Mathematics Grade 8 Part 1

Exercises Find the quotient of the following rational algebraic expressions. 1. 81xz3 ÷ 27x2z2 4. x2 + 2x + 1 ÷ x2 – 1 36y 12xy x2 + 4x + 3 x2 + 2x + 1 2. 2aa2 + 2b ÷ 4 5. x – 1 ÷ 1–x + ab a x+ 1 x2 + 2x + 1Answers to Activity 8 3. 16x2 – 9 ÷ 16x2 + 24x + 9 1. 5x − 50 2. 2x3 − 14x2 6 – 5x – 4x2 4x2 + 11x + 6 4 245 Activity 8 MISSING DIMENSIONThis activity may assess the learner’s understanding in dividing rational Find the missing length of the figures.algebraic expression. This may help learner consider the division of rationalalgebraic expressions in different context. 1. The area of the rectangle is x2 – 100 while the length is 2x2 + 20 . Find theLet them enumerate the steps in dividing rational algebraic expressions and 8 20identify the concepts and principle involved in every step height of the rectangle. Teacher’s Note and Reminders 2. The base of the triangle is 21 and the area is x2 . Find the height of the 3x – 21 35 triangle. Don’tForget! QU ?E S T I ONS 1. How did you find the missing dimension of the figures? 2. Enumerate the steps in solving the problems. 100

MAP OF CONCEPTUAL CHANGE Activity 9 CHAIN REACTION WebActivity 9: Chain ReactionAs the learner enumerates the steps in dividing rational algebraic Use the Chain Reaction Chart to sequence your steps in dividing Based Boosterexpression, his/her can identify mathematical concepts in each step. Place rational algebraic expressions. Write the process or mathematical Click on this web sitethe mathematical concept inside the chamber until he/she arrived at the final concepts used in each step in the chamber. Add another chamber, if below to watch videosanswer. This activity may be individual or collaborative work. necessary. in dividing rational Teacher’s Note and Reminders algebraic expressions http://www. onlinemathlearning. com/dividing-rational- expressions-help.html Chamber Chamber Chamber Chamber 1 2 3 4 Don’t __________________ __________________ __________________ __________________ Forget! __________________ __________________ __________________ __________________ _________________ _________________ _________________ _________________ _________________ _________________ _________________ _________________ _________________ _________________ _________________ _________________ __________________ __________________ __________________ __________________ _________________ _________________ _________________ _________________ __________________ __________________ __________________ __________________ QU ?E S T I ONS 1. Does every step have a mathematical concept involved? 2. What makes that mathematical concept important to every step?ANSWER TO REVIEW 3. Can mathematical concept in every step be interchanged? How? 4. Can you make another method in dividing rational algebraicPerform the operation of the following fractions. expressions? How? 1 3 5 9 7 9 3 =1521. 2 + 2 = 2 2. 4 + 4 = 2 3. 5 + 5 10 5 5 5 1 Activity 10 ADDING AND SUBTRACTING SIMILAR 13 13 13 4 4 RATIONAL ALGEBRAIC EXPRESSIONS4. − = 5. − =1Activity 10 Examine and analyze the following illustrative examples on the REVIEWThe illustrative examples in this topic also have ideas and questions to next page. Pause in a while to answer the check–up questions.guide the students in identifying concepts and principle involved in every Perform the operation of thestep. Before discussing and giving more examples in adding and subtracting In adding or subtracting similar rational expressions, add or following fractions.rational algebraic expressions, review them on how to add and subtract subtract the numerators and write it in the numerator of the resultfractions. over the common denominator. In symbols, 1. 1 + 3 4. 10 – 5 22 13 13 a + c = a + c, b ≠ 0 bb b 2. 5 + 9 5. 5 – 1 44 44 3. 9 + 3 55 101

Teacher’s Note and Reminders Illustrative example 6: Add x2 – 2x – 7 and 3x + 1 . x2 – 9 x2 – 9 x2 – 2x – 7 + 3x + 1 = x2 – 2x + 3x – 7 + 1 x2 – 9 x2 – 9 x2 – 9 = x2 + x – 6 Combine like terms in the x2 – 9 numerator. = (x + 3)(x – 2) Factor out the numerator and (x – 3)(x + 3) denominator. = (x – 2) Do we always factor out the (x + 3) numerator and denominator? Why yes or why not? = x – 2 ? x+3 Illustrative example 7: Subtract -10 – 6x – 5x2 from x2 + 5x – 20 . 3x2 + x – 2 3x2 + x – 2 x2 + 5x2 – 20 – -10 – 6x – 5x2 = x2 + 5x2 – 20 – (-10 – 6x – 5x2) 3x2 + x – 2 3x2 + x – 3 3x2 + x – 2 = x2 + 5x – 20 + 10 + 6x + 5x2 ? Why do we need to 3x2 + x – 2 multiply the subtrahend by – 1 in the numerator? = x2 + 5x2 + 5x + 6x – 20 + 10 3x2 + x – 2 = 6x2 + 11x – 10 3x2 + x – 2 Don’t = (3x – 2)(2x + 5) Forget! (3x – 2)(x + 1) = 2x + 5 Factor out the numerator and x+1 denominator. Exercises Perform the indicated operation. Express your answer in simplest form. 1. 6 + 4 4. x2 + 3x + 2 – 3x + 3 a–5 a–5 x2 – 2x + 1 x2 – 2x + 1 2. x2 + 3x – 2 + x2 – 2x + 4 5. x – 2 + x – 2 x2 – 4 x2 – 4 x–1 x–1ANSWER TO EXERCISE 3. 7 – 5 4x – 1 4x – 1Perform the operation of the following fractions.1. 10 2. 2x2 + x + 2 3. 2 1 4. x + 1 5. 2x − 4 a−5 x2 − 4 4x − x − 1 x−1 102

Activity 11 Activity 11 ADDING AND SUBTRACTING DISSIMILARBefore introducing the addition/subtraction of dissimilar rational algebraic RATIONAL ALGEBRAIC EXPRESSIONSexpressions, learners must review how to add/subtract dissimilar fractions. Letthem perform addition/subtraction of fraction and process their answers. Examine and analyze the following illustrative examples below. REVIEWANSWER TO REVIEW Pause in a while to answer the check–up questions. Perform the operation of thePerform the operation of the following fractions. following fractions. 1 4 11 3 2 17 3 1 5 In adding or subtracting dissimilar rational expressions 1. 1 + 4 4. 1 – 3 2 3 = 6 4 3 12 4 8 8 change the rational algebraic expressions into similar rational 23 42 2. 3 + 2 5. 1 – 2 43 691. + 2. + = 3. − = algebraic expressions using the least common denominator or 3. 3 + 1 LCD and proceed as in adding similar fractions. 484. 1 − 3 = - 5 5. 1 − 2 1 4 2 4 6 9 = 18 5 2. illustrative example 8: Find the sum of 18a4b and 27a3b2cIllustrative Example 8 5+ 2 = 5 + 2Each example in this topic has a box below the first step. Emphasize to them 18a4b 27a3b2c (32)(2)a4b (33)a3b2cthe process of finding the LCD between rational algebraic expressions. As muchas possible, link this process to how LCD of fraction is being derived so that Express the denominatorsthey can relate the process easily. If needed, before discussing the addition/ as prime factors. subtraction of rational algebraic expression, give them examples of finding LCD LCD of 5 and 2of rational algebraic expressions. (32)(2)a4b (33)a3b2cGive more examples in adding/subtracting dissimilar rational algebraic (32)(2)a4b and (33)a3b2c Denominators of the rationalexpressions if needed. In this topic, more examples are presented in thelearning guide. The LCD is (33)(2)(a4)(b2)(c) algebraic expressions Teacher’s Note and Reminders Take the factors of the denominators. When the same factor is present in more than one denominator, take the factor with the highest exponent. The product of these factors is the LCD. = 5 • 3bc + 2 • 2a (32)(2)a4b 3bc (33)a3b2c 2a = (5)(3)bc + (22)a (33)(2)a4b2c (33)(2)a4b2c Don’t = 15bc + 4a Forget! 54a4b2c 54a4b2c = 15bc + 4a Find a number equivalent to 1 that 54a4b2c should be multiplied to the rational algebraic expressions so that the denominators are the same with the LCD. 103

