040 the whole module

Mastering Fundamental Operation and Integers 1

ADRIEL G. ROMAN

MYRICHEL ALVAREZ

AUTHORS

NOEL A. CASTROMODULE CONSULTANT

FOR-IAN V. SANDOVALMODULE ADVISER


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.

MISSION AND MAIN THRUST

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.

GOALS

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.

OBJECTIVES OF BACHELOR OF SECONDARY EDUCATION (BSEd)

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.


Overview

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:

1. discuss the four fundamental operations;

2. perform the operations well;

3. check the answers in addition and multiplication using their inverse

operation.


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.

326 + 258 = 584 Sum

Addends


Lesson 1

WHAT IS ADDITION?

Objectives:

After this lesson, the students are expected to:

define what addition is; identify the different properties of addition; perform the operation (addition) correctly.

How well do you remember your basic addition facts? In addition sentence,

326 + 258 = 584, which are the addends and which is the sum?

Addition is a mathematical method on putting things together. Adding whole

numbers together is a method that requires placing the numbers in column to get the answer.

Addition is represented by the plus sign (+). The addends and the sum are the two parts of

addition. The sum is the total and the addends are the numbers needed to add.

Examples:


1. 27 +31=58 the addends are 27 and 31 and the sum is 58.

2. 11+21=32 the sum is 32 and the addends are 11 and 21.

WORKSHEET NO. 1

NAME: ___________________________________ DATE: _____________

YEAR & SECTION: ________________________ RATING: ___________

1. ADDITION-

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_______________________________________________________________.

2. ADDENDS-__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

3. SUM-

A. Define the following terms


__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

1. 6585+8793=__________________

2. 4333+9586=__________________

3. 423381+46537=_______________

4. 546263+9520=________________

5. 31481+369=__________________

6. 23634+12438=________________

7. 3497+6826=__________________

8. 81650+3897601=______________

9. 7333+62766=_________________

10. 178654321+236754=___________

B. ADD THE FOLLOWING

SOLUTION


Lesson 2

PROPERTIES OF ADDITION

Objectives:


define the properties of addition; use the different properties of addition in solving; perform an operation using the properties of addition.

PROPERTIES OF ADDITION

1. The 0 Property in Addition

Examples:

8 + 0 = 8 27 + 0 = 27

10 + 0 = 10 31 + 0 = 31

This property states that any

number added to 0 is the number

itself, that is, if “a” is any number,

a + 0 = a.


2. The Commutative Property of Addition

Examples:

6 + 8 = 14 8 + 6 = 14

11 + 27 = 38 27 + 11 = 3

3. Associative Property of Addition

Examples:

(4 + 3) + 8 = 4 + (3 = 8) = 15

9 + (8 + 6) = (9 + 8) + 6 = 23

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).

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.:

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,


WORKSHEET NO. 2

NAME: ___________________________________ DATE: _____________


A. Identify the properties of the following:

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_________________________________

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,


8. 65 + 498 = 498 + 65____________________________

9. 9 + 5 = 5 + 9__________________________________

10. (10+10) + 10 = 10+ (10+10) =30_________________

Lesson 3

MASTERING SKILLS IN ADDING WHOLE NUMBERS USING ADDITION TABLE

Objectives:


use the addition in table properly; mastering skills in addition using tables; discuss the use of addition table.

Addition Table

The Addition Table can help you to master the addition operation

+ 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 2122

How to use

Example: 3 + 5

Go down to the "3" row then along to the "5" column,and there is your answer! "8"

+ 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

You could also go down to "5"

+ 1 2 3 4 5 6 7

1 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9

Mastering Fundamental Operation and Integers

13

WORKSHEET NO. 3

NAME: ___________________________________ DATE: _____________


FOLLOW THE INSTRUCTION

1. Have your own addition table2. With your addition table, add the following

a. 1+4, 0+1, 3+4, 5+0, 5+4b. 6+4, 7+2, 8+0, 9+2, 10+4c. 1+6, 3+6, 5+6, 3+10d. 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.

