QUARTER 2 Mathematics G9 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and the authors do not represent nor claim ownership over them. This module was carefully examined and revised in accordance with the standards prescribed by the DepEd Regional Office 4A and CLMD CALABARZON. All parts and sections of the module are assured not to have violated any rules stated in the Intellectual Property Rights for learning standards. The Editors PIVOT 4A CALABARZON Math G9 PIVOT 4A Learner’s Material Quarter 2 First Edition, 2020 Mathematics Grade 9 Job S. Zape, Jr. PIVOT 4A Instructional Design & Development Lead Angelo D. Uy Content Creator & Writer Jhonathan S. Cadavido Internal Reviewer & Editor Lhovie A. Cauilan & Jael Faith T. Ledesma Layout Artist & Illustrator Jhucel A. del Rosario & Melanie Mae N. Moreno Graphic Artist & Cover Designer Ephraim L. Gibas IT & Logistics Published by: Department of Education Region IV-A CALABARZON Regional Director: Wilfredo E. Cabral Assistant Regional Director: Ruth L. Fuentes PIVOT 4A CALABARZON Math G9 Guide in Using PIVOT 4A Learner’s Material For the Parents/Guardians This module aims to assist you, dear parents, guardians, or siblings of the learners, to understand how materials and activities are used in the new normal. It is designed to provide information, activities, and new learning that learners need to work on. Activities presented in this module are based on the Most Essential Learning Competencies (MELCs) in Mathematics as prescribed by the Department of Education. Further, this learning resource hopes to engage the learners in guided and independent learning activities at their own pace. Furthermore, this also aims to help learners acquire the essential 21st century skills while taking into consideration their needs and circumstances. You are expected to assist the children in the tasks and ensure the learner’s mastery of the subject matter. Be reminded that learners have to answer all the activities in their own answer sheet. For the Learners The module is designed to suit your needs and interests using the IDEA instructional process. This will help you attain the prescribed grade-level knowledge, skills, attitude, and values at your own pace outside the normal classroom setting. The module is composed of different types of activities that are arranged according to graduated levels of difficulty—from simple to complex. You are expected to : a. answer all activities on separate sheets of paper; b. accomplish the PIVOT Assessment Card for Learners on page 38 by providing the appropriate symbols that correspond to your personal assessment of your performance; and c. submit the outputs to your respective teachers on the time and date agreed upon. PIVOT 4A CALABARZON Math G9 Parts of PIVOT 4A Learner’s Material Development Introduction K to 12 Learning Delivery Process What I need to know What is new What I know What is in What is it Engagement What is more What I can do Assimilation What else I can do What I have learned What I can achieve Descriptions This part presents the MELC/s and the desired learning outcomes for the day or week, purpose of the lesson, core content and relevant samples. This maximizes awareness of his/her own knowledge as regards content and skills required for the lesson. This part presents activities, tasks and contents of value and interest to learner. This exposes him/her on what he/she knew, what he/she does not know and what he/she wants to know and learn. Most of the activities and tasks simply and directly revolve around the concepts of developing mastery of the target skills or MELC/s. In this part, the learner engages in various tasks and opportunities in building his/her knowledge, skills and attitude/values (KSAVs) to meaningfully connect his/her concepts after doing the tasks in the D part. This also exposes him/her to real life situations/tasks that shall: ignite his/ her interests to meet the expectation; make his/her performance satisfactory; and/or produce a product or performance which will help him/her fully understand the target skills and concepts . This part brings the learner to a process where he/she shall demonstrate ideas, interpretation, mindset or values and create pieces of information that will form part of his/her knowledge in reflecting, relating or using them effectively in any situation or context. Also, this part encourages him/her in creating conceptual structures giving him/her the avenue to integrate new and old learnings. This module is a guide and a resource of information in understanding the Most Essential Learning Competencies (MELCs). Understanding the target contents and skills can be further enriched thru the K to 12 Learning Materials and other supplementary materials such as Worktexts and Textbooks provided by schools and/or Schools Division Offices, and thru other learning delivery modalities, including radio-based instruction (RBI) and TV-based instruction (TVI). PIVOT 4A CALABARZON Math G9 WEEKS 1-2 Variation Lesson I There are varied things around you that you should know how things are related to each other. Like how is time related to the speed of a vehicle or even as simple as the relationship of the length of your arm span with your height. Quantities may vary directly that is, as one quantity increases the other quantity must also increase or they may vary inversely, that is if one quantity increases the other quantity decreases. When two quantities increase at he same time or decrease at the same time , it shows direct variation. However if one quantity increases and the other decreases, it shows inverse variation. You encounter such situations in your everyday life. For instance, if walk slowly going to school it takes longer time but walking fast, takes a shorter