The Essence of Mathematics Skills Textbook- Math Primary_2

The Essence of Mathematics Skills Textbook Mathematics (BM11001) Primary Level Non-Formal and Informal Basic Curriculum of B.E. 2551 (A.D. 2008) Office of the Non-Formal and Informal Education Office of the Permanent Secretary Ministry of Education Not for sale This textbook is published with a national budget allocated for the purpose of people’s lifelong learning. Copyright owned by the Office of the Non-Formal and Informal Education, Office of the Permanent Secretary, Ministry of Education. Academic Document No. 10/2555

2 The Essence of Mathematics Skills Textbook Mathematics (BM11001) Primary Level Copyright owned by the Office of the Non-Formal and Informal Education, Office of the Permanent Secretary, Ministry of Education Academic Document No. 10/2554

3 Preface The Office of the Non-Formal and Informal Education (ONIE) has compiled this new textbook for teaching and learning according to the Non-Formal Basic Education Curriculum of B.E. 2551 which aims to instill into learners moral principles, ethics, wisdom and aptitude to carry out a profession, further their education and to enable them to live happily within their families, communities and society. Learners may use this textbook for self-learning and employ the activities and exercises in order to check their knowledge and understanding of its subject matter. Once learners have completed their study, it is possible for them to go back to review the content again if some understanding should still be lacking. Learners may also develop their knowledge further while studying with this textbook by exchanging views with classmates, studying local wisdom as well as learning from other sources of knowledge and by other means. The compilation of this textbook in line with the Non-Formal Basic Education Curriculum of B.E. 2551 is the result of the kind cooperation which the ONIE has received from academic experts and other resource persons who have reviewed various sources and written the content in a manner which has enabled the ONIE to provide a course in accordance with its curriculum and which will truly be beneficial to non-formal education learners. The ONIE would, therefore, like to express its gratitude to all the experts and writers as well as all those involved with the compilation of this textbook for their kind cooperation. The ONIE hopes that this textbook will duly benefit teachers and learners and would appreciate any recommendation for improvement. The Office of the Non-Formal and Informal Education

Table of Content 4 Preface Page Textbook Instruction Primary Level Mathematics Course Structure 1 Lesson 1: Number and Operation 76 Lesson 2: Fraction 99 Lesson 3: Decimal 117 Lesson 4: Percentage 126 Lesson 5: Measurement 168 Lesson 6: Geometry 192 Lesson 7: Introduction to Statistics and Probability 209 Lesson 1 Answer Key: Number and Operation 221 Lesson 2 Answer Key: Fraction 225 Lesson 3 Answer Key: Decimal 228 Lesson 4 Answer Key: Percentage 230 Lesson 5 Answer Key: Measurement 237 Lesson 6 Answer Key: Geometry 243 Lesson 7 Answer Key: Statistics and Elementary Probability

5 Textbook Instruction The Primary Level Basic Learning Skills Textbook in Mathematics (BM11001) is designed for non-formal education learners. To study the Basic Learning Skills in Mathematics, learners should proceed as follows: 1. Study the course structure so as to understand the main topics, the expected learning outcomes and the content scope. 2. Study content details of each lesson carefully and do the activities as assigned. Then, check your answers using the provided answer key. If your answers are wrong, review the content before proceeding onto the next topic. 3. Practice by doing the activities at the end of each topic in order to summarize relevant knowledge and understanding. Learners may check the result of such activities with their teachers and classmates . 4. This textbook consists of 7 lessons: Lesson 1: Number and Operation Lesson 2: Fraction Lesson 3: Decimal Lesson 4: Percentage Lesson 5: Measurement Lesson 6: Geometry Lesson 7: Introduction to Statistics and Probability

6 Mathematics Course Structure Primary Level (BM11001) Main content Knowledge and understanding relating to numbers and digits, fractions, decimals and percentages, measurements, geometry, statistics and elementary probability Expected learning outcome 1. Be able to identify or give examples of numbers and digits, fractions, decimals and percentages, measurements, geometry, statistics and elementary probability. 2. Be able to calculate and solve problems relating to numbers, fractions, decimals, percentages, measurements and geometry. Content scope Lesson 1: Number and Operation Lesson 2: Fraction Lesson 3: Decimal Lesson 4: Percentage Lesson 5: Measurement Lesson 6: Geometry Lesson 7: Introduction to Statistics and Probability Learning Media 1. Worksheet 2. Textbook

7 Lesson 1 Number and Operation Main content 1. Reading and writing numerals to represent numbers, estimations and additions, subtractions, multiplications and divisions as well as operations relating to numbers and their application in everyday life and integration with other sciences. 2. Property of counting numbers and zero, commutative property of addition and multiplication, associative property of addition and multiplication, property of zero addition, property of one time multiplication and distributive property which can be applied in calculations. Expected learning outcome After studying Lesson 1, learners shall be able to: 1. Read and write numerals to represent numbers 2. Identify place value and value of digits 3. Write numbers in an expanded form 4. Compare counting numbers 5. Round off numbers 6. Apply knowledge and properties of counting numbers and zero 7. Add, subtract, multiply and divide counting numbers 8. Find factors of counting numbers 9. Identify prime numbers and prime factors 10. Factor counting numbers 11. Identify the greatest common division (G.C.D.) and the least common multiple (L.C.M.) of given counting numbers Content scope Topic 1: Reading and writing numerals to represent numbers Topic 2: Place value and value of digits Topic 3: Expanded notation Topic 4: Ordering numbers Topic 5: Estimation and rounding Topic 6: Property of counting numbers and zero and application in problem solving

8 Topic 7: Addition, subtraction, multiplication and division of counting numbers and problem solving Topic 8: Factors of counting numbers and factorization Topic 9: Prime numbers and prime factors Topic 10: Factorization Topic 11: Greatest common division (G.C.D.) and least common multiple (L.C.M.) Topic 1: Reading and writing numerals to represent numbers Number is used to measure the quantity of people, animals or things. Numeral is a symbol used to express numbers. Digit is usually used to express numbers. There are ten basic digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. 1.1 One-digit number Number Words Thai Hindu- numerals Arabic numerals zero ๐ 0  one ๑ 1  two ๒ 2  three ๓ 3  four ๔ 4  five ๕ 5  six ๖ 6  seven ๗ 7  eight ๘ 8  nine ๙ 9

Exercise 1 9 a. Write Thai numerals and Hindu-Arabic numerals to represent the number of graphics in each case. Thai Hindu- nume- Arabic Number rals nume- (1) rals (2) (3) (4) (5) b. Practice writing Thai numerals and Hindu-Arabic numerals to represent numbers Thai numerals

10 1.2 Two-digit numbers Number Words Thai Hindu- numerals Arabic numerals  eleven ๑๑ ๕๐ 11  ๙๙ 50  99   fifty     ninety-nine    

11 Exercise 2 a. Write Thai numerals and Hindu-Arabic numerals to represent the number of graphics in each case. Number Thai Hindu-Arabic numerals numerals (1) (2) (3) (4) (5)

