Diwa Textbooks - Math for Smart Kids Grade 4 Math for Smart Kids is the grade school textbook which features online exercises in www.diwalearningtown.com to complement review of textbook lessons. The book addresses the learning needs in mathematics such understanding and skills in computing considerable speed and accuracy, estimating, communicating, thinking analytically and critically, and in solving problems using appropriate technology.

1. Math for Smart Kids 4

2. Math for Smart Kids
Grade 4
Textbook
Philippine Copyright 2010 by DIWA LEARNING SYSTEMS INC
All rights reserved. Printed in the Philippines
Editorial, design, and layout by University Press of First Asia
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ISBN 978-971-46-0121-5
The Editorial Board
Author
Dr. Estrella P. Mercado finished her PhD in Educational Management (with honor) and MA in Education at Manuel L. Quezon
University. She also holds an MEd in Special Education degree and a BS in Elementary Education degree from the Philippine
Normal University (PNU). She has been a classroom teacher, an Education supervisor, and an assistant chief of the Elementary
Division of the Department of Education, Culture and Sports (DECS-NCR). She was awarded as Outstanding Female Educator in
1998 by the Filipino Chinese Women Federation. She presently heads the Special Education Department at PNU.
Consultant-Author
Luzviminda M. Sibbaluca holds a master’s degree in Mathematics Education and a bachelor’s degree in Elementary Mathematics
Education (cum laude) both from the University of the Philippines-Diliman (UP-Diliman). She is one of the top ten successful
examinees of the Licensure Examination for Teachers (LET) for Elementary in 2003. She is currently an assistant professor at the
Mathematics Department of the University of the Philippines Integrated School (UPIS).
Reviewer
Margaret M. Diaz-Atela has been teaching Mathematics since 1990. She has a bachelor’s degree in Secondary Education, major
in Mathematics, and a master’s degree in Teaching, major in Mathematics, both from UP-Diliman. She is presently an assistant
professor at the Mathematics Department of UPIS.

3. Preface
Math for Smart Kids is a series of textbooks in Mathematics for
grade school, which is designed to help pupils develop appreciation and
love for mathematics. This series also aims to help the learners acquire the
skills they need to become computationally literate.
The lessons in each textbook present mathematics concepts and principles
that are anchored on the competencies prescribed by the Department of
Education. Each lesson starts with Let’s Do Math, where mathematics
concepts and principles are introduced through problems, stories, games,
or puzzles. This section is followed by Let’s Look Back, which lists questions
that will help the pupils to think critically on what has been introduced
in the lesson and allow them to discover things on their own. For easy
recall of important points or concepts taken up in a lesson, the section Let’s
Remember Our Learning has been included. Multilevel exercises are provided
in Let’s Practice and Let’s Test Our Learning that will assess how much the
pupils have learned from the lesson. The exercises will also determine if the
pupils are ready to learn new mathematics skills. The development of the
multiple intelligences of an individual is reflected in the different activities
that the pupils will perform—from concrete to semi-concrete, and from
semi-abstract to abstract kind of learning. Situations and real-life problems
are provided in Let’s Look Forward to give the pupils opportunities to apply
what they have learned to their daily life experiences.
This series of textbooks gives the learners the opportunity to explore
and enjoy Mathematics. Let’s have fun learning together!
The Authors

4. Table of Contents
Unit 1 Addition and Subtraction of Whole Numbers
Chapter 1 Whole Numbers through Billions
Lesson 1 Place Values through Billions ..............................................................2
2 Reading and Writing Numbers ............................................................6
3 Comparing and Ordering Whole Numbers ......................................... 10
4 Rounding off Numbers ...................................................................... 14
Chapter 2 Addition of Whole Numbers
Lesson 1 Properties of Addition ....................................................................... 17
2 Adding Whole Numbers ................................................................... 22
3 Adding Three or More Addends ………………………………....................26
4 Estimating Sums ................................................................................ 29
5 Adding Three-digit Numbers Mentally ............................................... 32
6 Solving Word Problems Involving Addition ....................................... 35
Chapter 3 Subtraction of Whole Numbers
Lesson 1 Subtracting Whole Numbers without Regrouping .............................. 40
2 Subtracting Whole Numbers with Regrouping ................................... 43
3 Estimating Differences ..................................................................... 46
4 Mental Subtraction ............................................................................ 49
5 Solving Word Problems Involving Subtraction ................................... 53
6 Combined Operations (Addition and Subtraction) .............................. 57
Unit 2 Multiplication and Division of Whole Numbers,
and Number Theory
Chapter 4 Multiplication of Whole Numbers
Lesson 1 Properties of Multiplication .............................................................. 62
2 Multiplying Whole Numbers without Regrouping ............................. 67
3 Multiplying Whole Numbers with Regrouping .................................. 70
4 Multiplying Whole Numbers That Are Multiples of 10 ...................... 73
5 Estimating Products ........................................................................... 76
6 Mental Multiplication ........................................................................ 78

