1. VMGOs MASTERING FUNDAMENTAL OPERATIONS AND INTEGERS A MODULAR WORKBOOK FOR 1ST YEAR HIGH SCHOOL ADRIEL G. ROMAN MYRICHEL ALVAREZ AUTHORS NOEL A. CASTRO MODULE CONSULTANT FOR-IAN V. SANDOVAL MODULE ADVISER

2. VISION . A premier university in CALABARZON, offering academic programs and related services designed to respond to the requirements of the Philippines and the global economy, particularly, Asian countries Title M G O Table of contents

5. OBJECTIVES OF BACHELOR OF SECONDARY EDUCATION (BSEd) V M G Produce graduates who can demonstrate and practice the professional and ethical requirements for the Bachelor of Secondary Education such as: 1. To serve as positive and powerful role models in the pursuit of learning thereby maintaining high regards to professional growth. 2. Focus on the significance of providing wholesome and desirable learning environment. 3. Facilitate learning process in diverse types of learners. 4. Use varied learning approaches and activities, instructional materials and learning resources. 5. Use assessment data, plan and revise teaching-learning plans. 6. Direct and strengthen the links between school and community activities. 7. Conduct research and development in Teacher Education and other related activities . Foreword

6. This teacher’s guide Visual Presentation Hand-out entitled: “MASTERING FUNDAMENTAL OPERATIONS AND INTEGERS (MODULAR WORKBOOK FOR 1st YEAR HIGH SCHOOL)” is part of the requirements in educational technology 2 under the revised Education curriculum based on CHEd Memorandum Order (CMO)-30, series of 2004. Educational technology 2 is a three (3) unit course designed to introduce both traditional and innovative technologies to facilitate and foster meaningful and effective learning where students are expected to demonstrate a sound understanding of the nature, application and production of the various types of educational technologies. FOREWORD Next VMGO Table of contents

8. The output of the group’s effort may serve as an educational research of the institution in providing effective and quality education. The lessons and evaluations presented in this module may also function as a supplementary reference for secondary teachers and students . FOR-IAN V. SANDOVAL Computer Instructor/ Adviser/Dean CAS NOEL A. CASTRO Engineer/Mathematics Instructor Back Preface

9. PREFACE This modular workbook entitled “Mastering Fundamental Operations and Integers (modular workbook for First Year High School)” aims you to become fluent in solving any mathematical expressions and problems. This instructional material will serve as your first step in entering to the world of high school mathematics. This modular workbook is divided into two units; the unit I consist of four chapters which pertains to the four basic operations in mathematics dealing with whole numbers and the unit II which pertains to the use of four fundamental operations in integers. In mastering the four fundamental operations, you will study the different parts of the four basic operations (addition, subtraction, division and multiplication), and their uses and the different shortcuts in using them. In this part, you will also learn on how to check one’s operation using their inverse operation. Foreword Next Table of contents

10. In the unit II, you may apply here all the knowledge that you have gained from the unit I. in this part, you may encounter several expressions where you need to use all the knowledge that you have gained from the unit I. you will also learn the nature of Integers, and also the Positive, Zero and Negative Integers. This instructional material was designed for you to have a further understanding about the four fundamental operations dealing with Whole Numbers and Integers. It was also designed for you to have a deep interest in exploring Mathematics. The authors feel that after finishing this lesson, you should be able to feel that EXPLORING MATHEMATICS IS INTERESTING AND FUN!!! THE AUTHORS Acknowledgement Back

11. ACKNOWLEDGEMENT The authors would like to give appreciation to the following: To Mr. For- Ian V. Sandoval , for his kind consideration and for his advice to make this instructional material more knowledgeable. To Mrs. Corazon San Agustin , for her guidance to finish this instructional modular workbook. To Prof. Lydia R. Chavez for her wonderful advice to make this instructional material becomes more knowledgeable. To Mrs. Evangeline Cruz for her kind consideration in allowing us to borrow books from the library. Back Next Table of contents

