Getting Started 
Misconception/Error The student is unable to provide an example of a nonlinear function. 
Examples of Student Work at this Level The student:
 Sketches a graph of a diagonal line and indicates that it is nonlinear.
 Writes a linear equation that is not in slopeintercept form.
 Writes a linear equation in one variable.
 Sketches a graph of a nonlinear relation that is not a function.

Questions Eliciting Thinking What does linear mean? What does the graph of a linear function look like? What does the equation of a linear function look like?
What does nonlinear mean? How might the graph of a nonlinear function look?
Does your equation represent a function? What are the inputs? What are the outputs?
What must be true of the graphs of all functions? 
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.
Be sure the student understands that the graph of a linear function is always a nonvertical line and every nonvertical line represents a linear function. Explain that a distinguishing feature of a linear function is that the rate of change is constant [e.g., any two values of x (or inputs) that differ by the same amount will have y values (or outputs) that differ by the same amount]. Use tables of values and graphs of both linear and nonlinear functions to compare rates of change. Ask the student to draw graphs of nonlinear functions and identify equal intervals of the domain for which the corresponding intervals of the range are unequal. 
Making Progress 
Misconception/Error The student is unable to justify his or her example. 
Examples of Student Work at this Level The student provides an equation, graph, or table of a nonlinear function. To justify the example, the student:
 Refers to the definition of a function or the vertical line test.
 States that the function is â€śnot increasing or decreasing.â€ť
 Provides an unclear explanation.
 States that input and output are not consistent.

Questions Eliciting Thinking What makes a function linear? What do you know about the equations and graphs of linear functions?
This is useful for telling me that this is a function, but how can you explain that your example is nonlinear?
What do you mean by â€śnot increasing or decreasing?â€ť 
Instructional Implications 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. Consequently, when a variable in the equation of a function is raised to a power other than one, the function is not linear. 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. Provide additional examples of equations of functions and ask the student to categorize the functions into those that are linear and those that are nonlinear and explain the reasoning for the determination. 
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 provides an example of a nonlinear function using either an equation, graph, or table. The student:
 Sketches a parabola or the graph of another nonlinear function and explains that it is nonlinear because it is not a line.
 Creates a table of values that includes ordered pairs such as [(1, 1), (2, 4), (3, 9), (4, 16)] and explains that equal differences in xvalues do not correspond to equal differences in yvalues.
 Provides an equation such as yÂ = and explains that it is nonlinear because the variable is raised to the second power or its graph is not a line.

Questions Eliciting Thinking What do you know about linear functions? What does the equation of a linear function look like? What does its graph look like? What can you say about the rate of change as you move from one point to another on the graph? 
Instructional Implications Ask the student to provide examples of nonlinear functions in forms that he or she did not use on this task (graph, table, or equation).
Describe realworld examples of nonlinear relationships (e.g., comparing distance to time for an accelerating vehicle). Ask the student to create a graph, chart, or equation for a nonlinear function with a realworld context. Ask the student to explain the significance of the different rates of change within the context. 