5. 7 Exponents ......................................................................................... 81
8 Solving Word Problems Involving Multiplication ............................... 85
Chapter 5 Division of Whole Numbers
Lesson 1 Dividing Whole Numbers without Remainder ................................... 89
2 Dividing Whole Numbers with Remainder ....................................... 93
3 Dividing Whole Numbers That Are Multiples of 10 .......................... 97
4 Estimating Quotients ...................................................................... 101
5 Mental Division .............................................................................. 104
6 Solving Word Problems Involving Division ..................................... 107
7 Series of Operations ........................................................................ 111
8 Using a Calculator .......................................................................... 115
Chapter 6 Number Theory
Lesson 1 Divisibility Rules ............................................................................ 119
2 Prime and Composite Numbers ...................................................... 123
3 Greatest Common Factor ................................................................ 127
4 Multiples and Least Common Multiple ........................................... 130
Unit 3 Fractions, Decimals, Ratio, and Proportion
Chapter 7 Fractions
Lesson 1 Kinds of Fractions .......................................................................... 134
2 Equivalent Fractions and Fractions in Lowest Terms ....................... 139
3 Comparing and Ordering Fractions ................................................. 144
4 Adding Similar Fractions ................................................................ 149
5 Subtracting Similar Fractions .......................................................... 153
6 Solving Word Problems Involving Addition and Subtraction
of Similar Fractions ........................................................................ 157
7 Adding Dissimilar Fractions ............................................................ 163
8 Subtracting Dissimilar Fractions ..................................................... 167
9 Multiplying Fractions ..................................................................... 172
10 Solving Word Problems Involving Multiplication
of Fractions ..................................................................................... 175
Chapter 8 Decimals
Lesson 1 Fractions and Decimals .................................................................. 179
2 Comparing and Ordering Decimals ................................................. 183

6. 3 Rounding off Decimals ................................................................... 187
4 Adding Decimals ............................................................................ 190
5 Subtracting Decimals ...................................................................... 194
6 Solving Word Problems Involving Addition and Subtraction
of Decimals .................................................................................... 198
7 Solving Word Problems Involving More Than One Operation ......... 202
8 Multiplying Decimals ..................................................................... 206
9 Solving Word Problems Involving Multiplication of Decimals ......... 209
Chapter 9 Ratio and Proportion
Lesson 1 Ratio .............................................................................................. 213
2 Proportion ...................................................................................... 218
3 Solving Word Problems Involving Ratio and Proportion .................. 222
Unit 4 Geometry, Measurement, and Graphs
Chapter 10 Geometry
Lesson 1 Basic Geometric Concepts .............................................................. 226
2 Angles ............................................................................................ 232
3 Polygons ......................................................................................... 238
4 Circles ............................................................................................ 243
5 Space Figures ................................................................................. 247
Chapter 11 Measurement
Lesson 1 Metric Conversion .......................................................................... 250
2 Perimeter of Polygons ..................................................................... 254
3 Area of a Rectangle and a Triangle .................................................. 258
4 Volume of Rectangular Prisms ........................................................ 263
Chapter 12 Graphs
Lesson 1 Bar Graphs ..................................................................................... 266
2 Constructing Bar Graphs ................................................................ 270
3 Line Graphs ................................................................................... 273
4 Constructing Line Graphs ............................................................... 278
Bibliography ....................................................................................................... 282

7. Unit
1
Addition and Subtraction
of Whole Numbers

8. Chapter 1
Whole Numbers through Billions
Lesson 1 Place Values through Billions
The following table shows the estimated population of five member
countries of the Asia-Pacific Economic Cooperation (APEC).
Estimated Population
Country
(2008)
Canada 33 212 696
China 1 330 044 544
Indonesia 237 512 352
Philippines 96 061 680
United States 303 824 640
Source: Time: Almanac 2009
Which of the countries have an estimated population that reached
billions?
From the table, you can see that China has an estimated population
that has already reached billions.
Study the place values of the digits in China’s population, that is,
1 330 044 544, in the place-value chart below.
Billions Millions Thousands Units
Period Period Period Period
Ten Millions
Ten Billions
Thousands
Thousands
Thousands
Hundreds
Hundred
Hundred
Hundred
Millions
Millions
Billions
Billions
Ones
Tens
Ten
1 3 3 0 0 4 4 5 4 4
Math for Smart Kids 4