time reaching the school. Your speed in walking varies inversely with the time. The faster you walk the shorter the time it takes you to reach your destination. Another situation is when your mother asked you to buy rice, which cost P 55.00 a kilo. If one kilo costs P55.00, how much 10 kilos of rice cost?. In this situation the more kilos of rice the more money you will pay. Hence this is an example of direct variation. Situations involving 3 or more quantities that varies from one another may show joint variation or combined variation. D Direct Variation In your previous lesson you were able to represent relationship between two quantities using graphs, table of values and equations. Examples: 1. Given: Graph: x 1 2 3 4 5 y 2 4 6 8 10 Table of values: Equation: y = 2x You can see from the table of values that as the value of x increases the value of y also increases. The graph is a line that rises to the right. This is an illustration of a direct variation. This means that y varies directly as x. In symbol: y k or y = kx, where k is the constant of variation or constant of x proportionality. In this case the constant of variation is 2. PIVOT 4A CALABARZON Math G9 6 2. You will be celebrating your 14th birthday and you want to have 14 balloons in your birthday party. The table below shows the number of balloons (x) and the corresponding cost (y). As you can see the values of x increase , the values of y also increase. Hence the cost of balloons vary 10 20 30 40 ... 70 ies directly to the number of balloons. y1 y2 10 20 Since y varies directly as x, find the constant of variation k: 5 x1 x2 2 4 The constant of variation (k) is 5. Thus the equation of the direct variation is y = 5x. x 2 4 6 8 ... 14 3. y varies directly as x. If y = 12 when x = 4, A. Find the constant of variation when y is 36? b. What is the equation? C. What is x Solution: y 12 A. k 3 x 4 B . y = 3x C. using the equation of variation: y = 3x 36 12 36 = 3x x= 3 When the value of y increases from 12 to 36, the value of x increases also from 4 to 12. Direct Square Variation The relationship between the area and the square of the radius is an example of a direct square variation. A = πr2 , the constant of variation is π or 3.1416. Direct square variation states that if y varies directly to the square of x, there is a non zero constant k such that y k x2 or y = kx2 . The graph of direct square variation is quadratic in nature it follows a quadratic curve. Examples. 1. y varies directly as the square of x. If y = 27 when x = 3, find x when y = 81. Solution: Find the constant of variation: k y 27 27 2 3 x2 3 9 Equation y = 3x2 Find x when y = 81 Using the equation y = 3x2 , find x 81 = 3x2 x2 = 81 27 3 x= 27 3 3 2. If p varies directly as the square of q and p = 256 when q = 8, find: A. k B. equation c. p when q = 2 3 Solution: A. k p 256 4 q2 82 B. p = 4q 2 C. p = 4q2 7 p = 4( 2 3)2 = 4(12) = 48 PIVOT 4A CALABARZON Math G9 Inverse Variation In real life there are different situations that shows inverse relation. Inverse variation states that y varies inversely as x or y is inversely proportional k to x if there is non zero constant k, such that y or xy = k. The decrease of x one quantity results to the increase of the other quantity. Examples: 1. The table below shows that number of persons working together (x) and the number of hours (y) the job is completed. Find the constant of variation and equation. Graph the given table of values. x 1 2 3 4 5 Constant of variation: xy = y 2 1 2 3 1 2 2 5 Equation xy = 2 or y 2 x 2 2 5 3 2(1) 2 5 3 Graph: 2. If y varies inversely as and y = 20 when x = 8, find A. constant of variation Solution: A. k = xy B. variation equation k = 8(20) = 160 B. xy = 160 or y = C. y = C. y when x = 16 160 160 10 x 16 160 x As the value of x increases from 8 to 16, the value of y decreses from 20 to 10. 3. The rate (r ) of the car varies inversely with the time (t). The car travels at a rate of 60kph for 1.5 hours. Find the rate of the car when it took 1 hour to travel the same distance (d), Solution: In this case, the faster the car travel, the shorter time it will take to cover the same distance. Hence the constant of variation here is the distance. Therefore d = rt = 60(1.5) = 90 km. To solve for the rate : r d 90 90 kph. t 1 The faster the car travels, the shorter the time it takes to travel the same distance. PIVOT 4A CALABARZON Math G9 8 Learning Task 1: A. Complete the table if y varies directly as x or y x 30 5 8 B. Complete the table if y varies inversely as x. x2 k Equation y x 5 6 4 10 18 y = 3x 72 y = 2x2 k Equation 36 20 y= 12 y= 50 x 24 x Joint and Combined Variation Joint and combined variation involves 3 or more quantities that may vary directly and or inversely to each other. Joint Variation If the ratio of one quantity to the product of the other two quantities is cony k stant, the they vary jointly. That is, if y varies jointly as x and z, then xz or y = kxz. Examples: 1. The surface area of a cylinder varies jointly as the radius and height . A = krh, where A is the surface area , r is the radius and h is the height of the cylinder. The constant of variation k = 2π. Hence A = 2πrh. Any of the variable r or h increases the area also increases. 2. Translate the following into variation equation: (a) p varies jointly as q and r. Equation: p = kqr (b) area of parallelogram varies jointly as its base and altitude. Equation: A = bh, k = 1 3. If m varies jointly as p and q and m = 50 when p = 5 and q = 2, find m when p = 10 and q = 6. 50 y k Solution: Determine the constant of variation: k = 5(2) 5 xz m = kpq = 5(10)(6) = 300. As p and q increase m also increases. Combined Variation This is a variation where one quantity varies directly to other quantity and inversely to the other quantity. The equation Examples: v kw means x the v varies directly as w and inversely as z. 1. v varies inversely to x and directly to w. If v = 12 when x = 4 and w = 8, find w when v = 24 and x = 2. 