Module 2: Rational Algebraic Expressions And Algebraic .
TEACHING GUIDEModule 2: Rational Algebraic Expressions and Algebraic Expressions with Integral ExponentsA.Learning Outcomes1.Grade Level StandardThe learner demonstrates understanding of key concepts and principles of algebra, geometry, probability andstatistics as applied, using appropriate technology, in critical thinking, problem solving, reasoning, communicating,making connections, representations, and decisions in real life.2.Content and Performance StandardsContent Standards:The learner demonstrates understanding of key concepts and principles of rational algebraic expressions andalgebraic expressions with integral exponents.Performance Standards:The learner is able to formulate real – life problems involving rational algebraic expressions and algebraicexpressions with integral exponents and solves these with utmost accuracy using a variety of strategies.63
UNPACKING THE STANDARDS FOR UNDERSTANDINGSUBJECT:Grade 8 MathematicsQUARTER:Second QuarterSTRAND:AlgebraTOPIC:Rational Algebraic Expressions andAlgebraic Expressions with IntegralExponentExponentsLESSONS:1. Rational Algebraic Expressions andAlgebraic Expressions with IntegralExponents2. Operations on Rational AlgebraicExpressionsLEARNING COMPETENCIESKnowledge: Describe and illustrates rational algebraic expressions. Interprets zero and negative exponents.Skill: Evaluates and simplifies algebraic expressions involving integral exponents. Simplifies rational algebraic expressions Performs operations on rational algebraic expressions Simplifies complex fractionsESSENTIAL UNDERSTANDING:ESSENTIAL QUESTION:Students will understand that rate – How can rate – related problems berelated problems can be modelled using modelled?rational algebraic expressions.TRANSFER GOAL:Students on their own, solve rate – related problems using models on rationalalgebraic expressions.B. Planning for Assessment1. Product/PerformanceThe following are the products and performances that students are expected to come up with in this module.a. Simplify rational algebraic expressions correctly.b. Perform operations on rational algebraic expressions correctly.c. Present creatively the solution on real – life problems involving rational algebraic expression.d. Create and present manpower plan for house construction that demonstrates understanding of rational algebraicexpressions and algebraic expressions with integral exponents.64
2. Assessment MatrixTYPEKNOWLEDGEPre - testPROCESS/SKILLSUNDERSTANDINGPERFORMANCEMatch It To Me,Egyptian FractionExplanation,InterpretationKWLH,Self – knowledgePerspectivePre – assessment/DiagnosticAnticipation guideSelf – knowledgeInterpretation,ExplanationPicture AnalysisInterpretation,Explanation, Self –knowledge, tation,ExplanationMy Definition ChartPerspective, Self knowledgeQuizInterpretation,Explanation3 – 2 – 1 ChartInterpretation,Explanation, Self –knowledge65
My ValueInterpretation,Explanation, Self –knowledgeMatch It DownInterpretation,Explanation, Self –knowledgeWho’s RightInterpretation,Explanation, Self –knowledge, EmpathyQuiz ConstructorInterpretation,Explanation, Self –knowledge, EmpathyCircle ProcessInterpretation,Explanation, Self –knowledge, EmpathyHow FastInterpretation,Explanation, Self –knowledge, Empathy,ApplicationChain ReactionInterpretation, Explanation,Self – knowledge, EmpathyFlow ChartInterpretation, Explanation,Self – knowledge, Empathy66
PresentationInterpretation,Explanation, Self –knowledge, ApplicationManpower planInterpretation,Explanation, emphaty,Self – knowledge,application, PerspectiveReaction GuideSelf – knowledge,Interpretation,ExplanationSummativePost – testInterpretation,Application, Self –knowledge, EmphatySelf - assessmentLearned – Affirmed –ChallengedInterpretation,Explanation, Self –knowledge, Empathy,PerspectiveWhat is Wrong WithMe?Interpretation,Explanation, Self –knowledge, Empathy,Perspective67