9. The digit 1 is in the billions place and its value is 1 000 000 000.
The digit 0 is in the millions and hundred thousands places. In both
place values, 0 acts as a placeholder.
The digits in a number are grouped by threes, starting from the right.
Each group of three digits is called period.
More examples:
1. The place values of the digit 2 in 237 512 352 are ones, thousands,
and hundred millions.
2. The values of the digit 6 in 96 061 680 are 600, 60 000, and
6 000 000.
1. How are the place values arranged? How are they grouped?
2. Do all digits in a number have place values? Explain.
3. Do all digits in a number have values? Explain.
Every digit in a given number has a place value and a value. The
value that each digit represents depends on its place value.
The digit 0 is used as a placeholder. It has a specific place value
but its value remains 0.
Digits are grouped into periods and each period is separated by a
space or a comma.
Addition and Subtraction of Whole Numbers

10. A. Write the place value and value of each underlined digit.
Place Value Value
1. 65 896 235 000 _______________ _______________
2. 7 329 136 495 _______________ _______________
3. 429 326 527 _______________ _______________
4. 3 829 425 612 _______________ _______________
5. 345 391 426 _______________ _______________
B. Write the value of the given digit of the number 7 390 426 815.
1. 1 ____________ 6. 7 ____________
2. 6 ____________ 7. 2 ____________
3. 8 ____________ 8. 9 ____________
4. 5 ____________ 9. 4 ____________
5. 0 ____________ 10. 3 ____________
Study the following table. Answer the questions on the next page.
Estimated Population
Country
(2008)
Brunei 381 371
Cambodia 14 241 640
Indonesia 237 512 352
Laos 6 677 534
Malaysia 25 274 132
Myanmar 47 758 180
Math for Smart Kids 4

11. Philippines 96 061 680
Singapore 4 608 167
Thailand 65 493 296
Vietnam 86 116 560
Source: Time Almanac 2009 and CIA World Fact Book
1. Which ASEAN countries have a population with a digit that has a
value of 100?
2. Which country has the largest population? What is the highest
place value in this country’s population?
A. Write the place value of each underlined digit.
1. 672 895 __________ 6. 3 000 425 __________
2. 4 826 321 __________ 7. 6 725 398 000 __________
3. 8 723 432 __________ 8. 72 829 426 125 __________
4. 24 829 000 __________ 9. 43 826 725 __________
5. 576 328 125 __________ 10. 49 000 126 345 __________
B. Do what is asked.
1. Find the number that is 20 000 more than 395 034 300.
2. Write 2 nine-digit numbers that have digits whose values are
50 000 and 3 000 000.
3. Find the number that is 200 000 000 less than 382 930 400 239.
4. Find the numbers before and after 385 300 294 000.
5. Write the biggest number that can be formed using the digits 0 to
9, without repeating any digit. What are the numbers before and
after this number?
Addition and Subtraction of Whole Numbers

12. Lesson 2 Reading and Writing Numbers
Asia is the largest continent in the
world. Its land area measures
31 699 257 square kilometers
(sq. km). It has an estimated total
population of 4 052 000 000 as of mid-
2008.*
How do you read the given numbers?
The place-value chart is a helpful tool
in reading and writing numbers through
billions in different forms.
Billions Millions Thousands Units
Period Period Period Period
Ten Millions
Ten Billions
Thousands
Thousands
Thousands
Hundreds
Hundred
Hundred
Hundred
Millions
Millions
Billions
Billions
Ones
Tens
Ten
3 1 6 9 9 2 5 7
Standard form: 31 699 257
Word form: thirty-one million, six hundred ninety-nine thousand, two
hundred fifty-seven
Expanded form: 30 000 000 + 1 000 000 + 600 000 + 90 000 + 9 000 +
200 + 50 + 7
*Sources: 2008 World Population Data Sheet, www.prb.org, and Time: Almanac 2009
Math for Smart Kids 4

