I need to find a better way to teach algebraic fractions in my math classes. I start by reviewing ordinary fractions because it builds on their previous knowledge but too many students are not fluent in the basics of fractions.
I follow a process when teaching algebraic fractions once I've reviewed basic fractions and focus on the most important rule for adding and subtracting fractions! The denominators must be exactly the same!
After reviewing basic fractions, I move on to having them solve simple fractions with the variable in the numerator so they can practice adding or subtracting fractions with a familiar denominator. I might give a problem like 3n/2 + 1n/2 so they gain experience. The next set of problems might be something like 3x/2 + 5x/7 so they have to find the common denominator.
At this point, I move the variable to the denominator and I make it a simple problem like 3/2m - 4/3m. This is where they start having issues because they do not like the variable in the denominator. It uses everything they've learned. Once they've become comfortable with this, I start throwing in more complex variables in the denominator such as 4/x + 3/x+1.
This last problem is the one that shows me who is really shaky on their algebraic expressions because they try to add one to the first term to make it 4 + 1/x + 1 = 5/x + 1 + 3/x + 1 = 8/x + 1. These are usually the same ones as the ones who do the 1/2 + 2/3 = 3/5.
If I have the time, I branch out to x+1/x + x/x+1 type problems with variables in both the numerator and denominator. I save solving actual algebraic equations for later once they have binomial multiplication down because I tend to teach that by multiplying through the whole equation using a common denominator to get rid of fractions.
If you want a couple of videos that are not YouTube - these sites offer videos that are accessible. I don't have access to Youtube at school so I've had to find alternative sites. Explaining Maths is a site out of the UK that has 7 different videos on everything from what are algebraic fractions to combining and simplifying algebraic fractions.
Finally, check out this power point presentation that talks about algebraic fractions and the four rules of fractions. It uses the multiply the numerator by the other denominator method of finding a common denominator but for algebraic fractions, that is the quickest way to do it. At the end, the author goes on to show how this is used to solve problems. This would be a great presentation to use as a review just before beginning to teach solving equations with algebraic fractions.