Assessment Matrix (Summative Test)Levels of ng30%What will I assess? How will I assess?How Will I Score?Describing and illustrating rational algebraicexpressions.Interpreting zero and negative exponents.Evaluating and simplifying algebraic expressionsinvolving integral.Simplifying rational algebraic expressionsPerforming operations on rational algebraicexpressionsSimplifying complex fractionsSolving problems involving rational algebraicexpressions.Paper and pen Test (refer to attachedpost – test)Students will understand that rate – relatedproblems can be modelled using rationalalgebraic expressions.MisconceptionPaper and pen Test (refer to attachedpost – test)Items 1, 2, and 3Paper and pen Test (refer to attachedpost – test)1 point for every correct responseItems 4, 5, 6, 7, and 81 point for every correct responseItems 9, 10, 11, 12, 13, and 14GRASPSApply the concepts of rational algebraicexpressions to model rate – related problemsPaper and pen Test (refer to attachedpost – test)Students will model rate–related problems usingrational algebraic expressions.A newlywed couple plans to constructa house. The couple has alreadya house plan from their engineerfriend. The plan of the house isRubric on manpower plan.illustrated below:1 point for every correct responseItems 15, 16, 17, 18, 19, and 20.LaboratoryProduct30%1 point for every correct response2mBedroom1mDining Room2mComfort1.5 m RoomLiving Room2.5 mMasterBedroomCriteria:3m3m3mAs a foreman of the project, you aretasked to prepare a manpower planto be presented to the couple. Theplan should include the following:number of workers needed tocomplete the project and their dailywages, cost and completion calityEfficiency68
C.Planning for Teaching-LearningIntroduction:This module covers key concepts of rational algebraic expressions and expressions with integral exponents. Thismodule is divided into lessons. The first lesson is the introduction to rational algebraic expressions and algebraic expressionswith integral exponents and the second lesson is on operations on rational algebraic expressions.The first lesson will describe the rational algebraic expressions, interpret algebraic expressions with negative and zeroexponents, evaluate and simplify algebraic expressions with integral exponents, and simplify rational algebraic expressions.In the second lesson, learner will perform operations on rational algebraic expressions, simplifies complex fraction, and solveproblems involving rational algebraic expressions.In this module, learner are given the opportunity to use their prior knowledge and skills in dealing with rational algebraicexpressions and algebraic expressions with integral exponents. They are also given varied activities to process their knowledgeand skills learned and deepen and transfer their understanding of the different lessons.To introduce the lesson, let the students reflect on the introduction and focus questions in the learner’s guide.INTRODUCTION AND FOCUS QUESTIONS:You have learned special products and factoring polynomials in Module 1. Your knowledge on these will help you betterunderstand the lessons in this module.Now, take a look at these -part-one/http://www.waagner-biro.com/images dynam/image zoomed/korea small103 01.jpgHave you ever asked yourself how many people are needed to complete a job? What are the bases for their wages?And how long can they finish the job? These questions may be answered using rational algebraic expression which you willlearn in this module.69
ExercisesEvaluate the following algebraic expressions1.40y-1, y 52.1, m -8m-2(m 4)2-2(p – 3) , p 13.4.145.(x – 1)-2 , x 2(x 1)-2y-3 – y-2, y 2A ctivity 14 BIN - GOMake a 3 by 3 bingo card. Choose a number to be placed in your bingo card fromthe numbers below. Your teacher will give an algebraic expression with integral exponentsand the value of its variable. The first student can form a frame wins the game.Activity 15: Quiz constructorThe learner will make his/her own algebraic expressions with integralexponents. The expression must have at least two variables and theexpressions must be unique from his/her classmates. The learner will alsoassign value to the variables and he/she must show how to evaluate thesevalues to his/her algebraic expressions.1174231- 811512934374251111332322155023443149012656The frame card must be likethis:A ctivity 15 QUIZ CONSTRUCTORBe like a quiz constructor. Write in a one – half crosswise three algebraic expressionswith integral exponents in at least 2 variables and decide what values to be assigned in thevariables. Show how to evaluate your algebraic expressions. Your algebraic expressionsmust be unique from your classmates.86
