This Warm-up prompts students to make sense of a new type of function, a piecewise linear function, before working with it and the mathematics that is involved (MP1). Students should notice that the points are not connected and wonder how well the lines model the sections of data they span, which is explored in a later activity. For that reason, the discussion should focus on collecting all the things students notice and wonder but not giving responses that explain the things they wonder about.
Launch
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
None
What do you notice? What do you wonder?
Scatterplot, horizontal, time in hours after midnight, 0 to 12 by ones, vertical, temperature in degrees Fahrenheit. Fifty points approximate a straight line from point 2 5 comma 50 increasing to 5 point 75 comma 59 and then decreasing from there to 12 comma 52 point 5.
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
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.
10.2
Activity
Standards Alignment
Building On
Addressing
8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
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 context.
Monitor for students who use the following numbers of line segments to represent the function, order here from fewest to greatest:
3 line segments
4 line segments
5 line segments
Launch
Arrange students in groups of 2. Display these two 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:
Give students 3–5 minutes of quiet work time and then time to compare their graphs and slope calculations with their partner. Follow with a whole-class discussion.
Activity
None
A scatterplot, horizontal, year, 1900 to 2015 by fives, vertival, percentage recycled, 15 % to 27 % by threes. Beginning with 1991 comma 16%, points trend linearly upward to 1996 comma 22%, then linearly upward at a less steep rate to 2011 comma 26 %, then two points trend down.
Approximate the percentage recycled each year with a piecewise linear function by drawing between 3 and 5 line segments to approximate the graph.
Find the slope for each piece. What do these slopes tell you?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to see different ways their peers created their models and to consider the benefits and drawbacks of using different numbers of line segments when making a piecewise linear function.
Invite previously selected groups to share their line segments. Sequence the discussion of the approaches 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:
“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.)
10.3
Activity
Standards Alignment
Building On
Addressing
8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
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.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Arrange students in groups of 2. Introduce the context of giving a dog a bath. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Display only the problem stem and related image, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Invite several partners to share one question with the class, and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “the slope of the line,” “Time is the independent variable, so it goes on the horizontal axis,” and “The tub was draining from Minute 21 to Minute 30.”
Reveal the questions, and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
“How do your questions relate to interpreting rates of change in context?"
Give students 3–5 minutes to answer the questions with their partner. Follow with a whole-class discussion.
Activity
None
Elena filled up the tub and gave her dog a bath. Then she let the water out of the tub.
The graph shows the amount of water in the tub, in gallons, as a function of time, in minutes. Add labels to the graph to show this.
When did she turn off the water faucet?
How much water was in the tub when she bathed her dog?
How long did it take for the tub to drain completely?
At what rate did the faucet fill the tub?
At what rate did the water drain from the tub?
Student Response
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Building on Student Thinking
Activity Synthesis
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.)
10.4
Activity
Optional
Standards Alignment
Building On
Addressing
8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
The purpose of this activity is for students to practice their skills interpreting a graph of a piecewise linear function and making sense of the situation the graph represents. Previously, students have had marked values to work from, but those are removed for this activity to encourage students to think more abstractly about what the changes in the graph represent and how they connect to the situation (MP2).
Launch
Give students 1–2 minutes of quiet work time. Select students describing the graph with different levels of detail, particularly students who state that the car got up to speed faster than the car slowed down to 0, to share during the Activity Synthesis. Follow with a whole-class discussion.
Activity
None
The graph shows the speed of a car as a function of time. Describe what a person watching the car would see.
Student Response
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Building on Student Thinking
Activity Synthesis
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. Ask previously selected students to share the situation they came up with. Sequence students from least descriptive to most descriptive. Ask students to 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.)
Engagement: Provide Access by Recruiting Interest. Invite students to generate a list of additional examples of graphs that represent speed over time and that connect to their personal backgrounds and interests. Supports accessibility for: Conceptual Processing, Attention
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to “Describe what a person watching the car would see.” Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
Lesson Synthesis
Conclude the lesson by inviting students to reflect on what they have learned about piecewise linear functions. Ask students, “How would you describe a piecewise linear function to someone who has never seen one?” (A piecewise linear function is a function whose graph is pieced together out of line segments. For different ranges of input, the output is changing at different approximately constant rates, so a different line is used for each range.) Give 1 minute of quiet think time and then time to share their response with a partner. Invite partners to share their responses with the class, and record them for all to see.
If students don’t include that there are different constant rates over different intervals of the independent variable, make sure that is made clear.
If time allows, invite students to think of other situations that change at different constant rates over time, and give partners 1 minute of think time before selecting groups to share their situations.
Student Lesson Summary
This graph shows Andre biking to his friend’s house, where he hangs out for a while. Then they bike together to the store to buy some groceries before racing back to Andre’s house for a movie night. Each line segment in the graph represents a different part of Andre’s travels.
Graph composed of 5 linear sections. Horizontal axis, time, vertical axis, distance from home. Beginning at the origin, first segment slopes up as it moves right. Second segment horizontal, third segment slopes up and right, more steeply than the first segment. Fourth segment horizontal, fifth segment slopes down and right back to the horizontal axis.
This is an example of a piecewise linear function, which is a function whose graph is pieced together out of line segments. It can be used to model situations in which a quantity changes at a constant rate for a while, then switches to a different constant rate.
We can use piecewise functions to represent stories, or we can use them to model actual data. In the second example, temperatures recorded at different times throughout a day are modeled with a piecewise function made up of two line segments. Which line segment do you think does the best job of modeling the data?
Scatterplot, horizontal, time in hours after midnight, 0 to 12 by ones, vertical, temperature in degrees Fahrenheit. Fifty points approximate a straight line from point 2 5 comma 50 increasing to 5 point 75 comma 59 and then decreasing from there to 12 comma 52 point 5.
Standards Alignment
Building On
Addressing
Building Toward
8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