Teacher’s Note and Reminders Illustrative example 9: Subtract t + 3 by 8t – 24 . t2 – 6t + 9 t2 – 9 Don’t Forget! t + 3 – 8t – 24 = t + 3 – 8t – 24 t2 – 6t + 9 t2 – 9 (t – 3)2 (t – 3)(t + 3) LCD of t + 3 and 8t – 24 Express the denominators (t – 3t)2 (t – 3)(t + 3) as prime factors. (t – 3)2 and (t – 3)(t + 3) The LCD is (t – 3)2(t + 3) = t + 3 • t + 3 – (8t – 24) • t – 3 ? What property (t – 3)2 t + 3 (t – 3)2(t + 3) t – 3 of equality is illustrated in this step? = (t + 3)(t + 3) – (8t – 24) (t – 3)2(t + 3) (t – 3)2(t + 3) = t2 + 6t + 9 – 8t – 48t + 72 t3 – 9t2 + 27t – 27 t3 – 9t2 + 27t – 27 = t2 + 6t + 9 – (8t2 – 48t + 72) ? What special products t3 – 9t2 + 27t – 27 are illustrated in this = t2 + 6t + 9 – 8t2 + 48t – 72 t3 – 9t2 + 27t – 27 step? = –7t2 + 54t – 63 t3 – 9t2 + 27t – 27 Illustrative example 10: Find the sum of 2x by 3x – 6 . x2 + 4x + 3 x2 + 5x + 6 2x + 3x – 6 = 2x + 3x – 6 x2 + 4x + 3 x2 + 5x + 6 (x + 3)(x + 1) (x + 3)(x + 2) LCD of (x + 2x + 1) and (x 3x – 6 2) ? What special products 3)(x + 3)(x + are illustrated in this step? (x + 3)(x + 1) and (x + 3)(x + 2) The LCD is (x + 3) (x + 1) (x + 2) ? What property of equality was used in this step? = (x + 2x + 1) • (x + 2) + (x (3x − 6) 2) • (x + 1) 3)(x (x + 2) + 3)(x + (x + 1) = (x + (2x)(x + 2) 2) + (x (3x − 6)(x + 1) 1) 3)(x + 1)(x + + 3)(x + 2)(x + = 2x2 + 4x + 3x2 − 3x − 6 x3 + 6x2 + 11x + 6 x3 + 6x2 + 11x + 6 104

ANSWER TO EXERCISE = 2x2 + 3x2 + 4x − 3x − 6Perform the operation of the following fractions. x3 + 6x2 + 11x + 61. 7x + 4 2. 4x2 + 2x + 20 3. x-x2 − 9 = 5x2 + x − 6 x2 + x x3 − 2x2 − 4x + 8 − 9 x3 + 6x2 + 11x + 64. x − 11 5. -x2 + 4 Exercises x3 − 4x2 + x + 6 2x Perform the indicated operation. Express your answer in simplest form.MAP OF CONCEPTUAL CHANGE 1. x +3 1 + 4 4. 3 – 2 –2Activity 12: Flow Chart x x2 –x x2 – 5x + 6Let them enumerate the steps in adding/subtracting rational algebraicexpressions, both similar and dissimilar expressions. Let them organize these 2. x + 8 + 3x − 2 5. x + 2 – x + 2steps by completing the flow chart below. You can validate their work by adding/ x2 − 4x + 4 x2 − 4 x2subtracting rational algebraic expressions using their flow chart. 3. 2x – 3 x2 − 9 x – 3 Activity 12 FLOW CHART Teacher’s Note and Reminders Now that you have learned adding and subtracting rational algebraic expressions. You are now able to fill in the graphic organizer below. Write each step in adding or subtracting rational algebraic expression in the boxes below. If similar rational Adding or subtracting If dissimilar rational algebraic expressions Rational Algebraic algebraic expressions Expressions STEPS STIO 1. Does every step have a STEPS QUE ? NS mathematical concept involved? 2. What makes that mathematical concept important to every step? Don’t 3. Can mathematical concept in Forget! every step be interchanged? How? 4. Can you make another method in adding or subtracting rational algebraic expressions? How? 105

Activity 13 Web – based Activity 13 WHAT IS WRONG WITH ME?This activity may help students to correct their misconceptions. This may also Booster:help you gauge whether the learners learned the concept or not. If necessary, Rewrite the solution of the first box. Write your solution in thegive more examples to strengthen their understanding. The response of the Watch the videos in second box and in the third box, write your explanation on how yourstudents in guided questions may be written in their journal notebook. these web sites for more solution corrects the original one . examples.Points to be emphasize in this activity http://www.For the solution in the first box: The error in this item is the (6 – x) becomes onlinemathlearning.(x – 6). The factor of (6 – x) is -1(x – 6). com/adding-rational- expressions-help.htmlFor the solution in the second box: The wrong concepts here are a – 5 (a) http://www.becomes a2 – 5a and the numerator of subtrahend must be multiplied by -1. onlinemathlearning.com/a – 5 (a) is equal to a – 5a. subtracting-rational- expressions-help.htmlFor the solution in the third box: 3 must not be cancelled out. The concept ofdividing out can be applied to a common factor and not to the common variable Original My Solution Myor number in the numerator and denominator. ExplanationFor the solution in the fourth box: b2 – 4b + 4 must be factored out as (b – 2) 2 x2 − 1 21(b – 2). The concept of factoring is essential in performing operations on rational 36 − x2 − 6x = (6 − x) (6 − x) − x(x + 6)algebraic expressions. = (x − 2 + 6) − 1 6) Teacher’s Note and Reminders 6) (x x(x + = 2 • x − 1 6) • x−6 (x − 6) (x + 6) x x(x + x−6 = 2x − 1(x − 6) 6) x(x − 6) (x + 6) x(x + 6)(x − = 2x − (x − 6) x(x − 6) (x + 6) = 2x − x + 6 6) x(x − 6) (x + = x(x x+6 + 6) − 6) (x = 1 6) x(x − = 1 x2 − 6x 2 − 3 =2• a −3• a−5 a−5 a a−5 a a a−5 Don’t = 2a − 3(a − 5)Forget! a − 5(a) a(a − 5) = a 2a − 3a − 15 − 5(a) a(a − 5) = 2a − 3a − 15 a(a − 5) = -a − 15 a2− 5a 106

Teacher’s Note and Reminders 3x + 9 = 3x + 9 2x − 3 3 − 2x 2x − 3 (-1)(2x − 3) Don’t Forget! = 3x − 9 2x − 3 2x − 3 = 3x − 9 2x − 3 = 3(x − 3) 2x − 3 = x−3 2x 4 + b2 − 4b = b2 − 4b + 4 b−2 b−2 b−2 = (b − 2)(b + 2) b−2 =b+2 QU ?E S T I ONS 1. What did you feel while answering the activity? 2. Did you encounter difficulties in answering the activity? 3. How did you overcome these difficulties?Activity 14. Complex Rational Expressions The previous activities deal with the fundamental operations on rational expressions.Like on the previous topics, each illustrative example has ideas and questions to Let us try these concepts in a different context.guide the learners in determining the concepts and principles in each step. Forthe students to relate the new topic, start the discussion by reviewing simplifying Activity 14 COMPLEX RATIONAL ALGEBRAIC EXPRESSIONScomplex fraction. You can also give more examples to give emphasis on theconcepts and principles involving in this topic. Examine and analyze the following illustrative examples on the next page. Pause in a while to answer the check – up questions.Answer to the Review: Rational algebraic expression is said to be in its simplest REVIEW form when the numerator and denominator are polynomialsPerform the operation of the following fractions. 5−4 with no common factors other than 1. If the numerator or Perform the operation of the 1+4 1−4 23 denominator, or both numerator and denominator of a rational following fractions. 23 23 2 +2 algebraic expression is also a rational algebraic expression, it is1. 1− 2 = 11 2. 3−2 = -10 3. =7 called a complex rational algebraic expression. To simplify the 1. 1 + 4 4. 1 + 5 2 16 complex rational expression, it means to transform it into simple 23 24 – 42 rational expression. You need all the concepts learned previously 1– 2 33 to simplify complex rational expressions. 3 3 43 3 2. 1 – 4 5. 5 + 4 23 93 1+5 = 21 5+4 = 17 3–2 1+ 2 24 8 93 15 43 3 4−2 1+ 24. 5. 3. 5 – 4 23 33 3 2 +2 3 107

Teacher’s Note and Reminders 2−3 Illustrative example 11: Simplify a b . Don’t 5 6 Forget! + b a2 2−3 ab 5+6 Main fraction a line that b a2 separates the main numerator and main denominator. ? Where does the b and a in the main b and ab numerator and the a2 in the main a2 b denominator came from? = 2b − 3a ÷ 5a2 + 6b ? What happens to the main numerator and ab a2b main denominator? = 2b − 3a • a2b ab 5a2 + 6b ? What principle is used in this = (2b − 3a)aab step? (5a2 + 6b)ab Simplify the rational algebraic = (2b − 3a)a expression. (5a2 + 6b) ? What laws of exponents are used = 2ab − 3a2 5a2 + 6b in this step? c −c Illustrative example 12: Simplify c2 − 4 c − 2 . 1+ 1 c+2 c −c c −c c2 − 4 c − 2 = (c − 2)(c + 2) c−2 1+ 1 1+ 1 c+2 c+2 c − c • (c + 2) = (c − 2)(c + 2) (c − 2) (c + 2) 1• c+2 + 1 c + 2 (c + 2) 108

