Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 56665
Estimating the Average Rate of Change
Students are asked to estimate the average rate of change of a nonlinear function over two different intervals given its graph.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, rate of change, slope, average, estimate, interval
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_EstimatingtheAverageRateofChange_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Estimating the Average Rate of Change worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not understand the concept of average rate of change or its calculation.
Examples of Student Work at this Level
When attempting to estimate the average rate of change, the student:
Subtracts the x-coordinates of the endpoints of the intervals.
Subtracts only the y-coordinates of the endpoints of the intervals.
Attempts to apply the slope formula but substitutes coordinates incorrectly.
Questions Eliciting Thinking
What is average rate of change? How does average rate of change differ from the constant rate of change of a linear function?
page 1 of 4 How do you calculate rate of change of a linear function?
How is the average rate of change of a nonlinear function calculated?
What are the coordinates of the endpoints of the given intervals? How can these coordinates be used to find the average rates of change over the intervals?
Instructional Implications
Provide additional instruction on the concept of rate of change. Initially, consider linear relationships and relate rate of change to the slope of the line that models a linear
relationship between two variables. Ask the student to calculate the rate of change of a linear function using several different ordered pairs and guide the student to
observe that the rate of change (like the slope) of a linear relationship is the same regardless of the ordered pairs used to calculate it. Remind the student that a defining
attribute of linear relationships is that the rate of change is constant. Emphasize that the average rate of change over any interval will always be the same as the rate of
change (or slope of the line) that represents the graph of the relationship.
Next, introduce the student to the concept of average rate of change in the context of nonlinear relationships. Begin with a relatively simple relationship such as
. Ask
the student to determine the change in y for several consecutive one unit intervals of x and to compare them. Relate the different rates of change in y to the steepness of
the graph. Provide instruction on calculating the average rate of change over larger intervals.
Provide the student with a nonlinear graph that models the relationship between two variables. Ask the student to calculate the average rate of change over several
different intervals. Ensure the student can identify the points on the graph represented by the endpoints of each interval. Have the student draw secant lines that contain
the endpoints of the intervals. Relate the average rate of change calculation to the calculation of the slopes of the secant lines.
Provide additional opportunities to calculate and interpret average rate of change over specified intervals for both linear and nonlinear functions.
Moving Forward
Misconception/Error
The student errs in identifying or justifying the interval for which the average rate of change is greater.
Examples of Student Work at this Level
The student correctly calculates the average rate of change of each interval. However, the student:
Incorrectly identifies
as the interval for which the average rate of change is greater.
Correctly identifies
as the interval for which the average rate of change is greater but provides an inadequate explanation.
Identifies the larger rate of change rather than the interval over which it occurs and provides an inadequate explanation.
Questions Eliciting Thinking
What value is greater, 4 or 3.5? So which interval has a greater average rate of change?
Does a nonlinear graph have slope?
Instructional Implications
Make clear that since 4 > 3.5, the average rate of change over the interval -
is greater than that of
. Assist the student in interpreting average rate
page 2 of 4 of change as the amount of change in the y-values associated with a corresponding change in the x-values of one unit, on average.
Explain that slope is a quality of a line rather than a curve. Ask the student to draw secant lines through the endpoints of the two intervals on the graph. Relate finding
average rates of change over these intervals to finding the slope of the secant lines that contain the endpoints of the intervals. Guide the student to observe how the
slopes of the secant lines relate to the steepness of the graph at each of the intervals.
Consider implementing MFAS task Air Cannon (F-IF.2.6), if not done previously.
Almost There
Misconception/Error
The student makes a minor mathematical error when estimating an average rate of change.
Examples of Student Work at this Level
The student:
Adds or subtracts incorrectly when estimating rate of change, for example, when subtracting -10 from -8.
Incorrectly converts
to a decimal.
Indicates a positive rate of change is negative.
Identifies the larger rate of change rather than the interval over which it occurs but provides an adequate explanation.
Questions Eliciting Thinking
You made a small mistake in your work. Can you find and correct it?
What does it mean for the rate of change to be positive? Negative?
You identified the larger rate of change but over what interval did it occur?
Instructional Implications
Assist the student in identifying and correcting any errors made. Provide the student with several examples of calculations of average rates of change that include errors
such as those listed above. Then have the student identify and correct the errors.
Provide additional opportunities to calculate and interpret average rates of change over specified intervals for both linear and nonlinear functions.
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student estimates the average rates of change as 4 and 3.5. The student identifies
as the interval for which the average rate of change is greater. The
student justifies this identification by explaining that 4 > 3.5. Upon questioning, the student can explain that over the interval
, the graph the change in y is
4 units, on average, for each one unit change in x.
Questions Eliciting Thinking
What does an average rate of change of 4 really mean?
How can the rate of change over the interval
be greater than the rate of change over the interval
when its maximum is less?
How does average rate of change over an interval of a nonlinear function differ from rate of change of a linear function?
Instructional Implications
Ask the student to draw secant lines through the endpoints of the two intervals on the graph. Relate finding average rates of change over these intervals to finding the
slope of the secant lines that contain the endpoints of the intervals. Guide the student to observe how the slopes of the secant lines relate to the steepness of the graph
over each of the intervals.
Have the student calculate the average rate of change for intervals with the same left endpoint but of decreasing length (e.g.
) for the function
, then
, then
. Have the student draw secant lines through the endpoints of each interval. Encourage the student to observe how the secant lines get
progressively closer to the line tangent to the graph at (1, 1). If available, use a graphing utility to illustrate this concept.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Estimating the Average Rate of Change worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
page 3 of 4 Related Standards
Name
MAFS.912.F-IF.2.6:
Description
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. ★
page 4 of 4