## Rational Expressions and Equations – Why So Hard?

For me, one of the toughest chapters to teach in elementary chapter is the one on rational expressions and equations. One problem is that many students have a weak foundation with numerical fractions. Simplifying rational expressions involve the same big ideas as simplifying numerical fractions, with the added abstraction of working with variables.

Students must employ a great deal of critical thinking when working with rational expressions and equations, and perhaps this is the area that needs more focus in class. For example, consider these 2 problems.

• Simplify: $\frac {3} {x^2 +6x +8} + \frac {5} {x^2 - 4}$
• Solve: $\frac {3} {x^2 +6x +8}=\frac {5} {x^2 - 4}$

Both problems involve finding a common denominator to a degree, but what we do with that expression is different depending on the problem. For those of you playing along at home, the common denominator is $(x+2)(x+4)(x-2)$. When adding two rational expressions, we rewrite each expression so that they share the same denominator. In other words, we need to build up each denominator. However, if we are trying to solve a rational equation, we multiply both sides of the equation by the common denominator to clear the equation of fractions.

As we review for the exam, I like to ask questions like “Which is the correct first step for this problem?” Simplify: $\frac {3} {x^2 +6x +8} + \frac {5} {x^2 - 4}$

Then I provide two possible first steps.

$\frac {3(x-2)} {(x+2)(x+4)(x-2)} + \frac {5(x+4)} {(x+2)(x-2)(x+4)}$

or

$(x+2)(x+4)(x-2)\frac {3} {(x+2)(x+4)} + (x+2)(x+4)(x-2)\frac {5} {(x+2)(x-2)}$

I also ask my students to tell me why the first approach is correct and the second is incorrect. I find that these opportunities to think and reflect help my students to develop the critical thinking necessary to learn and understand mathematics.

Another approach that I recommend to my students is to use note cards. On the front of a card students write a problem, and on the back they write the first step or two. These cards are easy to cycle through, and a great way to reinforce the subtle difference between the different problems.

If you have any tips for teaching rational expressions, or if you have any questions, I’d like to encourage you to share by leaving a comment, or reaching me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular math topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

• 1. Whit Ford  |  May 20, 2010 at 2:30 pm

If you are not already doing so, after showing them the two problems (simplify and solve), you may wish to ask them: “what kind of problem is each of these?”… looking for answers of “expression” and “equation” respectively.

That then provides the opportunity to ask:

“What kinds of things can I do to an expression?”, followed by “Why is it OK to do that to an expression?” for each… leading up to a final question of “so, if you were to generalize all the things that are OK to do to an expression, how could you do so?” (they all simplify to adding/subtracting zero or multiplying/dividing by 1)

Turning to the equation, you can repeat the process with a final generalization of “you can do everything you can do to an expression, plus you can also do exactly the same thing to THE ENTIRETY of each side”.

As to building student confidence in working with algebraic fractions, I recommend a worksheet of “ugly” problems that involve sums and differences of polynomial fractions with common factors… but some of the common factors cannot be cancelled until a common denominator has been found and the numerators combined. Learning to work these sorts of problems with confidence often requires 1-2 hours of tutoring time, so it might need 3-4 classes (re-introduce fractions, show how the same rules apply to both numerical and algebraic fractions, remind students that common factors can be numbers or polynomials, then gradually build an experience base of simplifying increasingly long/ugly expressions). I LOVE the problems in Chapter XV “Fractions” of “First Course in Algebra” by Herbert Hawkes, William Luby, and Frank Touton – Copyright 1909, 1910, and 1917 – Published by Ginn & Co.

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