Lesson Plan

## Operations with Rational Expressions

• Assessment Anchors
• Eligible Content
• Big Ideas
• Families of functions exhibit properties and behaviors that can be recognized across representations. Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations.
• Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations.
• Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
• Patterns exhibit relationships that can be extended, described, and generalized.
• Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
• There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
• Concepts
• Algebraic properties, processes and representations
• Exponential functions and equations
• Polynomial functions and equations
• Competencies
• Extend algebraic properties and processes to quadratic, exponential, and polynomial expressions and equations and to matrices, and apply them to solve real world problems.
• Represent a polynomial function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated polynomial equation to each representation.

### Objectives

In this lesson, students will perform operations with rational expressions. Students will:

·         find the sum of rational expressions.

·         find the difference of rational expressions.

·         find the product of rational expressions.

·         find the quotient of rational expressions.

·         solve applied problems involving operations with rational expressions.

#### Essential Questions

·         How can we extend arithmetic properties and processes to algebraic expressions and processes and how can we use these properties and processes to solve problems?

### Vocabulary

·         Polynomial: An algebraic expression that contains one or more polynomials. [IS.1 - Struggling Learners] [IS.2 - Struggling Learners]

·         Least common denominator (LCD): The least common multiple of the denominators in two or more fractions.

·         Expression: A variable, or any combination of numbers, variables, and symbols that represent a mathematical relationship (e.g., 24 × 2 + 5 or 4a – 9).

·         Rational expression: An expression that is the ratio, or quotient, of two polynomials.

·         Rational function: A rational expression that is written in function form, or expressed as a function, i.e., f(x) = 2 / ( x – 3).

·         Rational number: A number that can be expressed as a ratio of two integers. A rational number can be expressed in the form a/b, where a and b are integers and b is not equal to zero.

### Duration

200–270 minutes/3–4 class periods [IS.3 - All Students]

### Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

### Materials

·         copies of Multiplying Rationals Stations Problem Set (M-A2-5-1_Multiplying Rationals Stations Problem Set.doc)

·         copies of Stations Activity Records Sheet (M-A2-5-1_Stations Activity Records Sheet.doc)

·         copies of Level Up Problem Set (M-A2-5-1_Level Up Problem Set and KEY.doc)

·         copies of Multiplying Rationals IP (M-A2-5-1_Multiplying Rationals IP and KEY.doc)

·         copies of Operations Guided Notes (M-A2-5-1_Operations Guided Notes.doc)

·         copies of Common Denominators (M-A2-5-1_Common Denominators and KEY.doc)

### Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

·         copies of Multiplying Rationals Stations Problem Set (M-A2-5-1_Multiplying Rationals Stations Problem Set.doc)

·         copies of Stations Activity Records Sheet (M-A2-5-1_Stations Activity Records Sheet.doc)

·         copies of Level Up Problem Set (M-A2-5-1_Level Up Problem Set and KEY.doc)

·         copies of Multiplying Rationals IP (M-A2-5-1_Multiplying Rationals IP and KEY.doc)

·         copies of Operations Guided Notes (M-A2-5-1_Operations Guided Notes.doc)

·         copies of Common Denominators (M-A2-5-1_Common Denominators and KEY.doc)

### Formative Assessment

• View

·         Student responses on the Multiplying Rational Expressions independent practice worksheet can help students learn from their own errors. Collect common examples of misunderstanding and review each correction. [IS.9 - Struggling Learners]

·         Observations and responses during the Level Up! Activity on Dividing Rational Expressions should indicate where students are attempting division without using reciprocal multiplication.

·         Use student responses on the Exit Ticket as source material for reteaching.

·         Student responses on finding the common denominator worksheet are useful for diagnosing poor techniques for using LCM and GCF.

·         Student responses on the adding and subtracting rational expressions independent practice worksheet are particularly revealing when students attempt to add or subtract without finding common denominators.