You could also go down to "5"

+ 1 2 3 4 5 6 7

1 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9

A. MOTHER OF ALL SCIENCE!!!


14

Add the following numbers correctly.

SOLUTION1. 593423+4467=_____________________

2. 359+4843=________________________

3. 1297+4548=_______________________

4. 696493+266=______________________

5. 1898976+219876=__________________

6. 78589+66533=_____________________

7. 6485092+1764243=_________________

8. 828637+86464=____________________

9. 12379+2873=______________________

10. 53746+783579=___________________

11. 642578+325646=___________________

12. 12398+6327355=___________________

13. 563745+654689=___________________

14. 57684+8765358=___________________


15

15. 425778+87654=____________________


16

Lesson 4

DIFFERENT METHODS IN ADDING WHOLE NUMBERS

Objectives:


solve addition using other methods; discuss the different methods in adding whole numbers; solve mathematical problems using the other method.

There are some easy ways in adding whole numbers.

A. Adding the column separately. Let 326+258 use as our illustrative example.

1. Adding in reverse order

a. 326 300+20+6 +258 200+50+8

500

b. 300+20+6200+50+8 500+70

c. 300+20+6200+50+8 500+ 70+14

a. Add the numbers in the hundreds place.

b. Add the numbers in the tens place.

c. Add the numbers in the ones place.

d. Then add their sum to get the total sum.


17

d. 500+70+14=584

2. Adding in column separately

EXAMPLE:

526 + 278

14

+ 9

7

804

Tocheck;

1. Add it upward.

2. Subtract the sum to one of the addends.

3. Add the numbers in the addends and in the sum if your answer in the

sum is the same as in the addends, then your answer is correct.

1. Arrange the numbers vertically.2. Add the numbers in the ones

place.3. Then add the tens place and

place the sum under the tens place.

4. Then add the numbers in column.


18

WORKSHEET NO. 4

NAME: ___________________________________ DATE: _____________


A. Using any of the given ways, add the following and write the answer in the space provided. Show all your solutions.

1. 39, 28_________________

2. 43, 29_________________

3. 69, 51_________________

4. 70, 623________________

5. 890, 431_______________

6. 343, 86________________

7. 987, 652_______________

8. 6232, 7434_____________

9. 853 234, 578____________

10. 6 754 236, 643 123_______

SOLUTION:


19

B. Perform the operation using the procedure discussed. Check your answer by using the short method.

1. 225+264=____________________________

2. 367+201=____________________________

3. 9 632+2 330=_________________________

4. 1 423+54 673=________________________

5. 543 265+65 223=______________________

6. 673 895 462+54 289=___________________

7. 629 075+57823=_______________________

8. 642 890+57 829=______________________

9. 564 872 389+54 738=___________________

10. 12 345+42 321=________________________

11. 3255+6472865=________________________

12. 6437286+56387=_______________________

13. 54390+529=___________________________

14. 6348901+65890=_______________________

15. 7395+7598043=________________________

WRITE YOUR SOLUTION


20

Lesson 5

SOLVING WORD PROBLEM

Objectives


discuss how to solve a word problem; solve any given problems systematically; use problem solving plan in solving any given word problem.

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.


21

PROBLEM SOLVING PLAN

1. Understand the problem.2. Devise a plan. 3. Carry out the plan.4. 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.}

2. DEVISE A PLAN

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.

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.


22

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.

.

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


23

WORKSHEET NO. 5

NAME: ___________________________________ DATE: _____________


A. Discuss the different problem solving plan briefly.

1. Understand the problem

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________.

2. Devise a plan-

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________.

3. Carry out the plan

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________.

4. Check the answer


24

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________.

B. APPLICATION: Solve the given word problem correctly.

1. Maria has 17 candies and her friends gave her 3 candies each. If she has 3 friends, how many candies did she have?__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

2. If Maria has 17 candies and gave 8 to her friends, how many are left?

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

3. If Louie has 25 balloons and he gave 5 to Jesreel, how many are left?

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________


25

__________________________________________________________________

_________________________________________________________________.

Lesson 6

APPLYING ADDITION OF WHOLE NUMBERS IN WORD PROBLEM

Objectives


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 =?

