Introduction to Derivatives (Intro to Unit 9)
Unit 9 - Day 1
Explore rates of change in context
Estimate and compare rates of change from a graph
Interpret and calculate average rate of change
Quick Lesson Plan
After having students discuss the results of the Unit 8 test, we kicked off Unit 9 with a lab.
Today’s activity introduces much of the important ideas of this unit in a relevant context related to international COVID data. The goal of today’s activity is simply to get students to explore rates of change and how they relate to the slope of a function. Today’s intro activity and tomorrow’s lesson essentially cover the same learning targets, so if you are short on time you may choose to omit one of the days. We did both lessons because we wanted students to have a firm grasp on rates of change and slope before delving into the mechanics of calculating instantaneous rate of change using the limit definition of the derivative (Lesson 9.2).
The link given will take you to the worldometer site where students can get information about the number of COVID cases from a vast variety of countries. To avoid having multiple groups choose the US, we ask students to choose a country outside of the US.
To answer question 5, many students scrolled down on the site to see the “New Daily Cases” chart. While this was not necessarily the intent of the question, it brought up early conversations about the connections between f and f’. Ask your students how the site was able to calculate the “New Daily Cases” data and students should be able to say that they compared the number of cases today from the previous day. This is essentially finding a slope where the time interval is one day! To extend this thinking, ask students what kind of data would be needed to be able to find the rate of infection at a particular hour on that day. Push this even further to ask how they would calculate the rate of infection at exactly 3 PM. Students should notice that they would need data from two time points very close together, essentially building the concept of a derivative.
The ideas introduced in this activity range from average rate of change and slope all the way to concavity and inflection points. It is not necessary that students master these concepts or even discuss them formally. We simply want students to be able to analyze a function’s growth rate based on a graph and to interpret these values in context.
Note that students may suggest taking the daily rate and dividing by 24 to get the rate at a particular hour. This is a reasonable suggestion but it’s important to note that a “scaled” version of the rate does not actually add any more precision about the rate at a single hour or moment in time. Dividing by 24 simply takes the average daily rate and converts this to an average hourly rate.
The “How do you know?” in question 4 can prompt interesting discussion. Students should recognize that the slope of the cases graph is steepest (large increase in COVID cases in a short amount of time). Ask students what the graph of the Daily Cases would look like on this particular day. Students did a great job recognizing that a steep slope on the original graph represented a spike on the daily cases graph (an early nod to inflection points!).