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Lesson 3

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Course: Math 20-1 SS
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Date: Monday, 15 September 2025, 2:38 PM

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Table of contents

1. Lesson 3

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

Lesson 3: Adding and Subtracting Rational Expressions

 
Focus

 

This shows a photo of a real-estate board game.

© linno/16377673/Fotolia

In the game of Monopoly, the objective is to acquire and develop real-estate properties in order to build up wealth without going bankrupt. One of the rules of Monopoly is that houses or hotels cannot be constructed on a property unless a player owns all of the properties in the colour group where development is intended. There are different strategies for securing the properties in a particular colour group. For example, the properties can be purchased from the bank or through a real-estate transaction with another player.

 

In this lesson you will learn how to add and subtract rational expressions. Like the rules for Monopoly, you cannot add or subtract rational expressions until a certain condition is met.

 

Do you know what this condition is? You will soon. There are different strategies you can employ to meet the condition, including strategies you have previously learned.

 

Outcomes

 

At the end of this lesson you will be able to

  • compare the strategies for adding and subtracting rational expressions to the strategies for adding and subtracting rational numbers
  • determine the non-permissible values when adding and subtracting rational expressions
  • determine, in simplified form, the sum or difference of rational expressions with the same denominator as well as those in which the denominators are not the same and that may or may not contain common factors
  • simplify an expression that involves two or more operations on rational expressions

Lesson Questions

 

You will investigate the following questions:

  • How are adding and subtracting rational expressions similar to adding and subtracting rational numbers?
  • What patterns can be used to prepare rational expressions for addition or subtraction?
Assessment


Your assessment may be based on a combination of the following tasks:

  • course folder submissions from Try This and Share activities
  • additions to Module 6 Glossary Terms and Formula Sheet
  • work under Project Connection


1.1. Launch

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

Launch
 

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.

 

Before beginning this lesson you should be able to

  • determine the lowest common multiple of a set of numbers
  • add fractions
  • subtract fractions


1.2. Are You Ready?

Mathematics 20-1: Module 6

Module 6: Rational Expressions

 

Are You Ready?

 

Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

  1. Determine the LCM (lowest common multiple) of each set of numbers.

    1. 4 and 6 Answer

    2. 5 and 9 Answer

    3. 10, 12, and 15 Answer

  2. Add or subtract the following fractions. Simplify where possible.

    1. Answer

    2. Answer

    3. Answer

  3. Add or subtract the following fractions. Simplify where possible.

    1. Answer

    2. Answer

    3. Answer

  4. Evaluate the following expressions. Remember that the order of operations is important, so you may want to work each step out on paper.

    1. 8 × 16 ÷ 4 Answer

    2. 500 − 10 × 15 Answer

    3. Answer

How did the questions go? If you feel comfortable with the concepts covered in the questions, skip forward to Discover. If you experienced difficulties, use the resources in Refresher to review these important concepts before continuing through the lesson.


 

1.3. Refresher

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

Refresher

 

This is a play button that opens Lowest Common Multiple.

Determine the lowest common multiple of a set of numbers.

 

Open Lowest Common Multiple to review the definition and use an applet to find the lowest common multiple.

This is a play button that opens “Adding Fractions with Unlike Denominators.”

Khan Academy (CC BY-NC-SA 3.0)

Open “Adding Fractions with Unlike Denominators” for a review of addition techniques.

This is a play button that opens “Adding Mixed Fractions with Unlike Denominators.”

Khan Academy (CC BY-NC-SA 3.0)

Open “Adding Mixed Numbers with Unlike Denominators” for a review of addition techniques.

This is a play button that opens “Subtracting Fractions.”

Khan Academy (CC BY-NC-SA 3.0)

Review subtraction of fractions with “Subtracting Fractions.”

This is a play button that opens “Subtracting Mixed Numbers.”

Khan Academy (CC BY-NC-SA 3.0)

“Subtracting Mixed Numbers” will provide you with review of subtracting terms with mixed fractions.