9 PIVOT 4A CALABARZON Math G9 Solution: Equation: v = kw . x Find k: 6w x Solve for w: v = 12 24 6w 3 k (8) 4 12( 4) 6 8 24(3) w 12 6 k= 2. If p varies directly as the square of q and inversely as the square root of r, and p = 20 when q = 2 and r = 64, find p when q = 8 and r = 144. Solution: k (2) 2 kq 2 Solve for k: 20 64 r 2 = 40(8) 40(64) 640 12 3 144 Equation: p = Solve for p: p k= 20 64 20(8) 40 22 4 Learning Task 2 A. Supply the missing value if y varies B. Supply the missing value if y varies jointly as x and z y x z 30 3 5 80 k Equation 4 6 60 directly to v and w and inversely to z 8 Y= 2xz v w z 5 12 14 24 18 4 8 16 4 10 48 y Y =3xz 12 Y= 20 3 xy 4 12 24 6 6 2 7 15 k Equation 10 3vw 2z 8vw y z y 6 E Learning Task 3 A. Identify if the given equation is a direct, inverse, joint or combined variation with k as the constant of variation. 1. b = kd 3. m = 2. y = klm 4. a kc b kn p 5. mn = kpq B. Write the equation of variation and solve for the constant of variation. 1. 2. p 1 2 3 4 5 q 7 28 63 p 3 6 12 24 48 q 16 8 4 2 1 3. 112 175 PIVOT 4A CALABARZON Math G9 4. 10 p 3 6 12 24 48 q 16 8 4 2 1 p 6 12 18 24 30 q 2 4 6 8 10 A Learning Task 4 A. Solve the following: 1. The distance (d) from the center of the seesaw varies inversely as the weight (w) of a person. JB who weighs 50 kg sits 3 feet from the fulcrum. How far from the fulcrum must JP sit in order to balance with JB if he weighs 35 kg? 2. The number of pages (p) that Ethan reads varies directly as the number of hours (t) he is reading. A. write the variation equation B. If he can read 21 pages in 14 minutes, how may pages can he read in 21 minutes? 3. The pressure of the gas is directly proportional to the temperature and inverse ly proportional to its volume. A. Write the variation equation. B. What happened to the pressure if the volume is reduced to half and the temperature is doubled? 4. Given the equation y k statement is true or false. pq 2 , where k is the constant of variation, tell whivh r A. y and r varies directly. B. y and q2 are directly proportional C. y and pq2 varies jointly D. p and r are inversely proportional E. y and p varies directly. 5. The volume of a cylinder is given by the formula V = πr2 h. If r is increased by 50% and the height is reduced by 25%, what will happen to the volume? What is the constant of variation. B. Give at least three examples of quantities you know or you have or experienced that show different types of variation. 11 PIVOT 4A CALABARZON Math G9 WEEK 3 Integral and Zero Exponents I Lesson You have learned in previous lessons about laws of exponents. Let us recall these laws: (a) Product of a Power: am · an = am+n (b) Quotient of a Power: am amn an (c ) Power of a Power: (am)n = amn (d) Power of a product: (a · b)m = am bm m (e )Power of a Quotient: a b am bm In the process of performing the laws of exponents it may result to a positive, negative, or zero exponents. In this lesson you will learn on how to simplify expressions with integral (positive or negative integers) or zero exponent applying the different laws of exponents. D Zero Exponent Simplify the expression aa a2 1 . Write this expression in factored form aa a2 Applying the laws of exponent , quotient of a power, any variable/number a , where a ≠ 0, a0 = 1. a2 a22 a0 1 a2 . Thus, for Examples: 1. 12x0 = 12(1) = 12 , only variable x is raised to zero power 2. (12x)0 = (12)0 (x)0 = 1(1) = 1 applying the law of power of a product, both 12 and x are raised to zero power 3. (x0 + 3)(x + 3)0 = (1 + 3)( 1) = 4, the expression (x + 3) is raised to zero power so it is equal to 1. 4. Simplify: Solution: 45a 4 b 2 , apply laws of exponent in simplifying 5a 4 b 9a4 - 4 b2 - 1 = 9a0 b = 9b 5. Simplify: 2a0 + (2a)0 + 20a Solution: 2(1) + 1 + 1(a) = 2 + 1 + a = 3 + a 6. Simplify: 4m(n p ) 0 4m 0 Solution: 4m(1) m 4(1) PIVOT 4A CALABARZON Math G9 12 Integral Exponent An expression has an integral exponent if the exponent is either positive or negative integers. The expressions like 3x, 5x-2 + 2, m-8 , are examples or expressions with integral exponents. am a mn an One of the laws of exponent is quotient of a power that is : Apply this law of exponent in simplifying the following: 1. 2. x4 x 42 x 2 . You can use factoring to check the answer: x2 xxxx x x x2 xx x2 x 2 4 x 2 . The exponent is negative. Use factoring to simplify the expresx4 2 x x x 1 2 x x x x x4 x sion . 1 x2 . Hence the expression x -2 is equal to 1 The negative sign in the exponent means the reciprocal. Thus a n a n . Similarly if the expression with negative exponent is in the denominator , get its reciprocal to express the exponent as positive. Therefore 1 n a n a 3. Simplify: 3x3 y 6 x5 y 4 Solution: Observe the exponents of each variable. Let the variable with greater exponent remain in its position but subtract the smaller exponent from the larger exponent. 3x3 y 6 3 y64 3y2 5 4 53 x y x x2 4. Express all exponent to a positive integers. (a) 28a 5 b 6 c 4a 0 b 2 4b 2 7a 5b 4 c 3 c2 c2 (b) (5 - 2)5 · 3-3 + (5 + 2)0 = (3)5 · 3-3 + 1 = 32 + 1 = 9 + 1 = 10 (c ) [(8)2]-1 ⋅ (10 - 8)14 ⋅ 4-5 (8)-2 · (2)14 · 4-2 , apply the law of exponent power of a power (23 )-2 · 214 · (22 )-2 , change to the same base so can apply the law of 2-6 · 214 · 2-4 exponent. 2-6+14+(-4) = 24 = 16, apply the law of exponent product of a power. (d ) (5 - 2)5 · 3-8 + (5 + 2)0 (3)5 · 3-8 + 1 3-3 + 1 1 +1 33 1 27 28 27 27 27 apply product of a power get the reciprocal of the number with the negative exponent to make the exponent positive. Change to similar fractions before adding. 