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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
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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
The University shall primarily provide advanced education, professional, technological and vocational instruction in agriculture, fisheries, forestry, science, engineering, industrial technologies, teacher education, medicine, law, arts and sciences, information technology and other related fields. It shall also undertake research and extension services, and provide a progressive leadership in its areas of specialization.
In pursuit of the college vision/mission the College of Education is committed to develop the full potentials of the individuals and equip them with knowledge, skills and attitudes in Teacher Education allied fields to effectively respond to the increasing demands, challenges and opportunities of changing time for global competitiveness.
V M O Back
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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
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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
The students are provided with guidance and assistance of selected faculty members of the college through the selection, production and utilization of appropriate technology tools in developing technology based teacher support materials. Through the role and functions of computers especially the internet, the student researchers and the advisers are able to design and develop various types of alternative delivery systems. These kinds of activities offer a remarkable learning experience for the education students as future mentors especially in the preparation of instructional materials.
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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
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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
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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
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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
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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
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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
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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
In this unit, you will understand the concept of the basic fundamental operations dealing with whole numbers. This workbook will help you to master and to become skilled in the fundamental operations.
This modular workbook provides information about four operations and how to perform such kind of operation in solving word problem. It also provides exercises and activities that will help you become skilled and for you to master the fundamental operations.
Objectives:
After studying this unit, you are expected to:
discuss the four fundamental operations;
perform the operations well;
check the answers in addition and multiplication using their inverse operation.
Back Next Table of contents
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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
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This property states that any number added to 0 is the number itself, that is, if “ a ” is any number, a + 0 = a. This property states that changing the order of the addends does not change the sum. This means you need to remember only half of the basic facts. In symbols, the property says that a + b = b + a , for any numbers a and b. : Lesson 2
PROPERTIES OF ADDITION
Objectives :
After this lesson, the students are expected to:
define the properties of addition;
use the different properties of addition in solving;
perform an operation using the properties of addition.
The 0 Property in Addition Examples: 8 + 0 = 8 27 + 0 = 27 10 + 0 = 10 31 + 0 = 31 The Commutative Property of Addition Table of contents Back Next
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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
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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
WORKSHEET NO. 3 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ FOLLOW THE INSTRUCTION 1. Have your own addition table 2. With your addition table, add the following 1+4, 0+1, 3+4, 5+0, 5+4 6+4, 7+2, 8+0, 9+2, 10+4 1+6, 3+6, 5+6, 3+10 6+6, 10+6, 6+8, 10+10 3. After adding, try to put dots in every sum. Try to connect the dots by a line in every number to find what the mother of all science is. Back Next
discuss the different methods in adding whole numbers;
solve mathematical problems using the other method.
There are some easy ways in adding whole numbers. Lesson 4 Adding the column separately. Let 326+258 use as our illustrative example. Adding in reverse order 326 300+20+6 +258 200+50+8 500 300+20+6 200+50+8 500+70+14 500+70+14=584 Table of contents Table of contents Back Next
YEAR & SECTION: ________________________ RATING: ___________
Using any of the given ways, add the following and write the answer in the space provided. Show all your solutions.
WORKSHEET NO. 4
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WRITE YOUR SOLUTION B. Perform the operation using the procedure discussed. Check your answer by using the short method.
642 890+57 829=______________________
564 872 389+54 738=___________________
12 345+42 321=________________________
3255+6472865=________________________
6437286+56387=_______________________
54390+529=___________________________
6348901+65890=_______________________
7395+7598043=________________________
225+264=____________________________
367+201=____________________________
9 632+2 330=_________________________
1 423+54 673=________________________
543 265+65 223=______________________
673 895 462+54 289=___________________
629 075+57823=_______________________
Lesson 5 Back
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Throughout this lesson, we will be solving problems that deal with real numbers. In solving word problems, the following plan is suggested: This problem solving plan should be used every time we solve word problems. Careful reading is an important step in solving the problem. This lesson serves as an introduction to the next chapter. Lesson 5
Objectives
After this lesson, the students are expected to:
discuss how to solve a word problem;
solve any given problems systematically;
use problem solving plan in solving any given word problem.