Review how to evaluate expressions using the correct order of operations with the following two activities:

  • “Order of Operations”

    This is a play button that opens “Order of Operations.”
    Khan Academy (CC BY-NC-SA 3.0)


  • “Order of Operations 2”

    This is a play button that opens “Order of Operations 2.”
    Khan Academy (CC BY-NC-SA 3.0)

Go back to the Are You Ready section and try the questions again. If you are still having difficulty, contact your teacher.


 

1.4. Discover

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

Discover
 

This is an illustration of a cartoon man trying to decide on one of many paths to take.

© giz/27094086/Fotolia

 

To add and subtract fractions you often have to find the lowest common multiple (LCM) to use as the lowest common denominator. In this section you will explore two different methods of determining the lowest common denominator for a set of rational denominators.

 

Try This 1

 

Method 1: Finding the LCM Using the Factored Form

  1. Complete the following table. The first row has been completed to give you an idea of what is expected.

     
    Monomials Factored Form LCM

    x2

    2x

    x2 = x • x

    2x = 2 • x

    10xy2

    4x2y    		    

    6x
    4x2

    x3    		    

    8(a + b)3

    4(a2 − b2)    		    



Method 2: Finding the LCM Using Product and GCF

  1.  
    1. Discover an alternate method of finding the LCM by completing the table and then answering the questions that follow. In the table, treat binomials as a single factor. Do not expand (multiply out) any of the factors. The first row has been completed to give you an idea of what is expected.

       
      Expressions Product GCF LCM GCF × LCM

      x2

      2x

      x2 = x • x

      2x = 2 • x

       

      GCF = x

      x2 = x • x

      2x = 2 • x

       

      		 x • 2x2 = 2x3

      12xy

      15y2      		        

      (y + 2)

      2(y + 2)      		        

      (x + 3)

      (x − 3)      		        


    2. Look at the rows of your completed table. Do you notice any patterns as you look horizontally across the rows? From these patterns do you see another method you could use to find the LCM?

  2. What other previous math concepts have you learned that required you to work with lowest common multiples?

  3. Why do you think it is important to be able to have different strategies to determine the LCM of a set of expressions?

course folder Save your work in your course folder.

 

You will revisit these results later in the lesson.

 

1.5. Explore

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

Explore

 

This is a photo of a young man and woman playing backgammon.

Stockbyte/Thinkstock

Have you ever played backgammon? Backgammon is a classic board game for two players consisting of a special board, checkers, and dice. The object of the game is to be the first to move all of your checkers from their starting positions on the board around a track, and then off the board. The first player to remove all of his or her checkers off the board wins the game.

 

Before a player can begin clearing pieces off the board, the player must move all of the pieces to his or her home board, which is a quadrant of the backgammon board. In the photo, the young man’s home board is the quadrant by his left arm, and the young woman’s home board is the quadrant by her knee. The majority of the game is spent moving pieces to the home board. Once all of a player’s pieces are in the home board, the game ends.

 

In this lesson you will determine the sums and differences of rational expressions. Just like in backgammon, you will discover that the actual adding and subtracting of expressions is easy—it’s the preparation of the expressions and meeting of certain conditions that can be a time challenge.


 

1.6. Explore 2

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 
Adding and Subtracting Rational Expressions with Common Denominators

 

To add or subtract fractions with a common denominator, you must add or subtract the numerators and write the result over the denominator. Examples of how rational numbers are added and subtracted are shown below.

 

Example 1

 

Evaluate the following expressions.



Solution



The same method can be applied to the addition and subtraction of rational expressions. As usual with rational expressions, you must be sure to note the non-permissible values.

 

Try This 2
  1. Evaluate the following rational expressions and state the non-permissible values. Use the same pattern shown in Example 1.







  2. Verify your results by

    1. graphing the original expression and your result into a graphing calculator

    2. substituting a known value of the variable into the original expression and the result

course folder Save your work in your course folder.