12. Table of Contents VMGOs Foreword Preface Acknowledgement Table of Contents UNIT I- MASTERING BASIC FUNDAMENTAL OPERATIONS CHAPTER 1- ADDITION OF WHOLE NUMBERS Lesson 1- What is Addition? Lesson 2- Properties of Addition Lesson 3- Mastering Skills in Adding Whole Numbers Lesson 4- Different Methods in Adding Whole Numbers Lesson 5- How to solve a word problem? Lesson 6- Application of addition of whole numbers: WORD PROBLEM CHAPTER 2- SUBTRACTION OF WHOLE NUMBERS Lesson 7- What is Subtraction? Lesson 8- Mastering Skills in Subtraction Lesson 9- Problem Solving Involving Subtraction of whole numbers Back Next

13. CHAPTER 3- MULTIPLICATION OF WHOLE NUMBERS Lesson 10- What is Multiplication? Lesson 11- Properties of Multiplication Lesson 12- Mastering Skills in Multiplying Whole Numbers Lesson 13- “The 99 Multiplier” Shortcut in multiplying whole numbers Lesson 14- “Power of Ten Multiplication” Shortcut In Multiplying Whole Numbers Lesson 15- Problem solving involving Multiplication of Whole Numbers CHAPTER 4- DIVISION OF WHOLE NUMBERS Lesson 16- What is Division? Lesson 17- Mastering Skills in Division of Whole Numbers Lesson 18- “Cancellation of Insignificant Zeros” Easy ways in Dividing Whole Numbers Lesson 19- Problem Solving Involving Division of Whole Numbers Back Next

14. UNIT II- INTEGERS CHAPTER 5- WORKING WITH INTEGERS Lesson 20- What is Integer? Lesson 21- Addition of Integers Lesson 22- Subtraction of Integers Lesson 23- Multiplication of Integer Lesson 24- Division of Integers Lesson 25- Punctuation and Precedence of Operation MATH AND TECHNOLOGY REFERENCES About the Authors Back Next

17. CHAPTER-I ADDITION OF WHOLE NUMBERS Introduction In this chapter, you will learn deeply the addition operation, the different parts of it, the different properties and the use of this operation in solving a word problem. This chapter will serve as your first step in mastering the basic fundamental operations for this chapter will discuss how to solve a word problem using systematic ways. All the information you need to MASTER THE FUNDAMENTAL OPERATIONS DEALING WITH WHOLE NUMBERS is provided in this chapter. Back Next Table of contents

22. This property states that changing the grouping of the addends does not affect or change the sum, that is, if a, b and c are any numbers, (a + b) = c = a + (b + c). Remember to work in the parenthesis first. Summary: The 0 Property in Addition If “ a” is any number, a + 0 = a. The Commutative Property of Addition If a + b = b + a , for any numbers a and b. The Associative Property of Addition If a, b and c are any numbers, ( a + b) = c = a + (b + c). Examples: 6 + 8 = 14 8 + 6 = 14 11 + 27 = 38 27 + 11 = 3 Examples: (4 + 3) + 8 = 4 + (3 = 8) = 15 9 + (8 + 6) = (9 + 8) + 6 = 23 Associative Property of Addition Next Back

23. Identify the properties of the following WORKSHEET NO. 2 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ 1. 265 + 547 = 547 + 265___________________________ 2. 85 + 78 = 78 + 85_______________________________ 3. 15 + 0 = 15____________________________________ 4. 3 + (5 + 9) = (3 + 5) + 9 =17______________________ 5. 31+ (21+15) = (31+21) +15 = 67___________________ 6. 59 + 0 = 59____________________________________ 7. 100 + 0 = 100__________________________________ 8. 65 + 498 = 498 + 65_____________________________ 9. 9 + 5 = 5 + 9___________________________________ 10. (10+10) + 10 = 10+ (10+10) =30___________________ Next Back