1. Add the ones: 5 + 8 = 13 ones = 1 ten + 3 ones.

2. Write 3 in the ones column and bring the 1 ten to

the tens column.

3. Add the tens: 1 +7 +2 = 12 tens = 1 hundred + 2

tens.

4. Write 2 under the tens column and bring the 1

11

875 +648

1 523


26

hundred to the hundreds column.

5. 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.

Add: 5 986 + 3 759 =?

1. 6 + 9 = 15 =10 + 5

2. 1 ten + 8 tens + 5 tens = 14 tens = 1 hundred + 4 tens.

3. 1 hundred + 9 hundreds +7 hundreds = 17 hundreds = 1

thousand + 7 hundreds.

4. 1 thousand +5 thousands + 3 thousands = 9 thousands.

Thus, 5 986 + 3 759 = 9 745

Add: 5 326 + 1 456 =?

Here is another example:

1 11

5 986 +3 759

9 745

15 326

+ 1 456

6782


27

1. 6 + 6 = 12 =10 + 2

2. 1 ten + 2 tens + 5 tens = 8 tens

3. 3 hundred + 4 hundreds=7 hundreds

4. 5 thousand +1 thousands= 6 thousands.

Thus, 5 326 + 1 456 =678

WORKSHEET NO. 6

NAME: ___________________________________ DATE: _____________


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?

_______________________________________________________________

_______________________________________________________________


28

_______________________________________________________________

_______________________________________________________________

______________________________________________________________.

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


29

sheetrock are needed to complete the 3 jobs?

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

____________________________________________________________

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.


30

Lesson 7

WHAT IS SUBTRACTION?

Objectives


define what is subtraction; identify the parts in subtraction; differentiate the subtraction from addition.

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

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.

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.


31

Let us consider the notation below.

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

12 addend

+ 6 addend

18 sum Minuend 18

Subtrahend - 6

Difference 12


32

.

WORKSHEET NO. 7

NAME: ___________________________________ DATE: _____________


A. Give the meaning of the following words.

1. Subtraction-________________________________________________

2. Minuend-__________________________________________________

3. Subtrahend-________________________________________________

4. Difference-_________________________________________________

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


33

B. Name the following parts of the mathematical expression given below.

D. Solve the following to get the difference

1. 5637584-43675=________________

2. 5389-782=_____________________

3. 43674-768=____________________

4. 376598-5281=__________________

5. 67396-683=____________________

12638 _____

- 3630 _____

___ 9008

4. 10,000

-6,543

3. 5428

-2001

WRITE YOUR SOLUTION HERE:

2. 1243

-360

1. 349

-265

C. Find the difference

5 hrs + 17 mins

- 3 hrs + 28 mins


34

6. 57290-7849=___________________

7. 56284-6847=___________________

8. 683963-68363=_________________

9. 6254-978=_____________________

10. 654-87=______________________

Lesson 8

MASTERING SKILLS IN SUBTRACTING WHOLE NUMBERS

Objectives


enhance the knowledge in terms of subtracting whole numbers; develop the speed in solving subtraction; perform the steps in subtracting whole numbers.

Cain Kiblah typed 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?


5 hrs + 17 mins 77mins

- 3 hrs + 28 mins 28mins

1 hr + 49mins


35

To make the subtraction convenient, we borrow 1 minute so we have:

WORKSHEET NO. 8

NAME: ___________________________________ DATE: _____________


A. Solve and get the difference

1. Subtract 381 from 1895







8. Subtract670 from 2064


36

9. Subtract 739 from 1591 10. Subtract 754 from 772

B. Simplify the following numbers

a. 10327-1685=____________

b. 74577-7658=____________

c. 9443-99195=____________

d. 14652-9195=____________

e. 19919-8881=____________

f. 8322-4909=____________

g. 8851-8453=____________

h. 7609-6957=____________

i. 8858-182=_____________

j. 8905-18=______________

Lesson 9

PROBLEM SOLVING INVOLVING SUBTRACTION

Objectives


follow the steps correctly in problem solving involving subtraction;

discuss the different steps in problem solving; develop the knowledge in problem solving.