Teacher’s Note and Reminders = c − c(c + 2) (c − 2)(c + 2) (c − 2) (c + 2) c+2 + 1 c + 2 (c + 2) = c − c2 + 2c (c − 2)(c + 2) (c − 2) (c + 2) c+2 + 1 c + 2 (c + 2) c − (c2 + 2c) = (c − 2)(c + 2) c+2+1 c+2 -c2 − 2c + c = (c − 2)(c + 2) c+2+1 c+2 -c2 − c = (c − 2)(c + 2) c+3 c+2 -c2 − c = (c − 2)(c + 2) ÷ c+3 c+2 -c2 − c = (c − 2)(c + 2) • c+2 c+3 (-c2 − c)(c + 2) = (c − 2)(c + 2) (c + 3) Don’t = (c -c2 − c 3) Forget! − 2)(c + = -c2 − c c2 + c − 6 Exercises Simplify the following complex rational expressions. 1 − 1 3. b − 2b 5. 4− 4 1. x y 4. b−1 b−2 y2 1 1 x2 + y2 2b − 3b 2+ 2 yAnswer to Exercises x − yy b−2 b−3 2. x + y− x1. x2y + xy2 2. x2y − 2xy2 − y3 3. 2b − 3 x−y 1 −3 x + x+y a−2 a−1x2 + y2 x3 + 2x2y − xy2 y b 5+2 a−2 a−14. 2 − 2a 5. 2y − 2 7a − 9 y 109

Activity 15: Treasure Hunting Activity 15 TREASURE HUNTINGThis activity may strengthen the understanding of the learner regarding thetopic. Give extra points for correct answer. Find the box that contains treasure by simplifying rational expressions below. FindThe steps: the answer of each expression in the hub. Each answer contains direction. The correct1. Down 4 steps direction will lead you to the treasure. Go hunting now.2. 2 steps to the right3. Up 3 steps START HERELet them enumerate the steps they did in simplifying complex rational algebraicexpressions and identify the principles in each step. Teacher’s Note and Reminders 1. x2 − 4 2. x + x 3. 3 x2 2 3 x2 + 3x +2 x+ 2 1 x y 2 x+2 THE HUB 5x x2 − 2 1 x2 + 2 3 x2 + x − 6 x2 + x 3 x x−1 4 steps to the Up 3 steps 2 steps to the Down 4 steps 3 steps to the right right left Don’t QU ?E S T I ONS Based on the above activity, what are your steps in simplifying complexForget! rational algebraic expressions? 110

Activity 16: Vertical Chevron List Activity 16 VERTICAL CHEVRON LIST Web – basedIn the previous activity, the learner identified the steps in simplifying complex Booster:rational algebraic expressions. Let his/her organize these steps and principles Make a conceptual map in simplifying complex rational expressionusing vertical chevron list. using vertical chevron list. Write the procedure or important concepts in Watch the videos in every step inside the box. If necessary, add another chevron to complete these web sites for more Teacher’s Note and Reminders your conceptual map. examples http://www.wtamu.edu/ STEP 1 academic/anns/mps/math/ mathlab/col_algebra/col_ alg_tut11_complexrat.htm http://www.youtube.com/ watch?v=-jli9PP_4HA http://spot.pcc. edu/~kkling/Mth_95/ SectionIII_Rational_ Expressions_Equations_ and_Functions/Module4/ Module4_Complex_ Rational_Expressions.pdf STEP 2 STEP 3 Don’t STEP 4Forget! Activity 17 REACTION GUIDEActivity 17: Reaction Guide Revisit the second activity. There are sets of rational algebraic expressions in theIn activity 2, students were given anticipation guide. They will answer the same following table. Check agree if column I is the same as column II and check disagree if theitems in the anticipation guide, but this time they are expected to answer each two columns are not the same.item correctly. Let them compare their answer in the anticipation and reactionguide. Their answer on the questions may be written in the journal notebook. I II Agree DisagreeThis activity will enable the students to correct their initial understanding beforethe lesson was presented. Let them compare their response in the anticipation x2 − xy • x + y x-1 − y-1guide and their response in this activity. x2 − y2 x2 − xy 6y − 30 ÷ 3y − 15 2y y2 + 2y + 1 y2 + y y+ 1 5 +7 15 + 14x 4x2 6x 12x2 111

Teacher’s Note and Reminders a–b a+b b−a a−b b−a a+b −b a2 b a+b a+b 1+2 ba QU ?E S T I ONS Compare your answer from the anticipation guide to the reaction guide. Do they differ from each other? Why it so? In this section, the discussion was all about operations on rational algebraic expressions. How much of your initial ideas are found in the discussion? Which ideas are different and need revision? The skills in performing the operations on rational algebraic expressions is one of the key concepts in solving rate – related problems. Don’t WWhhaatt ttoo UUnnddeerrssttaannddForget! Your goal in this section is to relate the operations of rational expressions to a real – life problems, especially the rate problems. Activity 18 WORD PROBLEM Read the problems below and answer the questions that follow. 1. Two vehicles travelled (x + 4) kilometers. The first vehicle travelled for (x2 – 16)Activity 18: WORD PROBLEM hours while the second travelled for 2 hours.In this part, learner will be exposed more to how rational algebraic expressionscan modelled the rate–related problems. You can discuss and give more x–4examples similar to the items in this activity so that the students are guidedon how the concepts of rational algebraic expressions modelled rate–related a. Complete the table below.problems. Let them answer the activity individually or in collaborate work. Letthem also enumerate the steps in solving these problems. Vehicles Distance Time Speed Vehicle A Vehicle B 112

Activity 19: Accent Process b. How did you compute the speed of the two vehicles?Let the students enumerate the steps that they do in the previous activity. In this c. Which of the two vehicles travelled faster? How did you find your answer?activity, let them organize these steps using accent process chart. 2. Jem Boy and Roger were asked to fill the tank with water. Jem Boy can fill the Teacher’s Note and Reminders tank in x minutes alone while Roger is slower by 2 minutes compared to Jem Boy if working alone. a. What part of the job can Jem Boy finish in 1 minute? b. What part of the job can Roger finish in 1 minute? c. Jem Boy and Roger can finish filling the tank together within certain number of minutes. How will you represent algebraically, in simplest form, the job done by the two if they worked together? Activity 19 ACCENT PROCESS List down the concepts and principles in solving problems involving operations of rational algebraic expressions in every step. You can add a box in necessary. Step 1 Step 2 Step 3 ______________________ _____________________ _____________________ ______________________ _____________________ _____________________ ______________________ _____________________ _____________________ ______________________ ______________________ ______________________ ______________________ ______ ______ Don’t Activity 20 PRESENTATION Forget! Present and discuss to the class the process of answering the questions below. YourActivity 20: Presentation output will be graded according to reasoning, accuracy, and presentation.In preparation for the performance task in this module, let the learner perform Alex can pour a concrete walkway in x hours alone while Andy can pour the samethis activity. The learner is expected to present his/her output appropriately. walkway in two more hours than Alex. a. How fast can they pour together the walkway? b. If Emman can pour the same walkway in one more hours than Alex and Roger can pour the same walkway in one hour less than Andy, who must work together to finish the job with the least time? 113

Teacher’s Note and Reminders Rubrics for your output Don’t CRITERIA Outstanding Satisfactory Developing Beginning Forget! 4 3 2 1 Mathematical Explanation Explanation Explanation Explanation reasoning shows thorough shows shows gaps in shows illogical reasoning substantial reasoning. reasoning. and insightful reasoning justifications. Accuracy All All Most of the Some the computations computations computations computations are correct and are corrects. are correct. are correct. shown in detail. The presentation The The The uses presentation presentation presentation appropriate and uses uses some does not use Presentation creative visual appropriate visual materials. any visual materials. It is visual materials. It is delivered in materials. It is delivered in a It is delivered in a disorganized delivered in a very convincing a clear manner. manner. clear manner. manner. In this section, the discussion was about application of operations on rational algebraic expressions. It gives you a general picture of relation between the operations of rational algebraic expressions and 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? Copy the Learned, Affirmed, Challenged cards in your journal notebook and complete it. Affirmed Challenge LearnedBefore moving to the transfer part, let the learner fill in the LEARNED, What new connections What questions doAFFIRMED and CHALLENGED box. This activity will solicit ideas on what andhow the learner learned this lesson. Try to clear his/her thought by addressing What new realizations have you made? you still have? Whichthe questions regarding in this lesson. and learning do you Which of your old areas seem difficult have about the topic? ideas have been for you? Which do you confirmed/affirmed want to explore 114

WWhhaatt ttooTTrraannssffeerr WWhhaatt ttooTTrraannssffeerr This is the performance task in this module. Encourage the learner to interview skilled Your goal in this section is to apply your learning in real life situations. You will workers regarding their rate of work and the wage per worker. Encourage the learner be given a practical task which will demonstrate your understanding. to be resourceful in dealing with this performance task. They must present not only the manpower plan but also the process on how they transfer their understanding of ?E S T I O rational algebraic expressions to this performance task. Also, after the performance task, ask the learner what difficulties they encountered and how they manage these difficulties. QU NS A newly-wed couple plans to construct a house. The couple has already a house plan from their friend engineer. The plan of the house is illustrated CRITERIA Outstanding Satisfactory Developing Beginning below: Reasoning 4 3 2 1 Accuracy Explanation Explanation Explanation Explanation Laboratory 1mPresentation shows shows shows gaps in shows illogical thorough substantial reasoning. reasoning. 2 m Bedroom Dining Room Practically reasoning reasoning. Efficiency and insightful 2m justifications. 1.5 m Comfort Living Room 3 m All computa- All computa- Most of the Some of the Room tions are cor- tions are computations computations Master rect and shown correct. are correct. are correct. in detail 2.5 m Bedroom 3m 3m the The The The As a foreman of the project, you are task to prepare a manpower plan presentation presentation presentation presentation to be presented to the couple. Inside the plan is how many workers are uses uses uses some does not use needed to complete the project, daily wage of the workers, how many days appropriate appropriate visual any visual can they finish the project and how much can be spend for the entire job. and creative visual materials. It materials. It is The man power plan will be based on reasoning, accuracy, presentation, visual materials. It is is delivered in delivered in a practicality and efficiency. materials. it is delivered in a a disorganize clear manner. delivered in a clear manner. manner. very convincing manner The proposed The proposed The proposed The proposed plan will be plan will be project will be plan will be completed at completed in completed with completed the least time. lesses time. greater number with the most of days. number of days. The cost of the The cost of The cost of The cost of the plan is minimal. the plan is the plan is plan is very expensive. expensive. reasonable. 115

