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  1. 1. NEXTGENERATION I MATH (Textbook)
  2. 2. Next Generation Math ITextbookPhilippine Copyright 2011 by DIWA LEARNING SYSTEMS INCAll rights reserved. Printed in the PhilippinesEditorial, design, and layout by University Press of First AsiaNo part of this publication may be reproduced or transmitted in any form or by any means electronic ormechanical, including photocopying, recording, or any information storage and retrieval systems, withoutpermission in writing from the copyright owner. Exclusively distributed by DIWA LEARNING SYSTEMS INC 4/F SEDCCO 1 Bldg. 120 Thailand corner Legazpi Streets Legaspi Village, 1229 Makati City, Philippines Tel. No.: (632) 893-8501 * Fax: (632) 817-8700 ISBN 978-971-46-0180-2AuthorsMariel T. Atregenio earned her master’s degree in Mathematics and bachelor’s degree in Secondary Educationmajor in Mathematics, magna cum laude, from the University of the Philippines (UP)–Diliman. She has taughtMathematics at Malayan High School of Science, Claret School of Quezon City, and Francisco Homes College. Incollege, Ms. Atregenio was a presidential scholar, a UP College of Education Academic Excellence awardee, and aconsistent recipient of scholarship awards.Katrina Grace Q. Sumagit earned her master’s degree in Mathematics and bachelor’s degree in SecondaryEducation major in Mathematics and minor in Statistics from UP–Diliman. She is taking up her doctorate inMathematics also at UP–Diliman. She taught grade school and high school Mathematics at the University of thePhilippines Integrated School (UPIS) from 2006 to 2008. Ms. Sumagit has conducted seminar-workshops for highschool and grade school Mathematics teachers for Gurong Pahinungod (GP) and Ateneo Center for EducationalDevelopment (ACED). Currently, she is a full-time faculty at the Ateneo de Manila Grade School.Consultant-ReviewerLorelei B. Ladao-Saren obtained her master’s degree in Mathematics, with high distinction, from the De La SalleUniversity (DLSU)–Dasmariñas and her bachelor’s degree in Statistics from UP–Diliman. She is presently pursuingher doctorate in Mathematics Education at the Philippine Normal University. Ms. Ladao-Saren was a formerdirector for Research, Publication, and Community Extension Services at World Citi Colleges. She has also taughtMathematics at Asia Pacific College, Southville Foreign University, and DLSU–Dasmariñas. She currently teachesMathematics at DLSU–College of St. Benilde and at the Graduate School of Rizal Technological University.
  3. 3. Preface The Next Generation Math series covers topics and competencies that are aligned withthe Basic Education Curriculum (BEC) and the Engineering and Science Education Program(ESEP) of the Department of Education. It is composed of different mathematics disciplines:elementary algebra in first year; intermediate algebra in second year; geometry in third year;and advanced algebra, trigonometry, statistics, and calculus in fourth year. It tries to covernumerous important topics that will satisfy the needs of different groups of learners. The series supports the constructivist approach to teaching and learning process.Lessons are presented through meaningful activities which are designed to provide you anopportunity to make different connections between concrete situations and mathematics. Theactivities are designed to develop your skills in problem solving, critical thinking, decisionmaking, and creative thinking through exchange of ideas and your own discovery. Each bookin this series provides opportunities for you to discuss, explore, and construct mathematicalideas and interpret new information and knowledge at a different perspective. You willalso be able to structure and evaluate your own conjectures and apply previously acquiredknowledge and skills. The series has the following salient features: • Lessons are inquiry based, enriched with applicable technologies, and integrated with science and real-life applications. • Emphasis on the development of higher-order thinking skills is evident in the illustrative examples and exercises provided in every lesson. To enhance your mathematics skills, the degree of difficulty of the problems ranges from simple to more challenging ones. • Exercises include research work to emphasize the importance of research as a tool in satisfying the quest for knowledge and acquiring valuable insights about certain topics. • Historical notes, application of mathematical ideas in future careers, and pieces of trivia are presented in each chapter. It is with a sincere desire to provide a useful tool in enhancing appreciation and betterunderstanding of mathematics that the Next Generation Math series was conceptualized.