24. Lesson 3 Table of contents Next Back + 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 13 2 2 3 4 5 6 7 8 9 10 11 12 13 14 3 3 4 5 6 7 8 9 10 11 12 13 14 15 4 4 5 6 7 8 9 10 11 12 13 14 15 16 5 5 6 7 8 9 10 11 12 13 14 15 16 17 6 6 7 8 9 10 11 12 13 14 15 16 17 18 7 7 8 9 10 11 12 13 14 15 16 17 18 19 8 8 9 10 11 12 13 14 15 16 17 18 19 20 9 9 10 11 12 13 14 15 16 17 18 19 20 21 10 10 11 12 13 14 15 16 17 18 19 20 21 22

25. You could also go down to "5" and along to "3", or along to "3" and down to "5" to get your answer. How to use Example: 3 + 5 Go down to the "3" row then along to the "5" column, and there is your answer! " 8 " Back Next + 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 10 4 5 6 7 8 9 10 11 5 6 7 8 9 10 11 12 + 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 10 4 5 6 7 8 9 10 11 5 6 7 8 9 10 11 12

34. Solve the equations: 87+S=531 S=531-87 S=444 sack It is reasonable that the farmer harvested 444 sacks during the previous harvest. His harvest now which is 531 is more than the last harvest 3. CARRY OUT THE PLAN If step two of the problem solving plan has been successfully completed in detail, it would be easy to carry out the plan. It will involve organizing and doing the necessary computations. Remember that confidence in the plan creates a better working atmosphere in carrying it out . 4. CHECK THE ANSWER This is an important but most often neglected part of problem solving. There are several questions to consider in this phase. One is to ask if we use another plan or solution to the problem do we arrive at the same answer. . Next Back

38. Next Back WORKSHEET NO. 6 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ Answer the following problem solving 1. Mr. Parma spent Php.260 for a shirt and Php.750for a pair of shoes. How much did he pay in all? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 2. Miss Callanta drove her car 15 287 kilometers and 15 896 kilometers the next year. How many kilometers did she drive her car in two years? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

39. Next Back 3. Four performances of a play had attendance figures of 235, 368, 234, and 295. How many people saw the play during the period? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 4. The monthly production of cars as follows: January-4,356, February- 4,252, and March- 4425, June-4456, July-4287, August-4223, September-4265, October-4365, November-4109, and December- 4270. How many cars were produced in the whole year? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 5. If a sheetrock mechanic has 3 jobs that require 120 4x8 sheets, 115 4x8 sheets, and 130 4x8 sheets of sheetrock respectively. How many 4x8 sheets of sheetrock are needed to complete the 3 jobs? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

40. Next Table of Contents Back CHAPTER-II Introduction In this chapter, you will learn the subtraction operation, the different parts of it and the use of this operation in solving word problem. You will also learn the different ways on how to solve and check the answer or the difference which you can use in your everyday life. This chapter provides the information that will help you master the subtraction as one of the fundamental operation in Mathematics.

42. Next Back + 6 addend 12 addend 18 sum Minuend 18 Subtrahend - 6 Difference 12 Difficulties may arise in subtraction when a digit of the subtrahend is larger than the corresponding digit in the minuend. The process of doing a subtraction of this type is called barrowing or regrouping Let us consider the notation below. When we write 12 – 6, we wish to subtract 6 from 12 or to take away 6 from 12. To find the difference between two numbers, we have to look for a number which when added to the subtrahend , will give the minuend. The table shows the relation between addition and subtraction. One undoes the work of the other.

43. Next Back 12638 _____ - 3630 _____ 9008 _____ WORKSHEET NO. 7 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ A. Give the meaning of the following words. 1. Subtraction-________________________________________________ 2. Minuend-__________________________________________________ 3. Subtrahend-________________________________________________ 4. Difference-_________________________________________________ B. Name the following parts of the mathematical expression given below.

