on a concrete to abstract progression.

“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.

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

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