Getting Started 
Misconception/Error The student does not understand the basic form of a linear function and the role of its two parameters. 
Examples of Student Work at this Level Instead of writing an equation, the student:
 Describes some feature of the graph or the relationship between the two variables.
 Writes an expression or equation unrelated to the graph.
 Finds the slope (either correctly or incorrectly) without writing an equation.
The student writes an equation of the form y = mx. However, the student disregards the initial value and incorrectly calculates m.
The student writes an equation of the form y = mx + b. However, the student does not understand the role of the initial value and the rate of change in writing the equation. 
Questions Eliciting Thinking What is a linear function/equation? What are its two parameters?
What form does the equation for a linear function take?
What specifically do you need to know about the graph to write its equation? 
Instructional Implications Review:
 Independent and dependent variables and how functions that describe the relationship between them are represented by equations, tables, graphs, and realworld situations.
 The concept of a linear function and its representations (verbal description, equation, table, and graph).
 The slopeintercept form of a linear equation, y = mx + b, and its parameters (e.g., rate of change and initial value).Â
 Solutions of equations in two variables as ordered pairs of numbers.
Discuss what is needed to describe a linear relationship (e.g., rate of change and initial value) and to write its equation. Then provide the student with examples of linear functions presented in a variety of formats (verbal descriptions, tables of values, and graphs). Guide the student to identify the parameters of the relationship given each format and write the equation in slopeintercept form.
Provide additional opportunities for the student to write equations of linear functions given the graph of the function. 
Moving Forward 
Misconception/Error The student is unable to identify the rate of change of the function. 
Examples of Student Work at this Level The student understands the basic form of a linear function and may be able to identify the initial value as 200. However, the student is unable to correctly calculate the rate of change. For example, the student:
 Assumes the slope is 25 and then attempts to calculate theÂ yintercept.
 Disregards the scale on the axes when calculating the slope.
 Calculates the slope using the coordinates of a single point [e.g., uses the point (50, 200) to calculate the slope as ].

Questions Eliciting Thinking What is the rate of change in a function? If the rate of change were three, what would that mean?
How is rate of change calculated?
What is the difference between a positive rate of change and a negative rate of change? What sign should the rate of change in this equation have? 
Instructional Implications Review the concept of rate of change and how it is represented in graphs, equations, and tables. Ensure the student understands that rate of change is one of two parameters needed to write the equation of a linear function. Establish the definition of rate of change as the ratio of the amount change in the dependent variable to the amount of change in the independent variable. Explain that it can also be thought of as the amount of change in the dependent variable when the independent variable increases by one. Clarify that rate of change corresponds to the slope of the graph of the linear function.
Model the process of determining rate of change from a graph by highlighting the vertical and horizontal segments that correspond to the change in yvalues and the change in xvalues between two points on the graph. Emphasize that the lengths of these segments must be determined using the scales on the x and yaxes rather than by counting grid sections. Provide practice problems. Consider implementing the MFAS task Slope Triangles.
Review the formula for determining rate of change given two points (i.e., rate of change = ). Explain that this formula means that rate of change is the ratio of change in yvalues to the corresponding change in xvalues. Relate this formula to a graphical demonstration of slope.
Explain that typically rate of change (slope) cannot be determined from a single point [e.g., interpreting the point (50, 200) as indicating a slope of 4] since a single point provides no information about change.
Provide additional opportunities for the student to write equations of linear functions given their graphs. 
Almost There 
Misconception/Error The student is unable to determine the initial value of the function. 
Examples of Student Work at this Level The student recognizes that the function can be defined by an equation of the form y = mx + b and is able to determine the slope of the line but:
 Neglects to determine the initial value and writes the equation as y = 8x.
 Uses the xintercept (25) as the initial value.

Questions Eliciting Thinking What does the b represent in the equation y = mx + b?
What is a yintercept? How do you find it?
What is the initial value of a function? 
Instructional Implications Review the concept of initial value within a function. Discuss how it is represented on a graph (as the yintercept), in an equation (the value of y when x = 0), and in a table (the output value when the input is zeroâ€”note that this is not always included on tables and may have to be calculated). Discuss the significance of the initial value in realworld situations. For example, if a company charges $3 per item plus a flat $12 shipping fee, represented by the equation y = 3x + 12, the initial value is the price when zero items are purchased (e.g., the shipping fee). Consider implementing the MFAS task Competing Functions.
Provide additional opportunities for the student to write equations of linear functions given their graphs. 
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 is able to correctly determine both the slope and yintercept of the line and writes its equation as y = 8x â€“ 200. The student demonstrates understanding through shown work or a thorough explanation. 
Questions Eliciting Thinking What does a slope of eight mean in terms of Victoriaâ€™s store?
Does a slope of eight mean that Victoria charges $8 for each game? Might it mean something else?
What would it mean if the slope was 1? Would that be possible in this situation? 
Instructional Implications Ask the student to analyze the significance of a slope in this problem context and describe factors that might cause the slope to increase or decrease (increase in price per game or decrease in Victoriaâ€™s cost per game). Ask the student to explain why a slope of 1 is or is not possible in this context.
Challenge the student to find an example of a graph of a linear function in which a single point can be used to find the slope of the line. Ask the student to identify the point and the special type of linear relationship the graph represents. 