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# Modeling and Simplifying Rational Expressions

Lesson 7 of 15

## Objective: Students will be to model and simplify rational expressions.

## Big Idea: Students compare multiple models of rational expressions including algebraic, graphic and real world.

*50 minutes*

#### Warm Up and Homework Check

*10 min*

I include **Warm ups** with a **Warm up 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. This lesson’s Warm Up- Modeling and Simplifying Rational Expressions asks students to analyze to possible solutions to a simple division problem. I have found that students will often make mistakes when dividing a polynomial by a monomial so the goal of this warm up is to address those misconception prior to today's lesson. They key here is that neither possible solution is correct. One solution uses subtraction rather than division. The other, and most common mistake, divides only the first term and not the second. I think it is a valuable lesson, and a real test of students' critical thinking skills, to present two incorrect solutions and ask which is right (**Math Practice 1**).

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

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We begin our study of rational expressions through a simple, but rich, scenario (**Math Practice 4**).

*The 10 members of the local high school chess team raised some money for a year end trip to the county fair. After subtracting traveling costs, the team had $300 left to pay for admission to the fair and expenses*.

We begin by finding a couple of concrete examples and then an expression that models the amount of money left after paying for the fair admission. This is a great place to do the Note Card Activity** **as there will probably be some really rich mistakes (**Math Practice 3**). Next, the students simplify the problem and describe how it relates to the scenario (**Math Practice 2**).

The next several tasks ask students to find the width of a rectangle with a polynomial area given the length. The first two give a length that is a monomial while the third problem gives a length that is a binomial. This final problem will require the students to put their skills together to strive to solve a type of problem they haven’t really dealt with before (**Math Practice 1**).

Please see the PowerPoint for detailed presentation notes.

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Next, we generalize rational expressions simplify several. I planned them carefully in increasing levels of difficulty. I may add several if needed.

Once they have simplified an expression. I show them how to check their work using their calculator. If you graph the original and the simplified versions on the same coordinate plane, they will look the same. This can provide an amazing discussion (**Math Practice 7**). Are they exactly the same? This is a think-pair-share. If the students are convinced that they are the same, I will ask about the limits on the denominator of each problem. Why is there only one asymptote? I show the students that they can find the VALUE of a function at any x if they press 2^{ND} and then TRACE on the TI-84.

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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 simplify a rational expression.

#### Resources

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The first six problems on this assignment ask students to simplify rational expressions which will solidify the skill introduced in this lesson. Next, I give them two problems that extend the area model of rational expressions into a volume model (**Math Practice 1**). The final problem asks the students to help a fellow student make fewer mistakes when simplifying a rational expression. The goal of this problem is to help students identify their own strategies and hopefully cut down on future mistakes.

This assignment was created with Kuta Software which is an amazing resource for secondary mathematics teachers.

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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