Lesson 10

The purpose of this warm-up is for students to recall how to calculate an average rate of change from two points. Students will use this skill throughout the lesson and return to the context and data table in the last activity of the lesson.

Ask students to close their books or devices. Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time, and then select 3–5 students to share something they notice or wonder about the table  and record and display their responses for all to see. If not pointed out, ask students, "Describe how the data is changing as the value of \(t\) is increasing." (The values of \(p\) are decreasing at a slower rate.) Ask students to reopen their books or devices and complete the activity. Follow with a whole-class discussion.

Some students might confuse finding an average of the values in the table and finding an average rate of change. Help them see that average usually involves one unit, such as average number of cookies. Average rate of change involves comparing how one quantity changes when another quantity changes by 1, such as cost ($) per year.

The goal of this discussion is for students to recall the meaning of average rate of change for a function and how it is calculated. Select students to describe what the value of each expression represents in this context. For example, \(p(10)-p(0)\) is the difference in cost for a solar cell between 1977 and 1987 while \(\dfrac{p(10)}{p(0)}\) could be used to identify the percent change from 1977 to 1987. An important takeaway for students is that the actual expression for the average rate of change, \(\dfrac{p(10)-p(0)}{10-0} \approx \text-7.55\), tells us that the price decreased by -$7.55 each year on average since the expression looks at the total difference in price between the 1977 and 1987 divided by the total number of years that passed.

Earlier in the course, students looked at constant rates of change in linear functions. In this activity, they begin to explore average rates of change in an exponential growth function. In particular, the focus for this activity is on how exponential functions have different rates of change for different input intervals, which was not the case for linear functions.

In the next activity, they investigate average rates of growth in a decay function. In both cases, they revisit contexts from earlier lessons.

If students struggle with a rate of change that is not constant like the slope they saw in linear relationships, ask if they can draw a straight line between all the points on the graph in question 3. Spend time on the graph in the synthesis so the lines can help them see why the rate of change varies with an exponential function.

For the third question, make sure students see that when they calculated the average rate of change for each of the three time periods, they were in effect finding the slope of the line that goes through two points that represent the starting year and the ending year. Display a graph like the one shown here to help illustrate this point:

The line that connects the points for 1987 and 1990 fit the data for that period fairly well, so the slope of that line (the average rate of change between those two points) describes the growth in those three years fairly accurately. In contrast, the line that connects the points for 1987 and 1997, does not at all fit the data, so the slope of that line does not paint an accurate picture of how the company was growing that decade.

Here are some questions for discussion

• “Would the average rate of change between 1994 and 1997 accurately depict how the company was growing in the last three years in the data?” ( Yes, because the average rate of change from 1994 to 1997 is 329 shops per year since \(\dfrac{1412-425}{10-7}= 329\). The actual number of shops increased by 252, 338, and then 397 from 1994 to 1997, respectively.)
• “Is there a single period of time whose average rate of change would well summarize how the company was growing from 1987 to 1997?” (No, since the number of shops is increasing at an increasing rate, an average of two values is not enough to summarize the growth of the company.)

In this activity, students continue to explore average rates of change for exponential functions, this time focusing on a previously encountered exponential decay context. Unlike the previous activity, students are asked to calculate the average rate of change from a graph for two different intervals of time. Using those values, students then predict what the average rate of change is for an interval extending into the future beyond what it shown in the graph.

Monitor for students who draw in lines to visualize the average rate of change for the two intervals and ask them to share their visual and why they added it in during the discussion.

Select previously identified students to share their graphs and why they drew in the lines along with their interpretation of the meaning in the context of the problem. Invite other students who made slightly different calculations to share their reasoning, reminding students that, since everyone is making an estimate from the graph, answers are not likely to be identical, but they should be very close.

Conclude the discussion by selecting students to explain their reasoning about the last problem using their understanding of exponential functions.