Activity 16: Connect to my EquivalentThis activity will allow the learner to recall the steps and concepts in reducingfraction to its lowest term and relate these steps and concepts to simplifyingrational algebraic expressions.A ctivity 16CONNECT TO MY EQUIVALENTConnect column A to its equivalent simplest fraction to column B.Answer to this activityTeacher’s Note and RemindersAB52081213143412234851568QU?NSES TI O1.2.How did you find the equivalent fractions in column A?Do you think you can apply the same concept in simplifying arational algebraic expression?You might wonder how to answer the last question but the key concept of simplifyingrational algebraic expressions is the concept of reducing fractions to its simplest form.Examine and analyze the following examples. Pause once in a while to answercheck – up questions.Don’tForget!Illustrative example: Simplify the following rational algebraic expressions.1.4a 8b12?SolutionIllustrative Example4a 8b 4(a 2b)124 3You can have additional illustrative examples if necessary.87 a 2b3What factoring method isused in this step?
15c3d4e12c2d5w2.?SolutionWhat factoring method isused in this step?3 43 5c2cd4e15c2 d5 e 12c d w 3 4c2d4dw 5ce4dwx2 3x 2x2 – 1Solution3.7?2x 3x 2 (x 1)(x 2)x2 – 1(x 1)(x – 1) (x 2)(x – 1)Answer to Activity 17This activity may be a collaborative work or an individual performance.This may help in determining how far the learner understands the topic.QU?NSES TI OWhat factoring method isused in this step?Based on the above examples,1.What is the first step in simplifying rational algebraic expressions?2.What happen to the common factors of numerator anddenominator?ExercisesSimplify the following rational algebraic expressionsWebBased onal-expressions-2--simplifying881.2.3.y2 5x 44.y2 – 3x – 4-21a2b2 5.28a3b3x2 – 9x2 – x 12m2 6m 5m2 – m – 2x2 – 5x – 14x2 4x 4
CONCEPTUAL CHANGEActivity 18. Circle ProcessThe learner will write his/her understanding on the process of simplifyingrational algebraic expressions. This activity will gauge the learner if he/shecan really grasp the concept or not. If there are still difficulties in understandingthe concept, then give another activity.A ctivity 17 MATCH IT DOWNMatch the rational algebraic expressions to its equivalent simplified expression fromthe top. Write it in the appropriate column. If the equivalent is not among the choices, writeit in column F.a.Teacher’s Note and Remindersa 5-1 b. 1 c.d.3ae.a3a2 6a 5a 1a3 2a2 a3a2 6a 33a2 – 6aa–2a–11–a(3a 2)(a 1)3a2 5a 23a3 – 27a(a 3)(a – 3)a3 125a2 – 25a–8-a 818a2 – 3a-1 6a3a – 11 – 3a3a 11 3aa2 10a 25a 5AA ctivity 18BCDEFCIRCLE PROCESSWrite each step in simplifying rational algebraic expression using the circles below.You can add or delete circle if necessary.Don’tForget!In this section, the discussions were all about introduction on rational algebraicexpressions. How much of your initial ideas are found in the discussion? Which ideasare different and need revision? Try to move a little further in this topic through nextactivities.89