  4. 4. Table of ContentsUnit I Number Theory, Real Number System, and MeasurementChapter 1 Whole NumbersLesson 1 Addition and Subtraction of Whole Numbers ................................................ 2Lesson 2 Multiplication and Division of Whole Numbers .............................................. 8Lesson 3 Order of Operations ..................................................................................... 13IT Matters ................................................................................................................... 18Chapter 2 Number TheoryLesson 1 Divisibility Rules ........................................................................................ 20Lesson 2 Prime and Composite Numbers, and Prime Factorization ............................ 24Lesson 3 Greatest Common Factor and Least Common Multiple ................................ 28IT Matters ................................................................................................................... 32Chapter 3 SetsLesson 1 Set Notations and Kinds of Sets .................................................................. 34Lesson 2 Set Relations ............................................................................................... 38Lesson 3 Set Operations ............................................................................................ 43IT Matters ................................................................................................................... 50Chapter 4 IntegersLesson 1 The Signed Numbers .................................................................................. 51Lesson 2 Addition and Subtraction of Integers .......................................................... 57Lesson 3 Multiplication and Division of Integers ........................................................ 63IT Matters ................................................................................................................... 68Chapter 5 Rational and Irrational NumbersLesson 1 Squares and Square Roots .......................................................................... 70Lesson 2 Rational and Irrational Numbers ................................................................ 74Lesson 3 Addition and Subtraction of Rational Numbers ........................................... 78Lesson 4 Multiplication and Division of Rational Numbers ........................................ 84IT Matters ................................................................................................................... 91
  5. 5. Chapter 6 MeasurementLesson 1 Units of Length, Mass, and Capacity .......................................................... 93Lesson 2 Units of Time and Temperature ................................................................ 104Lesson 3 Angles ...................................................................................................... 109IT Matters ......................................................................................................................... 114Unit II Algebraic ExpressionsChapter 7 Fundamentals of Algebraic ExpressionsLesson 1 Simplifying Numerical Expressions ........................................................... 116Lesson 2 Evaluating and Simplifying Algebraic Expressions .................................... 122Lesson 3 Translating Verbal Phrases into Algebraic Expressions ............................. 128Lesson 4 Exponents and Scientific Notation ............................................................ 134IT Matters ................................................................................................................. 141Chapter 8 PolynomialsLesson 1 Introduction to Polynomials ...................................................................... 144Lesson 2 Adding and Subtracting Polynomials ........................................................ 150Lesson 3 Multiplying Polynomials ........................................................................... 155Lesson 4 Factoring Polynomials ............................................................................... 161Lesson 5 Dividing Polynomials ................................................................................ 168IT Matters ................................................................................................................. 178Chapter 9 Rational ExpressionsLesson 1 Simplifying Rational Expressions............................................................... 179Lesson 2 Multiplying and Dividing Rational Expressions .......................................... 185Lesson 3 Adding and Subtracting Rational Expressions ........................................... 192IT Matters ................................................................................................................. 200Unit III Equations and InequalitiesChapter 10 Linear Equations in One Variable and Literal EquationsLesson 1 Solving Linear Equations in One Variable ................................................. 203Lesson 2 Solving Linear Equations in One Variable Involving Absolute Value .......... 212Lesson 3 Solving Literal Equations .......................................................................... 219IT Matters ................................................................................................................. 224Chapter 11 Linear Inequalities in One VariableLesson 1 Solving Linear Inequalities in One Variable ............................................... 225Lesson 2 Solving Linear Inequalities in One Variable Involving Absolute Value ........ 233IT Matters ................................................................................................................. 240
  6. 6. Unit IV Systems of Linear Equations and Linear Inequalities in Two VariablesChapter 12 Linear Equations and Their GraphsLesson 1 The Cartesian Plane ................................................................................. 243Lesson 2 Graphing Linear Equations ...................................................................... 253Lesson 3 Point-slope and Slope-intercept Forms of Equations of Lines .................... 269Lesson 4 Parallel and Perpendicular Lines .............................................................. 278IT Matters ................................................................................................................. 284Chapter 13 Systems of Linear Equations and Linear Inequalities in Two VariablesLesson 1 Solving Systems of Linear Equations in Two Variables by Graphing ........... 285Lesson 2 Solving Systems of Linear Equations in Two Variables by Substitution ..... 292Lesson 3 Solving Systems of Linear Equations in Two Variables by the Addition or Elimination Method .............................................................................. 299Lesson 4 Graphing Linear Inequalities in Two Variables ........................................... 307IT Matters ................................................................................................................. 318Glossary ................................................................................................................. 321Bibliography ................................................................................................................. 326Index ................................................................................................................. 329
  7. 7. Number Theory, Unit I Real Number System,and Measurement Mathematics is a fi eld of study that deals with numbers and symbols. It is an essential part of a person’s everyday life experiences. As you progress in your study of mathematics, you will fi nd out and appreciate its many useful applications in your life. In this unit, you will learn about the real number system and measurement. This unit is divided into six chapters. In Chapter 1, you will study about whole numbers and review the basic mathematical operations on whole numbers. Number theory, which includes divisibility rules, prime numbers, composite numbers, prime factorization, greatest common factor, and least common multiple, will be tackled in Chapter 2. Chapter 3 includes lessons on sets and set notations, kinds of sets, set relations, and set operations. Chapter 4 is about integers where you will learn more about the signed numbers and the basic mathematical operations involving integers. Chapter 5 discusses rational and irrational numbers. Topics, such as squares, square roots, and operations involving rational and irrational numbers will be covered in this chapter.In Chapter 6, you will learn about measurement and angles.
