CLASS VII UNIT-3. CBSE-i. Introduction to. Numbers. Student's Section. Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi India

Transcription

1 CBSE-i CLASS VII UNIT-3 Introduction to Rational Numbers Student's Section Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi India

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3 CBSE-i CLASS VII UNIT-3 Mathematics Introduction to Rational Numbers Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi India

4 The CBSE-International is grateful for permission to reproduce and/or translate copyright material used in this publication. The acknowledgements have been included wherever appropriate and sources from where the material may be taken are duly mentioned. In case any thing has been missed out, the Board will be pleased to rectify the error at the earliest possible opportunity. All Rights of these documents are reserved. No part of this publication may be reproduced, printed or transmitted in any form without the prior permission of the CBSE-i. This material is meant for the use of schools who are a part of the CBSE-International only.

5 PREFACE The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the learning process in harmony with the existing personal, social and cultural ethos. The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It has about schools affiliated to it and over 158 schools situated in more than 23 countries outside India. The Board has always been conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in view. The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand, appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations and additions wherever and whenever necessary. The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them to incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of these requirements. The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all subject areas to cater to the different pace of learners. The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some non-evaluative components in the curriculum which would be commented upon by the teachers and the school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and learning curve. The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to millions of learners, many of whom are now global citizens. The Board does not interpret this development as an alternative to other curricula existing at the international level, but as an exercise in providing the much needed Indian leadership for global education at the school level. The International Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to their peers through the interactive platforms provided by the Board. I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr. Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the development and implementation of this material. The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion forums provided on the portal. Any further suggestions are welcome. Vineet Joshi Chairman

6 Advisory Conceptual Framework Shri Vineet Joshi, Chairman, CBSE Shri G. Balasubramanian, Former Director (Acad), CBSE Dr. Sadhana Parashar, Director (Training), Ms. Abha Adams, Consultant, Step Dr. Sadhana Parashar, Director (Training), Ideators VI-VIII Ms Aditi Mishra Ms Preeti Hans Ms. Charu Maini Ms. Malini Sridhar Ms Guneet Ohri Ms Neelima Sharma Dr. Usha Sharma Ms. Leela Raghavan Ms. Sudha Ravi Ms. Gayatri Khanna Prof. Chand Kiran Saluja Dr. Rashmi Sethi Ms. Himani Asija Ms. Urmila Guliani Dr. Meena Dhani Ms. Seema Rawat Ms. Neerada Suresh Ms. Anuradha Joshi Ms. Vijay Laxmi Raman Ms. Suman Nath Bhalla English : Ms Neha Sharma Ms Dipinder Kaur Ms Sarita Ahuja Ms Gayatri Khanna Ms Preeti Hans Ms Rachna Pandit Ms Renu Anand Ms Sheena Chhabra Ms Veena Bhasin Ms Trishya Mukherjee Ms Neerada Suresh Ms Sudha Ravi Ms Ratna Lal Ms Ritu Badia Vashisth Ms Vijay Laxmi Raman Chemistry Ms. Poonam Kumar Mendiratta Ms. Rashmi Sharma Ms. Kavita Kapoor Ms. Divya Arora ACKNOWLEDGEMENTS Material Production Groups: Classes VI-VIII Physics : Ms. Vidhu Narayanan Ms. Meenambika Menon Ms. Patarlekha Sarkar Ms. Neelam Malik Biology: Mr. Saroj Kumar Ms. Rashmi Ramsinghaney Ms. Prerna Kapoor Ms. Seema Kapoor Mr. Manish Panwar Ms. Vikram Yadav Ms. Monika Chopra Ms. Jaspreet Kaur Ms. Preeti Mittal Ms. Shipra Sarcar Ms. Leela Raghavan Mathematics : Ms. Deepa Gupta Ms. Gayatri Chowhan Ms. N Vidya Ms. Mamta Goyal Ms. Chhavi Raheja Hindi: Mr. Akshay Kumar Dixit Ms. Veena Sharma Ms. Nishi Dhanjal Ms. Kiran Soni CORE-SEWA Ms. Vandna Ms.Nishtha Bharati Ms.Seema Bhandari, Ms. Seema Chopra Ms. Madhuchhanda MsReema Arora Ms Neha Sharma Geography: Ms Suparna Sharma Ms Aditi Babbar History : Ms Leeza Dutta Ms Kalpana Pant Ms Ruchi Mahajan Political Science: Ms Kanu Chopra Ms Shilpi Anand Economics : Ms. Leela Garewal Ms Anita Yadav CORE-Perspectives M s. M a d h u c h h a n d a, RO(Innovation) Ms. Varsha Seth, Consultant Ms Neha Sharma Coordinators: Ms. Sugandh Sharma, E O Dr. Srijata Das, E O Dr Rashmi Sethi, E O Ms.S. Radha Mahalakshmi, (Co-ordinator, CBSE-i) E O Mr. Navin Maini, R O Ms. Madhu Chanda, R O (Inn) (Tech) Shri Al Hilal Ahmed, AEO Mr. R P Singh, AEO Ms. Anjali, AEO Shri R. P. Sharma, Consultant (Science) Ms. Neelima Sharma, Consultant (English) Mr. Sanjay Sachdeva, S O

