TERM 2 - CLASS 4 - PRIME YEARS

So, we define factors as: “The numbers that can divide a given number without leaving any remainder” Properties of factors: 1. Factors cannot be bigger than the given number. 2. Factors divide the given number without leaving any remainder. 3. Divisor and quotient are factors of a given number. 4. 1 is a factor of every number. 5. The number itself is a factor of a given number. Extended activity: Find the factors of the following numbers by arranging seeds in rows and columns. 1) 24 2) 36 Guided Check whether the first number is a factor of the second (apply divisibility rule) 1. 3, 15 2. 8, 34 3. 9, 135 4. 6, 216 5. 3, 20 6. 7, 36 7. 4, 192 51

1) Which of the following are factors of 112? 2, 3, 4, 5, 8, 6, 7 2) Check whether the first number is a factor of the second. a. 3, 115 b. 4, 428 c. 6, 125 d. 8, 132 e. 9,45 f. 7, 49 3) Find the factors of the following using multiplication table a. 15 b. 20 c. 21 d. 45 e. 90 (Eg: 15 = 1 x 15; 3 x 5 Factors of 15 are 1, 15, 3, 5) Independent Finding factors by division For large numbers, multiplication tables cannot be used to find factors. In such cases, we have to do it by division. The process is called factorization. e.g: 128 It is an even number. It is divisible by 2. So, 2 is a factor. It is divisible by 4 because the tens and units digit together form 28, which is divisible by 4. So, 4 is a factor too. It is not divisible by 3 as the sum of the digits is not divisible by 3 (1 + 2 + 8 = 11) It is not divisible by 6 as it is not divisible by 3. It is not divisible by 9 as the sum of the digits is not divisible by 9. 52

Let’s see whether it is divisible by 8. Step-1: Divide by 2. Quotient is 64. Thus, factors are 2 and 64. Step-2: Divide by 4. Quotient is 32. Thus, factors are 4, 32. Step-3: Divide by 8. Quotient is 16. Thus, factors are 8, 16. 1 2 8 2 6 4 -12 08 - 8 0 1 2 8 8 1 6 - 8 4 8 - 4 8 0 1 2 8 4 3 2 -12 0 8 - 8 0 From the above, we conclude that the factors of 128 are 1, 128, 2, 64, 4, 32, 8, and 16. Find the factors of the following by division: a. 112 b. 96 c. 180 Find the factors of the following numbers by division: a. 135 b. 240 c. 156 d. 320 Guided Independent Common Factors Take the number 6 and 4. Factors of are 6 1, 6, 3, 2. Factors of are 4 1, 4, 2. As you can see above, 1 and 2 are common factors for both 6 and 4. 53

Take the numbers 12 and 18. Factors of 12 are 1, 12, 6, 2, 3, 4 . Factors of 18 are 1, 18, 2, 9, 3, 6. The common factors of 12 and 18 other than 1 are 2, 3 and 6. Find the common factors of the following groups of numbers a. 8 and 12 b. 6, 8, 15 c. 6, 25, 17 d. 24, 96 e. 190, 112 Find common factors of the following sets of numbers. a. 12, 24, 36 b. 8,12, 22 c. 124, 138 d. 192, 112 e. 144, 96, 128 Independent Guided 1. Write the factors of 35. 2. Find the factors of 32. 3. Use division to find factors of 256 and 428. 4. Numbers which do not have common factors other than 1 are called Co-Prime. Eg: 3 and 4; 11 and 12. Find out which of the following sets of numbers are Co-Prime. a. 9, 12 b. 3, 8 c. 15, 10 d. 7, 6 e. 12, 18 f. 2, 9 g. 10, 20 h. 11, 33 54

Multiples Refer to the activity done earlier. 12 seeds can be arranged in rows in 3 different ways - 1 x 12, 6 x 2, 3 x 4. Writing the multiplication facts for it: 1 x 12 = 12 6 x 2 = 12 3 x 4 = 12 We say 12 is a multiple of 1 and 12. 12 is also a multiple of 6 and 2. 12 is a multiple of 3 and 4. In other words, 12 is a multiple of 1, 2, 3, 4, 6, and 12. Refer to the multiplication table of 2. 2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 2 x 4 = 8 ………. The numbers 2, 4, 6, 8 ……….. are multiples of 2. Similarly 3, 6, 9, 12, ………….. are multiples of 3. 3 x 1 = 3 3 x 2 = 6 3 x 3 = 9 3 x 4 = 12 ………. You find that multiple is the product of 2 numbers. The multiplicand and the multiplier are the factors. 55

Properties of Multiples: 1. Multiple is larger than the given number. 2. The number itself is a multiple of the given number. 3. Factors of a given number are fixed, but multiples extend upto infinity. Guided 1. Find the first 5 multiples of 6. Ans: 6 x 1, 6 x 2, 6 x 3, 6 x 4, 6 x 5. = 6, 12, 18, 24, 30. 2. Find first 6 multiples of 4. Independent 1. Find the first 10 multiples of 3. 2. Find the first 5 multiples of 10. 3. Find the first 6 multiples of 8. 4. Find the first 7 multiples of 9. Common Multiples: Multiples of 2 upto 12 terms = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24. Multiples of 3 upto 12 terms = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. From the above, we can see that common multiples of 2 and 3 among the first 12 terms are 6, 12 and 24. 56

