DECIMAL NUMBERS2

Learning to Solve DECIMAL NUMBERS (Modular Workbook for Grade VI) Student Researchers/ Authors: PAMN FAYE HAZEL M. VALIN RON ANGELO A. DRONA ASST. PROF. BEATRIZ P. RAYMUNDO Module Consultant MR. FOR – IAN V. SANDOVAL Module Adviser 7 8 1 04 3 6 90 5

L aguna u p s tate olytechnic niversity VMGOs

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. Vision

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. Mission

In pursuit of college mission/vision the college of education is committed to develop the full potential of the individuals and equip them with knowledge, skills and attitudes in teacher education allied fields effectively responds to the increasing demands, challenge and opportunities of changing time for global competitiveness. Goals of College of Education

Objectives of Bachelor of Elementary Education 1. Produce graduates who can demonstrate and practice the professional and ethical requirements for the Bachelor of Elementary Education such as: 2. Acquire basic and major trainings in Bachelor of Elementary Education focusing on General Education and Pre - School Education. 3. Produce mentors who are knowledgeable and skilled in teaching pre - school learners and elementary grades and with desirable values and attitudes or efficiency and effectiveness. 4. Conduct research and development in teacher education and other related fields. 5. Extend services and other related activities for the advancement of community life.

This Teacher’s Guide Module entitled “Learning to Solve Decimal Numbers (Modular Workbook for Grade VI)” is part of the requirements in Educational Technology 2 under the revised curriculum for Bachelor in Secondary Education 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.

The students are provided with guidance and assistance of selected faculty members of the university 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 kind of activities offer a remarkable learning experience for the education students as future mentors especially in the preparation of instructional materials.

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 Educational Technology 2 BEATRIZ P. RAYMUNDO Assistant Professor II / Consultant LYDIA R. CHAVEZ Dean College of Education

This instructional modular workbook entitled “Learning to Solve Decimal Numbers (Modular Workbook for Grade VI)”, which geared toward the objective of making quality education available to all and offers you a very interesting and helpful friend in your journey to the world of decimal numbers. Dear Learners,

This modular workbook offers you many experiences in learning decimal numbers. This time, you will study how to read, write, and name decimal numbers and how to compare order and round off decimal numbers. Of course you will also express the equivalent fractions and decimals.

You will also experience the four (4) fundamental operations (Addition, Subtraction, Multiplication and Division) dealing with decimal numbers. Lastly, you will also solve more difficult problems involving the four (4) fundamental operations using decimal numbers. Learning decimal content is much more skillful in drilling with the application of FUN WITH MATH which designed to achieve with outmost skill and convenience.

The authors feel that you can benefit much from this modular workbook if you follow the direction carefully. Be mindful as you lead yourselves to challenge the situation and circumstances and as you faces in every living as well as for the near future. If you do these, you will realize that indeed this modular workbook can be a very interesting and helpful companion. The Authors

We would like to express our sincerest gratitude for the following whom in are ways or another help us making this modular workbook become possible: To Prof. Corazon N. San Agustin , for her kindness and understanding to this modular workbook.

To Mr. For – Ian V. Sandoval , our instructor and adviser in Educational Technology 2, for giving sufficient technical trainings, suggestions, constructive criticism and unending support in our every needs. To Assistant Professor Beatriz P. Raymundo , our Module Consultant, for making her available most of the time for comments, suggestions and revision of the modular workbook.

To Professor Lydia R. Chavez , our Dean, College of Education, for inspiring advises and encouragement. To our classmates and friends for their never ending support.

To our beloved families , for unconditional love, emotional, spiritual and financial support all the way to used and for the filling up our duties in our home. And most importantly to Almighty God , for rendering abilities, wisdom, good health, strength, courage, source of enlightenment and inspiration to pursue doing this piece of material. The Authors

VMGO’s FOREWORD PREFACE ACKNOWLEDGEMENT TABLE OF CONTENTS

UNIT I Decimal Numbers Lesson 1 What is Decimal? Lesson 2 Reading and Writing Decimal Numbers Lesson 3 Reading and Writing Mixed Decimal Numbers Lesson 4 Reading and Writing Decimal Numbers Used in Technical and Science Work Lesson 5 Place Value Lesson 6 Comparing Decimal Numbers Lesson 7 Ordering Decimal Numbers Lesson 8 How to Round Decimal Numbers? Lesson 9 The Self-Replicating Gene

UNIT II Equivalent Fractions and Decimals Lesson 10 Expressing Fractions to Decimals Lesson 11 Expressing Mixed Fractional Numbers to Mixed Decimals Lesson 12 Expressing Decimals to Fractions Lesson 13 Expressing Mixed Decimals Numbers to Mixed Numbers (Fractions)

UNIT III Addition and Subtraction of Decimal Numbers Lesson 14 Meaning of Addition and Subtraction of Decimal Numbers Lesson 15 Addition and Subtraction of Decimal Numbers without Regrouping Lesson 16 Addition and Subtraction of Decimal Numbers with Regrouping Lesson 17 Adding and Subtracting Mixed Decimals Lesson 18 Estimating Sum and Difference of Whole Numbers and Decimals Lesson 19 Minuend with Two Zeros Lesson 20 Problem Solving Involving Addition and Subtraction of Decimal Numbers

UNIT IV Multiplication of Decimals Lesson 21 Meaning of Multiplication of Decimals Lesson 22 Multiplying Decimals Lesson 23 Multiplying Mixed Decimals by Whole Numbers Lesson 24 Multiplication of Mixed Decimals by Whole Numbers Lesson 25 Multiplying Decimals by 10, 100 and 1000 Lesson 26 Estimating Products of Decimal Numbers Lesson 27 Problem Solving Involving Multiplication of Decimal Numbers

UNIT V Division of Decimal Numbers Lesson 28 Meaning of Division of Decimals Lesson 29 Dividing Decimals by Whole Numbers Lesson 30 Dividing Mixed Decimals by Whole Numbers Lesson 31 Dividing Whole Numbers by Decimals Lesson 32 Dividing Whole Numbers by Mixed Decimals Lesson 33 Dividing Decimals by Decimals Lesson 34 Dividing Mixed Decimals by Mixed Decimals

OVERVIEW OF THE MODULAR WORKBOOK In this modular workbook, you will understand the concept of the language of decimal numbers. This modular workbook, will help you to read, write, and name decimal numbers for a given models, standard, mixed and technical and science work form. It provides the knowledge about place value, with the aid of a place - value chart. It also provides information on how to compare and order decimal numbers and also how to round off decimal numbers. This module will provide you a more difficult work in mathematics. Exercises will help the learners evaluate themselves to understand decimal numbers.

OBJECTIVES OF THE MODULAR WORKBOOK After completing this modular workbook, you expected to: 1. Know the language of decimal numbers. 2. Read, write, and name decimal numbers in different forms. 3. Read and write decimal numbers with the aids of place - value chart. 4. Compare and order decimal numbers. 5. Rounding off decimal numbers by following its rule.

Lesson 1 WHAT IS DECIMAL? Lesson Objectives After accomplishing the lesson, the students are expected to: . Define decimals. . Identify the terms in decimal numbers. . Familiarize the language of decimal numbers.

One important feature of our number system is the decimal. It involved many computational operations. It is very useful in the measurement of very thin sheets and in the computation involving in exact amount. But what is decimal? Look at the following examples:

.3 = 3 .03 = 3 10 100 .003 = 3 .0003 = 3 1000 10000 b. .5 = 5 .05 = 5 10 100 .005 = 5 .0005 = 5 1000 10000

From the example given above, a “ decimal ” may be defined as a fraction whose denominator is in the power of 10.

Numbers in the power of 10 are 10, 100, 1000, 1000, etc. The dot before a digit in a decimal is called “ decimal point ” which is an indicator that the number is a decimal. The place on the position occupied by a digit at the right of the decimal point is called a “ decimal place ”.

I. Give the meaning and explain the use of the following. 1. What are decimals? 2. What is decimal point? 3. What is decimal place? 4. Give some examples of decimal numbers. 1 Worksheet

Decimals ____________________________________________________________________________________ 2. Decimal Point ____________________________________________________________________________________ 3. Decimal Place ____________________________________________________________________________________ 4. Examples ____________________________________________________________________________________

II. Change the decimal numbers to fractional form. Example: 0.8 = 8 10 1. 0.9 =_______________ 2. 0.1 =_______________ 3. 0.04 =_______________ 4. 0.06 =_______________ 5. 0.09 =_______________ 6. 0.001 =_______________ 7. 0.009 =_______________ 8. 0.0071 =_______________ 9. 0.0009 =_______________ 10. 0.0003 =_______________

11. 0.0004 =________________ 12. 0.0005 =________________ 13. 0.00008 =________________ 14. 0.00009 =________________ 15. 0.148 =________________ 16. 0.79 =________________ 17. 0.1459 =________________ 18. 0.6 =________________ 19. 0.01 =________________ 20. 0.051 =________________

Lesson 2 READING AND WRITING DECIMAL NUMBERS Lesson Objectives After accomplishing the lesson, you are expected to: . Read and write decimal numbers. . Follow the rules in reading and writing decimal numbers. . Use the place value chart in order to read and write decimal numbers.

How to read and write decimals or decimal numbers? A decimal is read and write according to the number of decimal place it has. Here are the rules in reading and writing decimal numbers.

RULE I. A decimal of one decimal place is to be read and to be written as tenth. .4 is read as “4 tenths” and is to be written as “four tenths”; 4/10 .2 is read as “2 tenths” and is to be written as “two tenths”. 2/10

RULE II. A decimal of two decimal places is to be read and to be written as hundredth. .35 is read as “35 hundredths” and is to be written as “thirty – five hundredths”; 35/100 .43 is read as “43 hundredths” and is to be written as “forty – three hundredths”.43/100

RULE III. A decimal of three decimal places is to be read and written as thousandth. .261 is read as “261 thousandths” and is to be written as “two hundred sixty – one thousandths”; 261/1000 .578 is read as “578 thousandths” and is to be written as “five hundred seventy – eight thousandths”.578/1000

RULE IV. A decimal of four decimal places is to be read and to be written as ten thousandth. .4917 is read as “4917 ten thousandths” and is to be written as “four thousand, nine hundred seventeen ten thousandths”; 4917/10,000 .5087 is read as “5087 ten thousandths” and is to be written as “five thousand eighty - seven ten thousandths”.5078/10,000

A decimal is read and written like an integer with the name of the order of the right most digits added. tenths hundredths thousandths ten thousandths hundred thousandths Millionths ten millionths hundred millionths billionths ten billionths hundred billionths trillionths 0 . 4 3 5 7 8 9 6 1 2 5 3 4

Note: the names of the order of the different decimal places. Quadrillionths Pentillionths Hexillionths Heptillionths Octillionths Nonillionths Decillionths Undecillionths Dodecillionths Tridecillionths… SEQUENCES

Examples: 0.4 Read as four tenths. 0.43 Read as forty-three hundredths. 0.435 Read as four hundred thirty-five thousandths. 0.4357 Read as four thousand, three hundred fifty-seven ten thousandths.

0.43578 Read as forty-three thousand, five hundred seventy-eight hundred thousandths. 0.435789 Read as four hundred thirty-five thousand, seven hundred eighty nine millionths. 0.4357896 Read as four million, three hundred fifty-seven thousand, eight hundred ninety-six ten millionths.

0.43578961 Read as forty three million, five hundred seventy-eight thousand, nine hundred sixty-one hundred millionths. 0.435789612Read as four hundred thirty-five million, seven hundred eighty nine thousand, six hundred twelve billionths. 0.4357896125 Read as four billion, three hundred fifty seven million, eight hundred ninety six thousand, one hundred twenty five ten billionths.

0.43578961253 Read as forty-three billion, five hundred seventy eight million, nine hundred sixty-one thousand, two hundred fifty three hundred billionths. 0.435789612534 Read as four hundred thirty-five billion, seven hundred eighty-nine million, six hundred twelve thousand, five hundred thirty-four trillionths.