44. Next Back 3. 5428 -2001 4. 10,000 -6,543 2. 1243 -360 1. 349 -265 WRITE YOUR SOLUTION HERE: D. Solve the following to get the difference 1. 5637584-43675=________________ 2. 5389-782=_____________________ 3. 43674-768=____________________ 4. 376598-5281=__________________ 5. 67396-683=____________________ 6. 57290-7849=___________________ 7. 56284-6847=___________________ 8. 683963-68363=_________________ 9. 6254-978=_____________________ 10. 654-87=______________________

46. Next Back WORKSHEET NO. 8 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ Solve and get the difference Simplify the following numbers

48. Next Back 12 – 2 = n 12 – 2 = 10 N = 10 marbles left to Pedro. Checking: 2 + 10 = n 2 + 10 = 12 Another example: Mt. Everest, is 29 028 ft. high, while the Mt. McKinley is 20 320 ft. high. How much is Mt. Everest higher than Mt. McKinley? 1. What is asked? How much Mt. Everest higher than Mt. McKinley? 2. What are given? Mt. Everest, is 29 028 ft. high and Mt. McKinley is 20 320 ft. high. 3. What operation to be used? Subtraction 29 028 – 20 320 = n 29 028 – 20 320 = 8 708 ft. Checking: 8 708 + 20 320 = n 8 708 + 20 320 = 29 028

51. Next Back ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 8. If you born on 1953, how old are you now? ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 9. Mr. Fabre exported to other Asian countries P2 759 000 worth of furniture while Mr. Co exported P5 016 298 worth. How much more where Mr. Co ’ s exports than those of Mr. Fabre? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 10. The total number of eggs produced in the United States in 1993 was 71, 391, 000,000. The total number of eggs produced in 1992 was 70,541,000,000. How many more eggs were produced in the United States in 1993 than in 1992? ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

52. Next Table of Contents Back Multiplication of Whole Numbers Introduction In this chapter, you will learn about the multiplication operation, its different parts and ways in solving it and the use of this operation in word problem. This chapter provides lessons and exercises that will help you to master the multiplication of whole numbers. CHAPTER-III

59. In mastering the multiplication operation, knowing how to multiply using multiplication table helps you to become fluent in multiplying numbers. How to use multiplication table? Next Back

60. Multiplication Table Example: Remembering 9's What's 9 x 7? Use the 9-method! Hold out all 10 fingers, and lower the 7th finger. There are 6 fingers to the left and 3 fingers on the right. The answer is 6. Next Back X 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 2 0 2 4 6 8 10 12 14 16 18 20 22 24 3 0 3 6 9 12 15 18 21 24 27 30 33 36 4 0 4 8 12 16 20 24 28 32 36 40 44 48 5 0 5 10 15 20 25 30 35 40 45 50 55 60 6 0 6 12 18 24 30 36 42 48 54 60 66 72 7 0 7 14 21 28 35 42 49 56 63 70 77 84 8 0 8 16 24 32 40 48 56 64 72 80 88 96 9 0 9 18 27 36 45 54 63 72 81 90 99 108 10 0 10 20 30 40 50 60 70 80 90 100 110 120 11 0 11 22 33 44 55 66 77 88 99 110 121 132 12 0 12 24 36 48 60 72 84 96 108 120 132 144

62. Next Table of Contents Back Lesson 13 This lesson is concern in one of the easy ways in getting the product in multiplication. If the digits in the multiplier (or even multiplicand) are all 9 such as 9, 99, 999 … , annex to the multiplicand as many zeros as there are 9 ’ s in the multiplier and from it, subtract the multiplicand. Here some examples: 999×364= 364 000-364= 369 636 Why? 2834×99= 283 400-2834= 280566 Why? 31×999= 31 000-31= 30 969 Why?