13 PIVOT 4A CALABARZON Math G9 E Learning Task 1: Express the following expression into non zero and non negative exponents. Simplify your answer. 1 8 4 (ab) 24 x y 1. 7-1 6. 11. 9 6 x5 y 4 1 9 5x0 2. (14abc)0 7. 12. x 2 y 3. 10-9 8. 120a 5b 6 c 5 12a 2b 0 c 2 13. 4. 5(xy)0 9. (4 x) 0 y 5 z 2 (234 xyz ) 0 14. 5. 015 10. (5 xy ) o 10 2 15. 12b 3 a 4 1 a 5 n 1 2 A Learning Task 2 Find the single value of the following with- 1. 2-2 · 3-1 2. 100 + 10 3. 4(60) + 3(3-1 ) 4. 5-1 - 2-3 5. (120)- 10 11. 7 2 41 12. 9 21 13. 82 40 14. 3 15. 5 6. 24 + 2– 2 7. (5 + 5-2)0 (60 + 6)-2 8. 20 • 33 9. 5(-6)0 10. 10-3 ÷ 103 PIVOT 4A CALABARZON Math G9 1 ( 4) 3 14 2 out exponent. 5 Rational Exponent and Radical Expressions I WEEK 4 Lesson Your previous lesson is all about integral and zero exponents. This time, your lesson is on rational exponents and radicals numbers. When simplifying expressions with rational exponents you should remember the laws of integral exponents, the rule is the same. However, you need to remember the rules in adding fractions. In adding fraction, you can only add fractions if they are similar otherwise you change dissimilar fraction to similar fraction. Examples: Simplify the following expressions: 1. a 1 2 3 2 a a 1 3 2 2 , apply the product of a power law of exponent by adding the exponents of the same base. 4 = a 2 a2 2. 6 x 1 3 6 , apply the rule in adding fraction then simplify 6 6 x 3 , apply the power of a power law of exponents by multiplying the exponents. Use the rule in multi plying fractions = (6 x) 2 6 2 x 2 , apply power of a product law of exponent. = 36x2 , simplify 1 4 2 3. 9 = 1 4 2 1 9 2 2 3 2 1 2 2 1 2 , apply the quotient of a power law of exponents. , express 4 and 9 in exponential form 2 = 22 3 4. 2 1 12 3 12 2 2 2 2 , multiply the exponents , then simplify 3 2 12 3 4 = 1 2 3 126 6 , apply the product of a power law of exponent 7 12 6 = 12 6 , change the fractions to similar fraction then add. 12 1 6 6 1 1 , simplify by 12 6 12 6 1212 6 applying laws of exponents. Exponential expressions with rational exponents can be expressed into radical expressions. 15 PIVOT 4A CALABARZON Math G9 D Radicals An exponential number whose exponent is a rational number can be expressed as radicals Examples: Exponential 8 Base Radicals Exponent 1 3 radical sign index 3 8 radicand The denominator of the rational exponent becomes the index of the radical number. 3 This is read as “cube root of 8” 8 Exponential Radical 3 122 2 12 3 12 3 This is read as the “square root of 12 raised to the 3rd power” When changing the exponential number with rational exponent whose denominator is 2 to radical number you may or may not write the index 2. The exponential number a m n , when changed to radical is n am read as the nth root of a raised to m. The nth root of a number is a number that is taken n times as a factor of the radicand a. You can express exponential expression to radical expression and vice versa. Examples: 1. Change 5 1 4 5 = 1 4 to radical expression. 5 (read as the 4th root of 5) 4 3 2. Write 4 2 in radical expression 3 42 4. Write 4 ( read as the square root of 4 cube or 4 raised to the 3rd power) 3 8 in exponential form 8 8 , the index (3) is the denominator of the fractional exponent 3 3 4 4 4 3 and the power (4) is the numerator of the fractional exponent. PIVOT 4A CALABARZON Math G9 16 5. 5 Change the expression 32 to exponential form. 15 , the 5th root of 32 is equal to 32 raised to one fifth. 32 32 5 Laws of Radicals Laws of radicals are similar to the laws of integral exponents. Let a, b, m, n are integers: 1. 2. n a b a 1 n n a n n a 1 n n 3. n a b 4. n m n m a 1 b n ab a n b 1 a n 1 n Examples , the nth root of a raised to the power of n is equal to a. 1 n 1 1 a n 1 b n a n an a n a b n ab , the nth root of a times the nth root of b is equal to the nth root of the product of a and b. , the nth root of a divided by the nth root of b is equal to the nth root of the quotient of a and b. 1 1 m a mn mn a , the root of a root is the rth of a, where r = mn Apply the laws of radicals. 3 1.3 12 12 3 12 3 12 3 1 2. 5 3 4 5 8 5 ( 4)(8) 5 32 2 , multiply if the indices are the same, just multiply the radicand, copy the common index, then simplify. The fifth root of 32 is 2, meaning, when you multiply the root 2 by itself 5 times the answer is 32. The index 5 indicates the number of times the root be multiplied by itself to get the radicand. In this case, 2 5 = 32. Hence 5 32 2 3. 4. 18 2 3 18 2 9 3 , divide the radicands if the indices are the same, then simplify by extracting the square root of 9 . The 9 3, since 32 = 9 64 3 8 2 , the square root of 64 is 8 because the 82 = 64 and the cube root of 8 = 2, since 23 = 8. 5. 3 2 1 2 1 2 12 2 . 12 12 3 12 2 (12) 3 (12) 43 6 7 (12) 6 6 12 7 6 12 6 12 6 12 6 6 12 126 12 Apply the laws of rational exponents and radicals to simplify the expression. Learning Task 1 Use the Laws of rational exponent to simplify: 1. n 2. 3b 2 b 2 3 2 4 1 3. 3 a 1 2 1 3 4. 17 3 124 1 4 4 5. a b 3 1 4 12 PIVOT 4A CALABARZON Math G9 E Learning Task 2 A. Change exponential expression to radical expression. 2 1. 1253 3 3. 1 2. 5x 2 15 52 5 5 5. 12 4 16 2 xy 5 4. B. Write each expression in exponential form. 5 3 1. ( 7 ) 3. 5 11x 4 4m 2a 8 83 4 5. 3 228 4 A Learning Task 3 A. State the laws of Radical Exponents and give examples B. Apply the laws of rational exponents in simplifying the expression. 1. 1 2. 3. 12 14 5 x a 6 b 16 y8 3 8 2a 2a 2 1 2 2 8. x 1 10 5 12 3 7. 2 1 6 4m y 8 p 20 8 5. x 43 3 3 p q 12 4. 1 16m 9 n 3 26 6. 1 4 PIVOT 4A CALABARZON Math G9 9. 10. x4 5 1 3q 2 r 4 1 62 18 2 1 2 1 3 Simplifying Radical Expressions WEEKS 5-6 I A radical is in its simplest form if: 1. the radicand does not contain a perfect square, cube or nth power. Examples: 2 is in its simplest form since 2 is not a perfect square, meaning the square root of 2 is not exact. 