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# Solving Rational Equations

Lesson 4 of 8

## Objective: SWBAT solve equations that contain rational expressions.

## Big Idea: Rational equations can be solved through "cross-multiplication" as long as the equation is put into the form FRACTION = FRACTION.

*90 minutes*

#### Warm Up

*15 min*

Students warm up by solving a wide variety of equations including linear, quadratic, cubic, exponential, logarithmic and simple rational. Although we have not yet studied rational equations, I have confidence that my students will be able to tackle the simple one I included in the warm-up. I set this new type of equation amongst all the equations we have solved in this warm up because we are nearing the final exam and I want to help my students begin to prepare.

My students generally take about 20 minutes to complete WS Solving Equations Review.

#### Resources

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#### Direct Instruction

*15 min*

My students are familiar with "cross-multiplying" to solve equations of the form FRACTION = FRACTION, but there are many errors that they make unless I am very explicit in how to go about solving rational equations more generally. For this reason, I give students explicit notes about solving this type of equation.

Although there are multiple ways of thinking about solving rational equations, I have found that the one my students prefer is using operations to rewrite the equation in the form FRACTION = FRACTION and then cross multiply without "skipping steps." I instruct them not to skip any steps because many errors are made with the distributive property when students attempt to distribute as they cross multiply.

I run through about 5 problems on the board, labeling them with funny titles that indicate how hard I consider them. The first might be "child's play" and the last one might be "wicked hard."

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#### Math Train

*55 min*

Math Train is a really engaging way for my students to practice. It gets students talking about math, it's self-checking, it allows students to choose their level of difficulty, and students enjoy it.

To set up math train, arrange the desks in double rows, so that pairs of students are sitting across from each other. In my class I usually have 30 students, so I make one long table for 16 and another shorter one for 14. You'll also need to print out Math Train Cards - Rational Equations with the problem on one side and the answer on the back. Organize cards on different colored paper to indicate difficulty. Each student will select one card to become an expert in. The entire set of cards at each table must be unique, but the same card can be used at both of the tables.

Students are given about 10 minutes to figure out how to do their own problem. They should check the answer on the back and make sure they understand how to explain their process to a peer. During this part of the activity, I make myself available so that every student is comfortable with their own problem.

Set a timer for 8 minutes and instruct students to trade cards across the table. Each student gives the new problem a try and then asks the expert across the table for help if they get stuck. When the 8 minutes are up, partners get their original card back and then ONE of the rows gets up and moves down one seat (the other row will stay put the whole time). The timer is set for 8 more minutes and all students try a new problem, consulting the expert across from them.

As the activity progresses, students usually get a little faster at solving the equations, so they will not need 8 minutes after the first few rounds.

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#### Assignment

*5 min*

For homework, my students complete some independent practice in solving rational equations. I might assign a puzzle that I find online, some problems from our textbook, or a worksheet like WS Solving Rational Equations.

#### Resources

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- LESSON 1: Modeling Proportional Relationships
- LESSON 2: Simplifying Rational Expressions
- LESSON 3: Operations with Rational Expressions
- LESSON 4: Solving Rational Equations
- LESSON 5: Quiz and Intro to Inverse Functions
- LESSON 6: Inverse Functions
- LESSON 7: Review Stations
- LESSON 8: Unit Assessment: Rational and Inverse Functions