To master the application of subtraction in problem solving, here are some examples:


37

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

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?


38

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

WORKSHEET NO. 9

NAME: ___________________________________ DATE: _____________


1. 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?

A. Get one whole sheet of paper and

solve the following problem.


39

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

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?

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

4. Martial law was declared in 1972. Now, it is 2009, how many years ago it was?

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

5. If Clark was born on December 31 2009, how old is he now?


40

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

6. What number will make 2 816 to become 5229?

____________________________________________________________________

____________________________________________________________________

____________________________________________________________________

___________________________________________________________________.

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?

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

___________________________________________________________________.

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


41

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?

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

____________________________________________________________________.

Introduction

In this chapter, you will learn about the multiplication operation,


42

Lesson 10

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.

Introduction

In this chapter, you will learn about the multiplication operation,


43

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.

WORKSHEET NO. 10

NAME: ___________________________________ DATE: _____________


A. Identify the following.

7 __________

× 2 __________

14 ___________

3 multiplicand

×5 multiplier

15 products

Multiplication is a repeated addition. It can be thought

of as addition repeated a given number of times.


44

B. Get the product of the following.

1. 32 x 25= 6. 14 x 193=

2. 10 x10 = 7. 66 x 15=

3. 25 x 68= 8. 157 x 11=

4. 31 x1545= 9. 655 x 8=

5. 27 x 17781= 10. 856 x 18=

Lesson 11

PROPERTIES OF MULTIPLICATION

Objectives


review the different properties of multiplication;

develop the knowledge in the properties of multiplication;

apply the properties of multiplication in solving problem.

1. IDENTITY PROPERTY

The product of the 1 and any number a is a, that is, 1 x a = a for any

number.


45

Example:

21 x a = 21 27 x a =27 31 x a = 3111 x a = 11 5 x a = 5 13 x a = 13

Example:

0 x 87 = 0 0 x 98 = 0 15 x 0 = 045 x 0 = 0 14 x 0 = 0 58 x 0 = 0

Example:

7 x 4 = 28 = 4 x 7 5 x 12 = 60 = 12 x 55 x 6 = 30 = 6 x 5 4 x 11 = 44 = 11 x 4

3. 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.

2. ZERO PROPERTY

The product of 0 and any number a is 0, that is a x 0 = 0 for any

number a.

4. ASSOCIATIVE PROPERTY

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.


46

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 = 656 x (7 + 9) = 42 + 54 = 9

WORKSHEET NO. 11

NAME: ___________________________________ DATE: _____________


1. 6 x 7 = __ x 6 6. (7 x __) + (__ x 6) = 7 x (3 +6)

5. 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).

A. Fill on the blank and identify the

property of each.


47

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) = _____

Lesson 12

MASTERING SKILLS IN MULTIPLYING WHOLE

NUMBERS

Objectives


multiply whole numbers in easy way;

develop the speed in multiplying whole numbers;

B. Fill the missing number. Use the property of multiplication to get product


48

perform multiplication correctly.

1 1 Carries

2 4

3 5 8 Multiplicand

x 2 5 Multiplier

1 7 9 0 1st partial product

+7 1 6 2nd partial product

8 9 5 0 Product

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:

How to use multiplication table?

In mastering the multiplication operation, knowing how

to multiply using multiplication table helps you to become

fluent in multiplying numbers.


49

Multiplication Table

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


50

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

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.

WORKSHEET NO. 12

NAME: ___________________________________ DATE: _____________



51

A. 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 =_____________

Lesson 13

“THE 99 MULTIPLIER” SHORTCUT IN MULTIPLYING

WHOLE NUMBER

Objectives




52

multiply whole numbers mentally;

appreciate exploring the world of multiplication;

appreciate the multiplication operation.