7 Contents 1. Study Material 1 2. Student's Support Material 34 CSW 1: Warm Up Activity (W1) 35 Recall Fractions CSW 2: Warm Up Activity (W2) 36 Types of Fractions CSW 3: Warm Up Activity (W3) 37 Arithmetical Operations on Fractions CSW 4: Pre Content Worksheet (P1) 39 Need of Rational Numbers CSW 5: Pre Content Worksheet (P2) 40 Another Number System CSW 6: Content Worksheet (C1) 41 Defining Rational Numbers CSW 7: Content Worksheet (C2) 43 Rational Numbers and Fractions CSW 8: Content Worksheet (C3) 45 Standard form of Rational Numbers CSW 9: Content Worksheet (C4) 46 Rational Numbers on Number Line CSW 10: Content Worksheet (C5) 51 Comparison of Rational Numbers CSW 11: Content Worksheet (C6) 53 Rational Numbers Between given Rational Numbers CSW 12: Content Worksheet (C7) 55 Addition and Subtraction CSW 13: Content Worksheet (C8) 56 Skill Drill 1

8 CSW 14: Content Worksheet (C9) 58 Skill Drill 2 CSW 15: Content Worksheet (C10) 60 Multiplication of Rational Numbers CSW 16: Content Worksheet (C11) 61 Multiplication Skill Drill CSW 17: Content Worksheet (12) 64 Division of Rational Numbers CSW 18: Content Worksheet (C13) 67 Rational Numbers as Decimals CSW 19: Content Worksheet (C14) 69 Real Life Problems CSW 20: Post Content Worksheet (PC1) 73 Rational Numbers Drill CSW 21: Post Content Worksheet (PC2) 78 Test Your Progress 7. Suggested Videos/ Links/ PPT's 83

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10 Introduction INTRODUCTION TO RATIONAL NUMBERS You have already learnt about natural numbers, whole numbers, integers and fractional numbers (or fractions) along with fundamental operations on them. Recall that we had extended the collection of whole numbers to integers to represent the situations like profit and loss, temperatures below and above o C, height below and above sea level etc., by including negative numbers like.., 4, 3, 2, 1, etc. Recall that in daily life the quantities above cannot always be expressed in whole numbers above and that is why, we had to introduce fractions such as,,,, etc., In this unit, we shall further extend these numbers to new type of numbers called rational numbers by introducing numbers such as,,, etc., to handle opposite situations corresponding to situations represented by fractions,,, etc. We shall also study fundamental operations on rational numbers and their use in our day today life. 1. Fractions A Review Recall that fractions are numbers of the type,,,, etc, which can be written in the form, where p and q are natural numbers. In a fraction, p is called the numerator and q is called the denominator. Proper fraction A fraction is called a proper fraction if the numerator is less than the denominator. The fractions,,,, etc., are examples of proper fractions. Improper fraction A fraction is called an improper fraction if the numerator is greater than or equal to the denominator. The fractions,,,, etc., are example of improper fractions. Mixed fractions A combination of a natural number and proper fraction is called a mixed fraction or mixed number. Examples of mixed fractions are 2

11 2, 1, 69, 101, etc. 2 is same as or (improper fraction) 1 is same as or (improper fraction) etc. Thus, every mixed fraction can be converted into an improper fraction. Conversely, every improper fraction (with numerator greater than denominator) can be expressed as a mixed fraction. For example, = 2, = 2, = 1, etc Unit fraction A fraction whose numerator is 1, is called a unit fraction. For example,,,,, etc, are unit fractions. Note that fractions such as,, etc, are not unit fractions. Like fractions Fractions having the same denominators are called like fractions. For example,,,,, etc are like fractions. Similarly,,,,, are also like fractions. Unlike fractions Fractions having different denominators are called unlike fractions. For example,,,,, etc. 3

12 are unlike fractions. Simplest or Lowest form of a fraction A fraction in which numerator and denominator have no common factor other than 1 (i.e., numerator and denominator are coprime) is said to be in its simplest or lowest form. For example,,,,, etc., are fraction in simplest form. Equivalent fractions Fractions having same simplest form are called equivalent fractions. For example.,,,,, are equivalent fractions because each of there can be expressed in the simplest form as. = = = = = = = etc. Similarly, the fractions,,, 4

13 are equivalent fractions as = = = = = = etc, but,,, are not equivalent fractions as = = = but =, which is not equal to. Recall that we can obtain a fraction equivalent to a fraction dividing) p and q by same non zero numbers. by multiplying (or Operations on fractions (i) Addition Sum of two like fractions is a fraction whose numerator is the sum of numerators of given fractions and denominator is the same as that of given fractions, i.e., + = For example, + = = + = = For adding unlike fractions, we first convert them into like fraction and then add as above. = + + = + = = = 1 5

14 + = + = + = = [Note that denominator of each fraction has been made 60 which is LCM of 15 and 20] This can be extended to addition of more than two fractions. (ii) Subtraction Like fractions = [Subtracting numerators keeping denominator same] = Unlike fractions = [converting fractions into like fractions] = = (iii) Multiplication Product of two (or more) fractions = For example, = = = 1 = = = 1 = = = 6 (iv) Division To divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction. 6

