# Which Bicycle Shop Should I Use –A Lesson Introducing Proportional Relationships in Tables, Graphs, and Equations

“I’m thinking about purchasing a custom-made bicycle. Bicycle City charges \$160 plus \$80 per day that it takes to make the bicycle. Bike Town charges \$120 per day to make the bicycle. For what number of days will the charge be the same at both bicycle shops?”

I introduced the lesson with this question (taken from the NY Institute for Learning). We began with our usual process of making sense of the problem (What do you notice? What do you wonder? What is the question? What do you know?). Then, students set to work on the task individually.

Once everyone had solved the problem, I had three different students present their solutions (one who had used a table, one who had used a graph, and one who had used equations) to the class using the document camera. This gave us the chance to talk about how you find the solution using different solution paths.

At that point, I told students that a proportional relationship was one in which you could multiply the x value by a constant and get the y value. I asked them to figure out whether either bike shop’s pricing was proportional and how they saw that in their particular solution. This gave us the opportunity to discuss how they see the constant rate of change in each representation. It also gave students the chance to discover that a proportional relationship goes through the origin for themselves.

We wrapped up the lesson by completing a foldable summarizing the characteristics of a proportional relationship and how to see them in a table, graph, and equation. You can download the foldable by clicking on the link.

Proportional Relationships

The lesson is designed to give a whole-part-whole understanding of the concept. This is particularly important with students who are at the extremes on the conceptual/sequential dimension of cognitive style. I have several twice-exceptional students who are at one end of this spectrum or the other. I struggled with how to help them succeed for the entire first semester. I started building in the use of a whole-part-whole structure and the use of graphic organizers for note-taking more after reading Twice-Exceptional Gifted Children by Beverly Trail, EdD. It has had a pretty big impact for these students (test scores moved from the 50-60% range to the 98% range on the last test). I don’t know if that will continue to hold, but with that kind of improvement, I am going to keep incorporating this technique and find out.

The way that the foldable is designed, students can compare and contrast how one sees a proportional relationship in a graph, table, and equation by looking across the foldable.   The can also compare and contrast an example of relationship that is proportional and a relationship that is not proportional by looking down a column.   Marzano’s research shows that the use of comparing and contrasting can result in significant gains in student learning.

I also incorporated the use of color in the completion of the foldable to highlight the important points. This was not as intentional as it should have been. Early in the day, we worked through the foldable summarizing the ideas in the lesson. As the day progressed, I thought it would be a good idea to have kids highlight the key points in each representation (constant rate of change and going through the origin). By fourth period, I finally refined it so that kids used one color to highlight the rate of change and a different color to highlight that the relation goes through the origin. This seemed to really bring out the idea across representations, so I will do it this way from the start the next time I teach this lesson.