### Suggested Instructional Supports

• View
Scaffolding, Active Engagement, Modeling, Explicit Instruction W: This lesson introduces students to rational expression and how it can be combined using basic arithmetic. Students should be shown the relationship between the tasks presented in this lesson and how similar tasks are done with basic fractions. This knowledge will allow students to relate their prior knowledge to a new task. Previously, students have worked with operations with polynomial expressions; therefore, rational expression takes the same ideas one level further. The ideas presented in this lesson explain how to approach mathematical functions that involve fractions with variables in the denominator. Students will be assessed on their ability to perform such tasks using the appropriate techniques and on their ability to work with rational functions in applied situations. H: Prior to demonstrating each operation during the instructional process, it will be important to hook the students by pulling from their prior knowledge about operations with fractions. Guide students to recall how to perform addition, subtraction, multiplication, and division of basic fractions. Multiplication of fractions is assumed prior knowledge and students will likely need only a modest amount of practice to feel comfortable with their skills. The goal is to build upon students’ prior knowledge and recall important skills required to complete each of the skills within the lesson. E: Use the examples in this lesson to model the processes required to work through each type of problem. The goal is to equip students with the skills necessary for performing each operation with rational functions independently. The given problems build from basic to more difficult levels of problems. Through the modeling of these procedures students will have the experiences necessary to understand the procedures and perform them independently and with applied problems. R: Following the modeling of each process, this lesson includes a variety of activities and practice that are designed to allow students time to reflect, revisit, revise, and rethink the processes presented by the teacher. The varieties of activities are designed to meet many different learning styles and assist students in the attempt to master each topic (topic 1: interpersonal activity, topic 2: level up activity, topic 3: independent practice). E: Students will express their understanding of the topics in this lesson through a variety of activities and independent practices. Students will have the opportunity to receive guidance on the skill required for each topic and then eventually evolve to work through the problems individually. Through their work on the activities and independent practice, students should be able to evaluate their work and ask questions necessary to master the skills. T: Throughout each topic presented in the lesson there is a learning activity specifically designed to help assist students to reach mastery level. Included in the Extension section of the lesson are some more ideas to help differentiate for different students’ skill levels. This includes guided notes, extra practice sets, application extensions, and alternate directions for activities. O: This lesson is organized so that students move gradually from skill to skill, as well as from guided to more independent practice. Students will begin each skill with teacher-modeled examples, then move on to a learning activity, and then on to an independent practice activity. This level of organization is presented in the lesson for each topic: multiplication, division, and addition and subtraction.

 IS.1 - Struggling Learners Consider the following steps with regard to vocabulary for struggling learners: Use of a graphic organizer (e.g., Frayer Model, Verbal Visual Word Association, Concept Circles). Introduce new vocabulary using student friendly definitions and examples and non-examples. Review words with students. Provide opportunities for students to apply the new/reviewed terms. IS.2 - Struggling Learners Struggling learners may need to have the difference between “expression” and “equation” reinforced. IS.3 - All Students Consider pre-teaching the concepts critical to the lesson.  Use formative assessments throughout the lesson to determine the level of student understanding.  Use follow-up reinforcement as necessary. IS.4 - Struggling Learners Struggling learners may need to have multiplication of fractions reviewed prior to the lesson.  Consider providing them with a multiplication fact chart if they have difficulty with multiplying so that they can focus on the task at hand and not get stuck on remembering multiplication facts. IS.5 - Struggling Learners Struggling learners may need to have this reviewed and written examples provided. IS.6 - Struggling Learners Struggling learners may need to have these rules reviewed and be provided with written examples demonstrating the task. IS.7 - Struggling Learners Struggling learners may need to have division of fractions reviewed prior to the lesson.  Consider providing them with a division fact sheet to assist them. IS.8 - Struggling Learners Struggling learners may need to have addition and subtraction of fractions reviewed for them prior to this part of the lesson. IS.9 - Struggling Learners Consider using various forms of formative assessment for struggling learners.

### Instructional Procedures

• View

Part 1

Preface this section by explaining to students that the modeled solutions are simply one way to simplify the expressions. Encourage students to find and discuss other pathways to solving the problem.

“Recall the procedure for multiplying the following fractions: and take a moment to perform this task on your own paper.” [IS.4 - Struggling Learners]

Give students time to recall and perform the required task. Then discuss their findings.