SOLVING WORD PROBLEM Back Next Table of Contents
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One harvest season, a farmer harvested 531 sacks of rice. This was 87 more than his previous harvest. How many sacks did he harvest during the previous season?
What is the problem about?
What information is given?
What is being asked?
“ 87 more ” suggests addition and we can write a formula: 87+S=531.
PROBLEM SOLVING PLAN
Understand the problem.
Devise a plan.
Carry out the plan.
Check the answer.
Example: PROBLEM SOLVING PLAN 1. UNDERSTAND THE PROBLEM Understand the problem and get the general idea. Read the problem one or more times. Each time you read ask: Represent what is asked with a symbol. {The problem is about the number of sacks harvested. Let S be the number of sacks during the previous harvest . } This is a key part in the 4 step plan for solving problems. Different problem solving strategies have to be applied. A figure, diagram, chart might help or a basic formula might be needed. It is also likely that a related problem can be solved and can be used to solve the given problem. Another devise is to use the “ trial and learn from your errors ” process. There is a lot of problem solving strategies and every problem solver has own special technique. 2. DEVISE A PLAN Next Back
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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
YEAR & SECTION: ________________________ RATING: ___________
A. Discuss the different problem solving plan briefly. 1. Understand the problem ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
2. Devise a plan- ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
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Back Next Table of Contents 11 875 +648 1 523 Lesson 6
APPLYING ADDITION OF WHOLE NUMBERS IN WORD PROBLEM
Objectives
After this lesson, the students are expected to:
analyze the given problem;
to develop the skills and knowledge in solving word problems;
identify the different steps in word problems involving addition.
LOOK AT THE EXAMPLE A farmer gathered 875 eggs from one poultry house and 648 from another. How many eggs did he gather? We want the answer to 875 + 648 =?
Add the ones: 5 + 8 = 13 ones = 1 ten + 3 ones.
Write 3 in the ones column and bring the 1 ten to
the tens column.
Add the tens: 1 +7 +2 = 12 tens = 1 hundred + 2
tens.
Write 2 under the tens column and bring the 1
hundred to the hundreds column.
Add the hundreds: 1 + 8 + 6 = 15 hundreds = 1
thousand + h hundreds. Write 15 to the left of 2.
The farmer gathered 1 523 eggs.
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Next Back 1 11 5 986 +3 759 9 745 Here is another example: 1 5 326 + 1 456 6782
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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? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
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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? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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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.
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Next Back What is Subtraction? After learning and describing addition as a process of combining two or more groups of objects, we can now consider its opposite operation --- Subtraction. If addition is combining of group of object, subtraction is the process of taking away or of removing something. The symbol used for subtraction is the minus sign (-). Lesson 7
WHAT IS SUBTRACTION?
Objectives
After this lesson, the students are expected to:
define what is subtraction;
identify the parts in subtraction;
differentiate the subtraction from addition.
Table of Contents
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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.
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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.
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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=______________________
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Next Back Table of Contents 5 hrs + 17 mins - 3 hrs + 28 mins 1 hr + 49mins 5 hrs + 17 mins 77mins - 3 hrs + 28 mins Cain kiblah type his report in physics at the computer shop for about 5 hours and 17 minutes while Lane Margaret types her report for only 3 hours and 28 minutes. How fast does Lane Margaret type her report than Cain kiblah? Lesson 8
MASTERING SKILLS IN SUBTRACTING WHOLE NUMBERS
Objectives
After this lesson, the students are expected to:
enhance the knowledge in terms of subtracting whole numbers;
develop the speed in solving subtraction;
perform the steps in subtracting whole numbers.
To make the subtraction convenient, we borrow 1 minute so we have:
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Next Back WORKSHEET NO. 8 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ Solve and get the difference Simplify the following numbers
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Next Back Lesson 9 Table of Contents To master the application of subtraction in problem solving, here are some examples:
PROBLEM SOLVING INVOLVING SUBTRACTION
Objectives
After this lesson, the students are expected to:
follow the steps correctly in problem solving involving subtraction;
discuss the different steps in problem solving;
develop the knowledge in problem solving.
