5 Practices: Two Trains

5 Practices

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 6th post detailing my experience with the strategies discussed in the book. Here are the lessons I’ve written about so far:

  1. Filling it Up! (Solving Linear Equations)
  2. Nail Polish (Solving Linear Inequalities)
  3. Elevator or Stairs? (Systems of Equations)
  4. Stacking Cups (Linear Equations)
  5. Domino Skyscraper (Exponential Functions)
  6. Two Trains (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)

I had three learning goals for the lesson. (1) Re-Introduce systems of equations. (2)Be able to recognize when and why problems represent a system of equations. (3)Provide a context for the meaning of the point of intersection. I chose Two Trains because it sets up really well for the learning goals, and I’ve been intrigued by the task ever since Jon Orr published it. It just looks fun, so I wanted to use it!


“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.

5 Practices Two Trains Possible Solutions

5 Practices Two Trains Possible Mistakes


“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.

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

5 Practices Two Trains Filled In Monitoring Chart
While monitoring, I noticed a frequently occurring mistake that I didn’t anticipate. A lot of groups were finding the rate of change for the purple dot by dividing 77.3 cm by 3.87 seconds. They were forgetting to subtract 77.3 from 100 in order to get the correct distance for the 3.87 seconds.

Also, I was impressed by a visual / diagram that several groups were using (I stole Jon’s picture for the diagram).

Two Trains Diagram.png


“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)

During the monitoring phase, I noticed that most groups were creating two equations and then using a calculator to find the point of intersection. A little more than half of each class used this method. For the other groups who didn’t use this method, most used a version of the diagram shown above or some sort of list / table to estimate the intersection. Therefore, I chose these two methods for presentations when available. Also, in order to prevent dominance, I tried to choose students to present who aren’t common speakers in the class.


“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)

Since the equation method was the most common, I decided to have it presented first. This allowed the largest percentage of students to be immediately invested in the presentation process. Next, I had the table / diagram method presented in order to show a different perspective and also show some cool visuals.


“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 I also put sample state test questions in order to compare how they relate to the task.


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

2 thoughts on “5 Practices: Two Trains

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.