Average versus Instantaneous Rates of Change (Lesson 9.1)
Unit 9 - Day 2
Understand average rate of change as the slope over an interval and instantaneous rate of change as the slope at a point
Estimate instantaneous rate of change by calculating the average rate of change over a short interval
Quick Lesson Plan
Building on yesterday’s intro activity, students will discuss a Fitbit’s reporting of average pace and current pace. Students begin by exploring what these two measures mean and why they might differ. In question 2 and 3, students calculate Pamela’s speed from a Fitbit summary report and from a graph. The key take-away is that dividing the distance traveled by the elapsed time gives the miles Pamela traveled per minute, on average.
The fact that the graph is nonlinear demonstrates that Pamela’s speed is not constant. Students should be able to explain that a steeper slope represents a faster speed, since a lot of distance was covered in a short amount of time.
Question 7 is of course the big idea of the lesson, namely that by calculating the distance traveled over a very short interval of time, we can find Pamela’s current speed (i.e. the instantaneous rate of change). We are not yet using the formal limit notation, but the idea of ∆t going to 0 should send strong hints of what is to come!
Today’s margin notes focus on the ideas of average rate of change and instantaneous rate of change and how these are represented on a graph. We introduce the vocabulary of secant line and tangent line, which students may remember from Geometry. Remember that we are not yet calculating the instantaneous rate of change. Before introducing the limit definition of the derivative (lessons 9.2 and 9.3) we want students to have a firm grasp on how slope at a single point can be approximated by finding the rate of change over a very short interval of time.
Since we are only discussing forward motion, we chose to use the word “speed” instead of “velocity”. Students with a physics background were comfortable with the idea of velocity, but the goal of the lesson was to build the concept of a rate of change as a measure of “how fast” or “how quickly”, not necessarily to delve into motion and kinematics.