Getting Started 
Misconception/Error The student cannot correctly identify the type of function, describe its graph, or explain the equation values. 
Examples of Student Work at this Level The student describes the function in general terms such as:
 y = mx + b equation.
 â€śInput/output functionâ€ť or â€śindependent and dependent equation.â€ťÂ
The student describes the graph as:
 A parabola, curve or â€śvshape.â€ť
 Nonlinear.
The student describes the elements of the equation without proper context:
 Five is positive and is a fraction.Â
 â€śYou add fiveâ€ť and â€śmultiply by .â€ť
 â€śFive is the constant.â€ť

Questions Eliciting Thinking What does the question mean by â€śtype of function?â€ť What are some different types of functions? What are some different types of graphs?
How can you decide from an equation whether a graph is linear or nonlinear? What part of the equation do you look at to make that decision?
How can you use the equation to graph some points to check what the shape will be?
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? 
Instructional Implications Review the concept of a function emphasizing that a function is a relation in which each input value is paired with one output value. Provide examples of both functions and relations that are not functions described in a variety of ways (tables of values, mapping diagrams, algebraic rules, graphs, and verbal descriptions). Be sure to include many nonlinear examples of functions. Guide the student to carefully consider each example to determine whether or not it represents a function. Model explaining and justifying the reasoning behind the determination.
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 identify those that are linear. Then have the student justify his or her choices by writing the equations in the form ax + by = c. Next, focus on the slopeintercept form of the equation of a line. Provide examples of equations written in this form and ask the student to make a table of values for each equation. Then have the student use the table of values to graph each function. Use the graph as a means to explain the parameters of linear functions (e.g., rate of change or slope and initial value or yintercept). Guide the student to identify the slope and yintercept of an equation written in slopeintercept form and to use these parameters to graph the equation. Finally, provide examples of linear functions in context. Ask the student to find particular output values given their inputs and particular input values given their outputs. Challenge the student to explain the meaning of the intercepts and the slope in the context of the problem.
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]. Illustrate this idea using tables of values and graphs. 
Making Progress 
Misconception/Error The student does not use mathematically precise terminology to identify the type of function, describe its graph, or explain its parameters. 
Examples of Student Work at this Level The student uses imprecise terminology to describe the function as a:
 Slope function,
 Standard equation,
 Straight equation,
 Line, or
 Regular function.
The student describes the graph:
 As â€ślinearâ€ť or â€śa line function.â€ť
 Too exclusively, as a horizontal or vertical line.
 As â€śpositiveâ€ť or â€śincreasing upward,â€ť without specifying that it is a line.
The student cannot adequately describe the parameters in the equation:
 Reversing the meaning of the slope and yintercept.
 Listing 5 = b and Â = m without describing what m or b means.
 Saying the line will â€śstart at fiveâ€ť and means â€śit will go up two and over three.â€ť

Questions Eliciting Thinking What do you mean by a â€śslope function?â€ť What would the graph look like? How would you describe it?
What is a mathematical word for a function that graphs as a straight line?
How do you know from an equation if a line is horizontal? Can you draw a sketch of what you think the graph of this equation will look like?
Can you explain what you mean by the graph is increasing? What will that look like?
What does slope mean? What does yintercept mean?
What effect does each number have on the graph? How will the graph of the line change if you change each number? 
Instructional Implications Review the concept of a linear function, solutions of linear functions, and forms of the equations of a linear function. Model the use of appropriate terminology (e.g., coefficient, constant, linear, line, slope or rate of change, xintercept, and yintercept or initial value). Provide sample graphs, tables, equations, and verbal descriptions of linear functions, and ask the student to identify the xintercept, yintercept (initial value), slope (rate of change), an xvalue given a specific yvalue, and a yvalue given a specific xvalue. Be sure the student understands the significance of each of the two parameters of a linear function. Assist the student in identifying the slope and yintercept from the equation of a linear function and using these two values to graph the function. 
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 answers all questions completely and correctly. The student says:
 The function is linear.
 The graph is a line. The student may add that the line is rising from left to right or that it is a diagonal line that is increasing. The student may draw a picture of a straight line rather than describing it.
 Five is the yintercept which means (0, 5) is the place where the line crosses the yaxis.Â
 is the slope of the line, which describes its steepness or the amount of change in y compared to a corresponding change in x.

Questions Eliciting Thinking What about the equation indicated that it represented a line?
What would the equation of a horizontal or vertical line look like?
You said the graph of this function is a line. Are all lines examples of functions? Are all functions lines? Explain. 
Instructional Implications Engage the student in a discussion of the effects of changing parameters of a linear function on its graph. Have the student use a graphing utility to explore the effects of changing the slope and yintercept. Then ask the student to write a general statement that explains the effects of changing each parameter on the graph. Give the student an equation such as y = 3x â€“ 5 and ask the student to predict the effect of changing parameters in specific ways, such as changing the coefficient of x to 3 or 10 or changing the constant to 0 or 4.
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. 