CLASS 4 - TERM-1 PRIME YEARS

6 arrangements are possible with green fixed. In the same way, find other arrangements With each of the other colors fixed. Total number of arrangements = 24 G R Y B G R Y B G R B Y G R B Y G R B Y G R Y B How many 4-digit numbers can be formed without repeating any digit? ___________ Which is the largest number formed? ______________ Which is the smallest number formed? ______________ 5 2 8 4 Building numbers from digits. Activity1: Do this in groups. Have crayons or color pencils of 4 different colors. Example: red, green, blue and yellow. They should be of the same length. Make a square with these pencils. How many different ways can 4 pencils be arranged? Activity 2 Repeat the same experiment with number cutouts 51

1. Find the largest and smallest numbers formed by the digits without repeating any digit. a. 4,2,8,7 b. 1,3,8,4,7 c. 5,0,1,2 d. 1,0,9,2,5 e. 7,3,2,8,1 2. Study the pattern and complete the next two terms a. 94992, 94994, 94996, 94998, ___________, __________ b. 7325, 7330, 7335, ____________, ___________ c. 4304, 4314, 4324, __________, _________ d. 4526, 4626, 4726, ___________, ________ e. 5725, 6725, 7725, ___________, ________ 3. In the given numbers, write down:- a) 78147 How many thousands: __________ How many hundreds: ___________ b) 54801 How many thousands: ___________ How many hundreds: ____________ 4. How many 4 - digit numerals are there in all? How many 5- digit numerals are there in all? 5. How many 4- digit numerals can be made with 4, 0, 1, 2 ? 6. Arrange in ascending order and What is the place value of 8 in each of the above numbers? 28075, 81254, 73812, 17998 7. Make 5 numerals with the digits 1,0,8,5,4. using each digit only once. Arrange them in descending order. 52

Indian and International System of Numeration The system of numeration used all over the world is called Decimal System. In this system, the base is 10. Successive places increase by multiples of 10. Though the decimal system is followed in all countries, the names given to places are different in Indian and western system of numeration. Let’s first study the Indian System of numeration. 10 thousands Thousands Hundreds Tens Units 10 x 10 x 10 x 10 10 x 10 x 10 10 x 10 1 x 10 1 The number system now used worldwide is known as Indo Arabic Numerals. i.e. 1,2,3,4, ....... 53 Crore 10 Lakhs 10 Thou- Hundreds Tens Units lakhs Thousands sands 3 5 8 3 4 2 1 3 2 4 0 1 5 6 Here, the periods (places) are named as follows: 1. The numerals written in the table are read as three lakhs fifty eight thousand three hundred and forty two. 2. One crore thirty two lakhs forty thousand one hundred and fifty six.

The periods or places are separated using commas as shown: 1, 32, 40, 156 Units, tens and hundreds places are grouped together. The thousands places are grouped together. The lakhs are grouped together. The crores are grouped together. Guided 1. Write the given numbers in the Indian System of numeration and write the numerals using commas to separate the periods. a. 3 2 5 4 5 7 5 b. 8 7 5 1 0 5 7 4 Independent 1. Write the following numerals in the period chart using Indian System of Numeration and rewrite in figures using commas in the right place. a. 4 5 8 7 1 2 3 b. 1 5 3 8 4 c. 4 5 8 3 1 0 7 d. 8 1 3 0 1 3 e. 1 5 3 8 1 7 0 6 f. 3 2 8 1 7 g. 1 7 4 3 54

The numerals are read as follows: 1) Three million, two hundred and fifty four thousand, three hundred and twentyone. 2) Forty million, three hundred and seventeen thousand, four hundred and ninety two. 3) Three hundred twentyfive million, eight hundred and one thousand, five hundred and thirtyfour. 4) One billion, two hundred fiftyeight million, four hundred thirty two thousand, one hundred and sixtyseven. Commas to separate the periods are used as follows: 1. Units, tens and hundred are grouped together. 2. Thousands, 10 Thousands, and hundred thousands are grouped together. 3. Million, 10 Million and hundred million are grouped together. 4. The Billions are grouped together. 3, 254, 321 40, 317, 492 325, 801, 534 1, 258, 432, 167 55 The International System of Numeration In this system, the periods are named as shown: Billion 100 10 Million 100 10 Thou- Thou- Hund Tens Units Million Million Thousands sands sands reds 3 2 5 4 3 2 1 4 0 3 1 7 4 9 2 3 2 5 8 0 1 5 3 4 1 2 5 8 4 3 2 1 6 7

1. Write the given numerals in the International System of Numeration. Rewrite using commas. Write the numbers in words. a) 3 6 1 8 4 2 b) 1 0 3 5 3 1 6 4 0 c) 1 4 2 3 3 2 0 7 5 d) 1 5 3 8 1 4 0 2 5 1 e) 1 4 2 8 6 1 f) 4 2 0 3 4 0 2 g) 7 5 8 5 2 1 5 6 Independent Comparison between the two systems: How will you convert numbers in one system to another? This is essential for converting money in International System to Indian System and vice-versa. Study the table below: 100Cr. 10Cr. Cr. 10L L 10 Th. Th. Hun. Tens Units Billons 100M 10 M M 100Th. 10 Th. Th. Hun. Tens Units 56

