Rate of Change and Covariation in the College Algebra and Precalculus Pathway

A critical foundation for reasoning about rates of change is conceiving of changes in quantities as quantities in their own right and distinguishing such changes from the original quantities. From this foundation, students may begin to understand, distinguish, and use the meanings of constant rate of change and average rate of change in various contexts and representations. Students in College Algebra and Precalculus should develop a robust ability to  articulate, distinguish, and use the meanings of constant and average rates of change. In particular, constant rate of change entails a proportional relationship between changes in the two quantities (e.g., see the conceptual analysis in Thompson, 1994). Reasoning about these changes and their proportional relationship across multiple representations can build an important foundation for further development of average and instantaneous rates.

Faculty at the MIP College Algebra and Precalculus Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Rate of Change and Covariation for success in the Oklahoma Functions and Modeling Math Pathway should:

1. Help students conceive of changes in quantities as meaningful quantities in their own right (e.g., see early tasks involving describing and reasoning about changes in quantities in Carlson, Oehrtman, & Moore, 2016).

2. Engage students in interpreting average rates of change as a constant rate for an auxiliary scenario with the same total changes in both quantities. These materials could reinforce and motivate the use of function notation in algebraic representations of average rates, developing the difference
quotient.

3. Informally introduce instantaneous rates through a context that necessitates finding average rates over progressively smaller intervals.

4. Unpack rate of change statements in terms of coordinating amounts of change. Such tasks may ask students to analyze amounts of change in the function for constant increments of the independent variable (e.g., see MA3 reasoning in Carlson et al., 2002).

5. Draw diagrams that represent changes in the output variable corresponding to successive increments in the input variable to help students conceptualize varying rates more robustly. Students should subsequently represent these relationships graphically and algebraically and interpret them in terms of rate of change in the problem context.

Addressing Components of Inquiry

Participants of the MIP Workshop on College Algebra and Precalculus suggested the following ways modules about Rate of Change and Covariation could address the three MIP components of mathematical inquiry:

Active Learning: Students in a College Algebra or Precalculus course will have significant experience applying rate of change tools in proceduralized ways. Thus, it is particularly 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.

Meaningful Applications: Students should engage in rates of change as a natural entry point to understand, represent, and explain, how quantities change in actual situations. Correspondingly, identifying and applying key rate of change characteristics of various function types can help reinforce 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, 33(5), 352–378.

Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179–234). Albany, NY: SUNY Press.