Getting Started 
Misconception/Error The student does not understand the concept of a constant percentage rate of change. 
Examples of Student Work at this Level The student says the function is not increasing at a constant percentage rate because:
 The rate of change is not constant.
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 The cells are increasing randomly.
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The student says the function is increasing at a:
 Constant percentage rate of 1.5%, 15%, or 150% because he or she calculates the percent of increase by dividing the new amount by the original amount (e.g., ) and may or may not have correctly converted to a percent.
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 Constant rate of 1.5 and writes a linear function to model the relationship between the variables.
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Questions Eliciting Thinking What is a constant percentage rate of change? How is it different from a constant rate of change?
How can you calculate the rate of growth for each situation?
How would you describe the amount of increase in the cell population each minute in Cindyâ€™s experiment? Is there a pattern in the amount of increase each minute?
How would you describe the amount of increase in the cell population each minute in Simoneâ€™s experiment? Is there a pattern in the amount of increase each minute? 
Instructional Implications Have the student list the data for the first four minutes of Cindyâ€™s experiment in a table of values. Have the student use the table to verify that, even though the data was described as doubling in number every minute, the rate of change in Cindyâ€™s experiment is not constant. Point out to the student that the number of cells is increasing by a different amount every minute. If needed, have the student graph the data from the table of values and explain that if the number of cells were increasing at a constant rate of change, the graph would be a line modeled by a linear function.
Guide the student to determine the rate of change by calculating and comparing the ratio of each pair of successive functional values and observing that each ratio equals two. Explain that this constant ratio indicates that the number of cells in Cindyâ€™s experiment is growing exponentially. Review the basic form of an exponential function, its parameters and the relationship between the rate of growth, r, and the growth factor that appears in the equation 1 + r. Guide the student to model the growth of Cindyâ€™s cell population with an exponential function, Â where N is the number of cells after t minutes. Distinguish between the growth factor, 2, and the rate of growth, 1.00 or 100%.
Ask the student to analyze the values in Simoneâ€™s table to determine the constant ratio and write an exponential function that models the growth in the cell population. Ask the student to identify the constant percentage rate of change and to compare it to the rate of change of Cindyâ€™s cell population.
Consider implementing the MFAS tasks Exponential Functions â€“ 1 (FIF.3.8) or Exponential Functions â€“ 2 (FIF.3.8). 
Moving Forward 
Misconception/Error The student is unable to determineÂ in which experiment the cells are increasing more rapidly. 
Examples of Student Work at this Level The student determines that the cell population in Simoneâ€™s experiment is increasing more rapidly because:
 The daily numbers of cells are higher for the first four days.
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 She starts off with more cells at the beginning of the experiment.
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Questions Eliciting Thinking How would you compare the two patterns of growth?
Can one function increase by a larger factor than another function but have a smaller initial value?
Can one function increase by a larger factor than another function but have smaller output values? 
Instructional Implications Explain to the student that the initial value of the function does not determine the rate at which the function is increasing. Have the student consider linear functions. Remind the student that it is the slope (i.e., the rate of change) not the yintercept (i.e., the initial value) that determines how rapidly the function is increasing or decreasing.
Review the basic form of an exponential function and its two parameters (initial value and growth factor). Ask the student to model the growth of the cells in each experiment with exponential functions. Explain to the student that in order to determine whose cell population is increasing faster, one must compare growth factors rather than initial values. Have the student revise his or her response.
Provide other examples of linear and exponential functions given tables of values or verbal descriptions. Have the student analyze and describe the rates of change. Guide the student to describe each function as either linear or exponential based on the rates of change. 
Almost There 
Misconception/Error The student is unable to determine and explain what type of function can model the growth of Simoneâ€™s cell population. 
Examples of Student Work at this Level The student determines the constant percentage rate of growth of both Simoneâ€™s and Cindyâ€™s cell populations andÂ recognizesÂ that the cells in Cindyâ€™s population are increasing at a faster rate. However, the student does not understand that an exponential function can model the growth of Simoneâ€™s cell population. The student writes that the growth of Simoneâ€™s cell population is:
 Linear because it is increasing at a constant rate.
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 Quadratic because it is slowly increasing.

Questions Eliciting Thinking What kind of function can be used to model increases or decreases when the rate of change is a constant amount?
What kind of function can be used to model increases or decreases when the rate of change is a constant factor?
Is the rate of change in these two situations increasing by a constant amount or a constant factor? 
Instructional Implications Explain that the constant percentage rates of growth that the student previously described indicate that the number of cells in each experiment is growing exponentially. Review the basic form of an exponential function and its parameters. Ask the student to model the growth of the cells in each experiment with exponential functions.
Provide other examples of linear and exponential functions given tables of values or verbal descriptions. Have the student analyze and describe the rates of change. Guide the student to describe each function as either linear or exponential based on the rates of change. 
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 writes:
 The number of cells in Simoneâ€™s experiment is increasing at a constant percentage rate of 50% every minute. The student justifies this answer by showing appropriate work. For example, the student calculates ,Â and then verifies the percentage rate is constant by multiplying several other outputs in the table by 150% and comparing to the next output (e.g., ,Â ,Â ).
 The number of cells in Cindyâ€™s experiment is increasing at a faster rate (a constant percentage rate of 100%). Consequently, Cindyâ€™s cell population is increasing twice as fast as Simoneâ€™s cell population.
 The growth of Simoneâ€™s cell population can be modeled by an exponential function because the cell population is increasing by a constant percentage rate or factor.
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Questions Eliciting Thinking How would you describe the rate of change of a linear function?
How would you describe the rate of change of a quadratic function? 
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.
Consider implementing the MFAS task Prove Exponential (FLE.1.1). 