Refer to the table and make the conversions. One Lakh = _____________ Thousands 100 Million = _____________ Crore 1 Crore = _____________ Million 10 Lakh = _____________ Million 1 Billion = _____________ Crore Guided 57 1. Convert the following money in Indian System to International System. a. Rs. 32 lakhs b. Rs. 15 crores c. Rs. 125 crores d. Rs. 8 lakhs 50 thousand e. Rs. 525 crore f. Rs. 7 crores 35 lakhs 2. Convert the following amount to Indian System a. Rs. 250 million b. Rs. 7 million c. Rs. 85 million d. Rs. 754 thousand

1. Convert the following to International System a. 7 lakhs 25 thousand b. 82 lakhs c. 3 crores 45 lakhs d. 75 crores 2. Convert the following to Indian System a. 455 million b. 28 million c. 8 million d. 258 thousand 3. Write the given numbers, putting commas for both Indian and International System. Also write the number names in Indian and International System. a) 4 2 8 1 5 3 2 b) 6 3 8 7 2 5 4 c) 5 0 2 1 4 3 d) 8 3 2 1 4 0 e) 1 0 5 2 5 7 58

• Do it as group work. Cut out a strip from a chart paper. Prepare a table showing Indian and International System. • Both the systems are to be written in the same table for comparison. • Display it in the class. • Look for large numbers on hoardings or advertisements in newspapers or newspaper headlines. • Rewrite them putting commas in Indian and International Systems. • Also write their number names in both the systems. 4. Pick up the terms from both the boxes to match. 50L 70L 10Cr. 8 Cr20L 1Cr 90L 40L 2L46Th 7Cr 10L 2L 8L 100M 7M 82 M 4M 200Th 5 M 9M 1M 10M 70M 800Th 246Th 59

Place Value Place Value and Face Value Face value is the value of the digit irrespective of the position in a number. Place Value is the value of the same digit when it is in different places in a number. Take for example the name Arun. At school, he is a student. For his parents, he is a son. For his sister, he is a brother. For his playmates, he is a friend. In Cricket, he is a batsman. The same person takes different roles in different situations. In the same way, consider the digit 3 . When it is in Unit’s place, its value is 3. When it is in tens place, its value is 30, when it is in hundreds place, its value is 300 and so on. But the digit is always 3. It doesn’t change. Thus, we say that face value of 3 is always 3, but its place values change. In 47 3 Place value of is 3. 3 In 6 7 Place value of is 3 3 30 . In 60 Place value of is 3 3 300 . In the number 3 75 4, the place values of 3 3 are 30000 and 30 respectively Give the place value and face value of the underlined digits. (1) 871435 (2) 910347 (3) 257384 (4) 47600 Guided 60

Expanded form and place value When you split the number into units, tens, hundreds, thousands etc. we get the expanded form. The expanded form gives the place value of each digit in the number. Example: 43758 = 40000 + 3000 + 700 + 50 + 8 When added, you get the normal form 40000 + 3000 + 700 + 50 + 8 43758 Activity 1: Do it as group work. Take paper strips of different lengths. Take 9 strips of the same length. Write numbers 1,00,000, 2,00,000, 3,00,000…………… 9,00,000 as shown Take 9 strips of shorter length and write numbers 10000, 20000, 30000, …………. 90000 in the same way. Take 9 strips of still smaller length and write 1000, 2000, ……. 9000 and continue upto 1, 2, 3, 4, 5 ……9. Choose any number, say 567894 Take the 500000 strip, 60000 strip, 7000 strip, 800, 90 and 4. Keep one over the other. You will get the required number. Play this as a game. One asks for a number and the others choose the strips to make the number. 1 0 0 0 0 0 61

Staple the strips as shown on the right to get the normal form of the number. 5 0 0 0 0 0 6 0 0 0 0 7 0 0 0 8 0 0 9 0 4 5 6 7 8 9 4 Guided Write the place value and face value of the digits circled in the following numbers. a. 478346 - b. 732144 - c. 105283 - Independent 1. Write the place value and face value of the underlined digits. a. 8731495 - b. 1938143 - c. 4308143 - d. 2784153 - 62

2) Write the following in normal form. a. 900000 + 80000 + 7000 + 300 + 20 + 5 b. 700000 + 8000 + 700 + 30 + 6 c. 40000 + 700 + 8 d. 500000 + 60000 + 50 + 3 3) Write the following in expanded form. a. 570843 b. 340019 c. 170805 d. 630400 e. 900835 Successor and Predecessor 1. Give the successor and predecessor of the following: Predecessor Successor 29999 30000 15399 40090 Guided Independent 1. Give the successor and predecessor of the following: a. 47809 d. 28090 b. 39000 e. 17029 c. 10909 f. 49000 63

1. What is the difference in the place values of the 4 in the given numbers? a. 481485 b. 14324 2. What is the difference between the place value and face value of 5 in the number 65324? 3. What is the place value of 3 in the following numbers in the Indian and International System? a. 348579 b. 13254861 4. Write the expanded form of 8432174 5. Give the successor of the following: a. 45709 b. 15019 6. Give the predecessor of the following: a. 14080 b. 24000 1) How many 1000 rupee notes, 100 rupee notes and 10 rupee notes are needed to make a bundle containing Rs. 30980? 2) How many hundred rupee notes will make Rs. 18000? 64

