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* *Reflection: Debate
Modeling Rational Functions - Section 2: Modeling Rational Functions

This discussion technique has hands down become my favorite. While it does take time, the benefits make it entirely worthwhile. In this case, I asked my students to identify the domain representing the number of people who could have potentially gone to IHOP. I then collected the different variations and put the on the white board. For some classes, I inserted potential correct and incorrect versions that didn't come up. My students then discussed the variations in pairs. Once they had a few minutes, I asked for volunteers to share what they liked or don’t liked about any one possibility. I used a check or “x” to keep track on the Board as students voted for or eliminated particular versions. The conversation was fascinating. They debated over the potential maximum and minimum number of people who could go. Towards the end of the discussion, we talked about how mathematics isn't nearly as cut and dry when placed in a context. I let them know that I was open to multiple interpretations provided they had solid reasoning behind their choice (M**ath Practice 3**).

Every time I have done this activity, I get a high level of participation. My students have become more willing to take chances and share their reasoning during this activity (**Math Practices 1 and 3**). Both of these are important skills in the Common Core.

*Debate: Amazing Domain Discussion*

# Modeling Rational Functions

Lesson 2 of 15

## Objective: Students will be able model and graph rational functions.

#### Warm up and Homework Review

*10 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Modeling Rational Functions which asks students to describe a situation where the domain of a function is not all real numbers.

I also use this time to correct and record the previous day's Homework.

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#### Modeling Rational Functions

*23 min*

The goal of this lesson is to introduce students to the graphs and equations of rational functions by modeling real life situations (**Math Practice 4**).** **We begin with a problem on splitting the bill at a restaurant. Through this scenario, the students get an opportunity to investigate the domain, range, intercepts (or lack of intercepts) and asymptotes (**Math Practice 2**). This is run as a guided investigation. I pose a question to the students, have them discuss it with their partner, and then we discuss it as a class. Please read the amazing article on running effective classroom discussions.

The scenario then changes twice, once using a vertical translation and then a horizontal translation. The students compare the features of these graphs to the features of the original graph (**Math Practice 7**).

Please see the PowerPoint for detailed presentation notes.

#### Resources

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The next scenario involves splitting the cost of a Gift. We look at the inverse of this problem in addition to the general features introduced in the last scenario. This one is particularly interesting because the function and its inverse are equivalent. Referring the functions back to how they represent the scenario makes this activity richer.

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#### Exit Ticket

*2 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today's Exit Ticket asks students to list three things that they learned from the day’s lesson.

#### Resources

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This assignment presents the students with a new rational function scenario, asks them to model it, and then use the model to gain information about the scenario (**Math Practice 4**). The goal is to give students practice with modeling outside of the classroom environment.

#### Resources

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Quotients of Monomials
- LESSON 2: Modeling Rational Functions
- LESSON 3: Transformations of Rational Functions
- LESSON 4: Properties of Rational Functions Investigation Day 1 of 2
- LESSON 5: Properties of Rational Functions Investigation Day 2 of 2
- LESSON 6: Rational Functions Quiz
- LESSON 7: Modeling and Simplifying Rational Expressions
- LESSON 8: Multiplying and Dividing Rational Expressions
- LESSON 9: Adding and Subtraction Rational Expressions
- LESSON 10: Complex Rational Expressions
- LESSON 11: Equations with Rational Coefficients
- LESSON 12: Rational Equations
- LESSON 13: Rational Functions Review Day 1
- LESSON 14: Rational Function Review Day 2
- LESSON 15: Rational Functions Test