Getting Started 
Misconception/Error The student makes a decision based on an irrelevant aspect of the context of the problem. 
Examples of Student Work at this Level The student says the equation is linear because a triangle has:
 All straight sides.
 Lines that connect.
The student says the equation is nonlinear because a triangle has:
 Three lines that all connect, not just one line.
 Sides that are not all the same size if it is isosceles.
 An area, which is not a straight line.
The student is unable to answer the question and writes, â€śI donâ€™t know what linear or nonlinear means.â€ť

Questions Eliciting Thinking What does the equation of a linear function look like?
If you graphed this equation would you get a triangle?
If you were just given the equation and told nothing else, how would you be able to tell if it is linear or nonlinear? 
Instructional Implications Review the concept of a linear function. Describe the equation of a linear function as one that can be written in the form ax + by = c (where b 0), but explain that linear functions may be written in other forms, some of which are quite useful (such as slopeintercept form). Explain to the student that a distinguishing feature of a linear function in two variables is that the variables are both raised to the first power. Provide many examples of equations of functions and ask the student to categorize the functions into those that are linear and those that are nonlinear.
Use a graphing utility to graph examples of both linear and nonlinear functions. Assist the student in relating features of the equations of the functions (e.g., the power of the variables) to the graphs. Be sure the student understands that when a function is given in context, it is the equation of the function rather than the context that determines whether or not a function is linear. 
Moving Forward 
Misconception/Error The student makes a decision based on an irrelevant or incorrect feature of the equation. 
Examples of Student Work at this Level The student says the function is linear because:
 It has m = (and bÂ = 0).
 It would graph as a straight line.
The student says the function is nonlinear because the equation:
 Contains a fraction.
 Looks â€śunusual,â€ť â€śmessyâ€ť or â€śis not a regular equation.â€ť
 Does not contain an initial value (e.g., a constant).
The student confuses linear and nonlinear writing, â€śIt is linear because it has a squared sign.â€ť
The student writes â€śyes (or no) because of the exponentâ€ť without explaining whether the â€śyesâ€ť or â€śnoâ€ť represents â€ślinearâ€ť or â€śnonlinear.â€ť

Questions Eliciting Thinking What features of the equation did you look at to determine whether the function is linear or not?
What is the general form of a linear equation? What do you know about each element of that equation? What does the coefficient of x describe and what does the constant describe?
What do you mean that the equation is â€śunusualâ€ť? What are you comparing it to? How is it different? 
Instructional Implications Review slopeintercept form of the equation of a line. Guide the student to identify the slope and the yintercept from an equation written in this form and to relate these parameters to the graph of the equation. Explain to the student that a distinguishing feature of a linear function in two variables is that the variables are both raised to the first power. Provide many examples of equations of functions and ask the student to categorize the functions into those that are linear and those that are nonlinear. Use a graphing utility to graph examples of both linear and nonlinear functions. Assist the student in relating features of the equations of the functions (e.g., the power of the variables) to the graphs. 
Almost There 
Misconception/Error The student provides both a correct and incorrect justification. 
Examples of Student Work at this Level The student explains that the equation is nonlinear because of the exponent on the variable, but also because the equation relates to triangles.

Questions Eliciting Thinking What does it mean for a function to be linear? How did you make your decision? Does the coefficient affect that definition? Does the context change your decision?
If you were just given the equation and told nothing else, how would you be able to tell if it is linear or nonlinear? 
Instructional Implications Provide feedback to the student with regard to both the correct and incorrect parts of his or her response. Address any misconceptions the student might have about the context. Be sure the student understands that when a function is given in context, it is the equation of the function rather than the context that determines whether or not a function is linear. Provide the student with additional examples of equations of functions (without context) and ask the student to categorize the functions into those that are linear and those that are nonlinear. Then provide the student with a context for each function, some with geometric applications, and have the student explain whether the context changes the categorizations. 
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 explains the equation is nonlinear because:
 The variable has an exponent (of two).
 If you graph it, it will not be a straight line (or it will be a parabola).
 It cannot be written in the form y = mx + b.

Questions Eliciting Thinking Can you explain what you mean by, â€śit has an exponentâ€ť? Is this equation nonlinear: ?
What will the graph of this equation look like? Can you find some points to graph to show what it will look like?
If you were just given the equation and told nothing else, would you be able to tell if it is linear or nonlinear? 
Instructional Implications Engage the student in a discussion of the effects of changing the exponent of the variable on a function and its graph. Ask the student to make a table of values for y = x, y = , and y = using the same values of x. Then have the student graph each function and compare the graphs to their equations.
Engage the student in a discussion of what happens to the graph of a function as the value of the exponent changes.
Expose the student to a variety of graphs to include linear, quadratic, cubic, rational, and exponential. Include both horizontal and vertical lines. Have the student indicate which graphs represent linear functions and why. Then give the student equations of various types of functions and have the student categorize the functions into linear and nonlinear. 