15 For example, = reciprocal of = = = 3 = = x reciprocal of = = = 2. Rational Numbers Need for further extention of numbers Recall that sum and difference of two fractions is always a fraction. For example. + = = Fraction Again = = Fraction What about? Is it a fraction? Think!! If yes, find that fraction. If no, what to do then? Now, Suppose we have the equation 3x 6 = 0 You can verify that 2 is a solution of this equation which is a whole number. 7

16 If we have the equation 3x + 6 = 0 You can verify that 2 is a solution of this equation which is an integer but not a whole number. That is 3x + 6 = 0 cannot be solved in whole numbers but can be solved in integers. If we have the equation 3x -7 = 0 You can verify that is a solution of this equation, which is a fraction. Thus, this equation has no solution in integers but can be solved in fractions. If we have the equation 3x + 7 = 0 try to find a fraction which is its solution. Can you find such fraction? If yes, find the fraction. If no, what to do? In both the above situations, you are not able to find fractions providing answers. To overcome this difficulty there is a need to extend the number system further by including numbers like,,,, etc. You may call such numbers as negative fractions. The extended collection so obtained is known as collection of rational numbers. In general The numbers which can be expressed in the form q 0 are called rational numbers., where p and q are integers and p is called its numerator and q is called its denominator. 8

17 Thus, the numbers,,,,,,,, etc., are all rational numbers. From the above definition, you can observe: (a) Every whole number is a rational number because it can be written in the form, when p, q are integers and q 0 For example 0 =, 1=, 2 =, 3 =, 4 =, etc (b) Every integer is a rational number because it can be written in the form, where p, q are integers and q 0. For example, -4 =, -11 =, 3 =, 0 =, etc. (c) Every fractions is a rational number. What about converser of (a), (b) and (c)? Clearly, their converses are not true. That is (a ) Every rational number is not a whole number for example, but it is not a whole number. is a rational number (b ) Every rational number is not an integer. For example, is a rational number, is also a rational number but none of these is an integer. (c ) Every rational number is not a fraction. For example, is a rational number but not a fraction. Similarly, is a rational number but not a fraction. Positive and Negative Rational Numbers In a rational number, if both p and q are of the same sign i.e., both positive or both negative, then it is called a positive rational number, If p and q are of opposite signs, then rational number p/q is called a negative rational number. For example,,,,, etc., are positive rational numbers and,,,, etc, are negative rational numbers Rational Number 0 (i.e.,,,, etc., is neither positive nor negative. 9

18 Example 1: Which of the following rational numbers are fractions?,,,,,,,,, -5,,, Solution: Only following rational numbers are fractions:,,, 3. Standard form of a Rational Number A rational number is said to be in standard form if (i) Its denominator is positive, (q > 0) (ii) there is no common factor other than 1 of numerator and denominator, i.e., HCF of p, q = 1 For example rational numbers,,,, etc., are in standard form. The rational numbers,,, etc., are not in standard form as in, q is not positive although HCF of 3, 2 is 1. in, q > 0, but HCF of 8, 16 is 8 not 1 in, q > 0 but HCF of 15, 30 is 15 not 1 Example 2: Convert the following rational numbers in standard form: (i) (ii) (iii) (iv) 10

19 Solution: (i) = = [Denominator has been made positive by multiplying numerator and denominator by same number -1] (ii) In the rational number, the denominator is positive but HCF of 16 and 20 is 4 (not 1) So, we divide both numerator and denominator by 4. = =, which is in standard form. (iii) In, denominator is not positive. Also HCF of 10 and 2, not 1. So, we have = = = =, Which is in standard form (iv) = = = =, which is in standard form. 4. Equivalent Rational Numbers Two rational numbers are said to be equivalent rational numbers if they have the same standard form. For example and are equivalent rational numbers because = and = i.e., they have the same standard form. Similarly,,,,, are equivalent rational numbers because standard form of each of them is. 11

20 Conversion of a rational number into equivalent rational number Recall how to convert a fraction into an equivalent fraction. Likewise, a rational number can be converted into an equivalent rational number by multiplying (or dividing) both the numerator and denominator by the same non zero number. For example, = = So, is equivalent to the rational number Similarly, = = So, is also equivalent to Also, is equivalent to = Example 3 : Convert each of the following rational numbers into an equivalent rational number. (i) (ii) (iii) Solution: (i) = = (ii) = = There can be infinitely many equivalent rational numbers to a given rational number (iii) = = Example 4: Write in an equivalent rational number form so that. (i) its numerator is 60 (ii) its denominator is 15 12

