Getting Started 
Misconception/Error The student does not understand the concept of average rate of change and how it is calculated. 
Examples of Student Work at this Level The student:
 Finds the average cost of the five pizza sizes.
 Finds only the change in the prices between the two given sizes.
 Finds the change in price between each size and then averages the four changes.
 Divides the diameter by the cost of each size and then averages these five values.
 Multiplies the diameter and the cost of the two given sizes and then subtracts these values.

Questions Eliciting Thinking What is rate of change? How do you calculate rate of change?
What is the difference between an average and average rate of change?
What is the difference between constant rate of change and average rate of change? 
Instructional Implications Provide additional instruction on the concept of rate of change. Initially, consider linear relationships and ask the student to calculate the rate of change of a linear function using several different ordered pairs from a table. Guide the student to observe that the rate of change 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 a table of values for . 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 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 demonstrates some understanding of the concept of average rate of change but makes major errors calculating it. 
Examples of Student Work at this Level The student:
 Computes the average rate of change as Â instead of .
 Correctly uses Â but then divides incorrectly.
 Finds the difference in price between an 8inch pizza and a 16inch pizza but divides this value by five since there are five sizes.

Questions Eliciting Thinking Can you explain how you found your answer?
What formula did you use to find your answer? What values did you use for Â and ? 
Instructional Implications Review with the student how to find the average rate of change over a specified interval given a table of values of a function. Describe average rate of change as a change in output values over the corresponding change in input values. Discuss the similarities and differences in finding the average rate of change over an interval of a nonlinear function and finding the rate of change of a linear function. Provide additional opportunities to calculate and interpret average rates of change over specified intervals for both linear and nonlinear functions.
Consider implementing MFAS tasksÂ Identifying Rate of Change (FIF.2.6) or Air Cannon (FIF.2.6) if not previously used. 
Almost There 
Misconception/Error The student provides a correct response but with insufficient reasoning or imprecise language. 
Examples of Student Work at this Level The student calculates the average rate of change as $0.94 or $0.9425 but describes this value as:
 The slope of the table.
 How much the price changed from an 8inch pizza to a 16inch pizza.
 How much money is being saved if you get the larger pizza.

Questions Eliciting Thinking What does C(d) represent? What does d represent? So what would the rate of change that you found represent?
How would you write the average rate of change with appropriate units of measure? 
Instructional Implications Guide the student to consider the units of measure when interpreting and describing the average rate of change. For example, guide the student to write the average rate of change as $0.94 per inch and to interpret the meaning of the rate of change in context. Provide additional opportunities to calculate and interpret average rates of change over specified intervals for both linear and nonlinear functions and provide feedback.
Consider implementing MFAS tasks Identifying Rate of Change (FIF.2.6) or Air Cannon (FIF.2.6) if not previously used. 
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 calculates the average rate of change as $0.94 or $0.9425 per inch and explains that, on average, for each 1inch increase in diameter (from 8 inches to 16 inches), the cost of making a pizza increases by $0.94. 
Questions Eliciting Thinking Is this rate of change constant? Would you get the same value if you calculated the average rate of change over another interval?
How would you determine which pizza costs the least to make? 
Instructional Implications Ask the student to sketch a graph of the function using the ordered pairs given in the table. Then ask the student to draw a secant line through the endpoints of an interval on the graph. Relate finding average rate of change over the interval to finding the slope of the secant line that contains the endpoints of the interval. Guide the student to observe how the slope of the secant line relates to the steepness of the graph over the interval.
Consider implementing MFAS tasksÂ Identifying Rate of Change (FIF.2.6) or Air Cannon (FIF.2.6) if not previously used. 