Pedro had marbles. He gave away two of his marbles to Juan. If Pedro had twelve marbles, how many marbles left to Pedro after he gave two to Juan? We can use the problem solving plan: 1. Know what the problem is. a. What is asked? How many marbles left to Pedro? b. What are given? 12 marbles of Pedro and 2 to Juan c. What operation to be used? Subtraction
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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
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Next Back WORKSHEET NO. 9 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________
In 1992, William Clinton got 44 908 254 votes as the president of USA while George Bush got 39 10 343 votes and Foss Perot got 19 741 65 votes. How many more votes did Clinton have than Bush? Bush than Foss?
2. In May of 1994, there were 42 518 000 beneficiaries in the social security program while there were 41 784 000 beneficiaries on May 1993. How much was the increase of beneficiaries from 1993 to 1994?
3. In 1998, a school had an enrollment of 5908 pupils while there are 6519 pupils enrolled in 1999. How much more pupils enrolled in 1999 than in 1998?
7. A philanthropist donated P850 765 to an orphanage. The amount was used for some repairs and the purchase of some equipment worth P519 800. How much money was left for other projects?
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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? ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
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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
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Next Back Lesson 10 Table of Contents Multiplication is a repeated addition. It can be thought of as addition repeated a given number of times.
WHAT IS MULTIPLICATION?
Objectives
After this lesson, the students are expected to;
define what multiplication is.
identify the part of multiplication.
perform the multiplication operation properly.
For example, 3 x 5 = 15 can be solving as 5 + 5 + 5 =15. 3 mean that the 5 is to be used three times. The same problem can also be thought of as 5x 3, or 3 + 3 +3 + 3 + 3 =15. Written this way, the three is used as a total of five times in either case is 15. The number in the upper part is called the multiplicand and in the lower position is called the multiplier . The answer in the multiplication is called product . ×3 multiplicand 5 multiplier 15 product
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Next Back WORKSHEET NO. 10 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ A. Identify the following.
The product of the 1 and any number a is a, that is, 1 x a = a for any number.
ZERO PROPERTY
The product of 0 and any number a is 0, that is a x 0 = 0 for any number a.
PROPERTIES OF MULTIPLICATION
Objectives
After this lesson, the students are expected to:
review the different properties of multiplication;
develop the knowledge in the properties of multiplication;
apply the properties of multiplication in solving problem.
Example: 21 x a = 21 27 x a =27 31 x a = 31 11 x a = 11 5 x a = 5 13 x a = 13 Example: 0 x 87 = 0 0 x 98 = 0 15 x 0 = 0 45 x 0 = 0 14 x 0 = 0 58 x 0 = 0
Changing the grouping of the factors does not affect the product, that is, a x (b x c) = (a x b) x c for any number of a, b, and c.
DISTRIBUTIVE PROPERTY
If one factor is a sum of two numbers, multiply the addends to the multiplier before adding will not change the answer, that is a x (b + c) = (a x b) + (a x c).
Example: 7 x 4 = 28 = 4 x 7 5 x 12 = 60 = 12 x 5 5 x 6 = 30 = 6 x 5 4 x 11 = 44 = 11 x 4 Example: (7 x 4) x 5 = 140 = 7 x (4 x 5) (4 x 6) x 8 = 192 = 4 x (6 x 8) Example: 5 x (6 + 7) = 30 + 35 = 65 6 x (7 + 9) = 42 + 54 = 9
COMMUTATIVE PROPERTY
Changing the order of the factors does not change the product, that is, a x b = b x a for any number of a and b.
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WORKSHEET NO. 11 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ B. Fill the missing number. Use the property of multiplication to get product 1. 6 x 7 = __ x 6 6. (7 x __) + (__ x 6) = 7 x (3 +6) 2. 5 x 0 = __ 7. 27 x __ = 27 3. 8 x 1 __ 8. 8 x __ = 0 4. (4 x 5) x 7 = 4 x (__ x 7) 9. 6 x (3 x 4) = (6 x __) x 4 5. 8 x (2 + __) = (8 x 2) + (8 x __) 10. 4 x 9 =__ x 4 1. (8 x 4) + (8 x 6) = 8 x (__ + 6) = ______ 2. (7 x 5) x 2 = 7 x (__ x __) = ______ 3. (9 x 5) = 25 x__ = _______ 4.8 x 0 = ______ 5. (12 x 3) + (12 x 7) = _____
Fill on the blank and identify the property of each.
