Summary
In this lesson, students will work with rational functions that have 0, 1, or 2 vertical asymptotes and 0 or 1 horizontal asymptotes. Students will explore the relationship between the equation and graph of a rational function, learn what causes different types of asymptotes, and apply their knowledge to graph rational functions. This is the second lesson of two in the "Can't Touch This" lesson series—see "Can’t Touch This, Part 1" for prerequisite content.
Essential Question(s)
What can cause asymptotes?
Snapshot
Engage
Students organize graphs of rational functions that have 0, 1, or 2 vertical asymptotes and 0 or 1 horizontal asymptotes into groups through a Card Sort activity.
Explore
Students explore the relationship between the equation and the graph of a rational function.
Explain
Students complete guided notes with the class and formalize their understanding of graphing rational functions.
Extend
Students apply what they have learned to graph rational functions.
Evaluate
Students match graphs of rational functions with asymptotes and equations in a Card Matching activity.
Instructional Formats
The term "Multimodality" refers to the ability of a lesson to be offered in more than one modality (i.e. face-to-face, online, blended). This lesson has been designed to be offered in multiple formats, while still meeting the same standards and learning objectives. Though fundamentally the same lesson, you will notice that the different modalities may require the lesson to be approached differently. Select the modality that you are interested in to be taken to the section of the course designed for that form of instruction.
Materials
Lesson Slides (attached)
Card Sort document (attached; one set per pair; printed front only)
Exploring Rational Functions handout (attached; one per pair; printed front only)
Guided Notes handout (attached; one per student; printed front/back)
Graphing With Asymptotes, Part 2 handout (attached; one per student; printed front/back)
Card Matching document (attached; one set per pair; printed front only)
Pencils
Scientific calculators
Student devices with internet access
Guided Notes (Teacher Guide and Model Notes) (optional; attached)
Engage
10 Minute(s)
Introduce the lesson using the attached Lesson Slides. Display slide 3 to show the lesson’s essential question: What can cause asymptotes? Slide 4 identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.
Ask students to find a partner or assign partners yourself. Remind students to be kind and careful with the printed cards, then pass out one set of graph cards from the Card Sort document to each pair of students. Show slide 5. Share the instructional strategy Card Sort with the class, and have students use this strategy to group the nine graph cards into two, three, or four groups of their choosing. After students have had a chance to organize their cards into groups, ask students to find another pair with which to discuss their thinking.
As time allows, ask for volunteers to share with the class how they chose to organize their cards or how they would describe their groups of cards.
Collect each set of graph cards. These will be used later in the lesson (likely on a different day of class).
Explore
20 Minute(s)
Display slide 6 and provide students with the link to the GeoGebra activity: geogebra.org/m/d9ywzkrc. This interactive GeoGebra activity includes two GeoGebra applets: the first is focused on vertical asymptotes, while the second is focused on horizontal and slant asymptotes. Invite students to interact with both. This gives students a chance to explore rational functions that do not always have one vertical and one horizontal asymptote as they saw in the previous lesson, "Can't Touch This, Part 1."
After giving students time to explore the GeoGebra activity, pass out one of the attached Exploring Rational Functions handouts to each pair of students. Instruct students to work with their partner to complete the handout. In each applet, there is a reset button in the top-right corner that looks like two arrows making a circle. Encourage students to use this button if their exploration makes the graph difficult to see.
After students complete the GeoGebra activity, ask for volunteers to share their observations with the class.
Explain
25 Minute(s)
Go to slide 7. Give each student a copy of the attached Guided Notes handout. Distribute a scientific calculator to each student. Complete the handout as a class. Students should use the table feature of their scientific calculators to save time when calculating y-values.
Once finished, have students add the handout to their math notebook if that is a classroom norm.
Extend
25 Minute(s)
Have each pair of students find another pair and partner up to create groups of four students. Pass out a copy of the attached Graphing With Asymptotes, Part 2 handout to each student and display slide 8. Instruct students to work with their group to graph the first rational function.
As groups finish question 1, transition through slides 9–10 so students can check their work. Bring the class together for questions and discussion. Ask for volunteers to explain why question 1 does not have any vertical asymptotes. Ask students to explain how they determined the horizontal asymptote. Use student responses to help clarify any misunderstandings.
Instruct students to now work with only one person from their group of four to complete question 2.
As students work through and complete question 2, transition through slides 11–12 so students can check their work. Bring the class together once again for questions and discussion. Ask for volunteers to explain why question 2 has two vertical asymptotes. Ask students to explain how they determined the horizontal asymptote. Ask students if the curve crossed or touched an asymptote and why it is possible for a curve to touch a horizontal asymptote. Use student responses to help clarify any misunderstandings.
Challenge students to now work independently to complete question 3. Remind students that this is a great opportunity to reflect on what they know and what questions they may have.
As students complete question 3, transition through slides 13–14 so students can check their work. Give students an opportunity to ask questions and correct misunderstandings.
Evaluate
10 Minute(s)
Direct each student to find their original partner. Show slide 15. Return to each pair a set of graph cards from the Engage phase Card Sort activity. Pass out the asymptotes and equation cards from the Card Matching document as well. Direct students to complete their Card Matching activity by matching each graph card with an asymptotes card and an equation card together. In other words, students should end up with several groupings of three cards. Remind students that there are six sets of cards, so three of the graphs will not have matching asymptotes or equation cards. These three cards should not be matched together.
While students are working, walk around the classroom and ask students to share their thinking about certain matches you see they’ve made. Remember to encourage academic vocabulary. While circulating the room, use what you hear students talking about and what questions they ask to see what misconceptions still exist.
At the end of the class period, remember to collect students’ cards and prepare each set for the next class if necessary.
Resources
K20 Center. (n.d.). Card Matching. Strategies. https://learn.k20center.ou.edu/strategy/1837
K20 Center. (n.d.). Card Sort. Strategies. https://learn.k20center.ou.edu/strategy/147
K20 Center. (n.d.). Desmos Classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081
K20 Center. (n.d.). GeoGebra. Tech tools. https://learn.k20center.ou.edu/tech-tool/2352