Possible methods:

Multiply numerators; then denominators; then simplify:

OR simplify before multiplying:

“During this lesson we will be working with problems similar to the one shown here, but instead of working with rational numbers, we will be working with rational expressions. While a rational number is the ratio of two integers, [IS.5 - Struggling Learners] a rational expression is the ratio of two polynomials and would look similar to something like this:.

This lesson will take us through the skills required to multiply, divide, add, and subtract rational functions, which will be very similar to when we multiply, divide, add, and subtract rational numbers.”

“We will begin our lesson with multiplication.”

Model the following problems for students. Have students reflect on the type of factoring they would need to use for each part of the problem before beginning the steps.

“The rules for multiplying rational expressions require us to first make sure the numerators and denominators of the expressions are in factored form. Once they are in factored form, we should look for any common factors.” [IS.6 - Struggling Learners]

Note: This example can be done using the rules of exponents and simplifying fractions as well; the above work shows the common factors of the numerator and denominator.

Make sure to note to students that they cannot divide by the x-terms in this situation because the x-terms are not a factor in this answer; they are being added or subtracted. Suggest substituting a trial value for the x-term to make sure the calculation is correct.

Note: This example uses the greatest common factor type of polynomial factoring.

This example introduces the factoring of trinomials.

This example includes factoring of trinomials and difference of squares.

Cooperative Learning Activity: Use the following group activity to allow students to revisit and rethink through the steps of the problems. While working in groups, students will have the opportunity to work through the problems with their peers to reach the solutions to each problem. By the conclusion of the activity, students should have gained the skills necessary to perform these tasks independently.

Divide your classroom into six stations.

At each station have a problem printed out that displays the problem to be solved for the students (M-A2-5-1_Multiplying Rationals Stations Problem Set.doc).

Hand each student a Stations Activity Records Sheet (M-A2-5-1_Stations Activity Records Sheet.doc).

Divide students into groups at each station. Divide them in the manner that works best for your classroom dynamics.

Have students work at their station for an allotted amount of time (3–7 minutes). While working, monitor students’ progress and assist where necessary. Adjust timing as needed.

See M-A2-5-1_Multiplying Rationals Stations Problem Set.doc (p.7) in the Resources folder for solutions.

Part 2

Preface this section by explaining to students that the modeled solutions are simply one way to simplify the expressions. Encourage students to find and discuss other pathways to solving the problem.

“Recall the process for dividing the following fractions:  . We know there is a rule to follow when dividing fractions. However, let’s look at the reasoning behind the rule, before we state it here. The meaning of the statement is how many eight-ninths are in two-thirds? Since the reciprocal of eight-ninths is nine-eighths, because division is multiplication by the reciprocal. It is equivalent to two-thirds times nine-eighths:

[IS.7 - Struggling Learners]

Take time to complete this problem and discuss with the class two possible methods for reaching a solution. Once they use the reciprocal of the second fraction, the students should find the methods to be similar to what they did with multiplication of fractions. Solution: (3/4)

Model the following examples for students. Make sure to point out the relationship between the work they did for the multiplying section because once the reciprocal substitution is done, the problems become identical to what they did before.

Note: This problem can be done using rules of exponents and simplifying fractions as well; the above work shows the common factors of the numerator and denominator.

This problem involves the process of simplifying after factoring a greatest common factor and reminds students about factoring binomials.

This example is designed to practice using different factoring techniques: Greatest Common Factor, difference of squares, and integers.

This example exposes students to trinomial factoring and what happens when everything divides by a common factor.

Make it clear to students how it is possible to go from 1 − x to x − 1 by factoring out −1. This example is designed to show more advanced types of complex factoring.

Learning Activity: Level Up! This activity is designed to help students get to the point of completing the most advanced type of division problems independently. Students will begin with a simple problem, and as they work through the steps of the activity they will try to reach the ultimate goal of completing an advanced problem. This will help to address the needs of those students who need opportunities for additional learning while allowing students at or beyond the standards the opportunity to move forward.

Instructions: Prior to the activity, cut out the problems from the Level Up! Problem Set into strips (M-A2-5-1_Level Up Problem Set and KEY.doc).