67. Next Back Therefore, the screw machine can produce 5 700 crews in one hour. Therefore, there are 1 600 portable radios does the store have. Solution: 60 minutes = 1 hour 95 crews x 60 minutes = n N = 5 700 screws. Here is another example, A department store bought 32 crates of portable radios. Each crate contains 50 radios. How many portable radios does the store have? 1. What is asked? How many portable radios does the store have? 2. What are given? 50 portable radios in 1 crate and 32 crates 3. What operation to be used? Multiplication Solution: 1 crate = 50 radios 32 crates x 50 radios = n N = 1 600 portable radios

68. Next Back WORKSHEET NO. 15 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ Answer the following word problem. 1. Victoria and her brother, Daniel, deliver Sunday papers together. She delivers 58 papers and he delivers 49 papers. Each earns 75 cents for each paper delivered. How much more does Victoria earn than Daniel each Sunday? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 2. In one basketball stadium, a section contains 32 rows and each row contains 25 seats. If the stadium has 4 sections, how many seats it has? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 3. Season tickets for 45 home games cost P789. Single tickets cost P15 each. How much more does a season ticket cost than individual tickets bought of each game? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

69. Next Back 4. A store has 124 boxes of pencils with 144 pencils in each box. How many pencils they have? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 5. An eagle flies 70 miles per hour. How far can an eagle fly in 15 hours? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 6. Mandy can laid 65 bricks in 30 minutes. How many bricks can Mandy lay in 5 hours? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 7. Sound waves travels approximately 1 100 ft. per sec. in air. How far will the sound waves travel in 3 hours? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

70. Next Back SOLUTION: 9. If a worker can make 357 bolts in one hour, how many bolts he can make in eight hours? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 10. If 1cubic yard of concrete costs P55.00, how much would 13cubic yards cost? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 8. One cassette seller sold 650 cassettes. The cassettes cost her P15.00 each and sold them for P29.00 each. What was her total profit? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

71. Next Back Table of Contents Division of Whole Numbers Introduction In this chapter, you will learn about the division operation its different parts and uses in solving word problem. This chapter provides you the information you need to master one of the fundamental operations in mathematics which is division. CHAPTER- IV

73. Next Back quotient divisor dividend or dividend ÷ divisor = quotient Example: Suppose that we have twelve students in the class and we want to divide the class into three equal groups. How many should be in each group? Solution: We can ask the alternative question, "Three times what number equals twelve?" The answer to this question is four. We write 4 3 12 or 12 ÷ 3 = 4 we call the number 12 the dividend , the number 3 the diviso r , and the number 4 the quotient . In the above expression, a is called the dividend , b the divisor and c the quotient .

78. Next Back Table of Contents Lesson 17 In mastering the division operation, you should need to know all the things in this operation. When dividing numbers, it has not always given an exact quotient. This process is what we called division with remainder. MASTERING SKILLS IN DIVISION OF WHOLE NUMBERS Division with Remainder Often when we work out a division problem, the answer is not a whole number. We can then write the answer as a whole number plus a remainder that is less than the divisor. Example 34 ÷ 5 Solution Since there is no whole number when multiplied by five produces 34, we find the nearest number without going over. Notice that 5 x 6 = 30 and 5 x 7 = 35 Hence 6 is the nearest number without going over. Now notice that 30 is 4 short of 34. We write 34 ÷ 5 = 6 R 4 "6 with a remainder of 4“0

82. Next Back Use BODMAS to work out whether these statements are TRUE or FALSE. Work out the answers to the following questions (without a calculator). (a) 10 ÷ 2 = 2 ÷ 10 __________ (b) 12 + 8 ÷ 2 = 10 __________ (c) 3 + 12 ÷ 4 = 6 __________ (d) 6 ÷ 2 + 3 = 6 __________ (a) 3 + 4 × 8 __________ (b) 8 + 3 × 6 __________ (c) 8 × 6 - 4 __________ (d) 12 ÷ 2 + 5 __________ (e) 5 - 12 ÷ 3 __________ (f) 14 ÷ 2 + 8 __________ (g) 3 × 2 + 8 ÷ 4 __________

83. Next Table of Contents Back Lesson 18 The cancellation of Insignificant Zeros is one of the easy ways in performing division of whole numbers. It is done by cancelling the insignificant zeros in both the divisor and the dividend.