What toto UnderstandUnderstandWhatWhat toto UnderstandUnderstandWhatIn this part of the lesson, the learner should develop the key concepts ofrational algebraic expression to answer the essential question. To addressthe essential question, the learner should have background in solvingproblems involving the concept of rational algebraic expressions. He/she must be exposed to different scenarios where the rational algebraicexpressions involved especially rate–related problemsYour goal in this section is to relate the operations of rational expressions toa real – life problems, especially the rate problems.Work problems are one of the rate – related problems and usually deal with personsor machines working at different rates or speed. The first step in solving these problemsinvolves determining how much of the work an individual or machine can do in a given unitof time called the rate.Illustrative ExampleIllustrative example:As one way of solving problems, let the learner examine and analyze howthe table/matrix method works. Guide the learner on how to use on tableeffectively.A.Nimfa can paint the wall in 5 hours. What part of the wall is painted in 3 hours?Solution:1Since Nimfa can paint in 5 hours, then in one hour, she can paint 51of the wall. Her rate of work is 5 of the wall each hour. The rate of work is thepart of a task that is completed in 1 unit of time.Teacher’s Note and Reminders13Therefore, in 3 hours, she will be able to paint 3 5 5 of the wall.You can also solve the problem by using a table. Examine the table below.Don’tForget!90Rate of work(wall painted per hour)Time workedWork done(Wall painted)151 hour15152 hours25153 hours35
Illustrative ExampleAnother way of visualizing the problem is the part of the work done in certaintime. Let them examine and analyze how this method works.The learners should grasp the concept of rate – related problem(rate time work).You can add more examples to strengthen their ideas regarding solving raterelated problemsYou can also illustrate the problem.B.Teacher’s Note and Reminders1st hour2nd hour15153rd hour154th hour155th hour15So after 3 hours, nimfa3only finished painting5of the wall.Pipe A can fill a tank in 40 minutes. Pipe B can fill the tank in x minutes. What part ofthe tank is filled if either of the pipes is opened in ten minutes?Solution:1of the tank in 1 minute. Therefore, the rate is 1 of the tank per4040minute. So after 10 minutes,1110 40 of the tank is full.4Pipe A fills11of the tank in x minutes. Therefore, the rate is of the tank perxxminute. So after x minutes,Pipe B fills10 1 10 of the tank is full.xxIn summary, the basic equation that is used to solve work problem is:Rate of work time worked work done.r t wDon’tForget!A ctivity 19HOWS FAST 2Complete the table on the next page and answer question that follows.You printed your 40 – page reaction paper, you observed that the printer A inthe internet shop finished printing in 2 minutes. How long will it take printer A toprint 150 pages? How long will it take printer A to print p pages? If printer B canprint x pages per minute, how long will it take to print p pages? The rate of eachprinter is constant.Activity 19: How Fast 2 - RevisitedLearner will fill in necessary data in this table. This will assess the learnerif he/she grasps the concept of rational algebraic expressions in differentcontext.91
Teacher’s Note and RemindersPrinterPages40 pages45 pages150 pagesp pagesp pages30 pages35 pages40 pagesPrinter APrinter BQU?NSES TI O1.2.3.4.Time2 minutesRatex ppmHow did you solve the rate of each printer?How did you compute the time of each printer?What will happen if the rate of the printer increases?How do time and number of pages affect to the rate of the printer?The concepts of rational algebraic expressions were used to answer the situationabove. The situation above gives you a picture how the concepts of rational algebraicexpressions were used in solving rate – related problems.What new realizations do you have about the topic? What new connectionshave you made for yourself? What questions do you still have? Fill-in the Learned,Affirmed, Challenged cards given below.Don’toF rget!LearnedWhat new realizationsand learning do you haveabout the topic?To ensure the understanding of the learner, he/she will do this activity beforemoving to transfer stage. This will enable the learner to recall and reflectwhat has been discussed in this lesson and solicit ideas on what and howthe students learned in this lesson. Try to clear his/her thought by addressingthe questions regarding the topics in this lesson. Responses may be writtenin journal notebook.92AffirmedChallengedWhat new connectionshave you made?Which of your old ideashave veen confirmed/affirmed?What questions do youstill have? Which areasseem difficult for you?Which do you want toexplore
