Mathematics Grade 8 Part 1

After performing activity 19 allow the students to revisit IRF worksheets and c. 4x2y5 – 12x3y6 + 2y2 = 2y2 (2x2y3 – 6x3y4)discuss their answers as a group. You can ask them their thoughts in this d. 3x2 – 27 is not factorable or primelesson. 5. Are all polynomial expressions factorable? Cite examples to defend your Teacher’s Note and Reminders answer. Activity 20 IRF REVISIT Revisit your IRF sheet and revise your answer by filing in column 3 under FINAL column. Initial Revise Final Express the following as product of factors. 1. 4x2 – 12x = ___ 2. 9m2 – 16n2 = ___ 3. 4a2 + 12a + 9 = ___ 4. 2x2 + 9x – 5 = ___ 5. 27x3 – 8y3 = ___ 6. a3 + 125b3 = ___ 7. xm + hm – xn – hn = ___ QU ?E S T I ONS 1. What have you observed from your answers in your initial column? Is there a big difference? 2. What realization can you make with regard to the relationship of special products and factors? Don’t Activity 21 MATHEMAGIC! 2 = 1 POSSIBLE TO MEForget! Description: This activity will enable you to apply factoring to prove if 2 = 1. Instruction: Prove that 2 = 1 by applying your knowledge of factoring. You will need the guidance of your teacher in doing such. If a = b, Is 2 = 1?Challenge the students by doing Activity 21. Guide them in doing this activity a. Were you able to prove that 2 = 1?and help them realize that there is an error in this process. After the activity b. What different factoring techniques have you used to arrive at themake the students realize that 2 = 1 is not possible. Cite the mistake in the solution?activity given. As bring home activity, allow the students to complete their c. What error can you pinpoint to disprove that 2 = 1?journal to reflect their experiences in this module/lesson. d. What was your realization in this activity? 50

Activity 22 JOURNAL WRITING Description: This activity will enable you to reflect about the topic and activities you Instruction: underwent. Reflect on the activities you have done in this lesson by completing the following statements. Write your answers on your journal notebook. A. Reflect on your participation in doing all the activities in this lesson and complete the following statements: • I learned that I... • I was surprised that I... • I noticed that I... • I discovered that I... • I was pleased that I... Activity 23 LET’S SCALE TO DRAW! Description: In this activity you will draw plane figures to help you do the final project Directions: after this module. Using the skills you have acquired in the previous activity, follow your teacher’s instruction.To prepare the students for their final project do scaffold 1 and 2. This must 1. Drawing Plane Figuresbe a guided activity. Allow them to answer the process questions after and a. a square with a side which measures 10 cm.discuss it in class. b. a rectangle with a length 3 cm more than its width. c. draw any geometric figure and label its dimensions Teacher’s Note and Reminders algebraically. Don’t Forget! 2. A discussion on scale drawing will follow. After the discussion, the teacher will demonstrate the steps on how to do the following: a. A tree is five meters tall. Using a scale of 1m:2cm, draw the tree on paper. b. The school’s flag pole is 10 m high. Using a scale of 2.5m:1 dm, draw a smaller version of the tree. Give the height of your drawing. 3. The teacher will demonstrate how a cube can be made using a square paper. Follow what your teacher did. 51

Teacher’s Note and Reminders Activity 24 Model Making Description: Creation of a solid figure out of a given plane figure and expressing it in terms of factors of a polynomial. 1. Create a solid figure from the rectangular figure that was provided by following the steps given. a. Cut – out a 2 in by 2 in squares in all edges of a 12 in by 6 in rectangle. b. Fold all the sides upward. c. Paste/tape the edges of the new figure. QU ?E S T I ONS a. What is the area of the original rectangle if its side is x units? b. If the sides of the small squares is y, what expression represents Don’t Forget! its area? c. How will you express the area of the new figure in terms of theWWhhaatt ttooTTrraannssffeerr variables stated in letters a and b? Perform activity 25 to prepare the students for the final project. d. What is the dimension of the new figure formed? How about the For students culminating project ask them to do a packaging box and they will assume the role of member of a design department of a packaging volume of the solid? company. Present to the students the rubrics on how they will be graded in e. If the value of x = 4 cm and the value of y = 1 cm, what will be the this project. Students will present their output in front of the class. You may involve the class in evaluating their works. This culminating project must be dimension of the new figure? Its area? Its volume? done by group. You can have a display of their outputs. f. How did factoring help you find the dimensions of the new figure formed? the area? the volume? g. What did you learn from this activity? How can unknown quantities in geometric problems be solved? What new realizations do you have about the topic? What new connections have you made for yourself? Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. WWhhaatt ttooTTrraannssffeerr Your goal in this section is apply your learning to real life situations. You will be given a practical task which will demonstrate your understanding in special products and factoring. 52

SAMPLE RUBRIC FOR THE TRANSFER ACTIVITY Activity 25 I BRING MY TRASH HOME How did you find the performance task? How did the task help you see the real world application ofthe topic?CRITERIA Outstanding Satisfactory Developing Beginning RATING Description: Perform the activity in preparation for your final output in this module. 4 3 2 1 In response to the school’s environmental advocacy, you are Explanation of Explanations Explanations Explanations Explanations required to make cylindrical containers for your trash. This is in support the Proposal and presentation and and and presentation of the “I BRING MY TRASH HOME!” project of our school. You will of the lay-out is presentation of presentation of of the lay –out present your output to your teacher and it will be graded according to the (20%) detailed and clear. the lay-out is the lay-out is is difficult to following criteria: explanation of the proposal, accuracy of computations, clear. a little difficult understand and utilization of the resources and appropriateness of the models. Accuracy of The computations to understand and is missing Computations done are accurate The but includes several Activity 26 PACKAGING ACTIVITY and show computations critical components. (30%) understanding of done are components. Description: This activity will showcase your learning in this module. You will assume the concepts of accurate The the role of a member of a designing team that will present your proposal Utilization of special products and show a The computations Resources and factoring. wise use of computations done are to a packaging company. There is an the concepts done are erroneous and (20%) explanation for of special erroneous do not show The RER packaging company is in search for The first every computation products and and show wise use of the the best packaging for a new dairy product that they commercialAppropriateness made. factoring. some use of concepts of will introduce to the market. You are a member of the paperboard of the Model Resources are the concepts special products design department of RER Packaging Company. Your (30%) efficiently utilized Resources are of special and factoring. company is tapped to create the best packaging box that (not with less than fully utilized products and will contain two identical cylindrical containers with the corrugated) 10% excess. with less than factoring. Resources box’s volume set at 100 in3. The box has an open top. 10%-25% are not utilized The cover will just be designed in reference to the box’s box was The models excess. Resources are properly. dimensions. You are to present the design proposal for produced in are well-crafted utilized but a the box and cylinder to the Chief Executive Officer of England in and useful for The models lot of excess. The diagrams the dairy company and head of the RER Packaging understanding the are well- and models are department. The design proposal is evaluated according 1817 design proposal. crafted and The diagrams not useful in to the following: explanation of the proposal, accuracy They showcase useful for and models understanding of computations, utilization of the resources and the desired understanding are less the design appropriateness of the models. product and are the design useful in proposal. artistically done. proposal. They understanding showcase the design OVERALL the desired proposal RATING product. 53

POST - TEST1. Which statement is true? a. The square of a binomial is also a binomial. b. The product of a sum and difference of two terms is a binomial. c. The product of a binomial and a trinomial is the square of a trinomial. d. The terms of the cube of a binomial are all positive. Answer: B2. Which of the following is NOT a difference of two squares? a. 1 x4 – 1 4 b. x2 – 0.0001y4 c. 1.6(x – 1)2 – 49 d. (x + 1)4 – 4x6 Answer: C, it is the only binomial that is not a difference of two squares3. Which of the following can be factored? a. 0.08x3 – 27y3 b. 1.44(x2 + 1) – 0.09 c. 24xy(x – y) + 5 (x + y) d. 0.027(x2 + 1)3 – 8 Answer: D, it is factorable by difference of two cubes4. Which of the following values of k will make x2 – 5x + k factorable? a. 5 c. -10 b. 12 d. -14 Answer: D 54

5. If a square pool is to be made a rectangle such that the length is increased by 6 units and the width is decreased by 6 units, what will happen to its area? a. The area will increase by 12 b. The area will decrease by 12 c. The area will increase by 36. d. The area will decrease by 36 Answer: D6. What is the surface area of the given cube below? (x + 3y) cm A. (6x + 18y) cm2 B. (x2 + 6xy + 9y2)cm2 C. (6x2 + 26xy + 54y2)cm2 D. (x3 + 9x2y + 27xy2 + 27y3)cm2 Answer: C7. Factor 16x4 – 625y16 completely. a. (4x2 – 25y4)(4x2 + 25y4) b. (4x4 – 25y8)(4x4 + 25y8) c. (2x2 + 5y4)(2x2 – 5y4) (4x4 + 25y8) d. (2x2 + 5y4)(2x2 – 5y4)(2x2 + 5y4)(2x2 + 5y4) Answer: C, by factoring completely8. The area of a rectangular garden is (12x2 – 8x – 15)m2, what are its dimensions? a. (3x – 5)m by (4x + 3)m b. (6x + 5)m by (2x – 3)m c. (6x – 3)m by (2x – 5)m Answer: B d. (12x – 15)m by (x + 1) 55

