Getting Started 
Misconception/Error The student does not discern the difference in the rates of change. 
Examples of Student Work at this Level The student does not recognize the difference in the two rates of change and concludes that they are the same.
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Questions Eliciting Thinking What does rate of change mean? What would be evidence of a constant rate of change?
What is the difference in the numbers of seeds that Juan plants each day? What is the rate of change?
Did you make a table of values for Janie? What is the difference between the numbers of seeds that Janie plants each day? How could you describe this rate of change? 
Instructional Implications Review the concept of rate of change and illustrate a constant rate of change with Juanâ€™s table of values. Explain that the number of seeds planted by Juan each day can be described by the linear function N = 2d (where N represents the number of seeds and d represents the number of days of planting). Emphasize that the rate of change of a linear function is constant. Guide the student to create a table of functional values for both functions for x = 1, 2, 3, 4, and 5. Ask the student to describe and compare the rates at which the number of seeds planted by Juan and Janie is increasing. Explain to the student that the number of seeds planted by Janie is increasing each day by an ever increasing amount (rather than by the same amount), so it is not constant. Guide the student to observe that the function describing the number of seeds planted by Janie is exponential (i.e., can be described by the equation for d = 1, 2, 3, 4, and 5). Have the student use the table to analyze the differences in the number of seeds planted by each person each day.
Ask the student to make a table that describes the rate of change each day for each person:
Explain that the table shows the rate of change of the linear function is constant while the rate of change of the exponential function is not constant and is, in fact, increasing (exponentially). Guide the student to compare the rates of change and to identify when the rate of change of Juanâ€™s function exceeds that of Janieâ€™s and when the rate of change of Janieâ€™s function exceeds that of Juanâ€™s.
Provide other pairs of linear and exponential functions and ask the student to analyze and compare rates of change. Guide the student to describe the functions as either linear or exponential based on the patterns in the rates of change.
Consider implementing the MFAS task Comparing Linear and Exponential Functions (FIF.3.9). 
Making Progress 
Misconception/Error The student does not understand the concept of a constant rate of change. 
Examples of Student Work at this Level The student understands the differences in the rates of change but does not understand what is meant by a constant rate of change. The student explains:
 The number of seeds Janie plants is changing at a constant rate per day because the amount is doubling every time.
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 The number of seeds Juan plants is not changing at a constant rate because â€śhe added two each day.â€ť
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Questions Eliciting Thinking What does rate of change mean? What would be evidence of a constant rate of change?
What is the difference in the numbers of seeds that Juan plants each day? What is the rate of change?
What is the difference between the numbers of seeds that Janie plants each day? How could you describe this rate of change? 
Instructional Implications Clarify what is meant by a constant rate of change. Illustrate a constant rate of change with Juanâ€™s table of values by having the student write the change in each pair of consecutive values for both columns. Have the student make a table of values for Janie (if not done so already) and write the change in each pair of consecutive values for both columns. Be sure the student understands that when comparing changes in pairs of yvalues, the corresponding pairs of xvalues must change by the same amount. Explain that the number of seeds planted by Janie is increasing each day by an ever increasing amount (rather than by the same amount) so the rate of change cannot be described as constant. Guide the student to observe that the function describing the number of seeds planted by Janie is exponential while Juanâ€™s is linear.
Provide other pairs of linear and exponential functions and ask the student to analyze and compare rates of change. Guide the student to describe the functions as either linear or exponential based on the patterns in the rates of change.
Consider implementing the MFAS task Comparing Linear and Exponential Functions (FIF.3.9). 
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 explains that:
 The rate of change in the number of seeds that Janie plants is increasing (exponentially) each day and is, therefore, not constant.
 The rate of change in the number of seeds that Juan plants is increasing by the same amount every day (i.e., is increasing by two per day) and is, therefore, constant.
 The rates of change from the first to the second day are the same but after that, the rate of increase in the number of seeds planted by Janie exceeds that of Juan.
 The function that describes the number of seeds planted by Juan each day is linear.
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Questions Eliciting Thinking What factor describes the increase in the number of seeds planted each day by Janie?
In general, how is the rate of change of a linear function described? How is the rate of change of an exponential function described? 
Instructional Implications Introduce the concept of an average rate of change (see FIF.2.6). Give the student the equation of an exponential function, such as . Ask the student to describe the average rate of change as x increases from d to d + 2. Ask the student to use the result of this calculation to explain why the rate of change of an exponential function can be described by another exponential function. 