Understand that a linear function has a constant rate of change over any interval of its domain.
Explain why the rate of change of the average rates of change of a linear function is zero.
Interpret the slope of a linear function in terms of a rate of change.
Quick Lesson Plan
If you think a lesson on linear functions belongs in Algebra 1, wait until you see students wrestle with “How Much Does it Cost to Rent a U-Haul?”! In today’s activity, students look at how the price of renting a U-Haul changes as the number of miles driven on it changes. The context was chosen specifically so that the constant rate of change is not readily apparent from the table, though students may surmise it from the context if they have familiarity with renting moving trucks.
Instead of focusing on slope and y-intercept, students calculate the average rates of change over different intervals of the function’s domain. Are mileage increases charged equally for low-mileage customers than high-mileage customers? How do we know?
In questions 2 and 3, students connect a linear relationship with having a constant slope/rate of change AND with having a rate of change that has a rate of change of zero (since the rate of change is neither increasing nor decreasing)!
Questions 4 and 5 are a good opportunity for students to review writing linear functions and thinking about the base price conceptually rather than just reading a y-intercept from a graph. Be looking for any students that may have written their equation in question 5 in point slope form, and make sure to emphasize in the debrief that there is more than one way to write the equation of a linear function.
What does average rate of change mean in this context?
What do you think the average rate of change between m=10 and m=50 would be?
What values in the table could we use to check if there is a constant rate per mile?
How can you use the graph to figure out question 3? How can you use the table?
Could you have written this equation without knowing the base price?
The final two lessons in this unit introduce two function types (linear and quadratic) through the lens of their rate of change. While linear and quadratic functions are a staple of Algebra 1 and 2 courses, in AP Precalculus these functions are studied from a different perspective: the rate of change of a linear functions’ average rates of change is 0 whereas the rate of change of a quadratic functions’ average rates of change is constant.
Make sure that the debrief focuses on how the average rates of change of the cost are changing (or not changing as the case may be), rather than how the cost is changing. The focus here is not on constant first differences, though linear functions do exhibit this. The focus is on the rate of change!
As you debrief the graph, it is helpful to use the colors you used in Lesson 1.4 when differentiating between instantaneous rate of change (we used purple tangent lines) and average rate of change (we used a green secant line). The fact that both of these lines have the same slope is of course at the heart of a linear function. This is also a good place to point out that linear functions have no concavity–the rate of change is not increasing (as in the case of a concave up function) or decreasing (as in the case of a concave down function).
Question 2 of the Check Your Understanding further drives home the important ideas about a linear function’s rate of change as well as the connection between the change over an interval and the average rate of change over an interval. We suggest taking time to debrief parts a-c and having groups share out how they were thinking about these questions and what things they noticed.