I. Write each decimal numbers in words on the space provided. 1. 0.167213143____________________________ ______________________________________ 2. 0.52541876_____________________________ ______________________________________ 3. 0.263411859____________________________ ______________________________________ 4. 0.984562910____________________________ ______________________________________ 5. 0.439621512____________________________ _______________________________________ 2 Worksheet

II. Write the decimal number in standard form. 1. Nine tenths ______________________________________________ 2. Four hundredths ______________________________________________ 3. Two thousand, two hundred and two hundred thousandths ____________________________________________ 4. Four hundred seventy – six millionths ________________________________________________ 5. Forty thousand, one hundred forty – one millionths ________________________________________________

Lesson 3 READING AND WRITING MIXED DECIMAL NUMBERS Lesson Objectives At the end of the lesson, the students were expected to: . Read mixed decimal numbers. . Follow the rules in reading and writing mixed decimal numbers. . Write mixed decimal numbers.

Look at the following examples: a. 5.8 is read as “5 and 8 tenths” and is to be written as “five and eight tenths” b. 26.38 is read as “26 and 38 hundredths” and is to be written as “twenty – six and thirty – eight hundredths” c. 49.246 is read as “49 and 246 thousandths” and is to be written as “forty – nine and two hundred forty – six thousandths” d. 348.578 is read as “348 and 578 thousandths” and is to be written as “three hundred forty – eight and five hundred seventy – eight thousandths”

It is seen that the following rule has been followed in the above examples. RULE: In reading a mixed decimal numbers, read the integral part as usual “and” in place of the decimal point, the decimal point is read as usual also.

1.246.819_____________________________________________________________________________________ 2.65.42387____________________________________________________________________________________ 3.9023.145867_________________________________________________________________________________ 4.87.5843_____________________________________________________________________________________ 5.48.0089_____________________________________________________________________________________ Write the words of decimal number for each of the following: 3 Worksheet

II. Write decimal numbers for each of the following sentences: 1. Sixteen and sixteen hundredths _____________________________________________ 2.Two and one ten – thousandths _____________________________________________ 3.Ten thousand four and fourteen ten – thousandths _____________________________________________ 4. Ninety – nine billion and eight tenths _____________________________________________ 5. Twelve hundred two and seven millionths _____________________________________________

6. Ninety – nine and nine hundred nine thousand, nine millionths_________________________________ 7. Five billion and sixty – five hundredths ______________________________________________ 8. Three billion, six thousand and three thousand six millionths _____________________________________ 9. Seventy – one million, one hundred and fifty – five hundred thousandths ______________________________________________ 10. Two hundred two million, two thousand, two and two hundred two thousand two millionths ______________________________________________

Lesson 4 READING AND WRITING DECIMALS USED IN TECHNICAL AND SCIENCE WORK Lesson Objectives At the end of the lesson, the pupil should be able to: . Read and write decimals used in technical and science work. . Follow the rules in reading and writing decimals used in technical and science work. . Know the simple way of reading and writing decimals that can be used in technical and science work.

This method of reading decimals and mixed decimals is often used by people engaged in technical and science work. But this can be used by lay people especially if the part of the number has many digits. Observe the following examples:

a. 5.8 is read as “5 point 8” and is to be written as “five point eight” b. .9 is read as “point 9” and is to be written as “point nine” c. 6.893 is read as “6 point 893” and is to be written as “six point eight nine three” d. 348.09536 is read as “348 point 09536” and is to be written as “three four eight point zero nine five three six“ e. 8945.874205 is read as “8945 point 874205” and is to be written as “eight nine four five point eight seven four two zero five”

The rule followed in the above examples is as follows: To read decimals or mixed decimal numbers used in technical and science work or when the numbers of digits in the decimal is too many, just mention the values of the digits and separate the integral part by saying “point” instead of “and”. RULE:

0.009 Read:___________________________________________ Write:___________________________________________ 2. 45.78 Read:__________________________________________ Write:__________________________________________ 3. 3148 Read: ___________________________________________ Write:___________________________________________ 4. 3.456 Read:___________________________________________ Write:__________________________________________ I. Read and write the following in technical or science way. 4 Worksheet

4. 3.456 Read:______________________________________ Write:______________________________________ 5. 47.629 Read: ___________________________________________ Write:___________________________________________ 6. 5.78456 Read: ___________________________________________ Write:___________________________________________ 7. 0.491 Read:___________________________________________ Write:__________________________________________

8. 28.652 Read:__________________________________________ Write:_________________________________________ 9. 4928.95 Read:__________________________________________ Write:_________________________________________ 9. 4928.95 Read:__________________________________________ Write:_________________________________________ 10. 376.732 Read:__________________________________________ Write:_________________________________________

11. 841.50 Read:__________________________________________ Write:_________________________________________ 12. 3.62 Read:__________________________________________ Write:_________________________________________ 13. 0.03 Read:__________________________________________ Write:_________________________________________ 14. 97.5 Read:__________________________________________ Write:________________________________________ 15. 2.3148 Read:_________________________________________ Write:________________________________________

II. Write the following using decimal numbers. 1. one seven point three ___________________________________________ 2. point five four two nine ___________________________________________ 3. one two point zero nine ___________________________________________ 4. four three point one eight nine ___________________________________________ 5. two four point seven three two __________________________________________

6. three point seven six nine ______________________________________________ 7. two one seven point one five ____________________________________________ 8. point zero eight zero zero zero ___________________________________________ 9. nine point zero four zero ______________________________________________ 10. two point six seven two five ____________________________________________ 11. zero point nine eight nine ______________________________________________

12. zero point five two six eight two nine ____________________________________________ 13. five six zero point four zero one eight ____________________________________________ 14. one point one nine one eight ____________________________________________ 15. eight point five four three ____________________________________________

Lesson 5 PLACE VALUE Lesson Objectives In this lesson, the pupils are expected to: . Distinguish the relationship of place value in its place. . Write common fractions in decimal forms. . Give the place value for every digit.

PLACE VALUE CHART Place Value Names M I L L I O N S H T U H N O D U R S E A D N D S T T E H N O U S A N D S T H O U S A N D S H U N D R E D S T E N S O N E S T E N T H S H U N D R E D T H S T H O U S A N T H S T T E H N O U S A N T H S H T U H N O D U R S E A D N T H S M I L L I O N T H S Numerals 1 9 4 6 3 4 1 . 1 3 4 5 8 7 × × × × × × × . × × × × × × 10 6 10 5 10 4 10 3 10 2 10 1 1/10 0 1/10 1 1/10 2 1/10 3 1/10 4 1/10 5 1/10 6

What relationship exists in the diagram? What does the 1 in the tenths place mean? What does the 3 in the hundreds place represent? How about the 3 in the hundredths place?

Notice that: 0.1 = 1 × 1/10 = 1/10 (one tenth) 0.13 = 13 × 1/102 = 13/100 (thirteen hundredths) 0.134 = 134 × 1/103 = 134/1000 (one hundred thirty – four thousandths) 0.1345 = 1345 × 1/104 = 1345/10000 (one thousand three hundred forty – five ten thousandths) 0.13458 = 13458 × 1/105 = 13458/100000 (thirteen thousand four hundred fifty – eight hundred thousandths) 0.134587 = 134587 × 1/106 = 134587/1000000 (one hundred thirty – four thousand five hundred eighty – seven millionths)

Worksheet I. Complete the equivalent decimals to fractions. 5 Decimal Fraction 1. 0.23 2. 4.165 3. 0.937 4. 1.52 5. 0.041 6. 2.003 7. 0.1527 8. 16.775 9. 0.000658 10. 685.95

II. Answer the following. 1. In 246.819, what number is in each of the following place value? Example: __ 6 __ a. ones _ 246 _ c. hundreds _ 46 __ b. tens _ _.8 __ d. tenths _ .81 __ e. hundredths __ .819 _ f. thousandths 2. In 65.42387, tell what number is in each of the following places. _____a. tenths _____d. ten–thousandths _____b. hundredths _____e. hundred – thousandths _____c. thousandths _____f. ones _____g. tens

3. In 9023.45867, tell what number is in each of the following places. _____a. ones _____e. hundredths _____b. tens _____f. thousandths _____c. tenths _____g. thousands _____d. hundreds _____h. ten – thousandths _____i. hundred–thousandths _____j. millionths

Lesson 6 COMPARING DECIMAL NUMBERS Lesson Objectives At the end of the lesson, the pupils are expected to: . Compare decimal numbers. . Use fractional number to compare decimals. . Know the sign in comparing decimal numbers.

If there are two decimal numbers we can compare them. One number is either greater than (>), less than (<) or equal to (=) the other number.

A decimal number is just a fractional number. Comparing 0.7 and 0.07 is clearer if we compared 7/10 to 7/100. The fraction 7/10 is equivalent to 70/100 which is clearly larger than 7/1000.

Therefore, when decimals are compared start with tenths place and then hundredths place, etc. If one decimal has a higher number in the tenths place then it is larger than a decimal with fewer tenths. If the tenths are equal, compare the hundredths, then the thousandths, etc. Until one decimal is larger or there are no more places to compare. If each decimal place value is the same then the decimals are equal.

Worksheet I. Fill the frame with the correct sign (>) “less than”, (=) “equal to”, or (>) “greater than” between two given numbers. Example: 0.9 = 9/10 0.90 = 10/100 = 6

b. 9.004 0.040 f. 51.6 51.59 c. 20.80533 20.06 g. 50.470 50.469 d. 0.070 0.07 h. 0.90 0.9 e. 0.540 0.054 i. 0.003 0.03 j. 0.8000 0.080

Lesson 7 ORDERING DECIMAL NUMBERS Lesson Objectives After accomplishing the lesson, the pupils are expected to: Order decimal numbers. Know the terms in arranging decimal numbers. Understand how to arrange decimal numbers.

Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them.

Examples: If we start with numbers 4.3 and 8.78, the number 5.2764 would come between them, the number 9.1 would come after them and the number 2 would come before them. ( Descending- 9.1; 8.87; 5.2764; 4.3 ) 9.1> 8.87>5.2764>4.3 If we start with the numbers 4.3 and 4.78, the number 4.2764 would come before both of them; the number 4.5232 would come between them. ( Ascending- 4.2764; 4.3; 4.5232; 4.78) 4.2764< 4.3<4.5232<4.78

REMEMBER: The order may be ascending (getting larger in value) or descending (becoming smaller in value).

I. Write in order from ascending order and descending order by completing the table. 7 Worksheet Ascending Order Descending Order 1. 2.0342; 2.3042; 2.3104 Example: 2.0342 2.3042 2.3104 2.3104 2.3042 2.0342 2. 5; 5.012; 5.1; .502 3. 0.6; 0.6065; 0.6059;0.6061

5. 6.3942; 6.3924; 6.9342; 6.4269 6. 0.0990; 0.0099; 0.999; 0.90 7. 3.01; 3.001; 3.1; 3.001 8. 0.123; 0.112; 0.12; 0.121 9. 7.635; 7.628; 7.63; 7.625 4. 12.9; 12.09; 12.9100; 12.9150; 12 10. 4.349; 4.34; 4. 3600; 4.3560

FUN WITH MATH!!! Arrange the given decimal numbers from the least to greatest and you will find a famous quotation by Shakespeare.

Shakespeare (least) 7.301 All 8.043 climb 7.8 except 7.310 ambitious 8.88 or 7.84 those 9.100 of 7.911 which 10.5 mankind 7.33 are 8.43 up 8.513 upward 7.352 lawful 8.901 the 9.003 miseries

All ___ _______ ________ ________ ________ ________ 7.301 _______ ________ ________ ________ ________ _______ ________ ________ ________ ________ ________ _______ ________ ________ ________ ________ ________ _______ ________ ________ _______ ________ ________ . - Shakespeare II. Answer the following. a. The list below is the memory recall time of 5 personal computers. Which model has the fastest memory recall?

Answer: ___________________________________________ ___________________________________________ Model Recall Time Sterling PC 0.0195 sec. XQR 2000 0.01936 sec. Redi-mate 0.02045 sec. Vision 0.1897 sec. Sal 970 0.019 sec.

b. Arrange the memory recall time of computers in number 1 in ascending order. Answer: __________________________________________________________________________________ c. A carpenter uses different sizes of drill bits in boring holes. The sizes are in fractional form and their equivalent decimals. Arrange the decimal equivalent in descending order. 1/16 = 0.0625 ¼ = 0.25 1/8 = 0.125 5/6 = 0.3125

Answer: ___________________________________________ ___________________________________________ d. Which has the smallest decimal equivalent among the drill bits in item C? Answer: ________________________________________ ________________________________________

e. Which has the greatest decimal equivalent the drill bits in item C? Answer: ________________________________________ ________________________________________

Lesson 8 ROUNDING OFF DECIMALS Lesson Objectives After accomplishing the lesson, the pupils are expected to: 1. Round decimals. 2. Tabulate data in the chart. 3. Show rules in rounding decimal numbers.