12 b. Practice writing Thai numerals and Hindu-Arabic numerals in your notebook ๑๑ ๑๙ ๒๘ ๓๗ ๔๖ ๕๐ 11 19 28 37 46 50 c. Write 2-digit numbers in order in the blanks. Thai numerals ๑๐ ..... ๑๒ ๑๓ ...... ๑๕ ...... ....... ๑๘ ..... ..... ..... ๒๒ ...... ..... ...... ๒๖ ๒๗ ...... ...... ๓๐ Hindu-Arabic numerals 31 32 ..... ....... ....... 36 ...... ...... 39 ...... ....... 42 ...... 44 ...... ....... 47 ...... ....... 50 d. Write Hindu-Arabic numerals to represent the following numbers (1) thirty-eight ................ (2) sixty-five ................ (3) seventy-seven ................ (4) eighty-one ................ (5) ninety-six ................ (6) ninety- nine................ e. Write the following numbers in words (1) 35 ......................... (2) 53 ......................... (3) 68 ......................... (4) 86 ......................... (5) 79 ......................... (6) 97......................... 1.3 Three-digit numbers such as 238 2 is in the hundreds place. 3 is in the tens place. 8 is in the ones place. 2 3 8 Read: two hundred thirty-eight The first digit or the far left digit is the hundreds place value. The next digit to the right is the tens place value. The last digit or the far right digitis the ones place value. 1.4 Four-digit numbers such as 6,385 6 is in the thousands place. 3is in the hundreds place. 8 is in the tens place.

13 5 is in the ones place. 6 , 3 8 5 Read: six thousand three hundred eighty-five For ease of reading, a comma (,) is usually used to separate the hundreds’ place value from the thousands’ place value. 1.5 Numbers with five digits, six digits, seven digits and more 1) Five-digit numbers such as 76,432 7 is in the ten thousands place. 6 is in the thousands place. 4 is in the hundreds place. 3 is in the tens place. 2 is in the ones place. 7 6 , 4 3 2 Read: Seventy-six thousand four hundred thirty-two 2) Six-digit numbers such as 278,647 2 is in the hundred thousands place. 7is in the ten thousands place. 8is in the thousands place. 6is in the hundreds place. 4is in the tens place. 7 is in the ones place. 2 7 8,6 4 7 Read: Two hundred seventy-eight thousand six hundred forty-seven 3) 7-digit numbers such as 3,245,618 3 is in the millions place. 2 is in the hundred thousands place. 4 is in the ten thousands place. 5 is in the thousands place. 6 is in the hundreds place. 1 is in the tens place.

14 8 is in the ones place. 3 , 2 4 5, 6 1 8 Read: Three million two hundred forty-five thousand six hundred eighteen 4) Numbers with more than 7 digits such as 15,340,796 Read: Fifteen million three hundred forty thousand seven hundred ninety-six 421,674,081 Read: Four hundred twenty-one million six hundred seventy-four thousand eighty-one To count numbers with more than 7 digits, you can see that the digits on the left of the millions place are ten millions place value, hundred millions place value, thousand millions place value … respectively. Exercise 3 Write the following numbers in words. (1) 345 Read:_________________________________________________ (2) 8,017 Read:_________________________________________________ (3) 20,897 Read: _________________________________________________ (4) 302,466 Read: _________________________________________________ (5) 1,367,589 Read: _________________________________________________ (6) 703,970,500 Read:__________________________________________________

15 Topic 2: Place value and value of digits 2.1 Each digit in a number has a place value of 10 times the place to its right and the value of a digit in each place is computed by multiplying the given digit by its place value. 2.2 In reading numbers, read digits according to their corresponding place value in order starting from the digit with the largest place value to that with the smallest place value. For example, Number Million Hundred Ten Thousand Hundred Ten One Hun- Ten One thousand thousand dred 216,354,789 2 1 6 3 5 4 7 89 216,354,789 Read: Two hundred sixteen million three hundred fifty-four thousand seven hundred eighty-nine The place value and the value of a digit of a number are as follows: Place Place value The digit in each Value of the digit as per its Ones 1 place place value Tens 10 9 Hundreds 100 8 9x1 = 9 Thousands 1,000 7 Ten thousands 10,000 4 8 x 10 = 80 Hundred thousands 100,000 5 Millions 1,000,000 3 7 x 100 = 700 Ten millions 10,000,000 6 Hundred millions 100,000,000 1 4 x 1,000 = 4,000 2 5 x 10,000 = 50,000 3 x 100,000 = 300,000 6 x 1,000,000 = 6,000,000 1 x 10,000,000 = 10,000,000 2 x 100,000,000 = 200,000,000 From the above table, 9 is in the ones place and therefore has a value of 9. 8 is in the tens place and therefore has a value of 80. 5 is in the ten thousands place and therefore has a value of 50,000.

16 2 is in the hundred millions place and therefore has a value of 200,000,000. Example: 426,739 - What place is the underlined digit in and what is its value? Analysis: 426,739 2 is in the ten thousands place and has a value of 2 x 10,000 = 20,000. Exercise 4 Write the place and the value of the digit in  1. 115,116 ______________________________________________________ 2. 765,908 ______________________________________________________ 3. 9,235,776 ______________________________________________________ 4. 12,456,789 ______________________________________________________ 5. 420,831,546 ______________________________________________________ Topic 3: Expanded form of numbers A number can be shown as a sum of each digit multiplied by its corresponding place value. Example: Write 9,521,364 in an expanded form Analysis: 9,521,364 = (9x1,000,000) + (5x100,000) + (2x10,000) (1x1,000) + (3x100) + (6x10)+(4x1) Answer: 9,521,364 = 9,000,000 + 500,000 + 20,000 + 1,000 + 300 + 60 +4 Exercise 5 Write these numbers in an expanded form in your notebook. 1. 504,120 3. 19,754,830 2 468, 793 4. 562,849,321

17 Exception Reading numbers in everyday life In everyday life, digits are applied to different things with different reading methods as specified by agencies concerned or according to people’s preference. For example: 1) Reading digits according to their place value (1) Buddhist Era (B.E.) and Christian Era (or Anno Domini, abbreviated A.D.), for example B.E. 2552 Read: B.E. Two thousand five hundred fifty-two A.D. 2009 Read: A.D. Two thousand nine (2) House numbers can be read in 2 ways: 1. Read according to place value for digits in front of “/” symbol and read digits after the “/” symbol separately 2. Read each digit separately. For example, house number 377/13 can be read as “House number three hundred seventy-seven slash thirteen”, or “Three-seven-seven slash one-three”. House number 94/140 can be read as“House number ninety-four slash one hundred forty, or nine- four slash one-four-zero”. 2) Reading each digit separately (1) Postal code, for example 10510 Read: one-zero-five-one-zero 10300 Read: one-zero-three-zero-zero (2) Vehicle registration number, for example ธศ3041 Read: Thor Sor three-zero-four-one Remark: To avoid confusion, the name of Thai consonants is usually read, for example, Thor-Thong, Sor-Sa-La three-zero-four-one. (3) Telephone number, for example 02 – 571 – 4239 Read: zero-two-five-seven-one-four-two-three-nine 08 – 1480 – 3424 Read: zero-eight-one-four-eight-zero-three-four-two- four Remark: The digit 2 in telephone numbers is normally read as “Tho” in Thai for better identification.

