This topic is typically one of the most inaccessible to students. Algebra that involves rational equations and algebraic fractions are typically some of the most difficult topics students face. It is common that these topics alone cause students to fail classes and steer away from certain career paths because of it.
That does not need to be the situation. This page is written to help you, the teacher, provide the support students need to be able to successfully master these concepts and procedures. The pace and frequency of exposure to the topic has been developed over years of practice.
Each block below is a lesson (maybe more than a day long), complete with support for you and students, homework, and resources.
Findind and using the LCM and GCF including variables and exponents.
LCM and GCF
Note: This is the first lesson in a series that introduces students to algebraic fractions. This lesson can be powerful because students will often make connections with fractions and factoring that they never quite understood prior. As a result, it is important that throughout the lesson you remain as much of a tour-guide as possible. They need to do the heavy lifting and make sense of the information. Encourage collaboration between students as much as possible.
The first big idea we want students to understand from this lesson is the difference between a multiple and a factor. Factors are multiplicative parts of the original number. Multiples are products of the original number and another number.
The second big idea we want students to understand is that LCM is used to find common denominators and GCF is used to reduce.
There are multiple ways to find both of these, and typically, the easiest way for a particular problem is best. But students need to explore multiple ways. One method is shown explicitly in this lesson, one passively.
Students need to know how to find factors and prime factors. They also need to understand that the variable x is considered prime, where as x3 is composite.
Students need to know how to find the LCM and GCF of algebraic expressions.
Pro – Tip
If the small number in a set (that the LCM and GCF are being found for) divides evenly into the other number, that first number is the GCF. The second number is the LCM. Example, 4 and 12. Four is the GCF and 12 is the LCM.
Below is a screen shot from the lesson.
To download a printable version of the practice problems, please click here.
To download a printable copy of this reference material, please click here.
Reducing rational expressions by finding the common factor greater than one in all terms.
Note: This is one of the most difficult topics in all of mathematics. The reason it is so difficult for students is that they often have little to no conceptual understanding of fractions. Often, they do not even possess procedural fluency either. This is a good opportunity to address those short-comings.
It has been my experience that it is this topic that is the top reason that students fail College Algebra, which is required for most 4-year college degrees. There is no reason, related to cognitive and potential, that this topic should be inaccessible to students.
Algebraic fractions are fractions with variables. They can be reduced if all terms have a common factor greater than one (GCF). To reduce, the GCF is divided out of all terms.
Students must be able to find the GCF of algebraic expressions (especially dealing with exponents of like-bases). They must also know that terms are parts separated by addition, subtraction, and equal signs. However, fraction bars also separate groups from each other. So the numerator and denominator can have terms themselves.
Pro – Tip
Find the GCF and write each term as a product of the GCF and another term. The GCFs reduce to one, leaving just the other terms and their operations.
Below is a screen shot of the lesson.
To download a printable copy of this text, please click here.
Multiplication and division of algebraic fractions, with simple monomial expressions.
In day two we cover multiplication and division of rational expressions (algebraic fractions). The thing students will find trickiest is determining the GCF to see if reducing is possible.
When multiplying it is a best practice to reduce first. Division is multiplication by the reciprocal.
Students need to be able to factor basic algebraic expressions in order to find the GCF. They must also be familiar with basic properties of exponents for multiplication.
Pro – Tip
Reduce before multiplying!
Here is a screen shot of the lesson.
Addition and subtraction of rational expressions (algebraic fractions). This is a two-day topic.
Addition and Subtraction
of Algebraic Fractions
Day three is the first of two days devoted to addition and subtraction. Today is simpler with monomial denominators.
The LCM of the denominators is the lowest common denominator. We get denominators by multiplication, not addition. The reason is that multiplication will not be out of order with the order of operations and because multiplying by one does not change the value of another number.
Students must be familiar and proficient with LCM of algebraic expressions. The skill can be bolstered with this lesson. It is best NOT to have an isolated LCM lesson at this point, but instead remediate in context.
Pro – Tip
If you do not find the LCM, your final answer will need to be reduced.
Below are screen shots of the lessons.
To download a printable copy of the text, please click here.