Getting Started 
Misconception/Error The student does not understand linear or exponential growth. 
Examples of Student Work at this Level The student:
 Is unable to identify the functions or confuses the two.
 Describes the functions instead of naming them.
 Attempts to write an equation for each function.Â

Questions Eliciting Thinking What can you tell me about a linear function? What does the graph of a linear function look like?
What can you tell me about an exponential function? What does the graph of an exponential function look like?
How is Saraâ€™s pay changing in Plan 1? Plan 2? Can you show me the change in the table? Can you show me the change in the graph? 
Instructional Implications Review the basic forms of linear and exponential functions and their graphs. Provide opportunities for the student to explore, investigate, and compare linear and exponential functions by generating tables of values and graphs. Ask the student to compare two functions such as y = 2x 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 comparisons can be made that reference the context.
Consider using activities from Illustrative Mathematics such as Exponential Growth Versus Linear Growth I (http://www.illustrativemathematics.org/illustrations/366) or Population and Food Supply (http://www.illustrativemathematics.org/illustrations/645). 
Moving Forward 
Misconception/Error The student is unable to clearly describe similarities and differences in the two functions. 
Examples of Student Work at this Level The student correctly identifies Plan 1 as linear and Plan 2 as exponential. When describing the similarities and differences in the plans, the student says:
 One is curved and the other is straight.
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 One is increasing more than the other.
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Questions Eliciting Thinking How is Saraâ€™s pay changing in Plan 1? Plan 2? Can you show me the change in the table? Can you show me the change in the graph?
What do the two plans have in common? How are the tables similar? How are the graphs similar?
How do the two plans differ? Are they increasing at the same rate? How would you describe the increase in Plan 1? How would you describe the increase in Plan 2?
Is one increasing more than the other throughout the entire domain? When is Plan 1 increasing more rapidly? When is Plan 2 increasing more rapidly? 
Instructional Implications Model how to compare linear and exponential functions by describing and comparing features of their graphs such as intercepts, whether 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 and contrast linear and exponential functions given in context. Consider using activities from Illustrative Mathematics such as Exponential Growth Versus Linear Growth I (http://www.illustrativemathematics.org/illustrations/366) or Population and Food Supply (http://www.illustrativemathematics.org/illustrations/645). 
Almost There 
Misconception/Error The student cannot completely justify when it is appropriate to choose which plan. 
Examples of Student Work at this Level The student correctly identifies Plan 1 as a linear function and Plan 2 as an exponential function. In addition, the student is able to compare the plans by saying the plans are alike in that they are both increasing, but different in that Plan 1 is increasing at a constant rate, and Plan 2 is doubling with each hour. However, for Question 5, the student responds that Plan 1 is the better choice but is unable to support this answer. For Question 6, the student responds that Plan 2 is â€śbetter in the long runâ€ť without any further justification. 
Questions Eliciting Thinking You said that Plan 1 is better if Sara was babysitting for four hours. Why is it better?
What plan is better for one hour? Two hours? Seven hours? Eight hours? Nine hours? Ten hours? What about after 10 hours?
When is the first time that Plan 2 pays more than Plan 1? After that point, is Plan 1 ever going to pay more than Plan 2 again? Why? 
Instructional Implications Model for the student justifying why Sara should choose Plan 1 when she is babysitting for four hours and Plan 2 when she is babysitting for nine or more hours. Provide the student with additional examples of the graphs of two functions given in context and ask the student to identify and explain points of intersection in context. Have the student compare the two functions over intervals of x on either side of the points of intersection. Require the student to justify his or her response. 
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:
 Plan 1 is a linear function and Plan 2 is an exponential function.
 The plans are alike in that they are both increasing but different in that Plan 1 is increasing at a constant rate, and Plan 2 is doubling each hour or increasing exponentially.Â
 If babysitting four hours, Sara should choose Plan 1 because it would pay $20 while Plan 2 would only pay $2.Â
 Sarah should choose Plan 2 if she is going to babysit for nine or more hours because at that point, Plan 2 always pays more than Plan 1.
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Questions Eliciting Thinking What does it mean to increase exponentially?
Is there ever a time where both plans will pay the same?
Can you write an equation for each function? 
Instructional Implications Ask the student to compare an exponential decay function toÂ a linear function with a negative slope.
Consider using the MFAS task Compare Quadratic and Exponential Functions (FLE.1.3) if not previously used. 