84. To check multiply the quotient to the divisor then multiply also the place value of the removed zeros Remember that in cancelling both the dividend and divisor, the insignificant zeros are needed to be the same. If you cancelled 3 zeros in the dividend, you need also to cancel 3 zeros from the divisor. Next Back 50 5050 505÷5=101 ( both dividend and divisor) 50 050 050 0 101 210 2. 5 1050 105÷5=21(10) =210 (the insignificant zero in -10 dividend was cancelled) -50 50 0 300÷10=30 50÷50=1 1000÷10=100 Examples

88. Next Back Answer: The ski resort averaged 3,589 ticket sales per day in the month of January. Answer: Courtney can hang her 160 stars in 10 rooms Solution Since there are 31 days in January, we need to divide the total number of tickets by 31 3589 31 | 111259 93 31 x 3 = 93 182 111 - 93 = 18 and drop down the 2 155 31 x 5 = 155 275 182 - 155 = 27 and drop down the 5 248 31 x 8 = 248 279 275 - 248 = 27 279 31 x 9 = 279 0 Another example Courtney is hanging glow in the dark stars in each room of his house. If there are 160 stars in the box and she wants 16 in each room, how many rooms can she hang stars? Solution Since there are 160 stars in the box and she wants 16 in each room. And the problem is asking for how many stars in each room will be? 10 16 160 16x1=16 16 16-16=0 00 16x0=0 00 0

89. Next Back WORKSHEET NO. 19 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ A. Analyze and solve the following problems. 1. Jacinta has 5 pennies in a jar. If she divides it into 2 stacks of 50, how many stacks does she have now? ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________. 2. Harry has 300 pieces of chalk with the same amount in each box. There are 20 boxes how many pieces of chalk in EACH box? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

90. 3. The surface area of a floor is 150 square feet. How many 10 ft. square tiles will be needed (inside of 150 feet) to cover the floor? (How many 10's are inside of 150?) _____________________________________________________________________________ __________________________________________________________________________________________________________________________________ ______________________________________________________________________________________________________________________________________________________________________________________. 4. Billy was offered a job at the nearby golf course. The owner offered him $500.00 per seven day week or $50. the first day and agreed to double it for each following day. How could Billy make the most amount of money? Which deal should he accept and why? ______________________________________________________________________________________________ _____________________________________________________________________________ __________________________________________________________________________________________________________________________________________________________________________________________________________________________. 5. Sally is having a birthday party with 10 people. When everyone gets there she asks everyone to introduce themselves and shake everyone's hand. How many handshakes will there be? How do you know? __________________________________________________________________________________________________ _____________________________________________________ ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.

92. Next Table of Contents Back CHAPTER V INTEGERS Introduction You have finished Unit 1 of this modular workbook. You now already reviewed what you have taken in your Elementary level . Now, you are ready to proceed to the next chapter of this modular workbook, the INTEGERS. This chapter will give you a deep understanding about integers, the different kinds of integers, the uses of integers in Mathematics and the functions of integers in our real world. In studying high school math, integers are always present. It seems that you have already mastered the fundamental operations in whole numbers you may now proceed to the next chapter which is the application of the four fundamental operations that you have mastered.

94. Next Back We do not consider zero to be a positive or negative number. For each positive integer, there is a negative integer, and these integers are called opposites. For example, -3 is the opposite of 3, -21 is the opposite of 21, and 8 is the opposite of -8. If an integer is greater than zero, we say that its sign is positive. If an integer is less than zero, we say that its sign is negative. Example: Integers are useful in comparing a direction associated with certain events. Suppose I take five steps forwards: this could be viewed as a positive 5. If instead, I take 8 steps backwards , we might consider this a -8. Temperature is another way negative numbers are used. On a cold day, the temperature might be 10 degrees below zero Celsius, or -10° C . The Number Line The number line is a line labeled with the integers in increasing order from left to right, that extends in both directions: For any two different places on the number line, the integer on the right is greater than the integer on the left. Examples: 9 > 4, 6 > -9, -2 > -8, and 0 > -5

