Getting Started 
Misconception/Error The student cannot identify and interpret key features of the graph. 
Examples of Student Work at this Level The student is unable to identify most or all of the key features referenced in the task. For example, the student says:
 The intercepts are 0 and 10 (the lower and upper bounds of the domain); the intercept is the maximum, (5, 600); or there is an xintercept at 11.
 The graph is increasing by intervals of 2 and 100; by 150 meters every kilometer; or over the interval from 0 to 10.
 The function is periodic because it represents an interval of time; because it is continuous; or because it is both increasing and decreasing.

Questions Eliciting Thinking In general, what is an intercept? Whatâ€™s the difference between an xintercept and a yintercept? Where on the graph is the yintercept?
What does it mean for a function to be increasing? Decreasing? How are intervals over which the function is increasing or decreasing described?
What does it mean for a function to be periodic? 
Instructional Implications Review the meaning of intercepts. Describe intercepts geometrically as the point(s) where a graph crosses an axis, and algebraically as points of the form (a, 0) and (0, b). Assist the student in identifying the yintercept in this task and writing it as an ordered pair. Ask the student to consider whether or not there are any xintercepts. Next ask the student to identify the two variables that function f relates and their units of measure. Guide the student to interpret the intercept (0, 150) in terms of these variables and the problem context.
Review what it means for a graph of a function to be increasing, decreasing, or constant over an interval. Assist the student in identifying the interval of x in which the graph is increasing and the interval of x in which the graph is decreasing. Ask the student to also indicate the change in y over these intervals. Guide the student to interpret these intervals in terms of the context of the problem.
Review the meaning of periodic and provide examples of functions that are periodic. Ask the student to determine if f is periodic and to consider the context of the graph when explaining why or why not.
Provide additional examples of graphs of functions in context. Ask the student to identify and interpret key features of the graphs such as intercepts; intervals over which the graph is increasing, decreasing, or constant; relative maximums and relative minimums; symmetry; end behavior; and periodicity. 
Moving Forward 
Misconception/Error The student can identify key features of the graph but is unable to correctly interpret them. 
Examples of Student Work at this Level The student identifies an intercept at (0, 150), says f is increasing over the interval from x = 0 to x = 5, and states that the function is not periodic. However, the student is unable to correctly interpret these key features in the context of the problem.

Questions Eliciting Thinking What are the two variables related by function f? What are their units of measure?
You said there is a yintercept at (0, 150). What does the intercept mean in the context of this problem?
You said that f is increasing from 0 to 5. What about the context of this problem indicates that the graph should be increasing over this interval?
Why would the context of this graph indicate that it should not be periodic? 
Instructional Implications Ask the student to identify the two variables that function f relates and their units of measure. Encourage the student to always consider the variables in the problem context and their units of measure when interpreting features of a graph. For example, ask the student to rewrite the intercept with descriptive labels that include each variable and its unit of measure such as (0 kilometers of distance, 150 meters of elevation). Then ask the student to reread the problem and to consider what this intercept indicates in the context of the problem (e.g., that at the start of the trail, the elevation is 150 meters). Ask the student to describe the interval over which f is increasing in terms of the variables and their units of measure (e.g., from 0 kilometers to 5 kilometers in distance, the elevation is increasing). Again, ask the student to review the problem context and attempt to improve the description by relating it to the problem context (e.g., â€śthe hiking trail gains over 450 meters in elevation over its first five kilometersâ€ť). Ask the student to consider the context of the problem when explaining why f is not periodic.
Provide additional examples of graphs of functions in context. Ask the student to identify and interpret key features of the graphs such as intercepts; intervals over which the graph is increasing, decreasing, or constant; relative maximums and relative minimums; symmetry; end behavior; and periodicity. 
Almost There 
Misconception/Error The student does not understand what it means for a function to be periodic. 
Examples of Student Work at this Level The student correctly identifies and interprets the yintercept and the interval over which the function is increasing. However, the student does not understand what it means for a function to be periodic. The student says:
 The function is periodic for an incorrect reason such as it occurs over an interval of time.
 The function is not periodic because it is symmetric.

Questions Eliciting Thinking What does it mean for a function to be periodic?
How can you tell from the graph whether or not the function is periodic? 
Instructional Implications Discuss the importance of communicating mathematical ideas clearly, completely, and using correct notation. Provide feedback to the student with regard to his or her responses and ask the student to make revisions to improve them. Pair the student with another Almost There student so that they can exchange papers and give feedback to each other.
Review the vocabulary associated with features of graphs: increasing, decreasing, minimum, maximum, intercept, symmetry, end behavior, and periodicity. Provide additional opportunities to identify and interpret key features of graphs that model the relationship between two variables. 
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 one intercept at (0, 150) and explains that this indicates that the trail begins at an elevation of 150 meters.
 Says f is increasing in the interval (0, 5) which indicates that the trail increases in elevation for the first 5 kilometers.
 Says the graph is not periodic since it just displays the elevation along the trail for one roundtrip.

Questions Eliciting Thinking What kind of intercept is (0, 150)? Are there any xintercepts?
Is the graph symmetric? Why do you suppose that is?
What is happening during the interval from 5 to 10 kilometers?
Can one tell where the family is walking the fastest from this graph? 
Instructional Implications Challenge the student to describe a scenario in which the graph would be periodic.
Ask the student to construct a graph given certain properties (e.g., a trail is 4 km in length, it begins at sea level and ends at an elevation of 400 m, the section from 2.5 km to 3.0 km decreases in elevation from 380 m to 350 m after which it levels off until the final 0.5 km which is very steep). 