Here are the steps to solve the example problems: 1) How much is the total electric bill of the family for six months? - January: P1,500 - February: P2,000 - March: P2,500 - April: P1,000 - May: P500 - June: P1,000 - To find the total, add all the bills: P1,500 + P2,000 + P2,500 + P1,000 + P500 + P1,000 = P9,500 2) How much less is the electric bill of the family in May than in April? - April bill: P1,000

1. NEXT
GENERATION I
MATH
(Textbook)

2. Next Generation Math I
Textbook
Philippine Copyright 2011 by DIWA LEARNING SYSTEMS INC
All rights reserved. Printed in the Philippines
Editorial, design, and layout by University Press of First Asia
No part of this publication may be reproduced or transmitted in any form or by any means electronic or
mechanical, including photocopying, recording, or any information storage and retrieval systems, without
permission 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 978-971-46-0180-2
Authors
Mariel T. Atregenio earned her master’s degree in Mathematics and bachelor’s degree in Secondary Education
major in Mathematics, magna cum laude, from the University of the Philippines (UP)–Diliman. She has taught
Mathematics at Malayan High School of Science, Claret School of Quezon City, and Francisco Homes College. In
college, Ms. Atregenio was a presidential scholar, a UP College of Education Academic Excellence awardee, and a
consistent recipient of scholarship awards.
Katrina Grace Q. Sumagit earned her master’s degree in Mathematics and bachelor’s degree in Secondary
Education major in Mathematics and minor in Statistics from UP–Diliman. She is taking up her doctorate in
Mathematics also at UP–Diliman. She taught grade school and high school Mathematics at the University of the
Philippines Integrated School (UPIS) from 2006 to 2008. Ms. Sumagit has conducted seminar-workshops for high
school and grade school Mathematics teachers for Gurong Pahinungod (GP) and Ateneo Center for Educational
Development (ACED). Currently, she is a full-time faculty at the Ateneo de Manila Grade School.
Consultant-Reviewer
Lorelei B. Ladao-Saren obtained her master’s degree in Mathematics, with high distinction, from the De La Salle
University (DLSU)–Dasmariñas and her bachelor’s degree in Statistics from UP–Diliman. She is presently pursuing
her doctorate in Mathematics Education at the Philippine Normal University. Ms. Ladao-Saren was a former
director for Research, Publication, and Community Extension Services at World Citi Colleges. She has also taught
Mathematics at Asia Pacific College, Southville Foreign University, and DLSU–Dasmariñas. She currently teaches
Mathematics at DLSU–College of St. Benilde and at the Graduate School of Rizal Technological University.

3. Preface
The Next Generation Math series covers topics and competencies that are aligned with
the 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 cover
numerous 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 an
opportunity to make different connections between concrete situations and mathematics. The
activities are designed to develop your skills in problem solving, critical thinking, decision
making, and creative thinking through exchange of ideas and your own discovery. Each book
in this series provides opportunities for you to discuss, explore, and construct mathematical
ideas and interpret new information and knowledge at a different perspective. You will
also be able to structure and evaluate your own conjectures and apply previously acquired
knowledge 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 better
understanding of mathematics that the Next Generation Math series was conceptualized.

4. Table of Contents
Unit I Number Theory, Real Number System,
and Measurement
Chapter 1 Whole Numbers
Lesson 1 Addition and Subtraction of Whole Numbers ................................................ 2
Lesson 2 Multiplication and Division of Whole Numbers .............................................. 8
Lesson 3 Order of Operations ..................................................................................... 13
IT Matters ................................................................................................................... 18
Chapter 2 Number Theory
Lesson 1 Divisibility Rules ........................................................................................ 20
Lesson 2 Prime and Composite Numbers, and Prime Factorization ............................ 24
Lesson 3 Greatest Common Factor and Least Common Multiple ................................ 28
IT Matters ................................................................................................................... 32
Chapter 3 Sets
Lesson 1 Set Notations and Kinds of Sets .................................................................. 34
Lesson 2 Set Relations ............................................................................................... 38
Lesson 3 Set Operations ............................................................................................ 43
IT Matters ................................................................................................................... 50
Chapter 4 Integers
Lesson 1 The Signed Numbers .................................................................................. 51
Lesson 2 Addition and Subtraction of Integers .......................................................... 57
Lesson 3 Multiplication and Division of Integers ........................................................ 63
IT Matters ................................................................................................................... 68
Chapter 5 Rational and Irrational Numbers
Lesson 1 Squares and Square Roots .......................................................................... 70
Lesson 2 Rational and Irrational Numbers ................................................................ 74
Lesson 3 Addition and Subtraction of Rational Numbers ........................................... 78
Lesson 4 Multiplication and Division of Rational Numbers ........................................ 84
IT Matters ................................................................................................................... 91