  8. 8. Chapter 1 WHOLE NUMBERS Learning Objectives • Add, subtract, multiply, and divide whole numbers • Identify, illustrate, and use the basic properties of addition and multiplication of whole numbers • Interpret graphs and tables • Perform series of operations • Illustrate word problems • Solve word problems involving the four basic operations • Use a calculator to solve word problems Lesson 1 Addition and Subtraction of Whole Numbers Power Up The graph below shows the electric bills of the Reyes family for six months. Study the graph and answer the questions on the next page. Reyes Family’s Electric Bills 3 000 2 500 Electric Bill (pesos) 2 000 1 500 1 000 500 0 January February March April May June Month2 Next Generation Math I
  9. 9. 1. How much is the total electric bill of the family for six months? 2. How much less is the electric bill of the family in May than in April? 3. How much more is the electric bill of the family for the last three months compared to their electric bill for the first three months? Walk Through• Addition is the process of combining two or more numbers. The numbers that are added are called addends. The result or answer in addition is called the sum or total.• Basic properties of addition of whole numbers 1. Closure Property: The sum of two or more whole numbers is also a whole number. For any whole numbers x, y, and z: x + y = z. 2. Commutative Property: The order of the addends does not change the sum. For any whole numbers x and y: x + y = y + x. 3. Associative Property: The way the addends are grouped does not change the sum. For any whole numbers x, y, and z: (x + y) + z = x + (y + z). 4. Identity Property: The sum of any nonzero whole number and zero is always that non zero number. Zero (0) is the identity element for addition. For any nonzero whole number x: x + 0 = x.• Subtraction is finding the missing addends. The number from which you subtract is called the subtrahend. The number that is subtracted from the subtrahend is called the minuend. The result or answer in subtraction is called the difference.Example 1: Find the missing number n. a. 1 357 + 2 064 + 3 809 = n b. 2 706 – 1 354 = n c. 2 509 + 1 346 + n = 4 807 d. 7 500 – n = 6 453 e. n – 3 029 = 1 579Solution: a. 1 357 b. 2 706 c. 2 509 4 807 2 064 − 1 354 + 1 346 − 3 855 + 3 809 1 352 3 855 952 7 230 n = 1 352 n = 952 n = 7 230 Number Theory, Real Number System, and Measurement 3
  10. 10. d. 7 500 e. 1 579 − 6 453 + 3 029 1 047 4 608 n = 1 047 n = 4 608Example 2: Find the missing number n and identify the basic property of addition of whole numbers used. a. 1 305 + 2 604 + 3 987 = 1 305 + n + 2 604 b. 1 732 + 0 = n c. (2 048 + 3 609) + 4 507 = n + (3 609 + 4 507) d. (5 902 + 4 302) + 3 018 = 3 018 + (5 902 + n) e. 1 407 + 1 302 = n + 1 407 + 1 302 Solution: a. n = 3 987; Commutative Property b. n = 1 732; Identity Property c. n = 2 048; Associative Property d. n = 4 302; Commutative Property e. n = 0; Identity PropertyExample 3: Use a calculator to solve each problem. a. Mrs. Martinez sold 145 oranges on Monday, 102 oranges on Tuesday, and 213 oranges on Wednesday. How many oranges did she sell altogether for three days? b. John had 100 collectible cards. He gave 23 cards to Mark and 32 cards to Luke. He then sold the rest of his collectible cards. How many collectible cards did he sell?Solution: a. Let x total number of oranges sold for three days. = x = 145 + 102 + 213 = 460 Mrs. Martinez sold 460 oranges in all for three days. b. Let y = number of collectible cards sold. y = 100 – (23 + 32) = 100 – 55 = 45 John sold 45 collectible cards. Next Generation Math I
