##
* *Reflection:
Rational Approaches to Solving Rational Equations - Section 2: Explanation: Solving Rational Equations

During the notes today, my students and I had an interesting conversation about factoring. This probably would have been a much better placed conversation to have at the start of the polynomials unit; however, it’s better late than never. My students started by expressing concern when I mentioned that the trick with rational equations is to be sure to factor everything first and see if common factors divide out. There was that deep sigh that resonated throughout the classroom with that statement. I knew their factoring skills were lacking in the last unit. They are slow to get the factored form of a polynomial and aren’t confident in their answers. They often mix up signs and really mess up factoring when the leading coefficient is not 1. But my even bigger concern is that some students are still struggling with WHEN they should even factor. Many of my students are still trying to solve equations that require factoring or the quadratic formula by direct solving.

So for about 10 minutes of today’s lesson I turned it over to the kids. I asked students that felt comfortable factoring to show the rest of class how they factored and how they know when to factor. Here are some of the methods that arose during this conversation:

Generic Squares Method: Factoring Reflection, Factoring using Generic Squares

Factoring by Grouping: Factoring Reflection, Factoring by Grouping

Helicopter Method: Factoring Reflection, Helicopter Method

I am really curious what other structures teachers are using to help students to factor.

Furthermore, this whole conversation started making me think more deeply about the necessity of factoring polynomials. What is the purpose? Is it really necessary in math curriculum? Factoring Reflection, Necessary Evil

# Rational Approaches to Solving Rational Equations

Lesson 1 of 12

## Objective: SWBAT solve a rational equation for a specified variable.

#### Warm-up

*10 min*

To start off this unit, I want to first see what level of proficiency my students have with basic fraction operations. To do this, I am going to ask students to complete the three warm-up problems on slide 2 of the PowerPoint. I selected these problems to assess students’ knowledge of the rules of adding/subtracting fractions, multiplying/dividing fractions, and factoring. Question 3 is already simplified. I am expecting some students to still try to divide the terms. Once students have completed their warm-up problems. I plan to model how to simplify these for any students who got stuck.

Before we proceed with rational equations I want to talk with students about factoring, simplifying, and "canceling." There are often many misconceptions that surround factoring of rational expressions for students. They don’t know what ‘cancels’, when it ‘cancels,’ and what is left over. See Teaching Notes about Factoring for more detail about how I will lead students through this conversation.

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#### Closure: Here’s how…

*15 min*

As we finish today's lesson I will present Rational Approaches Closure Slide (slide 24 from the PowerPoint). I ask students to complete a Here’s How to close out today’s learning.

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I plan to assign Homework 1 - Rational Functions for homework this evening.

I also plan to start talking with my students about the presentations I would like them to do at the end of the week (4 lessons from now). I will begin this discussion as I present slide 14 in the PowerPoint. I want students to start researching an explanation that makes sense to them: **Why does dividing by zero causes a function to be undefined?**

#### Resources

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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