Getting Started 
Misconception/Error The student does not understand the significance of the two pairs of associated values given in the problem. 
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., 230 + 100 or 230 + 330).
 Divides a cost (c) by its associated rate (r) and writes an equation of the form .
 Writes an equation in one variable and attempts to solve it.
 Uses one pair of coordinates as parameters in the equation.

Questions Eliciting Thinking What is the independent variable? What is the dependent variable?
What values were given in the problem? Which is a value of the independent variable and which is a value 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).Â
 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 in each format and to write the associated equation in slopeintercept form.
Provide additional opportunities for the student to write equations of linear functions given a verbal description that includes two pairs of values. 
Moving Forward 
Misconception/Error The student is unable to correctly determine the rate of change. 
Examples of Student Work at this Level The student recognizes the given information as representing two ordered pairs but:
 Calculates the rate of change incorrectly.
 Generates additional ordered pairs but is unable to find the rate of change.

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? 
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 descriptions, graphs, tables, and equations) and interpreting the rate of change in context.
Review the formula for determining the rate of change given two points (i.e., Â 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 rate of change (slope) cannot be determined from a single point [e.g., interpreting the point (100, 230) as indicating a slope of 2.3] unless the relationship is proportional.
Provide additional opportunities for the student to write equations of linear functions given a verbal description that includes two pairs of values. 
Almost There 
Misconception/Error The student is unable to determine the initial value. 
Examples of Student Work at this Level The student recognizes the given information as representing two ordered pairs and correctly calculates the rate of change as or 1.8, but the student:
 Is unable to write the equation.
 Does not recognize that the function has an initial value and writes the equation as
 Recognizes the need for an initial value but is unable to calculate it, providing an answer such as y = .

Questions Eliciting Thinking 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). Provide instruction on calculating the initial value of a function given two ordered pairs. 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).
Provide additional opportunities for the student to write equations of linear functions given verbal descriptions that include two pairs 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 writes two ordered pairs from the given values, correctly calculates the rate of change as or 1.8, and correctly calculates the initial value as 50. The student writes the equationÂ . The student shows work clearly or writes an appropriate explanation. 
Questions Eliciting Thinking What does mean in the context of the problem?
What does 50 mean in the context of the problem? 
Instructional Implications Provide examples of linear functions in context given by verbal descriptions, tables of values, and graphs. Ask the student to write an equation to represent each linear function. Ask the student to interpret both the initial value and rate of change in context. 