18 (4) Official correspondence, for example, No. กท2013.2/27 reads “Number Gor Tor two-zero-one-three point two slash two-seven” Topic 4: Ordering number Ordering numbers is comparing pairs of numbers and putting them in an ascending order (from the smallest to the highest numbers) or a descending order (from the highest to the lowest number). In comparing numbers, compare the digits in the same place to see which one is higher. If the values of the digits are the same, compare the next pair of digits from the left. Repeat this until the pair of digits is different. Example: Compare 39,215 and 39,251 to see which one is higher and put them in an ascending order from the higher to the lower one. Analysis: Since the values of digits in the ten thousands, thousands and hundreds places are the same for both numbers, consider the next pair of digits, which is the tens place. The digit in the tens place of 39,251 is 5 and therefore has a value of 50, while the digit in the tens place of 39,215 is 1and therefore has a value of 10. So, 39,251 is greater than 39,215. The numbers can be written in descending order as 39,251 39,215 Exercise 6: Put these numbers in an ascending order from the lowest to the highest numbers. 1. 956,420 965,204 659,024 69,594 69,945 2. 10,050 10,500 1,001,001 110,001 111,100 3. 769,386 1,001,900 972,142 893,013 100,119 4. 2,403,107 2,460,710 2,471,613 2,498,789 999,991 4.1 Comparing numbers using > (greater than), < (less than), = (equal to), ≠ (not equal to) symbols (1) Use > symbol when the former number is greater than the latter number, for example: 85 > 58: Eighty-five is greater than fifty-eight. 72 > 48: Seventy-two is greater than forty-eight. (2) Use < symbol when the former number is less than the latter number, for example: 58 < 85: Fifty-eight is less than eighty-five. 48 < 72: Forty-eight is less than seventy-two. (3) Use = symbol when both numbers are equal, for example:

19 25 + 55 = 80: The sum of twenty-five and fifty-five is equal to eighty. 120 + 40 = 160: The sum of one hundred twenty and forty is equal to one hundred sixty. (4) Use ≠symbol when both numbers are not equal, for example: 1,031 ≠ 1,003: One thousand thirty-one is not equal to one thousand three. Exercise 7 Compare the following pairs of numbers by writing >, < or = symbol in the blanks (1) 89  98 (2) 1,181 1,811 (3) 1,888  8,881 (4) 355 553 (5) 1,001  1,100 (6) 1,500 1,005 (7) 202 + 28  230 (8) 23,870  23,807 (9) 495  385 + 110 (10) 7,605  7,650 Topic 5: Estimation and rounding Sometimes it is not necessary to give the exact size, amount or quantity of things in everyday life. In such case, the nearest number is estimated and used for ease of remembering. 5.1 Rounding to the nearest ten 110 111 112 113 114 115 116 117 118 119 120 121 122 114 is between 110 and 120 but it is closer to 110. So, 114 rounded to the nearest ten would be 110. 115 is equally close to 110 and 120, and 115 rounded to the nearest ten is 120. In rounding to the nearest ten, look at the digit in the ones place of the number If the digit in the ones place is less than 5, round down. If the digit in the ones place is equal to or more than 5, round up. 5.2 Rounding to the nearest hundred, thousand, ten thousand, hundred thousand The same rule applies to the rounding to the nearest hundred, thousand, ten thousand, hundred thousand. That is, look at the digit in the lower place value (the digit to the right). - Round 2,440 and 2,460 to the nearest hundred - 2,440 rounded to the nearest hundred is 2,400. - 2,460 rounded to the nearest hundred is 2,500.

20 Exercise 8 a. Round these numbers to the nearest ten 1) 54 _____________________________ 6) 718 ______________________________ 2) 129 ____________________________ 7) 895 ______________________________ 3) 381 _____________________________ 8) 919 ______________________________ 4) 562 _____________________________ 9) 1,045 ____________________________ 5) 675 _____________________________ 10) 2,655 ___________________________ b. Round these numbers to the nearest hundred 1) 109_____________________________ 6) 1,049 ______________________________ 2) 182 ____________________________ 7) 2,534 ______________________________ 3) 276 _____________________________ 8) 5,079 ______________________________ 4) 593 _____________________________ 9) 14,306 _____________________________ 5) 626_____________________________ 10) 203,148 ___________________________ c. Round the number of population of the following countries to the nearest hundred thousand 1) Japan 118,519,000 people _________________________________________ 2) France 55,239,000 people ________________________________________ 3) India 688,600,000 people________________________________________ 4) China 1,004,000,000 people_________________________________________ 5) Russia 279,900,000,000 people____________________________________

21 Topic 6: Property of counting numbers and zero, and application in problem exercises Counting numbers are positive integers which are 1, 2, 3, 4, 5, … and so on. The smallest counting number is 1. The largest counting number cannot be identified as the counting numbers are infinite. 0 (zero) is a digit, not a counting number. 6.1 Property of one 1) Multiply any number by one or multiply one by any number, and the product will be equal to that number. Example: 4 1=4 or 1  4 = 4 2) Divide any number by one and the quotient will be equal to that number. Example: 3÷1=3 or 7 ÷ 1 = 7 6.2 Property of zero 1) Adding any number to zero or adding zero to any number does not change that number. Example: 2+0=2 or 0 + 2 = 2 2) Any number multiplied by zero or zero multiplied by any number will be zero. Example: 2×0=0 or 0 × 2 = 0 3) Zero divided by any non-zero number will be zero. Example: 0÷6=0 or 0 ÷ 8 = 0 or 0 ÷ 15 = 0 Remark: In mathematics, 0 is not used as a denominator and so division by zero is undefined.

22 Example, 5 ÷ 0 is undefined or meaningless in mathematics. or 36 ÷ 0 is undefined or meaningless in mathematics or 790 ÷ 0 is undefined or meaningless in mathematics 4) If the product of a multiplication of any 2 numbers is equal to zero, at least one of the numbers must be zero. Example 4 × 0 = 0 or 0 × 9 = 0 or 0 × 0 = 0

23 Topic 7: Addition, subtraction, multiplication and division of counting numbers and problem exercises 1.1 Addition Definition of addition Addition is combining two or more numbers together and the result is called “total” or “sum”. The plus sign (+) is used as a symbol of addition. Forms of addition Horizontal addition Vertical addition 5+2 = 7 5 is known as Augend 2+ is known as Addend 7 is known as Sum Ones Tens Ones 50 + 10 = 60 5 + 0 + 1 0 60 Tens Tens Hundreds Tens Ones Ones 4 0 0 + 400 + 250 =650 2 5 0 Hundreds 650 Addition of zero 1) The sum of 0 and 0 is 0. 2) The sum of any number and 0 is equal to that number. For example, 5 + 0 = 5 or 0 + 5 = 5. Addition of two numbers and three numbers without carrying forward Addition of 2 numbers Addition of 3 numbers 123 + 543 = 6 6 6 2 , 3 1 2 + 2, 1 1 4 + 5, 3 2 1 = 9 , 7 4 7 123 + 2,3 1 2 + 54 3 2,1 1 4 5,3 2 1 666 9,7 4 7

