Lesson 6

The purpose of this warm-up is for students to realize there are different dependent variables that can be used when making a model of a context and the choice of which we use affects how a graph of a function looks. In the warm-up, students compare two graphs and then determine what the creator of each graph chose as their dependent variable. During the partner discussion, students should listen to their partner’s argument and decide if they agree or disagree with what they are saying (MP3).

Display the 5 pictures of the dog from the print statement for all to see. Ask “What is happening in this succession of pictures?” After a brief quiet think time, invite students to share their answers. (As time elapses, the dog is moving.) Tell students they are going to consider two graphs describing the dog’s movement.

Arrange students in groups of 2. Give 1–2 minutes of quiet work time and then have them share their responses with their partner to see if they agree or disagree with what variables Diego and Lin graphed. If partners do not agree, encourage students to make sense of their partner’s thinking and reach a consensus. Follow with a whole-class discussion.

The goal of this discussion is for students to understand that the same situation can be represented in different ways depending on what variables you choose to represent.

Select students to share the different variables that they think Lin and Diego used in their graphs. If any partners disagreed at first, ask those groups to share how they decided on their final response for what variables Diego and Lin were using.

The purpose of this activity is for students to sketch a graph showing the qualitative features of the function described in the problem. Building on the warm-up, in the first problem students decide what independent and dependent variables were used to create a given graph. 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.

Watch for groups choosing different graphs for the first problem and groups sketching different graphs for the last problem to share during discussion.

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Display the statement and graphs for all to see:

Elena filled up the tub and gave her small dog a bath. Then she let the water out of the tub.

Ask groups to decide which graph best fits the provided context and what independent and dependent variables were used to create it. Give 1–2 minutes for partners to discuss and then select 2–3 groups to share their graph pick and choice for variables. (The second graph shows the amount of water increasing, then staying steady, then decreasing over time. The amount of water in the bath tub is a function of time.)

Students may have a difficult time representing 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, to remind them that graphs do not have to be a single, connected line.

Select previously identified groups to share their responses and record and display their graphs for all to see.

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 1 minute to consider the graph. Invite 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 only have one input for each output, so relationships whose graphs have vertical lines cannot be functions.

One way to represent a function that steps in this way uses open and closed circles to show that the function has only one value at each particular time. Discuss with students how to redraw this graph with open and closed circles so that the graph represents a function.

The purpose of this task 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

Students may try and start graphing before they have clearly articulated and labeled their axes with their chosen variables. Encourage groups to make sure all are in agreement on what variables they are graphing before they create their display.

Select previously identified groups to share their visual displays for all to see throughout the discussion. Give students 2–3 minutes of quiet think time to review the 3 displays and identify differences and similarities. Invite students to share what they identified and record and display the responses for all to see. If not brought up by students, ask:

• “What is the same and different about the two quantities that each group chose to use in their visual display?”
• “Did each group use the same units of measure? Why does it make sense to use the unit of measure seen in any of these examples?”
• “Why does it make sense to have time be the independent variable in this situation?"
• “Do all of these examples make sense in relation to the situation?”

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.

Alternatively, have the class do a “gallery walk” in which students leave written feedback on sticky-notes for the other groups. Here is guidance for the kind of feedback students should aim to give each other:

• “What is one thing that group did that would have made your project better if you had done it?”
• “What is one thing your group did that would have improved their project if they did it too?”
• “Does their answer make sense in the situation?”
• “Is their answer about which house is closer to the park clear and correct?”
• “If there was a mistake, what could they be more careful about in similar problems?”