Activity: 1 Prepare strips of multiples of numbers 2 to 10. Paste one below the other. From this data, we can find common multiples of numbers from 2 to 10. Complete the table: Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24 26 Multiples of 3 Multiples of 4 Multiples of 5 Multiples of 6 Multiples of 7 Multiples of 8 Multiples of 9 Multiples of 10 3 6 9 12 15 18 21 24 2730 3336 39 From the table, find out 1) Common multiples of 2 and 5. Find the least common multiple. 2) Common multiples of 3 and 6. Find the least common multiple. 3) Common multiples of 3 and 4. Find the least common multiple. 4) Common multiples of 8 and 6. Find the least common multiple. 57

1. Write all the factors of 12. 2. Find 4 common multiples of the following sets of numbers and write the least common multiple. a. 3 and 9 b. 15 and 20 c. 25 and 30 3. Find the least common multiple of 12, 15 and 20. Prime and Composite Numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 S94 95 96 97 98 99 100 Sieve of Erasthosthenes: Activity 2: Prepare a 10 x 10 table for numbers 1 to 100. 1. Cut off 1. 2. Cut off all the multiples of 2 leaving 2 as shown. 58

3. Cut off the multiples of 3 leaving 3. 4. Continue the process cancelling multiples of 5 and 7. 5. Pick out / circle the numbers which are not cancelled. 6. Write them down in the note book. These are called Prime Numbers. They don’t have any factors other than 1 and itself. They don’t appear in any multiplication table. What are those numbers? 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Check whether you have done it correctly. Note: 1 is neither prime nor composite 7. How many prime numbers are there between 1 and 100? 8. Why is 1 not included in Prime Numbers? 9. From the given table write down Prime Numbers between: a. 1 and 20 ___________ b. 20 and 40 ___________ c. 40 and 60 ___________ d. 60 and 80 ___________ e. 80 and 100 ___________ In which period do you find maximum prime numbers? 10. How many Prime Numbers are there between 90 and 100? 11. Which is the only even Prime Number? Numbers which are crossed out are called Composite Numbers. Composite Numbers have factors other than 1 and itself. 12. Which is the smallest Composite Number? 59

Twin Primes: Prime Numbers separated by a composite number are called twin primes. Eg: 3 and 5. There is a composite number 4 in between them. 13. Pick out the twin primes from the table. 14. There is only one pair of consecutive prime numbers, that is, prime numbers without any composite number in between. Find out that pair. Prime factorization and its application Finding the prime number factors of a number is called prime factorization. The Factor Tree: Consider the number 32. Take any multiplication factor for 32. We take 32 = 4 x 8. Next, find the factor of each of those factors of 32. Continuously factorizing each number, we arrive at prime factors for 32. 32 = 2 x 2 x 2 x 2 x 2. 4 3 2 2 24 2 12 2 48 2 96 32 4 8 2 2 2 4 2 2 Let’s take another example. Consider 96. 96 = 2 x 2 x 2 x 2 x 2 x 3. 60

Do the prime factorization of the following numbers. a. 64 b. 75 c. 36 d. 24 Guided Independent Find the Prime Factors of the following numbers using factor tree method a. 27 b. 28 c. 56 d. 92 e. 108 Rapid fire 5 minutes 1. Circle the Prime Numbers 26, 23, 34, 37, 56, 59, 72, 83 2. Smallest prime number ____________ 3. Only even prime number ____________ 4. ____________ is neither prime nor composite. 5. Which of the following are twin primes? a. 12,14 b. 11,13 c. 37, 38 6. Consecutive primes are a. 2 and 5 b. 2 and 3 c. 3 and 5 7. The number of factors of 12 is ___________ 8. Prime factorization of 24 = 2 x 2 x ____ x _____ . 9. Least Common Multiple of 3 and 5 is _______ 10. Which of the following have no common factors? a. 15, 26 b. 17, 34 c. 8, 12 d. 10, 15 61

1. Find the factors of 147. 2. Find the prime factors of 84 by factor tree. 3. Draw the factor tree for 240 and 360. 4. Find the Common Factors of a. 15 and 12; b. 15 and 30. 5. Complete the Factor Tree. 3 135 96 8 2 4 15 3 Finding Highest Common Factor (HCF) Example: Find the Highest Common Factor of 12 and 24. Method-I: Factors of 12 = 1, 2, 3, 4, 6, 12 Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24. Common Factors = 2, 3, 4, 6, 12. Thus, highest common factor (H.C.F) is 12. Method-II: By Factor Tree Method Prime Factors of 12 = 3 x 2 x 2. Prime Factors of 24 = 2 x 2 x 2 x 3. a. b. 62

