Rates of Change (Lesson 1.4)
Unit 1 Day 6
CED Topic(s): 1.2
Find and interpret a function's average rate of change over an interval.
Estimate and interpret a function's rate of change at a point.
Compare rates of change at different intervals or values of a function's domain.
Quick Lesson Plan
As we have already seen in the first three lessons, covariation and rate of change continue to be big topics in the course (and will be helpful throughlines for studying various function types). Today we talk about how to calculate and interpret average rate of change and how to conceptualize instantaneous rate of change (which makes this Calculus teacher very happy!)
In today’s 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 6 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 do not use formal limit notation, but the idea of ∆t going to 0 is a nice nod to future learning!
Do any of you run with a Fitbit tracker (or smartwatch)? Have you ever seen these displays before?
Why is Pamela’s average pace not the same as her average speed?
Did Pamela run 0.107 miles per minute the whole time? Do you think she ever ran at this rate?
(Looking at graph) When did Pamela run faster than her average speed? When did Pamela run slower than her average speed? How do you know?
How would we find Pamela’s average speed on the first half of her run?
How would we find Pamela’s average speed on the first 5 minutes of her run?
Which of these measures do you think is the best estimate of how fast she was running 2 minutes into her run?
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 actually calculating the instantaneous rate of change. The goal is for students to understand how slope at a single point can be approximated by finding the rate of change over a very short interval of time.
Since Pamela’s distance traveled is a strictly increasing function, we chose to use the word “speed” instead of “velocity”. Students with a physics background may be comfortable with the idea of velocity but the goal of today’s activity is not to deep dive into topics of particle motion.