What toto TransferTransferWhatWhat toto TransferTransferWhatIn this part, students will show how to transfer their understanding in a real – lifesituation. They will be given a task as presented in the learning guide materials. Theywill present their work though presentation is not part of the criteria. This may be apractice for them in presenting an output because before they finish this learning guide,they have to present an output and one of the criteria is presentation.Your goal in this section is to apply your learning in real life situations. You willbe given a practical task which will demonstrate your understanding.A ctivity 20Teacher’s Note and RemindersHOURS AND PRINTSThe JOB Printing Press has two photocopying machines. P1 can print box ofbookpaper in three hours while P2 can print a box of bookpaper in 3x 20 hours.a.How many boxes of bookpaper are printed by P1 in 10 hours? In 25 hours? in65 hours?b.How many boxes of bookpaper can P2 print in 10 hours? in 120x 160 hours?in 30x2 40x hours?You will show your output to your teacher. Your work will be graded according tomathematical reasoning and accuracy.RUBRICS FOR YOUR OUTPUTCRITERIADon’toF 1Mathematical Explanationreasoningshowsthoroughreasoningand alreasoning.Explanationshows gaps inreasoning.Explanationshows illogicalreasoning.AccuracyAllcomputationsare corrects.Most of thecomputationsare correct.Some thecomputationsare correct.Allcomputationsare correctand shown indetail.OVERALLRATING93RATING
Lesson 2Operations of Rational Algebraic ExpressionsLessonWhat toto KnowKnowWhatBefore moving to the operation on rational algebraic expressions, review first operationsof fraction and the LCD.2Operations ofRational AlgebraicExpressionsWhat totoKnowKnowWhatIn the first lesson, you learned that rational algebraic expression is a ratio of twopolynomials where the denominator is not equal to zero. In this lesson, you will be ableto perform operations on rational algebraic expressions. Before moving to the newlesson, let’s look back on the concepts that you have learned that are essential to thislesson.Activity 1: Egyptian FractionThis activity will enhance the learner their capability in operating fractions.This is also a venuee for the learner to review and recall the concepts onoperations of fractions. Their response to the questions may be written ontheir journal notebook.In the previous mathematics lesson, your teacher taught you how to add andsubtract fractions. What mathematical concept plays a vital role in adding and subtractingfraction? You may think of LCD or Least Common Denominator. Now, let us take anotherperspective in adding or subtracting fractions. Ancient Egyptians had special rules in theirfraction. When they have 5 loaves for 8 persons, they did not divide it immediately by8, they used the concept of unit fraction. Unit fraction is a fraction with 1 as numerator.Egyptian fractions used unit fractions without repetition except 2 . Like 5 loaves for 83persons, they have to cut the 4 loaves into two and the last one will be cut into 8 parts. InAnswer to the activity:short:5 1 1828A ctivity 1 EGYPTIAN FRACTIONNow, be like an Ancient Egyptian. Give the unit fractions in Ancient Egyptian way.1.2.3.4.5.947 using 2 unit fractions.108 using 2 unit fractions.153 using 2 unit fractions.411 using 2 unit fractions.307 using 2 unit fractions.126.7.8.9.10.13 using 3 unit fractions.1211 using 3 unit fractions.1231 using 3 unit fractions.3019 using 3 unit fractions.2025 using 3 unit fractions.28
