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:
 Subtracts the ycoordinates of the endpoints of the given intervals and does not take into account the corresponding changes in x.
 Calculates the rate of change from (0, 0) to (20, 1).
 Says the rate of change is the same as the ycoordinate of the first ordered pair in the given interval getting $20 and $60 for questions 1 and 2, respectively.

Questions Eliciting Thinking What is rate of change? How do you calculate rate of change?
Can you identify the endpoints on the graph for the given intervals? Can you use these points to find the rate of change for each interval? 
Instructional Implications Provide additional instruction on the concept of rate of change. Initially, consider linear relationships and relate the 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. Consequently, 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 features of the graph. Provide instruction on calculating the average rate of change over larger intervals.
Provide the student with additional linear and nonlinear graphs that model the relationship between two variables. Ask the student to calculate the average rate of change over several different intervals. Provide feedback as needed. 
Moving Forward 
Misconception/Error The student demonstrates some understanding of the concept of average rate of change but makes major errors in calculating it. 
Examples of Student Work at this Level The student:
 Disregards the scales of the axes and calculates rate of change as if each axis were scaled by one.
 Calculates the average rate of change by calculating a difference in yvalues over a difference in xvalues but uses the wrong values.

Questions Eliciting Thinking Can you explain how you found the rate of change for these intervals? What is the unit of measure for the xcoordinates? What is the unit of measure for the ycoordinates?
What is the meaning of the rate of change you calculated, and how does it relate to the context of this problem? 
Instructional Implications Ask the student to describe the scale used on each axis. Discuss why different scales may have been used for the x and yaxes. Ask the student to reconsider his or her calculations. Guide the student to understand the relationship between the rates of change calculated in the first two questions and the features of the problem (e.g., the rate of change describes the rate that Jasmine is paid per child).
Describe the relationship between the number of children and Jasmineâ€™s earnings as linear and remind the student that a defining attribute of a linear relationship is that the rate of change is constant. Correct the studentâ€™s calculations for question #1 and question #2. Then, use these calculations to explain why the rate of change is the same regardless of the interval over which it is calculated. Emphasize that for a linear function, 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.
Provide additional opportunities to calculate and interpret average rates of change over specified intervals for both linear and nonlinear functions. 
Almost There 
Misconception/Error The student provides an inadequate rationale for the comparison of the rates of change. 
Examples of Student Work at this Level The student writes the answers are the same because:
 The rates of change are the same.
 The average rate of change never changes.

Questions Eliciting Thinking Why do you think the rates of change are the same?
How do you know the rate of change is constant?
Why does the average rate of change never change? 
Instructional Implications Describe the relationship between the number of children attending and Jasmineâ€™s earnings as linear, and remind the student that a defining attribute of a linear relationship is that the rate of change is constant. Use the studentâ€™s calculations for question #1 and question #2 to explain why the rate of change is the same regardless of the interval over which it is calculated. Emphasize that for a linear function, 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.
Provide the student with examples of linear and nonlinear functions graphed on the same set of axes. Ask the student to determine the rate of change for each over two different intervals and to compare the rates of change. Be sure the student understands that the rate of change for a nonlinear function is not necessarily the same over two different intervals.
Provide additional opportunities to calculate and interpret average rate 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 determines that the average rate of change in earnings per child for each of the two intervals is $8. The student explains that the rates of change are the same because the relationship between the number of children attending and Jasmineâ€™s earnings is linear. For each additional child attending, Jasmine earns an additional $8.

Questions Eliciting Thinking Suppose Jasmineâ€™s supervising fee is increased from $12 to $14. Will this affect the rate of change?
Would increasing her rate per child from $8 to $10 affect the rate of change? Why or why not?
Why is the line that includes the points on the graph not drawn for the problem?
What is the difference between continuous and discrete variables? 
Instructional Implications If necessary, guide the student to write his or her responses in a more precise manner. For example, suppose that in response to the third question, the student explains, â€śThis is because the graph is linear and the points increase at the same rate.' Indicate to the student that it is the earnings (rather than â€śthe pointsâ€ť)Â that increase at a constant rate for each additional child that attends.
Provide the student with opportunities to calculate and compare average rates of change over specified intervals for nonlinear functions.
Consider implementing MFAS task Air Cannon (FIF.2.6). 