K20 LEARN | One Piece at a Time

Summary

In this lesson, students will learn how to graph and analyze piecewise functions. The piecewise functions in this lesson will have a maximum of three branches, which could contain any combination of linear, quadratic, and exponential branches. (Note: This lesson is written so that it could be taught at the beginning of the school year or later, depending on your curriculum pacing. Interval notation is not required prerequisite content but would be helpful; calculations with exponential functions are limited to a base of one-half and two; and quadratics are written in vertex form.)

Essential Question(s)

How do we represent a single scenario when the pattern differs for different intervals?

Snapshot

Engage

Students write a story to match a given distance versus time graph.

Explore

Students match graphs of piecewise functions with their corresponding domains and ranges and increasing and decreasing intervals in a Card Matching activity.

Explain

Students review and add equation cards to complete the Card Matching activity.

Extend

Students apply what they have learned to model data with piecewise functions, graph piecewise functions, and analyze the graphs.

Evaluate

Students demonstrate their understanding of analyzing piecewise functions by creating their own piecewise function based on a given set of parameters.

Materials

Graphing Piecewise Functions handout (attached; one per student; printed front only)
Graphing Piecewise Functions (Sample Responses) document (attached; for teacher use)
Demos account
Create a Graph handout (attached; one per student; printed front only)
Pencils
Paper
Student devices with Internet access
Graph paper (optional)
Card Matching document (optional; attached; one per pair of students; printed front only)

Engage

15 Minute(s)

Teacher's Note: Desmos Activity Preparation

To use this Desmos Classroom activity, select the following link: "One Piece at a Time." Create an account or sign in under the "Activity Sessions" heading. After you log in, the green "Assign" dropdown button will be active. Click the arrow next to the word "Assign," then select "Single Session Code." After making some setting selections, select "Create Invitation Code" and give the session code to students. For more information about previewing and assigning a Desmos Classroom activity, go to https://k20center.ou.edu/externalapps/using -activities/.

For more detailed information about Desmos features and how-to tips, go to https://k20center.ou.edu/externalapps/desmos -home-page/.

To set up the activity’s pacing for students, select "View Dashboard" (next to the session code). In the upper-left corner of your screen, select the icon above the word "Pacing." Desmos Classroom should then prompt you to select the first and last screens that you want students to see. When prompted to set a range, select screens 1 and 3. Select "Restrict to Screens 1–3" to confirm your selection. This allows students to access only screens 1–3 at this time. For more information about teacher pacing, go to https://k20center.ou.edu/externalapps/pacing -activities/.

Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.

Introduce the lesson using screens 1–2 of the Desmos Activity. Screen 1 displays the lesson’s essential question. Screen 2 identifies the lesson’s learning objectives. Review each of these with students to the extent you feel necessary.

Instruct students to find a partner or assign students partners. Direct students’ attention to screen 3 and have them work with their partner to create a story that models the given piecewise distance versus time graph.

Once students are done, ask for volunteers to share their stories with the class. Make note of students’ stories and provide feedback, especially regarding the time values involving change: t = 3, 4, 5, 6, 7, and 10. Consider asking questions about why something changed in their story at one of those times. The stories should begin at t = 0 and end at t = 15. Students’ stories should also contain details showing that they noticed the different slopes, like running vs. walking vs. resting.

As time allows, ask for volunteers to share how they created their story. Use these responses to help check for misunderstandings. Also listening to a peer can help others better understand how to analyze a graph.

Explore

10 Minute(s)

On the Dashboard, press the orange plus sign twice to allow students to progress to screens 4–5. Have students work with their partner, using the Card Matching strategy, to match each graph card with a domain/range card and with an increasing/decreasing interval card. Students are provided 6 of each type of card.

Inform students that they can check their work on screen 5, where Desmos shows how many cards out of 18 are correctly matched. If the screen seems empty, it is because there are not yet any correct matches, whether that is from a lack of attempt, guessing, or misunderstanding.

Use student responses to determine if students need a quick refresh on domain and/or range. The idea of increasing and decreasing intervals may be new to students at this point. If students have never seen interval notation before, take a moment to explain that it is the starting and ending x-values for an interval, or segment, of the function. Otherwise, allow students to have a healthy struggle matching these cards; resist the temptation to give students the answer. If students are struggling with the idea of increasing and decreasing, consider asking guiding questions like, "What does it mean to increase?" or "What do you think it means to decrease?" or "Which direction do we read a graph – left to right or right to left?" Monitor student questions and address them during the Explain portion of the lesson.

Explain

15 Minute(s)

Direct students’ attention back to screen 4 and ask for volunteers to share how they matched their cards together. Use this time to make sure that the students understand the idea of increasing and decreasing and interval notation. Be sure to specifically ask about the two cards that only differ by brackets and parentheses at x = 4. Help students see the connection between open circles and parentheses and the relationship between closed circles and brackets.

On the Dashboard, press the orange plus sign to show students the definition of a piecewise function on screen 6. Press the orange plus sign on the Dashboard again to show students an illustrative example of why piecewise functions have their names on screen 7. Use this screen to explain to students where the transition numbers are for the piecewise function: x = –2 and x = 3. These are the x-values where the function transitions from the first piece to the second piece and from the second piece to the third piece.

On the Dashboard, press the orange plus sign to allow students access to screen 8. Review with students that they have seen open circles on number lines when graphing inequalities. Press the orange plus sign again to allow students to access screen 9. Guide students to use their prior knowledge of graphing inequalities to match an equation card with this graph card and set.

Press the orange plus sign on the Dashboard and review the correct answer with the students on screen 10. Ask for volunteers to share how they picked the k(x) function as the correct match.

Why Open Circles?

Students will likely need a reminder about open circles, especially since they have likely not seen them in the x-y-coordinate plane. Using the k(x) function, shown on screen 10, remind students that when we graph functions, we often pick integers for x-values to make the process easier and that often works out quite well. Looking at the first piece of this piecewise function where x < 4, this includes x-values such as 3.5, 3.9, 3.99, 3.999, etc. But these are not x-values that anyone really wants to plug in and evaluate, especially when the function is more complicated, so instead we plug in x = 4 and put an open circle to indicate where the function is tending towards, y = 5, but does not actually equal. Similarly, on the second piece where x > 4, we do not traditionally plug in 4.01, 4.001, 4.0001, etc. Instead, we plug in x = 4 and use an open circle to indicate where the function is tending towards as x approaches 4 from the right.

If students struggle with this idea, consider making a table with x-values: 3, 3.5, 3.9, 3.99, 3.999, 4, 4.001, 4.01, 4.1, 4.5, and 5 and use the piecewise function to get the corresponding y-values: 4, 4.5, 4.9, 4.99, 4.999, DNE, 4.999, 4.99, 4.9, 4.5, and 4. Plot these points. Show students how they would get the same result and how much easier it is to just use x = 3, 4, and 5 with an open circle at x = 4.

On the Dashboard, press the orange plus sign twice to allow students to progress to screens 11–12. Direct students to work with their partner to match the equation cards with the card sets. If students do not see cards on this screen, then they did not correctly complete the Card Matching activity on screen 4 and need to go back and correct their work.

Inform students that they can check their work on screen 12, where Desmos will show how many cards out of 12 are correctly matched. If the screen seems empty, it is because there are not yet any correct matches, whether that is from a lack of attempt, guessing, or misunderstanding.

Extend

30 Minute(s)

On the Dashboard, click the orange "Stop" button; now students can complete the Desmos activity at their own pace.

Instruct students to continue working with their partner to complete screens 13–18. Here they will use two scatter plots with provided data on Oklahoma population and median home values. Here students identify the correct shape of each interval of the data as linear, quadratic, or exponential. Then they analyze the graphs to determine when the function is increasing or decreasing and to determine the domain and range of the function. Questions 1–4 (on screens 11–12 and 14–15) provide students with feedback for their responses. Questions 5–6 (on screens 13 and 16) ask about domain and range and do not provide feedback. Tell students the notation that you prefer them to use for domain and range.

As students finish screen 18, pass out the Graphing Piecewise Functions handout to each student.

Direct students to graph the given piecewise functions, label the intercepts, identify the domain and range, and write the intervals where the function is increasing, decreasing, and constant. Remind students to pay extra attention to the open and closed circles at the transition numbers.

Evaluate

5 Minute(s)

Give each student the Create a Graph handout and direct students to work independently, so you can individually assess what students have learned from this lesson.

Resources

Bureau, U. S. C. (2021, October 8). Historical Population Change Data (1910-2020). Census.gov. https://www.census.gov/data/tables/time-series/dec/popchange-data-text.html
Historical home prices: US monthly median from 1953-2021. DQYDJ. (2021, November 6). https://dqydj.com/historical-home-prices/
K20 Center. (n.d.). Card Matching. Strategies. https://learn.k20center.ou.edu/strategy/1837
PublicDomainPictures. (2012, March 1). Hands [Photograph]. Pixabay. https://pixabay.com/photos/hands-puzzle-pieces-connect-20333/
K20 Center. (n.d.). Desmos Classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081

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