Getting Started 
Misconception/Error The student does not understand the significance of the ordered pairs described in the table. 
Examples of Student Work at this Level The student does not recognize that the given quantities represent ordered pairs that can be used to find the parameters (rate of change and initial value) needed to write the equation. The student:
 Conducts arithmetic operations on the quantities (e.g., sums coordinates).
 Understands the form of the equation but does not know how to find the parameters.
 Uses coordinates as parameters in the equation.

Questions Eliciting Thinking What is the independent variable? What is the dependent variable?
What is the significance of the values given in the table? Which are values of the independent variable and which are values of the dependent variable?
What is the form of a linear 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).
 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 a table of values. 
Moving Forward 
Misconception/Error The student is unable to correctly determine the value of the rate of change. 
Examples of Student Work at this Level The student recognizes the given information as representing ordered pairs but calculates the rate of change incorrectly. 
Questions Eliciting Thinking What were you trying to calculate? What is this value called and what does it mean? How will it help you write the equation?
How do you calculate rate of change?
What is the rate of change in the context of this problem?
Is the rate of change (or slope) going to be positive or negative in this function? 
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 of 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 various types of given information (verbal description, graph, table, equation) and interpreting the rate of change in context.
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 and remind the student to include the sign of the slope.
Explain that typically rate of change (slope) cannot be determined from a single point [e.g., interpreting the point (29, 3600) as indicating a slope of 122.9].
Provide additional opportunities for the student to write equations of linear functions given a table of values. 
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 correctly calculates the rate of change as 6, but the student:
 Does not understand how to determine the initial value and does not write an equation.
 Does not recognize that the initial value is given in the table and attempts to calculate it.
 Selects another value from the table as the initial value (e.g., 29 or 3600).
 Omits the initial value and writes the equation as y = 6x.
 Incorrectly identifies time as the dependent variable and determines the initial value is zero.

Questions Eliciting Thinking What is the basic form of a linear function?
What does the b represent in the equation y = mx + b?
Can you think of what the initial value might mean in the context of this problem?
What are the different ways you know to find an initial value? 
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 a table of values. 
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 correctly calculates the rate of change as 6, recognizes the initial value as 3776, and writes the equation y = 6x + 3776. The student shows work clearly or writes an appropriate explanation. 
Questions Eliciting Thinking What does 6 mean in the context of the problem?
What does 3776 mean in the context of the problem?
How long will it take Taro and Jiro to hike down the mountain? 
Instructional Implications Provide examples of linear functions in context given by verbal descriptions, tables of values, and graphs. Ask the student to write equations to represent each linear function. Ask the student to interpret both the initial value and rate of change in context. 