QUES TI O?NSActivity 2: Anticipation GuideThis activity aims to elicit background knowledge of the learner regardingoperations on rational algebraic expressions. You can use the response ofthe learner as benchmark.1.2.3.4.What did you do in giving the unit fraction?How do you feel giving the unit fractions?What difficulties do you encountered in giving unit fraction?What will you do in overcoming these difficulties?Teacher’s Note and RemindersA ctivity 2ANTICIPATION GUIDEThere are sets of rational algebraic expressions in the table below. Check agreeif the entries in column I is equivalent to the entry in column II and check disagree if theentries in the two columns are not equivalent.Don’toF rget!Ix2 – xy x yx2 – y2x2 – xyx-1 – y -16y – 30 3y – 15y2 2y 1y2 y2yy 15 74x2 6x15 14x12x2a – bb–a a–ba bb–aba b –a bb1 2baa2a bA ctivity 3Activity 3: Picture AnalysisLet the learner describe the picture. He/She may write his/her descriptionand response to the questions in the journal notebook.IIAgreeDisagreePICTURE ANALYSISTake a close look at this picture. Describe what you see.This picture may describe the application of operations on rational archives/2004/05/volunteers buil 1.php95
QU?NSES TI OWhat toto ProcessProcessWhatBefore moving to the topic, review them about operations of fraction. You cangauge their understanding on operation of fraction by letting them perform theoperation of fraction.1.2.3.4.What will happen if one of them will not do his job?What will happen when there are more people working together?How does the rate of each workers affect the entire work?How will you model the rate – related problem?The picture above shows how the operations on rational algebraic expressionscan be applied to real – life scenario. You’ll get to learn more rate – related problemsand how operations on rational algebraic expression associate to itANSWER TO REVIEWPerform the operation of the following fractions.1. 1 4 2 3. 8 33 3 5. 1 2 22 3311 40 56 9 272. 3 2 1 4. 1 3 34 24 328What toto ProcessProcessWhatYour goal in this section is to learn and understand key concepts in theoperations on rational algebraic expressions.As the concepts of operations on rational algebraic expressions become clear to youthrough the succeeding activities, do not forget to think about how to apply theseconcepts in solving real – life problems especially rate – related problems.Teacher’s Note and RemindersREVIEWPerform the operation of thefollowing fractions.Don’tForget!1.1 42 34.1 34 22.3 24 35.1 26 93. 8 3311 40A ctivity 4 MULTIPLYING RATIONALALGEBRAIC EXPRESSIONSExamine and analyze the illustrative examples below. Pause oncein a while to answer the check – up questions.The product of two rational expressions is the product of the numerators dividedby the product of the denominators. In symbols,a c ac , bd 0b d bdIllustrative ExampleIn every step in each illustrative example, there are ideas that are presentedand there are review questions and questions to ponder. These questions willunwrap the concept in every step in the solution. Let them analyze each step.You can also give more examples to emphasize the concept.Illustrative example 1: Find the product of 5t and825t 42 5t3 2 28 3t23t2) (5t)(2(22)(2)(3t2)964 .3t2Express the numerators anddenominators into primefactors as possible.
Teacher’s Note and Reminders5(2)(3t) 56t Simplify rational expressionusing laws of exponents.2 2Illustrative example 2: Multiply 4x and 3x y .3y104x 3x2y2 (22)x 3x2y23y103y(2)(5)2(2)(2)(x)(3)(x)(y)(y) (3)(y)(2)(5)3 (2)(x )(y)?(5)3 2x y5What laws of exponents wereused in these steps?x – 5 and 4x2 12x 9 ?(4x2 – 9)2x2 – 11x 52(2x 3)x–5 factoring(2x – 3)(2x 3) (2x – 1)(x – 5)? Whatmethods were(x – 5)(2x 3)(2x 3) used in this(2x – 3)(2x 3) (2x – 1)(x – 5)step?2x 3 (2x – 3)(2x – 1)? What are the rational algebraic 22x 3expressions equivalent to 1 in4x – 8x 4Illustrative example 3: What is the product ofx – 5 4x2 – 12x 94x2 – 9 2x2 – 11x 5this step?Don’toF rget!QU?NSES TI O1.2.What are the steps in multiplying rational algebraic expressions?What do you observe from each step in multiplying rationalalgebraic expressions?ExercisesFind the product of the following rational algebraic expressions.1.2.3.9710uv2 6x2y2 4.3xy25u2v2a2 – b2 a2 5.2aba–bx2 – 3x x2 – 4x2 3x – 10 x2 – x – 6x2 2x 1 y2 – 1y2 – 2y 1 x2 – 1a2 – 2ab b2 a – 1a– ba2 – 1