To round decimal numbers means to drop off the digits to the right of the place-value indicated and replace them by zeros. The accuracy of the place-value needed must be stated and it depends on the purpose for which rounding is done. We give rounded decimal numbers when we do not need the exact value or number. Instead, we are after an estimated value or measure that will serve our purpose. These are many instances in daily life when rounded numbers are what we need to use.

How well do you remember in rounding whole numbers? Study the example below. Round to the nearest 4935 ten 4940 hundred 4900 thousand 5000

See how the following decimals are rounded. Rounded to the nearest 0.31659 tenths 0.3 hundredths 0.32 thousandths 0.317 ten thousandths 0.3166

To round decimals, follow these rules: 1. Look at the digit immediately to the right of the digit in the rounding place. 2. All digits to the right of the place to which the number is rounded are dropped. 3. If the first of the digits to be dropped is 0,1,2,3 or 4, the last kept digit is not changed. 4. Increase the last kept digit by 1, when the first digit dropped is: a. 6,7,8 or 9;or b. 5 followed by non-zero digit(s); or c. 5 (alone or followed by zero or zeros) and the last kept digit is odd.

Example: Round off 78.4651 to the nearest hundredths. 7 8 . 4 6 5 1 = 78.47 Dropping digit Decimal number to be rounded off Examples: Round the following. a. 5.767 to the nearest tenths = 5.8 Since the digit to the right of 7 is 6.

b. 65.499 to the nearest hundredths = 65.50 Since the digit to the right of 9 is 9. c. 896.4321 to the nearest thousandths = 896.432 Since the digit to the right of 2 is 1. d. 32.28 to the nearest tenths = 32.3 Since the digit to the right of 2 is 8 e. 1000.756 to the nearest hundredths = 1000.80 Since the digit to the right of 5 is 6 f. 56.58691 to the nearest thousandths = 56.5870 Since the digit to the right of 6 is 9

1. 29.8492 to the nearest: a. tenths ___________________ b. ones ___________________ c. hundredths ___________________ d. thousandths ___________________ e. tens ___________________ Round off the following decimal numbers. 8 Worksheet

2. 3.097591 to the nearest: a. ones _______________________ b. tenths _______________________ c. hundredths _______________________ d. thousandths _______________________ e. ten-thousandths _______________________ 3. 6.152292 to the nearest: a. ones ______________________ b. tenths ______________________ c. hundredths ______________________ d. thousandths ______________________ e. ten-thousandths ______________________

4. 10.01856 to the nearest: a. ones ____________________ b. tenths ____________________ c. hundredths ____________________ d. thousandths ____________________ e. ten-thousandths ____________________ 5. 123.831408 to the nearest: a. ones ____________________ b. tenths ____________________ c. hundredths ____________________ d. thousandths ____________________ e. ten-thousandths ____________________

III. Select the best answer. A. 0.4278 rounded to the nearest thousandths a. 0.462 b. 0.46 c. 0.430 d. 0.464 B. 0.0042 rounded to the nearest thousandths a. 0.003 b. 0.0031 c. 0.004 d. 0.04

C. 0.6354 rounded to the nearest thousandths a. 0.635 b. 0.6 c. 0.630 d. 0.64 D. 0.635`4 rounded to the nearest hundredths a. 0.635 b. 0.6 c. 0.630 d. 0.64 E. 0.6354 rounded to the nearest tenths a. 0.635 b. 0.6 c. 0.630 d. 0.64

IV. Answer the following with TRUE or FALSE. ________________ 1. 0.32 rounded to the nearest tenths is 0.3. ________________ 2. 0.084 rounded to the nearest hundredths is 0.09. ________________3. 0.483 rounded to the nearest thousandths is 0.048. ________________4. 0.075 rounded to the nearest hundredths is 0.06. ________________5. 0.375 rounded to the nearest tenths is 0.4.

V. Round each of the following by completing the tables. Number 1 serves as an example. Decimals Round to the nearest Tenths Hundredths Thousandths Ten Thousandths Example: 1. 0.89432 0.9 0.89 0.894 0.8943 2. 5.09998 3. 2.96425 4. 5.2358 5. 5.39485 6. 0.86302 7. 28154 8. 42356

9. 2.38425 10. 0.56893 11. 2.9625 12. 62.84213 13. 29.04347 14. 85.42998 15. 1539485

FUN WITH MATH!!! I. Find the answer by rounding off to the nearest place value indicated. Draw a line to the correct rounded number. Each line will pass through a letter. Write the letter next to the rounded number. ONES 1.6 ● ● 1.63 __________ 5.38 ● ● 3.4 __________ 52.52 ● ● 2 __________ TENTHS 0.45 ● ● 3.433 __________ 3.421 ● ● 53 __________ 12.76 ● ● 0.35 __________ 88.55 ● ● 5 __________ HUNDREDTHS 0.345 ● ● 12.8 __________ 1.634 ● ● 0.044 __________ 13.479 ● ● 0.5 __________ 201.045 ● ● 11.68 __________ 11.677 ● ● 16.778 __________ THOUSANDTHS 0.0437 ● ● 88.6 __________ 3.4325 ● ● 105.312 __________ 16.7777 ● ● 13.48 __________ 23.40092 ● ● 23.401 __________ 105.31238 ● ● 201.05 __________ T V E H W G E N T O O L H M S E T What happened to the man who stole the calendar?

Lesson 9 FACTS AND FIGURES (The self-replicating Gene) or centuries, generations of scientists in Numerica had been working relentlessly on what had been dubbed as The Genetic Enterprise. It was founded for the purpose of controlling a runaway gene that had beleaguered the Decimal citizens of Numerica for millennia: the repeating decimal gene. F 4 ___ 44

Their history revealed that it all began when a woman named Four (4) united with a man named Forty-Four (44) positioned as 4/44. Their first offspring, a fraction, came out all right, and they named her Three-Thirty-thirds (3/33). Such a beautiful fraction she was. But their second child came out with the first problematic replicating gene--- the boy looked different and came with a long tail: 0.0909090909… Arbitrarily multiplying both sides of the equation by any power of ten does not change the value of the decimal nor does it destroy the balance of the equation. This is because of the Multiplication Property of Equality of the Real Numbers.

That wasn’t the end of it. Every week, the boy’s tail added a new segment of 09, and it just never ended! It wasn’t so much the unusual appearance of the boy that worried every one, as the difficulty of naming him. Point-Zero-Nine-Zero-Nine-Zero-Nine- Zero -Nine-Zero-Nine, going on forever, was just too long a name! There were many others in the community whose tails also continuously and regularly increased. Despite their peculiar form, though, those trouble with the repeating gene were never shunned, were treated equally with love, respect, and total acceptance. Still, the continually growing tail proved cumbersome for the Decimals, and they prayed to be changed to regular forms.

Arbitrarily multiplying both sides of the equation by any power of ten does not change the value of the decimal nor does it destroy the balance of the equation. This is because of the Multiplication Property of Equality of the Real Numbers. Remember

One fateful day, the newspaper headlines screamed: “The Genetic Enterprise: Finally Success!” All of Numerica was thrilled! At the state conference the next day, every citizen was present, especially those with continuously growing tails. “ And now, I must ask for a volunteer,” said Doctor One Half (½), head scientist of the project, over the microphone. Immediately, 0.33333…, friend of Point-Zero-Nine-Zero-Nine…, was up the stage. “ O-Point-Three-Three-Three…” One Half began, “do not be afraid. Go into that glass capsule to your left, for we must first clone you.”

The crowd was aghast. One Half reassured every one immediately. “Do not worry, it is only temporary.” When 0.33333… came out, his clone came out from the other capsule. The doctor spoke again. “I will run you all through the process as we proceed. First, we shall designate the clone as Ex = 0.33333… “ Now, 0.33333…, go back into the capsule---we will introduce a new gene into you. This gene is called Tenn (10), and this will change your appearance, so that we will temporarily call you Tennexx (10x). Do not be alarmed!” One Half added quickly at every one’s reaction.

When 0.33333… stepped out of the capsule, he had become 3.33333… 10x = 10 x 0.3333… = 3.3333… 10x = 3.3333… The crowd was mesmerized. “ Tennexx, go back into the capsule. Exx, go into the other capsule. This time, we shall remove the repeating gene from Tennexx---by taking out Exx!” 10x – x = 3.3333… - 0.333… 9x = 3 Now, Tennex come out! Let us all see what you have become…” One Half said dramatically.

The purpose of assigning a variable and multiplying both sides of the equation by 10 is to come up with whole numbers on both sides of the equation (on one side, with the variable, and on the other side of the equation, with just an integer). From this form we obtain a fraction equal to the original decimal. Note when finally making a subtraction, the digits in the decimal parts MUST be the same in order for the difference to be an integer. FACT BYTES

When Tennex came out, he had become 1/3. 9x = 3 x = 3/9 or 1/3 The applause was thunderous! 1/3 spoke on the microphone, tears of joy poring down his cheeks. “Once! Was Zero-point-Three- -Three-Three-Three-Three… now I am One-Third. Thank you, Doctor One Half!” he said.

LESSON LEARNED In the article, we see that there is still hope for repeating Decimal genes like Point O Nine O Nine O Nine and O Point Three-Three-Three. It is all about representing them using any variable, say, x, and taking away their never-ending tail. Any repeating decimal represents a geometric series 0.3333… is 0.3 + 0.03+ 0.03 +… The common ratio is 0.1, that is, the next term is obtained by multiplying the previous term by 0.1. The formula S= a1/1-r may be used if (r) < 1. S = a1/1-r S = 0.3/1-0.1 = 0.3/0.9 or 1/3 FACT BYTES 1 __ 3

PROBLEM BUSTER A DIFFERENT GENE May I have another Volunteer? Ah, yes. 0.83333 Do you think we can change 0.833333...in exactly the same way as what we did to 0.33333…? Notice that in this case, the numeral 8 does not repeat. If we introduced 10 like we did to0.33333…, by multiplying 0.833333…by 10, 0.833333…will become 8.33333… Following the process, we have, 10x –x = 8.33333… - 0.833333… which is the same as 9x =7.5 where the right side of the equation is not an integer! What are we to do? 1 / 2 1 / 2

1 / What we need to do is keep multiplying by 10, until we get two numbers whose digits or numerals in the decimal parts are exactly the same. Thus, x = 0.833333 – 10x = 10 x 0.833333… -- 10x = 8.33333… -- 10x (10) = 8.33333… x 10 -- 100x = 83.3333… With 0.33333…, it was enough to subtract x from 10x because the digits or numerals in their decimal parts are already exactly the same. Recall that when we subtracted them, we arrived at an integer on either side of the equation. This time, we subtract 10x from 100x, because the decimal parts of these two numbers have exactly the same digits or numerals. So that, 100x – 10x =83.3333… - 8.3333… -- 90x =75 -- x = 75/90 -- x = 5/6 -- 0.833333… 5/6 Therefore, 0.833333… is 5/6! 1 / 2

1. 0.88888…as a fraction is: 5/11 b. 7/8 c. 8/9 d. 10/11 2. 0.22222… in fraction form is: a. 3/7 b. 2/11 c. 3/10 d. 2/9 Choose the correct answer: 9 Worksheet

II. Change the following to fraction in simplest form. 3. 0.77777… 4. 0.9166666… 5. 0.9545454… 6. 0.891891891… 7. 0.153846153846153846… 8. 0.9692307692307692307…

Unit II EQUIVALENT FRACTIONS AND DECIMALS

This modular workbook provides knowledge about different form or ways of computing fractions to decimals and decimals to fraction. This will help you to understand better what equivalent fraction and decimal is and you can use it in your everyday life. OVERVIEW OF THE MODULAR WORKBOOK

After completing this Unit, you are expected to: 1. Transform fraction/mixed fractional numbers to decimals/mixed decimal. 2. Change decimal/mixed decimal to fraction /mixed numbers (fractions). 3. Follow the rules in expressing equivalent fractions and decimals. OBJECTIVES OF THE MODULAR WORKBOOK

Lesson 10 EXPRESSING FRACTIONS TO DECIMALS Lesson Objectives After accomplishing the lesson, you are expected to: . Change fractions to decimals. . Know the rules in changing fractions to decimals. . Understand the equivalent fractions and decimals.

Decimals are a type of fractional number. Let us now study how to write fractions to decimal form.

We will apply the principle of equality of fractions that is, if a/b =c/d then ad = bc .

