Rational Expressions & Equations

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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.

Looking Back

10 minutes

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)

Practice working with Rational Expressions

15 minutes

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.

Solving Rational Equations

20 minutes

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?.)