Rounding off numbers Rounding off numbers is a case of estimation. It helps in quick computation. Rounding off by tens: Consider the number 85. The digit in the units place is 5. We set the rule that if the digit in the units place is 5 or more than 5, the digit in the tens place is increased by 1 and the units place is replaced with 0. Thus, we write 85 90 Consider the number 84. Here, the units digit is less than 5. Hence, our estimation is 84 80. When the units place has a digit less than 5, we retain ten’s place as it is, and units place digit is replaced with 0. Independent Round off the following numbers to the nearest tens. a) 92 b) 72 c) 69 d) 58 e) 98 f) 23 g) 46 h) 34 i) 18 j) 55 Rounding off by hundreds: Consider the number 364 The number in the place lower to hundreds is 6. It is more than 5. So, 1 is added to the digit in the hundred’s place and the rest of the places are replaced by zero. 364 400 65

325 300 Similarly 480 500 429 400 Independent Round off the numbers to the nearest hundreds. a) 124 b) 184 c) 255 d) 228 e) 468 f) 709 g) 409 h) 752 Rounding off to ten in three digit numerals: Consider the number 328. The digit in the ten’s place is 2 and that in the unit’s place is 8, which is more than 5. So, add 1 to 2 and replace 8 by zero. Thus, we write 328 as 330 Similarly, 545 as 550 529 as 530 556 as 560 728 as 730 502 as 500 552 as 550 Independent Round off to the nearest tens. a) 345 b) 382 c) 402 d) 405 e) 861 f) 865 g) 738 h) 734 66 In 325, the digit in the place lower to hundreds is 2, which is less than 5. So, tens and units are replaced by zero, retaining the digit in the hundred’s place as it is.

Rounding off by thousands, hundreds and tens: Consider the number 4285. a) Digit in the hundred’s place is 2. Hence, the number when rounded off upto 1000’s place is 4000. b) When rounded off to hundred’s, it is 4300. c) When rounded off to tens, it is 4290. 1. Round off the following numbers to the highest place. a. 458 e. 42135 b. 5082 f. 28018 c. 1243 g. 37225 d. 1803 h. 44825 2. Estimate the sum of the following by rounding off the numbers to the nearest thousands. 4285 7013 1385 4325 4285 + 3902 + 5381 + 1815 + 3081 + 3902 3. Estimate the sum of the following by rounding off to the nearest hundreds. a. 4981 b. 2384 c. 1482 d. 3402 + 3472 +1925 +2034 +1957 4. Find the difference by rounding off to thousands. a. 4812 b. 6374 c. 7983 - 2417 - 2602 - 2310 67

1. Round off to nearest lakhs. a) 720185 c) 438631 b) 851284 d) 159301 2. Round off the same numbers to the nearest ten thousand. 3. Round off the same numbers to the nearest thousand. 4. Find the sum of the following by rounding off the numbers to the highest place. Calculate mentally. a) 34801 b) 23845 c) 4580 + 5623 + 3048 + 2610 + 169 + 2281 5. From a bundle containing Rs. 32540, Rs. 9585 were removed. How much money is left in the bundle now? Compute by rounding off to the highest place. Do it as a mental exercise. 6. In a travel company, the collections on three consecutive days are Rs. 8347, Rs. 3812 and Rs. 4590. Estimate the total amount collected, nearest to the thousands. 1) Organise a team quiz. Divide the class into two teams. One team gives a number. Any one from the other team can round off the number to the place asked for (nearest thousand, ten thousand, lakhs or even hundred). (Numbers can be written on the board) 2) Organise a similar quiz to tell the place value of the digit asked for in the given number. (Numbers can be written on the board). 68

Addition Do the following additions mentally. (Time 10 minutes) 1) 325 + 10 2) 4281 + 100 3) 4281 + 100 4) 825 + 5 5) 35 + 25 + 75 6) 4285 + 1000 7) 8 + 7 + 9 + 4 8) 5 + 4 + 6 9) 12 + 8 + 2 10) 100 + 210 Do the following word problems by writing relevant statements: 1) There are 5 baskets of apples containing 208, 250, 125, 70 and 50 apples respectively. What is the total number of apples? 2) In a cinema hall for a particular show, 120 first class tickets, 225 second class tickets and 180 third class tickets were sold. What is the total number of tickets sold? 3) In a factory, there are 1800 two wheelers and 160 four wheelers ready for sale. Can you find the total number of wheels for the vehicles kept for sale? 69

4) A train carried 4025, 8276 and 3025 passengers for three days respectively. What is the total number of passengers carried by the train in three days? Addition of large numbers 1. Add 1. Add ones and write below ones. 2. Add tens and write below tens. 3. Add hundreds and write below hundreds. 4. Add thousands and write below thousands. 5. Add 10 thousands and write below 10 thousands. 6. Add lakhs and write below lakhs. 2. Arrange in column and add 232125 + 41240 + 2134 Write as shown Add the units, tens, hundreds ………….. upto lakhs and write below their respective places. 2 1 3 2 5 1 +1 8 4 6 2 8 L 10Th Th H 3 9 7 8 7 9 T U 2 3 2 1 2 5 + 4 1 2 4 0 + 2 1 3 4 L 10Th Th H 2 7 5 4 9 9 T U Independent Arrange in column and add the following: a) 15042 + 30306 + 241211 b) 221103 + 12324 + 100221 Arrange the following in columns and add a) 24654 + 42031 + 2124 b) 12011 + 24024 + 113 c) 14141 + 202021 + 2104 d) 10254 + 132031 + 2014 Independent 70