POST - TEST1. Which of the following algebraic expressions could not be considered as rational algebraic expression? ab −+ b a. √50x b. 5x1/2 c. 4y2 – 9z2 d. a Answer: B. The exponent in the expression in B is a fraction. Rational algebraic expression has no fractional exponent.2. What is the rational algebraic expression equivalent to (48kk-p2p3-)30? a. 4k2p3 b. 2k2p3 c. k2p3 d. k5p6 4 4 Answer: C. The numerator is raised to 0 which means 1. The k and p are raised to a negative which means the multiplicative inverse of the expression.3. What is the value of the expression x-3yc8 when x is 2, y is 3 and c is -2? x2y-2c7 a. - 2167 b. 2176 c. 2327 d. -2327 x-3yc8 y3c (27)(-2) 27 Answer: A. x2y-2c7 = x5 = 32 = - 164. The area of the rectangle is x2 – 3x – 10. What is the length of the rectangle if the width is x + 2? a. (x + 5)(x – 2) b. (x + 5)(x – 2) c. x+5 d. x – 5 x–2 x–2 Answer: D. l = x2 – 3x – 10 = (x – 5)(x + 2) = x – 5 x+2 x+25. What must be added to 3x + 4 so that there sum is 3x2 + x – 4 ? x+2 x2 – 4 a. 3xx2 + 4 b. -x32x––44 c. xx2+–142 d. xx2––142 – 4 116

Answer: A. 3x2 + x – 4 – 3x + 4 = 3x2 + x – 4 – (x – 2)(3x + 4) = 3x2 + x – 4 – 3x2 + 2x + 8 = 3x + 4 x2 – 4 x+2 x2 – 4 (x + 2)(x – 2) x2 – 4 x2 – 46. If one of the factors of a 1 1 is a a–1 1, find the other factor. + – 2a + a. aa +– 1 b. a– 1 c. a2 – 2a + 1 d. a2 a2 – 1 1 1– a a2 – 1 – 2a + 1 Answer: A. a2 a –1 1 ÷ a 1 1 = a–1 • a+ 1 = a+1 – 2a + + (a – 1)(a – 1) 1 a–1 17. Which of the following rational algebraic expressions is equivalent to x2 + 5x + 6? 1 x+3 a. x 1 2 b. x 1 2 c. x 1 3 d. x 1 3 + – + – 11 1 x + 3 1 (x + 3)(x + 2) 1 + Answer: A. x2 + 5x + 6 = (x + 3)(x + 2) = • = 1 1 x 2 x+3 x+38. What is the difference between m and m ? 6 3 a. m6 b. - m c. m d. - m 2 2 2 Answer: A . m – m = 3m – 2m = m 2 3 6 69. A business man invested his money and was assured that his money will increase using the formula P(1 r )nt where P is money n invested; r is the rate of increase; n is mode of increase in a year and t is the number of years. If the business man invested Php 10 000, how much can he get at the end of the year if the rate is 50% and will increase twice a year? a. Php 15 652 b. Php 16 552 c. Php 15 625 d. Php 15 255 Answer: C. P(1 + r nt = 10000 (1 + 02.5)(2)(1) = 10000 (1 + 1 (2) = 10000 (1 + 5 (2) = 10000 ( 25 ) = 15 625 n 4 4 16 ) ) ) 117

10. Roger can do the project in x number hours. Concepcion can do the same job in 2 hours less than Roger does. Which of the choices below is the difference of their rate? a. x22x−−22x b. - x2 2 2x c. x2 2 2x d. - 2x − 2 − − x2 − 2x Answer: B. 1 − 1 = x−2−x =- x2 2 x x−2 x2 − 2x − 2x11. You have (x2 + 2) pesos to buy materials for your school project. You spent half of it in the first store, then you spent one – third of your money less than you spent in the first store. In the third store, you spent one – fourth of the remaining money from the two stores. What is the total cost of the materials? a. 4x24− 8 b. 3x2 + 6 c. 5x21+220 d. 7x2 + 14 4 12 Answer: B. x2 + 2 + x2 − 2 − x2 + 2 + x2 + 2 = 9x2 + 18 = 3x2 + 6 2 2 3 12 12 412. James were asked to simplify x2 + 2x − 8. His solution is presented below. x2 − 4 What makes the solution of James wrong? a. Cancelling 4. b. Crossing out the (x – 2). c. x2 – 4 being factored out. d. Dividing out the variable x. Answer: D. X in the (x + 4) and (x + 2) should not divided out because it is part of the term and it is not a common factor of the numerator and denominator. 118

13. Mary took the math exam. One of the problems in the exam is finding the quotient of x2 + 2x + 2 and 1 − x2 2 . Her solution is shown below. 4 − x2 x2 + x − Did Mary arrive at the correct answer? a. No, the dividend and divisor should be interchange. b. No, the divisor should be reciprocated first before factoring it out. c. No. (2 + x) is not the same as (x + 2). d. No. (x – 1) and (1 – x) is not equal to 1 Answer: D. (x – 1) is additive inverse of (1 – x). If the a term is divided by the its additive inverse, quotient is -1 y 2 1 + 3 +14. Greg simplify this way: 3 + 4 y + 1 Is there anything wrong in his solution? a. Something is wrong with the solution. He is not following the correct process of simplifying complex rational algebraic expression. b. None. Multiplying the numerator and denominator by the same quantity makes no difference on the given expression. c. Something is wrong with the solution. Numerator and denominator may be multiplied by a certain number but not an algebraic expression. d. None. The solution and answer of Greg is different but acceptable. Answer: B. In simplifying complex rational algebraic expression, numerator and denominator can be multiplied by their LCD 119

15. Your Project Supervisor ask you to make a floor plan of a house. As an engineer, what must be considered in completing the plan? a. Reasoning and accuracy b. Cost and design c. Feasible and accurate d. Practical and aesthetics Answer: C. Dividing the parts of the house must be accurate and it must be realistic.16. Your mother asked you to find for laborers in renovating your house. What will you look in choosing a laborer? a. His efficiency in doing the task. b. His attitude towards work. c. His perception in the job. d. His wage in a day. Answer: A. Though the rate/speed of the laborer counts but the quality of his work must not be compromised.17. You need a printer in your computer shop. The list of the printers and its capacities is presented in the table. Based in the table, what printer is best to buy?Printer Pages to print in a minute Capacity of the ink Average number of wasted paper per 500 pagesHD Turbo 16 450 pages 4IP Sun 7 500 pages 2Bazoka 23 350 pages 12Father’s 18 400 pages 6 a. Father’s. it has more pages to print and good capacity of ink. b. Bazoka. It has the most pages to print and nice capacity of ink. 120

c. IP Sun. It has the best ink capacity and least number of paper wasted. d. HD Turbo. It has lesser wasted paper and better ink capacity. Answer: D. Though the HD Turbo is slower compared to Father’s and Bazoka but it has the lesser wasted paper compared the other two printers. And the capacity of the ink is better compared to the other two printers.18. What qualities you must look in buying a printer for personal consumption? a. Brand and design b. Price and pages to print c. Cost of the printer and its efficiency. d . Brand and the quality of the output. Answer: C. It is better to consider the cost of the printer that will not compromise its efficiency.19. You were tasked, as a budget officer, to give comments regarding the work plan of the engineer. What aspect of the plan should you consider? a. The wage of the laborers and the rentals of the equipment. b. The number of laborers and equipment needed. c. The quality of work done by the laborers and efficiency of the equipment. d. The job done by the laborers in one day and appropriateness of the equipment. Answer: A. It is not necessary to look for the rate/speed and efficiency of the laborers as a budget officer because you will look for the financial aspect of the project.20. After you give comments in the work plan in number 19, what will you do next? A. Present a feasibility study. c. Look for financial resources B. Make a budget proposal. d. Give a financial statement Answer: B. A will be given by the engineer. C will be given after the budget plan. D will be given after the project. 121

SUMMARY Now that you have completed this module, let us summarize what have you learned: 1. Rate–related problems can be modeled using rational algebraic expressions. 2. Rational algebraic expression is a ratio of two polynomials where the denominator is not equal to one. 3. Any expression raised to zero is always equal to one. 4. When an expression is raised by a negative integer, it is the multiplicative inverse of the expression. 5. Rational algebraic expression is in its simplest form if there is no common factor between numerator and denominator except 1. 6. To multiply rational algebraic expression, multiply the numerator and denominator then simplify. 7. To divide rational algebraic expression, multiply the dividend by the reciprocal of the divisor then multiply. 8. To add/subtract similar rational algebraic expressions, add/subtract the numerators and copy the common denominator. 9. To add/subtract dissimilar rational algebraic expressions, express each expression into similar one then add/subtract the numerators and copy the common denominator. 10. Complex rational algebraic expression is an expression where the numerator or denominator, or both numerator and denominator are rational algebraic expressions.GLOSSARY OF TERMS USED IN THIS MODULE Complex rational algebraic expression – an expression where the numerator or denominator or both numerator and denominator are rational algebraic expressions. LCD – also known as Least Common Denominator is the least common multiple of the denominators. 122

