Getting Started 
Misconception/Error The student does not understand that the initial value (or yintercept) of each function must be identified in order to determine who paid the least for their coffee roasting machine. 
Examples of Student Work at this Level The student:
 Says Elizabeth’s coffee roasting machine costs less because there is no equation.
 Says the graph is a line so it costs less.
 Compares total cost for some portion of the domain or for a specific value of the domain.
 Compares the rates of change of the two functions.

Questions Eliciting Thinking What do you know about linear functions? If an equation of a linear function is in the form y = mx + b, what does the m tell you? What does the b tell you?
Can you explain the meaning of the 7 and the 55 in Daniel’s equation?
Can you explain what the graph is showing? 
Instructional Implications Review linear functions and the various ways that they can be described (with equations, tables, graphs, and verbal descriptions). Focus on the rate of change and initial value in a linear function, and relate these components of the equation to the slope and yintercept of the graph. Review how to calculate a value of one variable given a value of the other. Provide additional examples of linear functions that model the relationship between realworld quantities and ask the student to identify and compare properties of functions represented in different ways. 
Moving Forward 
Misconception/Error The student is able to correctly determine the cost of a roasting machine from only one of the representations. 
Examples of Student Work at this Level The student identifies the cost of:
 Elizabeth’s roasting machine as $60 or $60,000 but is unable to correctly determine the cost of Daniel’s roasting machine.
 Daniel’s roasting machine as $55 (or $55,000) but is unable to correctly determine the cost of Elizabeth’s roasting machine.

Questions Eliciting Thinking What do the coordinates of the points on the graph tell you?
What point on the graph tells you Elizabeth’s initial cost?
What part of the equation tells you Daniel’s initial cost?
What does the seven in Daniel’s equation tell you? What does the 55 in Daniel’s equation tell you? 
Instructional Implications Review the important properties of linear functions (e.g., rate of change and initial value) and how to identify and interpret them. Provide sample graphs, tables, equations, and verbal descriptions of functions, and ask the student to identify the xintercept, yintercept, rate of change, an xvalue when a yvalue is given, and a yvalue when an xvalue is given. Ask the student to describe in general terms the significance of each of these properties of a function (e.g., slope is the increase in a yvalue when an xvalue increases by one, yintercept is the yvalue when the xvalue is 0).
Provide the student with a linear function represented by a table. Have the student represent the same function with a graph and an equation. Help the student to identify the rate of change in all three representations. Then provide the student with two different functions (e.g., one represented by a graph and the other represented by a table). Challenge the student to determine and compare the rate of change of each function.
Consider implementing MFAS task Speed Reading (8.F.1.2) for further assessment. 
Almost There 
Misconception/Error The student understands that the initial value (or yintercept) of each function must be identified in order to determine who paid the least for their coffee roasting machine but does not correctly report the costs. 
Examples of Student Work at this Level The student correctly identifies both the initial value given in the equation and the yintercept of the graph but does not report the costs correctly. The student says that the costs are $55 and $60 rather than $55,000 and $60,000.

Questions Eliciting Thinking What do these values, 55 and 60, actually mean in this problem?
What are the units of measure for these values? 
Instructional Implications Remind the student that in both the equation and the graph, the costs are given in the thousands of dollars. Ask the student to revise his or her responses to reflect this.
Provide additional opportunities to interpret and explain the rate of change, initial value, and selected solutions of linear functions in the context of word problems. 
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 states that Daniel’s machine cost $55,000 and determined this by identifying the initial value in Daniel’s cost equation. The student states that Elizabeth’s machine cost $60,000 and determined this by reading the yintercept from the graph of her costs. The student concludes that Daniel’s machine cost the least.
The student may not initially describe how these costs were determined, but upon questioning can immediately explain.

Questions Eliciting Thinking What does the coefficient of x in Daniel’s cost equation represent?
What does the slope of Elizabeth’s graph represent? 
Instructional Implications Have the student graph Daniel’s equation on the same set of axes as Elizabeth’s graph. Challenge the student to determine the significance of the point of intersection of the two graphs and explain it in the context of the problem. 