The purpose of this activity is for students to sketch a graph showing the qualitative features of the function described in the problem (MP2).

In the first problem, students identify possible independent and dependent variables for a situation and then identify which of three graphs matches their choice of variables. In the second problem, students choose the variables and make their own sketch of the context.

The problems are designed to have multiple correct solutions based on which quantities students identify in the descriptions. For example, Jada's information could be viewed from the perspective that the time it takes her to swim a lap depends on how much she practices. Alternatively, we could also say that the number of seconds she takes off her time depends on how much she practices. Each of these choices would lead to students selecting a different graph.

If students are not sure how to represent the “jump” when money is added to the jar, remind them of some of the function graphs they have seen in the past, such as with only discrete points plotted, and that graphs do not have to be a single connected line.

Invite previously identified groups to share their responses. If possible, display their labeled graphs for all to see. Invite 2–3 groups per graph to share their reasoning about the variables they used and why they picked or sketched the graph they did.

If time allows, begin the discussion of the last problem by displaying this graph and asking students what they think of it as a possible representation of the amount of money in Andre’s savings jar:

Give students a brief quiet think time to consider the graph, then time to compare their answer with their partner. Invite 1–3 students to explain their thoughts about why it is or is not a good representation. (For example, the vertical lines would mean that at the same time the jar has two different amounts of money in it, which isn't possible, so this is not a good representation.) Remind students that functions have only one input for each output, so relationships whose graphs have vertical lines cannot be functions. A version of this graph with only horizontal lines does work.

The purpose of this activity is for students to sketch a graph from a story. In order to make the sketch, students must select two quantities from the story to graph, decide which is the independent variable and which is the dependent variable, and create and label their axes based on their decisions (MP4).

Monitor for displays that are correct but different from each other in important ways for students to focus on during the discussion. For example, they may differ by what variables were graphed, such as:

• Distance from home as a function of time
• Distance from park as a function of time
• Total distance traveled as a function of time

In the digital version of the activity, groups use an applet to create their graphs and add labels as needed. The applet allows groups to efficiently revise and refine their graphs. Consider using the digital version if displaying the digital graphs is possible.

The goal of this discussion is for students to understand that the same situation can be graphed in different ways depending on the variables chosen and assumptions made.

Display 2–3 graphs from previously selected groups for all to see. If time allows, invite students to briefly describe their graphs, then use Compare and Connect to help students compare, contrast, and connect the different graphs. Here are some questions for discussion:

• “What do the graphs have in common? How are they different?” (They are all about the same situation, but different groups thought about the situation from different points of view, so the graphs are not identical.)

• “How does time show up in each graph?” (Time is the independent variable for each graph.)

• “Why do the different approaches lead to different graphs?” (Our group assumed the park was farther from Noah’s house than his friend’s house, so the vertical height of our graph was shorter while he visited his friend’s house. Other groups assumed the opposite, so the vertical height of their graphs was shorter while Noah was at the park.)

Conclude the discussion by refocusing students on the input-output pairs described by the different graphs. For example, on a graph where distance from home is a function of time, there should be three inputs where the output is zero since he starts at home, returns home from his friend’s house, and ends at home.

The purpose of this activity is to give students more practice working with a situation that can be modeled with a piecewise linear function. Here, the situation has already been modeled, and students must calculate the rate of change for the different pieces of the model and interpret it in context (MP2). A main discussion point should be what the different rates of change mean in the situation and the connection between features of the graph and the events in the situation.

The purpose of this discussion is for students to make sense of what the rate of change and other features of the model mean in the context of this situation. Consider asking the following questions:

• “Did the tub fill faster or drain faster? How can you tell?” (The tub filled faster. The slope of the line representing the interval in which the water was filling the tub is steeper than the slope of the line representing the interval in which the water was draining from the tub.)
• “If you were going to write a linear equation for the first piece of the graph, what would you use for ? For ?” (I would use and , because the tub filled at a rate of 4 gallons per minute and the initial amount of water was 0.)
• “Which part of the graph represents the 2 gallons per minute you calculated?” (The last part of the piecewise function has a slope of -2, which is when the tub was draining at 2 gallons per minute.)