Teaching Algebraic Fractions

 I always find it challenging to teach algebraic fractions in high school, especially now.  Too many students have struggled with the past.  They had difficulty finding common denominators, remembering the rules to add, subtract, multiplication, or division so trying to transfer their knowledge to algebraic fractions can be problematic. 

One of the first things one should do is to review the concept of a fraction beginning with the type of fractions most people are used to. It is important they know how to work with fractions, to compare, to simplify, add, subtract, multiply, or divide.

The next step is to introduce students to algebraic fractions with variables in the denominator, numerator, or both. It is important to show how the rules concerning regular fractions also apply to algebraic fractions.

Once students have some comfort to algebraic fractions, it is time to how to simplify algebraic equations by canceling common factors.  Then take it a step farther by showing how to factor both numerators and denominators to find common factors so one can simplify.  This is an important step.

Then one needs to introduce students to adding and subtracting algebraic fractions with the same denominator, just like one does with numerical fractions.  Have them practice this so they become comfortable with the process, find  and don't forget to show how to simplify fractions so the answer is in the simplest form.

The next step in the process is to instruct students in adding or subtracting fractions that do not have the same denominator.  This requires showing them how to find a common denominator through the use of factoring, using the factors to determine the common denominator, and the process of changing the fractions so they both have the same common denominator. The first examples should be fractions with two different denominators and then move on to fractions that require factoring before finding the common denominator.  This is often where it gets harder for students because there are no real numbers to work with.

Once they learn to find common denominators and learn to change the fractions to have the same denominator, it is time to have them practice adding and subtracting fractions with different denominators.   One should keep reinforcing the idea to check the answer to see if it can be factored and terms crossed out to simplify the answer.

From adding and subtracting algebraic fractions, it is time to teach students to learn to multiply fractions. Before teaching this step, review the process of multiplying binomials, trinomials, and monomial terms so when they actually multiply, they can do it.  I admit, I tend to teach students to factor all the terms so they can eliminate common terms before they multiply.  Usually the books tell students to multiply first, then reduce but sometimes the final product can be hard to factor so if they factor first, then reduce, it becomes much easier.  Once everything is crossed out, students can multiply for a final answer or they can leave it in that form.

The final step is to help students learn to divide algebraic fractions.  Explain how division is actually multiplying by the terms reciprocal.  Again, I like having students factor the terms after they rewrite the equation so they can eliminate the common terms.  This leaves fewer terms to work with and a smaller, reduced answer at the end.

Finally, always have students practice the process at each step so they become proficient. I realize this can take time to do it right but it is important to give students time to learn.  Don't forget visual aids when possible and remind them to simplify, simplify, simplify.  Let me know what you think, I'd love to hear.