Here some examples:

1. 999×364= 364 000-364= 369 636 Why?

2. 2834×99= 283 400-2834= 280566 Why?

3. 31×999= 31 000-31= 30 969 Why?

WORSHEET NO. 13

NAME: ___________________________________ DATE: _____________


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.


53

A. Multiply the following using the “99 multiplier” method.

1. 99×99=________________________

2. 33×99=________________________

3. 47x99=________________________

4. 65x9=_________________________

5. 21x99=________________________

6. 81x99=________________________

7. 72x999=_______________________

8. 56x9999=______________________

9. 34x9=_________________________

10. 8x9=__________________________

B. Solve the following

1. Find the product of 873 and 9999=________________________

2. Find the product of 132 and 999=_________________________

3. Find the product of 665 and 99=__________________________

4. Find the product of 670 and 9=___________________________





9. Find the product of 493 and 999=_________________________


Lesson 14

“THE POWER OF TEN” MULTIPLICATION

Objectives


54


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

WORSHEET NO. 14

NAME: ___________________________________ DATE: _____________


When the factors are end in both zero, multiply the significant number and

used the number of zeros in both factors to the product.

When the factors are is 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.


55

A. 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 =________

B. 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=____________

Lesson 15

PROBLEM SOLVING INVOLVING MULTIPLICATION


56

Objectives


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.


Multiplication

Solution:

60 minutes = 1 hour

A screw machine can produce

95 screws in one minute. How many

screws it can produce in one hour?


57

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


Multiplication

Solution:

1 crate = 50 radios

32 crates x 50 radios = n

N = 1 600 portable radios

WORKSHEET NO. 15

NAME: ___________________________________ DATE: _____________

Therefore, there are 1 600 portable radios does the store have.

Therefore, the screw machine can produce 5 700 crews in one hour.


58


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?

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

4. A store has 124 boxes of pencils with 144 pencils in each box. How many

pencils they have?


59

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

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?

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

8. If a worker can make 357 bolts in one hour, how many bolts he can make in

eight hours?


60

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

9. If 1cubic yard of concrete costs P55.00, how much would 13cubic yards cost?

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

10. 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?

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

_________________________________________________________________.

SOLUTION:


61

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.


62

Lesson 16

WHAT IS DIVISION?

Objectives


define division; identify the parts of division; discuss the division operation.

Example

.

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

http://en.wikipedia.org/wiki/Addition

http://en.wikipedia.org/wiki/Subtraction

http://en.wikipedia.org/wiki/Multiplication

http://en.wikipedia.org/wiki/Arithmetic

http://en.wikipedia.org/wiki/Mathematics


63

In the above expression, a is called the dividend, b the divisor and c the 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 divisor, and the number 4 the

quotient.

quotient

divisor dividend or dividend ÷ divisor = quotient

http://en.wikipedia.org/wiki/Quotient

http://en.wikipedia.org/wiki/Divisor


64

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 the questions asks what number times 100 equals

100.

In general we conclude,

Any number divided by itself equals 1

Examples

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?

A. Division by Oneself

B. Division by 1


65

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.


Any number divided by 1 equals itself

Examples

12 ÷ 1 = 12 42 ÷ 1 = 42 33 ÷ 1 = 33

A.

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.


When Zero is the Dividend


66

Zero divided by any nonzero number equals zero

Examples

0 ÷ 4 = 0 0 ÷ 1 = 0 0 ÷ 24 = 0

B.

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.


