Rates of Change and Graph Behavior (Lesson 1.3)
Unit 1 - Day 3
Calculate and interpret an average rate of change over an interval using proper units
Identify the intervals on which a function is increasing, decreasing, or constant
Understand that the transition from increasing to decreasing results in a local maximum and the transition from decreasing to increasing results in a local minimum.
Find and interpret the zeros of a function in context
Quick Lesson Plan
This activity has students thinking critically about how prices are set in the food industry. They compare two supermarket purchases (baby formula and soft drinks) and consider how the demand for each product changes as the price changes. Students should notice from the slopes of both graphs that customers who buy soft drinks are more likely to give up this purchase when the price increases, but people who buy baby formula, while decreasing their consumption slightly, can not just give up buying formula for their children. There are many ties to economics here. Consider consulting your school’s econ teacher to make cross-curricular connections to price elasticity.
Although the math content on the first page is not new for students, it does help them establish the thinking skills and habits of mind that are needed to be critical consumers in today’s society. Throughout the activity, students consider the fairness of using revenue as the sole or primary factor in setting prices. Be ready for some debate as students might have strong opinions about this.
The second page of the activity has students thinking about graph behavior, specifically intervals of increasing/decreasing, maxima, and average rate of change. We want students to connect the idea of optimal with an extrema as this is an important tie in to AP Calculus once they learn about optimization. The idea of not yet having reached optimal conditions until we can no longer squeeze out any more revenue is an important one. Informally, we want students to arrive at the idea that they will reach optimal conditions once revenue has stopped increasing and is just about to start decreasing. Note that students may be using some estimation to find the maximum revenue and this is okay.
The work done in this lesson is great preparation for AP Calculus. At the end of the year we will revisit this lesson with the perspective of derivatives. Average rate of change is a key concept in students’ mathematical trajectory. Students are used to finding slopes of lines; in Precalculus we extend this idea to finding the slope of a secant line, which gives the change in the y-variable per unit change in the x-variable, on average. Repeatedly ask students whether the value they calculate is the actual change in revenue each time the price increases by $1. We want students to articulate that no, it is just the average increase, and sometimes the revenue changes more than that amount and sometimes it changes less.
A note about intervals of increasing/decreasing: Technically when there is a horizontal tangent line (at a max/min), the function is neither increasing nor decreasing. For this reason, we tend to use open intervals when discussing intervals of increasing/decreasing. Nevertheless, the AP Exam readers will not mark students down if they use brackets to include the endpoints. Similarly, for functions like the cubic function, it is okay for students to say the function is increasing on the interval from negative infinity to infinity, even though it is technically neither increasing nor decreasing at the instant when x=0.
A final note to emphasize to students is that intervals of increasing/decreasing are talking about intervals of the domain. Students should not be giving the y-values from lowest to highest. Conversely, when talking about a maximum or minimum value, the y-value is needed. Discourage students from giving the location of the extreme value (x-value) or even giving an ordered pair (we consider this students “covering their bases” and it’s hard to tell whether they truly understand that an extreme value is an output).