 

Share 1

 

Compare your answers to Try This 2 with a partner. Discuss any differences in your responses. Make any necessary revisions once a consensus has been reached.

  1. Describe a procedure that can be followed to determine the sum or difference, in simplest form, of rational expressions with the same denominator.

  2. The sum in 1.c. is equal to a constant. What did you have to do to simplify the expression to a constant?

  3. Refer to the two methods of verifying your answers in Try This 2 question 2.

    1. What do you have to watch out for when using each of these methods?

    2. Which method do you prefer and why?

  4. How would you add or subtract rational expressions with unlike denominators?

course folder Save your work in your course folder.


textbook

Work through “Example 1” on page 333 in the textbook. You will see three examples that are similar to those you investigated in Try This 2. As you work through the examples, you may find some things you might have overlooked in Share 1. Make any necessary revisions to your Share 1 responses. Focus on the following points as you work through “Example 1:”

  • In “Example 1.a.,” what would happen if brackets were not placed around “a – 1”? How do you know when brackets are necessary and when they are not?

  • “Example 1.b.” is similar to Try This 2 question 1.c. What is the important step here for simplifying the answer to a constant?

  • In “Example 1.c.,” why is there a non-permissible value, x ≠ 2, even though there is no denominator in the final result?
Self-Check 1

 

This is a play button that opens Adding and Subtracting Rational Expressions Self-Check: Common Denominators.

Complete Adding and Subtracting Rational Expressions Self-Check: Common Denominators.


 

1.7. Explore 3

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

You have discovered that adding or subtracting rational expressions is relatively straightforward when the expressions have common denominators. What do you do when the denominators are not common? It is time to learn strategies for evaluating sums and differences of rational expressions with unlike denominators.

 

Adding and Subtracting Rational Expressions with Unlike Denominators

 

To add and subtract fractions with unlike denominators, start by writing the fractions as equivalent fractions over a common denominator. Once the equivalent fractions are obtained, you can apply the same approach you previously used to add and subtract fractions with common denominators.

 

The lowest common denominator (LCD) can be found for any group of denominators using different methods. Retrieve your results from Try This 1. In that activity you explored two methods—the factored form method and the GCF method. Watch Making Equivalent Fractions: Factored Form and Making Equivalent Fractions: GCF × LCM to see these methods in action!

 

 
This is a play button that opens Making Equivalent Fractions: Factored Form.
 
This is a play button that opens Making Equivalent Fractions: GCF x LCM.



Now that you know the lowest common denominator (found using either of the methods in the preceding animations), you can evaluate the sum.

 

Example

 

Evaluate

 

Solution

 

 

Step 1: Find the LCD.

 

The LCD is 60.

 

Step 2: Convert the fractions to the LCD.

 

 

 

Step 3: Write the expression over one denominator.

 

 

 

Step 4: Simplify.

 

 

 

Share 2

 

With a partner, analyze the steps in the example and answer the following questions.

  1. Describe the mathematical ideas that are applied at each step.

  2. Explain why, in Step 2, the fractions are multiplied by and by , respectively. Why were those particular fractions chosen?

course folder Save your work in your course folder.


 

1.8. Explore 4

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

Try This 3

 

Share 2 should help you solidify the way to approach the addition and subtraction of rational numbers. In Try This 3 you will apply the same pattern you discovered to the addition and subtraction of rational expressions.

  1. Simplify

  2. Simplify

    Need some help? Ask yourself these questions.

    • How do I approach this problem? answer

    • What do I do first? answer

    • What do I do next? answer

  3. Compare your strategies for adding the rational numbers in question 1 and adding the rational expressions in question 2.

course folder Save your work in your course folder.

 

Self-Check 2

 

Adding and Subtracting Rational Expressions Self-Check: Unlike Denominators

Complete Adding and Subtracting Rational Expressions Self-Check: Unlike Denominators.

