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.

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 because  . The actual number of shops increased by 252, 338, and then 397 from 1994 to 1997, respectively.)
• “Would the average rate of change between any 2 years accurately reflect the growth for all of 1987 to 1997?” (No, because 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 is shown in the graph.

Monitor for students who:

• Use the graph to estimate values and calculate the average rates of change from these estimates.
• Look back at the data given in the Warm-up and calculate the average rates of change numerically.
• Draw in lines to visualize the average rate of change for the two intervals.

Display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “Did anyone solve the problem the same way, but would explain it differently?”