Getting Started 
Misconception/Error The student is unable to determine the rate of change of the linear function from its graph. 
Examples of Student Work at this Level The student:
 States the water is decreasing but cannot determine the rate of change.
 Counts grid increments to calculate the â€śriseâ€ť and the â€śrunâ€ť (disregarding the scale on the axes).
 Determines consecutive differences of the yvalues but is unable to determine the initial value.
 Determines the rate of change to be .
 Writes a ratio of the xvalue to the yvalue (e.g., ).

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 the water level decreases each hour? Does the level decrease by the same amount each hour?
What does initial value mean? How do you find the initial value?
Can you determine how much water is in the pool before it starts draining? 
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 descriptions, equations, tables, and graphs).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 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 descriptions, graphs, tables, and equations) and interpreting the rate of change in context.
Review the formula for determining 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.
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, explain that the initial value is the yvalue that corresponds to an xvalue of zero and indicates the amount of water in the pool before the pool is drained. Assist the student in developing a strategy for finding the initial value when it is not included in a table of values or on the graph.
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 rate of change as but is unable to determine the initial value.
The student writes the initial value as an ordered pair or a ratio. 
Questions Eliciting Thinking What does initial value mean? How do you find the initial value?
Can you determine how much water was in the pool before it started draining? 
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, explain that the initial value is the yvalue that corresponds to an xvalue of zero and indicates the amount of water in the pool before the pool is drained. Assist the student in developing a strategy for finding the initial value when it is not included in a table of values or on the graph.
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 and the initial value is 10,080. The student is unable to correctly interpret the meaning of one or both values in context. 
Questions Eliciting Thinking What do you mean by a rate of change of ? What does the 720 represent? What does the one represent?
What do you mean by an initial value of 10,080? What does that value 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., 720 gallons of water) associated with a corresponding amount of the independent variable (e.g., one hour). Then guide the student to interpret the rate of change in terms of the context of the problem. Model explaining, â€śThe water in the pool decreases by 720 gallons each hour.â€ť
Encourage the student to write the initial value as an ordered pair, including units [e.g., as (0 hours, 10,080 gallons)]. Then ask the student to consider the context of the problem to determine that the initial value, 10,080, represents the gallons of water in the pool before it started draining.
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 720 gallons/hour which means the water in the pool decreases by 720 gallons each hour.
 The initial value is 10080 which is the amount of water in gallons in the pool before it started draining.

Questions Eliciting Thinking At this rate, how many total hours would it take to completely drain the pool? In how many hours, would the pool be halfway drained?
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 positive?
What might cause this function to be nonlinear? 
Instructional Implications Challenge the student to write an equation to represent the linear function depicted in the graph. Ask the student to use the equation to determine the xintercept and explain its meaning in the context of the problem. Consider using the MFAS task Profitable Functions. 