Dividing by zero is impossible

Examples5 ÷ 0 = undefined 0 ÷ 0 = undefined 1 ÷ 0 = undefined

The Problem with Dividing by Zero


67

WORKSHEET NO. 16

NAME: ___________________________________ DATE: _____________


A. Give the name of the following unknown parts of division.

___________ 56÷8=7 _________

_______________

B. As far as you remember, try to divide the following.

1. 56÷7=

2. 54÷6=

3. 900÷100=

4. 64÷16=

5. 56÷8=

6. 122÷11=

7. 144÷12=

8. 256÷16=

9. 180÷9=

10. 360÷4=


68

Lesson 17

MASTERING SKILLS IN DIVISION OF WHOLE NUMBERS

Objectives


develop knowledge in dividing whole numbers;

follow the steps in dividing whole numbers;

master the 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

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.


69

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"

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

We can conclude that

4321 ÷ 6 = 720 R1

In general we write

Example

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

(Divisor x quotient) + Remainder = dividend


70

We can conclude that

18932 ÷ 37 = 511 R25

(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:

Brackets first O Divide Multiply Add Subtract

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 (Multiply before Add)

(b) 10 ÷ ( 2 + 3 ) = 10 ÷ 5 = 2 (Brackets before Division)

Take note: the remainder may also be expressed in decimals.

If you know your multiplication tables well, you should find it reasonably easy to do simple divisions in your head.


71

(c) 20 ÷ 2 ÷ 2 = 10 ÷ 2 = 5 (do operations left to right)

WORKSHEET NO. 17

NAME: ___________________________________ DATE: _____________


A. Work out the answers to the questions below and fill in the boxes.

Question 1

Use mental arithmetic to answer these questions (do not use a calculator). Then check.

(a) 16 ÷ 4 _________

(b) 12 ÷ 6 _________

(c) 15 ÷ 5 _________

(d) 20 ÷ 4 _________

(e) 18 ÷ 9 _________

(f) 40 ÷ 8 _________

(g) 36 ÷ 9 _________

(h) 15 ÷ 3 _________

(i) 64 ÷ 8 _________

(j) 42 ÷ 7 _________

(k) 24 ÷ 6 _________

(l) 32 ÷ 8 _________

http://www.cimt.plymouth.ac.uk/projects/mepres/book7/bk7i8/bk7_8i1.htm













72

B.

C.

(a) 10 ÷ 2 = 2 ÷ 10 __________

(b) 12 + 8 ÷ 2 = 10 __________

(c) 3 + 12 ÷ 4 = 6 __________

(d) 6 ÷ 2 + 3 = 6 __________

Work out the answers to the following questions (without a calculator).

(a) 3 + 4 × 8 __________

(b) 8 + 3 × 6 __________

(c) 8 × 6 - 4 __________

(d) 12 ÷ 2 + 5 __________

(e) 5 - 12 ÷ 3 __________

(f) 14 ÷ 2 + 8 __________

Use BODMAS to work out whether these statements are TRUE or FALSE.
















73

(g) 3 × 2 + 8 ÷ 4 __________

Lesson 18

“CANCELLATION OF INSIGNIFICANT ZEROS “EASY WAYS IN DIVIDING WHOLE NUMBERS

Objectives


divide whole numbers using other method;

perform division of whole numbers mentally;

define “ cancellation of insignificant zeros.”

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.



74

EXAMPLES:

101

1. 50 5050 505÷5=101 ( both dividend and divisor) -50

050

50

0

210

2. 5 1050 105÷5=21(10) =210 (the insignificant zero in

-10 dividend was cancelled)

50

-50

0

EXAMPLES:

1. 300÷100=3 or 300÷100=3 2. 30 000÷1 000=303. 2500÷100=25

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.

To check multiply the quotient to the divisor then multiply also the place value of the removed zeros


75

WORKSHEET NO. 18

NAME: ___________________________________ DATE: _____________


A. Divide the following using the Cancellation of Insignificant Method.

1. 640÷80=___________________

2. 140÷20=___________________

3. 36000÷600=________________

4. 700÷350=__________________

5. 3500÷70=__________________

6. 350÷ 100=__________________

7. 5600÷ 800=_________________

8. 600÷ 30=___________________

9. 100÷50=____________________

10. 800÷40=___________________


76

11. 1000÷ 100=_________________

12. 140÷ 70=___________________

13. 420 20=____________________

14. 14000÷ 70=_________________

15. 36000÷180=_________________

16. 4800÷ 240=_________________

17. 99000÷ 330=________________

18. 860÷ 20=___________________

19. 770÷ 770=__________________

20. 630÷ 30=___________________

B. CHALLENGE!!!

1. Copy the figure. Show how to divide it into 2 equal parts. Each part

must have the same size and shape.