Large additions with carrying: 3 4 5 8 7 1 + 2 7 5 3 8 4 L 10Th Th H 6 2 1 2 5 5 T U 1 1 1 1 1 1) Add ones & write below ones 1 + 4 = 5 2) Add 10’s 7 + 8 = 15. Write 5 below tens and carry 1 over to hundred. 3) Add the hundreds 1 + 8 + 3 = 12. Write 2 below hundreds and carry 1 over to thousand. 4) Add thousands 1 + 5 + 5 = 11. Write 1 under thousands and carry the other one over to 10 thousands. 5) Add the 10 thousands 1 + 4 + 7 = 12. Write 2 under 10 thousands and carry the 1 over to lakhs. 6) Add the lakhs and write down. 1 + 3 + 2 = 6 Ans: 621255 Guided Write in columns and add the following: a) 18215 + 3268 + 221347 + 167 b) 330275 + 127854 + 76521 + 8245 71 Eg: Add: 345871 + 275384

1. Arrange in columns and add the following: a) 128704 + 110817 + 84276 b) 448705 + 326507 + 15812 + 64066 c) 102841 + 28107 + 148275 + 1481 2. Do the following word problems, writing the required statements: a. Mr. X deposited Rs. 2,10,758 in a bank and invested Rs. 52,804 in shares. What are his total savings? b. Population of two villages is 2,18,264 and 1,52,984. What is the total population of the two villages? Independent Estimation: Estimate the sum of the following by rounding off to the nearest thousand. 1) 4852 + 3104 2) 1284 + 2502 3) 3600 + 1804 Estimate the sum of the following by rounding off to the nearest ten thousand. 1) 28174 + 31785 + 14824 2) 10025 + 35024 + 28608 Estimate the sum of the following by rounding off to the nearest thousand 1) 1284 + 20837 2) 12505 + 8094 3) 25843 + 34415 72

Rapid Fire 5 minutes 1. Sum of the largest 3 –digit number and smallest 4-digit number is __________. 2. 100 more than 6285 is _________ . 3. 1 lakh – 10 thousand = ___________ . 4. 3 hundreds + 3 tens = ____________ . 5. Sum of 725 and 836 rounded off to the nearest hundred is ________ . 6. The total number of 4-digit numerals is _________ . 7. 765 + _________ = 264 + ________ . 8. 1428 + 0 = ________ . 9. The estimated sum of 235 + 128 + 462 + 195 is _________ hundreds. 1. Find the sum of 827987 + 72814 + 1572. 2. Find the sum of 205740 + 653479 + 53899. 3. Estimate the sum, rounding off to the nearest lakhs: 727149 + 264705 + 589934. 4. Mr. X bought a bed for Rs. 34500, a dining table for Rs. 25875, and a refrigerator for Rs. 21847. What is the total amount he paid for all? 73

Discover: Digit Sum patterns: Digit sum is the sum of the digits in a given number. It can be used to check all mathematical operations. Eg. Add 78 + 25 = 103. To check whether it is correct, let’s use digit sum. 78 = 7 + 8 = 15 = 1 + 5 = 6 25 = 2 + 5 = 7 Adding the two digit sums: 6 + 7 = 13 = 1 + 3 = 4. The digit sum of the sum is: 103 = 1 + 0 + 3 = 4. As you can see, both are the same. Hence, the addition is correct. Sum of digit sums of numbers is equal to the digit sum of the sum. Add the following. Verify their addition by using digit sum property. 1. 125 + 315 2. 83 + 765 3. 123841 + 23402 Digit sum patterns: Take multiplication table for 2. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 …………………. Digit sums for that are: 2, 4, 6, 8, (1 + 0), (1 + 2), (1 + 4), (1 + 6), (1 + 8), (2 + 0), (2 + 2), (2 + 4), 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8 ………………… What do you observe? 74

You can make a design with digit sums for multiplication tables. Draw a circle and mark numbers 1 to 9 along the circle at equal distances. Join the digit sums of the multiplication table of 2 by drawing lines. You get a design as shown. Make designs for other multiplication tables too on the 9 – point circle. Observe the pattern in which the digits grow in successive steps. Complete the next two steps. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 75

Subtraction 1. Subtract 53 from 78. 2. Subtract the sum of 83 & 165 from 758. 3. From a bunch of 173 bananas, 85 were removed. How many are left? 4. Do the following subtractions: a. 4285 -1479 b. 2384 – 135 c. 5000 – 1924 d. 3285 – 1992 5. From a bundle of Rs. 2489, Rs. 1695 were removed. How much money is left in the bundle? 6. Two baskets had 325 and 485 apples respectively. They were mixed and put in the baskets equally. From one basket, 182 apples were removed. How many are left in the basket? 7. Estimate the difference by rounding off to tens: 92 - 65. 8. Rs. 1243 and Rs. 2563 were taken from a bag and put in the cash box. From that, Rs. 1283 was taken out. How much money is left in the cash box? 76

Subtraction involving large numbers Subtract 12128 from 32349. Subtract each digit in the respective places by backward counting. 3 2 3 4 9 - 1 2 1 2 8 2 0 2 2 1 10Th Th H T U Do the following subtractions: 1. 45879 – 24662 2. 58043 – 41021 3. 472345 – 50123 Guided Independent Solve the following: 1. 64784 – 51362 2. 48406 – 32002 3. 758232 – 423101 4. 199877 – 59875 5. Mr. X had Rs. 2 lakhs in a bank. He withdrew Rs. 1,72,583. How much money is left in his account? 6. A factory produced 27150 two wheelers in 2010 and 38995 two wheelers in 2011. How many more two wheelers were produced in 2011? 77