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Next Back Lesson 12 Table of Contents Since multiplication is a shortcut for repeated addition, we can get the product of a two factors without the use of a two factors without the use of repeated addition. Take a look at the example:
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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
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Next Back WORKSHEET NO. 12 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ WRITE YOUR SOLUTION HERE:
Find the product of the following. (You may use a multiplication table if you want).
1. 59x 8 =________________ 2. 48 x 3 =_______________ 3. 31 x 6 =_______________ 4. 27 x 21 =______________ 5. 11 x 15 =_______________ 6. 21 x 27 =_______________ 7. 14 x 17 =_______________ 8. 8 x 32 = ________________ 9. 78 x 45 =_______________ 10. 11 x 23 =_____________
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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?
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WORKSHEET NO. 13 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ Next Back
Multiply the following using the “ 99 multiplier ” method.
99×99=________________________
33×99=________________________
47x99=________________________
65x9=_________________________
21x99=________________________
81x99=________________________
72x999=_______________________
56x9999=______________________
34x9=_________________________
8x9=__________________________
B. Solve the following
Find the product of 873 and 9999=________________________
Find the product of 132 and 999=_________________________
Find the product of 665 and 99=__________________________
Find the product of 670 and 9=___________________________
Find the product of 154 and 9999=________________________
Find the product of 1063 and 999=________________________
Find the product of 948 and 9999=________________________
Find the product of 323 and 99=__________________________
Find the product of 493 and 999=_________________________
Find the product of 490 and 99=__________________________
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Next Back Lesson 14 Table of Contents When the factors are in the power of ten such as 10, 100, 1000, 10 000, 100 000 and so on and so fort, just multiply the digit that is form 1 to 9 and add the number of zeros. When the factors are end in both zero, multiply the significant number and used the number of zeros in both factors to the product.
“ THE POWER OF TEN ” MULTIPLICATION
Objectives
After this lesson, the students are expected to:
specializing skills in multiplication;
perform multiplication easily;
develop the speed in multiplying numbers.
Example: 31 x 100 = 3 100 270 x 10 = 2 700 15 000 x 100 = 1 500 000 Example: 2 380 x 40 = 95 200 2 380 x 400 = 952 000
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Next Back WORSHEET NO. 14 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________
Based to the power of ten, multiply the following.
1. 100 x 320 =_________ 6. 75 x 100 =_________
2. 10 x 27 = __________ 7. 56 x 10 = __________
3. 100 x 414 = ________ 8. 38 x 100 =__________
4. 176 x 100 = ________ 9. 68 x 10 000 =________
5. 39 x 1 000 = ________ 10. 59 x 1 000 =________
Find the product of the following.
1. 2 080 x 30 =____________ 6. 720 x 40 =____________
2. 3 150 x 60 =____________ 7. 7 230 x 50 =___________
3. 1 470 x 20 =____________ 8. 2 030 x 60=___________
4. 30 x 90 =____________ 9. 456 x 70=____________
5. 30 x 80 = ____________ 10. 86 x 690= ____________
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Next Table of Contents Back Lesson 15 A screw machine can produce 95 screws in one minute. How many screws it can produce in one hour?
PROBLEM SOLVING INVOLVING MULTIPLICATION
Objectives
After this lesson, the students are expected to:
describe how to use the multiplication in problem solving;
follow the steps correctly in multiplication of word problem;
discuss the use of multiplication in problem solving.
1. What is asked? How many screws a screw machine can produce in one hour? 2. What are given? Screw machine can produce 95 screws in a minute. 3. What operation to be used? Multiplication
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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
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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? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
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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? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
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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? _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
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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
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Next Back Table of Contents Lesson 16 In mathematics , especially in elementary arithmetic , division (÷) is the arithmetic operation that is the inverse of multiplication . Division can be described as repeated subtraction whereas multiplication is repeated addition . Division is defined as this reverse of multiplication. In high school, the process is also the same. since 64÷8=8 since 8 X 8=64
WHAT IS DIVISION?