24 Adding two numbers or three numbers without carrying forward is combining two numbers or three numbers together and the sum of the two or three digits of each digit place does not exceed 9. We can calculate the sum by means of horizontal addition as follows: Example: 423 + 215 = ? Method: 423 + 215 = 638 Answer: 638 Analysis 4 2 3 215 first number second number Add the digits in each place column from right to left starting with the rightmost digits of both numbers as follows: Ones Add 3 in the first number to 5 in the second number, and the result is 8. Put the sum in the ones place. Tens Add 2 in the first number to 1 in the second number, and the result is 3. Put the sum in the tens place. Hundreds Add 4 in the first number to 2 in the second number, and the result is 6. Put the sum in the hundreds place. Sum The sum is 638. Expanded method addition Example: 310 + 423 + 236 = ? Method: 310 + 423 + 236 = (300 + 10 + 0) + (400 + 20 + 3) + (200 + 30 + 6) = (300 + 400 + 200) + (10 + 20 + 30) + (0 + 3 + 6) = 900 + 60 + 9 = 969 Answer: 969

25 Exercise 9 a. Write > , < or = symbol in the  (1) 98  80 + 9 (2) 138 + 821  959 (3) 999 + 101  1,101 (4) 11,312  10,000 + 1,213 b. Find the sum of the following numbers by using the expanded method: (1) 62 + 6 (2) 43 + 34 (3) 1,234 + 2,103 (4) 312 + 213 + 101 (5) 2,311 + 3,042 + 506 Vertical addition Method 1: Expanded method Example1: 147 + 720 = ? Method: 147 100 + 40 + 7 720 + 700 + 20 + 0 + 800 + 60 + 7 = 867 Answer: 867 Example 2: Find the sum of 2,433 and 2,114 and 5,322 first number second number Method: 2,433 2,000 + 400 + 30 + 3 + sum third number 2,114 + 2,000 + 100 + 10 + 4 = 9,869 5,322 4,000 + 500 + 40 + 7 5,000 + 300 + 20 + 2+ sum 9,000 + 800 + 60 + 9 Answer: 9,869

26 Method 2: Short method Example: 147 + 720 = ? Method: 147 7 2 0 L+ine up the number by place value and add the digits in the same place 867 Answer: 867 Example: Find the sum of 2,433 and 2,114 and 5,322 Method: 2, 4 3 3 2, 1 1 4 + 4, 5 4 7 5, 3 2 2 + 9, 8 6 9 Answer: 9,869 Exercise 10 a. Find the sum of the following numbers by using the expanded method (1) 140 + 123 (2) 210 + 304 + 63 (3) 11,200 + 3,504 + 23,183 (4) 210,250 + 454,104 + 33,141 b. Find the sum by using the place value table and the short method (1) 121 + 47 (2) 132 + 325 (3) 12,100 + 454,104 + 33,141 (4) 1,152,113 + 2,112,421 + 1,320,260

27 Addition of two numbers and three numbers with carrying forward Addition of 2 numbers Addition of 3 numbers 7,665 + 5,247 = 12,912 22,452 + 76,258 + 50,864 = 149,574   7, 6 6 5 2 2,4 5 2 + 5, 2 4 7 76,258 + 12, 9 1 2 5 0, 8 6 4 1 4 9, 5 7 4 The method and analysis for the addition of two numbers and three numbers with carrying forward is the same as those for the addition without carrying forward. When the sum of digits in each place is 10 or more, the first digit of the sum is carried forward to the next digit place. Finding the sum by horizontal addition with carrying forward Method1: Short method Example: Find the sum of 7,665 and 5,247 Method: 7,665 + 5,247 = 12,912 Analysis Answer: 12,912 7,665 5,247 first number second number Ones Add 5 in the first number to 7 in the second number to get 12. Place 2 as the sum of digits in the ones place and carry 1 which is in the tens place forward to be added to the digits in the tens place. Tens Add 6 in the first number to 4 in the second number to get 10. Add the sum to 1 which is the carried forward number to get 11. Place 1 in the right as the sum of digits in the tens place and carry 1 in the left to be added to the digits in the hundreds place.

28 Hundreds Add 6 in the first number to 2 in the second number to get 8. Add the sum to 1 Thousands which is the carried forward number to get 9. Place 9 as the sum of digits in the Sum hundreds place. Add 7 in the first number to 5 in the second number to get 12. Place 2 as the sum of digits in the thousands place and place 1 as the sum of digits in the ten thousands place since there are no more digits to add. So, the sum is 12,912. Method 2: Expanded method Example: 7,665 + 5,247 = ? Method: 7,665 + 5,247= (7,000 + 600 + 60 + 5) + (5,000 + 200 + 40 + 7) = (7,000 + 5,000) + (600 + 200) + (60 + 40) + (5 + 7) = 12,000 + 800 + 100 + 12 = 12,000 + 900 + (10 + 2) = 12,000 + 900 + 10 + 2 = 12,912 Find the sum by Aconlsuwmenr:a1d2d,9i1t2ion with carrying forward Method 1: Expanded column method Example: Find the sum of 627,665 and 385,247 Method: 600,000 + 20,000 + 7,000 + 600 + 60 + 5 6 2 7, 6 6 5 3 8 5, 2 4 7 300,000 + 80,000 + 5,000 + 200 + 40 + 7 900,000 + 100,000 + 12,000 + 800 + 100 + 12 = 1,000,000 + (10,000 + 2,000) + 900 + (10 + 2) = 1,000,000 + 10,000 + 2,000 + 900 + 10 + 2 = 1,012,912 Answer: 1,012,912 Example: Find the sum of 31,562and 87,149and 60,975 MMetheotdho2:dS:h  3 1 ,5 6 2 + 8 7 ,1 4 9 6 0 ,9 7 5 1 7 9 ,6 8 6 Answer: 179,686

29 Exercise 11 a. Find the sum of the following numbers by using the expanded horizontal method (1) 54,623 + 93,545 (2) 871,496 + 247,308 b. Find the sum of the following numbers by using the place value table and short vertical method (1) 3,486,801 + 1,670,528 (2) 584, 169 + 958,782 + 321,456 Problem exercise: Addition Example: The first orchard collected 2,355 coconuts. The second orchard collected 4,020 coconuts. The third orchard collected 3,700 coconuts. How many coconuts were collected? Symbolic expression: 2,355 + 4,020 + 3,700 =  Method 1: The first orchard collected 2,355 coconuts The second orchard collected 4,020 coconuts The third orchard collected 3,700 coconuts The total number of coconuts collected is 2,355 + 4,020 + 3,700 = 10,075 Answer: 10,075 coconuts Method 2: The first orchard collected 2,355 coconuts The second orchard collected 4,020 + coconuts The third orchard collected 3,700 coconuts The total number of coconuts collected 10,075 coconuts Answer: 10,075 coconuts Solving addition problems is the same as performing normal addition of numbers depending on the method you want to use. Normally, the above 2 methods are preferred, especially Method 2 which is suitable for problems with several-digit numbers because it simplifies addition and minimizes errors. Exercise 12 Indicate the applied method: (1) In one sub-district, there are 1,323 old-aged people, 9,705 active age adults and 4,320 children. How many people are there in this sub-district? (2) Mr. Charlie gets Baht 18,257 for the first sale of rice, Baht 16,540 for the second sale and Baht 13,050 for the third sale. How much does Mr. Charlie get from all the three sales of rice?