Subtraction with regrouping: Consider the problem: 328075 – 187981 3 2 8 0 7 5 - 1 8 7 9 8 1 L 10Th Th H 1 4 0 0 9 4 T U 12 7 10 17 9 Solve the following: 1. 37148 – 28907 2. 428142 – 289528 3. 505107 – 326396 4. 700000 – 528174 5. The prices of 2 brands of car are Rs. 5,85,275 and Rs. 3,85,769 respectively. How much more does the first brand cost than the second? 6. In a village, there are 72812 males and 60987 females. How many more males are there in the village than females? Guided 2 78

Estimation: 1. Find the approximate difference between 128 and 75 by rounding off the numbers nearest to the highest place. 128 - Highest place is the hundreds digit. In the tens place is 2, which is less than 5. So the rounded figure is 100. 75 - Highest place – tens. Digit in the units place is 5. Thus, rounded figure is 80. Ans: 100 – 80 = 20. Independent Guided Estimate the difference by rounding off to the highest place. a) 87215 – 65021 b) 50281 – 52302 c) 158304 + 95005 – 125801 Estimate the difference by rounding off to the highest place. a) 8172 + 18872 – 20817 b) 180275 + 102895 – 115879 c) 702581 – 258714 d) 482574 + 350279 79

1. 8275 – 0 = ————————— (8275/ 0) 2. 0 – 1579 = —————————— (1579/ not possible) 3. 475 – 275 = 275 – 475 (Correct/ Wrong) 4. 875 – 432 = ———————————— approx. (300/ 400) 5. Approximate value of 4972 – 2603 = ———————— (8000/ 2000) 6. 50000 – 11111 = ———————— (38889/ 48999) 7. 23821 = —————— thousands. (20000/ 2000) 8. 1,82,824 is less than 1,84,728 by ——————— . 9. There are —————————— thousands in 26278. 10. The difference between the largest 5 –digit number and the largest 4-digit number is ——————————— . 11. 6 thousands - 6 hundreds = ——————— 1. Subtract 3,56,255 from 6,00,000. 2. Find the difference between 483695 and 329825. 3. Which is more? By how much? 1,75,328 or 1,69,395 4. Simplify: a. 2,51,743 + 6,21,301 – 1,19,254 b. 4,81,396 + 2,18,105 – 50,187. 5. From the sum of 45,347 and 3,25,175, subtract the difference of the two. 80

Multiplication Activity 1: You learned in the previous classes that multiplication is repeated addition. 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 10 9 8 7 6 5 4 3 2 1 = 20 = 18 = 16 = 14 = 12 = 10 = 8 = 6 = 4 = 2 Eg: 2 x 1 = 2 (one square) 2 x 2 = 4 (2 squares) 2 x 3 = 6 (3 squares) In the same way, draw patterns for multiplication tables of other numbers. Study the pattern obtained for multiplication table of 2. Adding the 2’s in each column will give the product. 81

Activity 2: 1) Complete the multiplication grid given on the right. 2) From the grid, find out the following: 1 1 2 3 4 5 6 7 8 9 10 1 1) 6 x 1 = 2) 7 x 8 = 3) 5 x 7 = 4) 6 x 8 = 5) 9 x 9 = 6) 9 x 10 = 7) 3 x 9 = 8) 8 x 8 = Multiplication by 10, 100, 1000 etc.: To multiply by 10, add one zero at the end of the number. Eg. 6 x 10 = 60 25 x 10 = 250 125 x 10 = 1250 To multiply by 100, add 2 zeroes to the number. 6 x 100 = 600 25 x 100 = 2500 125 x 100 = 12500 X 2 1 3 4 5 7 6 8 9 10 82

To multiply by thousand, add 3 zeroes to the number. 6 x 1000 = 6000 25 x 1000 = 25000 50 x 1000 = 50000 15 x 1000 = 15000 14 x 20 = (14 x 2) x 10 = 28 x 10 = 280 8 x 400 = (8 x 4) x 100 = 32 x 100 = 3200. Guided Do the following multiplications: a) 15 x 30 b) 25 x 20 c) 125 x 50 d) 16 x 200 e) 9 x 800 f) 25 x 300 Independent Do the following multiplications: 1) 13 x 20 2) 23 x 200 3) 5 x 800 4) 7 x 600 5) 28 x 20 6) 125 x 20 7) 8 x 800 8) 91 x 20 9) 82 x 200 10) 9 x 400 11) 124 x 40 12) 56 x 300 83

Properties of Multiplication: 1) Any number multiplied by zero is zero Eg: 2 x 0 = 0 215 x 0 = 0 12854 x 0 = 0 2) Any number multiplied by 1 is the number itself. 1 is called identity element in multiplication because multiplication by 1 does not change the value of the number. 25 x 1 = 25 125 x 1 = 125 4280 x 1 = 4280 3) Multiplication is commutative. Commutative property states that multiplication of 2 numbers can be done in any order. Eg: 2 x 3 = 3 x 2 45 x 5 = 5 x 45 85 x 64 = 64 x 85 4) Multiplication is associative. Associative property states that three or more numbers can be multiplied in any order. Eg: 5 x 3 x 2 = (5 x 3) x 2 = 5 x (3 x 2) = 3 x (2 x 5) = 15 x 2 = 5 x 6 = 3 x 10 = 30 = 30 = 30 Eg: 28 x 32 x 15 = (28 x 32) x 15 = 28 x (32 x 15) = 32 x (28 x 15) 84

