Lesson 10

Looking at Rates of Change

10.1: Falling Prices (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

Let \(p\) be the function that gives the cost \(p(t)\), in dollars, of producing 1 watt of solar energy \(t\) years after 1977. Here is a table showing the values of \(p\) from 1977 to 1987.

\(t\) \(p(t)\)
0 80
1 60
2 45
3 33.75
4 25.31
5 18.98
6 14.24
7 10.68
8 8.01
9 6.01
10 4.51

Which expression best represents the average rate of change in solar cost between 1977 and 1987?

  1. \(p(10)-p(0)\)
  2. \(p(10)\)
  3. \(\dfrac{p(10) - p(0)}{10-0}\)
  4. \(\dfrac{p(10)}{p(0)}\)

 

Student Response

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Anticipated Misconceptions

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.

Activity Synthesis

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.

10.2: Coffee Shops (20 minutes)

Activity

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.

Student Facing

Here is a table and a graph that show the number of coffee shops worldwide that a company had in its first 10 years, between 1987 and 1997. The growth in the number of stores was roughly exponential.

Graph of a roughly exponential function, origin O. Years since 1987 and number of coffee shops.
year number of stores
1987 17
1988 33
1989 55
1990 84
1991 116
1992 165
1993 272
1994 425
1995 677
1996 1,015
1997 1,412
  1. Find the average rate of change for each period of time. Show your reasoning.

    1. 1987 and 1990
    2. 1987 and 1993
    3. 1987 and 1997
  2. Make some observations about the rates of change you calculated. What do these average rates tell us about how the company was growing during this time period?
  3. Use the graph to support your answers to these questions. How well do the average rates of change describe the growth of the company in:

    1. the first 3 years?
    2. the first 6 years?
    3. the entire 10 years?
  4. Let \(f\) be the function so that\(f(t)\) represents the number of stores \(t\) years since 1987. The value of \(f(20)\) is 15,011. Find \(\dfrac {f(20) - f(10)}{20-10}\) and say what it tells us about the change in the number of stores.

Student Response

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Anticipated Misconceptions

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.

Activity Synthesis

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:

11 points on coordinate plane. Lines connect first point to fourth, seventh, and eleventh points. Horizontal axis from 0 to 10, time in years. Vertical axis from 0 to 1500, number of coffee shops.

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.)
Conversing, Representing: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to communicate how the rate of change differs during each specified time period. Encourage students to demonstrate mathematical thinking and problem solving by referencing two points on the graph in their explanation. Press for further detail by asking, "How can you use the graph to determine which time period is represented best by the rate of change?" This will help students understand how the rate of change differs in exponential functions through a whole-class discussion.
Design Principle(s): Support sense-making

10.3: Revisiting Cost of Solar Cells (10 minutes)

Activity

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.

Launch

Representing, Conversing: MLR7 Compare and Connect. Ask students to prepare a visual display that shows their graph and mathematical thinking in response to Clare's statement. Students should consider how to display their calculations so another student can interpret them. Some students may wish to add notes or details to their graphs or calculations to help communicate their thinking. When students find another person in the class that drew in similar lines on their graph or used the same points in calculations, provide 2–3 minutes of quiet think time for students to read and interpret each other's work. This will help students make connections between different representations of finding the average rate of change that produce similar results.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, students may use highlighters to color code each 5-year section a unique color.
Supports accessibility for: Visual-spatial processing

Student Facing

Here is a graph you saw in an earlier lesson. It represents the exponential function \(p\), which models the cost \(p(t)\), in dollars, of producing 1 watt of solar energy, from 1977 to 1988 where \(t\) is years since 1977.

Graph of an exponential function, origin O. t (years since 1977) and p (dollars per watt).
  1. Clare said, "In the first five years, between 1977 and 1982, the cost fell by about $12 per year. But in the second five years, between 1983 and 1988, the cost fell only by about $2 a year." Show that Clare is correct.
  2. If the trend continues, will the average decrease in price be more or less than $2 per year between 1987 and 1992? Explain your reasoning.

Student Response

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Student Facing

Are you ready for more?

Suppose the cost of producing 1 watt of solar energy had instead decreased by $12.20 each year between 1977 and 1982. Compute what the costs would be each year and plot them on the same graph shown in the activity. How do these alternate costs compare to the actual costs shown?

Student Response

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Activity Synthesis

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.

Lesson Synthesis

Lesson Synthesis

The purpose of this synthesis is for students to practice calculating the average rate of change for an exponential function over specific intervals and to consider when the average rate of change adequately describes the function over that interval.

Show students the table, which gives the value in dollars of a laptop originally priced for \$1,500 and losing \(\frac{3}{5}\) of its value every year after purchase.

years since purchase value in dollars
1500
1 600
2 240
3 96
4 38.40
5 15.36

Ask students

  • "About how much was the average rate of change for the 5 year period?" (almost \$300 per year)
  • "Does the average rate of change accurately describe how the value of the laptop changes over the 5 years?" (no, it loses almost \$1000 in value the first year and very little from the third year to the fifth)
  • "Are there some years where the average rate of change shows how the value of the laptop is changing?" (yes, from year 1 to year 2 it is a reasonable estimate)

10.4: Cool-down - An Average Rate of Change (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

When we calculate the average rate of change for a linear function, no matter what interval we pick, the value of the rate of change is the same. A constant rate of change is an important feature of linear functions! When a linear function is represented by a graph, the slope of the line is the rate of change of the function.

Exponential functions also have important features. We've learned about exponential growth and exponential decay, both of which are characterized by a constant quotient over equal intervals. But what does this mean for the value of the average rate of change for an exponential function over a specific interval?

Let's look at an exponential function we studied earlier. Let \(A\) be the function that models the area \(A(t)\), in square yards, of algae covering a pond \(t\) weeks after beginning treatment to control the algae bloom. Here is a table showing about how many square yards of algae remain during the first 5 weeks of treatment.

\(t\) \(A(t)\)
0 240
1 80
2 27 
3 9
4 3

The average rate of change of \(A\) from the start of treatment to week 2 is about -107 square yards per week since \(\dfrac{A(2)-A(0)}{2-0} \approx \text-107\). The average rate of change of \(A\) from week 2 to week 4, however, is only about -12 square yards per week since \(\dfrac{A(4)-A(2)}{4-2} \approx \text-12\).

These calculations show that \(A\) is decreasing over both intervals, but the average rate of change is less from weeks 0 to 2 than from weeks 2 to 4, which is due to the effect of the decay factor. If we had looked at an exponential growth function instead, the values for the average rate of change of each interval would be positive with the second interval having a greater value than the first, which is due to the effect of the growth factor.