Interpret the meaning of area under a rate of change function in contex
I can interpret the areas underneath the graph of a rate of change function as accumulated quantities.
I can use units to determine the relationship between rates of change and accumulated quantities.
Quick Lesson Plan
This lesson was engaging from the start --- we recently returned from an extended holiday/semester break and our students were already hoping for a “snow day!” A familiar context and constant rate functions provide an easy entry to the objective of Topic 6.1: relating the area between the graph of a rate of change function and the x-axis with accumulated quantities. Groups should readily make the connection between the graphical, analytical, and numerical representations of snow accumulation (both positive and negative!).
Topic 6.1 launches the study of the techniques and applications of integrations. This is a major conceptual shift from student work in Units 1 through 5. Today’s work is meant to redirect student thinking from describing the behavior of a function to investigating the area beneath its graph.
After groups complete the activity, take time during the debrief to emphasize the importance of correct units in computations and on solutions. Develop, if necessary, the concept that units on the accumulated quantity (area under the curve) are the product of rate of change units multiplied by the units of the independent variable. Allow adequate time for groups to discuss the Check Your Understanding problems as this section not only reviews previous concepts, but provides practice for Topic 6.1.
Topic 6.1 is intended to expose students to the concept of accumulation and its relationship to the area beneath a rate of change curve. Questions on the AP Test require students to take this foundational understanding of integration and apply the concepts to much more complex and advanced problems. The context for today’s activity is found in 2010 AB # 1 which illustrates how accumulation is accessed on the AP Test.
Knowing area formulas for familiar geometric regions will be valuable going forward. Encourage students to review and memorize area formulas for circles, rectangles, triangles, and trapezoids!