Objectives
After this lesson, the students are expected to:
define division;
identify the parts of division;
discuss the division operation.
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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 .
B. Division by 1 Example Suppose that you had $100 and had to distribute all the money to 100 people so that each person received the same amount of money. How much would each person get? Solution If you gave each person $1 you would achieve your goal. This comes directly from the identity property of one. Since the questions asks what number times 100 equals 100. In general we conclude, Example 100 ÷ 100 = 1 38 ÷ 38 = 1 15 ÷ 15 = 1 Example Now let’s suppose that you have twelve pieces of paper and need to give them to exactly one person. How many pieces of paper does that person receive? Any number divided by itself equals 1
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Next Back When Zero is the Dividend Solution Since the only person to collect the paper is the receiver, that person gets all twelve pieces. This also comes directly from the identity property of one, since one times twelve equals one. In general we conclude, Examples 12 ÷ 1 = 12 42 ÷ 1 = 42 33 ÷ 1 = 33 Example Now lets suppose that you have zero pieces of pizza and need to distribute your pizza to four friends so that each person receives the same number of pieces. How many pieces of pizza does that person receive?
Solution Since you have no pizza to give, you give zero slices of pizza to each person. This comes directly from the multiplicative property of zero, since zero times four equals zero.
In general we conclude,
Any number divided by 1 equals itself
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Next Back The Problem with Dividing by Zero Examples 5 ÷ 0 = undefined 0 ÷ 0 = undefined 1 ÷ 0 = undefined
Example Finally lets suppose that you have five bags of garbage and you have to get rid of all the garbage,
but have no places to put the garbage. How can you distribute your garbage to no places and still get rid of it all?
Solution You can't! This is an impossible problem. There is no way to divide by zero.
In general we conclude,
Examples 0 ÷ 4 = 0 0 ÷ 1 = 0 0 ÷ 24 = 0 Zero divided by any nonzero number equals zero Dividing by zero is impossible
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Next Back WORKSHEET NO. 16 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ ___________ 56÷8=7 _________ _______________
B. As far as you remember, try to divide the following.
56÷7=
54÷6=
900÷100=
64÷16=
56÷8=
122÷11=
144÷12=
256÷16=
180÷9=
360÷4=
A. Give the name of the following unknown parts of division.
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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
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Next Back (Divisor x quotient) + Remainder = dividend Take note: the remainder may also be expressed in decimals. Example 4321 ÷ 6 Solution 720 6 | 4321 42 6 x 7 = 42 12 43 - 42 = 1 and drop down the 2 12 6 x 2 = 12 01 12 - 12 = 0 and drop down the 1 0 6 x 0 = 0 1 1 - 0 = 1
511 37 18932 185 37 x 5 = 185 43 189 - 185 = 4 and drop down the 3 37 37 x 1 = 37 62 43 - 37 = 6 and drop down the 2 37 37 x 1 = 37 25 62 - 37 = 25
We can conclude that
18932 ÷ 37 = 511 R25 We can conclude that 4321 ÷ 6 = 720 R1 In general we write Example
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Next Back If you know your multiplication tables well, you should find it reasonably easy to do simple divisions in your head (SPECIAL TOPIC) Mental Division of Whole Numbers The process of division is just multiplication in reverse. This means that if 4 × 3 = 12 then 12 ÷ 3 = 4 and 12 ÷ 4 = 3 . For example: you want to work out 42 ÷ 7 , and you remember that 6 × 7 = 42 , so the answer is 6 . When there is more than one operation in a question, you need to remember the order in which operations are carried out. This can be summarized by BODMAS:
B rackets first
O
D ivide
M ultiply
A dd
S ubtract
If you see two of the same operation you just do them in the order they appear (left to right). Below are three examples of BODMAS used in a question. (a) 3 + 4 × 5 = 3 + 20 = 23 ( M ultiply before A dd) (b) 10 ÷ ( 2 + 3 ) = 10 ÷ 5 = 2 ( B rackets before D ivision) (c) 20 ÷ 2 ÷ 2 = 10 ÷ 2 = 5 (do operations left to right)
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Next Back WORKSHEET NO. 17 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ Work out the answers to the questions below and fill in the boxes. Question 1
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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 __________
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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.