Common Primes = 3 x 2 x 2 = 12 HCF = 12. Method-III: Division by Common Divisors. As there are no common divisors after the second step, Highest Common Factor is the product of the divisors. HCF = 2 x 6 = 12. Guided 1. Find HCF of the following pairs by finding common factors. a. 25, 30 b. 80, 45 c. 120, 48 d. 15, 20, 30 2. Find HCF of the following pairs by prime factorization a. 90, 42 b. 18, 63 c. 45, 30 3. Find HCF of the following pairs by division a. 45, 30 b. 36, 42 1. Find HCF of the following by all the three methods. a. 84, 72 b. 120, 25 c. 15, 30, 20 d. 12, 16, 18 2. Fill in the boxes with correct number Independent 96 4 8 6, 12 2 12, 24 1, 2 6 63

Least Common Multiple (LCM) by Common Multiple, Prime Factorization and Division method Method I: Find LCM of 12 and 15. Multiples of 12 = 12, 24, 36, 48, 60, 7 2, ……. Multiples of 15 = 15, 30, 45, 60, 75 ……. Common Multiple = 60 ……. Least Common Multiple (LCM) = 60. Method-II: Prime Factorization Prime Factorization of 12 = 3 x 2 x 2 Prime Factorization of 15 = 3 x 5 Common Primes: 3. LCM = Common Prime x Remaining Primes = 3 x 2 x 2 x 5 = 60 Method-III: Division Divide by Common Divisor LCM = Common Divisor x Quotients = 3 x 4 x 5 = 60. 12 3 4 2 2 15 5 3 12, 15 3 4, 5 1. Find LCM of the following sets by all the three methods. a. 6, 8, 10 b. 10, 15, 20 c. 8, 12, 16 Guided 64

1) Find the common factors of 15 and 18. Find the Highest Common Factor. 2) What are Co-Prime numbers? Show that 6, 8 and 15 are co-prime. 3) Find 5 Common Multiples of 5 and 9. Pick out the least common multiple. 4) Find the Least Common Multiple (LCM) of 15, 30 and 45 a. By finding multiples b. By division by Prime Numbers c. By Factor Tree Method. 5) Find the HCF of 24, 32 and 48 by a. Factor Tree Method b. Division by Prime Numbers. 6) Find LCM of a. 5 & 8 b. 6 & 7 c. 11 & 24 7) Find HCF of 240 and 360 by a. Factor Tree Method b. Division by Prime Numbers. 8) Prime Factorization of 24 = 2 x 2 x 2 x 3 Prime Factorization of 36 = 3 x 3 x 2 x 2 The HCF is ______________ ? 65

HCF of 12 and 15 = 3 LCM of 12 and 15 = 3 x 4 x 5 = 60. Product of HCF and LCM = 3 x 60 = 180. Product of the numbers = 12 x 15 = 180. Therefore, product of HCF and LCM = Product of the Numbers. Take any two other numbers and verify what you discovered. Eg: (1) 8 and 12 (2) 9 and 15 Problem: 1. HCF and LCM of two numbers are 3 and 120 respectively. If one number is 15, find the other number ? 2. Product of two numbers is 135. If their HCF is 3, find the LCM ? 12, 15 3 4, 5 Example: 1. 24 and 12 2. 15 and 20 Have some circles cut out from transparency sheet to show common factors of 24 and 12. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Factors of 12 are 1, 2, 3, 4, 6, 12 Write the factors of 24 on one circle as shown. Keep the other circle on the first as shown. The common factors are shown by the area between the two circles . Factors of 15, 20 15 = 1, 5, 3, 15; 20 = 2, 10, 5, 4 Show it using circles Try a few more from the exercises given in the text book. 2 3 4 6 12 24 8 common factors 5 3 15 2 10 1 4 66 12

Fractions Fraction is a part of a whole. In the fig. 1, one out of two parts is shaded. We write the shaded part as From the figure, we find that = half. Similarly in fig.2, one out of 4 parts is shaded. It is written as . From the figure, we find that the shaded part is half of half. It is called a quarter. One quarter 1 4 1 4 1 4 1 4 1 2 fig 1 fig 2 Ordering Fractions Experiment 1: Take 3 identical paper strips. Fold one into two equal parts. Shade one part. Fold the second one into 3 equal parts. Shade 1 part. Fold the third into 4 equal parts and shade one part. Keep one below the other. 1 2 1 2 1 2 1 4 = 1 4 67

Compare the shaded parts? Which is the most shaded and which is the least shaded? You find that is larger than which is larger than . We write them as 1 2 > 1 3 1 4 > Thus, we find that as the denominator increases (the numerator remaining constant), the value of a fraction decreases. We can say 1 2 1 3 1 4 1 2 1 4 1 2 > 1 4 1 8 > 1 16 > ............ 1 3 > 1 6 1 9 > 1 12 > ............ Experiment 2: 1 4 2 4 3 4 1 3 68

Take 3 identical paper strips. Fold each into 4 equal parts. Shade one part in the first one, 2 in the second and 3 in the third. Keep these strips one below the other as shown. Which of these is most shaded and which is least shaded? From the observation, we find that when denominator is the same, the value of the fraction increases as the numerator increases. Types of fractions 1. Like fractions: When denominators are same, the fractions are called like fractions. Eg: 3 4 > 2 4 1 4 > 2. Equivalent Fractions: Study the figure on the right. The shaded parts are equal in all of them. We write: 1 2 = 2 4 4 8 = These fractions are called equivalent fractions. Equivalent fractions are obtained by multiplying any fraction by any number (in the numerator and denominator) or by dividing the numerator and denominator by the same number. 1/8 1/2 1/4 69 1 5 , 2 5 3 5 , ............, 1 12 , 4 12 6 12 , 9 12 , ............