2. Copy the figure again. Show how to divide it in 3 equal parts.

3. 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.



77

Lesson 19

PROBLEM SOLVING INVOLVING DIVISION OF WHOLE NUMBERS

Objectives


solve the given problem critically;

follow the steps in problem solving ;

apply the division of whole numbers in solving mathematical

problem.

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?

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


78

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?

1016 160 16x1=16

16 16-16=0 00 16x0=0 00 0

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


79

WORKSHEET NO. 19

NAME: ___________________________________ DATE: _____________


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?_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_______________________________________________________________.


80

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?

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________


81

_________________________________________________________________

________________________________________________________________.

Overview

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:


1. discuss the integers;

2. use the fundamental operations in solving integers;

3. appreciate the integers as a part of your discussion;

4. gain more knowledge about integers that will guide you in the world of

algebra;

5. discuss the order of operation.


82

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.

Overview

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:


1. discuss the integers;

2. use the fundamental operations in solving integers;

3. appreciate the integers as a part of your discussion;

4. gain more knowledge about integers that will guide you in the world of

algebra;

5. discuss the order of operation.


83

Lesson 20

WHAT ARE INTEGERS?

Objectives


define what integers are; explain the difference between positive, zero and negative

integers; discuss the significance of integers.

Positive and Negative Integers

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, … . ]

http://en.wikipedia.org/wiki/%E2%88%921_(number)

http://en.wikipedia.org/wiki/Negative_and_non-negative_numbers

http://en.wikipedia.org/wiki/3_(number)




http://en.wikipedia.org/wiki/Natural_numbers


84

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


85

Absolute Value of an Integer

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


86

WORKSHEET NO. 20

NAME: ___________________________________ DATE: _____________


A. 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


87

Lesson 21

ADDITION OF INTEGERS

Objectives


add integers correctly; master the rules in adding integers; analyze the given expressions.

In adding integers, the following must be considered:

Examples:

2 + 5 = 7

(-7) + (-2) = - (7 + 2) = -9

(-80) + (-34) = - (80 + 34) = -114

1) When adding integers of the same

sign, we add their absolute values, and give the

result the same sign.


88

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.


89

WORKSHEET NO. 21

NAME: ___________________________________ DATE: _____________


A. Answer the following.

1. -56+90789=____________________

2. 1322+(-789)= __________________

3. 465+(-88976)= _________________

4. -6789+(-467)= _________________

5. 345+78=______________________

6. -2457+789=___________________

7. 2178+(-578) ___________________

8. 47+(-678)= ____________________

9. -678+(-98)= ___________________

10. 236+(-76)= ____________________

B. Solve the following.

1. 232+(-4567)+(-56)= _____________

2. 4523+7+(-789)= ________________

3. -978+(-789)+(-65)= _____________

4. 212+(-6)+67=__________________

5. 5679+(-432)+(-678)= ____________




90

Lesson 22

SUBTRACTION OF INTEGERS

Objectives


discuss how to subtract integers; perform the rules in subtracting integers; analyze the given expression.

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


91

Note: The result of subtracting two integers could be positive or negative.

WORKSHEET NO. 22

NAME: ___________________________________ DATE: _____________


A. Subtract the following integers.

1. Find the value of:

a. -312-12______________________

b. -433-6534____________________

2. -263-12=___________________________

3. 16287-(-678)= ______________________

4. -3647-(-67)= _______________________

5. 3764-879=_________________________

6. 345-(-768)= _______________________

7. 679-(-668)= _______________________

8. -6543-678=________________________

9. 3767-(-54)= _______________________

10. -456-578=_________________________



92

Lesson 23

MULTIPLICATION OF INTEGERS

Objectives


discuss how to multiply integers; master the rules in multiplying integers; analyze the given expression.