95. Next Back The number of units a number is from zero on the number line. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a number n by writing n in between two vertical bars: | n |. Examples: |6| = 6 |-12| = 12|0| = 0 |1234| = 1234 |-1234| = 1234 Absolute Value of an Integer

98. 2) When adding integers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the integer with the larger absolute value. Example: 8 + (-3) =? The absolute values of 8 and -3 are 8 and 3. Subtracting the smaller from the larger gives 8 - 3 = 5, and since the larger absolute value was 8, we give the result the same sign as 8, so 8 + (-3) = 5. Example: 8 + (-17) =? The absolute values of 8 and -17 are 8 and 17. Subtracting the smaller from the larger gives 17 - 8 = 9, and since the larger absolute value was 17, we give the result the same sign as -17, so 8 + (-17) = -9. Example: -22 + 11 = ? The absolute values of -22 and 11 are 22 and 11. Subtracting the smaller from the larger gives 22 - 11 = 11, and since the larger absolute value was 22, we give the result the same sign as -22, so -22 + 11 = -11. Example: 53 + (-53) = ? The absolute values of 53 and -53 are 53 and 53. Subtracting the smaller from the larger gives 53 - 53 =0. The sign in this case does not matter, since 0 and -0 are the same. Note that 53 and -53 are opposite integers. All opposite integers have this property that their sum is equal to zero. Two integers that add up to zero are also called additive inverses. Next Back

103. Back Next Examples: In the product below, both numbers are positive, so we just take their product. 4 × 3 = 12 In the product below, both numbers are negative, so we take the product of their absolute values. (-4) × (-5) = |-4| × |-5| = 4 × 5 = 20 In the product of (-7) × 6, the first number is negative and the second is positive, so we take the product of their absolute values, which is |-7| × |6| = 7 × 6 = 42, and give this result a negative sign: -42, so (-7) × 6 = -42. In the product of 12 × (-2), the first number is positive and the second is negative, so we take the product of their absolute values, which is |12| × |-2| = 12 × 2 = 24, and give this result a negative sign: -24, so 12 × (-2) = -24.

104. Counting the number of negative integers in the product, we see that there are 3 negative integers: -2, -11, and -5. Next, we take the product of the absolute values of each number: 4 × |-2| × 3 × |-11| × |-5| = 1320. Since there were an odd number of integers, the product is the opposite of 1320, which is -1320, so 4 × (-2) × 3 × (-11) × (-5) = -1320. To multiply any number of integers: 1 . Count the number of negative numbers in the product. 2 . Take the product of their absolute values. 3 . If the number of negative integers counted in step 1 is even, the product is just the product from step 2, if the number of negative integers is odd, the product is the opposite of the product in step 2 (give the product in step 2 a negative sign). If any of the integers in the product is 0, the product is 0. Example: 4 × (-2) × 3 × (-11) × (-5) = ? Next Back

107. Next Back In the division below, both numbers are positive, so we just divide as usual. 4 ÷ 2 = 2. In the division below, both numbers are negative, so we divide the absolute value of the first by the absolute value of the second. (-24) ÷ (-3) = |-24| ÷ |-3| = 24 ÷ 3 = 8. In the division (-100) ÷ 25 , both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |-100| ÷ |25| = 100 ÷ 25 = 4 , and give this result a negative sign: -4, so (-100) ÷ 25 = -4. In the division 98 ÷ (-7), both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |98| ÷ |-7| = 98 ÷ 7 = 14 , and give this result a negative sign: -14, so 98 ÷ (-7) = -14. LOOK AT THE EXAMPLES:

110. Next Back It seems that each student interpreted the problem differently, resulting in two different answers. Student 1 performed the operation of addition first, then multiplication; whereas student 2 performed multiplication first, then addition. When performing arithmetic operations there can be only one correct answer. We need a set of rules in order to avoid this kind of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation. The above problem was solved correctly by Student 2 since she followed Rules 2 and 3. Let's look at some examples of solving arithmetic expressions using these rules. Rule 1: First perform any calculations inside parentheses. Rule 2: Next perform all multiplications and divisions, working from left to right. Rule 3: Lastly, perform all additions and subtractions, working from left to right.

