Earlier this year, I searched for some newspaper headlines that dealt with rates. Here are a few that I came across:

- “Colorado unemployment rate among 10 lowest in the country”
- “Unintended pregnancy rate in U.S. is high, but falling”
- “Oil ends sharply higher. Logs 10% weekly gain as output draws focus”

Many of the headlines talked about rates, and not just rates, but *varying* rates. When I think about these headlines, I wonder how students make sense of *varying* rates. For example, what might a student think it means for a rate to be affected, for a rate to be low, or for a rate to be high, but falling? Furthermore, what does it mean for something to end “sharply” higher? Are there other kinds of ways to end higher? For example, what might ending “gradually” higher be like?

In my research, I investigate how students make sense of change, and more specifically, I study how students make sense of *variation* in change. Put another way, I want to know how—from a student’s perspective—an “increase” (or “decrease”) can be a *thing that can vary*. When I interact with students, I work to design learning experiences that can provide them opportunities to investigate different kinds of increases (and decreases).

When it comes to rate, I have found that it matters *how* students **form and interpret relationships between quantities**.

If we want students to think about rate as something that is capable of varying, we should help students **coordinate** change in two different quantities, such as the height and volume of liquid in a filling bottle. Specifically, we should provide opportunities for students to think about one quantity as continuing to change while another quantity is changing along with it. For example, students could think about how the height of the liquid in a filling bottle continues to change while the volume of liquid in the filling bottle changes along with it.

I developed a framework that explains how students’ different ways of forming relationships between quantities can impact how they think about rate. I wrote about this framework in a 2015 article published in the journal Mathematical Thinking and Learning. The published version of the article is available here: http://www.tandfonline.com/doi/pdf/10.1080/10986065.2015.981946

Here is the full citation for the article (APA 6th)

Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities. *Mathematical Thinking and Learning, 17*(1), 64-90.

Here is the accepted manuscript of the article that you can download: