Rate of Change in the Functions and Modeling Pathway

A rate of change is a measure of how much one quantity changes with respect to another. Rates of Change are an integral piece of understanding the nature of a function relationship between two quantities. Understanding rate of change provides students with tools with which they can analyze and make inferences about function behavior and hence gain insight into the real-life phenomena modeled by those functions. Rates of change that explicitly appear in Functions and Modeling include constant rate of change, average rate of change, and percentage change, but rates of change provide a means for developing quantitative understandings of function concepts as well – such as limiting value, maximum/minimum, concavity, and characterizations of function classes – suggesting that emphasizing rate of change is a primary goal throughout the entire Functions and Modeling course.

Faculty at the MIP Functions and Modeling Initiation Workshop in June, 2019 identified the following aspects of Rate of Change as critical for success in the Oklahoma Functions and Modeling Math Pathway.

Quantitative Reasoning: Activities should develop a quantitative understanding of rate of change (e.g. Thompson, 2011). Students should understand rate of change in a way that enables them to clearly articulate (1) what object is being measured, (2) what attribute of that object is being measured, and (3) what the unit of measurement is. Viewing changes in quantities as distinct quantities is key to developing a quantitative understanding of rate of change. Understanding function concepts in terms of amount of change is productive in itself and serves as a productive precursor to developing notions of rate of change – see Carlson et al.’s (2002) covariation framework for other examples of mental actions and imagery involving changes in quantities.

Tool for Function Relationships: Activities should establish rate of change as a conceptual tool to understand and analyze function relationships. There are several benefits to viewing rate of change as a unifying conceptual thread beyond just rates of linear functions. Focusing on quantitative meanings for function concepts can involve characterizations of the relevant properties in terms of rates of change (instead of relying solely on visual manifestations of the property – for examples, see Moore & Thompson, 2015). Students should conceive linear functions as those with a constant rate of change instead of only nonquantitative imagery such as ‘those that look like a straight line.’ Rate of change also allows students to develop a meaningful understanding of concavity (e.g. in terms of an increasing/decreasing average rate of change) instead of ‘up like a cup, down like a frown’, or characterizing limiting behavior in terms of the average rate of change (e.g. when a function approaches a limiting value, the average rate of change tends to 0).

Multiple Representations: Activities should develop the ability to reason flexibly about rates of change from each function representation, including formulas, graphs, tables, and context (see Oehrtman, Carlson, & Thompson, 2008). This is particularly useful for supplementing the perception-based understandings of function concepts (e.g. Moore & Thompson, 2015) that students hold with quantitative ones. For example, concavity might be visually evident in a graph based upon its shape, but prompting students to determine concavity for a function given in table form, or asking what attribute of a function relationship underpins the familiar “cup” shape encourages them to devise and rely on characterizations involving changes in quantities and rates of change. Recognizing and reflecting on the common structure(s) shared by amounts of change and rates of change across representations can also imbue otherwise procedural formulas and diagrams with meaning and anticipate characterizations of various function classes.

Technology: Activities should use technology as a tool in advancing students’ understanding of rate of change. Technology can be used to simplify the usually cumbersome task of generating function representations by hand, making it reasonable to prompt students to examine multiple representations frequently. Such use of technology can also aid visualization of rate of change. A productive conception of average rate of change involves considering the constant rate of change that would result in the same total change in output over the same input interval (see Musgrave and Carlson’s (2016) conceptual analysis of average rate of change). Students may thus be asked to imagine the ‘hypothetical’ linear function with the aforementioned constant rate over the interval in question.

Addressing Components of Inquiry

Participants of the MIP Workshop on Functions and Modeling suggested the following ways modules about Rate of Change could address the three MIP components of mathematical inquiry:

Active Learning: Students in a Functions and Modeling course may have significant experience applying rate of change in proceduralized ways. Thus, it is important that modules engage students in tasks that challenge these rote applications and require them to explore the underlying meanings, especially in terms of relationships between amounts of change. In particular, students should analyze amounts of change in ways that develops the approach as an analytic tool.

Meaningful Applications: Students should engage in rates of change as a natural entry point to understand, represent, and explain, how quantities covary in actual situations. Identifying and applying rate of change characteristics of various function types can help reinforce a broader understanding of these functions and their value in appropriate modeling scenarios. Varying the contexts promotes students’ development of a generalized concept of rate of change that is not bound to any single situation or representation.

Academic Success Skills: Exploring rate of change in-depth and in meaningful applications can help students reinforce their identity as STEM students. Modules could also attend to reinforcing a growth mindset and persistence by providing scaffolding that keeps students engaged without preempting their ability to develop significant solutions on their own.

References

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 352-378.

Moore, K. C., & Thompson, P. W. (2015). Shape thinking and students’ graphing activity. In Proceedings of the 18th meeting of the MAA special interest group on research in undergraduate mathematics education (pp. 782-789). Pittsburgh, PA.

Musgrave, S., & Carlson, M. (2016). Transforming graduate students’ meanings for average rate of change. In Proceedings of the 19th meeting of the MAA special interest group on research in undergraduate mathematics education. Pittsburgh, PA.

Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. Making the connection: Research and teaching in undergraduate mathematics education, 27, 42.

Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education. WISDOMe Mongraphs (Vol. 1, pp. 33- 57). Laramie, WY: University of Wyoming.

Additional Resources

Carlson, M., Oehrtman, M., & Moore, K. (2010). Precalculus: Pathways to calculus: A problem solving approach. Rational Reasoning.

Crauder, B., Evans, B., & Noell, A. (2013). Functions and change: A modeling approach to college algebra. Cengage Publishing.