Lesson 10

This warm-up connects to the previous lesson and is meant to elicit ideas that will be helpful for students in the next activity in this lesson. Students should notice that the points in the graph are not connected and wonder how well the lines model the sections of data they span, which is explored further in the next activity. For that reason, the discussion should focus on collecting all the things students notice and wonder but not explaining the things they wonder.

Tell students they will look at a graph and their job is to think of at least one thing they notice and at least one thing they wonder about the graph. Display the graph for all to see and give 1 minute of quiet think time.
Ask students to give a signal when they have noticed or wondered about something.

Ask students to share their ideas. Record and display the responses for all to see. If no one mentions that the dots are not connected or what they think the blue lines mean, bring these ideas to their attention and tell them they will be working more with these ideas in the next activity.

In this activity, students work with a graph that clearly cannot be modeled by a single linear function, but pieces of the graph could be reasonably modeled using different linear functions, leading to the introduction of piecewise linear functions (MP4). Students find the slopes of their piecewise linear model and interpret them in the context.

Monitor for students who:

• Choose different numbers of line segments to represent the function (e.g., 3, 4, and 5 segments).
• Choose different endpoints for the segments (e.g., two students have chosen 4 segments but different placement of those segments).

Arrange students in groups of 2. Display images for all to see. Tell students that sometimes we model functions with multiple line segments in different places. These models are called piecewise linear functions. For example, here are two different piecewise linear models of the same temperature data (note that the first image is not the same as the image in the warm-up):

Give students 3–5 minutes of quiet work time and then time to share their responses with their partners. Follow with a whole-class discussion.

Select previously identified students to share their responses and display each student’s graph for all to see. Sequence the student responses in order from fewest to greatest number of segments.

To highlight the use and interpretation of piecewise linear models, ask:

• “What do the slopes of the different lines mean?” (The slopes of the lines tell us the rate of change of the different linear pieces for the specific intervals of time.)
• “Can we use this information to predict information for recycling in the future?” (If the data continues to decrease as it does from 2011 to 2013, yes. If the data starts to increase again, our model may not make very good predictions.)
• “What are the benefits of having fewer segments in the piecewise linear function? What are the benefits of having more segments?” (Fewer segments are easier to write equations for and help show long-term trends. More segments give a more accurate model of the data.)

The purpose of this activity is to give students more exposure to 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 the context. A main discussion point should be around what the different rates of change mean in the situation and connecting features of the graph to the events in the context.

Students in groups of 2. Give 3–5 minutes of quiet work time and then ask them to share their responses with a partner and reach an agreement. Follow with a whole-class discussion.

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 the water was filling the tub is steeper than the slope of the line representing the interval 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 \(m\)? For \(b\)?” (I would use \(m=4\) and \(b=0\), 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.)

This activity is similar to the previous activity in that students are interpreting a graph and making sense of what situation the graph is representing. The difference here is that the specificity with numbers has been removed, so students need to think a bit more abstractly about what the changes in the graph represent and how they connect to the situation.

Monitor for students describing the graph with different levels of detail, particularly for any students who state that the car got up to speed faster than the car slowed down to 0.

Give students 1–2 minutes of quiet work time, followed by a whole-class discussion.

The purpose of this activity is for students to connect what is happening in a graph to a situation. Display the graph for all to see. Select previously identified students to share the situation they came up with. Sequence students from least descriptive to most descriptive. Have students point out the parts on the graph as they share their story about the situation.

Consider asking the following questions:

• “Did the car speed up faster or slow down faster? How do you know?” (The car sped up faster because the first part of the model is steeper than the third part of the model.)
• “How did you know that the car stayed that speed for a period of time?” (The graph stays at the same height for a while, so the speed was not changing during that time.)