Getting Started 
Misconception/Error The student is unable to correctly determine the rates of production from either the table or the equation of the functions. 
Examples of Student Work at this Level The student:
 Compares the quantities of vitamin C for the two methods at specific values of the domain [e.g., the student calculates the quantity of vitamin C after 15 hours using the equation for the old method (15, 558) and compares it to the corresponding quantity given in the table (15, 678)].
 Inconclusively investigates the rate of production for one method and then makes a decision.

Questions Eliciting Thinking What does production rate mean? How did you determine each production rate?
What do you suppose the 36 and the 18 in the equation represent?
Is the relationship between the quantities in the table linear?
What is the difference between rate of production and the variable Q? 
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 production rate from only one of the two representations. 
Examples of Student Work at this Level The student correctly identifies the rate of production of only one of the two methods.

Questions Eliciting Thinking How did you determine each production rate?
What do you think the 18 in the equation represents?
Do you think that the storage hopper was empty when the production run for the new method began? How could you tell? 
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 an equation). Challenge the student to determine and compare the rate of change of each function. 
Almost There 
Misconception/Error The student is unable to correctly interpret the rates of change in the context of the problem. 
Examples of Student Work at this Level The student correctly identifies the numerical values of the rates of production but is unable to assign a unit or explain what these values mean in the context of the problem.

Questions Eliciting Thinking What do these values, 34 and 36, actually mean?
What are the units of measure for the rates of production you found? 
Instructional Implications Guide the student to identify the units of each rate of production (i.e., kilograms per hour) and ask the student to explain what the rates mean in the context of the problem (e.g., the old method produces 36 kilograms of vitamin C per hour while the new method only produces 34 kilograms of vitamin C per hour). 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 identifies the production rate of the old method as 36 kilograms/hour and calculates the production rate of the new method as 34 kilograms/hour. The student concludes that the production rate is higher using the old method.

Questions Eliciting Thinking If the old method has a higher production rate, does that mean the old method will have a higher total production at any given time? Why or why not?
Can you write an expression (or equation) to represent the relationship in the table? 
Instructional Implications Encourage the student to explore the properties of these functions further. Have the student produce graphs for each method. Challenge the student to determine an approximate input value at which the output value for both functions would be the same. Ask the student to interpret and compare the yintercepts of the graphs. 