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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
Copy the figure. Show how to divide it into 2 equal parts.
Each part must have the same size and shape.
Copy the figure again. Show how to divide it in 3 equal parts.
Copy the figure again. Show how to divide it in 4 equal parts.
Draw a 2 dimensional clock. Then draw a line across the clock so that the sum of the numbers in each group is the same. Next Back
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Next Table of Contents Back Lesson 19 Like the first three operations, the division operation is very usable to our daily lives. We use also this operation to solve some problems. Take a look and study the examples given below
Example You are the manager of a ski resort and noticed that during the month of January you sold a total of 111,359 day ski tickets. What was the average number of tickets that were sold that month?
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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
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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? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
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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? __________________________________________________________________________________________________ _____________________________________________________ ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________.
In UNIT II, you will expect the concept of the basic fundamental operations dealing with the integers the concept, the nature and the difference between them. Likewise, the lessons provided in this unit will enable you to perform skillfully the four fundamental operations with integers.
You will think much critically to perform the activities and to solve the exercises that will be given to you in this unit. This unit also contains precedence of operations which you can use in Algebra II.
Objectives:
After studying this unit, you are expected to:
discuss the integers;
use the fundamental operations in solving integers;
appreciate the integers as a part of your discussion;
gain more knowledge about integers that will guide you in the world of algebra;
discuss the order of operation.
Working with Integers
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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.
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Next Table of Contents Back The Integers are natural numbers including 0 ( 0 , 1 , 2 , 3 , ...) and their negatives (0, −1 , −2, −3, ...). They are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2 ...}. Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5, ... . Negative integers are all the opposites of these whole numbers: -1, -2, -3, -4, -5, … . ] Lesson 20
WHAT ARE INTEGERS?
Objectives
After this lesson, the students are expected to:
define what integers are;
explain the difference between positive, zero and negative integers;
discuss the significance of integers.
Positive and Negative Integers
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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
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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
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Next Back WORKSHEET NO. 20 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________
Answer the following questions correctly.
Which integer represents this scenario?
A child grows 4 inches taller.
A loss of 3 dollars.
4 degrees above zero.
2 millimeter increase in volume.
4 kilogram increase in mass.
Weight gain 5 pounds.
5 gram decrease in mass.
Weight loss of 1 pound.
A child grows 9 inches taller.
7 millimeter decrease in volume
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Next Table of Contents Back Lesson 21 1) When adding integers of the same sign, we add their absolute values, and give the result the same sign. Examples: 2 + 5 = 7 (-7) + (-2) = - (7 + 2) = -9 (-80) + (-34) = - (80 + 34) = -114
ADDITION OF INTEGERS
Objectives
After this lesson, the students are expected to:
add integers correctly;
master the rules in adding integers;
analyze the given expressions.
In adding integers, the following must be considered:
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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
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Next Back WORKSHEET NO. 21 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ WRITE YOUR SOLUTION HERE: WRITE YOUR SOLUTION HERE:
Answer the following.
-56+90789= ____________________
1322+(-789)= __________________
465+(-88976)= _________________
-6789+(-467)= _________________
345+78= ______________________
232+(-4567)+(-56)= _____________
4523+7+(-789)= ________________
-978+(-789)+(-65)= _____________
212+(-6)+67= __________________
5679+(-432)+(-678)= ____________
-2457+789= ___________________
2178+(-578) ___________________
47+(-678)= ____________________
-678+(-98)= ___________________
236+(-76)= ____________________
Solve the following.