Example 1: Write the fraction 2/5 as a tenth decimal. In this case we are interested to find the value of x such that 2/5=x/10. Since the two fractions name the same rational number, we can proceed: 5x = 2(10) – applying equality principle 5x = 20 x = 20/5 or 4 Hence, 2/5 = 4/10 = 0.4

Example 2: Write the fraction 3 as a hundredth decimal. We are 4 interested to find the value of x such 3 that = x . 4 100 Applying the principle of equality we have 4x = 3(100) 4x = 300 x = 75 Hence, ¾ = 75/100 = 0.75

On the other hand, fractions can also be expressed as a decimal without using the equality principle. Instead we have to think of a fraction as a quotient of two integers that is a/b=a = a b. Example 3: Express 2/5 as a decimal. Expressing 2/5 as quotient of 2 and 5 we have 2/5 = 0.4

RULE To change a fraction to decimal, divide the numerator by the denominator up to the desired number of decimal places.

I. Give the meaning and explain the use of the following 1. How to change fractions to decimal? 2. What are the rules in changing fractions to decimals? 3. What is decimal? 4. Give some examples of fractions to decimals. 10 Worksheet

Change fractions to decimal __________________________________________ 2. Rules in changing fractions to decimals __________________________________________ 3. Decimal __________________________________________ 4. Examples of fractions to decimals __________________________________________

II. Change the following fractions to decimals. Limit the number to tree decimal places. 2/3 =_____________ 2. ¾ =___________ 3. 6/7 =_____________ 4. 8/9 =_____________ 5. 2/15 =_____________- 6. 1/9 =_____________ 7. 5/6 =_____________ 9. 4/5 =_____________ 10. 3/16 =_____________

11. 13/14 =__________ 12. ½ =__________ 13. 3/8 =__________ 14. 1/8 =__________ 15. 3/7 =__________ 16. 6/10 =__________ 17. 25/100 =__________ 18. 3/5 =__________ 19. 5/8 =__________ 20. 2/3 =__________

FUN WITH MATH!!! It was very fortunate that Sophie Germain , a woman mathematician was born at a time when people looked down on women. In 1776, women then were not allowed to study formal, higher level mathematics. Thus, this persistent woman reads books of famous mathematicians and studied on her own. Aware of her situation, she shared her theorems and mathematical formulae to other mathematicians and teachers through correspondence using a pseudonym.

Can you guess the pseudonym that she used? Yes, you can. Simply follow the instruction.

Select the right answer to the equation below. Write the letter of the correct answer on the respective number decode pseudonym that she used. You may use the letter twice. ______ ______ ______ ______ (1) (2) (3) (4) ______ ______ ______ ______ (5) (6) (7) (8) ______ ______ ______ (9) (10) (11) ______ ______ ______ ______ (12) (13) (14) (15)

Answers: A = 0.25 F = 0.65 K = 0.512 P = 0.27 B = 0.15 G = 0.28 L = 0.125 Q = 0.006 C = 0.6 H = 0.77 M = 0.333… R = 0.72 D = 0.54 I = 0.24 N = 0.40 S = 0.6 E = 0.76 J = 0.532 O = 0.75 T = 0.4113 U = 0.325

Lesson 11 EXPRESSING MIZED FRACTIONAL NUMBERS TO MIXED DECIMALS Lesson Objectives After accomplishing this lesson, you are expected to: . Express mixed fractional numbers to mixed decimals. . Know the rules in expressing mixed fractional numbers to mixed decimals. . Interpret the mixed fractional numbers to mixed decimals.

How can we change mixed fractional numbers to mixed decimals? See the following examples. 4 1/2 = 4.5 c. 21 1/8 = 21.125 14 3/8 = 14.375 d. 32 3/7 = 32.4285

From the examples given above, it can be seen that the rule in changing a mixed fractional number to mixed decimal is:

RULE To change a mixed fractional number to a mixed decimal, change the fraction to decimal up to the number of decimal places desired and then annex it to the integral part.

Worksheet I. Change the following mixed fractional numbers to mixed decimals. Limit the number to three decimal places. 4 2/5 = _____________________ 2. 3 4/5 = ______________________ 3. 7 3/16 = ______________________ 4. 10 13/14 = ______________________ 5. 12 9/17 = ______________________ 6. 21 14/19 = ______________________ 7. 32 21/41 = ______________________ 11

8. 2 ¼ = _______________ 9. 3 5/7 = _______________ 10. 4 ½ = _______________ 11. 8 ¼ = _______________ 12. 2 1/3 = _______________ 13. 5 4/6 = _______________ 14. 10 4/5 = _______________ 15. 3 ¼ = _______________ 16. 10 3/7 = _______________ 17.10 11/20 = _______________ 18. 8 3/10 = _______________ 19. 6 15/16 = _______________ 20. 8 1/10 =_______________

II. Copy the correct mixed decimal to mixed fractional numbers. 1 3/10 3. 31 503/100 a. 1.03 a. 31.0503 b. 1.30 b. 31.035 c. 1.013 c. 31.00503 d 1.13 d. 31.5030 2. 8 420/1000 4. 8 143/1000 a. 8.0420 a. 8.1430 b. 8.240 b. 8.0143 c. 8.420 c. 8.1043 d. 8.0042 d. 8.00143

5. 9 6/100 a. 9.16 b. 9.600 c. 9.006 d. 9.06

Lesson 12 EXPRESSING DECIMALS TO FRACTIONS Lesson Objectives At the end of the lesson, the students are expected to: . Change the decimals to fractions. . Follow the rule in expressing decimals to fractions. . Understand the equivalent decimals and fractions.

As what we have learned earlier, decimals are common fractions written in different way.

There are certain instances when it becomes necessary to change decimal into fraction. Hence, it is necessary to acquire skill in changing a decimal to faction. Now we will study how to write decimals in fractions.

Example 1: Write 0.5 in a faction form. 5 or 1 10 2 0.5 = 5(1/10) Example 2: Write 0.72 in a fraction form. 0.72 = 7(1/10) + 2(1/100) 18 25 = 72/100 or 18 25

On the other hand, a simple way of expressing decimal to factions is possible without writing the numeral in expanded form. What we need is only to determine the place value of the last digit as we read if from left to right.

Example 1: Write 0.5 in a faction form. Notice that the digit 5 is in the tenth place, we can write immediately: 0.5 = or 1 2 __ 5 __ 1000

The digit 2 is in the thousandths place so we write: 0.072 = 72/1000 = 9/125

Some Common Equivalent Decimals and Factions and 1/10 and 2/10 or 1/5 1.5 and 1 ½ or 1 5/10 or 1 ½ 0.25 and 25/100 or ¼ 0.50 and 50/100 or ½ 0.75 and 75/100 or ¾

Identifying Equivalent Decimals and Fractions Decimals are a type of fractional number. The decimal 0.5 represents the fraction 5/10. The decimal 0.25 represents the fraction 25/100. Decimal fractions always have a denominator based on a power of 10. We know that 5/10 is equivalent to 1/2 since 1/2 times 5/5 is 5/10. Therefore, the decimal 0.5 is equivalent to 1/2 or 2/4, etc.

It can be seen from the examples above the rule in changing a decimal to fraction is as follows:

RULE To change a decimal number to a fraction, discard the decimal point and the zeros at the left of the left-most non-zero digit and write the remaining digits over the indicated denominator and reduce the resulting fraction to its lowest terms. (The number of zeros in the denominator is equal to the number of decimal places in the decimal number.

Worksheet Change the following decimals to factional form and simplify them. 1. 0.4 = ________________ 2. 0.007 = ________________ 3. 0.603 = ________________ 4. 0896 = ________________ 5. 056 = ________________ 6. 0.06 = ________________ 7. 0.125 = ________________ 8. 0.5 = ________________ 9. 0.42857 = ________________ 10. 0.375 = ________________ 12

11. 0.54 = ________________ 12. 0.14 = ________________ 13. 0.8187 = ________________ 14. 0.956 = ________________ 15. 0.3567 = ________________ 16. 0.578 =_________________ 17. 0.34878 =_________________ 18. 0.47891 =_________________ 19. 0.12489 =_________________ 10. 0.14789 =_________________

FUN WITH MATH!!! How can you make a tall man short? To find the answer, change the following decimal number to lowest factional form. Each time an answer is given in the code, write the letter for that exercise.

1. 0.6 = A 6. 0.24 = _______ O 2. 0.5 = __ _____ B 7. 0.125 = _______ H 3. 0.7 = _______ N 8. 0.55 = _______ L 4. 0.4 = _______ I 9. 0.3 = _______ W 5. 0.75 = _______ O 10. 0.048 = _______ R 11. 0.25 = ______ O 12. 0.75 = _____ L 13. 0.2 = _____ E 14. 0.225 =______ O 15. 0.24 = _____ Y 16. 0.8 = _____ S 17. 0.5688=______ R

_____ _____ _____ ______ ______ _____ ½ 6/25 6/125 711/1250 225/ 1000 3/10 __ A ___ ______ ______ 3/5 ¾ 11/20 _____ ______ ______ 1/8 4/10 12/15 _____ _____ _____ _____ _______ 8/32 12/16 14/20 18/90 36/150

Lesson 13 EXPRESSING MIXED DECIMAL NUMBERS TO MIXED FRACTIONAL NUMBERS Lesson Objectives At the end of the lesson, the pupils should be able to . Express mixed decimal numbers to mixed fractional numbers. . Follow the rules in expressing mixed decimal numbers to mixed fractions. 3. Identify mixed decimals to mixed fractions.

How can we change mixed decimals to mixed fractions? Study the following examples:

5.03 = 5 3/100 b. 6.2 = 6 2/10 = 6 1/5 24.75 = 24 75/100 = 24 ¾ d. 37.248 = 37 248/1000 = 37 31/125 The rule applied to the above example is: RULE To change a mixed decimal number to a mixed fractional number, do not change the integral part, change the decimal part to a fraction according to the rule, and write the result as a mixed fractional number.

Worksheet Change the following mixed decimals to mixed fractional numbers. (First is an example.) 1. 3.06 = 3 6/10 6. 67.7362 = ___________ 2. 5.72 = ________ 7. 62.72 = ___________ 3. 11.302 = ________ 8. 71.4684 = ___________ 4. 10.642 = ________ 9. 92.5896 = __________ 5. 51.136 = ________ 10. 4.789 = __________ 13

II. Identify the following by writing D if it is mixed decimals and F if it is mixed fractional numbers. _____1. 1 217/100 _____ 11. 14.3245 _____ 2. 1.0124 _____ 12. 18 18/24 _____ 3. 1.4568 _____ 13. 9.28 _____ 4. 32 8/18 _____ 14. 1.0406 _____ 5. 2.510 _____ 15. 4 235/1000 _____ 6. 10.01 _____ 16. 450 11 /111 _____ 7. 39 45/100 _____ 17. 1.5345 _____ 8. 45 105/265 _____ 18. 143.445254 _____ 9. 101 81/411 _____ 19. 12 34/91 _____ 10. 1.01123 _____ 20. 653 185/1124

Unit III ADDITION AND SUBTRACTION OF DECIMALS NUMBERS

OVERVIEW OF THE MODULAR WORKBOOK This modular workbook provides you greater understanding in all aspects of addition and subtraction of decimal numbers. It enables you to perform the operation correctly and critically. It includes all the needed information about the addition and subtraction of decimal numbers, its terminologists to remember, how to add and how to subtract decimals with or without regrouping, how to estimate sum and differences, and subtracting decimal numbers involving zeros in minuends. This modular work will help you to enhance your minds and ability in answering problems deeper understanding and analysis regarding all aspects of adding and subtracting decimal numbers.

OBJECTIVES OF THE MODULAR WORKBOOK After completing this Unit, you are expected to: 1. Familiarize the language in addition and subtraction. 2. Learn how to add and subtract decimal numbers with or without regrouping. 3. Know how to check the answers. 4. Estimate the sum and differences and how it is done. 5. Know how to subtract decimal numbers with zeros in the minuend. 6. Develop speed in adding and subtracting decimal numbers. 7. Analyze problems critically.

Lesson 14 MEANING OF ADDITION AND SUBTRACTION OF DECIMAL NUMBERS Lesson Objectives: After accomplishing this lesson, you are expected to: . Define addition and Subtraction. 2. Identify the parts of addition and subtraction. 3. Familiarize the language in addition and subtraction.

Addition is the process of combining together two or more decimal numbers. It is putting together two groups or sets of thing or people.