Example: 1 x 2 2 x 2 = 2 4 3 x 4 8 x 4 = 12 32 = 1 2 4 8 4 2 x 3 3 x 3 = 6 9 5 x 2 8 x 2 = 10 16 = 1 2 3 6 3 3 Extended Activity: By folding and shading the paper strips, find 4 equivalent fractions for . Paste the strips in your note book. Independent 1. Find 3 equivalent fractions for the given fractions: 2 5 a. 4 9 b. 1 10 c. 3 5 d. 1 11 e. 8 10 f. 2. Find 3 equivalent fractions for the given fractions: 16 32 a. 18 27 b. 12 36 c. 4 1 3 ; ; 70

Checking for equivalent fractions Example: Check whether and are equivalent. 2 3 6 9 Cross Multiply: 2 x 9 = 18 Numerator of the first Denominator of the second. X 3 x 6 = 18. Numerator of the second Denominator of the First. X Both the products are same. Hence, they are equivalent fractions. 2 3 6 9 Check whether the following are equivalent fractions: Guided 3 8 , 9 32 1. 4 5 , 20 25 2. 6 7 , 42 80 3. 5 6 , 25 30 4. 71

Proper, Improper and Mixed Fractions In proper fractions, the numerator is less than the denominator. For example: In improper fractions, the numerator is greater than the denominator. Eg: In mixed fractions, there is a combination of a whole number and a proper fraction. 1 4 , 2 3 5 8 , 6 7 , etc. 4 3 , 5 2 8 3 , 10 3 , , 15 4 etc. 1 whole and of a whole make 1 1 4 1 4 1 Example: 1 2 , 2 1 1 3 , 3 3 4 etc. 1 2 means 1+ 1 1 2 1 3 means 2+ 2 1 3 ; ; 3 4 means 3 + 3 3 4 etc. 1 4 72

Conversion of improper fraction into mixed fraction and vice versa An improper fraction can be converted to a mixed fraction by dividing the numerator by the denominator. Example: Divide 15 by 2. Q = 7, Remainder = 1. We write it as a mixed fraction as 7 15 2 1 2 Remainder Divisor Q+ To convert mixed fraction into improper fraction Multiply the whole number by the denominator, add the numerator, this is the new numerator. Keep the denominator as it is. Remainder Divisor Q Quotient X Divisor + Remainder Divisor = Example: 3 5 2 = 2 x 5 + 3 5 = 13 5 1 3 4 = 4 x 3 + 1 3 = 13 3 1. Convert the following to improper fractions: a. b. c. d. Independent 1 3 4 2. Convert the following to mixed fractions: a. b. c. d. e. 4 7 3 7 8 2 3 20 5 15 7 54 5 23 4 18 7 43 6 Remainder Divisor Q or 73

1 Circle the improper fractions. 2. Pair the equivalent fractions. 3. Find 3 equivalent fractions larger than each of the given fractions. 4. Find 2 equivalent fractions smaller than the given fractions 5. Pick out like fractions 2 3 , 3 2 4 3 , 3 4 , , 5 3 , 3 5 8 3 , 3 8 , 2 3 , 1 5 4 12 , 5 25 , , 8 15 , 3 4 8 3 , 3 8 , 2 3 a. 5 8 b. 2 7 c. 15 24 a. 24 48 b. 12 18 c. 20 60 d. 1 8 , 1 4 2 3 , 3 5 , , 2 8 , 7 8 5 3 , 8 7 , 6. Convert the following mixed fractions to improper fractions. 7. Convert the following improper fractions to mixed fractions. 18 4 a. 121 3 b. 83 5 c. 27 4 d. 125 7 e. 74 a. 2 b. 10 c. 18 d. 4 1 5 1 2 2 3 11 12

Study the given table of fractions and answer the questions that follow. 1. Find 4 equivalent fractions for . 2. How many are equal to four ? 3. Which is the largest among the fractions? 4. Which is the smallest among the fractions? 5. Two = 6. 7. Find 2 equivalent fractions larger than . 8. Find one equivalent fraction larger than . 9. Arrange the fractions in the ascending order. 1 3 1 4 1 12 , 1 3 1 4 , 1 8 1 6 , 1 2 , , 10. = 3 6 8 1 12 12 12 12 12 12 1 1 1 1 1 1 12 12 12 12 12 12 1 1 1 1 1 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 6 1 6 1 6 1 6 1 6 1 6 1 4 1 4 1 4 1 4 1 3 1 3 1 3 1 2 1 2 1 6 1 8 1 2 75 1 6 1 12 = 1 3 6 = 12

