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# Evolving Rational Functions

Lesson 5 of 12

## Objective: SWBAT explain why dividing by zero causes a function to become undefined and will be able to solve for these domain restrictions algebraically.

## Big Idea: Students build a rational function to investigate the relationships between a graph, an equation, and the undefined values that restrict the function's domain.

*50 minutes*

At the start of class today, I give students two minutes to practice their presentations about why a function is undefined when we divide by zero (See Homework Section of Lesson 1 in this unit for more information about the initial assignment). After these two minutes are up, I want each team to work together to pick what they think is "the best explanation".

As students debate which explanation is the best, there may be some **critiquing of reasoning and constructing arguments occurring (Math Practice 3)**. I help students by praising and highlighting good conversations that I hear around the room. When necessary, I prompt students to challenge a table mate's explanation. I might ask, "Does that make sense to you?" or "Does anyone think it is not fully correct?"

This activity concludes with team presentations. If nobody on a team wants to present, I will make a general statement like, “The person with the most colorful shoes will present.” Normally, I find that students aren't necessarily going to jump up and volunteer to present. I allow team time before presentations to give students an opportunity to gain confidence. A student who doesn’t know what to say or what to do, can at least have some time to learn from their team before they stand in front of the class.

Today, I expect a variety of different explanations about why dividing by zero is undefined. It is important that students understand this fact before they continue on to the rest of today’s lesson.

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After students have completed the *Building Rational Functions* worksheet, I formally define asymptotes for students by presenting the definition on page 3 of the Flipchart: Evolving Rational Functions. Students should add this to their personal dictionaries. Then, I have students demonstrate their understanding of finding vertical asymptote. I ask students match graphs to equations (see pages 4 and 5 of the Flipchart).

The closure activity should be completed without a calculator.

During the final phase of class, I plan to review students' responses to the question on page 4 of the Flipchart by discussing vertical asymptotes, intercepts, test points, and horizontal asymptotes. I also want to point out the fact that 1/x will continue to give us a smaller and smaller numbers as x gets larger and larger, but it will never evaluate as zero. I use this example to help my students to develop an informal understanding of the idea of a limit.

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Rational Approaches to Solving Rational Equations
- LESSON 2: Speed Dating Rationally
- LESSON 3: Another Rational Approach to Solving: Graphing
- LESSON 4: Ahoy again! What can you see now? Building Rational Functions
- LESSON 5: Evolving Rational Functions
- LESSON 6: Rationalized Transformations: Shifting Rational Functions
- LESSON 7: Light it Up - Day 1: Modeling with Rational Functions
- LESSON 8: Light it Up – Day 2: Modeling with Rational Functions
- LESSON 9: You can’t Get There from Here: Finding the Asymptotes
- LESSON 10: Rational Functions Review Day
- LESSON 11: Rational Functions Test Review
- LESSON 12: Rational Functions and Equations Test