Getting Started 
Misconception/Error The student does not understand the concept of rate of change. 
Examples of Student Work at this Level The student may be able to identify the function types (e.g., as linear and exponential) or calculate functional values. However, the student is either unable to describe or misrepresents the rates of change. For example, the student explains:
 The number of messages the daughter receives is increasing by two every day.
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 The slope of the motherâ€™s equation is four so hers is increasing more rapidly.
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 The number of messages the mother receives over time can be represented by a linear function and the number of messages the daughter receives over time by an exponential function but does not address the rate of change of either.

Questions Eliciting Thinking Can you explain how you determined that the daughter is receiving two additional messages each day?
What does the slope tell you about the rate of change? Does a nonlinear function have slope?
How many messages were received by each on first day? How many messages were received by each on the second day? 
Instructional Implications Review the concept of rate of change and illustrate with the linear function, y = 4x. 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 = 0, 1, 2, 3, 4, and 5. Ask the student to describe and compare the rates at which the mother and daughter are receiving messages each day. Explain to the student that the rate at which the daughter is receiving messages each day varies since the number of messages is increasing by an ever increasing amount. Guide the student to observe that the number of messages receivedÂ each day by the daughter can be modeled by an exponential function. Have the student create a ttable to analyze the differences in the number of messages the daughter is receiving for each of four or five days.
Ask the student to make another table that describes the average rate of change each day for each function:
Explain that the table shows that 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 the motherâ€™s function exceeds that of the daughterâ€™s and when the rate of change of the daughterâ€™s function exceeds that of the motherâ€™s.
Provide other pairs of functions represented in different ways and ask the student to compare rates of change as well as other features of the functions and their graphs.
Consider implementing the MFAS task How Does Your Garden Grow? (FLE.1.1). 
Making Progress 
Misconception/Error The student does not provide a complete explanation of the comparison. 
Examples of Student Work at this Level The student demonstrates an understanding of rate of change. However, the student does not provide a complete explanation of the difference in the two rates of change or a complete description of whose number of daily messages is increasing more rapidly. For example, the student:
 Explains that the number of messages the mother receives is increasing at a constant rate while the number of messages the daughter receives is not constant.
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 Describes the rate at which the mother receives messages as constant while the daughterâ€™s is increasing but does not address whose number of daily messages is increasing more rapidly.
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 Describes the motherâ€™s function as linear and the daughterâ€™s as nonlinear and explains that the number of daily messages received by the daughter is increasing more rapidly. The student does not acknowledge that the number of daily messages received by the mother is increasing more rapidly during the first two days.
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Questions Eliciting Thinking Can you describe in more detail how the number of messages the daughter receives are increasing each day? Would you say the number of messages the daughter receives are increasing at a constant rate? Why or why not?
How many messages were received by each on the first day? How many messages were received by each on the second day? What is happening on the third day? The fourth day? 
Instructional Implications Guide the student to specifically describe the rates of change each day for each function to better compare the two functions. Ask the student to make a table to summarize this information. Guide the student to describe the rate at which the mother is receiving messages as constant (four per day) and the rate of change in the number of messages the daughter is receiving as exponential. Ask the student to compare the rates of change by describing when the rate of change of the motherâ€™s function exceeds that of the daughterâ€™s and when the rate of change of the daughterâ€™s function exceeds that of the motherâ€™s. Allow the student to revise his or her original response.
Consider implementing the MFAS tasks How Does Your Garden Grow? (FLE.1.1) or Exponential Growth (FLE.1.1). 
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 the difference in the rates of change as follows: The rate of change in the number of messages the mother receives is constant (four messages per day). The rate of change in the number of messages the daughter receives is increasing since the number of messages she receives doubles each day.
When explaining whose numbers of daily messages is increasing more rapidly, the student acknowledges that it varies. The student explains that during the first two days, the mother receives messages at a faster rate (four messages per day) than the daughter (one and two messages per day, respectively). The rate of increase from the second to third day is the same for both the mother and daughter (four messages per day). But after the third day, the rate of increase in the number of messages received by the daughter exceeds four per day. For example, for the interval from three to four days, the daughter receives messages at an average rate of eight per day while the mother still receives messages at a rate of four per day. 
Questions Eliciting Thinking Can you write an equation that describes the number of messages the daughter receives as a function of time?
Can you describe the domain and range of each function? What about the context of the situation might limit the domains and ranges? 
Instructional Implications Explain to the student the concept of average rate of change over a given interval. Have the student calculate and compare the average rates of change forÂ Â over the following intervals: [1, 2], [1, 3] and [1, 4].
Discuss with the student the end behavior of the graph ofÂ . Ask the student to describe what happens to the graph of the function as x approaches positive infinity. Have the student consider changes in the values of y as x increases. Ask the student to describe the end behavior of the graph and, if needed, model explaining that as x increases, f(x) increases without bound.
Ask the student to describe what happens to the graph ofÂ Â as x approaches negative infinity. Have the student consider the average rate of change of the function over the intervals [2, 1], [3, 2], and [4, 3]. Ask the student to describe the end behavior of the function as x approaches negative infinity. If needed, explain to the student that the graph of the function gets closer and closer but never actually intersects the horizontal line, y = 0. Have the student explain how the continued decrease in the average rate of change of the function relates to the end behavior of this graph and why f(x) can never equal zero. 