Operations on Fractions Addition and subtraction involving like fractions Activity 1: Have circles or rectangles cut out from a transparent sheet (available in stationery shops), or use thick polythene sheets. Divide one sheet or circle into 4 equal parts and shade 1 part of it using a marker. Divide another sheet or circle into 4 equal parts and shade two parts of it. Note: All rectangles or circles are to be of the same size. a) Put one over the other. What do you see? You find 3 shaded parts. We write the mathematical statement as b) You have 3 parts out of 4 shaded now. Remove the circle with one fourth shaded portion. What is left is 1 4 + 2 4 = 3 4 3 4 - 1 4 = 2 4 2 4 Keep the two circles together. Remove the circle with 2 shaded portions. What is left is 3 4 - 2 4 = 1 4 In this activity, you find that in addition or subtraction, it is the numerators that are added or subtracted. 1 4 76

Addition and subtraction involving unlike fractions Example 1: Consider Here, the denominators are different. Addition is possible only if denominators are same. We have to equalize the denominators for adding the two fractions. How do we do it? Follow these steps. 1. Find the L.C.M of the denominators. L.C.M = 2 x 1 x 2 = 4. 2. Multiply both the numerator and denominator of each fraction by the L.C.M. 1 2 + 1 4 2, 4 2 1, 2 1 2 x x 4 4 2 2 4 = 1 4 x x 4 4 1 1 4 = 1 2 x 4 2 2 + 1 4 4 3 4 = 1 4 x 4 1 + Method 2: Now add the two fractions 2 4 + 1 4 = 3 4 In method 2, instead of multiplying the denominator by LCM, LCM is taken as the common denominator. Example 2: L.C.M. of 3 and 5 3 and 5 do not have common factors to divide. LCM of 3 and 5 = 3 x 5 = 15 1 3 + 2 5 Method 1: 77

Multiply the numerator and denominator of each fraction by the LCM. 1 3 x x 15 15 5 1 x 5 15 = 1 3 x 15 5 5 + 6 15 15 11 = 2 5 x 15 3 + Method 2 5 15 = 2 5 x x 15 15 3 2 x 3 15 = 6 15 = 5 15 Adding 11 15 + = 6 15 1 x 5 2 x 3 15 = + = Method 1 2 12 3 12 x x 4 2 x 4 12 = 2 12 3 x 4 8 -3 12 12 12 = 1 12 4 x 3 - Method 2 8 12 = 1 12 4 12 x x 3 1 x 3 12 = 3 12 = 8 12 5 12 - = 3 12 2 x 4 3 12 = - = Method 1 Example 3: LCM of 3 and 4 = 12 2 3 - 1 4 5 15 78

1. Add the given fractions Guided 2. Do the following subtractions 2 5 + 1 4 a. 3 4 + 1 8 b. 3 18 + 5 12 c. 2 5 - 1 8 a. 3 5 - 1 15 b. Add the given fractions Independent Addition and subtraction of mixed numerals 5 9 + 2 3 a. 7 12 + 5 6 b. 3 7 + 4 21 c. 3 4 + 3 5 d. 4 15 + 3 12 e. Example: 1. Add the whole numbers 2 + 3 = 5. Add Whole + Fraction = 1 3 + 3 1 2 2 1 3 + 1 2 = 1 x 6 6 2. Add the fractions 3 + 1 x 6 2 2 3 = 1 x 2 + 1 x 3 6 = 5 6 5 + 5 6 = 5 5 6 79

Simplifying Fractions Reducing fractions to the simplest form In the simplest form, there are no common factors for the numerator and denominator. Example: Guided 2 3 + 1 3 3 a. 1 4 + 3 1 5 3 b. 3 4 + 1 1 2 2 c. Independent 1 3 +3 1 2 2 a. 1 2 + 2 1 3 1 b. 1 2 - 2 3 4 3 c. 4 5 + 4 2 3 3 d. 3 4 + 3 1 5 2 e. 3 4 - 1 2 3 2 f. In this fraction, 4 is a common factor for both the numerator and the denominator. 4 x 1 = 4; 4 x 2 = 8. Such fractions can be simplified by dividing the numerator and the denominator by the common factor. 4 8 4 4 8 4 = 1 2 The division can be carried out mentally and the dividends can be crossed out. Example: 3 is a common factor for both. 12 15 12 3 15 3 = 4 5 (or) = 4 5 12 15 4 5 80 Add the following Solve the mixed fractions