5 Is in its simplest form since 5 is not a perfect cube, meaning, the cube root of is not exact. 3 2. there is no fraction in the radicand or no radical in the denominator Example: 3 4 3 3 , the fractions is simplified such that 2 4 there is no radicand in the denominator. 3. the index of the radical id the lowest possible index. Example: 6 8 is not yet in simplest form since the index 6 can be expressed as the product of 3 and 2. 3 8 2 since the cube root of 8 To simplify express it as is 2 and square root of 2 is not exact. You have to simplify radical expressions with: 1. a perfect nth factor 2. a fraction radicand or denominator 3. reducible index. Recall that : m n a a n m a. n a m , is a radical expression which indicates the nth root of am . b. n is the index which gives the order of radicals; and c. am is the radicand, the number within radical sign. D The properties of radicals which can be useful in simplifying radical expressions are as follows: 1. 2. 3. n ab n m n p p q n nm n p n q a n b , the nth root of a product is equal to the nth root of each factor. p , the nth root of the mth root of a number id equal to the nmth root of the number. , the nth root of the quotient is equal to the nth root of the numerator and the nth root of the denominator. 19 PIVOT 4A CALABARZON Math G9 Simplifying Radicals by Reducing the Radicand Reducing radicand is finding a factor of a radicand whose indicated roots can be found. Examples: 1. Simplify 50 Solution: 50 is not a perfect square, therefore you have to find a factor of 50 which is a perfect square. 50 25 .2 , the factors of 50 are 25 and 2. 25 is a factor which a perfect square, meaning you can extract the square root 25 2 , the square root of the product is equal to the = square root of the factors. = 5 2 , the square root of 25 is 5 , square root of 2 is already in simplified form. Therefore 2. Simplify 3 50 5 2 in simplest form 81 Solution: The factor of 81 which is a perfect cube is 27. 3 Therefore 3. Simplify 3 5 81 3 27 3 3 27 3 3 33 3 81 3 3 since 3 3 3 3 3 3 13 3 3 27(3) 81 3 32 x 7 y 5 z Solution: To get the root of variables with integral exponent, the exponent must be divisible by the index. 5 32 x 7 y 5 z 5 32 x 5 y 5 x 2 z 5 32 x 5 y 5 5 x 2 z 2 xy 5 x 2 z Take note that when you divide the exponent of x which is 7 by the index 5 there is a remainder of 2 and the exponent of z which is 1 cannot be divided by 5. Hence the variables tha will remain under the radical sign are x 2 and z. 4 5 6 4 4. Simplify 5 48a b c Solution: Factor the radicand such that one factor is the 4th power of a certain number ad the variable such the the exponent is divisible by the index 4. 54 48a 4b10 c 5 54 16a 4b8c 4 3b 2 c 54 16a 4b8c 4 4 3b 2 c 5(2ab 2 )4 3b 2 c 10ab 2 c 4 3b 2 c The factors of 48 is 16 and 3 where 16 is a perfect 4th power of a number. The factors of b10 are b8 where the exponent is divisible by 4 and b2 . The factors of c5 are c4 and c. 5 is the coefficient of the radical expression. Multiply the coefficient 5 with the 4th root of 16a4 b8 c4 . PIVOT 4A CALABARZON Math G9 20 Simplifying Radicals by Reducing the Order of Radicals To reduce the order of radicals is to reduce the index to its lowest possible number. Examples: 1. Simplify a12 8 Solution: 8 a 12 8 a 8 .a 4 , find a factor of radicand that is a power of 8 8 = a 8 8 a 4 , find the 8th root of each factor 4 8 = a a 4 = a a8 = aa = a 2. Simplify 4 , change the radical to exponential form 1 2 , reduce the rational exponent to lowest term a , change the exponential form to radical form. 25 Solution: 25 is not a perfect power of 4 but a perfect power of 2 . Hence you can express 25 as 52 25 4 4 52 2 1 = 5 4 5 2 , change to exponential form and reduce the rational exponent to lowest term 1 = 52 5 Therefore 3. Simplify , change the exponential to radical form. 25 5 4 3 , in simplest form 64 Solution: Simplify the innermost radical 3 64 3 = 2 4. Simplify , the square root of 64 is 8, since 82 = 64. 8 , the cube root of 6 is 2, since 2 3 = 8 27m 3 n 3 6 Solution: 6 27m 3 n 3 6 33 m 3 n 3 6 (3mn) 3 , express 27 as power of 3: 33 , then apply the laws of exponent 3 6 1 (3mn) 3 (3mn) 6 (3mn) 2 , change radical to exponential form and re duce the rational exponent to lowest term (3mn) 1 2 3mn , change exponential form to radical form Therefore the simplest form of 5. Simplify Solution: 8 6 27m 3 n 3 is 3mn 81 1 8 1 1 2 814 81 818 4 81 21 3 , the exponent 1 1 1 8 4 2 PIVOT 4A CALABARZON Math G9 Simplifying Radicals by Rationalizing Denominators In a radical expression, id there is radical number in the denominator, the expression is not in simplest form. To eliminate the radical number in the denominator you should apply the rationalization process. To rationalize denominator, you have to multiply the numerator and denominator by a number such that the resulting radicand in the denominator is a perfect nth power. Examples: 3 2 1. Simplify Solution: Apply quotient law of radical 3 2 3 2 3 2 , express as quotient of two radicals 2 2 6 , multiply the radical fraction by a radical which will 4 make the radicand in the denominator a perfect square. = 6 , simplify 2 Remember that a radical fraction is in its simplest form if there is no radical number in the denominator. 2. Simplify Solution: 3. Simplify: 3 3 3 48 6 48 3 82 6 32 54 3 3 Solution: , apply the quotient law of radicals then find the quotient of the radicand. The quotient 8 is a perfect cube. Thus, the cube root of 8 is 2. 