Getting Started 
Misconception/Error The student is unable to determine when the values of one function exceed those of another. 
Examples of Student Work at this Level The student:
 Identifies an approximate xcoordinate of a point of intersection of the two graphs.
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 Describes years or intervals over which one function exceeds the other incorrectly.
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Questions Eliciting Thinking Did you inspect the values in the table to see which park has the greater attendance?
How can you tell from the graphs when each park had greater attendance? 
Instructional Implications Review the basic forms of quadratic and exponential functions and their graphs. Provide opportunities for the student to explore, investigate, and compare quadratic and exponential functions by generating tables of values and graphs. Ask the student to compare two functions such as Â and . Provide a table of xvalues and ask the student to calculate the corresponding values of each function and then graph each function. Encourage the student to analyze both the tables and the graphs to identify intervals over which the functional values of one function exceed those of the other and to speculate about the end behavior of the graphs. Provide additional examples in context so that comparisons can be made that reference the context.
Consider using the Illustrative Mathematics activity Exponential Growth Versus Polynomial Growth (http://www.illustrativemathematics.org/illustrations/367). 
Moving Forward 
Misconception/Error The student is unable to clearly explain the differences between the two functions. 
Examples of Student Work at this Level The student correctly determines that Park A had greater attendance in the first year and during the third through approximately eighth year, while Park B had greater attendance in the second year and after the eighth or ninth year. However, the student is unable to correctly and clearly describe differences in the two functions.
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Questions Eliciting Thinking What kind of function represents the attendance at Park A? What kind of function represents Park Bâ€™s attendance?
How do the graphs of these two types of functions differ? How do their rates of change differ? 
Instructional Implications Model how to compare quadratic and exponential functions by describing and comparing features of their graphs, such as intercepts, where the functions are increasing or decreasing, and intervals over which the functional values of one exceed those of the other. Provide the student with additional opportunities to compare quadratic and exponential functions given in context.
Consider using the Illustrative Mathematics activity Exponential Growth Versus Polynomial Growth (http://www.illustrativemathematics.org/illustrations/367). 
Almost There 
Misconception/Error The student cannot completely describe the end behavior of each function. 
Examples of Student Work at this Level The student correctly determines that Park A had greater attendance in the first year and during the third through approximately eighth year, while Park B had greater attendance in the second year and after the eighth or ninth year. When describing the differences in the functions, the student writes that Park Aâ€™s attendance is represented by a quadratic function, while Park Bâ€™s attendance is represented by an exponential function. When asked whether Park Aâ€™s attendance will ever surpass Park Bâ€™s attendance, the student suggests that it is possible.
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Questions Eliciting Thinking Do both graphs always increase?
If the trends continue, what do you think is going to happen in year 10? What about in year 15? 
Instructional Implications Review with the student the end behavior of quadratic and exponential functions. If possible, allow the student to use graphing technology to compare the graphs of the two given functions as x increases. Assist the student in understanding that the rate of change of the quadratic function can never surpass that of the exponential function for values of x greater than nine.
Provide the student with additional examples of graphs of two functions given in context and ask the student to compare the functions at given intervals of x. 
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 correctly determines that Park A had greater attendance in the first year and during the third through approximately eighth year, while Park B had greater attendance in the second year and after the eighth or ninth year. When describing the differences in the functions, the student writes that Park Aâ€™s attendance is represented by a quadratic function, while Park Bâ€™s attendance is represented by an exponential function. When asked whether Park Aâ€™s attendance will ever surpass Park Bâ€™s attendance again, the student explains that once an exponential function exceeds a quadratic function, the exponential function will always have a greater value.
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Questions Eliciting Thinking Is there ever a time when the attendance is the same for both parks? 
Instructional Implications Ask the student to calculate and compare the average rates of change of each function over an interval such as [100, 110] in order to further investigate why the quadratic function will never exceed the exponential function for values of x greater than nine.
Consider using MFAS task Compare Linear and Exponential Functions (FLE.1.3) if not previously used. 