Example: 0.5 + 0.3 = 0.8 Addends Sum or Total Addends are the decimal numbers that are added. Sum is the answer in addition. The symbol used for addition is the plus sign (+).

The process of taking one number or quantity from another is called Subtraction . It is undoing process or inverse operation of addition. It is an operation of taking away a part of a set or group of things or people. Note: Decimal points is arrange in one column like in addition of decimals.

Example: 14. 345 Minuend - 3.120 Subtrahend 11.232 Difference Minuend is in the top place and the bigger number in subtraction. The number subtracted from the minuend is called subtrahend . It is the smaller number in subtraction. The subtrahend is subtracted or taken from the minuend to find the difference. Difference is the answer in subtraction. The symbol used for subtraction is the minus sign (-).

Worksheet I. Give the meaning and explain the use of the following. 1. What is addition? 2. What is subtraction? 3. What are the parts of addition? 4. What are the parts of subtraction? 14

1. Addition ______________________________________________ 2 Subtraction ______________________________________________ 3. Parts of addition ______________________________________________ 4. Parts of subtraction ______________________________________________

II. Identify the following decimal numbers whether it is addends, sum, minuend, subtrahend or difference. Put an if addends, if sum, if minuend, if subtrahend and if difference. 1. 0.9 _______ + 0.8 _______ 1.7 _______ 2. 2.24 _______ + 2.38 _______ 4.62 _______ 3. 12.85 _______ - 0. 87 _______ 11.98 _______ 4. 7.602 _______ - 2.664 _______ 4.938 _______

5. 0.312 _______ + 0.050 _______ 0.362 _______ 6. 6.781 _______ - 1.89 _______ 8.676 _______ 7. 0.215 _______ + 0.001 _______ 0.216 _______ 8. 0.156 _______ + 1.811 _______ 1.967 _______ 9. 0.113 _______ + 0.009 _______ 0.122 _______ 10. 0.689 _______ - 1.510 _______ 2.199 _______

III. Answer the following by completing the letter in each box which indicate the parts of addition and subtraction of decimals. 1. It is the numbers that are added. 2. The answer in addition. 3. It is the process of combining together two or more numbers.

4. Sign used for addition. 5. It is undoing process or inverse operation of addition. 6. Sign used for subtraction. 7. It is the answer in subtraction.

8. It is in the top place and the bigger number in subtraction. 9. It is the smaller number in subtraction. 10. Subtraction is an operation of _________ a part of a set or group of things or people.

Lesson 15 ADDITION AND SUBTRACTION OF DECIMAL NUMBERS WITHOUT REGROUPING Lesson Objectives: After finishing the lesson, the students are expected to: . Know how to add and subtract decimal numbers without regrouping. 2. Develop speed in adding and subtracting decimal number. 3. Follow the steps in adding and subtracting decimal numbers.

Add the following decimals: 28. 143 and 11.721. If you added them this way, you are right. 28. 143 + 11. 721 39. 864 Let us add the decimals by following these steps.

STEP 1 STEP 2 Add the thousandths place 3+ 1 = 4 28. 143 + 11. 721 4 Add the hundredths place 4 + 2 = 6 28. 143 + 11. 721 64

STEP 3 STEP 4 Add the tenths place 7 + 1 = 8 28. 143 + 11. 721 864 Add the following up to the ones. 8 + 1 = 9 28. 143 + 11. 721 9. 864

STEP 5 Add the following up to the tens. 2 + 1 = 3 28. 143 + 11. 721 39. 864

Now subtract 39. 864 to 11. 721. 39. 864 minuend - 11. 721 subtrahend 28. 143 difference

2 Ways of Checking the Answer 1. minuend – difference = subtrahend 39. 864 minuend - 28. 143 difference 11. 721 subtrahend 2. difference + subtrahend = minuend 28. 143 difference + 11. 721 subtrahend 39. 864 minuend

If you subtract the difference from minuend and the answer is subtrahend the answer is correct. Also, adding the difference and subtrahend will the result to the minuend: it is also correct.

As a procedure for adding or subtracting decimal numbers, we have the following: Write the decimal numbers with the decimal points falling in one column. 2. Add or subtract as if they were whole numbers. 3. Place the decimal point of the result in the same column as the other numbers.

Worksheet Add and subtract as fast as you can. 15

FUN WITH MATH!!! Add and subtract the following to find the mystery words and write the letter of each answer in the code below. This appears twice in the Bible (In Matthew VI and Luke II).

85. 367 2. 645. 987 + 16. 252 - 314.625 R P 74. 617 + 21. 721 O 2,936. 475 - 1,421.061 S 51. 437 6. 658.325 + 18. 042 - 137.210 Y L 895. 399 8. 945. 374 - 471. 287 + 33. 161 A R 9. 32. 511 + 11. 621 R

7,649.251 11. 66.341 - 36.030 + 12.412 E D _______ 521. 115 _______ 96. 338 _______ 44. 132 _______ 78. 753 _______ 1515. 414 _______ 331.362 _______ 101.619 _______ 424.112 _______ 69.478 _______ 7613.221 _______ 978.535

Lesson 16 ADDITION AND SUBTRACTION OF DECIMAL NUMBERS WITH REGROUPING Lesson Objectives: After accomplishing the lesson, the students are expected to be able to: 1. Define regrouping. 2. Learn how to add and subtract decimal numbers with regrouping. 3. Answer and perform the operation critically and correctly.

In the past lesson, you’ve learned how to add and subtract decimal numbers without regrouping. The only difference in this lesson is that it involves regrouping and borrowing. It is easy to add and subtract decimal numbers without regrouping.

Regrouping is a process of putting numbers in their proper place values in our number system to make it easier to add and subtract. Here’s how to add decimal numbers with regrouping.

Example 1: 0. 7 + 0. 5 0.7 + 0.5 = 12 10 tenths is regroup as ( 1 ) one. Ones . Tenths 1 0 + 0 . . 7 5 1 . 2

Example 2: 0.09 + 0.06 0.9 + 0.6 = 15 hundredths 10 hundredths is 1 regrouped as 1 tenth. O . T H 0 0 . . 0 0 9 6 0 . 1 5

Example 3: 0.065 + 0.008 5 + 8 = 13 thousandths 10 thousandths is regrouped as 1 hundredth. O T H Th 0. + 0. 0 0 6 0 5 8 0. 0 7 3

Subtract decimals like you were subtracting whole numbers.

Example 4: 0. 93 - 0. 28 9 is renamed as 8 + 1 tenths. 1 tenth is regrouped as 10 hundredths. 0. 9 3 - 0. 2 8 0. 6 5 Check: 0. 28 + 0. 65 0. 93 ones tenths hundredths 0. 9 3 0. 8 - 1 10 0. 8 3 0. 8 13

Example 5: 0.730 - 0.518 2 10 0.730 - 0.518 0.212 Check: 0.518 + 0.212 0.730 Answer ones tenths hundredths thousandths 0. 7 3 0 0. 7 2+1 10 0. 7 - 5 2 - 1 0 - 8 0. 0. 2 1 2

Worksheet I. Answer the following. A. Add the following and check your answer on the Check Box below. 0.6 2. 0.07 + 0.8 + 0.49 0.36 4. 0.746 + 0.56 + 0.235 16

B. Subtract the following and check your answer on the Check Box below. 1. 0.62 2. 0.762 - 0.58 - 0.325 3. 0.850 4. 0.452 - 0.328 - 0.235

II. Write on the blank ( + ) or ( - ) sign to make the statement TRUE. 1. 4.793 ___ 3.549 = 8.342 2. 72.685 ___ 45.726 ___ 13.493 = 104.918 3. 1.45 ___ 0.50 ___ 3.95 ___ 5.66 = 11.56 4. 36.58 ___ 35.789 ___ 354.587 = 426.956 6.57 ___ 0.456 ___ 236.5 ___ 5 ___ 213.66 = 34.866 6. 28. 625 ___ 25.361 = 3.264 7. 57.54 ___ 0.25 = 57.29 8. 86.3 ___ 0.456 ___ 32.58 = 118.424 9. 39 ___ 5.65 = 33.35 10. 53.654 ___ 5.236 = 48.418

Lesson 17 ADDING AND SUBTRACTING MIXED DECIMALS Lesson Objectives: After finishing the lesson, the students are expected to: 1. Understand and know how to add and subtract mixed decimal numbers. 2. Follow the rules in adding and subtracting mixed decimal numbers. 3. Perform the operation correctly.

Ramon traveled from his house to school, a distance of 1.39845 kilometers. After class, he traveled to his friend’s house 1.85672 kilometer away in another direction. From his friends to his own house, he rode another 1.23714 km over. How many kilometers did Ramon traveled? 3 . T H Th T Th H Th 1 1 1 +1 . . . 1 3 8 2 2 9 5 3 1 8 6 7 1 4 7 1 5 2 4 4 . 4 9 2 3 1

He traveled a total of 4.49231 km. The following day, he traveled to the school and the seashore for a total of 6.35021 km. How many more kilometers did Ramon traveled than previous day? O T H Th T Th H Th 5 6 -4 . . 12 3 4 14 5 9 9 0 2 12 2 3 1 1 1 . 8 5 7 9 0

Ramon traveled 1.85790 kilometers more. In adding and subtracting mixed decimals, remember to align the decimal points and regroup when necessary.

Worksheet I. Add or subtract these mixed decimals. 4.59804 2. 3.14879 3. 5.11788 7.81657 5.37896 1.93523 + 1.30493 + 2.95321 + 3.40175 4. 2.42814 5. 7.20453 6. 9.57128 - 1.19905 - 4.35712 - 2.89340 17

II. Rewrite with the correct alignment of decimal points on the space provided. Find the sum and difference. 1. 4.930000 4. 18.17932 57.5244 + 2.41256 + 637.3672 73.59203 5. 12.48004 + 154.38762 - 9.86327 3. 142.567021 6. 42.20239 - 85.791503 - 2.34876

4. 18.16532 9. 5.306321 - 4.01985 002.7509 + 4.952005 5. 951.235 7.18902 10. 103.93284 + 00.3 + 43.76895

Lesson 18 ESTIMATING SUM AND DIFFERENCE OF WHOLE NUMBERS AND DECIMALS Lesson Objectives: After understanding the lesson, you must be able to: . Define estimation. 2. Know the two methods in making estimates. 3. Learn how to estimate sum and difference and how it is done.

Estimation is a way of answering a problem which does not require an exact answer. An estimate is all that is needed when an exact value is not possible. Estimation is easy to use and or to compute. Rounding is one way of making estimation. Each decimal number is rounding to some place value, usually to the greatest value and the necessary operation is performance on the rounded decimal numbers.

Two methods are used in making estimation, the rounding off the desired digit one and finding the sum of the first digit only. We have learned how to round decimal numbers in this section, first only the front digits are used. If an improved or refined estimate is desired, the next digits are used.

When large decimal numbers are involved, it is wise to estimate before computing the exact and user is expected to be about or close to the estimate. Method 1: Sum of the First Digit only Estimate in Addition 3.455 + 2.672 + 5.135

Rounded off to the nearest ones 3 .455 3.000 2 .672 3.000 + 5 .134 + 5.000 11.000 first estimate Rounded off to the nearest tenths 3. 4 55 0.500 2. 6 72 0.700 + 5. 1 34 + 0.100 1.300 to be added the first estimate if desired or required.

Thus the sum 3.455 + 2.672 + 5.134 can be roughly estimated by 11.000. If a better estimate is required or desired, then add 1.300 to get 11.300.

Estimate 5.472147 – 2.976543 Rounded to the nearest ones Actual Subtraction 5.472147 5.000000 5.472147 - 2.976543 - 3.000000 - 2.976543 2.000000 2.495604

Method 2: Rounding Method Estimate the sum by rounding method in place of whole numbers. Example: 6.567 7.000 5.482 5.000 + 4.619 + 5.000 17.000

b. Estimate the difference by rounding method. Example : 14.525 15.000 - 11.018 - 11.000 4.000 By the rounding method, the first example is estimated by 17.000 and the second one by 4.000. The actual value of the sum of example no.1 is 16.668 and the difference of example no. 2 is 3.507 respectively. Both methods give a reasonable estimate.

Remember: In estimating the sums, first round each addend to its greatest place value position. Then add. If the estimate is close to the exact sum, it is a good estimate. Estimating helps you expect the exact answer to be about a little less or a little more than the estimate. However, in estimating difference, first round the decimal number to the nearest place value asked for. Then subtract the rounded decimal numbers. Check the result by actual subtraction.