Answers to Activity 5: What’s My Areab11. - 2.433.A ctivity 5y 23WHAT’S MY AREA?Find the area of the plane figures below.This activity is multiplying rational algebraic expressions but in a differentcontext. After this activity, let them sequence the steps in multiplying rationalalgebraic expression. Let them identify the concepts and principles for everystep.a. b.c.Teacher’s Note and Reminders?A ctivity 6Don’toF rget!1.2.NSQUES TI OHow did you find the area of the figures?What are your steps in finding the area of the figures?THE CIRCLE ARROW PROCESSBased on the steps that you made in the previous activity, make a conceptual mapon the steps in multiplying rational algebraic expressions. Write the procedure or importantconcepts in every step inside the circle. If necessary, add a new circle.Step 1Web – basedBooster:Step 2Watch the videos in thisweb – sites for tiplying-rational-Step 3Step 4expressions-help.htmlFinal StepQUES TI OAs the learner sequences the steps, he/she will identify the mathematicalconcepts behind each step. Place the mathematical concept inside the circleuntil he/she arrived at the final answer.?NSCONCEPT CHANGE MAPActivity 6: The Circle Arrow Process1.2.3.4.98Does every step have a mathematical concept involved?What makes that mathematical concept important to every step?Can the mathematical concepts used in every step beinterchanged? How?Can you give another method in multiplying rational algebraicexpressions?
Activity 7: Dividing Rational Algebraic ExpressionsThe same as the illustrative examples in multiplying rational algebraicexpressions, each illustrative example in this operation has key ideas, reviewquestion to unveil the concept on each step. But before they begin dividingrational algebraic expressions, they have to review how to divide fractions.A ctivity 7 Dividing Rational Algebraic ExpressionsExamine and analyze the illustrative examples below. Pause oncein a while to answer the check – up questions.REVIEWPerform the operation ofthe following fractions.1. 1 3242. 5 9243. 9 3244. 10 51645. 1 124The quotient of two rational algebraic expressions is the product of the dividendand the reciprocal of the divisor. In symbols,a c a d ad , bc 0bbcdbc22 2Illustrative example 4: Find the quotient of 6ab and 9a b2 .4cd8dcTeacher’s Note and Reminders6ab2 9a2b2 6ab2 8dc24cd8dc24cd 9a2b2232 (2)(3)ab (22 )dc22 2(2) cd(3 )a b222 (2 2)(2 )(3)ab dcc2(2 )(3)(3)cdaab2 (2) c(3)a 4c3aMultiply the dividend by thereciprocal of the divisor.Perform the steps in multiplyingrational algebraic expressions.22Illustrative example 5: Divide 2x x – 6 by x – 2x – 8 .2x2 7x 52x2 – 3x – 202x2 x – 6 x2 – 2x – 82x2 7x 5 2x2 – 3x – 20? Why do we need to factorout the numerators and222x x–62x–3x–20 2 denominators?22x 7x 5x – 2x – 8 (2x – 3)(x 2) (x – 4)(2x 5)(2x 5)(x 1)(x 2)(x – 4)(2x–3)(x 2)(x–4)(2x 5) (2x 5)(x 1)(x 2) (x – 4)? What happens to the commonfactors between numerator(2x–3) and denominator?(x 1) 2x – 3x 1Don’toF rget!99
ExercisesFind the quotient of the following rational algebraic expressions.81xz3 27x2z2 4.x2 2x 1 x2 – 136y12xyx2 4x 3 x2 2x 12a 2b4x–1 1–x 5.a2 abax 1 x2 2x 116x2 – 9 16x2 24x 96 – 5x – 4x2 4x2 11x 61.2.Answers to Activity 8321. 5x 502. 2x 14x43.245A ctivity 8This activity may assess the learner’s understanding in dividing rationalalgebraic expression. This may help learner consider the division of rationalalgebraic expressions in different context.Let them enumerate the steps in dividing rational algebraic expressions andidentify the concepts and principle involved in every stepMISSING DIMENSIONFind the missing length of the figures.22The area of the rectangle is x – 100 while the length is 2x 20 . Find the208height of the rectangle.1.Teacher’s Note and Reminders2The base of the triangle is 21 and the area is x . Find the height of the3x – 2135triangle.2.Don’toF rget!QU100?NSES TI O1.2.How did you find the missing dimension of the figures?Enumerate the steps in solving the problems.