To begin the activity, pass out problem 1 to each student (most basic problem). Students are to work on the problem independently while the teacher addresses students’ needs where appropriate. Once a student has satisfactorily completed the problem, he/she receives a problem one level harder and repeats the process. The teacher may consider adding a rewards system for each level that is passed to help motivate students. Solutions to this activity can be found on page three of the Level Up! Problem Set document (M-A2-5-1_Level Up Problem Set and KEY.doc).

Part 3

Preface this section by explaining to students that the modeled solutions are simply one way to simplify the expressions. Encourage students to find and discuss other pathways to solving the problem.

“When adding and subtracting fractions, what is the rule that we have to follow?” Students should answer that a common denominator is needed. [IS.8 - Struggling Learners]

“Looking at the following sum,, what would the common denominator be ?” (15)

“What do we have to do to make these two fractions have a common denominator?” Multiply the numerator and denominator of the first fraction by 5 and then multiply the numerator and denominator of the second fraction by 3, thereby creating the new equivalent sum: . Ask students whether they can think of another way to derive equivalent fractions with the same denominator.

“Once we have a common denominator we can add these two fractions, but make sure when you add or subtract that you are only doing so with the numerators; the denominator stays as it is. Therefore, the sum is (22/15).”

“The same rules that are necessary when adding or subtracting numbers are also required when adding or subtracting rational expressions. In other words, we must have common denominators. We multiply by (5/5)  which is really multiplying by 1.”

“To begin a problem involving the adding or subtracting of rational expressions, the first thing we have to do is make sure the denominators are in factored form. This will allow us to find a common denominator more easily.”

Prior to modeling the examples of how to add and subtract rational expressions, use the Common Denominator Practice Sheet to help students get used to the process of determining the LCM of the two denominators so they can perform this task more efficiently when completing the addition and subtraction problems (M-A2-5-1_Common Denominators and KEY.doc).

Model the following examples for students, carefully guiding them through the process of finding the common denominator and then the process required to combine the two fractions.

When discussing this example with students, make sure to emphasize the important steps involved (factoring, distributing, and combining like terms).

Use this example as an opportunity to discuss the relationship between the two denominators and how factoring out −1 will help reach the LCD.

Note that this example begins with factoring. Make sure to emphasize that this must be done before finding a common denominator. Be sure to remind students about multiplying the numerators before combining like terms.

This example is more advanced with the level of factoring, multiplying, and combining like terms.

Part 4

Introduce students to the idea of a rational function, i.e., a function given by a rational expression. A rational number is one that can be expressed as the ratio (or the quotient) of two integers.

Solve the following problem:

Billy’s Boat Repair Shop manager wants to determine his profit for last month. He has concluded that the revenue can be modeled by the function
and the costs can be modeled by the function ,
where x is the number of engines repaired.
Find a function P(x) that can model profit for Billy’s Boat Repair Shop. Determine the profit for repairing 12 boats.

[Solution: P(x) = ; \$29.96]

·         As an Exit Ticket: have students write a brief statement responding to the following: How are multiplying and dividing rational expressions the same/different?

·         After going through topic 1, use the independent practice worksheet (M-A2-5-1_Multiplying Rationals IP and KEY.doc) to provide students with extra practice. The problems range from basic to most advanced. If necessary, assign specific problems to students based on their needs. Solutions can be found on page 2 of the document.

·         Use guided notes packets provided in the Resources folder for students who need opportunity for additional learning (M-A2-5-1_Operations Guided Notes.doc).

Extension:

·         Use the Level Up! Problem Set (M-A2-5-1_Level Up Problem Set and KEY.doc) as a homework assignment if it is not possible to complete in class. Adjust required problems to be done by students depending on their levels, if necessary.

·         Division Application Extension: If the area of a rectangle is and the width is , what is the length? Solution: (3x + 7)

·         After going through topic 3, use the independent practice worksheet (M-A2-5-1_Adding Subtracting IP and KEY.doc; solutions on page 2 of document) to provide students with extra practice. The problems range from basic to most advanced. If necessary, assign specific problems to students based on their needs.

### Related Instructional Videos

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DRAFT 11/05/2010