 

Write equivalent expressions over the same denominator.
Find the LCD.
The approach is very similar to the approach you used to answer question 1.

 

1.9. Explore 5

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 
Simplifying Rational Expressions with Binomial Factors in the Denominator

 

The factors of the denominator of a rational expression can be

  • a constant
  • a monomial
  • a binomial
This is a play button that opens Example: Simplifying Complex Expressions.

In any of these cases, consider factoring the denominator to its simplest form to help determine the lowest common denominator. Watch Example: Simplifying Complex Expressions to see how you can add and subtract rational expressions with binomial factors in the denominator.



textbook

Read through “Example 3” on page 326 and “Example 2.a.” on page 334 to review other instances where you must simplify expressions involving multiple operations on rational expressions. As you work through each example, focus on finding the answers to the following questions:

  • In “Example 3,” how do you decide which operation to evaluate first?

  • In “Example 3,” what is the relationship between the expression (2m − 3) and (3m − 2)?

  • In ”Example 2.a.,” what is the denominator of an expression when a denominator is not given?

  • In ”Example 2.a.,” the first term, , could have been simplified to What is the danger in doing this at the beginning?
Self-Check 3


textbook
Complete “Apply” questions 8, 10, and 15 on pages 336 to 338 in the textbook. Answer

 

1.10. Connect

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

Connect
 

This is a photo of fingers typing on a keyboard.

iStockphoto/Thinkstock

 

Lesson 3 Assignment


assessment

Open your copy of Lesson 3 Assignment, which you saved in your course folder at the start of this lesson. Complete the assignment.

 

course folder Save your work in your course folder.

 

Project Connection


assessment

Go to Module 3 Project: Game Challenge. Complete Step 3.

 

course folder Save your work in your course folder.

 

Going Beyond

 

This is a photo of a college-aged woman working on a complex math problem at the blackboard.

© lightpoet/27708030/Fotolia

Abstraction is the process of thinking about a concept in the general sense, without talking about specific examples. For example, in a previous module you used the process of abstraction to generate the quadratic formula,

 

 

 

The ability to abstract is an important skill in mathematics, and one that is developed throughout your schooling. To abstract well you must be able to think clearly, logically, and with precision. Abstraction allows an individual to apply concepts to a wide variety of examples, not just one. The ability to abstract also enables one to describe new phenomena.

 

One way to practise abstraction is to apply mathematical principles to expressions involving variables. Rearranging formulas is an example of abstraction. In this lesson you simplified sums and differences of rational expressions. In this section you will go beyond the lesson by adding and subtracting rational expressions with multiple variables.

 

Simplify the following expressions:





course folder Save your work in your course folder.

 

As you advance to higher levels of math through high school, college, or university, you will  have more opportunities to develop your ability to abstract concepts.



1.11. Lesson 3 Summary

Mathematics 20-1 Module 6

Module 6: Rational Expressions

 

Lesson 3 Summary

 

This is a photo of a Chinese abacus.

Photodisc/Thinkstock

 

In this lesson you investigated the following questions:

  • How are adding and subtracting rational expressions similar to adding and subtracting rational numbers?

  • What patterns can be used to prepare rational expressions for addition or subtraction?

You discovered that adding and subtracting rational expressions is similar to adding and subtracting rational numbers. In both cases, you must do the following:

 

Step 1: Determine the lowest common denominator (LCD).

 

Step 2: Use the property of 1 to write equivalent rational expressions over the LCD.

 

Step 3: Add or subtract the numerators, and then simplify.

 

You discovered that the steps in the procedure cannot be applied out of order if the correct answer is desired. You learned two strategies for determining the lowest common denominator. One method is based on a strategy similar to prime factorization. The other method is based on the pattern that the lowest common denominator is equal to the product of the denominators divided by the greatest common factor:

 

 

 

In the next lesson you will learn how to solve rational equations. You will continue to see how useful the lowest common denominator is in solving these types of equations.