Simplify (reduce to the lowest terms) the fractions. Guided 24 30 a. 32 48 b. 12 20 c. Independent 18 24 a. 25 30 b. 20 40 c. 15 30 d. 24 30 e. 121 132 f. 72 81 g. 56 63 h. Multiplication of Fractions Multiplication of a fraction by a whole number Example: 1 3 x 2 What does the statement mean? 1 3 x 2 means 2 times 1 3 = 1 3 1 3 + = 2 3 1x 2 3 2 3 = In multiplication by a whole number, only the numerator is multiplied. Example: Guided 2x 7 5 a. 1x 8 3 b. 1x 5 2 c. Simplify the fractions. 81 2x 5 3 10 3 = 2x 5 3 = Multiply the following

Multiplication of a mixed number by a whole number Example: Independent 2x 8 3 a. 1x 5 4 b. 2x 7 3 c. First, convert the mixed numeral into an improper fraction, and then carry out the multiplication. 1 3 x 5 2 1 3 = 2 7 3 7 3 x 5 = 35 3 1 4 x 3 2 a. 1 2 x 8 3 b. 1 5 x 3 1 c. 1 2 x 8 3 d. Multiplication of a fraction by a fraction Guided Example: The statement means one third of half. One half is again divided into 3 parts. What will be the value of each part? 1 3 x 1 2 If half is divided into 3 parts, it is the same as the whole divided into 6 parts. 82 Multiply the following Find the product of the following:

Each part is 1 6 Example: One fourth of 1 3 x 1 2 = 1 6 1 4 x 3 5 = 3 5 Value of one part = In multiplication by fractions, the numerators and denominators are multiplied. Example: 1 4 of 3 5 = 3 20 2 3 x 3 4 = 6 12 Reduce this fraction to lowest terms: (Dividing by 6) You can also do this way Note: When two fractions are multiplied, the product is smaller than either of the fractions. 6 12 = 1 2 2 1 2 3 x 3 4 1 1 2 1 = 1 2 How do we justify? 1 2 x 1 4 = 1 8 1 3 x 1 4 = 1 12 83

Do this activity Draw a rectangle. Divide it into 3 equal parts horizontally. Shade 1 part. 1 3 x 2 5 2 5 1 3 The shaded part is Divide the same rectangle into 5 equal parts vertically. Shade 2 parts in a different colour. This shaded region is 1 3 2 5 How many parts are shaded in both the colours? __________ What is the total number of parts? __________ What is the fraction shaded in both the colours? ________ Ans: 2 15 Multiplication involving mixed numerals Example: 2 1 3 x 1 4 Convert the mixed numeral into an improper fraction. 2 1 3 = 7 3 7 3 x 1 4 = 7x1 12 = 7 12 2 1 4 x 3 1 3 = 9 4 10 3 = 3x5 2x1 = 15 2 = 7 1 2 3 2 x 5 1 84 Example:

Multiply and reduce to lowest terms. Guided Multiply and reduce to lowest terms. Division of Fractions Reciprocal of a Fraction Consider the fraction Its reciprocal is Reciprocal is obtained by reversing the numerator and denominator. Reciprocal of = 2. Reciprocal of Independent 2 5 x4 a. 1 4 x 2 5 b. 3 4 x 2 3 c. 1 9 x d. 1 5 x 1 1 3 e. 2 3 15 3 4 x 2 2 3 f. 1 2 5 x3 1. x8 2. 1 3 x4 3. 2 7 x5 4. 2 5 2 3 x 5. 1 4 3 8 x 6. 4 5 3 4 x 7. 2 5 3 7 x 8. 1 6 3 4 x 1 1 2 9. 2 1 5 x 2 1 3 10. 3 2 5 x 3 1 3 11. 1 1 2 x 1 2 5 12. 3 2 3 3 2 1 2 7 8 Division by a fraction is multiplication by its reciprocal. 8 7 = 85

Division Statement and its meaning Example: Three wholes are each divided into half. How many halfs will be there? Obviously 6 How do you get 6? You multiply 3 by the reciprocal of to get 6. 3 1 2 3x 2 1 = 6 1 2 It says half is divided into 3 parts. What is the value of each part? You know it is How do you get it? 3 1 2 1 6 One third of half is 1 2 x 1 3 = 1 6 1 6 1 2 1 4 Example: The Statement says “How many one fourths are there in a half? Obviously 2 onefourths make 1 half. How do you get 2? Guided 1 2 x 4 1 = 2 86 Example: Divide: 3 1 2 a. 4 c. 2 3 b. 2 5 1 3 3 5 d. 1 4 1 6 e. 3 5

Independent 5 1 4 a. 1 2 f. 1 3 1 4 g. 3 4 1 3 h. 3 6 3 2 3 b. 5 1 2 c. 4 3 5 d. 2 1 6 e. 1 3 i. 2 1 2 1. 2 x means a 2 added times . b added twice . c multiplied twice . d None of these . 1 5 1 5 1 5 1 5 2. 1 3 1 4 = 3. a. 1 12 b. 12 c. 4 3 d. 3 4 4. 5 1 7 = a. 5 7 b. 7 c. 35 d. 1 35 5. means 1 3 4 = a. 4 3 b. 12 c. 1 12 d. 3 4 2 1 3 1 3 a. divided into 2 equal parts b. added 2 times 1 3 1 3 1 3 c 2 multiplied by . d . multiplied by 2 5 87 Divide:

