Getting Started 
Misconception/Error The student is unable to determine the rate of change and the initial value of a linear function represented by a table of values. 
Examples of Student Work at this Level The student:
 Attempts to graph the data to determine the rate of change and initial value, and provides incorrect answers.
 Determines consecutive differences of the yvalues (dependent values) and determines the initial value to be $255.
 Attempts to find the rate of change and determines the initial value to be $160.

Questions Eliciting Thinking What does rate of change mean? How do you find the rate of change? Is the rate of change constant?
Can you determine how much Brent saves each month? Does Brent save the same amount each month?
What does initial value mean? How do you find the initial value?
Can you determine how much money Brent had in his account before he started putting $85 in each month? 
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 twovariables as ordered pairs of numbers.
Review the concept of rate of change and how it is represented in graphs, equations, and tables. 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 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.
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 zero—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, explain that the initial value is the yvalue that corresponds to an xvalue of zero and indicates the amount of money in Brent’s account at zero months. Assist the student in developing a strategy for finding the initial value when it is not included in a table of values.
Provide additional opportunities for the student to identify and interpret the rate of change and the initial value in linear relationships. 
Moving Forward 
Misconception/Error The student is able to determine the rate of change but unable to determine the initial value of the linear function represented in a table of values. 
Examples of Student Work at this Level The student determines the correct rate of change but identifies the first value of the dependent variable given in the table as the initial value (e.g., $160).

Questions Eliciting Thinking What does initial value mean? How do you find the initial value?
Can you determine how much money Brent had in his account before he started putting $85 in each month? 
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 zero—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, explain that the initial value is the yvalue that corresponds to an xvalue of zero and indicates the amount of money in Brent’s account at zero months. Assist the student in developing a strategy for finding the initial value when it is not included in a table of values.
Provide additional opportunities for the student to identify and interpret the rate of change and the initial value in linear relationships. 
Almost There 
Misconception/Error The student is unable to interpret the rate of change and/or initial value in terms of the situation it models. 
Examples of Student Work at this Level The student determines the rate of change is 85 and the initial value is 75. The student is unable to correctly interpret the meaning of one or both of these values in context.

Questions Eliciting Thinking What do you mean by a rate of change of 85? What does the 85 represent?
What do you mean by an initial value of 75? What does the 75 represent? 
Instructional Implications Encourage the student to assign units of measure to each part of the rate of change ratio. Guide the student to explain the meaning of the rate of change as an amount of the dependent variable (e.g., $85.00) associated with a corresponding amount of the independent variable (e.g., one month). Then guide the student to interpret the rate of change in terms of the context of the problem. Model explaining, “Brent’s account balance increases $85 each month.”
Encourage the student to write the initial value as an ordered pair, including units [e.g., as (0 months, $75)]. Then ask the student to consider the context of the problem to determine that the initial value, $75, represents the amount of money in Brent’s account when he started saving for the television.
Provide additional opportunities for the student to identify and interpret the rate of change and the initial value in linear relationships. 
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 determines:
 The rate of change is $85/month, which means that Brent’s account balance increases by $85 every month.
 The initial value is $75, which means that there was $75 in Brent’s account when he started saving.

Questions Eliciting Thinking What ordered pair represents the initial value? If this linear function is graphed, where would you find this ordered pair?
What would it mean if the rate of change were negative?
What might cause this function to be nonlinear?
Why is the difference in consecutive amounts of money in the table sometimes $255 and sometimes $170? 
Instructional Implications Challenge the student to construct an equation to model the linear relationship between the two quantities. Have the student use his or her equation to determine how much Brent will have in his account after one year and the number of months Brent must save in order to save for a Smart TV that costs $2,115.
Consider using the MFAS task Trekking Functions (8.F.2.4) or the MFAS task Construction Function (8.F.2.4). 