The purpose of this activity is for students to interpret coordinates on graphs of functions and nonfunctions as well as understand that context does not dictate the independent and dependent variables.

In the first problem, time is a function of distance since each input of meters ran has one and only one output of seconds past. The graph and table help determine how long it takes for Kiran to run a specific distance. In the second problem, time is not a function of Priya’s distance from the starting line since this graph includes her distance from the starting line as she returns to the starting line. This results in a graph in which each input does not give exactly one output.

The purpose of this discussion is for students to understand that independent and dependent variables are not determined by the context (and specifically that time is not always a function of distance). Select students to share their strategies for calculating the answers for the first set of problems. For each problem, ask students whether the graph or table was more useful. Further the discussion by asking:

• “When are tables useful for answering questions?” (We can get exact values from the numbers in the table, while with the graph, we can only approximate.)
• “When are graphs more useful for answering questions?” (A line graph shows more input-output pairs than a table can list easily.)
• “Why does it make sense to have time be a function of distance in this problem?” (The farther Kiran runs, the longer it will take, so it makes sense to represent time as a function of distance.)
• “Does time always have to be a function of distance?” (No, this graph could be made the other way with time on the horizontal axis and distance on the vertical axis, and then it would show distance as a function of time, which makes sense since the longer Kiran runs, the farther he will travel.)

For the second graph, ask students to indicate if they think it represents a function or not. If there are students who say yes and no, invite students from each side to share their reasoning and try to persuade the rest of the class to their side. If all students are not persuaded that the graph is not a function, remind them that functions can only have one output for each input, yet the answer to the problem statement “Estimate when she was 100 meters from her starting point” has two possible responses, 18 seconds or 54 seconds. Since that question has two responses, the graph cannot represent function.

The purpose of this activity is for students to begin using a graph of a functional relationship between two quantities to make quantitative observations about their relationship. For some questions, students must identify specific input-output pairs, while in others, they can use the shape of the graph. For example, when asked for which time the temperature was warmer, students need only compare the relative height of the graph at the two different times (MP2). Similarly, students can identify another time the temperature was the same as at 4:00 p.m. without actually knowing the temperature at 4:00 p.m.

Display the graph for all to see during the discussion. Invite 1–2 previously identified students to share how they found their answers on the displayed graph for the first four questions. If not mentioned by students, demonstrate how to find the solution to the fourth problem by either identifying the temperature values at each time and subtracting or by measuring the vertical change for each time interval.

For the final question, ask students to plot the point on their graphs if they did not already do so and describe what the point means in the context.

If time allows, give 1 minute quite think time for each group to come up with their own question that someone else could answer using the graph. Invite each group to share their question, and ask a different group to give the answer.

The purpose of this activity is for students to identify where a function is increasing or decreasing from a graphical representation. In the previous activity, students focused more on single points. In this activity they focus on collections of points within time intervals and what the overall shape of the graph says about the relationship between the two quantities (MP2).

Monitor for groups who use different strategies for identifying increasing or decreasing intervals. Some strategies, ordered here from simplest to most involved, might be:

• Using a finger to show that, as they move from left to right on the graph, the function trends upward or downward.
• Drawing line segments between discrete points and observing whether the line segment slants up or down as they move from left to right on the graph.
• Finding the amount of garbage that corresponds to different years and comparing their values using subtraction.

The purpose of this discussion is for students to use and listen to the language used to describe intervals of increasing and decreasing for the graphs.

While discussing each graph, display for all to see. For the first graph, invite previously selected students to share how they identified when the amount of garbage produced was increasing or decreasing. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “Which method seems most efficient?” (Drawing in a line segment between the points at the start and end of the time period makes it quick to tell if it is an increase or a decrease.)
• “Which method seems the most precise?” (Estimating the output value of the points and using subtraction to find the difference between the start and end of the time periods.)

Conclude the discussion by displaying the second graph and asking, “How might you generally describe this graph to someone who couldn't see it?” (The percentage of garbage that was recycled increased overall from 1990 to 2011 but began decreasing from 2011 to 2013.) Invite students to share their descriptions with their partner, and then select 2–3 students to share with the class.