30 (3) A foster home receives a first donation of Baht 351,279 and a second donation of Baht 131,217. How much does the foster home receive from all donations? (4) If Mr. Pong has to pay monthly installments of Baht 2,500 for a refrigerator, Baht 3,500 for a TV set and Baht 500 for an electric rice cooker. How much does Mr. Pong pay to the shop each month? Commutative property of addition Example 1 403 + 326 = 729 326 + 403 = 729 Therefore 403 + 326 = 326 + 403 Example 2 234 641 6 4 1+ 234 + 875 875 ===== ===== When two numbers are added, the sum is the same regardless of the order of the addends. For example: 12 + 36 = 36 + 12 This concept is called the “commutative property of addition”. Associative property Associative property of addition 3 + 5+ 2 = (3 + 5) + 2 3 + 5 + 2 = 3 + (5 + 2) = 8 +2 = 3+7 = 10 = 10 Therefore (3 + 5 ) + 2 = 3 + (5 + 2) 121 + 122 + 321 = (121 + 122) + 321 121 + 122 + 321 = 121 + (122+321) = 121 + 443 = 564 = 243 + 321 = 564 Therefore (121 + 122) + 321 = 121 + (122 + 321) When three numbers are added, any two numbers are first added and the result is then added to the remaining number. The sum is the same regardless of the grouping of the addends. This

31 concept is called the “Associative property of addition”. Usually, a parenthesis ( ) is used to separate the first two numbers from the last number. The associative property of addition can be illustrated as follows: Example 41 + 12 + 34 = (41 + 12) + 34 41 + 12 + 34 Method 1: = 53 + 34 = 87 Method 2: = 41 + (12 + 34) = 41 + 46 = 87 Normally, the associative property of addition is applied by adding the two numbers with the lower values before adding the result to the highest-value number such as in Method 2. For ease of calculation, if the sum of any two numbers ends with 0, those two numbers will be added first and the result will be added to the remaining number. 7.2 Subtraction Definition of subtraction Subtraction is to deduct one number from another number or comparing two numbers. The remaining amount or the difference between these two numbers is called the “difference”. The minus sign (-) is used as a symbol of subtraction. Forms of addition Horizontal subtraction 7 Vertical subtraction 7- 2 = 5 2– Minuend Minuend Subtrahend Difference 5 Subtrahend Difference 405 - 200 = 205 Hundreds Tens Ones 40 20 5– 20 0 5 1. Like addition, there are two forms of subtraction, i.e. horizontal subtraction and vertical subtraction. The method and analysis are the same as those for addition meaning that subtraction is performed on a pair of numbers in each digit place starting from those in the ones place to the left.

32 2. Zero and subtraction 2.1 Subtracting zero from zero gives zero. 2.2 Subtracting zero from a number always equals that number. For example, 5 – 0 = 5. Subtraction without carrying over Subtraction without carrying over is the subtraction of two numbers where the value of each digit of the subtrahend does not exceed the value of the digit in the same digit place of the minuend. Subtraction without carrying over can be illustrated as follows: Method 1: Short method Example: By subtracting 214 from 465, how much is left? Method: Symbolic expression: 465 –214 =  465 - 214 = 251 Answer 251 Analysis: Step 1: Start by subtracting the two ones’ place digits, i.e. Subtract 4 from 5 gives 1 Step 2:Subtract the two tens’ place digits, i.e. Subtract 1 from 6 gives 5 Step 3: Subtract the two hundreds’ place digits, i.e. Subtract 2 from 4 gives 2 Therefore, the difference is 251. Method 2: Expanded method Example: Malai has Baht 255 and spends Baht 120. How much money does she have left? Symbolic expression: 255 – 120 =  Method: Malai has 255 baht She spends 120 baht She will have 255 – 120 = (200 + 50 + 5) – (100 + 20 + 0) = (200 – 100) + (50 – 20) + (5 – 0) = 100 + 30 + 5 = 135 baht Answer: 135 baht Analysis: Step 1:Expand the minuend and the subtrahend according to the place value. Separate each place value with a parenthesis ( ) and put the minus sign (-) between two parentheses.

33 Step 1: Rearrange by putting the numbers in the same digit place in the same parenthesis with a minus sign (-) between the numbers. The plus sign (+) is placed between the parentheses. (See Line 2) Step 2: Subtract the numbers in each parenthesis and put a plus sign (+) between the differences. (See Line 3) Step 3: Notice that numbers in Line 3 are in the expanded form. Sum them up to get a single number that is 135. (See Line 4) Vertical subtraction of two numbers without carrying over With vertical subtraction, the minuend must always be placed on the top of the subtrahend, and the digits must be in the correct column. The difference can be obtained in the following manner: Method 1: Expanded method Example: 756 –302 =  Method: 7 5 6 700 + 50 + 6 302 – 300 + 0 + 2 – 400 + 50 + 4 = 454 Answer: 454 Method 2: Short method As the short method applies the concept and method of the place value table, the calculation is illustrated as follows: Example: Find the difference between 578 and 453 Symbolic expression: 578 - 453 =  Method: 578 453 – 125 Answer: 125 Subtraction with carrying over Subtraction with carrying over is used when a digit in the minuend represents a value which is less than the value of the corresponding digit in the subtrahend. In such case, the minuend is carried over by taking or borrowing 1 from the next higher place and adding the value to the place directly to

34 the right. After that, the subtrahend is subtracted from the minuend. Subtraction with carrying over can be illustrated as follows: Horizontal subtraction of two numbers with carrying over For horizontal subtraction, the short method is preferred to the expanded method because it is simpler. As a result, only the short method is shown here. Example: 56 – 38 =  Method: 56 – 38 = 18 Answer: 18 Analysis 56 – 50 + 6 – 38 30 + 8 Based on the expanded vertical method, we can see that the minuend, which is 6, is less than the subtrahend, which is 8. So, borrow 1 from the tens digit and add 10 to 6 to get 16 before subtracting 8 from 16. 50 + 6 – 40 + 16 – 30 + 8 30 + 8 10 + 8 = 18 Subtraction numbers with several digits also applies the same concept. That is, if the tens digit in the minuend represents a value which is less than the value of the tens digit in the subtrahend, carry over by borrowing 1from the hundreds digit and adding 100 to the tens digit before subtracting. If the hundreds digit in the minuend represents a value which is less than the value of the hundreds digit in the subtrahend, carry over by borrowing 1 from the thousands digit and adding 1,000 to the hundreds digit before subtracting, and so on. Method 1: Expanded method Exercise: 724 - 467 =  Method: 724 700 + 20 + 4 – 600 + 110 + 14 467 – 400 + 60 + 7 400 + 60 + 7 – 200 + 50 + 7 = 257 Answer: 257

35 Analysis 724 – 700 + 20 + 4 467 400 + 60 + 7 Since the ones digit in the minuend, which is 4, is less than the corresponding digit in the subtrahend, which is 7, borrow 1 from the tens digit and add 10 to 4 to get 14. 700 + 20 + 4 700 + 10 + 14 400 + 60 + 7 400 + 60 + 7 As for the tens digit, after carrying over, the remaining value is only 10 which is less than the value of the tens digit in the subtrahend, which is 60, borrow 1 from the hundreds digit and add 100 to get 110 before subtracting. 700 + 10 + 14 – 600 + 110 + 14 – 400 + 60 + 7 400 + 60 + 7 200 + 50 + 7 = 257 Likewise, the same carry over pattern applies to subtraction of numbers in the hundreds place. Method 2: Short method Example: Find the difference between 7,151 and 6,249 Method: 6 7 4 1111 7 15 1 9– 2 6 24 90 Answer: 902 Exercise N13ote that the difference of the leftmost digit is zero. Subtract the following numbers: (1) 900 - 400 (2) 888 - 727 (3) 15,280 - 10,270 (6) 27,648 – (4) 63 – (5) 6,248 – 25 41 9,806 (7) 3,000 + 500 + 40 + 5 (8–) 50,000 + 4,000 + 500 + 60 – 20,000 + 3,000 + 1,000 + 400 + 30 + 2 300 + 30 Relationship between addition and subtraction Subtraction Addition Minuend – Subtrahend = Difference Difference + Subtrahend =Minuend 7– 2 =5 5+ 2 =7 Since subtraction is deducting one number from another number, it is the inverse operation of addition or the opposite of addition. Addition is combining two numbers together to