5) Multiplication is distributive. Distributive property states that multiplication can be distributed over different terms, and the products can be added up. Eg.1: 2 x (5 + 3)= (2 x 5) + (2 x 3) = 10 + 6 = 16 Eg.2: 3 x 18 = 3 x (20 -2) = 3 x 20 – 3 x 2 = 60-6 = 54 Eg.3: 5 x 24 = 5 x (20 + 4 ) = 5 x 20 + 5 x 4 = 100 + 20 = 120 Eg.4: 6 x 98 = 6 x (100 -2 ) = 6 x 100 – 6 x 2 = 600 – 12 = 588 Eg.5: 4 x (103) = 4 x (100 + 3) = 4 x 100 + 4 x 3 = 400 + 12 = 412 1. Show that 5 x 8 = 8 x 5 by repeated addition. 2. Show that 3 x 4 x 7 = (3 x 4) x 7 = 3 x (4 x 7) = 4 x (3 x 7) 3. Use distributive property to do quick multiplication. a) 3 x 99 b) 15 x 12 c) 25 x 102 d) 3 x 23 Guided 85

1. Fill in the blanks: a. 284 x _____ = _____ x 54 b. 115 x 205 = _____ x 115 c. 303 x _____ = _____ x 92 d. 3 x 4 x _____ = _____ 3 x 7= _____ x 4 x 7 e. 21 x 12 x _____ = 15 x _____ x 21 = 12 x _____ x 15 2. Use associative property to find the product quickly for the following: a. 2 x 5 x 3 b. 4 x 5 x 8 c. 3 x 50 x 2 Use distributive property to find the following products quickly: 1) 5 x 12 2) 8 x 18 3) 7 x 101 4) 9 x 99 Multiplication Multiplication by single digit: 245 x 3 1. Multiply the one’s place by 3. 3 x 5 = 15 Write 5 under one’s and carry over 1 to ten’s place. 5) 125 x 11 6) 23 x 22 7) 126 x 19 8) 103 x 50 86 2 4 5 x 3 7 3 4 H T U 1 1

2. Multiply ten’s place by 3 and add 1. 3 x 4 = 12 + 1 = 13 Write 3 under one’s and carry over 1 to ten’s place. 3. Multiply hundred’s and add 1 2 x 3 + 1 = 7 Write 7 under hundreds. Multiplication of 4, 5 and 6 digit numerals. Example: 15275 x 4. Units place – 5 x 4 = 20 i.e., 2 carried over. Tens place 7 x 4 = 28 + 2 = 30. 3 carried over. Hundred’s place 2 x 4 = 8 + 3 = 11. 1 carried over. Thousand’s place 5 x 4 = 20 + 1 = 21. 2 carried over. 10 Thousand’s place 1 x 4 = 4 + 2 = 6. th Guided Do the following multiplications: a) 2584 x 5 b) 32586 x 7 c) 153895 x 9 1 5 2 7 5 x 4 6 1 1 0 0 HT U 10 Th Th 2 1 3 2 Independent Do the following multiplications: a) 5623 x 6 b) 2508 x 7 c) 15202 x 7 d) 78138 x 9 e) 152832 x 8 f) 25316 x 5 87

Multiplication by 2 digit numbers: 1. 236 x 32 Here, 236 is the multiplicand and 32 is the multiplier. Consider the multiplier 32. It can be written as 30 + 2. 4 7 2 +7 0 8 0 7 5 5 2 2 3 6 ( 30 + 2) 2 3 6 x 3 7 0 8 1 Using distributive property of multiplication, we can write 236 x 32 = 236 x (30 + 2). = 236 x (2 + 30) = 236 x 2 + 236 x 30. Let us see the multiplication process. First, multiply 236 by 2 and write down the product, that is, 472. Next, multiply 236 by 30. 236 x 30 = 236 x 3 x 10 = 7080. Write it below 472 and add. Ans: 7552 (2) 4214 x 23 4214 x (20 + 3) = 4214 x 3 + 4214 x 20. = 12642 + 84280 = 96922. The short method of multiplying is shown on the right. Instead of adding the Zero in the unit’s place, start writing the product of multiplication from the ten’s place onwards. The * sign says not to write anything in that place. 1 2 6 4 2 +8 4 2 8 0 9 6 9 2 2 4 2 1 4 x 2 3 1 88 x 1 2 6 4 2 + 8 4 2 8 * 9 6 9 2 2 4 2 1 4 x 2 3 1

Do the following multiplications: 1) 326 x 34 2) 125 x 25 3) 2432 x 23 4) 2547 x 87 Guided Independent Do the following multiplications: 1) 256 x 16 2) 125 x 96 3) 485 x 24 Multiplication by 3-digit numbers 1. 826 x 214 214 = 200 + 10 + 4 Step-I: Multiply by 4 826 x 4 = 3304 Step-II: Multiply by 10 826 x 10 = 8260 Step-III: Multiply by 200 826 x 200 = 165200 Adding, we get 176764. Instead of putting zeros, it can be done like this 8 2 6 x 2 1 4 by 10 by 200 by 4 3 3 0 4 1 7 6 7 6 4 826 214 2 + 1 6 5 2 * * + 8 2 6 * 3304 +8260 +165200 176764 8 2 6 2 1 4 2 89 4) 2523 x 36 5) 3483 x 35 6) 2888 x 74 x 1 x 1

