Lesson 2

Lesson 2: Multiplying and Dividing Rational Expressions

Focus

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What’s your favourite game or sport that is played on a court? Basketball, volleyball, and tennis are played on courts. Most court sports are team games, where members of the same team co-operate to score points against the other team. Some of these games may also be played without the co-operative aspect and instead be played one-on-one.

Jai alai is a sport with Spanish origins played on a large rectangular court called a fronton. The court consists of a floor that can extend up to 200 ft in length and includes three wall surfaces. Players in jai alai bounce a special ball off the front wall using a custom-made basket that is attached to the wrist. A player earns a point if the opponent fails to catch the ball when it rebounds. Known as the fastest sport on Earth and also described as “ballet with bullets,” jai alai is a sport that can be played one-on-one or with pairs.

In this lesson you will learn how to multiply and divide rational expressions. You will use rational expressions to model the dimensions of the courts in various sports and games, including jai alai.

Outcomes

At the end of this lesson you will be able to

• compare the strategies for multiplying and dividing rational expressions to the strategies for multiplying and dividing rational numbers
  
  
• determine the non-permissible values when multiplying and dividing rational expressions
  
  
• determine, in simplified form, the product or quotient of rational expressions

Lesson Questions

You will investigate the following questions:

• How are multiplying and dividing rational expressions similar to multiplying and dividing rational numbers?
  
  
• How are non-permissible values determined when multiplying and dividing rational expressions?

Assessment

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

• completion of the Lesson 2 Assignment (Download the Lesson 2 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Try This and Share activities
  
  
• additions to Module 6 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

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

• multiply fractions
  
  
• divide fractions

Are You Ready?

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

1. How do you multiply fractions? Answer
   
   
2. Multiply the following fractions. Simplify where possible.
   
   
   1. Answer
      
      
   2. Answer
      
      
   3.  Answer
      
      
3. How do you divide fractions? Answer
   
   
4. Divide the following fractions. Simplify where possible.
   
   
   1. Answer
      
      
   2. 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.

Refresher

Work through the interactive multimedia piece Fractions – Multiply and Divide to review how to multiply and divide fractions.

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

Discover

An illustration of a jai alai court is shown. The court consists of three walls shown in green, an in-bounds floor area divided into four sections, and an out-of-bounds area shown in red. The length and width of the court can be modelled by the illustration shown.

The width of each section of the floor area is seven-tenths the width of the court. The length of each section is approximately one-quarter the length of the court.

Try This 1

1. Write an expression that represents the total court area, including the out-of-bounds area. Remember area = length × width.
   
   
2. Write an expression that represents each of the following dimensions of the court:
   
   
   1. the length of a blue section of the court floor (matches the length of a short green wall)
      
      
   2. the width of one blue section (matches a part of the long green wall)
      
      
   3. the area of one blue section
      
      
3.  
   1. Write a rational expression that represents the fraction of the total court area occupied by one blue section.
      
      
   2. Simplify this expression.
      
      
4. If x = 30 ft in the illustration, determine the area of one blue section using
   
   
   1. the expression in 2.c.
      
      
   2. the expressions for the dimensions (length and width) of the entire court and applying the original information about the length and width of one blue section
      
      
5. Compare your results in 4.a. and 4.b.

Save your responses in your course folder.

Explore

You may often find familiar elements in science fiction books and films that depict futuristic worlds. For example, in the film Tron: Legacy (2010), silicon-based gladiators battle each other in a game that is reminiscent of jai alai or racquetball. The gladiators throw glowing discs at each other with the intention of terminating the opponent. Like jai alai, the discs rebound off the surrounding walls until they are either caught or hit their mark. The discs are thrown with great speed, and, again as in jai alai, the gladiators undergo dazzling acrobatic maneuvers in order to score points.

In the Launch section of this lesson, you reviewed how to multiply rational numbers. In the general case, recall that

Notice that the numerators are multiplied together and that the denominators are multiplied together. The result is then reduced to lowest terms, where possible.

Retrieve your results from Try This 1. The answers to questions 4.a. and 4.b. should have been the same. That result leads to the conclusion that rational expressions are multiplied and divided in the same way as rational numbers. In other words, numerators are multiplied together and denominators are multiplied together. The resulting expression is then simplified, where possible.

In Try This 2 you will multiply each set of rational expressions. Remember to include the non-permissible values!

Try This 2

Multiply the rational expressions as you would rational numbers. Reduce to lowest terms, where possible. Include the non-permissible values with your final answer.

Save your responses in your course folder.

Share 1

Compare your answers and strategies from Try This 2 with a partner. Discuss the following questions with your partner.

1. What do all of the expressions in Try This 2 have in common?
   
   
2. How are the expressions in Try This 2 different from the following expression?
   
   
    
   
   
   
3. How could you simplify the multiplication process in order to obtain the product of the expression in question 2?

Save your responses in your course folder.

You have now learned how to multiply and divide rational expressions. In the final section of this lesson you will consider an application of rational expressions in the sport of lacrosse.

Lacrosse, like jai alai, is a sport where players toss the game ball using sticks with baskets on the ends. Lacrosse has origins in aboriginal culture. In Self-Check 3 you will analyze the dimensions of a lacrosse field using rational expressions.

Self-Check 3

1. The diagram shows the layout of a lacrosse field. The area outlined in white is called the goal area. The location of the goal is marked by the circle in the goal area.
   
   
   
   
   1. Write an expression for the area in factored form. Answer
      
      
   2. Write an expression for the width of the goal area in factored form. Answer
      
      
   3. Use the two expressions to determine the length of the goal area. Answer
      
      
   4. If x = 18, determine the dimensions of the goal area. Report the dimensions in yards. Answer

Connect

Lesson 2 Assignment

Lesson 2 Summary

In this lesson you investigated the following questions:

• How is multiplying and dividing rational expressions similar to multiplying and dividing rational numbers?

• How are non-permissible values determined when multiplying and dividing rational expressions?

In this lesson you learned about the similarities between multiplying and dividing rational expressions and multiplying and dividing rational numbers. In both cases, you must multiply numerators with numerators and denominators with denominators. At some point during the process, you must also simplify the result. You discovered that obtaining a simplified result is more easily done when the polynomial expressions in the numerator and denominator are factored first.

One aspect of multiplying and dividing rational expressions that does not emerge in multiplying and dividing rational numbers is the need to determine non-permissible values. You must identify those values of the variable(s) that would result in a zero denominator as non-permissible.

When dividing rational expressions, you must also consider the numerator of the divisor (or the second fraction). Since the numerator is reciprocated in the course of division, you must also identify as non-permissible the value(s) of the variable(s) that would render the numerator equal to zero.

In the next lesson you will compare the addition and subtraction of rational expressions to the addition and subtraction of rational numbers.