36 get a higher value but subtraction is deducting one number from another number to get a lower value. From the above example, it can be seen that: Minuend+ Subtrahend = Difference On the other hand, Difference - Subtrahend = Minuend As a result, based on the inverse relationship between addition and subtraction, we can use addition to check whether the difference is correct or not. Example: Find the difference and check the answer 465 – Check answer 485 Check answer 251 + – 271 + 214 214 214 214 485 251 465 271 271 is the correct answer. Mixed25a1disditthieoncoarnredctsuanbstwraecrt.ion In addition to the above relationship, sometimes the symbolic expression of a math problem may contain both plus and minus signs. In such cases, the operation in the parenthesis is done first. Example: (3,237,596 + 242,456) – 366,530 =  Method:  3 , 2 3 7, 5 9 6 + 2 4 2, 4 5 6 – 3 , 4 8 0, 0 5 2 3 6 6, 5 3 0 3 , 1 1 3, 5 2 2 AnalysAisnswer: 3,113,522 As the first number given here has the highest value, we may choose one of the following two methods. First, add the second number to the first number before subtracting the third number from the sum as shown in Method 1. Alternatively, the third number may be subtracted from the first number before the second number is added as per Method 2. However, if the first number and the

37 second number which is the addend are less than the subtrahend, we cannot perform subtraction first. In such case, only Method 1 is applicable. Mixed addition and subtraction also applies to matters in our everyday life as will be discussed in the next topic. Problem exercises Like addition, subtraction problems are related to everyday life. Example 1: A seller sells 350 pomelos and 270 mangosteens. How many more pomelos are sold? Symbolic expression: 350 – 270 = – 350 pomelos Method: The seller sells and sells 270 mangosteens The seller sells 80 pomelos more than mangosteens Answer: 80 pomelos Example 2: Last month, Somchai had Baht 3,456. He earns Baht 2,009 this month. He spends Baht 1,750 to buy a closet. How muc+h money does he have left? Symbolic expression: (3,456 + 2,009) – 1-,750 =  3,456 Method: Last month, Somchai had – baht This month, he earns 2,009 baht Analysis He has a total of 5,465 baht InHExeabmupyles 2a, caldodsietitonfomrust be perfo1r,m75e0d firbstbtoahsete how much Somchai earns in two months. AftHerethnaot,wsuhbatrsact the closet cost fro3m,71a5ll the mbaohntey he has since he will have less money after he paysAfnorswtheerc:los3e,7t.15 baht Multiplication Definition of multiplication Multiplication is a mathematical operation where a number is added to itself a number of times. The operation can be illustrated by multiplying 2 numbers, which are the added number and the times the number is added. The number obtained from multiplying 2 numbers is called the “product” and the multiplication sign (×) is written between those two numbers. In other words, multiplication is a shortcut to repeated addition. An example of the symbolic expression of multiplication is 2 × 9 = 18. The statement reads 2 times 9 equals 18 2 is known as Multiplicand

38 9 is known as Multiplier 18 is known as Product Therefore, Multiplicand × Multiplier = Product As multiplication is a shortcut to repeated addition, multiplication tables or the so-called times tables are developed to make repeated addition easier and faster. The time tables are usually memorized. Examples of times tables are as follows: Times Tables Multiplied by Multiplied by Multiplied by Multiplied by Multiplied by 23456 2×1 =2 3×1 =3 4×1 =4 5×1 =5 6×1 =6 2×2 =4 3×2 =6 4×2 =8 5×2 = 10 6×2 = 12 2×3 =6 3×3 =9 4×3 = 12 5×3 = 15 6×3 = 18 2×4 =8 3×4 = 12 4×4 = 16 5×4 = 20 6×4 = 24 2×5 = 10 3×5 = 15 4×5 = 20 5×5 = 25 6×5 = 30 2×6 = 12 3×6 = 18 4×6 = 24 5×6 = 30 6×6 = 36 2×7 = 14 3×7 = 21 4×7 = 28 5×7 = 35 6×7 = 42 2×8 = 16 3×8 = 24 4×8 = 32 5×8 = 40 6×8 = 48 2×9 = 18 3×9 = 27 4×9 = 36 5×9 = 45 6×9 = 54 2 × 10 = 20 3 × 10 = 30 4 × 10 = 40 5 × 10 = 50 6 × 10 = 60 2 × 11 = 22 3 × 11 = 33 4 × 11 = 44 5 × 11 = 55 6 × 11 = 66 2 × 12 = 24 3 × 12 = 36 4 × 12 = 48 5 × 12 = 60 6 × 12 = 72 Multiplying by 0 0 is 0 × 1 = 0 0 + 0 is 0 × 2 = 0 0+0+0 is 0 × 3 = 0 0+0+0+0 is 0 × 4 = 0 Any number multiplied by 0 equals 0. Based on the above times tables, a simple multiplication chart can be developed to assist those who cannot memorize the times tables. Multiplication chart 1 2 3 4 5 6 7 8 9 10 11 12 1 2 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84

39 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144 How to use a multiplication chart: 1. The first row of the table, from left to right, contains the multiplicand. 2. The first column of the table, from top to bottom (vertical column), contains the multiplier. 3. Read across and down to find the product of any two numbers. The number in the square where the row and column meet is the product. For example, to find the product of 6 × 8, find 6 in the first row and read down. Then, find 8 in the first column and read across from left to right. The row and column meet at the square containing 48. So, the product of 6 × 8 is 48. To find the product of other numbers, repeat the steps mentioned above. For example, 3 × 7 = 21, 9 × 4 = 36. Exercise 14 Fill the products in the following multiplication charts. (1) × 1 2 3 4 5 6 7 8 9 10 11 12 44 20 40 (2) × 9 10 11 12 ×3 4 5 6 7 6 54 66 13 4 5 6 7 7 70 84 2 8 12 3 9 15 21 8 72 88 4 16 24 5 15 25 35 9 90 108 10 90 110 Forms of multiplication Vertical multiplication Horizontal multiplication 7 Multiplicand 5× Multiplier 7× 5 = 35 35 Product

40 86× 4 = (80 + 6) × 4 86 4× = (80 ×4) + (6 × 4) = 320 + 24 344 = 344 Finding the product of numbers with 3 digits or less When the multiplier is a one-digit number: When multiplying two numbers where the multiplicand may contain one, two or three digits but the multiplier is a one-digit number, the product can be calculated as follows: Method 1: Simple multiplication This method is suitable for a single-digit multiplicand. We can find the product by referring to the multiplication chart or using times tables to give an immediate answer. Example: 7 × 5 =  Vertical multiplication Horizontal multiplication Method: 7 Method: 7 × 5 = 35 5× Answer: 35 35 Answer: 35 Note that the horizontal multiplication is a multiplication using a symbolic expression. Method 2: Expanded method This method is applicable to multiplication where the multiplicand contains 2 digits or more. Example: Find the product of 37 and 4 Horizontal multiplication Vertical multiplication Method: 37 × 4 = (30 + 7) × 4 Method: 37 30 + 7 = (30 ×4) + (7 × 4) 4 × 4× 120 + 28 = = 120 + 28 148 = 148 Answer: 148 Answer: 148