Do the following multiplications: a. 628 x 277 b. 4234 x 234 c. 6724 x 192 Guided Independent Do the following multiplications: a. 245 x 164 b. 679 x 234 c. 708 x 321 d. 4324 x 274 e. 3217 x 184 f. 2936 x 198 Rapid Fire: 5 minutes 1. 2487 x 0 = ——————————— 2. 1583 x 1 = ——————————— 3. 253 x _____ = ____ x 182 4. 21 x _____ x 43 = 43 x 15 x ____ = 21 x 15 x ____ 5. 5 x 50 = —————————— 6. 153 x 200 = ——————————— 7. 325 x 100 = —————————— 8. 25 x 101 = —————————— 9. 2 x 19 x 5 = —————————— 10. 5 x 24 = —————————— 90

1. Find the products of the following quickly by suitable rearrangement. a. 5 x 25 x 2 b. 125 x 20 c. 3 x 8 x 5 d. 125 x 12 e. 870 x 102 f. 915 x 98 g. 998 x 25 h. 99 x 252 2. Find the products of the following by long multiplication method. a. 873 x 42 b. 2432 x 98 c. 521 x 325 d. 2537 x 342 e. 5672 x 346 f. 6724 x 234 g. 1825 x 192 h. 288 x 175 1. Cost of one bicycle is Rs. 2345. What is the cost of 28 such bicycles? 2. How many seconds are there in a day? 3. The daily collection in a cinema theatre is Rs. 42578. What is the collection in a month? 4. A Syntex tank can hold 4825 litres of water. What is the capacity of 125 such tanks put together? 5. A book contains 286 pages. How many pages are required to print 156 such books? 91

Division Activity: 1. You are given 24 seeds and asked to arrange them into rows. a. If there are 6 seeds in each row, how many rows are there? b. If there are 4 seeds in each row, how many rows are there? 2. Write the division facts for both the arrangements. 3. If 24 seeds are arranged in 3 rows, how many seeds are there in each row? Show the division fact. 4. If there are two rows, how many seeds are there in each row if 24 seeds are arranged in rows? Show the division fact. Draw the sketch for the arrangement of seeds. Division by 10, 100, 1000 etc. Rules: 1. When a number is divided by 10, the unit's digit in the number becomes the remainder and the rest is the quotient. Eg: 1256 ÷ 10 2480 ÷ 10 24800 ÷10 Q = 125 Q = 248 Q = 2480 R = 6 R = 0 R = 0 92

2. When a number is divided by 100, the unit's and ten's digits become remainders and the remaining is the quotient. 1256 ÷ 100 2480 ÷ 100 24800 ÷ 100 Q = 12 Q = 24 Q = 248 R = 56 R = 80 R = 0 3. When a number is divided by 1000, unit's, ten's and hundred's digits become remainder. 1256 ÷ 1000 Q = 1 R = 256 In general, the remainder will have as many places as the number of zeros in the divisor, if the divisor is a multiple of 10. Find quotients and remainders for the following without actual division. 1) 480 ÷ 10 2) 2500 ÷ 100 3) 481534 ÷ 1000 4) 12305 ÷ 100 5) 83140 ÷ 10 6) 148270 ÷ 1000 7) 123 ÷ 10 8) 160205 ÷ 10000 Independent 93

Properties of division: 1. 125 ÷ 0 = doesn’t exist. 0 ÷ 125 = 0. Zero divided by any number gives '0' as quotient. 2. 125 ÷ 1 = 125 Any number divided by 1 gives the number itself as the quotient. 3. When a number is divided by itself, the quotient is 1. 125 ÷ 125 = 1 4. Dividend = Divisor x Quotient + Reminder 1. Write the division facts for the given arrangement. 2. Find the quotient and remainder for the following without actual division. a. 485 ÷ 10 b. 1258 ÷ 100 c. 1428 ÷ 1000 d. 1258 ÷ 10 e. 1428 ÷ 100 f. 1428 ÷ 10 94

3. Quotient = ————————— ÷ Divisor 4. Dividend = Divisor x (Quotient + Remainder). What is wrong with this statement? 5. In an Orchard, 105 mango trees are grown in 7 rows. How many trees are there in each row? 6. If 8 pens cost Rs. 96, what is the cost of 1 pen? 7. A train covers 456 km in 6 hours. What is the speed of the train in km/hr? Division a. Division by a single digit: Eg.1: 4285 ÷ 3 1. Check the digit in the highest place. It is more than the divisor. First divide that digit 4. From the multiplication table of 3, find the digit just less than 4. 3 x 1 = 3. 2. Write it below 4 and subtract. 3. Bring down 2. From the multiplication table of 3, we find that 4 x 3 = 12. Write 3 in the quotient and 12 below 12. Subtract. 4. Bring down 8. From multiplication table of 3, we find that 3 x 2 = 6. Write 2 in the quotient & 6 below 8. Subtract. 5. Bring down 5. 6. From the table of 3: 3 x 8 = 24. 7. Write 8 in the quotient and 24 below 25. Subtract. Thus, Quotient = 1428. Remainder = 1. 4 2 8 5 3 1428 -3 1 2 - 1 2 0 8 - 6 2 5 - 2 4 1 95

