Unit 9, Lesson 10
Looking at Rates of Change
Integrated Math 1
10.1
Warm-up
Standards Alignment
Building On
Addressing
HSF-IF.B.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
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
Display the table for all to see. Ask students to describe how quickly is decreasing between 1977 () and 1978 () and then how it is decreasing between 1986 () and 1987 (). (In the first year on the table it’s decreasing by 20. In the last year on the table, it decreases only by 1.5.)
Activity
NoneLet be the function that gives the cost , in dollars, of producing 1 watt of solar power years after 1977. Here is a table showing the values of from 1977 to 1987.
| 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 can be used to calculate the average rate of change in solar cost between 1977 and 1987?
Student Response
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Building on Student Thinking
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, is the difference in cost for a solar cell between 1977 and 1987 while 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, , tells us that the price decreased by -\$7.55 each year on average, because the expression looks at the total difference in price between 1977 and 1987, divided by the total number of years that passed.
10.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
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.
Launch
NoneActivity
None10.3
Activity
Instructional Routines
MLR7: Compare and ConnectMaterials
NoneActivity Narrative
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.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
Select work from students with different strategies, such as those described in the Activity Narrative, and ask them to share later.
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 of its value every year after purchase.
| years since purchase | value in dollars |
|---|---|
| 0 | 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.)
Student Lesson Summary
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 that we studied earlier. Let be the function that models the area, , in square yards, of algae covering a pond 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.
| 0 | 240 |
| 1 | 80 |
| 2 | 27 |
| 3 | 9 |
| 4 | 3 |
The average rate of change of from the start of treatment to Week 2 is about -107 square yards per week because . The average rate of change of from Week 2 to Week 4, however, is only about -12 square yards per week because .
The negative average rates of change show that is decreasing over both intervals, but the average rate of change for the time during Weeks 0 to 2 indicates that the values are decreasing more rapidly than during Weeks 2 to 4 due to the effect of the decay factor. For an exponential function with a growth factor greater than 1, the values for the average rate of change of each interval are positive, with the second interval increasing more quickly due to the effect of the growth factor.