Inside Collection: Advanced Algebra II: Teacher's Guide
Summary: A teacher's guide to rational expressions.
Begin by explaining what rational expressions are, and making the points I made above—we are going to put together some of our old skills in a new way, which will require us to be good at fractions. So, we’re going to start by reviewing how to work with fractions.
Then pair the students up. We’re going to do a sort of doityourself TAPPS exercise. One partner is the student, one is the teacher. The teacher’s job is to add
Then they switch roles. The former student becomes the new teacher, and gives two lessons: how to multiply
The original teacher takes over again, and shows how to simplify a fraction.
Finally, you give a brief lesson on multiplying fractions—they’ve already done that, but the key point to emphasize here is that you can cancel before you multiply. For instance, if you want to multiply
and then try to simplify that. But it’s a lot easier to simplify before you multiply. The
Of course, you can only do this trick—canceling across different fractions—when you are multiplying. Never when you are adding, subtracting, or dividing!
So why are we going through all this? Because, even though they know how to do it with numbers, they are going to get confused when it comes to doing the exact same thing with variables. So whenever they ask a question (“What do I do next?” or “Do I need a common denominator here?” or some such), you refer them back to their own notes on how to handle fractions. I have had a lot of students come into the test and immediately write on the top of it:
They did this so they would have a “template” to follow when adding rational expressions—it’s a very smart move.
Other than basic fraction manipulation, there is only one other big thing to know about rational expressions—always factor first. Factoring shows you what you can cancel (especially when multiplying), and how to find the least common denominator (when adding or subtracting).
So, walk through some sample problems for them on the blackboard. Put up the problem, and ask them what the first step is…and what the second step is…and so on, until you have something like this on the blackboard.
Emphasize over and over that this is just the same steps you would take to add
Now walk through something harder on the blackboard, like this:

The original problem 

Always factor first! 

Simplify (cancel the “ 

Get a common denominator. This step requires a lot of talking through. You have to explain where the common denominator came from, and how you can always find a common denominator once you have factored. 

Now that we have a common denominator, we can combine. This step is a very common place to make errors—by forgetting to parenthesize the 

Done! Of course, we could multiply the bottom through, and many students want to. I don’t mind, but I don’t recommend it—there are advantages to leaving it factored. 
Once again, have them try numbers (on their calculators) to confirm that
Whew! OK, you’ve been lecturing all day. If there are 10 minutes left, they can begin the exercise. They should work individually (not in groups or pairs), but they can ask each other for help.
Finish the inclass exercise and do “Homework—Rational Expressions”
"This is the "teacher's guide" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing […]"