32 54 3 3 84 Simplify the radicand by finding a factor which is a perfect cube. 27 2 23 4 = 3 3 2 = 2 3 3 extract the cube root of the factors that are perfect cube. 2 PIVOT 4A CALABARZON Math G9 simplify by dividing the radicands 22 3. Simplify 64 3 4 2 Solution: Find smallest possible number where you can extract the 4th root. Since the radicand in the denominator is 2, you need to find a num ber when multiplied by 2 will give you a number where you can find the 4th root. In this case the, radicand in the denominator must be 16. Thus multiply both radicand by 8. 64 3 8 64 24 64 24 4 34 24 4 2 2 8 16 , simplify by extracting the 4th root of 16. then divide 6 by 2. Conjugate of Radical Expression Examples: Radical Expression Conjugate 3 2 3 2 5 2 5 2 The product of the radical expression and its conjugate is an integer. In finding its product is the same procedure in multiplying the sum and difference of 2 binomials. Like, (x + y)(x– y) = x2 - y2 Finding the product of radical expression and its conjugate: A. ( 3 B. ( 5 2)( 3 2) = (3)2 - ( 2 ) 2 9 2 7 2 )( 5 2 52 3 4. Rationalize the expression 3 2 3 Solution: Find the conjugate of the denominator then multiply the numerator and the denominator by the conjugate. 3(2 3 ) 3(2 3 ) 3(2 3) 43 (2 3 )(2 3 ) 23 Simplify the denominator. PIVOT 4A CALABARZON Math G9 E Learning Task 1 A. Simplify the radical expressions 1. 63 6. 2. 48 3. 75 4. 5. 3 11. 24 4 625 5 96 7. 3 81 12. 6 8. 3 128 13. 4 243 99 9. 3 40 14. 3 92 10. 3000 3 15. 135 128 B. Simplify radicals by reducing the order or index of radical. 1. 2. 6 16 3 3. 64 10 4. 5. 32 12 12 729 81 A Learning Task 2 A. Find the product of the radical expression and its conjugate. Radical Expression Conjugate 1. 3 5 2. 3 2 3. 6 3 B. Simplify by rationalizing the denominators. 8 2 1. 6. 1 5 2 2. 3 5 4 7. 4 62 3. 4 5 8 8. 2 7 3 14 7 9. 5 4 5 9 6 10. 2 2 1 4. 5. 3 3 PIVOT 4A CALABARZON Math G9 24 Product Operations of Radicals WEEK 7 Lesson I In this lesson you have to perform operations of radicals such as: A. adding and subtracting similar and dissimilar radicals. B. multiply and/or divide radicals Radical terms are similar if it has the same radicand and index. Examples: 5 ,3 5 , 5 . These three radical numbers are similar since the radicand are the same which is 5 and the index is 2 or all square root. Radicals are unlike or dissimilar if the radicand are not the same and/or the index are also different. Examples: 5 , 3 5 Though the radicals have the same radicand but the index are different. 5 , 6 The index are the same but the radicand are different. D Addition and Subtraction of Radicals In adding or subtracting radicals, you can only add or subtract similar radicals. To add or subtract similar radicals, add or subtract only the coefficients of the radical number and copy the common radicals. Examples: 1. 4 2 2. 5 4 2 4 (5 2) 4 3 4 3( 2) 6. Subtract the numerical coeffi 2 2 2 ( 4 1 2) 2 3 2 . Add and/or subtract the numerical coefficients. If there is no number before the radical number it is understood that the coefficient is 1. cients. Since the radicand is a perfect square, then get the square root. Multiply the root to the coefficient. 3. 3 3 5 3 2 5 (3 5) 3 2 5 8 3 2 5 . Only add the coeffi cients of terms that are similar. 4. 2 50 8 3 2 . In this case we can simplify first the radicand. Since the lowest radicand is 2 find two factors of the other radicand in such a way that one factor is 2 and the other is a perfect square. 2 25 2 4 2 3 2 . The factors of 50 are 25 and 2, where25 is a perfect square and the factors of 8 are 4 and 2, where 4 is also a perfect square. 25 PIVOT 4A CALABARZON Math G9 2(5) 2 2 2 3 2 Simplify by getting the square root of numbers which are perfect square. This time the terms are now similar. Hence you can already combine them. 10 2 2 2 3 2 (10 2 3) 2 9 2 , simplify. 5. Simplify: Simplify the radicand by finding a factor 2 x3 8 3x 53 27 x 3 3x which are perfect cube. 2 x( 2)3 3 x 5(3 x)3 3 x Extract the cube roots 4 x 3 3 x 15 x 3 3 x 19 x 3 3 x Simplify and combine similar terms. Multiplication of Radicals There are three cases to be considered in multiplying radicals. 1. Multiplying radicals of the same order or index. Apply the rule n a n b n ab Examples: A. Find the product of 3 5 2 Solution: 3 5 4 3(1) 5 2 , multiply the coefficients and the radicands. = 3 10 B. Find the product of 43 2 33 4 Solution: 43 2 33 4 (4)(3)3 2 4 123 8 . Multiply both coefficients and both radicands. = (-12)(2), extract cube root of 8 = -24 C. Find the product of 32 xy 3 18 x 2 Solution: Simplify first the radicand. 32 xy 3 18 x 2 16 y 2 2 xy = 4y 2 xy 3 x 9x2 2 2 4 y (3 x ) Multiply the coefficients and radicands . = 12xy Extract the square root of 4. = 12xy(2) Simplify PIVOT 4A CALABARZON Math G9 = 24xy 26 2 xy ( 2) 4 xy xy xy 2. Multiplying Radicals with different indices but same radicands. If the radicands are the same but different indices, you have to change them into similar index. Examples: 33 3 1. Find the product of 1 1 3 3 3 3 2 33 , change to exponential form Solution: 1 1 3 = 32 = 3 , apply the law of exponent 3 2 6 6 , change the exponential fraction to similar fractions 5 = 3 6 , add the fractions = = 6 2. What is the product of Solution: 3 2 4 4 35 , change to radical form 243 , find the power of 35 6 3 3 2 4 4 2 .4 2 2 , rename 4 to make the radicands the same 1 2 23 24 , change to exponential form 1 1 2 = 2 3 2 , reduce to lowest term and apply the law = 4 of exponent. = 2 3 26 6 5 2 6 , change the exponential fractions to similar fractions, then add. = 6 25 , change to radical form = 6 32 , the power of 25 Binomials can be multiplied using distributive property or applying the rule of special products of algebraic expressions or binomials. 3. Find the product and simplify: 3 2 4 2 Solution: 3 2 4 2 3 3 2 4 2 3 2 3 3 , apply distributive property = 3(4) 2 2 3(1) 2 3 , multiply both coefficients and both radicands = 12(2) - 3 6 = 24 - 3 6 , simplify 4. Find the difference of the sum and difference of radical binomials 2 3 3 2 2 3 3 2 Solution: Apply the rule in multiplying sum and difference of binomials. 