Getting Started 
Misconception/Error The student does not understand how to use either function representation to answer questions about properties of the functions. 
Examples of Student Work at this Level The student makes assumptions about the context without using or referencing the functions. For example, the student states:
 The bus travels farther because it has to make stops.
 It is faster for Jeremy to ride his bike because it is a more direct route.
 The bus is faster because it can travel faster than a bike.
The student can interpret some features of the graphed function but is unable to effectively relate them to the problem context.
The student confuses the two variables in the problem and attempts to calculate distances to answer the question about time.

Questions Eliciting Thinking What does the graph represent in the context of this problem?
What do the components of the equation represent in the context of the problem?
If you graph the equation, will it be the same line as the graphed line? Why or why not? 
Instructional Implications Ask the student to identify the two variables in the problem and explain what each represents in the context of the problem. Have the student graph the function given as an equation on the graph provided. Assist the student in interpreting the intercepts and the slopes of each function in the context of the problem.
Review linear functions and the various ways they can be described (with equations, tables, graphs, and verbal descriptions). Focus on the rate of change and initial value of a linear function and relate these components of the equation to the slope and yintercept of the graph. Review how to find the xintercept both graphically and algebraically. 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 attempts to answer the questions using properties of the functions but makes significant errors. 
Examples of Student Work at this Level The student:
 Attempts to answer the question about time by graphing the bus equation and comparing the slopes and yintercepts.
 Uses the distance travelled when riding the bike (e.g., four miles) to calculate the time it takes to ride the bus.
 Compares the rates of the two modes of transportation without taking into account the distances travelled.

Questions Eliciting Thinking What does the slope of the graph represent in the context of this problem?
What does the yintercept of the graph represent in the context of this problem?
What does the coefficient of t in the equation represent in the context of this problem?
What does the constant in the equation represent in the context of this problem?
What does the xintercept represent in the context of this problem?
Is it possible for a vehicle to travel faster than a second vehicle, but arrive later than the second vehicle? Why or why not? 
Instructional Implications Discuss with the student the relationship between distance, rate, and time. Then provide linear graphs on the same set of axes that represent distance as a function of time. Draw the graphs so each has a different slope but the same yintercept. For each line, discuss the meaning of the slope and the x and yintercepts in terms of distance, rate, and time. Provide a second set of graphs, each with a different slope but with the same xintercept. For each line, discuss the meaning of the slope and the x and yintercepts in terms of distance, rate, and time.
Reverse the roles of distance and time (i.e., represent time as a function of distance) and provide two more graphs. Challenge the student to interpret and compare the slopes and the x and yintercepts in terms of distance, rate, and time. 
Almost There 
Misconception/Error The student does not clearly justify his or her answers with mathematical support. 
Examples of Student Work at this Level The student provides correct answers but no supporting work or explanation. When asked, the student is unable to justify his or her answers.

Questions Eliciting Thinking How do you know the bus travels six miles to the school and the bike travels four miles to the school?
How do you know the bus takes 12 minutes to get to school and the bike takes 16 minutes to get to school?
How did you find your answers? 
Instructional Implications Guide the student to consider what he or she did in order to answer each question. Encourage the student to compose explanations and verbalize them before writing them on paper. Have the student rewrite his or her responses and provide feedback.
Give the student more experience explaining mathematics and justifying answers. Allow the student to work with a partner in order to compare interpretations and explanations before finalizing them on paper.
Share examples of complete, correct, and wellwritten responses prepared by other students. 
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 distance that Jeremy travels to school on his bike (four miles) by reading the yintercept of the graph and the distance that Jeremy travels to school on the bus (six miles) by identifying the initial value in the equation or by substituting zero for t and solving for d.
 The student determines that it takes Jeremy 16 minutes to travel to school on his bike by reading the xintercept of the graph and that it takes Jeremy 12 minutes to travel to school on the bus by substituting zero for d and solving for t in the equation.

Questions Eliciting Thinking How much faster would Jeremy have to pedal to arrive at the school in the same amount of time as the bus?
Is there a time at which both the bus and the bike are the same distance from the school? 
Instructional Implications Have the student graph the equation on the same set of axes as the given 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.
Ask the student to calculate the point of intersection algebraically. 