21 Solution: (i) = = [Multiplying both numerator and denominator by 30 so that numerator becomes 60] (ii) = = [Multiplying both numerator and denominator by ( 3), so that denominator becomes -15] (iii) 5. Operations on Rational Numbers The operation of addition, subtraction, multiplication and division of rational numbers are similar to that of integers and fractions which you have already studied. We explain these through examples. (i) Addition (a) Addition of rational numbers with same denominator. Let two rational number be and + = Adding numerators Keeping denominator same = [Adding integers (-19) and 5] = [Dividing numerator and denominator by 2] = [Rational number in standard form] Similarly + = Adding numerators Keeping denominator same = In general, to add two rational numbers and + = The above rule can be extended to more than two rational numbers. 13

22 Example 5: Add and Solution: We write + = + = = = Example 6: Add, and Solution: We write = Now, + + = + + = = = (b) Addition of rational numbers with different denominators Let two rational numbers be and + = + [Making the denominators same] = + = [Adding numerators and keeping denominator same] = Thus, + = 14

23 Similarly, let us add and + = + = = + [Making the denominator = 15 = LCM of 15 and 3] = + = = = Again consider two rational numbers, + = + [Making the denominators same i.e., 30 which is LCM = + of 10 and 15] = = You may also do it as follows: + = + = + = = = You may note that it is always convenient to add two rational numbers with different denominators, by converting their denominators into LCM of two denominators. 15

24 In general, To add two (or more) rational numbers with different denominators, we first convert them into equivalent rational numbers with common denominator equal to their LCM and then add as done before for adding rational numbers with same denominator. Example 7: Add, and Solution: = + + = + + [LCM of 3, 9 and 12 = 36] = + + = = = (ii) Subtraction of Rational Numbers (a) Subtraction of rational numbers with same denominator Let two other rational numbers be and = [Subtracting 2 nd numerator from first numerator keeping = same denominator] Let us go back to the problem of subtracting from faced by us in (2) above. We have = [Subtracting 2 nd numerator from first numerator = keeping the denominator same] Similarly, = [ Subtracting 2 nd numerator from first numerator, keeping denominator same] 16

25 = = and - = = = = -1 In general, if and are two rational numbers = Example 8: Subtract from Solution: = = = Example 9: Subtract from Solution: = (Making denominator positive) So, = = = = (b) Subtraction of rational numbers with different denominators Here again, we convert the given rational numbers into equivalent form with same denominator, preferably their LCM. We explain it through examples. Example 10: Subtract from. Solution: = [Making denominator same i.e., 15 = LCM of 5 and 3] 17

26 = = = Example 11: Subtract from Solution: = [LCM of 24 and 36 is 72] = = = Suppose we add and. We have + = = = 0 Similarly, + = = 0 In such case, we say that is additive inverse of rational number and is additive inverse of. Similarly, additive inverse of is and that of is, as + = = = 0 + = = 0 In general, if and are two rational numbers such that + = + = 0 then is called the additive inverse of and vice versa. In Example 11, = + additive inverse of = + 18

27 = + = = Thus, to subtract from, we can add the additive inverse of to. In general, = + additive inverse of = + (iii) Multiplication of Rational Numbers You already know that for two fractions product of fractions = For example, = = = = i.e., 90] = [diving numerator and denominator by HCF of 450 and 180 = The above rule of multiplication of two rational numbers can be extended to more than two rational numbers. Example 12: Multiply -2, and Solution: We have 19

28 = = = = = 4 (iv) Division of Rational Numbers Recall that to divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction. We follow the same rule for dividing a rational number by a non-zero rational number. As = 1 and 4 = 1., is the For example, to divide reciprocal of 4 and 4 is the reciprocal by, we multiply of. by the reciprocal of i.e., = reciprocal of Similarly, as = = 1; is the reciprocal of and is reciprocal of. = = = = Reciprocal of is as =1 Example 13: Divide Solution: = reciprocal of = = = = 20

29 6. Representation of a Rational Number as a Decimal Recall that a fraction can be represented as a decimal For example, = 0.5 = 0.75 = Similarly, a rational number can represented in the form of a decimal. For example, = -0.5 = = 0.75 = 1.8 Thus, if a rational number is negative, we first find decimal representation of corresponding fraction and then assign sign ( ) to the decimal. Let us find decimal representation of rational number. By long division, 21

30 = Here the decimal representation is not ending as each time, we get remainder 1. Also, the digit 3 in the decimal representation is repeating. We say that decimal representation of rational number is (i) (ii) Non terminating and repeating Clearly, decimal representation of,,, are terminating and so is the case with,,. Let us find decimal representation of some more rational numbers. Example14: Final decimal representation of (i) (ii) (iii) (iv) (ii) (vi) Solution: (i) = Here, decimal representation is terminating 22

31 (ii) = Here, decimal representation non terminating and repeating Here, digit 6 is repeating we write = = 3. 23

32 (iii) = Here, decimal representation is non terminating and repeating = (iv) = = -0.3 Here, decimal representations is non terminating and repeating

33 (v) = Here, decimal representation is non-terminating and repeating. A block of is repeating. We write = (vi) = 0.12 Decimal representation is terminating Observe that the decimal representation of a rational number is either terminating or non-terminating repeating. 7. Representation of Rational Numbers on the Number Line Recall that integers can he represented on the number line and so are the fractions. In the same way, rational number can also be represented on the number line. We proceed as follows: 25