41 Example: Horizontal multiplication Vertical multiplication Method: Method: 578 × 6 = (500 + 70 + 8) × 6 = (500 × 6) + (70 × 6) + (8 578 500 + 70 + 8 4× ×6 × 6) = 3,000 + 420 + 48 3,000 + 420 + 48 = = 3,468 3,468 Answer: 3,468 Answer: 3,468 Method 3: Short method Normally, this method is preferred to Method 2 for multiplication where the multiplicand contains 2 digits or more. Example: Find the product of 45 and 9 Horizontal multiplication Vertical multiplication Method: 45 × 9 = 405 Method: 45 9× Answer: 405 405 Answer: 405 Concept: Multiply each pair of numbers starting from digits in the ones place Ones 5 ×9 = 45 Put 5 in the ones column and carry over 4 which is in the tens place to the next column for addition to the digit in the tens place. Tens 4 ×9 = 36 Add the carried over 4 to the result to get 40. Put 0 in the tens place and 4 in the hundreds place as there are no more digits to multiply. Product Therefore, the product is 405. Example: Find the product of 578 and 6 Horizontal multiplication Vertical multiplication Method: 578 × 6 = 3,468 Method: 578 6× Answer: 3,468 3,468 Answer: 3,468 Concept : In Example 2, the concept is the same as Example 1. Just additionally multiply digits in the hundreds place. Ones 8× 6 = 48 Put 8 in the ones column and carry over 4 which is in the tens place to the next column for addition to the digit in the tens place.

42 Tens 7× 6 = 42 Add the carried over 4 to the result to get 46. Put 6 in the tens place and carry over 4 to the next column for addition to the digit in the hundreds place. Hundreds 5×6 = 30 Add the carried over 4 to the result to get 34. Put 4 in the hundreds place and 3 in the thousands place as there are no more digits to multiply. Product Therefore, the product is 3,468. Exercise 15 a. Find a number which makes the statements true. (1) 5× 37 = 5 × (30 +  ) (2) 65× 3 = (60 × 3) + (5× ) (3) 47× 8 =( +  ) × 8 (4) 123× 7 = (  + 20 +  ) × 7 (5) (300 + 40 + 6)× 9 = (300 × 9) + (  × 9) + (6 ×  ) b. Find the product by using the short method. (1) 28× 3 (2) 78× 4 (3) 64× 7 (4) 90× 8 (5) 328× 8 When the multiplier is a two-digit number: When multiplying two numbers where the multiplicand may contain one, two or three digits but the multiplier is a two-digit number, multiplication is performed by multiplying each multiplier digit by the multiplicand starting from the ones digit. The products of each multiplier digit and multiplicand are added together to get the overall product. There are several multiplication methods but the preferred ones are as follows: Method 1: Short method Vertical multiplication is preferred to horizontal multiplication because of ease of examination and clarity. Horizontal multiplication applies the same concept as vertical multiplication, but is not illustrated here. Only the product is shown. Only examples of vertical multiplication are displayed below. Example: 234× 36 Form 1 Form 2 Method: Method: 234 30 + 6 234 3 6× 234× 6 3 6× 234 × 140 4 140 4 7 0 2 0+ 7 0 2+

43 30 8,4 2 4 8,4 2 4 Answer: 8,424 Answer: 8,424 Analysis: According to this method, the place Analysis: According to this method, the last value of each multiplier digit multiplies by the digit of the product of each multiplier digit is in multiplicand, and the product of each multiplier the same column as that multiplier digit, and digit is added together. the product of each multiplier digit is added together. Method 2: Factorization of multiplier Finding factors of a multiplier is decomposing the multiplier to get the one-digit numbers which multiply together to get the multiplier. For example, 21 = 3 × 7. So, 3 and 7 are called factors of 21. With this method, the multiplier is a single-digit number and the multiplication is simpler regardless of how many digits the multiplicand has. There is no need to sum up the product of each multiplier digit. Just multiply each factor of the multiplier by the multiplicand. Example: Find the product of 274 and 21 Horizontal multiplication Vertical multiplication Method: 21 = 3 × 7 Method: 21 = 3 × 7 274 × 21 = 274 × (3 × 7) 274 × 274 = (274 × 3) × 7 21 × 3 = 822 × 7 822 × = 5,754 7 Answer: 5,754 5,754 Answer: 5,754 Analysis 1. Find factors of 21 which are 3 × 7 2. Multiply 3, which is smaller, by 274 to get 822 (Multiplying a smaller number gives a smaller product which makes the next multiplication easier) 3. Multiply 7 by 822 to get the product of 5,754 Method 3: Factorization of multiplier which is a multiple of 10 This method is used when the multiplier is a multiple of 10 or ends with 0.

44 Example: Find the product of 324 and 30 Horizontal multiplication Vertical multiplication Method: 30 = 3 × 10 Method: 30 = 3 × 10 324 × 30 = 324 × (3 × 10) 324 = (324 × 3) × 10 × 3 = 972 × 10 972 × = 9,720 10 Answer: 9,720 9,720 Answer: 9,720 Method 4: Expanded method This method makes multiplication easier and is suitable for multiplication of numbers with several digits. The number with more digits, either the multiplicand or the multiplier, is expanded according to place value before multiplying by another number. Then, the products are added together as in Method 1 – Short method. Example: Find the product of 382 and 23 Horizontal multiplication Vertical multiplication Method: Method: 382 × 23 = (300 + 80 + 2) × 23 382 × 300 + 80 + 2 = (300 × 23) + (80 × 23) + (2 × 23) 23 × 23 = 6,900 + 1,840 + 46 6,900 + 1,840 + 46 = 8,786 = 8,786 Answer: 8,786 Answer: 8,786 Exercise 16 a. Find the product by using the short method: (1) 36 × 17 (2) 45 × 22 (3) 55 × 40 (4) 79 × 30 (6) 123 × 21 b. Find the product by using factorization of multiplier: (1) 54 × 20 (2) 63 × 21 (3) 154 × 24 (4) 583 × 32 c. Find the product by using the horizontal expanded method

45 (1) 78 × 60 (2) 98 × 72 (3) 825 × 56 (4) 999 × 80 When the multiplier is a three-digit number There are several multiplication methods for multiplying three-digit numbers but the appropriate and convenient ones are as follows: Method 1: Short method Vertical multiplication is preferred for this method. The calculation is performed in the same way as multiplying 2-digit numbers. Only examples of vertical multiplication are given here. Example: 267 × 125 Form 1 Form 2 Method: Method: 267 × 267 × 125 125 1335 267× 5 1335 267 × 20 + + 5340 534 26700 267 × 100 267 3 3,3 7 5 3 3,3 7 5 Answer: 33,375 Answer: 33,375 Method 2: Factorization of multiplier which is a multiple of 10 This method is used when the multiplier is a multiple of 10 in the same manner as the two-digit multiplier. Example 1: Find the product of 372×250 Horizontal multiplication Vertical multiplication Method: Method: 250 = 25 × 10 = 5 × 5 × 10 250 = 25 × 10 = 5 × 5 × 10 372 × 250 = 372 × (5 × 5 × 10) 372 × = (372 × 5) × 5 × 10 5 = (1,860 × 5) × 10 1,860 × = 9,300 × 10 5 = 93,000 9,300