6. reduced to lowest term is 7. Simplify the following fractions. (Reduce to lowest terms) 10 15 a. 2 5 b. c. 5 15 d. 2 15 2 3 8. Do the following multiplications. Reduce the fractions to lowest terms. a. 18 54 b. 27 63 9. Do the following division and simplify the fraction obtained. 3 x a. 2 15 x b. 2 15 2 19 x 2 c. 1 6 6 13 x 1 d. 3 1 10 1 3 13 a. 1 3 6 b. 3 4 c. 6 15 3 5 d. 2 1 4 1 3 1 e. 2 2 3 3 f. 3 4 1 3 1 88

Decimals Decimals are fractions with multiples of 10 in the denominator. Decimals are represented in a way different from other fractions. Activity 1: 1 10 = 0.1 5 10 = 0.5 7 10 = 0.7 1 100 = 0.01 5 10 = 0.05 7 10 = 0.07 1 1000 = 0.001 5 1000 = 0.005 7 1000 = 0.007 Things needed: 10 x 1 strips. 1. To represent 0.1 on the strip, colour one square in the strip. 1 out of 10 squares is coloured. Similarly, show 0.2, 0.3, 0.4, 0.5 ……………. 0.9 on paper strips. 0.1 0.2 89

2. To represent 0.01, 0.02 …………… Cut off a 10 x 10 grid in your square ruled note book or the given grid paper. Colour 1 square in the grid. 1 out of 100 squares is colored. It is represented as 0.01. 1 100 = 0.01 10 100 = 0.1 20 100 = 0.2 25 100 = 0.25 10 100 = 0.1 0.01 Extended Activity: In the 10 x 10 grid, show the following decimals: 1) 0.39 2) 0.28 3) 0.45 4) 0.04 5) 0.09 Decimals on Abacus: Activity 2: Things needed: 5 sets of abacus, red beads and blue beads. Divide students into groups. Each one is given an abacus and beads of two different colours (if there are enough abacuses, each one can be given one). The red beads are used to represent whole numbers and blue beads are used to represent decimals. 90

Example: 25.4 Tell the number represented on the abacus. The number is represented as shown: Guided Independent Represent these numbers on the abacus. 1. 12.34 2. 285.325 T 1 10 1 100 1000 1 u T 1 10 1 100 1000 1 u T 1 10 1 100 1000 1 u H T 1 10 1 100 u 10 7.24 is represented as shown: T 1 10 1 100 u 10 T 1 10 1 100 u 10 1. Represent these numbers on the abacus. a. 20.125 b. 185.025 c. 24.008 d. 12.05 2. Tell the number shown on the abacus. 91

Representing decimals in the Place Value Chart How do we read these decimals? 1. One point two five eight 2. Twenty three point zero one five 3. One hundred twentythree point zero two five 4. Eight point zero zero eight Note: Remember how decimals are read and written in words. Thousands Hundreds Tens Unit Tenth Hundredth Thousandth 1 10 1 100 1 1000 1. 2. 3. 4. 1 . . . . . 3 2 1 0 5 5 2 8 3 2 0 0 5 8 0 0 8 1 The Place Value Chart for decimals is as shown: Writing decimals in expanded notation: 125. 286 can be written as 100 + 20 + 5 + + + 28.106 = 20 + 8 + + 92 2 10 8 100 1000 6 1 10 6 1000

1. Read the following decimals: a. 2.387 b. 38.085. 2. Write the following in expanded form. Also write them in words. a. 42.089 b. 112. 208 c. 1216.57 Guided 1. Write the given decimals in words. a. 423.125 b. 8. 487 c. 638.308 d. 45.007 2. Express the above decimals in expanded form. Independent Expressing as decimals 28 10 = 2.8 28 100 = 0.28 28 1000 = 0.028 5 10 = 0.5 5 100 = 0.05 5 1000 = 0.005 93

For division of a number by 10, put the point after unit’s digit. For division of a number by 100, put the point after units and tens digit. If tens place is absent, put a zero to the left of the units digit before putting the decimal. For division by thousand, put the point after unit, tens and hundreds digit. If tens and hundreds are absent, put two zeros to the left of the units digit before putting the decimal. Guided Express the following as decimals. 847 10 a. 847 100 b. 847 1000 c. Express as decimals. Independent 75 10 a. 75 100 b. 75 1000 c. 125 10 a. 125 100 b. 125 1000 c. 8 10 a. 8 100 b. 8 1000 c. 112 10 a. b. 112 112 1000 c. 25 10 a. 25 100 b. 25 1000 c. 10 78 10 a. 78 100 b. 78 1000 c. 1. 2. 3. 420 10 a. 420 100 b. 420 1000 c. 4. (Note: Zero at the end is not valid. For example 1.20 is same as 1.2) 2. 4. 1. 3. 94

