Getting Started 
Misconception/Error The student is unable to differentiate between the linear and exponential function descriptions. 
Examples of Student Work at this Level The student cannot correctly distinguish between the linear and exponential descriptions and:
 Provides no reasoning or inaccurate reasoning.
 Describes each rate of change as constant.

Questions Eliciting Thinking What is a linear function? How can you recognize a linear function from an equation? From a graph? From a table? From a description?
What is an exponential function? How can you recognize an exponential function from an equation? From a graph? From a table? From a description? 
Instructional Implications Provide instruction on linear and exponential functions. Guide the student to recognize that linear functions grow by equal differences over equal intervals, and exponential functions grows by equal factors over equal intervals. Provide tables of values as examples and assist the student in locating the tables that represent linear or exponential functions by examining how the function changes over equal periods. When the student is proficient differentiating these two types of functions from tables of values, reintroduce verbal descriptions of functions. Emphasize that both kinds of relationships are characterized by an initial amount but are distinguished by the rate of change. Guide the student to identify terms that suggest equal differences over equal intervals and terms that suggest equal factors over equal intervals.
Provide an opportunity for students to explore the graphs of linear and exponential functions using a graphing calculator or a computer software program. Encourage the student to compare function growth over equal intervals. 
Moving Forward 
Misconception/Error The student confuses a percent reduction with a constant rate of change. 
Examples of Student Work at this Level The student concludes that a 50% reduction every four hours represents a constant rate of change.
Note: These students also provide incomplete justifications. 
Questions Eliciting Thinking What does a 50% reduction mean? What operation corresponds to 'taking off 50%'?
What distinguishes a linear relationship from an exponential relationship? What do you know about the rate of change of each? 
Instructional Implications Ask the student to calculate and record in a table the number of bacteria for several four hour periods. Guide the student to observe that the change in the number of bacteria is not the same for equal intervals of time. Indicate that there is a constant ratio of 0.5 or Â which indicates that the relationship is exponential. Model explaining that linear functions grow by equal differences over equal intervals and exponential functions grow or decay by equal factors over equal intervals. Ask the student to identify the rate of change as a constant difference (e.g., d = 1.5) or a constant ratio (e.g., r = 0.5) for each scenario. 
Almost There 
Misconception/Error The student understands the distinction between linear and exponential but provides inadequate or incomplete justifications. 
Examples of Student Work at this Level The student provides correct responses but does not explain each one completely.

Questions Eliciting Thinking How do you know the function is linear or exponential?
How do linear functions differ from exponential functions?
What distinguishes a linear relationship from an exponential relationship? What do you know about the rate of change of each? 
Instructional Implications Assist the student in elaborating and describing the characteristics of linear and exponential functions. Model explaining that linear functions grow by equal differences over equal intervals and exponential functions grow or decay by equal factors over equal intervals. Ask the student to identify the rate of change as a constant difference (e.g., d = 1.5) or a constant ratio (e.g., r = 0.5) for each scenario. 
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 is able to distinguish between descriptions of linear and exponential relationships. The student provides justifications that reference the constant difference of the linear function and the constant ratio of an exponential function. For example, the student explains:
 This is a linear function because there is a constant rate of change or a constant difference in the total cost. For each mile traveled, you add a $1.50 to the charge.
 This is an exponential function since the value of the function is changing by a constant ratio. Every four hours the number of bacteria decreases by 50%. Every four hours, there are half as many bacteria.
 This is an exponential function since the volume is changing by a constant ratio. Every three years, the volume is doubled or multiplied by two.
 This is a linear function because there is a constant rate of change or a constant difference in the altitude. The balloon rises 120 feet every minute.

Questions Eliciting Thinking Which tends to grow faster  the linear or exponential function? Why?
Can you tell a linear from an exponential function based on the initial value? 
Instructional Implications Ask the student to graph the functions fÂ (x) = 2x and Â on the same set of axes. Then ask the student to identify the intervals for which fÂ (x) = gÂ (x), f (x) < g (x),Â andÂ fÂ (x) >Â gÂ (x).
Consider implementing the MFAS tasks Prove Linear (FLE.1.1), Prove Exponential (FLE.1.1), How Does Your Garden Grow? (FLE.1.1), and Exponential Growth (FLE.1.1). 