9. How much wood is needed in the window frame illustrated below? m e a. (m + e)2 square units b. (m2 + e2) square units c. (m – e)(m – e) square units d. (m + e)(m – e) square units Answer: D10. If the area of a square garden is (4x2 – 12x + 9) square units, is it possible to solve its sides? a. Yes, using factoring difference of two squares. b. No, one of the sides must be given c. Yes, the area is a perfect square trinomial d. No, the area is not factorable Answer: C11. Which of the following is a possible base of a triangle whose area is (2x2 – 6x + 9) square meters? A. (2x – 9) meters B. (4x – 6) meters C. (2x + 1) meters D. (4x – 3) meters Answer: B 56

12. Liza factored the expression 15x2y3 + 10x4y + 5xy as 5xy (3xy2 + 2x3). Did Liza factor it correctly? a. No, because 5xy is not the common factor. b. Yes, because the last term is cancelled out c. Yes, there exist a common factor on all terms d. No, because the last term when factored is 1 and should not be omitted Answer: D13. Anne squared 3x + 4y as 9x2 + 16y2, which of the following statement is correct with the answer of your classmate? a. The answer is correct, because to square a binomial distribute the exponent b. The answer is wrong, because the product of squaring a binomial is a trinomial. c. The answer is correct, because the product of squaring a binomial is another binomial d. The answer is wrong, because to square a binomial is to multiply the expression by 2. Answer: B14. Below is the solution of Rogelio in factoring 3x4 – 243: 3(x4 – 81) (x2 – 9)(x2 + 9) (x + 3)(x – 3) Is the solution of Rogelio correct? A. No, because the other factors was omitted. B. No, because it lacks 3 as its factor. C. Yes, because 3x4 – 243 is divisible by x + 3. D. Yes, because the complete factorization of the expression is (x + 3)(x – 3) Answer: AUSE THE DATA BELOW TO ANSWER THE QUESTIONS THAT FOLLOW.15. A driver asked you to create a utility box with no top from a 12 in by 10 in piece of metal by cutting identical squares from each corner and turning up the sides. The box must have a capacity of 96 in3. If you are the driver, what are the standards you will look into the box? I. Appropriateness of the dimension II. Artistic 57

III. Durability IV. Innovations a. I & II c. I & III b. III & IV d. II & IV Answer: C16. Which of the following is the appropriate thing to do to ensure that the correct dimension of the box will be obtained? a. Find a model for the box and measure it. b. Measure the sides of the squares thoroughly to create a box. c. Make a trial and error until the desired capacity is obtained. d. Find the dimension of the square to be cut through factoring and scale drawing. Answer: D17. Marie Fe ask your advice on what to do so that her heterozygous blue eyed dog will have a big chance of having a blue eyed offspring, what advice could you give? a. Bring her dog to an ob – gyne b. Pair it with another blue eyed dog c. Pair it with a homozygous blue eyed dog d. Pair it with a heterozygous blue eyed dog. Answer: C, using the concept of Punnet square, a homozygous creature paired with another homozygous will have a high chance of resulting into the desired genes.18. As finance officer of RTN plantation, you were asked by the company to prepare a budget to fence 120 hectares of your company’s lot. What will you do to minimize the use of fencing materials, knowing that the length is 1 hectare less than twice the width? a. Estimate the dimension of the lot. b. Measure the dimension of the lot manually. c. Solve the dimension of the lot. d. Hire an engineer to survey the lot. Answer: C 58

19. Your friend an event organizer approach you to seek for your help to arrange 80 chairs in a weeding and suit it in the demand of the couple that the number of chairs in each rows is two less than the number of rows. How will you help your friend as to not to consume too much time in arranging? a. Make a trial and error of arrangement. b. Make a plan of arrangement of the chairs. c. Ask chair renting company to resolve the problem d. Used the data given and make an appropriate plan. Answer: D20. As the principal of a school, you asked an architect to prepare a blue print for new classroom that you plan to build. The square classroom should have different areas for utilities (lavatory, CR, storage room and locker). What criteria will you use to approve the blue print? I. Maximizing the area II. Appropriateness of the location utilities. III. Dimensions of classroom utilities. IV. Uniqueness of design a. I, II & III c. I, III & IV b. I, II & IV d. II, III & IV Answer: A 59

SUMMARY After completion of this module the students should have learned that products of some polynomials are obtained using thedifferent patterns, and these products are called special products. They must also learn the different examples of special products,such as, the square of binomials, sum and difference of two terms, squaring trinomials, and cubing a binomial. Students must have also realized that factor of different products can be obtained through the use of different patterns andrules. They should already learned the different types of factoring such as: (1) Factoring by greatest common monomial factor, (2)Factoring difference of two squares, (3) Factoring perfect square trinomials, (4) Factoring general trinomials, (5) Factoring the sumor difference of two cubes, and (6) Factoring by grouping. And at this point student must already understand and used the concepts of special products and factoring in the context ofreal – life situations.GLOSSARY OF TERMS USED IN THIS LESSON:AREA – the amount of surface contained by a figureCOMPOSITE FIGURE – a figure that is made from two or more geometric figuresFACTOR – an exact divisor of a number.GENETICS – is the area of biological study concerned with heredity and with the variations between organisms that result from it.GEOMETRY – the branch of mathematics that deals with the nature of space and the size, shape, and other properties of figuresas well as the transformations that preserve these properties.GREATEST COMMON MONOMIAL FACTOR – is the greatest factor contained in every term of an algebraic expression.HETEROZYGOUS – refers to having two different alleles (group of genes) for a single trait.HOMOZYGOUS – refers to having identical alleles (group of genes) for a single trait.PATTERN – constitutes a set of numbers or objects in which all the members are related with each other by a specific rule.PERFECT SQUARE TRINOMIAL – result of squaring a binomial.PERIMETER – the distance around a polygon.POLYNOMIAL – is a finite sum of terms each of which is a real number or the product of a numerical factor and one or more variablefactor raised to a whole – number powers.PRODUCT – the answer of multiplicationPUNNET SQUARE - is a diagram that is used to predict an outcome of a particular cross or breeding experiment. And is used by 60

biologist to determine the chance of an offspring's having a particular genotype.SCALE DRAWING – a reduced or enlarged drawing whose shape is the same as an actual object that it represents.VOLUME – the measure of space occupied by a solid bodyREFERENCES AND WEBSITE LINKS USED IN THIS LESSON:Oronce, O & Mendoza, M. (2003). Exploring Mathematics. Rex Book Store. Manila, Philippines.Oronce, O & Mendoza, M. (2007). E – Math: Worktext in Mathematics First Year High School. Rex Book Store. Manila, Philippines.Gamboa, Job D. (2010). Elementary Algebra. United Eferza Academic Publications. Bagong Lipa, Batangas City.Ho, Ju Se T., et al. 21st Century Mathematics: First Year (1996). Quezon City: Phoenix Publishing House, Inc.,2010 Secondary Education Curriculum: Teaching Guide for Mathematics II. Bureau of Secondary Education. Deped Central Officehttp://en.wikipedia.org/wiki/Punnett_squarehttp://www.youtube.com/watch?v=u5LaVILWzx8http://www.khanacademy.org/math/algebra/polynomials/e/factoring_polynomials_1http://www.xpmath.com/forums/arcade.php?do=play&gameid=93http://www.quia.com/rr/36611.htmlhttp://www.coolmath.com/algebra/algebra-practice-polynomials.htmlhttp://www.onlinemathlearning.com/algebra-factoring-2.htmlhttp://www.youtube.com/watch?v=3RJlPvX-3vghttp://www.youtube.com/watch?v=8c7B-UaKl0Uhttp://www.youtube.com/watch?v=-hiGJwMNNsMwww.world–english.orghttp://www.smashingmagazine.com/2009/12/10/how-to-explain-to-clients-that-they-are-wrong/http://www.mathman.biz/html/sheripyrtocb.htmlhttp://blog.ningin.com/2011/09/04/10-idols-and-groups-pigging-out/http://k-pop-love.tumblr.com/post/31067024715/eating-sushihttp://www.apartmenttherapy.com/tile-vault-midcentury-rec-room-39808http://onehouseonecouple.blogzam.com/2012/03/ master-shower-tile-progress/http://www.oyster.com/las-vegas/hotels/luxor-hotel-and-casino/photos/square-pool-north-luxor-hotel-casino-v169561/#http://www.youtube.com/watch?v=PcwXRHHnV8Y 61

http://www.youtube.com/watch?v=bFtjG45-Udk http://www.youtube.com/watch?v=OWu0tH5RC2Mhttp://math123xyz.com/Nav/Algebra/Polynomials_Products_Practice.phphttp://worksheets.tutorvista.com/special-products-of-polynomials-worksheet.html#http://www.media.pearson.com.au/schools/cw/au_sch_bull_gm12_1/dnd/2_spec.html.http://www.wikisori.org/index.php/Binomial_cubehttp://www.kickgasclub.org/?attachment_id949http://howardnightingale.com/fencing-lot-lines-and-municipal-requirements-some-real-estate-advise-from-poet-robert-frost/http://www.sulit.com.ph/index.php/view+classifieds/id/6268534/CONDO+MID+RISE+Paranaque+near+Airport 62