111. Next Back In Example 1, each problem involved only 2 operations. Let's look at some examples that involve more than two operations. In Example 1, each problem involved only 2 operations. Let's look at some examples that involve more than two operations. Order of Operations Expression Evaluation Operation 6 + 7 x 8 = 6 + 7 x 8 Multiplication = 6 + 56 Addition = 62 16 ÷ 8 - 2 = 16 ÷ 8 - 2 Division = 2 - 2 Subtraction = 0 (25 - 11) x 3 = (25 - 11) x 3 Parentheses = 14 x 3 Multiplication = 42

112. Next Back Step 1: 3 + 6 x (5 + 4) ÷ 3 - 7 = 3 + 6 x 9 ÷ 3 - 7 Parentheses Step 2: 3 + 6 x 9 ÷- 7 = 3 + 54 ÷ 3 - 7 Multiplication Step 3: 3 + 54 ÷ 3 - 7 = 3 + 18 - 7 Division Step 4: 3 + 18 - 7 = 21 - 7 Addition Step 5: 21 - 7 = 14 Subtraction Example 2: Evaluate 3 + 6 x (5 + 4) ÷ 3 - 7 using the order of operations.

113. Next Back In Examples 2 and 3, you will notice that multiplication and division were evaluated from left to right according to Rule 2. Similarly, addition and subtraction were evaluated from left to right, according to Rule 3. Solution: Step 1: 9 - 5 ÷ (8 - 3) x 2 + 6 = 9 - 5 ÷ 5 x 2 + 6 Parentheses Step 2: 9 - 5 ÷ 5 x 2 + 6 = 9 - 1 x 2 + 6 Division Step 3: 9 - 1 x 2 + 6 = 9 - 2 + 6 Multiplication Step 4: 9 - 2 + 6 = 7 + 6 Subtraction Step 5: 7 + 6 = 13 Addition Example 3: Evaluate 9 - 5 ÷ (8 - 3) x 2 + 6 using the order of operations.

114. Next Back When two or more operations occur inside a set of parentheses, these operations should be evaluated according to Rules 2 and 3. This is done in Example 4 below. Solution: Step 1: 150 ÷ (6 + 3 x 8) - 5 = 150 ÷ (6 + 24) - 5 Multiplication inside Parentheses Step 2: 150 ÷ (6 + 24) - 5 = 150 ÷ 30 - 5 Addition inside Parentheses Step 3: 150 ÷ 30 - 5 = 5 - 5 Division Step 4: 5 - 5 = 0 Subtraction Example 4: Evaluate 150 ÷ (6 + 3 x 8) - 5 using the order of operations.

115. Next Back Example 5: Evaluate the arithmetic expression below: Solution: This problem includes a fraction bar (also called a vinculum), which means we must divide the numerator by the denominator. However, we must first perform all calculations above and below the fraction bar BEFORE dividing. Thus Evaluating this expression, we get:

118. We can use these digits to make a word in the calculator. Let ’ s try to make words using our calculator . Next Table of Contents Back MATH AND TECHNOLOGY Calculator Puzzle PUZZLE 1 Press each digit from 0-8 one at a time. After pressing each digit, turn the calculator upside down. What letters of the alphabet resemble the digits? DIGIT LETTER 0 0 1 I 3 E 4 H 5 S 6 G 7 L 8 B

120. Next Back Solution How far do you understand the lesson about the basic fundamental operation? In this part, all you have to do is just to fill up the missing numbers in the puzzle to get the appropriate equation. PUZZLE 2

121. Next Back Solution PUZZLE 3