46 Answer: 93,000 10 93,000 Answer: 93,000 Example 2: Find the product of 362 and 100 Horizontal multiplication Vertical multiplication Method: Method: 100 = 10 × 10 100 = 10 × 10 362 × 100 = 362 × (10 × 10) 362 × = (362 × 10) × 10 10 = 3,620 × 10 3,620 × = 36,200 10 Answer: 36,200 36,200 Answer: 36,200 When the multiplier is 100, which is a multiple of 10, we can see that any number multiplied by 100 equals that number followed by two zeroes (00). From Example 2, we can find the product of 362 and 100 by using the short method which is easier. Horizontal multiplication Vertical multiplication Method: 362 × 100 = 36,200 Method: 362 Answer: 36,200 × 100 36,200 Answer: 36,200 Exercise 17 a. Find the product by using the short method. (1) 136 × 111 (2) 275 × 165 (3) 398 × 234 (4) 764 × 491 b. Find the product by using the expanded vertical method. (1) 247 × 200 (2) 624 × 120 (3) 879 × 240 (4) 917 × 320

47 Mathematical problems As multiplication is a shortcut for addition, math problems about multiplication are thus related to our everyday life as in the case of addition. However, multiplication makes problem Example 1: Dried garlic costs Baht 18 per kilogram. How much money do we get from selling 9 kilograms of dried garlic? solMvientghofads:teSry. mbolic expression: 18×9 =  One kilogram of dried garlic costs 18 baht Sell 9 kilograms We will get 18 × 9 = 162 baht Answer: 162 baht Example 2: A barrel of rice costs Baht 130. A bottle of fish sauce costs Baht 18. How much does it cost to buy 4 barrels of rice and 14 bottles of fish sauce? Method: Symbolic expression: (130 × 4) + (18 × 14) =  A barrel of rice costs 130 baht 4 barrels of rice cost 130 × 4 = 520 baht A bottle of fish sauce costs 18 baht 14 bottles of fish sauce cost 18 × 14 = 252 baht Total cost is 520 + 252 = 772 baht Answer: 772 baht In solving multiplication problems, both vertical and horizontal multiplications can be performed. However, vertical multiplication in the form of symbolic expression is normally preferred because it is simpler than other methods. Exercise 18 (1) Orange costs Baht 15 per kilogram. How much does it cost to buy 10 kilograms of orange? (2) Eight groups of villagers build a road to their village. There are 9 people in each group. How many people build the road?

48 (3) An orchard farmer plants 9 rows of mango trees. Each row has 20 mango trees. He also plants 7 rows of guava trees. Each row has 22 guava trees. How many trees are there in the orchard? (4) First, a woman buys 15 bags of cookies. Each bag has 5 cookies. Then, she buys 20 bags of cookies. Each bag has 6 cookies. If she gives 1 cookie to 1 child, how many children will get the cookie? (5) A farmer sells 43 carts of paddy rice at Baht 4,500 per cart. How much money does the farmer get? 1.2 Commutative property of multiplication Horizontal multiplication Vertical multiplication 3×2 = 6 32 2× 3 = 6 × 3× 2 Therefore 3 × 2 = 2 × 3 6= 6 10 × 9 = 90 10 9× 9× 10 = 90 × 9 10 Therefore10 × 9 = 9 × 10 90 = 90 When multiplying two numbers, the product is the same regardless of the order of the multiplicand and the multiplier. For example, 3 × 2 = 2 × 3 or 10 × 9 = 9 × 10. We call this concept the “Commutative property of multiplication”. Associative property of multiplication 3×5×6 = (3 × 5) × 6 3×5×6 = 3 × (5 × 6) = 15 × 6 = 3 × 30 = 90 = 90 Therefore (3 × 5) × 6 = 3 × (5 × 6) When multiplying three numbers, the product is still the same regardless of their grouping. That is, any two numbers can be multiplied before multiplying by the third number and the product is the same. We call this concept the “Associative property of multiplication”. Based on this property, the following guidelines make calculation easier: 1. Multiply two smaller numbers first before multiplying by the third number.

49 2. Multiply two numbers that give the product ending with zero before multiplying by the third number. 3. When there is one number having 3 digits or less and ending with zero, multiply the other two numbers first before multiplying by the number ending with zero. 3. Distributive property of multiplication (5 + 10) × 4 = 15 × 4 (5 + 10)× 4 = (5 × 4) + (10 × 4) = 60 = 20 + 40 = 60 Therefore (5 + 10) × 4 =(5 × 4) + (10 × 4) Multiplying the sum of two numbers by a third number is the same as multiplying each of the two numbers by that third number and adding up the products. This concept is called the “Distributive property of multiplication”. This property of multiplication often applies in the multiplication of 2 numbers having 2 digits or more by the horizontal expanded method. As for this course, it is preferably used in multiplying numbers with up to 2 digits which learners have already learned in the multiplication of single-digit numbers. Exercise 19 Fill a number in the  to make the statement true. 1.  ÷ 7 = 0 2.  × 1 = 4 3. 10 ÷ = 10 4. 46 +  = 46 5.  + 0 = 0 + 8 6. 0 × 9 = 9 ×  7. 716 +  = 210 + 716 8. 50 × 70 = 70 ×  9. (9 +7) + 26 = 9 + (  + 26) 10. (40 × 17) × 69 = 40 × (17 ×  ) 11. (5,040 + 1,460) × 445 = + (1,460 × 445)

50 1.4 Division Definition of Division Division is splitting a number into equal groups or deducting equal amounts repetitively which can be shown by the division of just two numbers. The number resulting from the division of two numbers are called “dividend” and the symbol ÷ is used for division. For example, 8 ÷ 2 Example 1 How many times can 3 be subtracted from 15? 1st time: 15 – 3 remains 12 2nd time: 12 – 3 remains 9 3rd time: 9 – 3 remains 6 4thtime: 6 – 3 remains 3 5th time: 3 – 3 remains 0 It can be seen that 3 can be subtracted from 15 a total of 5 times until the remains becomes zero. Hence, 15 ÷ 3 = 5. Example 2 How many plates with 4 pieces each–are used to distribute 10 pieces of snacks? Total snack 10 pieces Put into the first plate 4 pieces Remaining 6 pieces Put into the second plate 4 – pieces Remaining 2 pieces Thus, the snack can be put onto 2 plates with a remainder of 2 pieces. Hence, 10 ÷ 4 = 2 remainder 2. Repeated subtraction of the same amount until the result of the subtraction becomes 0 as shown in Example 1 is called “exact division”. But if the final result of subtraction is not zero, as shown in Example 2, it is called “inexact division”, and the number left over from the last subtraction is called “remainder”. From the subtraction example above, it can be seen that division is a shortcut approach of subtraction and the expression showing the division, for example, 15 ÷ 3 = 5 is called division expression. It is read as 15 divided by 3 equals 5. 15 is called dividend. 3 is called divisor.