## Adding Rational Expresions – A Little Story To Help

Today I’ll share the way I teach my students how to add rational expressions with unlike denominators. As with many topics I teach, I stop in the middle to tell what appears to be a totally unrelated story, only to tie it in with the current topic. These stories really seem to help my students remember what to do with a given problem.

### Setup

The day before we covered adding rational expressions with like denominators, so my students know the finishing steps for adding rational expressions. To start the class off, I will put a problem like this on the board.

Add: $\frac{4}{x^2+4x-32} + \frac{7}{x^2-64}$

The students know that they cannot add these expressions yet, because the denominators are not the same. I ask them what they think we should do first, and based on their previous experience with rational expressions, they say that we should factor the denominators. So I ask them to do that and I write their result on the board.

$\frac{4}{x^2+4x-32} + \frac{7}{x^2-64}=\frac{4}{(x+8)(x-4)} + \frac{7}{(x+8)(x-8)}$

### The Story

I then turn to one student and say something along the lines of “Last night I got home and walked into the backyard. I saw Mrs. Woodbury peering over the fence into our neighbors’ yard and I knew I was in trouble.” Students start to chuckle, or tilt their heads out of curiosity.  Then I continue the story, now talking to the whole class.

“My wife has a little rivalry going on with our neighbors. If they have something in their yard, it’s only a matter of time before she runs out to buy the same exact thing. You know, keeping up with the Jones. Last week they bought a new grill. Next thing I know my wife is bringing home a new grill. The funny thing is that my neighbor’s wife is the same way. I see her peeking over the fence all the time. Last week we bought a hammock for the backyard, and sure enough, two days later there’s a new hammock next door! Unbelievable! Wait, where were we? Oh, right, back to math.”

I then point at the first denominator and say “Suppose this is our backyard. Does our neighbor have any factors that we are missing?” The students say that the second fraction has a factor of (x-8) in its denominator. I say “We need one of those” and write (x-8) in the numerator and denominator of the first fraction, like this.

$\frac{4}{x^2+4x-32} + \frac{7}{x^2-64}=\frac{4(x-8)}{(x+8)(x-4)(x-8)} + \frac{7}{(x+8)(x-8)}$

Then I ask “What does our neighbor need?” The students reply x-4. I write (x-4) in the numerator and denominator of the second fraction, like this.

$\frac{4}{x^2+4x-32} + \frac{7}{x^2-64}=\frac{4(x-8)}{(x+8)(x-4)(x-8)} + \frac{7(x-4)}{(x+8)(x-8)(x-4)}$

We finish the problem by distributing in each numerator, adding the numerators over the common denominator, and simplifying if possible.

### Wrap Up

We do spend some time talking about why I was able to write the missing factor in the numerator and denominator. And I do mention that my wife never actually looks into our neighbors’ yard, although I do see our neighbors looking in our yard from time to time. I think the story helps them to remember the process, and my students do pretty well with this topic. It’s a little like Aunt Sally, a way to remember. When we review for the exam, I ask my students “What do we do when we are trying to add rational expressions?” Someone always says “Check out your neighbor!”

• 1. Stephanie  |  July 5, 2010 at 12:12 pm

This is adorable! I’ll definitely try this with my class.

• 2. Meghan  |  September 28, 2012 at 12:41 pm

Oh my goodness!! Right now, I’m thinking “you’ve just saved my life!” I’m going from teaching my honors class rational expressions, to teaching my lower level Algebra 2, and if they can stop talking long enough to listen to the story I think it will work! Then again, whatever I think will work, totally flops, and vice versa! Hm….

• 3. esteiner51  |  March 24, 2013 at 5:45 am

This is a great idea. My students struggle with this idea. We talk about FFOO (fancy form of one) all year long, so they get the whole multiple the top by what is missing. This is also seen when we start and I give them fractions without variables.

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