Getting Started 
Misconception/Error The student is unable to show that linear functions grow by equal differences over equal intervals for specific values of a given function or in general terms. 
Examples of Student Work at this Level The student states that fÂ (8)  fÂ (6) = fÂ (3)  fÂ (1) because 8  6 = 2 and 3  1 = 2.

Questions Eliciting Thinking What does f denote?
What does fÂ (8) mean?
Can you find the value of fÂ (8) when fÂ (x) = 3x + 2? 
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 fÂ (8) for the student. Allow the student to find fÂ (6), fÂ (3), and fÂ (1). Then ask the student to consider if fÂ (8)  fÂ (6) = fÂ (3)  fÂ (1) and to explain why. Ensure the student shows his or her calculations for fÂ (8), fÂ (6), fÂ (3), and fÂ (1) separately and neatly before attempting to conclude that fÂ (8)  fÂ (6) = fÂ (3)  fÂ (1).
Ask the student to make a table of functional values given a linear function such as fÂ (x) = x  1. Instruct the student to complete the table for at least five values of x. Then have the student compare the differences in the fÂ (x) values over equal intervals of x [e.g., compare the difference between fÂ (2) and fÂ (0) to the difference between fÂ (0) and fÂ (2)]. Guide the student to observe that these differences 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 a linear function grows by equal differences over equal intervals in general. 
Examples of Student Work at this Level The student can show that a linear function grows by equal differences over equal intervals for specific values of a given function, but not in general.

Questions Eliciting Thinking What do you know about a function that is linear?
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 linear. Remind the student that every linear function can be written in the form fÂ (x) = mx + b.
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 representingÂ Â as , and then have the student write equivalent expressions for , , andÂ . Guide the student through the process of reasoning from the assumptionÂ Â to the conclusionÂ .
Consider implementing the CPALMS problem solving task Equal Differences Over Equal Intervals 1Â (ID43645). 
Almost There 
Misconception/Error The student proves the converse of the statement. 
Examples of Student Work at this Level The student reasons algebraically from the conclusion: Â to the assumption,Â .

Questions Eliciting Thinking What does â€śif A then Bâ€ť mean? What is the assumption and what is the conclusion in this statement?
If you are going to prove an ifthen statement such as â€śif A then B,â€ť what do you assume is true? What are you trying to prove?
What did you assume is true when you began your proof? What should you have assumed is true? 
Instructional Implications Explain to the student the structure and meaning of a conditional statement. Be sure the student understands that when proving a statement of the form, â€śIf A, then B,â€ť A is the assumption (or what can be assumed to be true) and B is the conclusion (or the statement to be proven). Ask the student to write the proof so he or she is assuming thatÂ Â is true. Guide the student to reason from this statement to the conclusion:Â .
Consider introducing the converse, contrapositive, and inverse of the conditional statement. Discuss the structure of each and which are logically equivalent. Be sure the student understands that a conditional statement and its converse are not logically equivalent.
Consider implementing the MFAS tasks Compare Linear & Exponential Functions (FLE.1.3) and Comparing Linear and Exponential Functions (FIF.3.9). 
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 if fÂ (x) = 3x + 2, then fÂ (8)  fÂ (6) = fÂ (3)  fÂ (1). The student calculates fÂ (8) = 26, fÂ (6) = 20, fÂ (3) = 11, and fÂ (1) = 5 and shows that fÂ (8)  fÂ (6) = 6 = fÂ (3)  fÂ (1).
The student then proves that linear functions grow by equal differences over equal intervals by showing that ifÂ Â thenÂ Â as follows: Suppose f is a linear function such that fÂ (x) = mx + b. IfÂ Â thenÂ . ButÂ . Likewise, . Therefore,Â . 
Questions Eliciting Thinking Where in the proof did you use the fact that f is linear? How did you use the assumption thatÂ ?
Can you draw a diagram on the coordinate plane to illustrate this proof? 
Instructional Implications Consider implementing the MFAS tasks Compare Linear & Exponential Functions (FLE.1.3) and Comparing Linear and Exponential Functions (FIF.3.9). 