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.


93

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.


94

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) = ?

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.


95

WORKSHEET NO. 23

NAME: ___________________________________ DATE: _____________


Solve the following

1. -2344x-65=_______________________

2. 5423x(-7)= _______________________

3. 56x(-67)= ________________________

4. -576x(-67)= _______________________

5. -54x7=___________________________

6. 768x(-753)= ______________________

7. -432x(-67)= _______________________

8. 754x(-67)= _______________________

9. 123x(-664)= ______________________

10. 6788x(-7)= _______________________

11. 12x(43)(-8)= ______________________

12. 54x(-65)(5)= ______________________

13. 56x8(-78)= _______________________

SOLUTION


96

14. 45x(-65)(45)= _____________________

15. 56x(-97)(45)= _____________________

Lesson 24

DIVISION OF INTEGERS

Objectives


discuss how to divide integers; master the rules in dividing integers; analyze the given expression.

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.


97

LOOK AT THE EXAMPLES:

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 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 below, both numbers are positive, so we just

divide as usual.

4 ÷ 2 = 2.

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.


98

WORKSHEET NO. 24

NAME: ___________________________________ DATE: _____________


A. Solve the following.

1. 56÷(-8)= ______________________

2. 54÷(-6)= ______________________

3. -99÷9=________________________

4. -144÷72=______________________

5. 24÷(-24)= ______________________

6. 81÷9=_________________________

7. 100÷(-4)= ______________________

8. -35÷7=________________________

SOLUTION


99

9. -124÷2=_______________________

10. 64÷(-32)=______________________

Lesson 25

PUNCTUATION AND PRECEDENCE OF OPERATION

Objectives


describe the use of punctuations in mathematics; solve expressions using some rules in solving integers; discuss the series of operation.

Problem: Evaluate the following

arithmetic expression shown in

the picture:

Solution: Student 1 Student 2

3 + 4 x 2 3 + 4 x 2

= 7 x 2 = 3 + 8

= 14 = 11


100


101

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.

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.


102

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.


103

Example 1: Evaluate each expression using the rules for order of operations.

Solution: 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

In Example 1, each problem involved only 2 operations. Let's look at some examples

that involve more than two operations.


104

Example 2: Evaluate 3 + 6 x (5 + 4) ÷ 3 - 7 using the order of operations.

Solution: 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 3: Evaluate 9 - 5 ÷ (8 - 3) x 2 + 6 using the order of operations.

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


105

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.

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.

Example 4: Evaluate 150 ÷ (6 + 3 x 8) - 5 using the order of operations.

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 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


106

Evaluating this expression, we get:

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:

1. Simplify all operations inside

parentheses.

2. Perform all multiplications and

divisions, working from left to right.

3. Perform all additions and

subtractions, working from left to

right.


107

WORKSHEET NO. 25

NAME: ___________________________________ DATE: _____________


A. Try to solve the following then explain.

1. 43+5786-57=______________

2. (6754-65+64)(7)=___________

3. 78÷39+5-65=______________

4. 234x3-56+8=______________

5. (136-56+65)÷5=____________

Solution


108

6. 343-65=___________________

7. 1234+(-87)x8=______________

8. 84-8+54(6)= _______________

9. 4638-870=_________________

10. 543+(-8)+(-78)(8)= __________

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

We can use these digits to

make a word in the calculator.

Let’s try to make words using

our calculator.

Solution


109

QUESTION:

3. What did Tony see

in his bonnet when

he woke up

grumpy?

(38208÷48)-458 4. What will your money be

if you spend part of it?

(1725243+68745)÷324

1. 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

2. What does the dog

do if it needs

food? 6272-5634


110

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

5. What pimples do you

have when you shiver?

(1495153÷43)+235

Solution


111

PUZZLE 3

Solution

Solution


112

Solution