34 We draw a line and take a point 0 on it to represent the rational number 0. The positive rational numbers will be on the right side of 0 and the negative rational number will be on the left side of 0. Fig.1 Recall that consecutive points representing number 1, 2, 3,, 1, 2, 3, are at unit distance from each other. To represent a rational number say, OA into two equal parts OM and MA. The mid-point M represents the rational number. on the number live, we divide the line segment To represent rational number parts, ON and NA., we divide the line segment OA into two equal The point N represents the rational number. Similarly, we can represent the rational number,,, by dividing the each unit distance into two equal parts as shown in Fig.2. Fig.2 For representing the rational number say, we divide each unit distance into four equal parts and take the 5 th point from O towards right as shown in Fig.3. Clearly, the rational number will be represented by the 5 th point to the left of O. 26

35 Fig.3 In the same way, we can represent rational number,,,,,,, etc on the number line. Example 15: Represent the rational numbers and on the number line Solution: Divide each unit distance into three equal parts. The second point. from O to the right represent. Fig.4 Redraw: Divide the line segment (0,1) into 3 equal parts. Similarly (0, (-1) into 3 equal parts. Note: Distance (0,1), (-1, -2), (-2, -3) or (0,1), (1, 2), (2, 3) all are equal. Similarly, the second point from O to the left represents. In the same way, we can represent,,...,,,... on the number line. In this manner, every rational number can be represented on the number line by dividing each unit distance into q equal parts and taking the pth point from O toward right (for positive) and left (for negative) rational number. 8. Comparison of Rational Numbers (a) Graphically: For comparing two rational number graphically, we represent them on the number line. 27

36 The number on the right is greater than the other (See Fig. 5) and the number on the left is smaller than the other. Fig. 5 Redraw: Divide the line segment (0,1) into 3 equal parts. Similarly (0, (-1) into 3 equal parts. Note: Distance (0,1), (-1, -2), (-2, -3) or (0,1), (1, 2), (2, 3) all are equal. and so on. Clearly, <, < >, > > Algebraically: Let us compare two rational numbers say and. = = and = = [Converting to equivalent rational numbers having same denominator. HCF of 3 and 4 = 12] As denominator of and is the same, so we compare their numerators 9 and 8. Since, 9 > 8, so, > Or, > In fact, every positive rational number is greater than every negative rational number. Now, let us compare and = = and = = = = [Making denominators same and positive] 28

37 Denominators of and are the same. So, we compare their numerators 25 and 6. Here, 6 > 25 (Why?) So, > Hence, > Example 16: Which is smaller? (i), (ii), Solution: (i) = = [LCM of 11 and 7 = 77] and = = Here, denominators are the same, we compare numerators 55 and 42. Clearly, 55 > 42. So, >. Hence, is smaller. (ii) = = and = = Here, we compare numerators 55 and 42, because denominators are the same. 55 < 42 So, < Hence, is Smaller. To compare two rational numbers. (i) (ii) (iii) First make their denominator positive. If they are of opposite signs, positive rational number is always greater than the negative rational number. If both are positive, compare them as fractions. 29

38 (iv) (v) If both are negative, compare their corresponding fractions negative (ignoring the sign). If one fraction is greater than the other, then the corresponding rational number is less than the other. Every positive rational number is greater than 0 and every negative rational number is less than Finding Rational Numbers between any two Rational Numbers. Take any two integers say 7 and 2. Recall that there are only eight integers 6, 5, 4, 3, 2, 1, 0, and 1 between these two integers. Since 6, 5, 4, 3, 2, 1, 0 and 1 are rational numbers also, so we can say that there are eight rational numbers between rational numbers 7 and 2. Can you find some more rational numbers between the two given rational numbers. 7 can be written as and 2 can be written as. So, rational numbers,,,,,, and, are between and or between 7 and 2. How many number are these? They are 89 such numbers. Similarly 7 = and 2 = Clearly, rational numbers,,,, and are between rational number 7 and 2. In this way, we can say that there are infinitely many rational numbers between any two rational numbers. Example 16: Find three rational numbers between and. Solution: = and = 30

39 Clearly, rational numbers,,,, and are between and. We can choose any three of them. Alternate Method Let us find: = = = Let us compare and and and. = and i.e., > Further < So, lies between and. In general, if and are two district rational numbers then, their average (or mean) i.e., + is always a rational number lying between and By this method also, we can find infinitely many rational numbers, between two rational number. Another rational number between and will be = = Similarly, = = is another rational number between and. 10. Applications of Rational Numbers in Problem solving. Example 18: A person travels a distance of km towards west and then a distance of km towards east on the same road. Find the distance and direction of the person from the starting point. Solution: Let the person start from a point, say, O toward west. Distance travelled towards west = Distance travelled towards east = km km 31

40 Position of the person is given by the expression = + = + = = So, the person is at a distance of km from his original point in west direction. Example 19: As a part of a km long marathon race, km was to be covered by cycles, km by boat and rest on foot. How much is the distance travelled on foot? Solution: Distance to be covered by cycles and boat = km Distance to be covered on foot = km = km = km = km = km = km = km Example 20: A field is km long and km wide. It is prosposed to sow wheat in part of the field and grow vegetable in the remaining part. Find the area of the field in which vegetables are to be grown? Solution: Area of the field = 32