2 3 3 2 2 3 3 2 2 3 3 2 2 2 = 4(3) - 9(2) = 12 - 18 = - 6 27 PIVOT 4A CALABARZON Math G9 Dividing Radicals If the nth root of a number is divided by the nth root of another number, then the result is the nth root of the quotient. p q n n n p q , where q≠0 Dividing radicals with the same indices Examples: 3 16 2 16 2 1. Find the quotient: 3 Solution: 2. Divide 2 16 2 8 by Solution: 2 3 3 8 2 , find the quotient of 16 and 2. the quotient is 8 which is a perfect cube. Hence the cube root of 8 is 2. 5 8 5 8 5 , rationalize the denominator by a number that will 5 5 make the denominator a perfect square. 40 25 = , multiply 4 10 25 , simplify = 2 10 5 3 6 3. Find the quotient 6 3 = Solution: Change first to same index. 3 6 1 = = 4. Divide 8 by 2 6 3 1 36 6 3 6 62 (3 2) 2 6 3 3 3 2 3 2 6 6 6 1 36 2 , simplify 6 3 4 6 12 , find the quotient then simplify 3 3 Solution: To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator. 8 3 3 3 3 3 3 = 8(3 3 ) (3 3)(3 3 ) 8(3 3 ) 8(3 3 ) 4 (3 3 ) 93 6 3 Multiply sum and difference of binomial radicals them simplify. PIVOT 4A CALABARZON Math G9 28 E Learning Task 1 A. Make each pair of radicals similar by reducing the radicand. 1. 2. 2, 3 3. 4. 5. 3 , 3 24 6. 48 , 12 7. 28 , 63 125 8. 4 32 , 4 162 3 , 5 96 9. 3 40 , 3 135 45 , 5 8 27 , 75 10. 8 , 200 B. Referring to each pair in letter A, do the following. A. Find the sum of each pair of radicals. B. Subtract the first radical from the second radical. C. Find the product of each pair of radicals D. Divide the second radical by the first radical. A Learning Task 2. A. 1. What is/are the rule(s) in adding or subtracting radicals? 2. What is/are the rule(s) in multiplying radicals? 3. How do you divide or rationalize radicals? 4. How do you rationalize if the denominator is a binomial radicals. B. Perform the indicated operations. Express your answer in simplest radical form. 1. 11. 9 25 12. 2 4 8 2. 3. 4. 5. 13. 33 2 3 3 6. 7. 5 14. 5 2 8. 9. 15. 3 4 5x 2 4 10. 29 3x 3 PIVOT 4A CALABARZON Math G9 WEEK 8 Radical Equations I Lesson A radical equation is an equation whose unknown quantity is in the radicand. Below are examples of radical equations and not radical equations. Radical Equations Not Radical Equations 3 x 5 20 1. 1. x 3 5 20 3 3 2. x 3 2 3 2x 3 x 2 2. Variable x is not part of radicand. Some variable x are found in the radicand. In this lesson you have to solve radical equations. Hence you have to assume that if two numbers are equal, then the square, cube or n th power are also n equal. If x = y, then xn = yn or if n x y then n x y n and x = y n . D Solving Radical Equations In solving radical equations, you have to follow the following steps: 1. Write the equation such that the radical containing the unknown is on one side of the equation. 2. Combine similar terms. 3. Raise both sides of the equation to a power same as the index of the radical. The equation should be free of radical to complete the solution. 4. Check if the value or values obtained will make the original equation true. Examples: 1. Solve for the value of x in x 7 Solution: x 2 72 Since the index is 2, raise both sides of the equation to power of 2. x = 49 Check: x 7 49 7 7 = 7 , the square root of 49 is 7. PIVOT 4A CALABARZON Math G9 30 3 2. Solve: x6 3 Solution: x6 3 3 33 , cube both sides of the equation since the index of the radical is 3 x - 6 = 27 x = 27 + 6, by APE combine similar terms = 33 Check: Substitute the value of x to the original equation 33 6 3 3 3 27 3 3 = 3 , the cube root of 27 is 3 3. Solve : 4 x2 x Solution: x2 x4 x2 2 , by APE , radical should be in one side of the equation x 4 2 , square both sides of the equation since the index of the radical is 2. x - 2 = x2 - 8x + 16, square of a binomial x 2 9 x 18 0 , combine similar terms x 6( x 3) 0 , solve quadratic equation by factoring. x - 6 = 0 or x - 3 = 0 , by zero product property x = 6 or x=3 Check: For x = 6 4 For x = 3 x2 x 4 62 6 4 4 x2 x 4 32 3 4 4 6 13 4 1 3 426 6= 6 5 ≠ 3 Only 6 makes the equation true. Hence, 6 is the solution of the equation. 3 is an extraneous root of the quadratic equation. 31 PIVOT 4A CALABARZON Math G9 4. The square root of the sum a number and 9 is 6. Find the number. Solution: Let x = the number x + 9 = the sum of the number and 9 x 9 6 , is the equation x9 2 2 6 , square both sides of the equation since the index is 2 x + 9 = 36 ,simplify x = 36 - 9, by APE x = 27 Check: x9 6 27 9 6, replace x by its value 36 6 6= 6 Common problem involving radical is the solution of right triangle applying the Pythagorean Theorem. This theorem is about the relationships among the sides of a right triangle. The theorem states the “sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Thus, if a and b are the legs and c is the hypotenuse of the right triangle, then c 2 a 2 b 2 or c a2 b2 Examples: 1. A man walks 8 meters to the east ,then turn south and walk 6 meters. How far is he from his starting point? Solution: Visualize the problem and sketch the figure. Let A be the starting point , B the ending point and C the turning point. The distance of the man from the starting point is the hypotenuse of the right triangle. c a2 b2 c 6 2 82 c 36 64 c 100 c = 10 The distance of the man from his starting point is 10 meter. PIVOT 4A CALABARZON Math G9 32 2. A ladder 12 ft tall leans against the wall of a building. The ladder touches a point of the wall 8 ft from the ground. How far is the foot of the ladder from the building? Solution: Sketch the figure. Let l = 12, length of the ladder which becomes the hypotenuse of the right triangle. h = 6, distance of the top of the ladder from the foot of the build ing. d = ? , distance of the foor of the ladder from the foot of the building. l 2 h2 d 2 d 2 l 2 h2 d by APE l 2 h2 d 122 62 d 144 36 d 108 d 36 3 6 3 The distance of the distance of the foot of the ladder from the foot of the building is 6 3 ft. 3. The area of a square lot is 162 m2 . How long is the side of the square? Solution: A = s2 , the area of a square 162 = s2 , substitute the given are to the formula. s= s= 162 , solving for s. 81 2 s9 2 , simplify the radicand by finding a factor that is a perfect square m, the side of the square 4. The diagonal of the rectangle is 5 x . If the width of the rectangle is x and the length is twice the width, what is the dimension of the rectangle. Solution: Using the Pythagorean theorem 5 c2 a 2 b2 5 x x 2 2 x x 2 25 x 4 x 2 x 2 2 The width is 5 units, length is 2(5) = 10 units and the diagonal is 5 5 units. 