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 xcoordinates of the endpoints of the intervals.
 Subtracts only the ycoordinates 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?
How do you calculate rate of change ofÂ 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 Â as the interval for which the average rate of change is greater.
 Correctly identifies Â 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 of change as the amount of change in the yvalues associated with a corresponding change in the xvalues 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 (FIF.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 Â 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 severalÂ 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 Â 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 Â be greater than the rate of change over the interval Â 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 overÂ 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. , then , then )Â for the function . 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. 