41 Part of the filed for sowing wheat = So, the part of the field left for vegetables = 1 = Area of the field for vegetable = km = = = km. Example 21: The length of a ribbon is metres. Its part is used for wrapping gift packets each of length is metre. Find the number of gift packets. Solution: Length of ribbon used for gift packets = of metres = x metres = metres. Length of the piece used for one gift pack = metres So, number gift packets = = x = 50 33

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43 Student s Worksheet 1 Recall Fractions Warm Up W1 Name of the student Date Activity I recall The tiling shown below is the basic unit that makes an entire floor pattern. What fraction of the tiling is? a) BLUEb) PINKc) WHITE- 35

44 Student s Worksheet 2 Types of Fractions Warm Up W2 Name of the student Date Activity Types of Fractions Q1. Rearrange the letters to form meaningful words in fractions a) KIEL: b) ROPPRE: c) NIUT: d) EMDXI: e) RRPPEOMI: f) INEUKL: g) UEEVIQLTAN: Q2. Write a fraction for the shaded part in each case and answer the questions: a) d) b) e) 36

45 c) (i) (ii) (iii) (iv) The like fractions are: The unit fractions are: The unlike fractions are: Give two equivalent fractions of each: Student s Worksheet 3 Arithmetical Operations on Fractions Warm Up W3 Name of the student Date Activity Add or subtract Weight= 4 7 kg Weight= 3 7 kg Weight= 2 7 kg 37

46 Now answer the following:- a) The total weight of the teddy and the car is: b) Whose weight is less, Rattle or Car, and by how much? d) How much more does the teddy weigh than the car? e) Together the weight of the car and the rattle is: 38

47 Student s Worksheet - 4 Need of Rational Numbers Pre Content Worksheet P1 Name of the student Date Activity I feel the need - I Give 4 real life situations to express need of : WHOLE NUMBERS Counting the number of RED tiles in a RUBRICS puzzle. INTEGERS FRACTIONS Solve given fraction problem: Is your answer a fraction? So, we need more numbers which are neither an integer nor a fractional number. 39

48 Student s Worksheet - 5 Another Number System Pre Content Worksheet P2 Name of the student Date Activity I feel the need - II Find out the definitions of given terms by following the given link, to understand and appreciate the difference between the various number systems. Whole numbers Integers Decimal numbers and Fractions 40

49 Rational numbers Task 2: Write the given rational numbers in the given arrow boxes as per their type. Negative Rational Numbers Positive Rational Numbers Student s Worksheet -6 Defining Rational Number Content Worksheet C1 Name of the student Date Activity- Defining Dialogues Mr. Int is a member of INTEGER CLUB and Mr. Rat is a member of RATIONALS CLUB. 41

50 Given below are 4 sets of dialogues(in outer rectangles) and 4 options (in the inner circle). Read the dialogues and write the correct option in the dialogue box. 42

51 Student s Worksheet 7 Rational Numbers and Fractions Content Worksheet C2 Name of the student Date Activity 1 I see I understand Observe the given Venn Diagram.... Classify the given numbers as per your observations : 1. 56/3 a) fraction b) rational c) rational, whole, integer d) fraction, rational, integer /4 a) fraction b) rational c) rational, whole, integer d) fraction, rational, integer 43

52 3. Classify a) rational, whole b) rational, integer c) whole d) decimal, rational 4. 1/3 a) rational b) rational, integer c) rational, whole, natural d) rational, whole, natural, integer a) rational, whole, natural, integer b) rational, integer c) whole, integer d) rational Its 54 when expressed in standard form..! a) rational, whole, natural, integer b) whole, integer c) rational, whole, integer d) none of these a) not rational b) rational c) rational, integer d) rational, whole, integer, natural a) not rational b) rational, whole, natural, integer c) rational, whole, integer d) rational, integer 44

53 a) not rational b) rational c) rational, integer d) rational, whole, integer, natural a) not rational b) rational, whole, natural, integer c) rational, whole, integer d) rational, integer Student s Worksheet 8 Standard form of Rational Numbers Content Worksheet C3 Name of the student Date Activity- Standard Forum A rational number is said to be in its standard form if its numerator and denominator have no common factor other than 1, and its denominator is a positive integer. Which of the following rational numbers are not in the standard form? (Circle them) Filter out the circled rational and write their standard form in the space provided. 45

54 Student s Worksheet 9 Rational Numbers on Number Line Content Worksheet C4 Name of the student Date Activity - Mirror Images On the number line: Positive rational numbers are represented to the right of 0. Negative rational numbers are represented to the left of 0. 46

55 A) Number Line 1: Take 1 as 7/7. Given Shape Rational number for given shape Rational number for image of given shape Repeat the same on Number Line 2 and 3 and fill the given tables. 47

56 Number Line 2: Given Shape Rational number for given shape Rational number for image of given shape 48

57 Number Line 3: Given Shape Rational number for given shape Rational number for image of given shape B) Study the given number lines and write the rational number marked. 49