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Next Table of Contents Back Lesson 22 Subtracting an integer is the same as adding it ’ s opposite. Examples: In the following examples, we convert the subtracted integer to its opposite, and add the two integers. 7 - 4 = 7 + (-4) = 3 12 - (-5) = 12 + (5) = 17 -8 - 7 = -8 + (-7) = -15 -22 - (-40) = -22 + (40) = 18
SUBTRACTION OF INTEGERS
Objectives
After this lesson, the students are expected to:
discuss how to subtract integers;
perform the rules in subtracting integers;
analyze the given expression.
Note: The result of subtracting two integers could be positive or negative.
101.
WRITE YOUR SOLUTION HERE: WORKSHEET NO. 22 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________
-6543-678=________________________
3767-(-54)= _______________________
-456-578=_________________________
-263-12=___________________________
16287-(-678)= ______________________
-3647-(-67)= _______________________
3764-879=_________________________
345-(-768)= _______________________
679-(-668)= _______________________
-312-12______________________
A. Subtract the following integers. Next Back
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Next Table of Contents Back Lesson 23 To multiply a pair of integers if both numbers have the same sign, their product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative). If one or both of the integers is 0, the product is 0.
MULTIPLICATION OF INTEGERS
Objectives
After this lesson, the students are expected to:
discuss how to multiply integers;
master the rules in multiplying integers;
analyze the given expression.
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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.
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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
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SOLUTION WORKSHEET NO. 23 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________ Solve the following
-54x7=___________________________
768x(-753)= ______________________
-432x(-67)= _______________________
754x(-67)= _______________________
123x(-664)= ______________________
6788x(-7)= _______________________
12x(43)(-8)= ______________________
54x(-65)(5)= ______________________
56x8(-78)= _______________________
45x(-65)(45)= _____________________
56x(-97)(45)= _____________________
-2344x-65=_______________________
5423x(-7)= _______________________
56x(-67)= ________________________
-576x(-67)= _______________________
Next Back
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Next Table of Contents Back Lesson 24 To divide a pair of integers if both integers have the same sign, divide the absolute value of the first integer by the absolute value of the second integer. To divide a pair of integers if both integers have different signs, divide the absolute value of the first integer by the absolute value of the second integer, and give this result a negative sign.
DIVISION OF INTEGERS
Objectives
After this lesson, the students are expected to:
discuss how to divide integers;
master the rules in dividing integers;
analyze the given expression.
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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:
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Next Back SOLUTION WORKSHEET NO. 24 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________
solve expressions using some rules in solving integers;
discuss the series of operation.
Problem: Evaluate the following arithmetic expression shown in the picture: Student 1 Student 2 3 + 4 x 2 3 + 4 x 2 = 7 x 2 = 3 + 8 = 14 = 11
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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.
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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
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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.
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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.
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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.
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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:
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Next Back Example 6: Write an arithmetic expression for this problem. Then evaluate the expression using the order of operations. Mr. Smith charged Jill $32 for parts and $15 per hour for labor to repair her bicycle. If he spent 3 hours repairing her bike, how much does Jill owe him? Solution: 32 + 3 x 15 = 32 + 3 x 15 = 32 + 45 = 77 Jill owes Mr. Smith $77. SUMMARY:
When evaluating arithmetic expressions, the order of operations is:
Simplify all operations inside parentheses.
Perform all multiplications and divisions, working from left to right.
Perform all additions and subtractions, working from left to right.
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Next Back Solution WORKSHEET NO. 25 NAME: ___________________________________ DATE: _____________ YEAR & SECTION: ________________________ RATING: ___________
234x3-56+8=______________
(136-56+65)÷5=____________
343-65=___________________
1234+(-87)x8=______________
84-8+54(6)= _______________
4638-870=_________________
543+(-8)+(-78)(8)= __________
43+5786-57=______________
(6754-65+64)(7)=___________
78÷39+5-65=______________
A. Try to solve the following then explain.
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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
What did Tony see in his bonnet when he woke up grumpy? (38208÷48)-458
What will your money be if you spend part of it? (1725243+68745)÷324
What pimples do you have when you shiver? (1495153÷43)+235
What part of the body do you have below the knee? To find the answer do 704625÷125 then turn the calculator upside down and read the answer.
What does the dog do if it needs food ? 6272-5634
QUESTION :
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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
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