Eg.2: 4285 ÷ 6 1) When the digit in the highest place is less than the divisor, take the next digit along with the highest digit. Here, it happens to be 42. Put a comma on the top of 2 to indicate that first step of division is done. For 42: 6 x 7 = 42. Write 7 in the quotient and 42 below 42. Subtract. 2) Bring down 8. 6 x 1 = 6. Write 6 below 8 and 1 in the qutient. Subract 6 from 8. 3) Bring down 5. 6 x 4 = 24. Write 4 in the quotient and subtract 24 from 25. 4 2 8 5 6 7 1 4 4 2 0 8 - 6 2 5 - 2 4 1 Quotient = 714 Remainder = 1 Eg.3: 4285 ÷ 9 1. As done earlier, first divide 42 by 9. 9 x 4 = 36; 9 x 5 = 45. We choose 9 x 4 = 36. Write 4 in the quotient and subtract 36 from 42. 2. Bring 8 down. 7 x 9 = 63. Write 7 in the quotient and subtract 63 from 68. 3. Bring down 5. 9 x 6 = 54. Write 6 in the quotient and subtract 54 from 55. Q = 476; R = 1. Eg.4: 9818 ÷ 9 1. Take 9. Divide by 9. 9 x 1 = 9. Write 1 in the quotient & subtract 9 from 9. 2. Bring down 8. Now, it is smaller than the divisor. So we need to bring the next digit down too. When we do so, we put a zero in the quotient. 4 2 8 5 9 4 7 6 -3 6 6 8 - 6 3 5 5 - 5 4 1 96

Rule: When two digits are brought down at the same time after the first step in the division, a zero is added to the quotient. Now, for 81, 9 x 9 = 81. Write 9 in the quotient and subtract 81. 3. Bring down the 8. But it cannot be divided. As we did in step-2, put a zero in the quotient. The Remainder is 8. Quotient = 1090. Remainder = 8. 9 8 1 8 9 1 0 9 0 -9 0 8 1 - 8 1 0 8 Do the following divisions: 1. 1284 ÷ 8 2. 2442 ÷ 4 3. 1818 ÷ 9 Guided Independent Do the following divisions: 1. 6724 ÷ 6 2. 23804 ÷ 8 3. 12875 ÷ 8 4. 43280 ÷ 9 5. 26847 ÷ 7 97 4. 6572 ÷ 5 5. 3563 ÷ 7 6. 18275 ÷ 3

b) Division by 2-digit numerals: Eg.1: 5384 ÷ 25 Step-1: Start from the left. Take the first two digits, ie. 53. Find multiplication sequence of 25 till you get a number equal to or less than 53. 25 x 1 = 25. 25 x 2 = 50. 25 x 3 = 75. We will take 25 x 2 = 50. Write 2 in the quotient and subtract 50. Step-2: Bring down 8. 25 x 1 = 25. Write 1 in the quotient and 25 below 38 and subtract. Step-3: Bring down 4. 25 x 4 = 100 25 x 5 = 125 25 x 6 = 150 We take 25 x 5 = 125. Write 5 in the quotient & subtract 125. Quotient = 215. Remainder = 9. Checking: Divisor x Quotient + Remainder = Dividend. 25 x 215 + 9 = 5384. 5 3 8 4 25 2 1 5 -5 0 3 8 - 2 5 1 3 4 - 1 2 5 9 2 1 5 x 2 5 1 0 7 5 4 3 0 5 3 7 5 + 9 5 3 8 4 Eg.2: 18 books cost Rs. 2034. What is the cost of 1 book? Working: Cost of 18 books = Rs. 2028. Cost of 1 book = 2034 ÷ 18 = Rs. 113. 2 0 3 4 18 1 1 3 -18 2 3 - 1 8 5 4 - 5 4 0 98

Divide the following and check the answer using the rule for division. 1. 607 ÷ 17 2. 13865 ÷ 21 3. 9289 ÷ 32 Guided Divide the following and check the answer using rule for division. 1. 90 ÷ 18 2. 375 ÷ 12 3. 4652 ÷ 21 4. 5290 ÷ 36 5. 42905 ÷ 28 6. 2. 67925 ÷ 25 7. 7842 ÷ 27 8. 13481 ÷ 42 1. How many packets are required to pack 11287 biscuits if each packet can hold 75 biscuits? No biscuit is to be left out. All should be packed. 2. Divide Rs. 7800 among 65 people equally. How much money will each get? 3. In a school assembly, 646 students stand in 7 rows. How many students are there in each row? 4. 645 saplings are to be planted in rows of 25 each. How many rows are there? How many saplings will be there in the last row? (The question is tricky.) 99

Test of Divisibility How do we know whether a number is divisible by a given digit? Let us see. Divisibility by 2 All even numbers are divisible by 2. Any number which has 0, 2, 4, 6, or 8 in the unit’s place is divisible by 2. Eg: 120, 342, 1594, 546, 1028 Divisibility by 3 If the sum of the digits of a number is divisible by 3, the number is divisible by 3. Example: 1728 Sum of the digits = 1 + 7 + 2 + 8 = 18 18 is divisible by 3. (3 x 6 = 18) Hence, 1728 is divisible by 3 Check whether the given numbers are divisible by 3: (1)1050 (2) 282 (3) 1382 (4) 4284 (5) 866 Divisibility by 4 If the digits in the units and tens place are together divisible by 4, the number is divisible by 4. Example: 1728 The last two digits are 28. It is divisible by 4. (4 x 7 = 28). Hence, 1728 is divisible by 4. 1324: 24 is divisible by 4 (4 x 6 = 24), hence the number is divisible by 4 too. 185: 85 is not divisible by 4. Hence, 185 is not divisible by 4. Find which of these numbers are divisible by 4 (1)132 (2) 2748 (3) 1622 (4) 15314 (5) 153 100