13. Billions Millions Thousands Units
Period Period Period Period
Ten Millions
Ten Billions
Thousands
Thousands
Thousands
Hundreds
Hundred
Hundred
Hundred
Millions
Millions
Billions
Billions
Ones
Tens
Ten
4 0 5 2 0 0 0 0 0 0
Standard form: 4 052 000 000
Word form: four billion, fifty-two million
Expanded form: 4 000 000 000 + 50 000 000 + 2 000 000
From the two examples given, you can see that you multiply each
digit of a number by its value to get the expanded form of a number.
1. How does a place-value chart help in writing numbers in word
form?
2. How do you write numbers in expanded form?
To read large numbers, read the number in each period, starting
with the highest period, followed by the period name (without the –s).
Do not mention the period name when reading the numbers in the
units period.
To get the expanded form of a number, multiply each digit of the
number by its value.
Addition and Subtraction of Whole Numbers

14. A. Write each number in word form.
1. 2 342 504 162
____________________________________________________________
____________________________________________________________
2. 4 403 927 243
____________________________________________________________
____________________________________________________________
3. 62 730 902 842
____________________________________________________________
____________________________________________________________
4. 153 642 140 803
____________________________________________________________
____________________________________________________________
5. 275 407 129 430
____________________________________________________________
____________________________________________________________
B. Write the following in standard form.
1. three billion, nine hundred twenty-two million, three hundred
twelve thousand, one hundred forty-five
_____________________________________
2. forty-four billion, eight hundred fifty-three million, two hundred
thousand, seven hundred sixty-five
_____________________________________
3. eight hundred billion, seven hundred fifteen million, six hundred
twenty-seven thousand, four hundred two
_____________________________________
4. seventy billion, eight hundred forty-five million, nine hundred
two thousand, five hundred fifty-one
_____________________________________
Math for Smart Kids 4

15. 5. thirty-nine billion, seven hundred fifty million, one hundred five
thousand, six hundred seventy-three
_____________________________________
Write the orbit (in kilometers) of each planet in word and expanded
forms.
1. Mercury – 57 050 000
____________________________________________________________
____________________________________________________________
2. Venus – 108 110 000
____________________________________________________________
____________________________________________________________
Write the following numbers in expanded form.
1. 68 307 003 _______________________________________________________
__________________________________________________________________
2. 593 695 330 693 _________________________________________________
__________________________________________________________________
3. 492 403 206 202 ____________________________________________
__________________________________________________________________
4. twenty billion, seven hundred million, ninety-nine thousand,
eighteen _________________________________________________________
__________________________________________________________________
5. five hundred seventy-three billion, eight million, five hundred
thousand, five ___________________________________________________
__________________________________________________________________
Addition and Subtraction of Whole Numbers

16. Lesson 3 Comparing and Ordering Whole Numbers
Land Areas of the Different Continents
Continent Land Area (sq. km)
Africa 30 246 121
Antarctica 14 200 000
Asia 31 699 257
Europe 23 039 279
North America 24 293 585
Oceania 8 515 144
South America 17 824 370
Source: Time: Almanac 2009
Which continents have a greater land area than Europe?
To find out, compare the land area of each continent with that
of Europe. Look at the underlined digits. These are the digits being
compared.
Europe and Africa Europe and North America
23 039 279 30 246 121 23 039 279 24 293 585
Europe and Antarctica Europe and Oceania
23 039 279 14 200 000 23 039 279 8 515 144
Europe and Asia Europe and South America
23 039 279 31 699 257 23 039 279 17 824 370
Hence, the continents that have a greater land area than Europe are
Africa, Asia, and North America.
10 Math for Smart Kids 4

17. To compare whole numbers with the same number of digits, compare
the digits starting from the highest place value. If the digits are the same,
compare the digits in the next place value until you find different digits in
the same place value.
When comparing numbers with different number of digits, the
number with more digits is the greater number.
The land areas given in the table can be arranged either in ascending
(least to greatest) or descending (greatest to least) order.
The following table shows the arrangement of the seven continents in
ascending order.
Continent Land Area (sq. km)
Oceania 8 515 144
Antarctica 14 200 000
South America 17 824 370
Europe 23 039 279
North America 24 293 585
Africa 30 246 121
Asia 31 699 257
To arrange numbers in ascending or descending order, compare the
numbers two at a time. Then, arrange them in the specified order.
1. How do you compare whole numbers?
2. How do you arrange numbers in ascending order? in descending
order?
Addition and Subtraction of Whole Numbers 11