Manpower plan – a plan where the number of workers needed to complete the project, wages of each worker in a day, how many days can workers finish the job and how much can be spend on the workers for the entire project. Rate–related problems – Problems involving rates (e.g., speed, percentage, ratio, work) Rational algebraic expression – ratio of two polynomials where the denominator is not equal to one.REFERENCES AND WEBSITE LINKS USED IN THIS MODULE: Learning Package no. 8, 9, 10, 11, 12, 13. Mathematics Teacher’s Guide, Funds for assistance to private education, 2007 Malaborbor, P., Sabangan, L., Carreon, E., Lorenzo, J., Intermediate algebra. Educational Resources Corporation, Cubao, Quezon City, Philippines, 2005 Orines, F., Diaz, Z., Mojica, M., Next century mathematics intermediate algebra, Pheoenix Publishing House, Quezon Ave., Quezon City 2007 Oronce, O., Mendoza, M., e – math intermediate algebra, Rex Book Store, Manila, Philippines, 2010 Padua, A. L, Crisostomo, R. M., Painless math, intermediate algebra. Anvil Publishing Inc. Pasig City Philippines, 2008 Worktext in Intermediate Algebra. United Eferza Academic Publication Co. Lipa City, Batangas, Philippines. 2011 123

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut11_complexrat.htmhttp://www.youtube.com/watch?v=-jli9PP_4HAhttp://www.onlinemathlearning.com/adding-rational-expressions-help.htmlhttp://www.onlinemathlearning.com/subtracting-rational-expressions-help.htmlhttp://www.onlinemathlearning.com/dividing-rational-expressions-help.htmlhttp://www.onlinemathlearning.com/multiplying-rational-expressions-help.htmlhttp://spot.pcc.edu/~kkling/Mth_95/SectionIII_Rational_Expressions_Equations_and_Functions/Module4/Module4_ Complex_Rational_Expressions.pdfImage creditshttp://www.portlandground.com/archives/2004/05/volunteers_buil_1.php 124

TEACHING GUIDEModule 3: RELATIONS AND FUNCTIONSA. Learning Outcomes Content Standard: The learner demonstrates understanding of key concepts of linear functions. Performance Standard: The learner is able to formulate real-life problems involving linear functions and solve these with utmost accuracy using variety of strategies. UNPACKING THE STANDARDS FOR UNDERSTANDINGSUBJECT: LEARNING COMPETENCIESGrade 8 Mathematics 1. describe and illustrate the Rectangular Coordinate System and its uses;QUARTER: 2. describe and plot positions on the coordinate plane using the coordinate axes;Second Quarter 3. define relation and function;STRAND: 4. illustrate a relation and a function;Algebra 5. determine if a given relation is a function using ordered pairs, graphs andTOPIC: equations; differentiate between dependent and independent variables;Relations and Functions 6. describe the domain and range of a function. define linear function;LESSONS: 7. describe a linear function using its points, equation and graph;1. Rectangular Coordinate System 8. 2. Representations of Relations and Functions 9. 3. Linear Function and Its Application 125

10. identify the domain and range of a linear function; 11. illustrate the meaning of the slope of a line; 12. find the slope of a line given two points, equation and graph; 13. determine whether a function is linear given the table; 14. write the linear equation Ax + By = C in the form y = mx + b and vice-versa; 15. graph a linear equation given (a) any two points, (b) the x-intercept and y-intercept, (c) the slope and a point on the line, (d) the slope and y-intercept; 16. describe the graph of a linear equation in terms of its intercepts and slope; 17. find the equation of a line given (a) two points; (b) the slope and a point; (c) the slope and its intercept; and 18. solve real-life problems involving linear functions and patterns. ESSENTIAL UNDERSTANDING: ESSENTIAL QUESTION: Students will understand that problems How can the value of a quantity given involving constant rate of change can the rate of change be predicted? be solved using linear function. TRANSFER GOAL: Students will on their own formulate and make representations of quantitative relationships in real-life situations and use these to solve problems.B. Planning for AssessmentProduct/Performance The following are products and performances that students are expected to come up with in this module:a. CoordinArt and Constellation Art making where Rectangular Coordinate System is applied by locating significant points in xy-plane;b. A gallery walk of informative and creative leaflets whose contents are representations of relations and functions; andc. A creative leaflet illustrating that electricity bill is a function of its power consumption. 126

Assessment Matrix KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCE TYPE Pre-test Written Exercises IRF Worksheet Pre–assessment/ Oral Questioning Explanation Self-Knowledge Diagnostic Explanation Application Self-Knowledge Self-Knowledge IRF Worksheet Gallery Walk Written Exercises Quiz Self-Knowledge (Making Informative Explanation Explanation Leaflets) Application Perspective Spotting Erroneous Application Self-Knowledge Self-Knowledge Coordinates Explanation Explanation Self-Knowledge Quiz Graph Analysis Perspective Explanation Explanation Self-Knowledge Making a Relation Perspective Interpretation Empathy Application Self-Knowledge Perspective Quiz Explanation Explanation PerspectiveFormative Flash Card Drill Perspective Interpretation Self-Knowledge Finding Steepness Self-Knowledge of an Inclined Object Graph Analysis (Steep Up!) Explanation Explanation Interpretation Perspective Perspective Self-Knowledge 127

Story Telling Interpretation Application Self-Knowledge Perspective Unit Test Unit Test Unit Test CoordinArt Making Explanation Explanation Explanation Explanation Interpretation Interpretation Interpretation Application Application Application Application Interpretation Self-Knowledge Self-Knowledge Self-Knowledge PerspectiveSummative Writing the Steps Constellation Art of Graphing Linear Making (Optional) Equations Explanation Self-Knowledge Application Interpretation Explanation Perspective Periodical Periodical Periodical Periodical Examination Examination Examination Examination Explanation Explanation Explanation Explanation Application Application Application Application Self-Knowledge Self-Knowledge Self-Knowledge Self-Knowledge Perspective Perspective Perspective Perspective Interpretation Interpretation Interpretation InterpretationSelf-assessment IRF Worksheet Self-KnowledgeLegend:Six Facets of Understanding: Explanation, Interpretation, Application, Perspective, Empathy, Self-Knowledge 128

Assessment Matrix (Summative Test)Levels of Assessment What will I assess? How will I assess? How Will I Score? 1 point for every correct response • describe and illustrate the Rectangular Coordinate Paper and Pen Test (Refer to 1 point for every correct response Knowledge System and its uses; attached post-assessment) 15% 1 point for every correct response • describe and plot points on the coordinate plane 129Process/Skills 25% using the coordinate axes; Items 1, 2 and 3 • define relation and function; Paper and Pen Test (Refer to • illustrate relation and function; attached post-assessment) • determine if a given relation is a function using ordered pairs, graphs and equations; • differentiate between dependent and independent Items 4, 5, 6, 7 and 8 variables; • describe the domain and range of a function; • define linear function; • describe a linear function using its points, equation and graph; • identify the domain and range of a linear function; • illustrate the meaning of the slope of a line; • find the slope of a line given two points, equation and graph; • determine whether a function is linear given the table; • write the linear equation Ax + By = C in the form y = mx + b and vice-versa; • graph a linear equation given (a) any two points, (b) the x-intercept and y-intercept, (c) the slope and a point on the line, and (d) the slope and y-intercept; • describe the graph of a linear equation in terms of its intercepts and slope; • find the equation of a line given (a) two points; (b) the slope and a point; (c) the slope and its intercept; and • solve real-life problems involving linear functions and patterns.Understanding • Students will understand that problems involving Paper and Pen Test (Refer to 30% constant rate of change can be solved using linear attached post-assessment) function; and • The value of y increases as the value of x increases. Items 9, 10, 11, 12, 13 and 14 (Misconception)

GRASPS Paper and Pen Test (Refer to 1 point for every correct response Students will, on their own, formulate and make attached post-assessment) representations of quantitative relationships in real Rubric on Problem Posing / life situations and use these to solve problems. Items 15, 16, 17, 18, 19 and 20 Formulation and Problem Solving Criteria: The learner is able to formulate real-life problems Your student is a barangay councilor involving linear functions and solve these with in San Sebastian. Every month he Relevant utmost accuracy using a variety of strategies. conducts information drive on the Creative different issues that concern every InsightfulProduct 30% member in the community through Authentic the use of leaflets. For the next Clear month, his focus is on electricity consumption of every household. Rubric on CoordinArt Making He is tasked to prepare a leaflet Criteria: design which will clearly explain about electricity bill and consumption. Creative Include recommendations that will Accurate help lessen electricity utilization. He is Authentic expected to orally present your design Neatness to the other officials in your barangay. He will be assessed according to Rubric on Leaflet Design the following criteria: (1) use of Criteria: appropriate mathematical concepts and accuracy, (2) organization, Use of mathematical concepts and (3) quality of presentation, and (4) accuracy practicality of recommendations. Organization Quality of presentation Practicality of recommendations 130