MAP OF CONCEPTUAL CHANGEActivity 9: Chain ReactionAs the learner enumerates the steps in dividing rational algebraicexpression, his/her can identify mathematical concepts in each step. Placethe mathematical concept inside the chamber until he/she arrived at the finalanswer. This activity may be individual or collaborative work.Teacher’s Note and RemindersA ctivity 9Use the Chain Reaction Chart to sequence your steps in dividingrational algebraic expressions. Write the process or mathematicalconcepts used in each step in the chamber. Add another chamber, ifnecessary.Don’toF rget!Chamber3Chamber4QU?1.2.3.4.Does every step have a mathematical concept involved?What makes
b. Perform operations on rational algebraic expressions correctly. c. Present creatively the solution on real – life problems involving rational algebraic expression. d.Create and present manpower plan for house construction that demonstrates understanding of rational algebraic expressions and algebraic expressions with integral exponents. 64
Lesson 4: Introduction to Rational Expressions Define rational expressions. State restrictions on the variable values in a rational expression. Simplify rational expressions. Determine equivalence in rational expressions. Lesson 5: Multiplying and Dividing Rational Expressions Multiply and divide rational expressions.
Multiplying and Dividing Rational Expressions Find the product of rational expressions Find the quotient of rational expressions Multiply or divide a rational expression more than two rational expressions 3.2 Add and Subtract Rational Expressions Adding and Subtracting Rational Expressions with a Common Denominator
64. Reduce rational expressions. 65. Multiply and divide rational expressions. 66. Find the least common multiple of polynomial expressions. 67. Add and subtract rational expressions. 68. Simplify complex rational expressions. 69. Solve rational equations. 70. Solve applied problems using rational equations, including proportions. Chapter 7 (7 .
Unit 2: Algebraic Expressions Media Lesson Section 2.4: Simplifying Algebraic Expressions Steps for Simplifying Algebraic Expressions Step 1: Simplify within parentheses Step 2: Use distributive property to eliminate parentheses Step 3: Combine like terms. Example 1: Simplify the following algebraic expressions. Show all possible steps.
Rational Rational Rational Irrational Irrational Rational 13. 2 13 14. 0.42̅̅̅̅ 15. 0.39 16. 100 17. 16 18. 43 Rational Rational Rational Rational Rational Irrational 19. If the number 0.77 is displayed on a calculator that can only display ten digits, do we know whether it is rational or irrational?
Lesson 9-1 Multiplying and Dividing Rational Expressions Pages 476–478 2. To multiply rational numbers or rational expressions, you multiply the numerators and multiply the denominators. To divide rational numbers or rational expressions, you multiply by the reciprocal of the divisor. In either case, you can reduce your answer by dividing the .
Rational Team Concert Rational DOORS NG Rational Collaborative Lifecycle Management Rational Developer for System z Worklight Studio Rational Quality Manager Rational Test Virtualization Server Rational Test Workbench Rational Test Workbench – Mobile Test Edition Rational Development and Test En
The American Revolution, 1763-1783 By Pauline Maier This essay excerpt is provided courtesy of the Gilder Lehrman Institute of American History. INDEPENDENCE The Seven Years’ War had left Great Britain with a huge debt by the standards of the day. Moreover, thanks in part to Pontiac’s Rebellion, a massive American Indian uprising in the territories won from France, the British decided to .