5. Chapter 6 Measurement
Lesson 1 Units of Length, Mass, and Capacity .......................................................... 93
Lesson 2 Units of Time and Temperature ................................................................ 104
Lesson 3 Angles ...................................................................................................... 109
IT Matters ......................................................................................................................... 114
Unit II Algebraic Expressions
Chapter 7 Fundamentals of Algebraic Expressions
Lesson 1 Simplifying Numerical Expressions ........................................................... 116
Lesson 2 Evaluating and Simplifying Algebraic Expressions .................................... 122
Lesson 3 Translating Verbal Phrases into Algebraic Expressions ............................. 128
Lesson 4 Exponents and Scientific Notation ............................................................ 134
IT Matters ................................................................................................................. 141
Chapter 8 Polynomials
Lesson 1 Introduction to Polynomials ...................................................................... 144
Lesson 2 Adding and Subtracting Polynomials ........................................................ 150
Lesson 3 Multiplying Polynomials ........................................................................... 155
Lesson 4 Factoring Polynomials ............................................................................... 161
Lesson 5 Dividing Polynomials ................................................................................ 168
IT Matters ................................................................................................................. 178
Chapter 9 Rational Expressions
Lesson 1 Simplifying Rational Expressions............................................................... 179
Lesson 2 Multiplying and Dividing Rational Expressions .......................................... 185
Lesson 3 Adding and Subtracting Rational Expressions ........................................... 192
IT Matters ................................................................................................................. 200
Unit III Equations and Inequalities
Chapter 10 Linear Equations in One Variable and Literal Equations
Lesson 1 Solving Linear Equations in One Variable ................................................. 203
Lesson 2 Solving Linear Equations in One Variable Involving Absolute Value .......... 212
Lesson 3 Solving Literal Equations .......................................................................... 219
IT Matters ................................................................................................................. 224
Chapter 11 Linear Inequalities in One Variable
Lesson 1 Solving Linear Inequalities in One Variable ............................................... 225
Lesson 2 Solving Linear Inequalities in One Variable Involving Absolute Value ........ 233
IT Matters ................................................................................................................. 240

6. Unit IV Systems of Linear Equations and Linear Inequalities
in Two Variables
Chapter 12 Linear Equations and Their Graphs
Lesson 1 The Cartesian Plane ................................................................................. 243
Lesson 2 Graphing Linear Equations ...................................................................... 253
Lesson 3 Point-slope and Slope-intercept Forms of Equations of Lines .................... 269
Lesson 4 Parallel and Perpendicular Lines .............................................................. 278
IT Matters ................................................................................................................. 284
Chapter 13 Systems of Linear Equations and Linear Inequalities in Two
Variables
Lesson 1 Solving Systems of Linear Equations in Two Variables by Graphing ........... 285
Lesson 2 Solving Systems of Linear Equations in Two Variables by Substitution ..... 292
Lesson 3 Solving Systems of Linear Equations in Two Variables by the Addition
or Elimination Method .............................................................................. 299
Lesson 4 Graphing Linear Inequalities in Two Variables ........................................... 307
IT Matters ................................................................................................................. 318
Glossary ................................................................................................................. 321
Bibliography ................................................................................................................. 326
Index ................................................................................................................. 329

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. 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
Month
2 Next Generation Math I

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 = 1 579
Solution:
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. d. 7 500 e. 1 579
− 6 453 + 3 029
1 047 4 608
n = 1 047 n = 4 608
Example 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 Property
Example 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. 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. Move Up
I. 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. 7 026
6 Next Generation Math I

13. 10. n – 3 792 = 2 071
a. 1 721 c. 5 863
b. 2 721 d. 6 863
II. 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