##
* *Reflection: Real World Applications
Rational Expressions & Equations - Section 2: Practice working with Rational Expressions

**The question might be asked, "Wouldn't it be better to have some real-world application for these problems?"**

We've seen in the Cell Phone problem that rational functions do arise in the real world, so the utility of these functions is clear. However, now it seems that it's important to consider them as mathematical objects in their own right, to study them in their natural habitat, so to speak. Later, we'll be able to take this deep conceptual understanding back into the real world.

**On another note, I've used the analogy of "false positives" in my explanation of extraneous solutions. Perhaps that needs some explanation.**

When solving these equations, we are often left at the end of the process with two different *x*-values. Ordinarily, we'd move forward with confidence to say that there are two solutions. But our faith in the algebraic process has been shaken; we know that sometimes one of these apparent solutions is really a fake.

Our algebraic solution process has ruled out all numbers except these two. What if we ruled out a number that actually *is* a solution? That would be a "false negative" result. What if we failed to rule out a number that actually *is not* a solution? That would be a "false positive", an extraneous solution. We know for sure that we sometimes get a false positive. How can we be certain that our solution process *never* results in a false negative?

*Thoughts on the Content*

*Real World Applications: Thoughts on the Content*

# Rational Expressions & Equations

Lesson 7 of 17

## Objective: SWBAT simplify and solve rational algebraic expressions and equations. SWBAT make use of the structure of the equation to identify points of discontinuity.

## Big Idea: Examining the structure of an equation before solving it helps students avoid pitfalls and recognize extraneous solutions.

*45 minutes*

#### Looking Back

*10 min*

I use the opening few minutes of this lesson to discuss the most important points from the previous lesson. In particular, it's important to discuss why the simplified form of a rational algebraic expression might not be perfectly equivalent to the original. The pair of graphs that the students all generated should illustrate this nicely. The original and simplified forms are perfectly equivalent at all but one point, and that point is determined by the common factor that was eliminated in the simplification process.

So, is it appropriate to simplify rational expressions? Yes, but we have to pay attention to what's been eliminated because it might be important.

The other focus of the discussion should be on how students were able to use the structure of the expression to simplify it. Students should have noticed things like the difference of two squares, the usefulness of factoring out a -1, etc. This is a good opportunity to emphasize the ways in which we can make use of structure in mathematics. (**MP 7**)

*expand content*

Keeping with the rhythm that's been established over the previous two lessons, it's now time for students to begin solving a handful of problems individually. These are somewhat more challenging; some involve cubic polynomials, some contain complex fractions.

The aim during these 15 minutes is for students to successfully simplify the four expressions on the front of Simplifying Rational Expressions 3. I'll circulate around the room and help students to quickly catch errors. If necessary, we'll discuss appropriate strategies for simplifying the expressions more efficiently. For instance, once students have factored the denominators, it's only necessary to check the numerators for *common* factors. If common factors do not exist, there is no need to completely factor the numerators.

*expand content*

#### Solving Rational Equations

*20 min*

Now, students should work either individually or in small groups to solve the problems on the back of the worksheet. Rather than simplifying expressions, they are now solving equations.

- Watch out! Some students will look for common factors and then try to "cancel" them across the equals sign.
- You'll notice that I've tried to ask the same question in three different ways: what x-values are excluded? This may be the most important question regarding rational expressions and equations, and it's important for students to see it from slightly different angles.
- Do extraneous solutions arise? Yes, but only in the final problem. Students should have already identified the excluded value, so they should recognize the problem immediately. When it's time to discuss this solution, I'll define the term
*extraneous solution*and we'll discuss*why*this false answer arises. - An excellent extension: If the process of solving a rational equation can result in
*extra*solutions, can it also*miss*or*eliminate*valid solutions? (Please see the video, Can we trust the process?.)

*expand content*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: The Cell Phone Problem, Day 1
- LESSON 2: The Cell Phone Problem, Day 2
- LESSON 3: The Cell Phone Problem, Day 3
- LESSON 4: Discontinuity in Rational Functions
- LESSON 5: Simplifying Rational Expressions Day 1
- LESSON 6: Simplifying Rational Expressions, Day 2
- LESSON 7: Rational Expressions & Equations
- LESSON 8: Solving Rational Equations
- LESSON 9: Solving Rational Equations, Day 2
- LESSON 10: Asymptotic Behavior, Day 1 of 2
- LESSON 11: Asymptotic Behavior, Day 2 of 2
- LESSON 12: Practice with Asymptotes
- LESSON 13: Egyptian Fractions
- LESSON 14: Combined Fuel Economy, Day 1 of 2
- LESSON 15: Combined Fuel Economy, Day 2 of 2
- LESSON 16: The Tin Can Model
- LESSON 17: A Medical Model