C. Planning for Teaching-Learning Introduction: This module covers the key concepts of linear functions. It is divided into three lessons, namely: Rectangular Coordinate System, Representations of Relations and Functions and Linear Function and its Applications. In Lesson 1, the students will plot points on the xy-plane. The students will also describe and illustrate the Rectangular Coordinate System and its uses. In Lesson 2, the students will illustrate the difference of relations and functions, and of independent and dependent variables, then give the domain and the range of a function. In Lesson 3, the students will describe a linear function using its points, equation and graph, illustrate the meaning of slope, find the slope, write the linear equation in any form, draw the graph of the linear equation, and solve real-life problems involving linear functions and patterns. In most lessons, students are encouraged to visit the links provided in the module. They are also encouraged to use software such as GeoGebra to graph the linear equation easily. They are also allowed to use any graphing materials, sharp edge and other tools. They are provided with varied activities to process the knowledge and skills acquired, deepen their understanding and transfer it to new context. As an introduction to the main lesson, ask the students the following questions: Have you ever asked yourself how the steepness of the mountain affects the speed of a mountaineer? How does the family’s power consumption affect the amount of the electric bill? How is a dog’s weight affected by its food consumption? How is the revenue of the company related to number of items produced and sold? How is the grade of a student affected by the number of hours spent in studying? Hook the students to find out the answers to these questions leading to the essential question: “How can the value of a quantity given the rate of change be predicted?” 131

Objectives: After the learners have gone through the lessons contained in this module, they are expected to: 1. describe and illustrate the Rectangular Coordinate System and its uses; 2. describe and plot positions on the coordinate plane using the coordinate axes; 3. define relation and function; 4. illustrate a relation and a function; 5. determine if a given relation is a function using ordered pairs, graphs and equations; 6. differentiate between dependent and independent variables; 7. describe the domain and range of a function. 8. define linear function; 9. describe a linear function using its points, equation and graph; 10. identify the domain and range of a linear function; 11. illustrate the meaning of the slope of a line; 12. find the slope of a line given two points, equation and graph; 13. determine whether a function is linear given the table; 14. write the linear equation Ax + By = C in the form y = mx + b and vice-versa; 15. graph a linear equation given (a) any two points, (b) the x-intercept and y-intercept, (c) the slope and a point on the line, (d) the slope and y-intercept; 16. describe the graph of a linear equation in terms of its intercepts and slope; 17. find the equation of a line given (a) two points; (b) the slope and a point; (c) the slope and its intercept; and 18. solve real-life problems involving linear functions and patterns. 132

Pre–testDirection: Read the questions carefully. Write the letter that corresponds to your answer on a separate sheet of paper.1. What is the Rectangular Coordinate System? a. It is used for naming points in a plane. b. It is a plane used for graphing linear functions. c. It is used to determine the location of a point by using a single number. d. It is a two-dimensional plane which is divided by the axes into four regions called quadrants. Answer: D y2. Which of the following is true about the points in Figure 1? JH C a. J is located in Quadrant III. D b. C is located in Quadant II. F c. B is located in Quadrant IV. x d. G is located in Quadrant III. Answer: D3. Which of the following sets of ordered pairs does NOT define a function? G a. {(3, 2), (-3, 6), (3, -2), (-3, -6)} b. {(1, 2), (2, 6), (3, -2), (4, -6)} B c. {(2, 2), (2, 3), (2, 4), (2, -9)} d. {(4, 4), (-3, 4), (4, -4), (-3, -4)} Figure 1 Answer: B4. What is the domain of the relation shown in Figure 2? a. {x|x ∈ ℜ} c. {x|x > -2} b. {x|x ≥ 0} d. {x|x ≥ -2} Answer: D Figure 2 133

5. Determine the slope of the line 3x + y = 7. a. 3 c. 13 - 31 b. -3 d. Answer: B6. Rewrite 2x + 5y = 10 in the slope-intercept form. a. y = 2 x + 2 c. y= 2 x + 10 5 5 2 2 b. y = 5 x + 2 d. y= 5 x + 10 Answer: A7. Find the equation of the line with the slope -2 and passing through (5, 4). a. y = 2x + 1 c. y = 2x + 14 b. y = -2x + 1 d. y = -2x + 14 Answer: D8. Which line passes through the points (3, 4) and (8, -1)? a. y = -x + 7 c. y = x + 7 b. y = -x − 1 d. y = x − 1 Answer: A9. Jonathan has a job mowing lawns in his neighborhood. He works up to 10 hours per week and gets paid Php 25 per hour. Identify the independent variable. a. the job c. the lawn mowing b. the total pay d. the number of hours worked Answer: D 134

10. Some ordered pairs for a linear function of x are given in the table below. x13579 y -1 5 11 17 23 Which of the following equations was used to generate the table above? y = -3x + 4 a. y = 3x – 4 b. y = 3x + 4 c. y = -3x – 4 d. Answer: A11. As x increases in the equation 5x + y = 7, the value of y Figure 3 a. increases. b. decreases. c. does not change. d. cannot be determined. Answer: B12. What is the slope of the hill illustrated in Figure 3? (Hint: Convert 5 km to m.) a. 4 c. 14 y 1215 l b. 125 d. Answer: D x m13. Which line in Figure 4 is the steepest? d. line p a. line l b. line m c. line n p n Figure 4 Answer: C14. Joshua resides in a certain city, but he starts a new job in the neighboring city. Every Monday, he drives his new car 90 kilometers from his residence to the office and spends the week in a company apartment. He drives back home every Friday. After 4 weeks of this routinary activity, his car’s odometer shows that he has travelled 870 kilometers since he bought the car. Write a linear model which gives the distance y covered by the car as a function of x number of weeks since he used the car. a. y = 180x + 150 b. y = 90x + 510 c. y = 180x + 510 d. y = 90x + 150 Answer: A 135

For item numbers 15 to 17, refer to the situation below. A survey of out-of-school youth in your barangay was conducted. From year 2008 to 2012, the number of out-of-school youths was tallied and was observed to increase at a constant rate as shown in the table below. Year 2008 2009 2010 2011 2012 Number of 30 37 44 51 58out-of-school youth, y15. If the number of years after 2008 is represented by x, what mathematical model can you make to represent the data above? a. y = -7x + 30 b. y = -7x + 23 c. y = 7x + 30 d. y = 7x + 23 Answer: C16. If the pattern continues, can you predict the number of out-of-school youths by year 2020? a. Yes, the number of out-of-school youths by year 2020 is 107. b. Yes, the number of out-of-school youths by year 2020 is 114. c. No, because it is not stipulated in the problem. d. No, because the data is insufficient. Answer: B17. The number of out-of-school youths has continued to increase. If you are the SK Chairman, what would be the best action to minimize the growing number of out-of-school youths? a. Conduct a job fair. c. Let them work in your barangay. b. Create a sports project. d. Encourage them to enrol in Alternative Learning System. Answer: D 136

18. You are a Math teacher. You gave a task to each group of students to make a mathematical model, a table of values, and a graph about the situation below. A boy rents a bicycle in the park. He has to pay a fixed amount of Php 10 and an additional cost of Php 15 per hour or a fraction of an hour, thereafter. What criteria will you consider so that your students can attain a good output? I. Accuracy II. Intervals in the Axes III. Completeness of the Label IV. Appropriateness of the Mathematical Model a. I and II only c. II, III and IV only b. I, II and III only d. I, II, III and IV Answer: D19. If y refers to the cost and x refers to the number of hours, what is the correct mathematical model of the situation given in item 18? a. y = 15x + 10 b. y = 10x + 15 c. y = 15x – 10 d. y = 10x – 15 Answer: A20. You are one of the trainers of a certain TV program on weight loss. You notice that when the trainees run, the number of calories c burned is a function of time t in minutes as indicated below: t 12345 c(t) 13 26 39 52 65 As a trainer, what best piece of advice could you give to the trainees to maximize weight loss? a. Spend more time for running and eat as much as you can. b. Spend more time for running and eat nutritious foods. c. Spend less time for running. d. Sleep very late at night. Answer: A 137