Worksheet I. Estimates the sum and difference to the greatest place value. Check how close the estimated sum (E.S.) / estimated difference (E.D.) by getting the actual sum (A.S.) and actual difference (A.D.) . A. Actual Sum/ Estimated Sum 1. 3.417 3.000 2. 36.243 36.000 2.719 3.000 29.641 30.000 + 1.829 + 2.00 + 110.278 + 110.000 A.S. E.S. A.S. E.S. 18

3. 648.937 649.000 4. 871.055 871.000 214.562 215.000 276.386 276.000 + 450.211 + 450.000 + 107.891 + 108.000 A.S. E.S. A.S. E.S. 5. 374.738 375.000 6. 342.165 342.000 469.345 469.000 178.627 179.000 + 213.543 + 213.500 + 748.715 + 749.000 A.S. E.S. A.S. E.S.

B. Actual Difference/ Estimated Difference 7. 14.255 14.000 8. 28.267 28.000 - 11.812 - 12.000 - 16.380 - 16.000 A.D. E.D A.D. E.D. 9. 345.678 346.000 10. 92.365 92.000 - 212.792 - 213.000 - 75.647 - 76.000 A.D. E.D. A.D. E.D. 11. 62.495 62.000 12. 9.2875 9.0000 - 17.928 - 18.000 - 6.8340 - 7.0000 A.D. E.D. A.D. E.D.

FUN WITH MATH!!! Match a given decimals with the correct estimated sum / difference to the greatest place – value. The shortest verse in the Bible consists of two words.

To find out, connect each decimals with he correct estimated sum / difference to the greatest place – value. Write the letter that corresponds to the correct answer below it. 1. 36.5+18.91+55.41 U. 939.00 2. 639.27-422.30 S. 216.00 3. 48.21+168.2 P. 2.0000 4. 285.15+27.35+627.30 E. 146.000 5. 8.941-8.149 W. 28.10 6. 18.95+9.25 J. 111.00 7. 129.235+16.41 T. 537.00 8. 9.2875-6.834 S. 1.000 9. 989.15-451.85 E. 217.00

_____ ______ ______ ______ ______ 1 2 3 4 5 _____ ______ ______ ______ 6 7 8 9

Lesson 19 MINUEND WITH TWO ZEROS Lesson Objectives: After accomplishing the lesson, the students are expected to be able to: . Know how to subtract decimal numbers with two zeros in minuend. 2. Follow the steps in subtraction of numbers involving zeros. 3. Check the answer and perform the operation correctly.

You always have to regroup in subtracting decimal numbers with zeros. You will have to regroup from one place to the next until all successive zeros are renamed and ready for subtraction.

STEPS IN SUBTRACTION OF DECIMAL NUMBER INVOLVING ZEROS . Arrange the digits in column. . Regroup from one place to the next until all successive zeros are renamed. . Subtract to find the answer. . Check the answer.

Example: 0.8005 - 0.6372 O T H Th T Th 0. 8 0 0 5 0. 7+1 10 9+1 10 0. 7 9 10 5 0. 6 3 7 2 0. 1 6 3 3

Rewriting: 0.8005 - 0.6372 Difference 0.1633 Checking: 0.6372 + 0.1633 0.8005

Worksheet I. Subtract the following and check. 1. 16.004 - 2.875 2. 28.009 - 11.226 3. 18.003 - 5.739 4. 11.001 - 9.291 5. 4.0075 - 2.9876 0.10013 - 0.00011 7. 2.00143 - 0.88043 0.7008 - 0.5383 9. 0.8008 - 0.0880 10. 0.14003 - 0.03333 19

FUN WITH MATH!!! Answer the following to find the mystery words. In what type of ball can you carry? To find the answer, draw a line connecting each decimal number with its equal difference. The lines pass through a box with a letter on it. Write what is in the box on the blank next to the answer.

Lesson 20 PROBLEM SOLVING INVOLVING ADDITION AND SUBTRACTION OF DECIMALS Lesson Objectives: After accomplishing the lesson, the students are expected to be able to: . Follow the step of solving problem. 2. Analyze the problem critically. 3. Develop interest in solving word problem.

Kristina saves her extra money to buy a pair of shoes for Christmas. Last week she saved Php. 82.60; two weeks ago, she saved Php. 100.05. This week she saved Php. 92.60. How much did she save in three weeks? Steps in Solving a Problem 1. Analyze the problem 2. What is asked? Total amount did Kristina save in three weeks. 3. What are the given facts? Php. 82.60, Php. 100.05, and Php. 96.10 Know

3. What is the word clue? Save. What operation will you use? We use addition. 4. What is the number sentence? Php. 82.60 + Php. 100.05 + Php. 96.10 = N 5. What is the solution? Php. 82.60 Php. 100.05 + Php. 96.10 Php. 278.75 Solve Decide Show

Check 6. How do you check your answer? We add downward. Php. 82.60 Php. 100.05 + Php. 96.10 Php. 278.75 “ Kristina saves Php. 278.75 in three weeks.” It is easy to solve word problems by simply following the steps in solving word problem.

Worksheet I. Read the problem below and analyze it. A. Baranggay Maligaya is 28.5 km from the town proper. In going there Angelo traveled 12.75 km by jeep, 8.5 km by tricycle and the rest by hiking. How many km did Angelo hike? 1. What is asked? __________________________________________________________________________________________ 2. What are the given facts? __________________________________________________________________________________________ 20

3. What is the process to be used? ______________________________________________________________________________________________ 4. What is the mathematical sentence? ______________________________________________________________________________________________ 5. How the solution is done? 6. What is the answer? ______________________________________________________________________________________________

7. How do you check the answer? B. Faye filled the basin with 2.95 liters of water. Her brother used 0.21 liter when he washed his hands and her sister used 0.8 liter when she washed her face. How much water was left in the basin?

1. What is asked? __________________________________________________________________________________________ 2. What are the given facts? __________________________________________________________________________________________ 3. What is the process to be used? __________________________________________________________________________________________ 4. What is the mathematical sentence? __________________________________________________________________________________________ 5. How the solution is done?

6. What is the answer? __________________________________________________________________________________________ 7. How do you check the answer?

C. Ron cut four pieces of bamboo. The first piece was 0.75 meter; the second was 2.278 meters; the third was 6.11 meters and the fourth was 6.72 meters. How much longer were the third and fourth pieces put together than the first and second pieces put together? 1. What is asked? __________________________________________________________________________________________ 2. What are the given facts? __________________________________________________________________________________________

3. What is the process to be used? __________________________________________________________________________________________ 4. What is the mathematical sentence? __________________________________________________________________________________________ 5. How the solution is done? 6. What is the answer? _________________________________________________________________________________________

7. How do you check the answer? D. Pamn and Hazel went to a book fair. Pamn found 2 good books which cost Php. 45.00 and Php. 67.50. She only had Php.85.00 in her purse but she wanted to buy the books. Hazel offered to give her money. How much did Hazel share to Pamn?

E. Marlene wants to buy a bag that cost Php. 375.95. If she has saved Php. 148.50 for it, how much more does she need? 1. What is asked? __________________________________________________________________________________________ 2. What are the given facts? __________________________________________________________________________________________ 3. What is the process to be used? __________________________________________________________________________________________

4. What is the mathematical sentence? ______________________________________________________________________________________________ 5. How the solution is done? 6. What is the answer? ______________________________________________________________________________________________ 7. How do you check the answer?

UNIT IV MULTIPLICATION OF DECIMALS

OVERVIEW OF THE MODULAR WORKBOOK This modular workbook provides you with the understanding of the meaning of multiplication of decimals, multiply decimals in different form and how to estimate products. It will develop the ability of the students in multiplying decimal numbers. This modular workbook will help you to solve problems accurately and systematically.

OBJECTIVES OF THE MODULAR WORKBOOK After completing this Unit, you are expected to: 1. Define multiplication, multiplicand, multiplier, products and factors. 2. Know the ways of multiplying decimal numbers. 3. Learn the ways of multiplying decimal numbers involving zeros. 4. Learn how to make an estimate and know the ways of making estimates.

Lesson 21 MEANING OF MULTIPLICATION OF DECIMAL NUMBERS Lesson Objectives: After learning this lesson, you are expected to: Define multiplication. 2. Locate where the multiplicand, multiplier and product are. 3. Familiarize the terms in multiplication.

Multiplication is a short cut for repeated addition. It is a short way of adding the same decimal number. It is the inverse if division. .4 + .4 + .4 + .4 + .4 + .4 = 2.4 In multiplication, it is written as: .4 -> multiplicand x 6 -> multiplier 2.4 -> product (answer in multiplication) factors

The decimal numbers we multiply are called multiplicand and multiplier is the decimal number that multiplies. The answer in the multiplication is the product . The decimal numbers multiplied together are factors . Another examples: 9 0.08 1.24 0.007 x 0.5 x 3 x 2 x 4 4.5 0.24 2.48 0.028

1. What is multiplication? 2. What are factors? 3. What are products? 4. Give some examples of multiplication decimals. I. Give the meaning and explain the use of the following. 21 Worksheet

multiplication ________________________________________________________________________________ 2. factors ________________________________________________________________________________ 3. products ________________________________________________________________________________ 4. Examples of multiplication decimals ________________________________________________________________________________

Identify the words by looping vertically ,horizontally and diagonally directions. (Word – Puzzle)

____________ 1. The number we if multiply. ____________ 2. The numbers multiplied together. ____________ 3. The number that multiplies. ____________ 4. It is a short way of adding the same number of number times. ____________ 5. Multiplication is the inverse of _____________

Complete the following that corresponds to the missing answer. 1. 0.42 - ______ 6. 0.183 - ______ x 0.34 - ______ x 0.141 - ______ _____ - product _____ - product 2. 0.12 - ______ 7. 12.55 - ______ x ____ - multiplier x 21.45 - ______ 0.0132 - ______ _____ - product 3. ____ - multiplicand 8. ____ - multiplicand x 4.62 - _______ x 0.96 - _______ 0.1848 - _______ 0.1848 - _______

4. 56.08 - ______ 9. 1.45 - ______ x 31.901 - ______ x 6.56 - ______ _______ - product ______ - product 5. 8.08 - multiplicand 10. 8.145 - multiplicand x 8.14 - multiplier x 6.001 - multiplier _____ -________ _____ -________

Lesson 22 MULTIPLYING DECIMALS Lesson Objectives: After finishing the lesson, the students are expected to: 1. Learn how to multiply decimal numbers. . Follow the steps in multiplying decimal numbers. . Know how to place the decimal point in the product.

Study these examples. Where do you place the decimal point in the product? 0.432 0.614 × 0.15 × 0.37 2160 4298 + 432 + 1842_ 0.06480 0.22718

Remember: In multiplying decimals, the placement of the decimal point in the product is determined by the total number of decimal places in the factors. Count the number of decimal places from the right. To check, divide the product by either factors.

6480 four digits 22718 five digits Add a zero to make Additional zeros is five decimal places in the product. not needed. 0.06480 0.2271 Additional Zero Add the decimal places in the factors. Then see how many decimal places the product has. 0.432 × 0.15 Five decimal places 0.614 × 0.37 Five decimal places

PRACTICE: Find the product by fill in the boxes for the correct answer. 0.3 0.2 0.4 0.1 0.5 0.6 0.4 0.7 0.3 0.5 0.4 0.3 0.7 0.6 0.8 0.4 0.2 0.1 0.5 0.4 0.3 0.7 0.6 0.8 0.4 0.2 0.1

1.9 1.5 1.8 2.5 3.5 2.0 3.5 3.8 3.1 0.1 0.44 0.87 0.54 0.53 0.09 0.9 0.76 0.36 1.90 1.2 2.9 1.8 2.2 2.99 1.66 0.8 1.5 2.2 1.4 1.9 1.4 1.7 1.9 1.7 2.0 2.7 1.6 1.8 1.7 1.89 1.89 1.7 2.7 2.6 2.9

Put the decimal point on the product for the correct places. 1. 0.192 x 0.428 1536 384 + 768__ 82176 2. 0.342 x 0.153 1026 1710 + 342 52326 3. 0.208 x 0.274 832 1456 + 416 56992 22 Worksheet

4. 0.263 x 0.29 2367 + 526 7627 5. 0.1594 x 0.37 11158 + 47852 58978 Multiply the following decimal numbers and put the decimal point. 1. 0.987 x 0. 270 2. 0.158 x 0.258 3. 0.4789 x 0.1247

4. 0.2547 x 0.2479 5. 0.3647 x 0.1248

What did the big flower say about the little flower? FUN WITH MATH!!! To find the answer, write each of the following products in multiplying decimals.