18. To compare whole numbers with the same number of digits,
compare the digits in the same place value, starting from the highest
place value. If the digits are the same, compare the digits in the next
place value until you find different digits in the same place value.
When comparing numbers with different number of digits, the
number with more digits is the greater number.
To arrange numbers either in ascending or descending order,
compare the numbers two at a time. Then, arrange them in the
specified order.
A. Compare each pair of numbers. Write , , or = in the box.
1. 483 399 386 483 399 836
2. 789 302 388 002 789 320 388 002
3. 40 309 204 40 000 000 + 3 000 000 + 9 000 + 200 + 4
4. 700 000 000 + 5 000 000 + 100 + 30 57 000 130
5. three billion, twenty million, five hundred two 3 020 000 502
B. Arrange the numbers from greatest to least.
1. 1 430 239, 1 421 695, 1 422 426
2. 2 561 326, 2 560 291, 2 563 427
C. Arrange the numbers from least to greatest.
1. 5 634 807, 5 189 346, 5 635 887
2. 79 847 000, 79 826 000, 79 835 000
1 Math for Smart Kids 4

19. Arrange the given estimated population of some countries from
greatest to least.
Country Population
Venezuela 28 439 218
Malaysia 27 468 000
Saudi Arabia 25 721 000
Uzbekistan 27 488 000
Afghanistan 28 150 000
(Based on July 2009 estimate by the UN Department
of Economic and Social Affairs)
A. Compare each pair of numbers. Write , , or = in the box.
1. 8 275 302 8 285 427
2. 782 321 000 783 324 000
3. 93 467 000 000 90 000 000 000 + 3 000 000 +
400 000 + 60 000 + 7
4. 84 672 923 000 eighty-four billion, six hundred
seventy-two million, nine
hundred twenty-three thousand
B. Arrange the given numbers in the box. Use the given numbers at the
start and end of each set as your guide.
1. 5 483 490 _______ _______ ________ 5 549 035
5 488 394 5 510 359 5 499 294
2. 14 593 299 ______ ________ _______ 14 101 304
14 405 395 14 123 493 14 443 395
Addition and Subtraction of Whole Numbers 1

20. Lesson 4 Rounding off Numbers
The following table shows the number of livestock raised in 2006.
About how many goats were raised?
Livestock Number of Live Animals
pigs 13 046 680
goats 3 735 816
buffalo 3 357 956
Source: Time: Almanac 2009
To answer the question, round off the given number of goats to the
nearest millions.
Identify the digit in the place value you are rounding off to (3). Look
at the digit to its right (7). If it is 5 or more, round up. Add 1 to the digit
in the place value you are rounding off to. Then, replace all the digits to
the right with zeros. If it is less than 5, retain the digit in the place value
you are rounding off to and replace all the digits to the right with zeros.
Since 7 5, round up.
Therefore, 3 735 816 rounded off to the nearest millions is 4 000 000.
Example:
Round off P3 296 000 000 to the nearest billions.
The digit in the billions place is 3. The digit to its right is 2. Since
2 5, round down or retain the digit in the place value you are rounding
off to.
Thus, P3 296 000 000 rounded off to the nearest billions is
P3 000 000 000.
1 Math for Smart Kids 4

21. How do you round off a number to a specified place value?
To round off a number to a specified place value, look at the
digit to the right of the place you are rounding off to. If the digit is 5
or more, add 1 to the digit in the place value you are rounding off
to and replace all the digits to its right with zeros. If the digit is less
than 5, retain the digit in the place value you are rounding off to and
replace all the digits to its right with zeros.
Round off each number to the highest place value.
1. 7 405 385 724 ______________________
2. 38 486 406 493 ______________________
3. 90 394 593 294 ______________________
4. 6 394 395 305 ______________________
5. 8 538 395 302 ______________________
6. 503 495 694 364 ______________________
7. 824 386 399 912 ______________________
8. 80 143 586 392 ______________________
9. 65 388 200 372 ______________________
10. 823 495 385 385 ______________________
Addition and Subtraction of Whole Numbers 1

22. Answer the following:
1. What is the least number that can be rounded off to 5 000 000?
2. What is the greatest number that can be rounded off to 64 280 000?
3. Give two numbers that can be rounded off to 390 000 000.
A. Round off each number to the indicated place value.
1. 8 905 346 millions _________________
2. 58 394 724 146 hundred thousands _________________
3. 49 385 284 124 millions _________________
4. 723 781 545 365 hundred billions _________________
5. 185 234 524 567 ten millions _________________
6. 5 584 816 127 thousands _________________
7. 6 389 878 392 billions _________________
8. 562 493 827 921 ten billions _________________
9. 43 678 453 ten millions _________________
10. 267 289 054 125 hundred billions _________________
B. Answer the following word problems.
1. There were 573 405 people who joined an online group. About
how many hundred thousands of people joined the group?
2. Last year, 1 995 394 passengers preferred to take economy flights.
This year, the number of passengers decreased by 500 000. About
how many million people took economy flights this year?
1 Math for Smart Kids 4