WWhhaatt ttoo KKnnooww 1Lesson Rectangular Coordinate System Provide students the opportunity to recall the binary operations and the cardinality of sets as well as the number line. Ask them to perform WWhhaatt ttoo KKnnooww Activities 1 and 2. In Activity 1, you may use colorful chips, cartolinas, or any tangible objects to represent colors of each set. This is an opening Let’s start this module by reviewing the important lessons on “Sets.” As you go activity so allow them to be motivated for them to be eager to learn more through this part, keep on thinking about this question: How can the Rectangular in this module. You may modify the activity based on the availability Coordinate System be used in real life? of materials. As you go through with the activities, let them realize the importance of Rectangular Coordinate System in real life. Pose the topical Activity 1 RECALLING SETS Essential Question: How can the Rectangular Coordinate System be used in real life? Description: This activity will help you recall the concept of sets and the basic Direction: operations on sets. Answers Key Let A = {red, blue, orange}, B = {red, violet, white} and C = {black, blue}. Find the following.Activity 11. A ∪ B = {red, blue, orange, violet, white} 1. A ∪ B 4. n(A ∪ B) 7. A ∩ B ∩ C2. A ∩ B = {red} 2. A ∩ B 5. n(A ∩ B) 8. A ∩(B ∪ C)3. A ∪ A ∪ C = {red, blue, orange, violet, white, black} 3. A ∪ B ∪ C 6. A ∩ C 9. n(A ∩ (B ∪ C))4. n(A ∪ A) = 5 QU ?E S T I ONS Have you encountered difficulty in this lesson? If yes, what is it?5. n(A ∩ B) = 1 Activity 2 BOWOWOW!6. A ∩ C = {blue} Description: This activity is in the form of a game which will help you recall the concept Direction: of number line.7. A ∩ B ∩ C = { } Do as directed the given tasks. 1. Group yourselves into 9 or 11 members.8. A ∩ (B ∪ C) = {red, blue} 2. Form a line facing your classmates. 3. Assign integers which are arranged from least to greatest to each9. n(A∩(B∪C)) = 2 group member from left to right. Teacher’s Note and Reminders 4. Assign zero to the group member at the middle. 5. Recite the number assigned to you. Don’t Forget! 138

Teacher’s Note and Reminders 6. Bow as you recite and when the last member is done reciting, all of you bow together and say Bowowow! QU ?E S T I ONS 1. What is the number line composed of? 2. Where is zero found on the number line? 3. What integers can be seen in the left side of zero? What about on the right side of zero? 4. Can you draw a number line? Activity 3 IRF WORKSHEET Don’t Description: Below is the IRF Worksheet in which you will give your present Forget! Direction: knowledge about the concept. Give your initial answers of the questions provided in the first column and Elicit students’ present knowledge of Rectangular Coordinate System by write them in the second column. answering the “Initial Answer” column in the IRF Worksheet. Questions Initial Answer Revised FinalWWhhaatt ttoo PPrroocceessss Answer Answer Provide students enabling activities/experiences that they will have to go through 1. What is a rectangular to validate understanding of Rectangular Coordinate System. These would correct some coordinate system? of their misconceptions on this topic that have been encountered in real-life situations. After letting the students give their initial answers to the questions in the IRF Worksheet, 2. What are the different tell them that at the end of the lesson, they are expected to do the CoordinArt Making as parts of the rectangular a demonstration of their understanding about the Rectangular Coordinate System. coordinate system? Let the students read and understand some important notes on Rectangular Coordinate System before they perform the succeeding activities. Tell them to study 3. How are points plotted carefully the examples provided. on the Cartesian plane? 4. How can the Rectangular Coordinate System be used in real life? You just tried answering the initial column of the IRF Sheet. The next section will enable you to understand what a Rectangular Coordinate System is all about and do a CoordinArt to demonstrate your understanding. WWhhaatt ttoo PPrroocceessss Your goal in this section is to learn and understand the key concepts of Rectangular Coordinate System. 139

Teacher’s Note and Reminders Rectangular Coordinate System is introduced using the concept of sets. You have learned the binary operations of sets: union and intersection. Recall that A ∪ B and A ∩ B Don’t are defined as follows: Forget! A ∪ B = {x│x ∈ A or x ∈ B} A ∩ B = {x│x ∈ A and x ∈ B} The product set or Cartesian product of nonempty sets A and B, written as A × B and read “A cross B,” is the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B. In symbols, . Illustrative Examples: Let A = {2, 3, 5} and B = {0, 5}. Find (a) A × B and (b) B × A. Solution: A × B = {(2, 0), (2, 5), (3, 0), (3, 5), (5, 0), (5, 5)} B × A = {(0, 2), (5, 2), (0, 3), (5, 3), (0, 5), (5, 5)} The cardinality of set A is 3, symbolized as n(A) = 3. The cardinality of a set is the number of elements in the set. The cardinality of A × B, written as n(A × B), can be determined by multiplying the cardinality of A and the cardinality of B. That is, n(A × B) = n(A) • n(B) Illustrative Examples: Let A = {2, 3, 5} and B = {0, 5}. Find (a) n(A × B), and (b) n(B × A). Questions: Solution: Is n(A × B) = n(B × A)? n(A × B) = 3 ∙ 2 = 6 Why? n(B × A) = 2 ∙ 3 = 6 Answers to the Questions: Yes, n(A × B) = n(B × A). QU?E S T I ONS What can you conclude? It is because n(A × B) = n(B × A) implies n(A) • n(B) = n(B) • n(A) and it holds by Multiplication Property of Equality. 140

Answer State your conclusions by competing the statements below using the correct relation For any nonempty sets A and B, symbol = or ≠. 1. n(A × B) = n(B × A). 2. A × B ≠ B × A. For any nonempty sets A and B, 1. n(A × B) ___ n(B × A).Answers to the Exercises: 2. A × B ___ B × A.Exercise 1. Given that A = {4, 7, 8} and B = {5, 6}, find the following: Exercise 11. A × B = {(4, 5), (4, 6), (7, 5), (7, 6), (8, 5), (8, 6)} Given that A = {4, 7, 8} and B = {5, 6}, find the following:2. B × A = {(5, 4), (6, 4), (5, 7), (6, 7), (5, 8), (6, 8)} 1. A × B 3. n(A × B)3. n(A × B) = 6 2. B × A 4. n(B × A)4. n(B × A) = 6Exercise 2. Find (a) X × Y, (b) Y × X, (c) n(X × Y), and (d) n(Y × X) given the Exercise 2following sets X and Y: Find (a) X × Y, (b) Y × X, (c) n(X × Y), and (d) n(Y × X) given the following sets X and Y:1. X = {2, 3} and Y = {8, 3} 1. X = {2, 3} and Y = {8, 3} (a) X × Y = {(2, 8), {2, 3), (3, 8), (3, 3)} 2. X = {1, 3, 6} and Y = {1, 5} (b) Y × X = {(8, 2), {3, 2), (8, 3), (3, 3)} 3. X = {2, 5, 8, 9} and Y = {0, 8} (c) n(X × Y) = 4 4. X = {a, e, i, o, u} and Y = {y│y is a letter of the word paper}. (d) n(Y × X) = 4 5. X = {x│1 < x < 10, x is a prime number} and Y = {y│y ∈ N,1 < y < 3}2. X = {1, 3, 6} and Y = {1, 5} (a) X × Y = {(1, 1), {1, 5), (3, 1), (3, 5), (6, 1), (6, 5)} Let ℜ be the set of real numbers. The notation ℜ2 is the set of ordered pairs (x, y), (b) Y × X = {(1, 1), {5, 1), (1, 3), (5, 3), (1, 6), (5, 6)} where x and y ∈ ℜ; that is, (c) n(X × Y) = 6 (d) n(Y × X) = 6 ℜ2 = ℜ × ℜ = {(x, y)│x ∈ ℜ, y ∈ ℜ}.3. X = {2, 5, 8, 9} and Y = {0, 8} (a) X × Y = {(2, 0), {2, 8), (5, 0), (5, 8), (8, 0), (8, 8), (9, 0), (9, 8)} ℜ2 is also called the xy-plane or Cartesian plane in honor of the French mathematician (b) X × Y = {(0, 2), {8, 2), (0, 5), (8, 5), (0, 8), (8, 8), (0, 9), (8, 9)} René Descartes (1596 – 1650), who is known as the “Father of Modern Mathematics.” (c) n(X × Y) = 8 (d) n(Y × X) = 8 The Cartesian plane is composed of two perpendicular number lines4. X = {a, e, i, o, u} and Y = {y│y is a letter of the word paper}. that meet at the point of origin (0, 0) and divide the plane into four regions called quadrants. It is composed of infinitely many points. Each point in the (a) X × Y = {(a, p), {a, a), (a, e), (a, r), (e, p), (e, a), (e, e), (e, r), (i, p), (i, a), (i, e), coordinate system is defined by an ordered pair of the form (x, y), where x (i, r), (o, p), (o, a), (o, e), (o, r), (u, p), (u, a), (u, e), (u, r)} and y ∈ℜ. The first coordinate of a point is called the x-coordinate or abscissa and the second coordinate is called the y-coordinate or ordinate. We call (b) Y × X = {(p, a), {a, a), (e, a), (r, a), (p, e), (a, e), (e, e), (r, e), (p, i), (a, i), (x, y) an ordered pair because it is different from (y, x). The horizontal and (e, i), (r, i), (p, o), (a, o), (e, o), (r, o), (p, u), (a, u), (e, u), (r, u)} vertical lines, typically called the x-axis and the y-axis, respectively, intersect at the point of origin whose coordinates are (0, 0). The signs of the first and (c) n(X × Y) = 20 second coordinates of a point vary in the four quadrants as indicated below. (d) n(Y × X) = 85. X = {x│1 < x < 10, x is a prime number} and Y = {y│y ∈ N,1< y < 3} Quadrant I x > 0, or x is positive y > 0, or y is positive or (+, +); (a) X × Y = {(2, 2), {3, 2), (5, 2), (7, 2)} Quadrant II x < 0, or x is negative y > 0, or y is positive or (−, +); (b) X × Y = {(2, 2), {2, 3), (2, 5), (2, 7)} Quadrant III x < 0, or x is negative y < 0, or y is negative or (−, −); (c) n(X × Y) = 4 Quadrant IV x > 0, or x is positive y < 0, or y is negative or (+, −). (d) n(Y × X) = 4 There are also points which lie in the x- and y-axes. The points which lie in the x-axis have coordinates (x, 0) and the points which lie in the y-axis have coordinates (0, y), where x and y are real numbers. 141