__________ ___________ ___________ __________ 0. 7537344 0.0132 0.0003 0.08537832 ___________ ___________ 0.001445 0.290523 _________ __________ _________ ________ ________ 0.0000195 0.0044902 0.000492 0.05626725 0.0006

Lesson 23 MULTIPLYING MIXED DECIMALS BY WHOLE NUMBERS Lesson Objectives: After finishing the lesson, the students are expected to: 1. Multiply mixed decimals by whole numbers. 2. Find the partial products. 3.Understand the rules in multiplying mixed decimals by whole numbers.

Christopher can save Php. 18.65 in one month. How much money can he save in four months? 18.6 -> two decimal places x 4 74.60 Decimals are multiplied the same way as whole number.

Remember: In multiplying mixed decimals by whole numbers, count the decimal places in the mixed decimal to determine the placement of the decimal point in the product. Start counting the number of decimal places from the right.

Study other examples. 23.729 -> three decimal places x 47 166103 + 94916 1115.263 ↑ Partial product

6.3572 -> four decimal places x 158 508576 317860 + 63572 1004.4376 ↑ Partial product

In the following problems, the final product is given. Find the partial products. Place the decimal points in the correct position. 1. 81.83 2. 62.872 3. 7.0194 × 57 × 34 × 271 + + 466431 2137648 + 19022574 23 Worksheet

4. 17.59 5. 48.723 6. 8.0035 × 83 × 52 × 179 + + 145997 2533596 + 14326265

II. Find the product. 7. 934.04 8. 282.5601 9. 37.5852 × 251 x 49 × 784 10. 51.207 11. 4672.397 12. 693.3521 × 490 × 268 × 922

13. 75.373 14. 149.1811 15. 10.1496 x 44 x 1012 x 189

Lesson 24 MULTIPLICATION OF MIXED DECIMALS BY MIXED DECIMALS Lesson Objectives: After accomplishing this lesson, you are expected to: 1. Multiplying mixed decimals by mixed decimals. 2. Perform the operation correctly. 3. Understand the rules in multiplying mixed decimals by mixed decimals.

What is the area of Ariel’s backyard if it is 12.932 m long and 8.45 m wide? NOTE: Area = length x width = 12.93m x 8.45m = 109.27540 sq.m² = m x m = m² 12.932 -> three decimal places × 8.45 -> two decimal places 64660 51728 + 103456 109.27540 -> five decimal places The backyard is 109.27540 square meters.

NOTE: Area = length x width = 12.93m x 8.45m = 109.27540 sq.m² = m x m = m² When multiplying mixed decimals by mixed decimals, the decimal point of the product is determined in this manner.

Decimal Decimal Decimal Places of first Places of second Places of Factor Factor the product

Worksheet I. Rewrite and arrange the partial products properly. Find the product and place the decimal points in the correct position. 1. 4.9526 2. 9.18234 × 3.215 × 75.68 247630 7345872 49526 5509404 99052 451170 + 148578 + 6427638 25

3. 57.6012 4. 2.01938 × 4.765 × 36.24 2880060 807752 3456072 403876 4032084 1211628 + 2304048 + 605814

Find the product. 5. 15.6027 6. 92.46355 7. 8.932682 × 8.306 × 1.728 × 9.1865 8. 743.9516 9. 268.924 10. 5.1367 × 4.321 × 4.321 × 9.824

Lesson 25 MULTIPLYING DECIMALS BY 10, 100 and 1000 Lesson Objectives: At the end of the lesson, you are expected to: 1. Multiply decimals by 10, 100 and 1000. 2. Write the product correctly. 3. Observe the rules in multiplying decimals by 10, 100 and 1000.

Take a decimal, 0.7568. Multiply it by 10, by 100 and by 1,000. What are the products? Look at the following: 0.7568 0.7568 0.7568 × 10 × 100 × 1000 7.5680 75.6800 756.8000

You see that the number of zeros contained in the factors 10, 100 and 1,000 tells how many places the decimal point in the other factor must be moved to the right to get the product. Examples: 10 × 0.75 = _______ 100 × 0.75 = _______ 1,000 × 0.75 = _______

Observe: Move 1 place to the right. Move 2 place to the right. Move 3 place to the right. 750. 75. 7.5 0. 750 0. 75 0.75 0.750 0.75 0.75 0.75 × 1,000 × 100 × 10 Decimal

Worksheet Complete the following equations. 1. 3.67 × 10 = ______ 2. 100 × _____ = 4521 3. 1000 × _____ = 0.0049 4. _____ × 100 = 854.8 5. 2.918 × _____ = 2918 6. 35.66 × _____ = 35660 7. 0.0074 × _____ = 7.4 8. _____ × 10 = 0.163 9. 0.089 × 10 = _____ 10. _____ × 100 = 100.78 25

II. Complete the table by multiplying each factor by 10, 100 and 1,000.

III. Multiply the following. Write your answers in the blanks provided: 1. 0.386 × 10 = ________ 2. 0.86 × 100 = ________ 3. 0.36 × 1000 = ________ 4. 0.473 × 1000 = ________ 5. 0.496 × 10 = ________ 6. 0.85 × 1000 = ________ 7. 0.7 × 1000 = ________ 8. 0.512 × 100 = ________ 9. 0.93 × 100 = ________ 10. 0.603 × 10 = ________

Lesson 26 ESTIMATING PRODUCTS OF DECIMAL NUMBERS Lesson Objectives: After understanding the lesson, you must able to: 1. Learn how to estimate the products correctly. 2. Learn how to make an estimates product in fastest way. 3. Follow the steps in estimating products.

The fastest way of solving problem is to estimate. In estimating the products: Round the given decimal numbers to the highest place value. Estimate and multiply. Compute the exact answer. Products can be estimated in the same way as sum and difference.

1. Rounding Method 4.52 × 6 27.12 Actual Value Rounded Value 5.00 × 6 30.00 2. Front End Method 4 .56 4.00 4. 5 2 .50 450 × 6 × 6 × 6 × 6 × 6 24.00 + 3.00 = 27.00

The front – end method with adjustment is usually closer to the actual value.

Worksheet Estimate the product using rounding and front-end with adjustment. 1. 3.754 2. 48.263 3. 28.169 × 8 × 5 × 7 26

4. 38.721 5. 28.765 6. 75.814 × 3 × 9 × 13 7. 96.250 8. 18.263 9. 927.231 × 42 × 41 × 507

10. 36.287 11. 76.298 12. 28.183 × 206 × 304 × 543

Lesson 27 PROBLEM SOLVING INVOLVING MULTIPLICATION OF DECIMAL NUMBERS Lesson Objectives: After understanding the lesson, you must able to: Solve word problem involving multiplication of decimals. Write the numbers sentence. Solve word problems correctly and accurately.

Example 1: A cone of ice cream costs Php. 16.25, how much in all did the 6 children spend for ice cream?

Example 2: What is the area of a rectangle with a length of 9.72 cm and width of 6.34 cm?

Worksheet Read, analyze and translate these problems to number sentence then solve. 1. Mrs. Hernandez baked 1,000 pineapple pies for a party of her daughter Kiana. If each pie costs Php. 17.85, how much did the 1,000 pies cost? 27

2. If a car travels 55.6 km an hour, how far will it travel in 8 hours?

3. Mang Freddie sold 46 cotton candies at Php. 2.15 each. How much altogether is the cost of the cotton candies?

4. A rope measures 4.63 m. How long is it in centimeters?

5. If 1 meter of cloth costs Php. 72.95, how would 6.5 meters cost?

6. Mang John, a balot vendor bought 120 new duck eggs at Php. 3.85 each. How much did he pay all the eggs?

7. A can of powdered milk has a mass of 0.345 kilogram. What is the mass of 12 cans of milk?

8. Mr. Gelo Drona bought a residential lot with an area of 180.75 m at Php. 650.00 per square meter. How much did he pay for the lot?

9. Niña works 40 hours a week. If his hourly rate is Php. 640.25, how much is she paid a week?

10. The rental for a Tamaraw FX is Php. 3,500 a day. What will it cost you to rent it in 3.5 days?

OVERVIEW OF THE MODULAR WORKBOOK This modular workbook provides you’re the language of division of decimal numbers and how to divide decimals in different ways. OBJECTIVES OF THE MODULAR WORKBOOK After finishing this unit, you are expected to: 1. Understand the language of division of decimals. 2. Know how to divide decimal numbers. 3. Follow the steps in division of decimal numbers. 4. Participate actively in division of decimal numbers. 5. Learn the different form of dividing decimal numbers.

Lesson 28 MEANING OF DIVISION OF DECIMAL NUMBERS Lesson Objectives: After accomplishing this lesson, you are expected to: 1. Define division. 2. Understand the language in division of decimals. 3. Know the parts in dividing decimal numbers.

Division is the process of finding out how many times one number is contained in another number. 0.09 -> quotient 9 0.81 -> dividend - 0 81 - 81 0 Divisor

The number that contains another number a number of times is called the dividend . The number that is contained in another number a number of times is called the divisor . The number that indicated how many times a number contained in another number is called the quotient . Division may also be defined as the process of separating a number into as many equal parts as indicated by another number. The symbol for division ( ÷ ), which is read as “divided by”. Thus, 0.81 ÷ 9 = 0.09 is read as “eight-one hundredths divided by nine equals nine thousandths.”

Another symbol is a line written over and above the dividend and a slanting line connecting it at the left of the dividend and at the right of the divisor. Another symbol is a line over which the dividend is written and the divisor below. 0.81 9

Worksheet I. Give the meaning and explain the use of the following: 5points each. What is division? What is divisor? What is dividend? What is quotient? 28

1. Division ____________________________________________________________________________________ 2. Divisor ____________________________________________________________________________________ 3. Dividend ____________________________________________________________________________________ 4. Quotient ____________________________________________________________________________________

II. Enumeration… A. What are the parts of division? B. What symbols that can be used in dividing numbers? A._____________________________________________________________________________________________________________________________ B._____________________________________________________________________________________________________________________________

Lesson 29 DIVIDING DECIMAL BY WHOLE NUMBERS Lesson Objectives: After accomplishing this lesson, you are expected to: 1. Divide decimals by whole numbers. 2. Follow the rules in dividing decimals by whole numbers. 3. Find the quotient correctly. 4. Using two methods in dividing decimals.

Dividing decimals by a whole number is the same as dividing a whole number by another whole number. Observe the following examples. 0.15 0.05 4 0.60 9 0.45 - 4 - 0 20 45 - 20 - 45 0 0

To check the answer, multiply the quotient by the divisor: 0.15 x 4 0.60 0.05 x 9 0.45 In dividing decimals by whole numbers, the number of decimal places in the quotient equals the number of decimal in the dividend.

Look at the other examples: Example 1: 0.6 ÷ 3 = _____ We used 2 methods in dividing decimals by whole numbers.

1. Using a region 0.6 A whole is divided into 10 equal parts. Each part is called 1/10 or 0.1. 6/10 or 0.6 are shaded.0.6 is divided into 3 groups. How many tenths are in each group? 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2

2. Using computations 6 ÷ 3 = 6 ÷ 3 = 2 10 1 10 ÷ 1 = 10 0.2 3 0.6 6 0 Let us check by using reciprocals. Fractional Division: 6 ÷ 2 = 6 x 3 = 6 x 3 = 18 10 3 10 2 10 x 2 20

6 ÷ 3 = 6 × 1 = 6 = 1 = 0.2 and 1 10 10 3 30 5 5 is equivalent to 0.2 Why? Explain. 6 ÷ 6 = 1 30 6 5 0.2 5 1.0 1.0 0

Let us divide hundredths by a whole number. Example 2: 6 0.18 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1. Using a region A whole is divided into 100 equal pairs. Each part is called 1/100. Eighteen parts are called 18/100 or 0.18. We divide 0.18 in 6 groups.

How many hundredths are in each group? Using computation 18 ÷ 6 = 18 ÷ 6 = 3 100 1 100 ÷ 1 100 to get the tenths place. 0.03 -> Quotient 6 0.18 - 18 0 Check: 6 × 0.03 0.18

Worksheet Find the quotient by using region and computations. 3 0.12 2. 5 0.35 3. 7 0.14 4. 3 0.42 5. 8 0.64 6. 4 0.56 29

7. 2 0.6 8. 9 0.81 9. 4 0.424 Find the quotients. Answer the question that follows. 1. 6 0.732 2. 4 0.524 3. 2 0.236 4. 5 0.655 5. 4 0.435 6. 7 0.851

How many decimal places are in the dividends of 1 to 6? ________________________________________________________________________________ How many decimal places should there be in the quotient? ________________________________________________________________________________ How many we use zero in the quotient? ________________________________________________________________________________

Lesson 30 DIVIDING MIXED DECIMAL BY WHOLE NUMBERS Lesson Objectives: After accomplishing this lesson, you are expected to: Divide mixed decimals by whole numbers. 2. Understand the rule in dividing mixed decimals by whole numbers. 3. Answer the operation correctly.

Divide mixed decimals in the same way as in dividing whole numbers. To check, multiply the quotient by the divisor. Long method of division: To check, multiply the quotient by the divisor. 1.5734 x 5 7.8670 1.5734 5 7.8670 - 5 28 - 25 36 - 35 17 - 15 20 - 20 0

Remember that zeros added to a number to the right of the decimal point does not affect the value of the number. 7.8670 = 7.867 When dividing mixed decimals by whole numbers. The number of decimal places in the quotient is equal to the number of decimal places in the dividend. Align the decimal points of the quotient with that of the dividend.

Do another division. To check, multiply… Answer (quotient by division) 5.1268 x 14 205072 51268 71.7752 5.1268 14 71.1152 - 70 17 - 14 37 - 28 95 - 85 112 - 112 0

Worksheet Find the quotient. Check by multiplication. CHECKING: 48.102 ÷ 21 = 56.381 ÷ 87 = 30

3. 140.722 ÷ 9 = 4. 28.6134 ÷ 5 = 5. 189.526 ÷ 32=

II. Solve for the quotient and check by long multiplication method. CHECKING: 83.7169 ÷ 21 = 2. 92.0314 ÷ 37 =

3. 152.51 ÷ 28 = 4. 293.763 ÷ 48 = 5. 451.306 ÷ 89 =

Lesson 31 DIVIDING WHOLE NUMBERS BY DECIMALS Lesson Objectives: After accomplishing the lesson, the students are expected to: 1. Divide whole numbers by decimals. 2. State the rule for dividing a whole number by a decimal. 3. Find the quotient correctly.

Let us divide whole numbers by decimals in tenths. Example 1: 0.8 72 Here are the steps in dividing whole numbers by decimals... STEP 1 Before we divide, we must change the divisor to a whole number. We multiply 0.8 by 10. We have 8.

STEP 2 We multiply the dividends by 10 also. 10 x 72 = 720, we have 720 as dividend. STEP 3 Then we begin to divide. 90 -> quotient 8 720 - 72 0 We check: 90 × 0.8 72.0 STEP 4

How we divide a whole number by a decimal in the hundredths? Follow this step to find the quotient.

Example 2: 0.14 588 STEP 1 Make the divisor a whole number. Multiply it by 100. 0.14 x 100 = 14 0.14 588 STEP 2 Multiply the dividend by 100. 14 588.00

STEP 3 Then divide as if dividing whole numbers. 4200 14 58800 -56 28 - 28 0 STEP 4 We check: 4200 x 0.14 16800 4200 58000

Worksheet I. Find the quotient and check. 0.3 ÷ 936 = 2. 0.8 ÷ 856 = CHECKING: 31

0.9 ÷ 756 = 0.5 ÷ 485 = 0.4 ÷ 348 =

6. 0.6 ÷ 911 = 7. 0.2 ÷ 613 = 8. 0.7 ÷ 518 =

9. 0.2 ÷ 434 = 10. 0.5 ÷ 775 = 11. 0.7 ÷ 434 =

12. 0.7 ÷ 714 = 13. 0.8 ÷ 872 = 14. 0.6 ÷ 846 = 15. 0.5 ÷ 305 =

Lesson 32 DIVIDING WHOLE NUMBERS BY MIXED DECIMALS Lesson Objectives: At the end of the lesson, the students are expected to: 1. Divide whole numbers by mixed decimals 2. Follow the rule in dividing whole numbers by mixed decimals. 3. Study division where the quotient is found to the ten thousandths place.

Let us observe the rule in dividing whole numbers by mixed decimals. Example 1: Move the decimal point in the divisor to make it a whole number. The number of places the decimal has been moved to the right in the divisor is the same as the number of places the decimal point is to be moved in the dividend. Add the appropriate zeros to the dividend. Align the decimal point of the dividend.

In dividing decimals always try to divide to the last digit. When there are too many digits to divide, you can stop at the division by multiplying the quotient by the divisor. Divide 84 by 1.25. 67.2 1.25 84.00.0 - 750 900 - 875 250 - 250 0 67.2 x 1.25 33.60 13.44 + 67.2 84.000

Check the divisor by multiplying the quotient by the divisor. Study another division where the quotient is found to the ten thousandths place.

Example 2: 16.0516 5.42 87.00.0000 - 542 3280 - 3252 280 - 000 2800 - 2710 900 - 542 3580 - 3252 328 To check: 16.0516 × 5.42 0.321032 6.42064 80.2580 86.999672 + 0.000328 87.000000

Worksheet I. Find the quotient to the ten thousandths place then check. CHECKING: 15 ÷ 4.7 = 2. 9 ÷ 5.28 = 3. 86 ÷ 7.245 = 32

4. 16. ÷ 7.32 = 5. 23 ÷ 8.16 = 6. 43 ÷ 15.8 =

7. 8 ÷ 1.43 = 8. 15 ÷ 3.786 = 9. 74 ÷ 16.37 = 10. 22 ÷ 5.61 =

Lesson 33 DIVIDING DECIMAL BY DECIMALS Lesson Objectives: At the end of the lesson, the students are expected to: 1. Divide decimals by decimals. . Follow the step in dividing decimals by decimals . Use fraction in checking the division of decimals.

How is division done with decimals? What do we do with the decimal points? Let’s observe the following example . Example: 0.5 0.75 STEP 1 STEP 2 STEP 3 0.5 0.75 5 0.7.5 1.5 5 7.5 - 5 25 - 25 0 Multiply 0.5 by 10 to make it a whole number. Multiply 0.75 by 10 also. What we do with the divisor, we do to the dividend. Divide just like whole numbers. The quotient has the same number of decimal places as the dividend.

Let us check by using fractions. 75 ÷ 5 = 75 ÷ 5 = 15 or 1 5 or 1.5 100 10 100 ÷ 10 10 10

Worksheet I. Divide the following and check it by using fractions. 0.72 ÷ 0.3 = 2. 0.96 ÷ 0.4 = 3. 0.387 ÷ 0.09 = CHECKING: 33

4. 0.516 ÷ 0.6 = 5. 0.81 ÷ 0.9 = 6. 0.96 ÷ 0.8 =

7. 0.441 ÷ 0.7 = 8. 0.558 ÷ 0.06 = 9. 0.36 ÷ 0.3 = 10. 0.72 ÷ 0.8 =

II. Try to analyze. Check your answer. 0.56 ÷ 0.008 = 2. 0.90 ÷ 0.090 = CHECKING :

3. 0.72 ÷ 0.4 = 4. 0.26 ÷ 0.2 = 5. 0.9015 ÷ 0.5 =

Lesson 34 Dividing Mixed Decimals by Mixed Decimals Lesson Objectives: At the end of the lesson, the students are expected to: Divide mixed decimals by mixed decimals. 2. Observe the rule in dividing mixed decimals by mixed decimals. 3. Perform the operation correctly.

A full-grown Philippine eagle can grow to length of 102.6 cm including its tail. The tail can reach 49. 8 cm. When one of its wings is spread out, it can reach 63.2 cm. The length of its tail is what part of its whole length? Divide 102.6 cm (total length) by 49.8 cm (tail) to the hundredths place.

Move the decimal point one place to the right. The number of decimal places point has been moved in the divisor determines the number of decimal places it is moved in the dividend. 2. 06 49.8. 102. 6. 00 - 996 300 - 000 3000 - 2988 12 To check, multiply it. 49.8 x 2.06 2.9 88 0.00 99.6 __ 102.5 88 + 0.0 12 102.6 00 remainder In the case like this, when the remainder is added, the sum is equal to the dividend.

Mixed Decimals are divided in the same way as whole numbers. In dividing mixed decimals by mixed decimals, remember that the decimal point in the divisor is moved to the right to make it a whole number.

Worksheet I. Find the quotient to the tenths place. Check it through multiplication. 8.376 ÷ 1.942 = 2. 7.801 ÷ 2.334 = 3. 9.482 ÷ 4.7636 = CHECKING: 34

4. 10.857 ÷ 6.135 = 5. 23.154 ÷ 5.719 = 6. 41.028 ÷ 12.149 =

7. 183.945 ÷ 20.132 = 8. 151.932 ÷ 46.741 = 9. 273.921 ÷ 87.553 = 10. 491.72 ÷ 78.521 =

Curriculum Vitae of the Authors and Module Consultant

PAMN FAYE HAZEL M. VALIN Brgy Bagong Pook Sta. Maria, Laguna January 8, 1991 E-mail Address : herzelle_0108@yahoo.com

EDUCATIONAL ATTAINMENT Elementary Santa Maria Elementary School Secondary Santa Maria National High School Tertiary Laguna State Polytechnic University (Formerly LSPC) Siniloan Host Campus Course Bachelor of Elementary Education Major General Education

RON ANGELO A. DRONA Patricio Street, Brgy. San Jose Pangil, Laguna June 04, 1991 E-mail Address : [email_address]

EDUCATIONAL ATTAINMENT Elementary Pangil Elementary School Secondary Laguna State Polytechnic University (Formerly LSPC) Siniloan Host Campus Tertiary Laguna State Polytechnic University (Formerly LSPC) Siniloan Host Campus Course Bachelor of Elementary Education Major General Education

BEATRIZ P. RAYMUNDO Brgy Bagong Pook Sta. Maria, Laguna April 21, 1951 E-mail Address : [email_address]

EDUCATIONAL ATTAINMENT Elementary Santa Maria Elementary School Secondary Santa Maria Academy Tertiary Immaculate Conception College Course Bachelor of Secondary Education

Major M athematics Minor English Master’s Degree MA Teaching (National Teacher’s College, Manila) Teacher in Values Education and Filipino

FOR-IAN V. SANDOVAL Siniloan, Laguna April 5, 1979 E-mail Address : [email_address]

EDUCATIONAL ATTAINMENT Elementary Palasan Elementary School Secondary Union College of Laguna Tertiary Far Eastern University Course Bachelor of Science in Mathematics Bachelor of Secondary Education (Unit Earner)

Major Computer Science Master’s Degree Master of Arts in Education Major in Educational Management (with units)

Books Benigno, Gloria D. Basic Mathematics for College Students. Manila: REX Bookstore. 1993. Calderon, Jose F. Basic Mathematics I. Quezon City: Great Books Trading. 1994. Del Fiero, Jong. Power in Numbers 6. Manila: Saint Mary’s Publishing Corporation. 1999. Department of Education Culture and Sports. Mathematics in Everyday Life (textbook for Grade V) Revised edition. Manila: Cacho Hermanos Inc., 1993.

Department of Education. Lesson Guides in Elementary (Mathematics for Grade VI). Bureau of Elementary Education in coordination with Ateneo de Manila University., 2003 Ibe, Milagros D. et. al. Highschool Mathematics-Concept and Operation, 3rd edition, First Year. Manila: DIWA Learning Systems Inc., 1999. Jovero, Natividad V. Power in Numbers IV (Teachers Manual, Mathematics 4). Manila: Saint Mary’s Publishing Corporation. 1999. Llanes, Estrelita M. and Li, Bernardino Q. Living with Math VI. Revised Edition. Quezon City: FNB Educational Inc. 1988.

Lopez, Kelli L., The Self-replicating Gene. Tatsulok. Vol. 14 No. 2 1st year. Pp 4-6,15. Mendoza, Marilyn O. Workbook in Mathematics. Manila: Gintong Aral Publication. 1997. Roxas, Mia P. and Zara, Evelyn F. Elementary Algebra. High School Mathematics. (Worktext I). EFEREZA Pulbication House. 2003. http://www.321know.com http://whyslopes.com

Photo Credits http://www.ask.com/pictures http://images.google.com.ph http://images.search.yahoo.com http://photobucket.com/images

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