58 C) Draw the number lines and represent given rational numbers on it: 1) 2) 3) 4) 50

59 Student s Worksheet 10 Comparison of Rational Numbers Content Worksheet C5 Name of the student Date Activity - Mirror Distance A) Use the three number lines given in the previous worksheets and compare and arrange their equivalent rational numbers as per their placement on number line in the table given below. Number Shapes as per their Rational number for Rational number for Line position on given shapes in images of given shapes number line(in increasing order in increasing order increasing order)

60 Observe that rational numbers having like positive denominators can easily be arranged and compared. B) Convert to rational numbers having like denominators and put =, <, or > : 1. -ve numbers < +ve numbers

61 C) Arrange in descending order: Student s Worksheet -11 Rational Numbers Between given Rational Numbers Content Worksheet C6 Name of the student Date Activity- Search Is On.. Little Winnie is working on after numbers. She went to Ms. Rational to clear her doubts. Please help her search for more rational numbers that can be inserted between two given rational numbers. Winnie : Write 5 rational numbers inserted between 4/7 and 5/7. Ms. Rational : Rewrite them in standard form. 53

62 1. Find 5 rational numbers between given rational numbers: a) Make use of the gap between the numerators. b) c) 2. Find 5 rational numbers between given rational numbers by converting them to like rational numbers:

63 Student s Worksheet -12 Addition and Subtraction Content Worksheet C7 Name of the student Date Activity- Complete the circle Use the cut outs having the question and answer texts running parallel with the top and bottom edges of the segments. Place the answer text next to the question text to make a complete circle and colour it. 55

64 Student s Worksheet -13 Skill Drill 1 Content Worksheet C8 Name of the student Date Activity- Solution Search Step 1: Use the answers given in the box and write them in front of the questions(without solving). Step 2: Solve the questions and check the answers written in the step

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66 Student s Worksheet 14 Skill Drill 2 Content Worksheet C9 Name of the student Date Activity- Matching Cards In the following cards activity use the given cards and find their correct order. For e.g. Card 1 follows card 7 and one of the given cards shall follow card 1 and so on. Card No. Front Back 7 44 I have 44. Who has less than it? I have 6 2. Who has 2 less than it? 5 5 Now try it out yourself. Card No. Front Back 1 19 I have19.who has less than it? 35 2 ½ I have ½. Who has 7 less than it? I have3 1. Who has 1.25 less than it? I have 2 3. Who has double of 1 more? I have 4 3. Who has less than it? I have Who has 3.5 more than it? I have 44. Who has less than it? 7

67 8 19 I have Who has 2 more than it? I have 2. Who has 3 1 less than it? I have1 3. Who has less than it? I have 4. Who has 5/8 less than it? 12 6/7 I have 6/7. Who has 2/5 more than it? I have11. Who has 1/14 more? I have 5 1. Who has more? I have Who has 8 23 more? I have 7 5. Who has 1.5 more? 7 59

68 Student s Worksheet 15 Multiplication of Rational Numbers Content Worksheet C10 Name of the student Date Activity - Catch the butterfly Solve correctly and catch the butterfly in the net by colouring them in same the shade. 60

69 Student s Worksheet 16 Multiplication Skill Drill Content Worksheet C11 Name of the student Date Activity - Mission Multiplication Solve the given multiplication problems that come on your way to MISSION MULTIPLICATION

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72 Student s Worksheet 17 Division of Rational Numbers Content Worksheet C12 Name of the student Date Activity I- Divide and join the correct jigsaw pieces with a line. 64

73 Activity II - Division Drill 1. Evaluate: i) 1 ( 14) ii) iii) iv)

74 2. The product of two numbers is -28/27. If one of the numbers is -4/9, find the other. 3. By what number should we divide 27/16 so that the quotient is -15/8? 4. A small scale company pays Rs per week for advertising in the local paper. What is the total cost of advertising for the company for one year? 5. What number must be divided by so that the quotient is 60? 6. What number must be multiplied by so that the product is ? 7. Sam buys a laptop for Rs.45000, and pays for it with a two year interest-free loan. If he makes equal monthly payments, how much are his monthly payments? 66

75 Student s Worksheet 18 Rational Numbers as Decimals Content Worksheet C13 Name of the student Date Activity: Convert rational numbers to decimals Q1. Divide and write the decimal equivalent beside each of the following rational numbers. i) 4 10 ii) 2 3 iii) iv) 1 4 v)

76 vi) 7 3 vii) viii) 1 6 ix) 2 11 x) 5 4 Find a relation between a rational number and its decimal representation: 68

77 Q2. Give five examples of rational numbers that fit between each of the following sets of rational numbers. a and b and c and 3.46 Student s Worksheet 19 Real Life Problems Content Worksheet C14 Name of the student Date Activity- Solve the real life problems 3 1. The price of oranges is Rs. 65 per kilogram. What quantity of oranges can be 4 1 bought for Rs

78 1 2. The price of eleven cricket balls is Rs Find the price 5 of one ball. 3. A car is moving at a speed of distance it will cover in hours km per hour. Find the 4. Divide the sum of 1 4 and 1 1 by the sum of 5 10 and By what number should we multiply to get the product as -1? 70