TEACHING GUIDEModule 2: Rational Algebraic Expressions and Algebraic Expressions with Integral ExponentsA. Learning Outcomes 1. Grade Level Standard The learner demonstrates understanding of key concepts and principles of algebra, geometry, probability and statistics as applied, using appropriate technology, in critical thinking, problem solving, reasoning, communicating, making connections, representations, and decisions in real life. 2. Content and Performance Standards Content Standards: The learner demonstrates understanding of key concepts and principles of rational algebraic expressions and algebraic expressions with integral exponents. Performance Standards: The learner is able to formulate real – life problems involving rational algebraic expressions and algebraic expressions with integral exponents and solves these with utmost accuracy using a variety of strategies. 63

UNPACKING THE STANDARDS FOR UNDERSTANDINGSUBJECT: LEARNING COMPETENCIESGrade 8 Mathematics Knowledge:QUARTER: • Describe and illustrates rational algebraic expressions.Second Quarter • Interprets zero and negative exponents.STRAND: Skill:Algebra • Evaluates and simplifies algebraic expressions involving integral exponents. • Simplifies rational algebraic expressionsTOPIC: • Performs operations on rational algebraic expressionsRational Algebraic Expressions and • Simplifies complex fractionsAlgebraic Expressions with IntegralExponent ESSENTIAL UNDERSTANDING: ESSENTIAL QUESTION:ExponentsLESSONS: Students will understand that rate – How can rate – related problems be1. Rational Algebraic Expressions and related problems can be modelled using modelled? Algebraic Expressions with Integral Exponents rational algebraic expressions.2. Operations on Rational Algebraic Expressions TRANSFER GOAL: Students on their own, solve rate – related problems using models on rational algebraic expressions.B. Planning for Assessment1. Product/Performance The 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 algebraic expressions and algebraic expressions with integral exponents. 64

2. Assessment Matrix KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCE TYPE Pre - test Match It To Me, Egyptian Fraction Explanation, InterpretationPre – assessment/ Anticipation guide KWLH, Diagnostic Self – knowledge Self – knowledge Interpretation, PerspectiveFormative Excercises Explanation Picture Analysis Interpretation, Interpretation, Explanation My Definition Chart Explanation, Self – Perspective, Self - knowledge, Application, knowledge Perspective Quiz 3 – 2 – 1 Chart Interpretation, Interpretation, Explanation Explanation, Self – knowledge 65

My Value Who’s RightInterpretation, Interpretation,Explanation, Self – Explanation, Self –knowledge knowledge, EmpathyMatch It Down Quiz ConstructorInterpretation, Interpretation,Explanation, Self – Explanation, Self –knowledge knowledge, EmpathyHow Fast Circle ProcessInterpretation, Interpretation,Explanation, Self – Explanation, Self –knowledge, Empathy, knowledge, EmpathyApplication Chain Reaction Interpretation, Explanation, Self – knowledge, Empathy Flow Chart Interpretation, Explanation, Self – knowledge, Empathy 66

Presentation Manpower plan Interpretation, Interpretation, Explanation, Self – Explanation, emphaty, knowledge, Application Self – knowledge, application, Perspective Summative Post – test Reaction GuideSelf - assessment Interpretation, Self – knowledge, 67 Application, Self – Interpretation, knowledge, Emphaty Explanation What is Wrong With Learned – Affirmed – Me? Challenged Interpretation, Interpretation, Explanation, Self – Explanation, Self – knowledge, Empathy, knowledge, Empathy, Perspective Perspective

Assessment Matrix (Summative Test)Levels of Assessment What will I assess? How will I assess? How Will I Score? Paper and pen Test (refer to attached 1 point for every correct response Knowledge • Describing and illustrating rational algebraic post – test) 15% expressions. 1 point for every correct response Items 1, 2, and 3Process/Skills • Interpreting zero and negative exponents. Paper and pen Test (refer to attached 25% • Evaluating and simplifying algebraic expressions post – test) involving integral. Items 4, 5, 6, 7, and 8 • Simplifying rational algebraic expressions • Performing operations on rational algebraic expressions • Simplifying complex fractions • Solving problems involving rational algebraic expressions.Understanding • Students will understand that rate – related Paper and pen Test (refer to attached 1 point for every correct response 30% problems can be modelled using rational post – test) algebraic expressions. Items 9, 10, 11, 12, 13, and 14 • Misconception Paper and pen Test (refer to attached GRASPS post – test) 1 point for every correct response Apply the concepts of rational algebraic Items 15, 16, 17, 18, 19, and 20. expressions to model rate – related problems Students will model rate–related problems using A newlywed couple plans to construct rational algebraic expressions. a house. The couple has already a house plan from their engineer friend. The plan of the house is Rubric on manpower plan. illustrated below: Laboratory 1m Criteria: Dining Room 2m 2 m Bedroom 1. ReasoningProduct 2. Accuracy 30% Comfort Living Room 3. Presentation 1.5 m Room 3m 4. Practicality 5. Efficiency Master 2.5 m Bedroom 3m 3m As a foreman of the project, you are tasked to prepare a manpower plan to be presented to the couple. The plan should include the following: number of workers needed to complete the project and their daily wages, cost and completion date. 68

C. Planning for Teaching-Learning Introduction: This module covers key concepts of rational algebraic expressions and expressions with integral exponents. This module is divided into lessons. The first lesson is the introduction to rational algebraic expressions and algebraic expressions with 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 zero exponents, 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 solve problems involving rational algebraic expressions. In this module, learner are given the opportunity to use their prior knowledge and skills in dealing with rational algebraic expressions and algebraic expressions with integral exponents. They are also given varied activities to process their knowledge and 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 better understand the lessons in this module. Now, take a look at these pictures.http://www.newroadcontractors.co.uk/wp-content/ http://planetforward.ca/blog/top-10-green- http://www.waagner-biro.com/images_dynam/gallery/road-construction/dscf1702.jpg building-trends-part-one/ image_zoomed/korea_small103_01.jpg Have 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

After you finished the module, you should be able to answer the following questions: a. What is a rational algebraic expression? b. How will you simplify rational algebraic expressions? c. How will you perform operations on rational algebraic expressions? d. How will you model rate – related problems?Objectives: At the end of the module, learner will be able to: 1. describe and illustrate rational algebraic expressions. 2. interpret zero and negative exponents. 3. evaluate and simplify algebraic expressions involving integral exponents. 4. simplifie rational algebraic expressions 5. perform operations on rational algebraic expressions. 6. simplifie complex fractions. 7. solve problems involving rational algebraic expressions.Pre – test1. Which of the following expressions is a rational algebraic expression? a. x b. √(a3+c-31)0 c. 4y-2 + z-3 d. a − bb √3y a + Answer: D. Rational algebraic expression is a ratio of two polynomials2. What is the value of a non – zero polynomial raised to 0? a. constant b. zer o c. undefined d. cannot be determine Answer: A. Any expression raised to 0 is 1 and 1 is a constant.3. What will be the result when a and b are replaced by 2 and -1, respectively, in the expression (-5a – 2b)(-2a – 3b2)? a. 1276 b. -156 c. 3 d. - 2 7 7 Answer: B. (-5a-2b)(-2a-3b2) = 10b3 = 10(-1)3 = -10 = -156 a5 25 32 70

?4. What rational algebraic expression is the same as a. x + 1 b. x – 1 c. 1 d. -1 Answer: A. x2 – 1 = (x – 1)(x + 1) =x+1 x–1 x–15. When a rational algebraic expression is subtracted from x 3 5 , the result is -x – 10 . What is the other rational algebraic expression? – x2 – 5x a. 4x b. x –x 5 c. 2x d. x -–25 Answer: C. -x – 10 + 3 = -x – 10 + 3(x) = -x – 10 + 3x = 2x – 10 = 2(x – 5) = 2 x2 – 5x x–5 x2 – 5x (x – 5)(x) x2 – 5x x2 – 5x x(x – 5) x6. Find the product of a2 – 9 and a2 – 8a + 16. + a – 20 3a – 9 a2 a. a –a 1 b. a12 – a1 c. a2 3–a7+a 1+512 d. a2 – 1 – a2 – a + 1 Answer: C. a2 – 9 • a2 – 8a + 16 = (a – 3)(a + 3) • (a – 4)(a – 4) = (a – 3)(a – 4) = a2 – 7a + 12 a2 + a – 20 3a – 9 (a – 4)(a + 5) 3(a + 3) 3(a + 5) 3a + 15 27. What is the simplest form of b– 3 ? – b 2 1 –3 2 b + 5 1 1–b a. 5 – b. 4 c. – d. 3 b b 1 2 2 2–b+3 2 b–3 2 b–3 b–3 b–3 5–b 5–b Answer: A. b–3 = ÷ = • = 2 – 1 b–3 x –2 x + 28. Perform the indicated operation 3 – 2 . a. x + 5 b. x + 1 c. x – 6 d. -x – 10 Answer: D. x–2 – x+2 = 2x – 4 – 3x – 6 = -x – 10 3 2 6 71

9. The volume of a certain gas will increase as the pressure applied to it decreases. This relationship can be modeled using the formula: V2 = V1P1 P2 where V1 is the initial volume of the gas, P1 is the initial pressure, P2 is the final pressure and the V2 is the final volume of the is 1 atm, what is the final volume of the gas if the final gas. If the initial volume of the gas is 500ml and the initial pressure 2 pressure is 5 atm? a. 10ml b. 50ml c. 90ml d. 130ml Answer: B. V2 = V1P1 = (500ml)(1/2) = 250ml = 50ml P2 5 510. Angelo can complete his school project in x hours. What part of the job can be completed by Angelo after three hours? x 3 a. x + 3 b. x – 3 c. 3 d. x Answer: D. w = rt = 1 (3) = 3 x x11. If Maribel, a groupmate of Angelo in number 10, can do the project in three hours, which expression below represents rate of Angelo and Maribel working together? a. 3 + x b. x – 3 c. 1 – 1 d. 1 + 1 3 x 3 x 1 1 Answer: D. Rate of Angelo + rate of Maribel: 3 + x12. Aaron was asked by his teacher to simplify a2 – 1 on the board. He wrote his solution on the board this way: a2 – a a2 – 1 (a + 1) (a – 1) a2 – a = a(a – 1) =1 Did he arrive at the correct answer? a. Yes, the expressions that he crossed out are all common factors. b. Yes, the LCD must be eliminated to simplify the expression. 72

c. No, a2 must be cancelled out so that the answer is 1 . a d. No, a is not a common factor of numerator Answer: D. In simplifying rational algebraic expression, we can only divide out the common factor but not the common variable.13. Your friend multiplied x – 1 and 1 + xx. His solution is presented below: 2 – x 1 – x–1 • x+1 = (x – 1) (x + 1) = x+1 2–x 1–x (2 – x) (1 – x) 2–x Is his solution correct? a. No, there is no common factor to both numerator and denominator. b. No, the multiplier must be reciprocated first before multiplying the expressions . c. No, common variables must be eliminated. d. No, dividing an expression by its multiplicative inverse is not equal to one. Answer: D. (x – 1) is additive inverse of (1 – x). If the a term is divided by the its additive inverse, quotient is - 114. Laiza added two rational algebraic expressions and her solution is presented below. 4x2+ 3 + 3x – 4 = 4x + 3 + 3x – 4 = 7x + 1 3 2+3 5 Is there something wrong in her solution? a. Yes, solve first the GCF before adding the rational algebraic expressions. b. Yes, cross multiply the numerator of the first expression to the denominator of the second expression. c. Yes, she may express first the expressions as similar fractions. d. Yes. 4x – 4 is equal to x Answer: C. We may express first the expressions into similar rational algebraic expressions and follow the concepts in adding/subtracting rational expressions. 73

15. Your father, a tricycle driver, asked you regarding the best motorcycle to buy. What will you do to help your father? a. Look for the fastest motorcycle. c. Find an imitated brand of motorcycle. b. Canvass for the cheapest motorcycle. d. Search for fuel – efficient type of motorcycle. Answer: D. A, B and C are not good qualities of a motorcycle for livelihood.16. The manager of So – In Clothesline Corp. asked you, as Human Resource Officer, to hire more tailors to meet the production target of the year. What will you look in hiring a tailor? a. Speed and efficiency c. Time conscious and personality b. Speed and accuracy d. Experience and personality Answer: A. To meet the deadline, you need a fast worker but an efficient one.17. You own three hectares of land and you want to mow it for farming. What will you do to finish it at a very least time? a. Rent a small mower c. Do kaingin b. Hire three efficient laborers d. Use germicide Answer: B. Germicide cannot kill weeds. Kaingin is prohibited according to law. Small mower is not effective for wide area.18. Your friend asked you to make a floor plan. As an engineer, what aspects should you consider in doing the plan? a. Precise and realistic c. Logical and sufficient b. Layout and cost d. Creative and economical Answer: A. The size of the parts must be realistic and should be accurate19. Your SK Chairman planned to construct a basketball court. As a contractor, what will you do to realize the project? a. Show a budget proposal c. Present a feasibility study b. Make a budget plan d. Give a financial statement Answer: C. Budget proposal is for budget approval. Budget plan is like a budget proposal. Financial statement will be given after the project is completed.20. As a contractor in number 19, what is the best action to do in order to complete the project on or before the deadline but still on the budget plan? a. All laborers must be trained workers. c. Add least charge equipment and machines. b. Rent more equipment and machines. d. Trained and amateur workers must be proportionate. Answer: D. A and B are expensive; C could not give the best quality of work. 74

Learning Goals and Targets: In this module, learners will have the following targets: • Demonstrate understanding of the key concepts of rational algebraic expressions and algebraic expressions with integral and zero exponents. • Formulate real–life problems involving rational algebraic expressions and algebraic expressions with integral and zero exponents and solve these with utmost accuracy using a variety of strategies. Teacher’s Note and Reminders Don’t Forget! 75

WWhhaatt ttoo KKnnooww 1Lesson Rational Algebraic Expressions Activity 1 elicits prior knowledge of the learner in translating verbal phrases to mathematical phrases which is one of the key concepts that WWhhaatt ttoo KKnnooww the student should learned in solving word problems in algebra. The result of this activity may become a benchmark on how to start facilitating word Let’s begin the lesson by reviewing some of the previous lessons and gathering problems later on. your thoughts in the lesson. Aside from that, this also assesses the learner regarding the concepts Activity 1 MATCH IT TO ME in polynomial. They should have a firm background regarding concepts in polynomial for rational algebraic expression. There are verbal phrases below. Look for the mathematical phrase in the figures that corresponds to the verbal phrases. Answers Key 1. The ratio of number x and four added by two.Activity 1 pq 2. The product of square root of three and the number y. x 3 3. The square of a added by twice the a.1. 4 + 2 2. √3y 3. a2 + 2a 4. b2 – (b+2) 5. 4. The sum of b and two less than the square of b. 8. z3 – 9 5. The product of p and q divided by three6. c2 7. 10y + 6 9. w – √9 10. h4 6. One – third of the square of c. 3 7. Ten times a number y increased by six 8. Cube of the number z decreased by nine. Teacher’s Note and Reminders 9. Cube root of nine less than number w. 10. Number h raised to four. x +2 2 – 2 pq b2 9 – 1 4 x x2 3 (b + 2) w2 √3y √3y Don’t y Forget! x2 – 1 a2 + 2a w – ∛9 3 x2 – 2x + 1 b2 – (b + 2) 2 c2 c2 z3 1 3 n3 10x + 6 10 z3 – 9 h4 y + 4 76

Their responses in these questions may be written in their journal notebook. QU QU?E S T I ONS NS 1. What did you feel in translating verbal phrases to mathematicalAs to its purpose, this activity is not meant for giving grades but a benchmark phases?for your lesson in this module. If ever the learner has difficulty in theseprerequisite concepts, try to have a short review in these concepts. 2. What must be considered in translating verbal phases to mathematical phrases? Teacher’s Note and Reminders 3. Will you consider these mathematical phases as polynomial? Why yes or why not? 4. How will you describe a polynomial? The above activity deals with translating verbal phrases to polynomial and you encountered some of the examples of non - polynomials. Translating verbal phases to polynomial is one of the key concepts in answering worded problem. All polynomials are expressions but not all expressions are polynomials. In this lesson you will encounter some of these expressions that are not polynomials. Activity 2 HOW FAST Don’t Suppose you are to print you 40 – page research paper. You observed that printer AForget! in the internet shop finished printing it in 2 minutes. a. How long do you think printer A can finish 100 pages? b. How long will it take printer A finish printing the p pages? c. If printer B can print x pages per minute, how long will printer B take to print p pages? ?E S T I O 1. Can you answer the first question? If yes, how will you answer it? If no, what must you do to answer the question? 2. How will you describe the second and third questions? 3. How will you model the above problem?Activity 2 How Fast Before moving to the lesson, you have to fill in the table below regarding your ideasThe learner is not expected to have correct answers in this activity. The aim on rational algebraic expressions and algebraic expressions with integral exponents.of this activity is to find out whether he/she has a background on rationalalgebraic expressions applied in a real-life situation. The response to thisactivity could help the teaching – learning process more efficient and effectiveas basis for teaching – learning process. The answers may be written in aclean sheet of paper. 77

MAP OF CONCEPTUAL CHANGE Activity 3 KWLHActivity 3: KWHL Write your ideas on the rational algebraic expressions and algebraic expressionsAside from Activity 2, KWHL is also an activity eliciting the background of with integral exponents. Answer the unshaded portion of the table and submit it to yourthe learner regarding the rational algebraic expressions. He/She could use teacher.his/her understanding in activity 2 in doing this activity. Keep their responsebecause at the end of this lesson, they will continue to answer this activity What I Know What I Want to What I Learned How Can I Learnto track their learning. Find Out MoreWWhhaatt ttoo PPrroocceessss Topic: Introduction to Rational Algebraic Expressions You were engaged in some of the concepts in the lesson but there are questions in your mind. The next lessons will answer your queries and clarify your thoughtsTeacher’s Note and Reminders regarding to our lesson. Don’t WWhhaatt ttoo PPrroocceessss Forget! Your goal in this section is to learn and understand the key concepts on rational algebraic expressions and algebraic expressions with integral exponents. As the concepts on rational algebraic expressions and algebraic expressions with integral exponents become clear to you through the succeeding activities, do not forget to think about how to apply these concepts in real – life problems especially rate – related problems.Activity 4: Match It to Me – Revisited Activity 4 MATCH IT TO ME – REVISITEDGoing back to activity 1, let them distinguish the polynomials from the (REFER TO ACTIVITY 1)non–polynomials in this activity by describing it. Give emphasis on thenon–polynomial examples in the activity. Remind them that these non– 1. What are the polynomials in the activity “Match It To Me”? List these polynomialspolynomials in the activity are not the only non–polynomials. Be guided under set P.that these non–polynomials are just rational expressions and that not allnon–polynomials are rational algebraic expressions. 2. Describe these polynomials. 3. In the activity, which are not polynomials? List these non – polynomials under set R. 4. How do these non – polynomials differ from the polynomial? 5. Describe these non – polynomials. 78

Activity 5: Compare and Contrast Activity 5 COMPARE AND CONTRASTAs they describe the polynomials and non–polynomials in Activity 4, they willsummarize their work by completing the given graphic organizer. This activity Use your answers in the activity “Match It To Me – Revisited” to complete thewill enable them to describe rational algebraic expressions and distinguish graphic organizer compare and contrast. Write the similarities and differences betweenit from polynomials. The learner may present his/her output to the class but polynomials and non – polynomials in the first activity.this is not meant for rating the learner. This activity will guide the learner todescribe the rational algebraic expressions. After the presentation, discuss POLYNOMIALS NON -that these non–polynomials are rational algebraic expressions. This activity POLYNOMIALSmay be done individually or by group. How Alike? Teacher’s Note and Reminders ____________________________________ ____________________________________ ____________________________________ ____________ How Different? ______________________ In terms of ... ______________________ ______________________ ______________________ ______________________ _________________ ______________________ ______________________ _________________ ______________________ ______________________ _________________ ______________________ _________________ ________________ _________________ ________________ _________________ _________________ _______ Don’t In the activity “Match It to Me”, the non – polynomials are called rational algebraicForget! expressions. Your observations regarding the difference between polynomials and non – polynomials in activities 4 and 5 are the descriptions of rational expression. Now, can you define rational algebraic expressions? Write your own definition about rational algebraic expressions in the chart below. 79

Activity 6: My Definition Chart Activity 6 MY DEFINITION CHARTAfter they have described the rational algebraic expressions, let them definerational algebraic expression on their own. Their response may be different Write your initial definition on rational algebraic expressions in the appropriate box.from the axiomatic definition of rational algebraic expressions but let it be. Your final definition will be written after some activities.The purpose of this activity is to generate their ideas on rational algebraicexpressions based on the examples and illustrations of rational algebraic _____________________________________ _________________________________expressions given. They can exchange their initial definitions with their My_In__iti_a_l________________________________ _____________________________________classmates and discuss how they are alike or different. My_F_i_n_a_l ________________________________ Def_in_i_ti_o_n_________________________________ Def_in_i_ti_o_n_________________________________ Teacher’s Note and Reminders _____________________________________ _____________________________________ Try to firm up your own definition regarding the rational algebraic expressions by doing the next activity. Activity 7 CLASSIFY ME Classify the different expressions below into rational algebraic expression or not rational algebraic expression. Write the expression into the appropriate column. Don’t m+2 k Rational Algebraic Not Rational Forget! √2 3k2 – 6k Expressions Algebraic y+2 Expressions y–2 1 a a6 1 – m y2 – x9 m3 c c4 a–2 8√5 Activity 7: Classify Me ?E S T I Om + 2 and c4 are the only expressions that belong to the Not Rational QU NS 1. How many expressions did you place in the rational algebraic √2 3√5 expression column?Algebraic Expressions column. After they classify the expressions, let them 2. How many expressions did you placed in the not rational algebraicdescribe the expressions in each column and compare and contrast the expression column?expressions in the two columns. This activity may guide them in formulating 3. How did you classify a rational algebraic expression from a not rational algebraic expression?definition similar to the axiomatic definition of rational algebraic expressions. 4. Were you able to place each expression to its appropriate column? 5. What difficulty did you encounter in classifying the expressions? 80

Activity 8: My Definition Chart - Continuation In the previous activities, there might be some confusions to you regardingAfter Activity 7, they can now finalize their initial definitions on rational rational algebraic expressions, but this activity firmed up your idea regarding rationalalgebraic expressions. Let them exchange their final definition and discuss algebraic expressions. Now, put into words your final definition on rational algebraicit with their classmate. In this stage, you can discuss further if there are expression.questions that need to be answered. Activity 8 MY DEFINITION CHARTProcess their final definition. You may give emphasis on the axiomaticdefinition of rational algebraic expression. After they defined rational Write your final definition on rational algebraic expressions in the appropriate box.algebraic expressions, let them illustrate it and give at least three examples.You can discuss rational algebraic expression for clarification purposes. ______________________________________ ______________________________________Mathematical Investigation: Learner may investigate the concept, My_I_n_it_ia_l_________________________________ My_F_i_n_a_l_________________________________“polynomial divided by zero”. Ask the learner why the denominator should notbe equal to zero. Let him/her investigate the clue given. You can give more Def_in_i_ti_o_n_________________________________ Def_in_i_ti_o_n_________________________________clues if needed to generate the pattern and will lead them to the concept of _____ _____undefined numbers. Compare your initial definition and your final definition on rational algebraic Teacher’s Note and Reminders expressions. Is your final definition clears your confusions? How? Give at least 3 rational algebraic expressions differ from your classmate. MATH DETECTIVE Rational algebraic Remember: expression is a ratio of two polynomials where the denominator is not equal Rational algebraic expression is a ratio of two polynomials to zero. What will happen P when the denominator of Q provided that the numerator is not equal to zero. In symbols: a fraction becomes zero? Clue: Start investigating in ,where P and Q are polynomials and Q ≠ 0. 4 = 2 ≫≫ 4 = (2)(2) 4 = 4 21 ≫≫ 4 = (1)(4) Don’t In the activities above, you had encountered the rational algebraic expressions. YouForget! might encounter some algebraic expressions with negative or zero exponents. In the next activities, you will define the meaning of algebraic expressions with integral exponents including negative and zero exponents . 81

Before moving to the next activity, review the laws of exponents RECALL Activity 9 LET THE PATTERN ANSWER IT LAWS OFActivity 9: Let the Pattern Answer It EXPONENTSThis activity will serve as a review on laws of exponents. Let the learnercomplete the table to recall the concept on laws of exponents. Let the learner I – Product of Powersexamine and analyze the pattern in this activity. If the expressionsThe pattern in this activity: the first row under in column III is divided by the multiplied have the same base,base of the expression. add the exponents.This activity may be done by group or individual work. xa•xb = xa+b Teacher’s Note and Reminders II – Power of a Power Complete the table below and observe the pattern. If the expression raised to Don’t a number is raised by another Forget! number, multiply the exponents. (xa)b = xab III – Power of a Product A B A BC A BC A B If the multiplied expressions x5 2•2•2•2•2 25 3•3•3•3•3 35 243 4•4•4•4•4 45 1,024 x•x•x•x•x is raised by a number, multiply the exponents then multiply the 2•2•2•2 3•3•3•3 4•4•4•4 x•x•x•x expressions. (xa yb)c = xac ybc (xy)a = xaya IV – Quotient of Power 2•2•2 3•3•3 4•4•4 x•x•x If the ratio of two expressions is raised to a 2•2 3•3 4•4 x•x number, then 23 4 x Case I. xa = xa-b, where a > b Case II. xb xa = 1 , where a < b xb xb-a QU?E S T I ONS 1. What do you observe as you answer the column B? 2. What do you observe as you answer the column C? 3. What happens to its value when the exponent decreases? 4. In the column B, how is the value in the each cell/box related to its upper or lower cell/box? Now, use your observations in the activity above to complete the table below. A BA B A B A B 25 32 35 243 45 1,024 x5 x•x•x•x•x 24 34 44 x4 23 33 43 x3 22 32 42 x2Activity 9: Let the Pattern Answer It 23 4 xBased on the pattern that they observe in the first table in this activity, let themcomplete the table. This will enable the learner to interpret the expressions 20 30 40 x0with negative exponents. He/she will discover that the implication of negativeexponents is the multiplicative inverse of the expression. 2-1 3-1 4-1 x-1 2-2 3-2 4-2 x-2 2-3 3-3 4-3 x-3 82

Teacher’s Note and Reminders QU ?E S T I ONS 1. What do you observe as you answer the column A? 2. What do you observe as you answer the column B? Don’t 3. What happen to its value when the exponent decreases? Forget! 4. In the column A, how is the value in the each cell/box related to its upper or lower cell/box? 5. What do you observe when the number has zero exponent? 6. Do you think that when a number raised to zero is the same to another number raised to zero? Justify your answer. 7. What do you observe to the value of the number raised to a negative integer? 8. What can you say about an expression with negative integral exponent? 9. Do you think it is true to all numbers? Cite some examples? Exercises Rewrite each item to expressions with positive exponents. l0 1. b-4 5. de-5f 9. p0 c-2 x + y 2 2. d-8 6. (x – y)0 10. (a – b+c)0 ( ( 3. w-3z-2 7. a6b8c10 0 a5b2e8 4. n2m-2o 8. 14t0 Activity 10 3 – 2 – 1 CHARTAnswer to Exercises Complete the chart below. 1 2. dc28 3. b14 4. mno2 5. de5f1. b4 6. x + y 7. 1 8. 14 9. 1 10. 2 ____________________________________________________________________________________ 3 things you ________________________________________________________________________________ found out _____________________________________________________________________________ _________________________________________________________________________ _____________________________________________________________________ ____________________________________________________MAP OF CONCEPTUAL CHANGE 2 interesting ______________________________________________________Activity 10: 3 – 2 – 1 Chart things ___________________________________________________Before moving to the next lesson, the learner should complete the 3 – 2 – 1 _______________________________________________chart. This activity will give the learner a chance to summarize the key concepts ___________________________________________in algebraic expressions with integral exponents. Address the question of the _______________________________________learner before moving to the next activity. ____________________________________ _________________________ ________________________ 1 question ____________________ you still ________________ have ____________ _________ _____ 83

Activity 11: Who is Right Activity 11 WHO IS RIGHT?Let the learner examine and analyze the solution of Allan and Gina. Let him/her decide who is correct and explain how this solution is correct and what Allan and Gina were asked to simplify n3 . There solutions are shown below togethermakes the other solution wrong. with their explanation. n-4After this, explain to the learner that there is no wrong solution between thetwo. Explain how the concepts of laws of exponents applied to the solution. Allan’s Solution Gina’s Solution Teacher’s Note and Reminders n3 n3 n4 n-4 n-4 1 = n3–(-4) = n3+4 = n7 = n3 = n3 = n7 1 n-4 Quotient law was used Expressing the exponent of in my solution the denominator as positive integer, then following the rules in dividing polynomials. Who is right? Write your explanation in a sheet of paper. You have learned the some concepts of rational algebraic expression as you performed the previous activities. Now, let us try to put these concepts in different context. Activity 12 SPEEDY MARS Don’t Mars finished the 15 – meter dash within 3 seconds. Answer the questionsForget! below. RECALL 1. How fast did Mars run? Speed is the rate of 2. At this rate, how far can Mars ran after 4 seconds? 5 seconds? moving object as it transfers from one point to 6 seconds? another. The speed is the 3. How many minutes can Mars run 50 meters? 55 meters? 60 ratio between the distance and time travelled by the meters? object. QU ?E S T I ONS How did you come up with your answer? Justify your answer. Activity 12: Speedy Mars What you just did was evaluating the speed that Mars run. Substituting the value ofThis activity aims to recall the evaluation of linear equation in grade 7. the time to your speed, you come up with distance. When you substitute your distance toExpounding the ways of solving the problem will help in evaluating rational the formula of the speed, you had the time. This concept of evaluation is the same withalgebraic expressions. evaluating algebraic expressions. Try to evaluate the following algebraic expressions in the next activity. 84

Activity 13. My Value (Answer) Activity 13 MY VALUEYou can discuss the examples in this activity to the class and give moreexamples, if necessary. The activity may be done in group or individual. a-2 27 1 Find the value of each expression below by evaluation. b-3 4 4a2 + b3 ----- 32 + 43 = 73 ------ a-1b0- My Value of a Value of b My solution My -----22 + 43 = 68 Expression 2 Value ------ 8 9 13 3 Example:Teacher’s Note and Reminders a2 + b3 a2 + b2 = 22 + 33 3 = 4 + 9 = 13 4 Your solution here: 2 4 Example: a-2 -2 3 a-2 = (-2)-2 b-3 b-3 3-3 33 27 = (-2)2 4 = 27 4 Don’t a-2 3 Your solution here Forget! b-3 2 2 a-1b0 3 QU ?E S T I ONS 1. What have you observed in the solution of the examples? 2. How these examples help you to find the value of the expression? 3. How did you find the value of the expression? 85

Exercises Evaluate the following algebraic expressions 1. 40y-1, y = 5 2. 1 , m = -8 m-2(m + 4) 3. (p2 – 3)-2, p = 114 4. (x – 1)-2, x = 2 (x + 1)-2 5. y-3 – y-2, y =2 Activity 14 BIN - GO Make a 3 by 3 bingo card. Choose a number to be placed in your bingo card from the numbers below. Your teacher will give an algebraic expression with integral exponents and the value of its variable. The first student can form a frame wins the game. 1 17 2 - 31 1 4 8 15 2 3 37 1 9 4 4 25 1 1 3 32 2 11 3 2 1 0 23 4 55 43 1 0 126 The frame card must be like 49 5 6 this:Activity 15: Quiz constructor Activity 15 QUIZ CONSTRUCTORThe learner will make his/her own algebraic expressions with integralexponents. The expression must have at least two variables and the Be like a quiz constructor. Write in a one – half crosswise three algebraic expressionsexpressions must be unique from his/her classmates. The learner will also with integral exponents in at least 2 variables and decide what values to be assigned in theassign value to the variables and he/she must show how to evaluate these variables. Show how to evaluate your algebraic expressions. Your algebraic expressionsvalues to his/her algebraic expressions. must be unique from your classmates. 86

Activity 16: Connect to my Equivalent 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 simplifying Connect column A to its equivalent simplest fraction to column B.rational algebraic expressions. ABAnswer to this activityTeacher’s Note and Reminders 51 20 3 81 12 4 43 84 51 15 2 62 83 QU?E S T I ONS 1. How did you find the equivalent fractions in column A? 2. Do you think you can apply the same concept in simplifying a rational algebraic expression? Don’t Forget! You might wonder how to answer the last question but the key concept of simplifying rational algebraic expressions is the concept of reducing fractions to its simplest form. Illustrative Example Examine and analyze the following examples. Pause once in a while to answerYou can have additional illustrative examples if necessary. check – up questions. Illustrative example: Simplify the following rational algebraic expressions. 1. 4a1+28b ? What factoring method is Solution used in this step? 4a1+28b = 4(a + 2b) 4•3 = a + 2b 3 87

2. 1125cc23dd54we ? What factoring method is Solution used in this step? 15c3d4e = 3•5c2cd4e 12c2d5w 3•4c2d4dw = 5ce 4dw 7 3. x2 + 3x + 2 ? What factoring method is x2 – 1Answer to Activity 17 used in this step?This activity may be a collaborative work or an individual performance. SolutionThis may help in determining how far the learner understands the topic. x2 + 3x + 2 = (x + 1)(x + 2) x2 – 1 (x + 1)(x – 1) = (x + 2) (x – 1) QU?E S T I ONS 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 and denominator? Exercises Simplify the following rational algebraic expressions Web 1. y2 + 5x + 4 4. m2 + 6m + 5 Based Booster y2 – 3x – 4 m2 – m – 2 http://mathvids.com/ 2. -21a2b2 5. x2 – 5x – 14 lesson/mathhelp/845- 28a3b3 x2 + 4x + 4 rational-expressions-2--- simplifying 3. x2 – 9 x2 – x + 12 88

CONCEPTUAL CHANGE Activity 17 MATCH IT DOWNActivity 18. Circle ProcessThe learner will write his/her understanding on the process of simplifying Match the rational algebraic expressions to its equivalent simplified expression fromrational algebraic expressions. This activity will gauge the learner if he/she the top. Write it in the appropriate column. If the equivalent is not among the choices, writecan really grasp the concept or not. If there are still difficulties in understanding it in column F.the concept, then give another activity. a. -1 b. 1 c. a + 5 d. 3a e. a Teacher’s Note and Reminders 3 a2 + 6a + 5 a3 + 2a2 + a 3a2 – 6a a–1 a+1 3a2 + 6a + 3 a–2 1–a (3a + 2)(a + 1) 3a3 – 27a a3 + 125 a–8 3a2 + 5a + 2 (a + 3)(a – 3) a2 – 25 -a + 8 18a2 – 3a 3a – 1 3a + 1 a2 + 10a + 25 -1+ 6a 1 – 3a 1 + 3a a+5 ABCDE F Don’t Activity 18 CIRCLE PROCESSForget! Write each step in simplifying rational algebraic expression using the circles below. You can add or delete circle if necessary. In this section, the discussions were all about introduction on rational algebraic expressions. How much of your initial ideas are found in the discussion? Which ideas are different and need revision? Try to move a little further in this topic through next activities. 89

WWhhaattttooUUnnddeerrssttaanndd WWhhaatt ttoo UUnnddeerrssttaanndd In this part of the lesson, the learner should develop the key concepts of Your goal in this section is to relate the operations of rational expressions to rational algebraic expression to answer the essential question. To address a real – life problems, especially the rate problems. the essential question, the learner should have background in solving problems involving the concept of rational algebraic expressions. He/ Work problems are one of the rate – related problems and usually deal with persons she must be exposed to different scenarios where the rational algebraic or machines working at different rates or speed. The first step in solving these problems expressions involved especially rate–related problems involves determining how much of the work an individual or machine can do in a given unitIllustrative Example of time called the rate.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 table Illustrative example:effectively. A. Nimfa can paint the wall in 5 hours. What part of the wall is painted in 3 hours? Teacher’s Note and Reminders Solution: 1 Don’t 5 Forget! of Since Nimfa can paint in 5 hours, then in one hour, she can paint the wall. Her rate of work is 1 of the wall each hour. The rate of work is the 5 part of a task that is completed in 1 unit of time. Therefore, in 3 hours, she will be able to paint 3 • 1 = 3 of the wall. 5 5 You can also solve the problem by using a table. Examine the table below. Rate of work Time worked Work done (wall painted per hour) 1 hour (Wall painted) 2 hours 1 3 hours 1 5 5 1 2 5 5 1 3 5 5 90

Illustrative Example You can also illustrate the problem.Another way of visualizing the problem is the part of the work done in certaintime. Let them examine and analyze how this method works. 1st hour 2nd hour 3rd hour 4th hour 5th hourThe learners should grasp the concept of rate – related problem 11 11 1 So after 3 hours, nimfa (rate ● time = work). 55 55 5 3You can add more examples to strengthen their ideas regarding solving rate- only finished painting 5related problems of the wall. Teacher’s Note and Reminders B. Pipe A can fill a tank in 40 minutes. Pipe B can fill the tank in x minutes. What part of the tank is filled if either of the pipes is opened in ten minutes? Solution: Pipe A fills 1 of the tank in 1 minute. Therefore, the rate is 1 of the tank per 40 40 minute. So after 10 minutes, 10 • 1 = 1 of the tank is full. 40 4 Pipe B fills 1 of the tank in x minutes. Therefore, the rate is 1 of the tank per x x minute. So after x minutes, 10 • 1 = 10 of the tank is full. x x In summary, the basic equation that is used to solve work problem is: Rate of work • time worked = work done. r • t =w Don’tForget! Activity 19 HOWS FAST 2Activity 19: How Fast 2 - Revisited Complete the table on the next page and answer question that follows.Learner will fill in necessary data in this table. This will assess the learner if he/she grasps the concept of rational algebraic expressions in different You printed your 40 – page reaction paper, you observed that the printer A incontext. the internet shop finished printing in 2 minutes. How long will it take printer A to print 150 pages? How long will it take printer A to print p pages? If printer B can print x pages per minute, how long will it take to print p pages? The rate of each printer is constant. 91

Teacher’s Note and Reminders Printer Pages Time Rate Printer A 40 pages 2 minutes x ppm 45 pages Printer B 150 pages p pages p pages 30 pages 35 pages 40 pages QU ?E S T I ONS 1. How did you solve the rate of each printer? 2. How did you compute the time of each printer? 3. What will happen if the rate of the printer increases? 4. 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 situation above. The situation above gives you a picture how the concepts of rational algebraic expressions were used in solving rate – related problems. What new realizations do you have about the topic? What new connections have you made for yourself? What questions do you still have? Fill-in the Learned, Affirmed, Challenged cards given below. Don’t Learned Affirmed Challenged Forget! What new realizations What new connections What questions do youTo ensure the understanding of the learner, he/she will do this activity before and learning do you have have you made? still have? Which areasmoving to transfer stage. This will enable the learner to recall and reflect seem difficult for you?what has been discussed in this lesson and solicit ideas on what and how about the topic? Which of your old ideas Which do you want tothe students learned in this lesson. Try to clear his/her thought by addressing have veen confirmed/the questions regarding the topics in this lesson. Responses may be written explorein journal notebook. affirmed? 92

WWhhaatt ttooTTrraannssffeerr WWhhaatt ttooTTrraannssffeerr In this part, students will show how to transfer their understanding in a real – life Your goal in this section is to apply your learning in real life situations. You will situation. They will be given a task as presented in the learning guide materials. They be given a practical task which will demonstrate your understanding. will present their work though presentation is not part of the criteria. This may be a practice for them in presenting an output because before they finish this learning guide, Activity 20 HOURS AND PRINTS they have to present an output and one of the criteria is presentation. The JOB Printing Press has two photocopying machines. P1 can print box of Teacher’s Note and Reminders bookpaper in three hours while P2 can print a box of bookpaper in 3x + 20 hours. Don’t a. How many boxes of bookpaper are printed by P1 in 10 hours? In 25 hours? in Forget! 65 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 to mathematical reasoning and accuracy. RUBRICS FOR YOUR OUTPUT CRITERIA Outstanding Satisfactory Developing Beginning RATING 4 3 2 1 Mathematical Explanation Explanation Explanation Explanation shows shows gaps in shows illogical reasoning shows substantial reasoning. reasoning. reasoning. thorough Most of the Some the All computations computations reasoning computations are correct. are correct. are corrects. and insightful OVERALL RATING justifications. Accuracy All computations are correct and shown in detail. 93

Lesson 2 Operations of Rational Algebraic Expressions 2 Operations of Rational AlgebraicWWhhaatt ttoo KKnnooww Lesson ExpressionsBefore moving to the operation on rational algebraic expressions, review first operations WWhhaattttooKKnnoowwof fraction and the LCD.Activity 1: Egyptian Fraction In the first lesson, you learned that rational algebraic expression is a ratio of two polynomials where the denominator is not equal to zero. In this lesson, you will be ableThis activity will enhance the learner their capability in operating fractions. to perform operations on rational algebraic expressions. Before moving to the newThis is also a venuee for the learner to review and recall the concepts on lesson, let’s look back on the concepts that you have learned that are essential to thisoperations of fractions. Their response to the questions may be written on lesson.their journal notebook. In the previous mathematics lesson, your teacher taught you how to add andAnswer to the activity: subtract fractions. What mathematical concept plays a vital role in adding and subtracting fraction? You may think of LCD or Least Common Denominator. Now, let us take another perspective in adding or subtracting fractions. Ancient Egyptians had special rules in their fraction. When they have 5 loaves for 8 persons, they did not divide it immediately by 8, 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 8 3 persons, they have to cut the 4 loaves into two and the last one will be cut into 8 parts. In short: 5 = 1 + 1 8 2 8 Activity 1 EGYPTIAN FRACTION Now, be like an Ancient Egyptian. Give the unit fractions in Ancient Egyptian way. 1. 170 using 2 unit fractions. 6. 13 using 3 unit fractions. 2. 185 using 2 unit fractions. 12 3. 34 using 2 unit fractions. 11 4. 3110 using 2 unit fractions. 7. 12 using 3 unit fractions. 5. 172 using 2 unit fractions. 8. 31 using 3 unit fractions. 94 30 19 9. 20 using 3 unit fractions. 10. 25 using 3 unit fractions. 28

Activity 2: Anticipation Guide QU ?E S T I ONS 1. What did you do in giving the unit fraction?This activity aims to elicit background knowledge of the learner regarding 2. How do you feel giving the unit fractions?operations on rational algebraic expressions. You can use the response of 3. What difficulties do you encountered in giving unit fraction?the learner as benchmark. 4. What will you do in overcoming these difficulties? Teacher’s Note and Reminders Activity 2 ANTICIPATION GUIDE There are sets of rational algebraic expressions in the table below. Check agree if the entries in column I is equivalent to the entry in column II and check disagree if the entries in the two columns are not equivalent. I II Agree Disagree x2 – xy • x + y Don’t x2 – y2 x2 – xy x-1 – y -1Forget! 6y – 30 ÷ 3y – 15 2y y2 + 2y + 1 y2 + y y+1 15 + 14x 5 +7 12x2 4x2 6x a+b a –b b–a b–a a–b a2 a+b – b a+b b a+b 1+2 baActivity 3: Picture Analysis Activity 3 PICTURE ANALYSISLet the learner describe the picture. He/She may write his/her descriptionand response to the questions in the journal notebook. Take a close look at this picture. Describe what you see.This picture may describe the application of operations on rational algebraicexpression. http://www.portlandground.com/archives/2004/05/ volunteers_buil_1.php 95

WWhhaatt ttoo PPrroocceessss QU ?E S T I O NS 1. What will happen if one of them will not do his job? 2. What will happen when there are more people working together?Before moving to the topic, review them about operations of fraction. You can 3. How does the rate of each workers affect the entire work?gauge their understanding on operation of fraction by letting them perform the 4. How will you model the rate – related problem?operation of fraction.ANSWER TO REVIEW The picture above shows how the operations on rational algebraic expressions can be applied to real – life scenario. You’ll get to learn more rate – related problemsPerform the operation of the following fractions. and how operations on rational algebraic expression associate to it1. 1 • 4 = 2 3. 8 • 33 = 3 5. 1 • 2 = 223 3 11 40 5 6 9 27 WWhhaatt ttoo PPrroocceessss2. 3 • 2 = 1 4. 1 • 3 = 3 Your goal in this section is to learn and understand key concepts in the43 2 42 8 operations on rational algebraic expressions. As the concepts of operations on rational algebraic expressions become clear to youTeacher’s Note and Reminders through the succeeding activities, do not forget to think about how to apply these concepts in solving real – life problems especially rate – related problems. REVIEW Activity 4 MULTIPLYING RATIONAL ALGEBRAIC EXPRESSIONS Perform the operation of the following fractions. 1. 1 • 4 4. 1 • 3 Examine and analyze the illustrative examples below. Pause once 23 42 in a while to answer the check – up questions. 2. 3 • 2 5. 1 • 2 43 69 Don’t 3. 8 • 33 Forget! 11 40Illustrative Example The product of two rational expressions is the product of the numerators dividedIn every step in each illustrative example, there are ideas that are presented by the product of the denominators. In symbols,and there are review questions and questions to ponder. These questions willunwrap the concept in every step in the solution. Let them analyze each step. a • c = ac , bd ≠ 0You can also give more examples to emphasize the concept. b d bd Illustrative example 1: Find the product of 5t and 4 . 8 3t2 22 5t • 4 = 5t • 3t2 Express the numerators and 8 3t2 23 (5t)(22) denominators into prime = (22)(2)(3t2) factors as possible. 96

Teacher’s Note and Reminders = 5 Simplify rational expression (2)(3t) using laws of exponents. Don’t Forget! = 5 6t Illustrative example 2: Multiply 4x and 3x2y2 . 3y 10 4x • 3x2y2 = (22)x • 3x2y2 3y 10 3y (2)(5) = (2)(2()3(x)()y(3)()2(x)(25)()y)(y) = (2)((x53))(y) ? What laws of exponents were = 2x53y used in these steps? Illustrative example 3: What is the product of x–5 and 4x2 + 12x + 9 ? (4x2 – 9) 2x2 – 11x + 5 x–5 4x2 (2x + 3)2 4x2 – 9 • 2x2 – 12x + 9 = (2x x–5 + 3) • (2x – 1)(x – 5) ? What factoring – 11x + 5 = – 3)(2x methods were = used in this = (x – 5)(2x + 3)(2x + 3) step? (2x – 3)(2x + 3) (2x – 1)(x – 5) 2x + 3 (2x – 3)(2x – 1) 2x + 3 ? What are the rational algebraic 4x2 – 8x + 4 expressions equivalent to 1 in this step? QU ?E S T I ONS 1. What are the steps in multiplying rational algebraic expressions? 2. What do you observe from each step in multiplying rational algebraic expressions? Exercises Find the product of the following rational algebraic expressions. 1. 10uv2 • 6x2y2 4. x2 + 2x + 1 • y2 – 1 3xy2 5u2v2 y2 – 2y + 1 x2 – 1 2. a2 – b2 • a2 5. a2 – 2ab + b2 • a – 1 2ab a – b a2 – 1 a– b 3. x2 – 3x • x2 – 4 x2 + 3x – 10 x2 – x – 6 97

Answers to Activity 5: What’s My Area Activity 5 WHAT’S MY AREA?1. - b 2. 1 3. y−2 Find the area of the plane figures below. 4 3 3 a. b. c. 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. Teacher’s Note and Reminders QU QU?E S T I O NS NS 1. How did you find the area of the figures? 2. What are your steps in finding the area of the figures? Activity 6 THE CIRCLE ARROW PROCESS Don’t Based on the steps that you made in the previous activity, make a conceptual map Forget! on the steps in multiplying rational algebraic expressions. Write the procedure or important concepts in every step inside the circle. If necessary, add a new circle.CONCEPT CHANGE MAPActivity 6: The Circle Arrow Process Web – based Step 2 Step 1As the learner sequences the steps, he/she will identify the mathematical Booster: Step 4 Step 3concepts behind each step. Place the mathematical concept inside the circle Final Stepuntil he/she arrived at the final answer. Watch the videos in this web – sites for more examples. http://www. onlinemathlearning. com/multiplying-rational- expressions-help.html ?E S T I O 1. Does every step have a mathematical concept involved? 2. What makes that mathematical concept important to every step? 3. Can the mathematical concepts used in every step be interchanged? How? 4. Can you give another method in multiplying rational algebraic expressions? 98

Activity 7: Dividing Rational Algebraic Expressions Activity 7 Dividing Rational Algebraic Expressions REVIEWThe same as the illustrative examples in multiplying rational algebraicexpressions, each illustrative example in this operation has key ideas, review Examine and analyze the illustrative examples below. Pause once Perform the operation ofquestion to unveil the concept on each step. But before they begin dividing in a while to answer the check – up questions. the following fractions.rational algebraic expressions, they have to review how to divide fractions. 1. 1 ÷ 3 4. 10 ÷ 5 24 16 4 2. 5 ÷ 9 5. 1 ÷ 1 2 4 2 4 3. 9 ÷ 3 24 The quotient of two rational algebraic expressions is the product of the dividend and the reciprocal of the divisor. In symbols, a c a d ad b ÷ d = b • c = bc , bc ≠ 0 Illustrative example 4: Find the quotient of 6ab2 and 9a2b2 . 4cd 8dc2Teacher’s Note and Reminders 6ab2 ÷ 9a2b2 = 6ab2 ÷ 8dc2 Don’t 4cd 8dc2 4cd 9a2b2 Forget! Multiply the dividend by the = (2)(3)ab2 ÷ (23)dc2 reciprocal of the divisor. (2)2cd (32)a2b2 Perform the steps in multiplying = (22)(22)(3)ab2dcc rational algebraic expressions. (22)(3)(3)cdaab2 = (2)2c (3)a = 4c 3a Illustrative example 5: Divide 2x2 + x – 6 by x2 – 2x – 8 . 2x2 + 7x + 5 2x2 – 3x – 20 2x2 + x – 6 ÷ x2 – 2x – 8 ? Why do we need to factor 2x2 + 7x + 5 2x2 – 3x – 20 out the numerators and denominators? = 2x2 + x – 6 • 2x2 – 3x – 20 2x2 + 7x + 5 x2 – 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) ? What happens to the common (2x­ + 5)(x + 1)(x + 2) (x – 4) factors between numerator and denominator? = (2x – 3) (x + 1) = 2x – 3 x+1 99