Discuss the Rectangular Coordinate System connecting it with the concepts Illustrated below is a Cartesian plane.of sets. Start the discussion with the founder of the Cartesian plane, René y axisDescartes followed by the different parts of the Cartesian plane such asaxes, quadrants, origin, points, abscissa, and ordinate. Quadrant II 7 Quadrant I (−, +) 6 (+, +) Teacher’s Note and Reminders 5 Positive direction is upward and 4 to the right 3 2 1 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 -1 -2 -3 origin -4 (−, −) -5 (+, −) Quadrant III -6 Quadrant IV -7 Don’t How do you think can we apply this in real life? Let’s try the next activity.Forget! Example Suppose Mara and Clara belong to a class with the following seating arrangement. C1 C2 C3 C4 C5 C6 R5 R4 R3 Mara R2 Clara R1 Teacher's Table 142

Teacher’s Note and Reminders Questions: Don’t 1. Using ordered pairs, how do we describe Mara’s seat? How about Clara’s seat? Forget! 2. Using ordered pairs, how do we locate the seat of any classmate of Mara and Clara? 3. Can we make a set of ordered pairs? If yes, state so. Solutions: 1. Mara’s seat is at the intersection of Column 2 and Row 3. Clara’s seat is at the intersection of Column 4 and Row 2. In symbols, we can write (2, 3) and (4, 2), respectively, if we take the column as the x-axis and the row as y-axis. 2. We locate the seat of Mara’s and Clara’s classmates’ by using column and row. We can use ordered pair (Column #, Row #) to locate it. 3. Here is the set of ordered pairs: {(C1, R1), (C2, R1), (C3, R1), (C4, R1), (C5, R1), (C6, R1), (C1, R2), (C2, R2), (C3, R2), (C4, R2), (C5, R2), (C6, R2), (C1, R3), (C2, R3), (C3, R3), (C4, R3), (C5, R3), (C6, R3), (C1, R4), (C2, R4), (C3, R4), (C4, R4), (C5, R4), (C6, R4), (C1, R5), (C2, R5), (C3, R5), (C4, R5), (C5, R5), (C6, R5)} Activity 4 LOCATE YOUR CLASSMATE! Description: This activity will enable you to locate the seat of your classmate in your Direction: classroom using ordered pairs. This can be done by groups of five members each. Locate your seat and the seats of groupmates in the classroom. Complete the table below: Name LocationLet the students locate seats of their classmates using rows and columns. QU ?E S T I ONS How do you locate the seat of your classmate in the classroom?Ask them to perform A ctivity 4. See to it that the chairs are arranged properly.You may also extend this activity outside the classroom by forming lines. Seeto it that each student is equidistant to one another. 143

Teacher’s Note and Reminders Activity 5 MEET ME AT THIRDY’S RESIDENCE Don’t y Forget! Description: Finding a particular point such (1, 4) in Aurora 5th St. Direction: the coordinate plane is similar to finding Aurora 4th St. a particular place on the map. In this Aurora 3rd St. activity, you will learn how to plot points Aurora 2nd St. on the Cartesian plane. Aurora 1st St. With the figure at the right above, find the x following locations and label each with QU Mabini 1st St.letters as indicated. Mabini 2nd St. Mabini 3rd St. Mabini 4th St. Mabini 5th St. a. Mabini 4th corner Aurora 1st Streets – A b. Mabini 2nd corner Aurora 2nd Streets – B c. Mabini 3rd corner Aurora 5th Streets – C d. Mabini 5th corner Aurora 4th Streets – D e. Mabini 1st corner Aurora 1st Streets – E ?E S T I ONS 1. How do you find each location? 2. Which axis do you consider first? next? 3. If (1, 4) represents Mabini 1st Street corner Aurora 4th Street, then how could these points be represented? a. (3, 1) d. (4, 2) b. (4, 5) e. (5, 3) c. (1, 2) 4. If you are asked to plot those points mentioned in item number 3 in the Cartesian plane, can you do it? If yes, plot them. 5. How can Rectangular Coordinate System be used in real life? Activity 6 HUMAN RECTANGULAR COORDINATE SYSTEM Description: This activity is a form of a game which will enable you to learn the Direction: Rectangular Coordinate System. Form two lines. 15 of you will form horizontal line (x-axis) and 14 for the vertical line (y-axis). These lines should intersect at the middle. Others may stay at any quadrant separated by the lines. You may sit down and will only stand when the coordinates of the point, the axis or the quadrant you belong is called.Let the students experience decribing the coordinates of locations in real lifeby performing Activities 6 and 7. 144

Teacher’s Note and Reminders ?E S T I O 1. What is the Rectangular Coordinate System composed of? 2. Where do you see the origin? QU NS 3. What are the signs of coordinates of the points in each quadrant? a. Quadrant I b. Quadrant II c. Quadrant III d. Quadrant IV Activity 7 PARTS OF THE BUILDING Description: This activity will enable you to give the coordinates of the part of building. Direction: Describe the location of each point below by completing the following table. An example is done for. Note that the point indicates the center Don’t of the given part of the building. Forget! Parts of the Coordinates Quadrant Parts of the Coordinates Quadrant Building II Building Example: (-11, 8) Morning Room 1. Gilt 8. Marble Room HallParts of the Parts of the 2. Terrace 9. Reception Building Building Hall Office Coordinates Quadrant Coordinates Quadrant 3. Old 10. Drawing II Kitchen Room Example: IIMorning Room (-11, 8) IV 4. Billiard 11. Entrance IV Room1. Gilt I 8. Marble 12. library Room (-11, 5) I Hall (-5, 2) II 5. Salon III (-11, -10) III 13. Spa2. Terrace (12, -3) IV 9. Reception I 6. Reception Hall Office (2, 8) III Hall 14. Harborough (-13, -2) II Room3. Old (12, -6) 10. Drawing (-6, 7) IV 7. Grand Kitchen Room (7, -7) I Staircase4. Billiard (12, 8) 11. Entrance (7, 7) Room 12. library5. Salon (6, 2) 13. Spa6. Reception (-11, -4) Hall 14. Harborough Room7. Grand (1, -1) Staircase 145

Teacher’s Note and Reminders Don’t ?E S T I OForget! QU NS 1. What is the Rectangular Coordinate System composed of? 2. How can the Rectangular Coordinate System be used in real life? 146

Let the students find the coordinates of the point and identify the quadrant/ Activity 8 OBJECTS’ POSITIONaxis where it is located by performing Activity 8. Description: This activity will enable you to give the coordinates of the point where theAnswers to the Activity 8: Direction: object is located. Describe the location of each point below by the completing the following Object Coordinates Quadrant/Axis table. An example is done for you.Example: ball (4, 2) I1. spoon (6, -5) IV Object Coordinates Quadrant/Axis2. television set (-5, 6) II3. laptop (2, -4) IV Example: ball (4, 2) I4. bag (-4, -3) III5. pillow (1, 5) I 1. spoon6. camera (0, 0)7. table (-2, 2) x-axis and y-axis 2. television set II 3. laptop 4. bag 5. pillow 6. camera 7. tableTeacher’s Note and Reminders Don’t Forget! QU ?E S T I ONS How can the Rectangular Coordinate System be used in real life? 147

Answer to Exercise 3 Exercise 3 Indicate the name of each point in the Cartesian plane. Name each point by writing the letter beside it. The coordinates are provided in the box below. An example is done for you.E y I 12 11 C 10 FD 9 8 B 7 GH 6 5 J 4 3 2 x 1 1 2 3 4 5 6 7 8 9 10 11 12 13 0Teacher’s Note and Reminders -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 A -6 -7 -8 -9 -10 -11 -12 -13 Don’t 1. A(-2, -6) 6. F(-4, 0) Forget! 2. B(3, -3) 7. G(0, -5) 3. C(-1, 3) 8. H(6, -5) 4. D(0, 0) 9. I(6, 5) 5. E(-9, 11) 10. J(13, -8) 148

Answer to Exercise 4 Exercise 4 Write the coordinates of each point. Identify the quadrant/axis where each point lies. Teacher’s Note and Reminders Complete the table below. Don’t Forget! Coordinates Quadrant / Axis 1. B( __ , __ ) 2. C( __ , __ ) y 3. F( __ , __ ) 4. G( __ , __ ) 5 5. H( __ , __ ) 6. L( __ , __ ) 7. K( __ , __ ) 4F 3 H G 2 1 C x -5 -4 -3 -2 -1 01 2 3 4 5 B -1 K -2 -3 L -4 -5 QU ?E S T I ONS 1. In what quadrant/axis does a point lie? 2. How do you locate points on the Cartesian plane? 3. Have you had an experience in your daily life where a Rectangular Coordinate System is applied? If yes, cite it. 4. How can the Rectangular Coordinate System be used in real life? 149