25 x 5 x 2 By MPE, 25 x 5x2 5x 5x x= 5 33 PIVOT 4A CALABARZON Math G9 E Learning Task 1: Solve the following equations. x 9 1. 3x 6 12. 3. 4 x 6 13. 4. 7 x 12 5. 4 2. 3 3x 5 8 2 x 2 x 10 11. x 2 144 0 x x 2 3 3x 3x 1 4 x 14. 2 15. 2 x 1 x 5 0 3x 2 5 6. 7. 5 x 5 7 8. 3 x 33 7 9. 84 x 7 33 x 12 10-. A Learning Task 2: Solve the following problems. 1. Twice the square root of a number is 12. Find the number. 2. The square root of a number increased by 9 is 27. find the number. 3. The square root of the sum of twice a number and 3 is 6. Find the number. 4. The perimeter of the square is 25 and the side is x 3 . Find the number. 5. The circumference of the circle( C = 2πr) is 24 cm and the radius is Find x and its radius. x2 6. Find the side of the square if the area is 64m 2 . 7. The shorter l6g of a right triangle is one half of the longer leg and the hypote nuse is 3 cm. How long is each leg of the triangle? 8. Five times the square root of a number is five. Find the number? 9. The cube root of a number decreased by 3 is zero. Find the number. 10. The square root of the cube root of a number is 2. find the number. PIVOT 4A CALABARZON Math G9 34 5. 5.a b 12 3 4 432 4. 3. 6 n12 2. 3b2 5. 4. 5 5 4 3. a 1. 2. 5 A. 1. Learning Task 1 3 5 5 52 xy 2 16 2. B. 1. 2 35 1 6. 1284 5. 4. 3. 3 x 125 Learning Task 2 1 3 PIVOT 4A CALABARZON Math G9 5. 4. 2a 1 84 3 3. 1 4 11x 2 8 4m 5 2. 1 7 B. 1. 5 3 m2 y 2 p5 2 2 a b 2 5 x q3 p 10 m n 6 5 10. 9q r 6 9. 2x2 8. 6 a 2a 7. x6 y3 6. 5 3 m3n 2 Learning Task 3 Week 4 Week 3 Learning Task 1 1. Learning Task 2 6. 4x3 1 7. 49 4 2. 9. 3. 11. 49 1 4 16 1 5 1. 13. 14. a5n 7. 3. 5 8. 27 4. 15. 32 11. –64 6. 2. 11 12. 9x 2 8. 4. 10. 100 12 1 b3 12a 4 64 1 9 25 1 ab 9 5 y 9. 6 5. 1 12. c5 10a 7 b 6 40 3 1000000 1 5. 10. 7 1 13. 18 10 9 1 y5 z 2 1 14. 15. Week 1 and 2 Learning Task 1 Learning Task 2 A. A. y x Learning Task 3 z p 48 q q 48 k 10 2. w z x 6 k k Equation 3 2 2. a. B. 21.5 y x k Equation 30 9 b. True C. True y = 6x y = 10x 6 3 Learning Task 4 12 1. 4.3 ft v 2 3 z 10vw z 6vw 50 d. False 2 E. True 8 y 68.75% 2. Joint y y = 2xz 6 2 80 4. y p= =4xz 3q 2 48 B. y Equation 3. a. B. It will increase 4 times 4. a. False y B. A. 1.Direct Equation 4. Combined 3. Combined 5. Combined2 B. 1. q = 7p2 3. 192 p V kT 2 p t 3 P 7z 7 y 5vw 5 5 18 96 x 36 x y 30 y 2 5 6 5 24 5. k = π, Volume will in crease by 5 168 Answer Key Week 5-6 Learning Task 1 Learning Task 2 A. Week 7 Learning Task 2 Learning Task 1 PIVOT 4A CALABARZON Math G9 36 Week 8 Learning Task 2 Learning Task 1 References 37 PIVOT 4A CALABARZON Math G9 Personal Assessment on Learner’s Level of Performance Using the symbols below, choose one which best describes your experience in working on each given task. Draw it in the column for Level of Performance (LP). Be guided by the descriptions below. - I was able to do/perform the task without any difficulty. The task helped me in understanding the target content/lesson. - I was able to do/perform the task. It was quite challenging but it still helped me in understanding the target content/lesson. - I was not able to do/perform the task. It was extremely difficult. I need additional enrichment activities to be able to do/perform this task. Distribution of Learning Tasks Per Week for Quarter 2 Week 1 LP Learning Task 1 Week 2 LP Learning Task 1 Week 3 LP Learning Task 1 Week 4 Learning Task 1 Learning Task 2 Learning Task 2 Learning Task 2 Learning Task 2 Learning Task 3 Learning Task 3 Learning Task 3 Learning Task 3 Learning Task 4 Learning Task 4 Learning Task 4 Learning Task 4 Learning Task 5 Learning Task 5 Learning Task 5 Learning Task 5 Learning Task 6 Learning Task 6 Learning Task 6 Learning Task 6 Learning Task 7 Learning Task 7 Learning Task 7 Learning Task 7 Learning Task 8 Learning Task 8 Learning Task 8 Learning Task 8 Week 5 Learning Task 1 LP Week 6 Learning Task 1 Week 7 LP Learning Task 1 LP LP Week 8 LP Learning Task 1 Learning Task 2 Learning Task 2 Learning Task 2 Learning Task 2 Learning Task 3 Learning Task 3 Learning Task 3 Learning Task 3 Learning Task 4 Learning Task 4 Learning Task 4 Learning Task 4 Learning Task 5 Learning Task 5 Learning Task 5 Learning Task 5 Learning Task 6 Learning Task 6 Learning Task 6 Learning Task 6 Learning Task 7 Learning Task 7 Learning Task 7 Learning Task 7 Learning Task 8 Learning Task 8 Learning Task 8 Learning Task 8 Note: If the lesson is designed for two or more weeks as shown in the eartag, just copy your personal evaluation indicated in the first Level of Performance in the second column up to the succeeding columns, i.e. if the lesson is designed for weeks 4-6, just copy your personal evaluation indicated in the LP column for week 4, week 5 and week 6. PIVOT 4A CALABARZON Math G9 38 PIVOT 4A CALABARZON For inquiries or feedback, please write or call: Department of Education Region 4A CALABARZON Office Address: Gate 2, Karangalan Village, Cainta, Rizal Landline: 02-8682-5773, locals 420/421 Email Address: lrmd.calabarzon@deped.gov.ph