Ordering Decimals: As we have already learned, when the denominator increases, and the numerator remains the same, the value of the fraction decreases. Thus, 1. 7 1.07 1.007 > > 4. 5 4.05 4.005 > > 3. 1 3.09 2.99 > > To compare decimals, compare the whole number and the decimal part. Write the given decimals in ascending order. 1. 3.5, 2.55, 3.55, 3.7, 2.95 2. 12.5, 12.58, 12.095, 12.005 Guided 1. Decimal form of : _______________ 2. = ___________________ 3. 155.072 = 100 + 50 + 5 + 4. Place value of 3 in 107.03 is ___________ 5. Say true or false: 5 100 270+ 5 10 1000 + 2 1 2 a. = 0.5 1 5 b. = 0.02 9 100 c. =0.90 20 100 d. = 1 5 95

6. Write the following in figures. a. Twentythree and three thousandths b. Eight and five hundredths. 7. Write in words a. 12.624 b. 0.547 8. Write the following in expanded form: a.12.084 b. 112.005 c. 125.5 9. Write the place value of the underlined digit in the following: a.1.235 b. 8.075 c. 12.500 10. Put or > < a. 28.008 21.997 b. 0.99 1.09 c. 1.80 1.08 11.Write the following in ascending order. 21.28, 20.95, 20.09, 20.099 12. Write the following in descending order. 8.25, 10.225, 0.75, 0.058 By shading in the 10 x 10 grid, find which is greater. a. 0.4 and 0.04 b. 0.5 and 0.06 1. Are 0.5 and 0.05 equal ? Why or why not ? 2. Are 0.2 and 0.20 equal ? Why or why not ? 3. Are 2.00 and 0.002 equal ? Why or why not ? 96

Inter Conversion of fractions and decimals Example: Convert the denominator into multiples of 10, 100, 1000 etc. by multiplying both the numerator and denominator by the same number. 8 10 = 0.8 2 10 = 0.2 2 5 = 2 x 2 5 x 2 = 4 10 = 0.4 2 25 = 2 x 4 25 x 4 = 8 100 = 0.08 Alternate Method: Divide the numerator by the denominator. Example: 2 cannot be divided by 5. Hence, insert a zero beside 2 and insert a decimal point in the quotient. We have to divide 20 by 5. 4 x 5 = 20. Quotient is 0.4 Divide: (1)insert 2 zeros beside 2. insert a point and 0 in the quotient. (2) 25 x 8 = 200. Quotient is 0.08 0.4 5 20 - 20 4 4 10 2 25 In the case of mixed numerals, convert them into improper fraction and then divide. 0.08 25 200 - 200 00 2 5 97

Convert the following fractions into decimals. 4 5 a. Guided Convert into decimals 8 32 b. 3 4 c. 1 250 d. 3 4 e. 1 5 f. 4 15 Independent Converting decimals to fractions 9 20 a. 7 20 b. 21 250 c. 1 8 d. 47 4 e. 5 9 125 f. 1 4 g. 6 1 2 h. 7 1 20 e. Example: 0.5 = (reducing to lowest terms) 5 10 = 1 2 1 2 5 100 = 1 20 1 20 0.05 = 6 10 = 60 3 5 3 5 60.6 = 60 (Cancelling by 2) 28 100 = 7 25 7 25 0.28 = (Cancelling by 4) 98

Convert the following into fractions: a. 0.96 b. 5.25 c. 20.8 d. 0.125 Guided Independent Convert into fractions. a. 0.35 b. 2.4 c. 0.836 d. 5.6 e. 3.75 f. 5.25 g. 0.04 h. 0.783 i. 1.236 j. 1.014 1. Convert to decimals 2. Convert to fractions and reduce to lowest terms 2 5 a. 8 10 b. 4 25 c. 6 120 d. a. If Can we say 0.2 = 0.4? a. 0.5 b. 0.25 c. 0.3 d. 0.02 e. 0.45 f. 0.6 g. 6.12 h. 16.02 2 5 = 4 10 2 5 = 4 b. If is = ______ 20 5 = 40 10 1 2 = 3 2 1 = 3 c. If is = ______ 1 1 d If 0.5 = , what is = ________ 5 50 100 99 20 20 10 10

1. Express as a decimal without division. 2. Express as a decimal without division. 1 4 2 3 4 1 Money and Metric Measures in terms of decimals 100 paise = 1 rupee 1 Paisa = rupee = 0.01 rupee. 25 Paisa = = 0.25 rupee. 15 rupee 25 paise = Rs. 15.25 125 rupee 80 paise = Rs. 125.80 5 rupee 8 paise = Rs. 5.08 Length: 100 cm = 1 metre 1000m = 1 km 1cm = m = 0.01 m 1 m = = 0.001 km 5 cm = 0.05m 50 m = = 0.05 km 50 cm = 0.5 m 500 m = = 0.5 km 75 cm = 0.75 m 250 m = = 0.250 km 12m. 50 cm = 12.5 m 12 km . 250 m = 12.250 km 8 m. 75 cm = 8.75 m 8 km. 125 m = 8.125 km 10m.5 cm = 10.05m 1 100 25 100 1 100 1 1000 50 1000 500 1000 250 1000 100 Money: