### Summary

In this lesson, students will use an investigation to explore rational functions. Students will formalize their understanding of a rational function and continue to investigate the relationship between the equation of a simple rational function and its graph.

### Essential Question(s)

What is a rational function? What might we use a rational function to model?

### Snapshot

**Engage**

Students are provided prompts that encourage them to imagine the graphs of data relationships.

**Explore**

Students investigate a relationship that can be modeled as a rational function.

**Explain**

Students formalize their understanding of rational functions and are introduced to the vocabulary of hyperbolas, branches, and asymptotes.

**Extend**

Students further investigate rational functions through a Desmos Classroom Marbleslides activity.

**Evaluate**

Students draw conclusions and analyze the standard form of a rational function with the It Says, I Say, and So strategy.

### Materials

Lesson Slides (attached)

Pasta Branches handout (attached; one per group; printed front only)

Note Catcher handout (attached; one per student; printed front only)

1 bag or box of dry spaghetti noodles

1 bag of dried beans

Plastic cups (2–3 oz., one per group)

String (one 10–12-in. piece per group)

Permanent markers (one per group)

Rulers (one per group)

Tape (painter's tape is recommended)

Pencils

### Engage

10 Minute(s)

Introduce the lesson using the attached **Lesson Slides**. Display **slide 3 **to share the lesson's essential question with students. Go to **slide 4 **to share the lesson's learning objectives. Review each of these with students to the extent you feel necessary.

Show **slide 5** and ask students if they agree or disagree with the given prompt: *A house cat can crawl farther out onto a tree branch than a firefighter.* Help the class come to the consensus that this is true because the further away from the tree trunk, the less weight a branch can support.

Display **slide 6** and introduce the idea of wanting to know the relationship between the length (distance from the tree trunk to the breaking point of the branch) and the mass it would take to break the branch.

Show **slide 7** and then ask the class to consider what the graph of this relationship would look like, where *x* is the length and *y* is the mass. Allow students time to think individually and use the options on the slide to predict what shape the graph would most likely be.

As time allows, have students share with an Elbow Partner why they picked the curve they picked. Then ask for volunteers to share with the class.

Show **slide 8** and ask, *Why might this be good information to know?* Engineers need to know how much weight a beam could safely support. Allow students to share their ideas. Tell them that we are going to explore this kind of relationship today.

### Explore

45 Minute(s)

Assign or have students choose groups of three to work. Display **slide 9** and preview the activity with the class. Pass out the attached **Pasta Branches** handout to each group of students.

Show **slide 10** and review the following roles: counter, recorder, and catcher with the students. Direct students to decide within their groups who should take on which role.

Display **slide 11** and direct students to where they can gather their supplies for this activity.

Show **slide 12** and go through the steps of the activity on the Pasta Branches handout with the students. Use the picture on this slide to point out the markings and how to prevent the container string from sliding off of the pasta noodle.

As students complete their investigation and return their materials, show **slide 13**. Direct students to go to desmos.com and click "Graphing Calculator." Have students add a table by clicking the plus sign in the top-left corner of their screen. Guide students to enter their data into the table.

### Explain

15 Minute(s)

After students have completed entering their data into the Desmos Studio graphing calculator, show **slide 14**. Ask students to talk with their group about which graph most closely matches their data points and to reflect if this graph is the same one as they selected earlier in the lesson. Facilitate a class discussion on their data and ask students to use the data trend to complete the following sentence: *As the distance from the tree trunk (length) increases, the mass it takes to break the branch ____.*

Show **slide 15** and use this slide to direct students on how to generate a curve that models their data.

Display **slide 16** and explain that the graph on their screen is a hyperbola with two branches and two asymptotes. Explain these vocabulary terms to your students. Clarify to students that the asymptotes on the slide are not visible on the graphing calculator because they represent where the function is approaching but not what the function equals. Explain that we often draw them by hand when sketching hyperbolas. Then share the general equation for a simple rational function: y = a/(x–h) + k. Review this equation, the equation of the parent graph y = 1/x, and the definition of a rational function to the extent you see necessary but wait to explain the direct relationship between the equation and the graph, as students discover this later in the lesson.

### Extend

45 Minute(s)

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

Display **slide 17** and give each pair of students a copy of the **Note Catcher** handout to use as they progress through the Desmos Classroom activity.

Screen 1 gives a preview to Desmos Classroom Marbleslides, which is a creative way for students to explore the relationship between equations and their graphs by trying to get a marble to follow the path of the curve to roll or go through a series of stars on the screen.

Screens 2 and 4–7 ask students to make one change to the given rational function to complete the Marbleslides challenge. Screen 3 gives directions on how to reset their graphing calculator screen within the Desmos Classroom activity.

Screens 10–15 ask students to make predictions about how changing a specific value will affect the graph.

The activity continues with less scaffolding in place, continuing to challenge students.

### Evaluate

5 Minute(s)

Display **slide 18** and introduce students to the It Says, I Say, and So strategy. Direct students' attention to the bottom of the Note Catcher handout and ask students to use their notes from the Desmos Classroom Marbleslides activity, where they circled what they changed (It Says) and described the change in the graph (I Say), to explain how *a*, *h*, and *k* of y = a/(x–h) + k each affect the graph (and So).

### Resources

Desmos Classroom. (n.d.). Marbleslides: Rationals [Interactive activity]. Desmos. https://teacher.desmos.com/activitybuilder/custom/566b31794e38e1e21a10aae8?collections=featured-collections%2C5e73b3a1bb8b0c7628d2809c

K20 Center. (n.d.). Desmos Classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081

K20 Center. (n.d.). Desmos Studio. Tech tools. https://learn.k20center.ou.edu/tech-tool/2356

K20 Center. (n.d.). Elbow partners. Strategies. https://learn.k20center.ou.edu/strategy/116

K20 Center. (n.d.). It says, I say, and so. Strategies. https://learn.k20center.ou.edu/strategy/1514

Murdock, J., Kamischke, E., & Kamischke, E. (2010). Introduction to Rational Functions. In

*Discovering advanced algebra: An investigative approach*(Second section, pp. 490–491). Key Curriculum Press.Sheppard, V. (n.d.). Homemade Tomato Sauce [Photograph]. The Vegetarian Ginger. https://thevegetarianginger.com/2012/09/06/homemade-tomato-sauce/