Sometimes, I learn more from my lessons than my students do. Of course, just as is true form my students, I have to be paying attention. This week, as we were reviewing proportional relationships, I noticed a few things that I didn’t expect.

**Setting the Stage – The Prequel**

First, this is a review. My students learned about proportional relationships last year and this was intended as a refresher leading into work with linear relationships a little later in the year.

This lesson was the second in a sequence of four on proportional relationships. The previous day, we had worked with proportional relationships in tables. It had gone about as I had expected. After a few minutes and a little bit of discussion, students recalled the criteria as being a constant rate of change and containing the point (0,0) (even if it is only implied). They had also recognized that one could get the y value by multiplying the x value by a constant. I was pretty happy.

**The Lesson**

Students were going to build on their experience from the previous lesson using three different representations. Each table group was given a set of two word problems. For each word problem, they were to create a table and a graph. Finally, they were to determine whether the relationship was proportional and explain why or why not.

Problem 1: Evie built a robot that can put ping pong balls in a cup. It took it three minutes to travel the course, put the ball in the cup and return to the start point. Evie’s robot was going to compete in a 12 minute race. Create a table and a graph showing the robot’s performance. Determine if the relationship is proportional.

Problem 2: Sofia was really excited about all the things she was seeing in Italy. She called Natalia to tell her all about the volcano she had visited. It cost her four dollars for the first minute and one dollar for every minute after that. Create a table and a graph showing the cost of the phone call.

I chose to give them two problems with a constant rate of change but with different y-intercepts because this seemed to be their sticking point in determining whether something was proportional. They were successfully determining whether the rate of change was constant but were sometimes hitting a wall on the y-intercept. Hence, I wanted to focus there.

As each group worked, they were to document their result for each problem on a poster. The groups would then do a gallery walk to see each other’s work. This is some of what they saw.

Clearly, there are some errors in thought in this work. However, the gallery walk served it’s purpose. Students noticed the differences in the different solutions and they started talking.

“Time is the independent variable. It should be in the left column.”

“What is it when you connect the points? or don’t connect them?” “Continuous or discrete.”

“Should the points be connected for the ping pong balls? You can’t have part of a ping pong ball.” “We were weren’t counting ping pong balls, we were counting laps and you can have part of a lap.” “Should it be time or should it be laps?”

The conversations were not about the question that I posed. Everyone got which graphs were proportional. They did so without relying on the table (which I had made part of the task as a scaffold in case anyone needed it and to also have them make connections across the two representations). They were important conversation reviewing the finer points of making graphs that I hadn’t expected to be necessary. They were though, and using student work to drive the conversation was the best thing that could have happened.

After a class debrief, students worked with table groups to complete a proportional relationship card sort. (You can get the card sort here.)

**What I Learned **

- Most of my students easily translated the word problems into graphs.
- Most of my students had a good grasp of proportional relationships in graphs. I should have added more layers of difficulty in my problem set. I focused primarily on the y-intercept as the tell-tale indicator of their mastery because I expected they would get the “constant rate of change” very readily in a graph. Perhaps I should have added a scenario with exponential growth and perhaps I should have added a scenario in which part of the scenario is linear and part of it is not.
- My students are rusty with regard to the finer points of making tables and graphs (where does the independent variable belong, what does it mean to be continuous or discrete, what exactly should the variables be). This proved to be an important lesson for my students, but not for the reasons that I expected. I’m thankful that I took the time to review a little bit rather than just rushing forward with new material, which is a somewhat natural inclination for me. Taking a little bit of time to review important ideas was time well spent.

**Coming Attractions **

Lesson three in this sequence will explore proportional relationships in equations. Lesson four will look at proportional relationships in all four representations.