Getting Started 
Misconception/Error The student is unable to show that exponential functions grow by equal factors over equal intervals for specific values of a given function or in general terms. 
Examples of Student Work at this Level The student reasons:

Questions Eliciting Thinking What does f denote?
What does f(4) mean?
Can you find the value of f(4) when ?Â 
Instructional Implications Review the definition of a function and function notation. Remind the student how to find the value of a function for a given input. Model finding the value of f(4) for the student. Allow the student to find f(2), f(7), and f(5). Then ask the student to evaluate each of Â andÂ Â to determine if they are equal. Guide the student to determine the equality by using properties of exponents. For example, show the student that each ratio is equal to Â rather than completely evaluating each ratio.
Ask the student to make a table of functional values given an exponential function such as . Instruct the student to complete the table for at least five values of x. Then have the student compare ratios of values of f(x) over equal intervals [e.g., compareÂ Â toÂ ]. Guide the student to observe that the ratios are equal.
Consider implementing the CPALMS problem solving task Equal Differences Over Equal Intervals 1Â (ID 43645). 
Moving Forward 
Misconception/Error The student is unable to show that an exponential function grows by equal factors over equal intervals in general. 
Examples of Student Work at this Level The student can show that an exponential function grows by equal factors over equal intervals for specific values of a given function, but not in general.

Questions Eliciting Thinking What do you know about exponential functions?
Can you show the two expressions in the first problem are equal by using properties of exponents rather than evaluating each?
Do you see any similarity between the first problem and the second problem? 
Instructional Implications Discuss with the student what it means for a function to be exponential. Remind the student that every exponential function can be written in the form . Review properties of exponents and guide the student to determine the equality of the two expressions in the first problem by using properties rather than evaluating. For example, show the student thatÂ .
Review the meaning of the notation used in the statement to be proven. Guide the student to see the similarity between the first problem and the second problem. Assist the student in representing ,Â and then have the student write equivalent expressions for ,Â Â andÂ . Guide the student through the process of reasoning from the assumption, Â to the conclusion .
Ask the student to use the equation for an exponential function and write an alternate form forÂ Â using the substitution property. Guide the student to recognize thatÂ Â and then have the student write equivalent formsÂ forÂ ,Â Â andÂ . Guide the student through the simplification process including factoring out aÂ and using the quotient of powers property to relate the exponents of the b terms.
Consider implementing the CPALMS problem solving task Equal Differences Over Equal Intervals 1Â (ID 43645). 
Almost There 
Misconception/Error The student develops an essentially correct proof that contains some errors or omissions. 
Examples of Student Work at this Level The student can show that an exponential function grows by equal factors over equal intervals for specific values of a given function and writes an essentially correct proof for the general statement. However, the student:
 Omits an important step. The student does not make explicit the reason that Â (i.e., becauseÂ ).
 Includes unnecessary steps. The student states that â€śSince b = b (Reflexive Property) and we are givenÂ , thenÂ .â€ť
 Writes a statement that is not true (e.g., the student states that ).

Questions Eliciting Thinking How do you know thatÂ ? What allowed you to conclude this?
What property of exponents allows you to reason fromÂ Â toÂ ?
Can you show me how you rewroteÂ Â as ? 
Instructional Implications Provide feedback concerning any errors or omissions and allow the student to revise his or her proof.
Consider implementing the MFAS tasks Compare Linear & Exponential Functions (FLE.1.3), Comparing Linear and Exponential Functions (FIF.3.9), and 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 shows that:
 Â andÂ Â soÂ .
 Â andÂ . The student reasons that sinceÂ , thenÂ Â so thatÂ .

Questions Eliciting Thinking Where in the proof did you use the fact that f is exponential? How did you use the assumption thatÂ ?
What property of exponents allows you to conclude thatÂ ?
Is there a property of exponents that allows you to reason fromÂ Â toÂ ?
Would a diagram on the coordinate plane help you to explain this concept? 
Instructional Implications Consider implementing the MFAS tasks Compare Linear & Exponential Functions (FLE.1.3), Comparing Linear and Exponential Functions (FIF.3.9), and Exponential Growth (FLE.1.1). 