  11. 11. Example 4: Use a calculator to solve the problem. There are 5 318 nonfiction books in the library. There are 2 405 fewer fiction books than nonfiction books. How many nonfiction and fiction books are there in the library?Solution: Let z = number of fiction books in the library. z = 5 318 – 2 405 2 913 = There are 2 913 fiction books in the library. Let w = total number of nonfiction and fiction books in the library. w = 5 318 + 2 913 = 8 231 There are 8 231 nonfiction and fiction books in the library.Example 5: Katrina has 40 stamps. Grace has 18 more stamps than Katrina. Mae has 12 more stamps than Grace. How many stamps do they have altogether?Solution: Let x = number of stamps Grace has. x = 40 + 18 = 58 Grace has 58 stamps. Let y = number of stamps Mae has. y = 58 + 12 = 70 Mae has 70 stamps. Let z total number of stamps Katrina, Grace, and Mae have. = z 40 + 58 + 70 = = 168 Katrina, Grace, and Mae have 168 stamps altogether. Number Theory, Real Number System, and Measurement
  12. 12. Move UpI. Encircle the letter that corresponds to the correct value of n. 1. 1 006 + 2 090 + 3 135 = n a. 3 096 c. 5 225 b. 4 141 d. 6 231 2. 3 901 – 2 046 – 1 308 = n a. 547 c. 1 855 b. 738 d. 2 593 3. 4 203 − 3 013 + 2 192 = n a. 1 190 c. 5 024 b. 3 382 d. 7 216 4. 5 310 + 2 345 – 3 875 = n a. 2 965 c. 7 655 b. 3 780 d. 11 530 5. 6 385 – (4 784 – 2 129) = n a. 1 601 c. 3 730 b. 2 655 d. 4 256 6. 6 724 – (3 384 + 1 120) = n a. 2 220 c. 4 504 b. 3 340 d. 5 604 7. 2 189 + n = 4 305 a. 2 016 c. 6 484 b. 2 116 d. 6 494 8. n + 3 709 + 1 247 = 9 656 a. 4 700 c. 5 700 b. 4 956 d. 5 947 9. 4 142 – n = 2 984 a. 1 058 c. 7 126 b. 1 158 d. 7 0266 Next Generation Math I
  13. 13. 10. n – 3 792 = 2 071 a. 1 721 c. 5 863 b. 2 721 d. 6 863II. Identify the basic property of addition used in each equation. 1. 1 286 + 714 = 714 + 1 286 2. (579 + 312) + 608 = 579 + (312 + 608) 3. 459 + 367 + 0 = 459 + 367 4. 802 + (0 + 315) = 802 + 315 5. (3 218 + 3 201) + 7 042 = 7 042 + (3 218 + 3 201) 6. 2 907 + (1 704 + 3 416) = (2 907 + 1 704) + 3 416 7. 0 + 2 379 = 2 379 8. (3 560 + 0) + 5 077 = (0 + 3 560) + 5 077 9. 1 025 + 7 035 + 0 = 1 025 + 0 + 7 035 10. (1 306 + 4 509) + (2 064 + 0) = (1 306 + 4 509) + 2 064 III. Read and analyze each problem carefully. Use a calculator to solve each problem. 1. Manned Apollo flights to the moon used the giant Saturn V rocket. The rocket had three stages. The first stage produced 7 500 000 pounds (lb) of thrust (power), the second stage produced 6 500 000 lb of thrust less than the first stage, and the third stage produced 800 000 lb of thrust less than the second stage. How many pounds of thrust was produced in all? 2. Mrs. Santiago baked 1 500 muffins. She sold 360 muffins on Monday. She sold 143 more muffins on Tuesday than on Monday and 132 fewer muffins on Wednesday than on Tuesday. How many muffins did she have left? 3. Raphael, George, and Manuel have a total of 1 000 ice cream sticks. Raphael has 432 ice cream sticks. George has 124 fewer ice cream sticks than Raphael. How many ice cream sticks does Manuel have? Number Theory, Real Number System, and Measurement

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