79 6. A farmer has a rectangular piece of land of dimensions (in meters). He distributes it among his five children. How much area will each child get? 7. The product of two rational numbers is. One of the rational numbers is 1/6. What is the other rational number? 8. A boy has a cardboard of length 10 7 m and breadth 5 4 m.he makes five pieces of it to form play cards. What is the area of three such cards? 71

80 9. Divide by the sum of and. 10. A train travels a distance of 1000km400m in hours30minutes.find the average speed of the train. 11. By what number should we divide so that the quotient is? 72

81 Student s Worksheet -20 Rational Numbers Drill Post Content Worksheet PC1 Name of the student Date 1. Draw the number lines and represent given rational numbers on it: 1) 2) 2. The correct increasing order of the fractions is. 73

82 3. The numbers can be arranged in descending order as. 4. Arrange in descending order:

83 5. Solve the following a) b) 6. Which number should be subtracted from -1/5 to obtain -2/5? 7. Simplify a) b) 8. The product of two rational numbers is. One of the rational numbers is 1/6. What is the other rational number? 75

84 9. By what number should we multiply to get the product as? 10. Divide the sum of and by. 11. Solve 12. Solve 76

85 13. Find 14. Simplify 15. Find ) 16. Solve 77

86 Student s Worksheet 21 Test Your Progress Post Content Worksheet PC2 Name of the student Date Activity- Test your progress. Q1. Multiple choice questions: (5 X 1=5) i) If x = 1 3 and y = 1 3,then A) x + y =1 B) x + y -3 C) x y = 1 D) x /y = 0 ii) Which of the following rational numbers are in standard form? A) 3 16 B) C) 105 D) iii) The value of is A) 1 3 B) C) 1 6 D) 1 6 iv) The reciprocal of a negative rational number A) Is a positive rational number B) Doesnot exist rational number C) Is a negative rational number D) can be either positive or negative 78

87 v) What number should be subtracted from 3 5 to get -3? A) 7 5 B) 12 5 C) 7 5 D) 13 5 Q2. TRUE/FALSE : (5 X 1 = 5) i) All rational numbers can be represented on a number line. ii) We can insert finite number of rational numbers between -2 and -5. iii) The product of two rational numbers is always a positive rational number. iv) v) 3 5 lies to the right of 0 on a number line. Q3. Fil in the blanks: (5 X 1 = 5) i) If 3 7,then x =.. 5 x ii) iii) The denominator of a rational number cannot be. 79

88 iv).. is not reciprocal of any number. v) The additive inverse of is Q4. Match the following: (5 X 1 = 5) i) ii) A rational number between 1 2 and iii) 11 4 iv) 1 6 v) Standard form of Q5. Arrange in ascending order: (2 X 2 = 4) i) 5, 2, 3, ii) 3 7 1,,0, Q6. Write four rational numbers between: (3 X 2 = 6) i)-1 and 1 ii) -3 and -2 iii) 4 5 and

89 (Q7 to Q16 each carry three marks) Q7. Express in decimal form i) 19 8 Q8. Multiply: ii) by Q9. Divide: by Q10. Add, and Q11. Evaluate: Q12. Subtract the sum from the sum Q13. Simplify: Q14. The product of two rational numbers is 5 18.If one of them is 3,find the other

90 Q15. Represent on number line: i) 5 6 ii) 11 3 Q16. How many pieces of rope of each metres long? metres long can be cut from a rope of Q17 to Q21 each carry four marks) Q17. Find two rational numbers whose sum is 1 and if greater is divided by the 2 smaller, the result is Q18. A basket contains three types of fruits weighing kg in all. If kg of these be pineapples, kg be plums and the rest pears. What is the weight of the pears in the basket? Q19. Riya had Rs.300. She spent 1/3 of her money on notebooks and 1/4 of the remainder on stationery items. How much money is left with her? 20. Sandy wanted to go to his friend s house from school. He walked 5 km towards 4 east and then from there he walked 12 7 km towards west to reach his friend s house. What is the distance between the school and his friend s house? 21. Aryan earns Rs per month. He spends 1/4 of his income on food; 3/10 of the remainder on house rent and 5/21 of the remainder on the education of children. How much money is still left with him? 82

91 Suggested Video Links Name Title/Link Video Clip 1 Rational Numbers rational-numbers-entertainment-ppt-powerpoint/ Video Clip 2 Rational numbers Video Clip 3 Adding and Subtracting Rational Numbers Video Clip 4 Multiplying and Dividing Rational Numbers teachers.henrico.k12.va.us/math/int10405/.../mult_div_fracnt2.ppt Video Clip 5 Order Rational Numbers GB&v=KTR10AmibwM Weblink1 Solve online problems on Rational Numbers Weblink 2 Comparing and Ordering Rational Numbers iadmin.bismarckschools.org/.../alg-2-ch-8-add-and-subtractrational-numbers.ppt Numbers/comparing-rational-numbers.html 83

92 CENTRAL BOARD OF SECONDARY EDUCATION Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi India