CLASS VI CBSE-i PARTS OF WHOLES FRACTIONS & DECIMALS UNIT 4 Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India CBSE-i PARTS OF WHOLES FRACTIONS & DECIMALS UNIT 4 CLASS - VI Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India 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 has been taken duly mentioned. In case anything 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. 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 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. 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 nonevaluative 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 Acknowledgements Advisory Shri Vineet Joshi, Chairman, CBSE Shri Shashi Bhushan, Director(Academic), CBSE Ideators Ms. Aditi Misra Ms. Amita Mishra Ms. Anita Sharma Ms. Anita Makkar Dr. Anju Srivastava Dr. Indu Khetarpal Ms. Vandana Kumar Ms. Anju Chauhan Ms. Deepti Verma Ms. Ritu Batra Conceptual Framework Shri G. Balasubramanian, Former Director (Acad), CBSE Ms. Abha Adams, Consultant, Step-by-Step School, Noida Dr. Sadhana Parashar, Head (I & R),CBSE Ms. Anuradha Sen Ms. Jaishree Srivastava Ms. Archana Sagar Dr. Kamla Menon Ms. Geeta Varshney Dr. Meena Dhami Ms. Guneet Ohri Ms. Neelima Sharma Dr. Indu Khetrapal Dr. N. K. Sehgal Material Production Group: Classes I-V Ms. Rupa Chakravarty Ms. Anita Makkar Ms. Anuradha Mathur Ms. Kalpana Mattoo Ms. Savinder Kaur Rooprai Ms. Monika Thakur Ms. Seema Choudhary Mr. Bijo Thomas Ms. Kalyani Voleti Dr. Rajesh Hassija Ms. Rupa Chakravarty Ms. Sarita Manuja Ms. Himani Asija Dr. Uma Chaudhry Ms. Nandita Mathur Ms. Seema Chowdhary Ms. Ruba Chakarvarty Ms. Mahua Bhattacharya Material Production Groups: Classes VI-VIII Mathematics : Ms. Seema Rawat Ms. N. Vidya Ms. Mamta Goyal Ms. Chhavi Raheja Political Science: Ms. Kanu Chopra Ms. Shilpi Anand English : Ms. Rachna Pandit Ms. Neha Sharma Ms. Sonia Jain Ms. Dipinder Kaur Ms. Sarita Ahuja Science : Dr. Meena Dhami Mr. Saroj Kumar Ms. Rashmi Ramsinghaney Ms. Seema kapoor Ms. Priyanka Sen Dr. Kavita Khanna Ms. Keya Gupta English : Ms. Sarita Manuja Ms. Renu Anand Ms. Gayatri Khanna Ms. P. Rajeshwary Ms. Neha Sharma Ms. Sarabjit Kaur Ms. Ruchika Sachdev Geography: Ms. Deepa Kapoor Ms. Bharti Dave Ms. Bhagirathi Ms. Archana Sagar Ms. Manjari Rattan Mathematics : Dr. K.P. Chinda Mr. J.C. Nijhawan Ms. Rashmi Kathuria Ms. Reemu Verma Science : Ms. Charu Maini Ms. S. Anjum Ms. Meenambika Menon Ms. Novita Chopra Ms. Neeta Rastogi Ms. Pooja Sareen Political Science: Ms. Sharmila Bakshi Ms. Srelekha Mukherjee Economics: Ms. Mridula Pant Mr. Pankaj Bhanwani Ms. Ambica Gulati Geography: Ms. Suparna Sharma Ms. Leela Grewal History : Ms. Leeza Dutta Ms. Kalpana Pant Material Production Groups: Classes IX-X Dr. Sadhana Parashar, Head (I and R) Shri R. P. Sharma, Consultant Ms. Seema Lakra, S O History : Ms. Jayshree Srivastava Ms. M. Bose Ms. A. Venkatachalam Ms. Smita Bhattacharya Coordinators: Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi, E O (Com) E O (Maths) E O (Science) Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO Ms. Preeti Hans, Proof Reader Preface CONTENT Acknowledgment 1. Syllabus 1 2. Scope Document 2 3. Teacher’s Support Material 8 2 Teacher’s Note 9 2 Activity Skill Matrix 18 2 Warm up W1 : Importance of Numbers 20 2 Warm up W2 : Tangrams 20 2 Pre Content Worksheet P1 21 A Quality as a part of Whole 2 Pre Content Worksheet P2 22 What is it any way 2 Content Worksheet C1 22 Understanding and Representing Fractions and its Components 2 Content Worksheet C2 23 Locating Fractions on the Number Line 2 Content Worksheet C3 25 Introducing Decimals 2 Content Worksheet C4 25 Placing Decimals on the Number Line 2 Content Worksheet C5 26 Equivalent Fractions 2 Content Worksheet C6 27 Converting Fractions to Decimals 2 Content Worksheet C7 28 Simplification of Fractions 2 Content Worksheet C8 29 Independent Practice Content Worksheet C9 2 Place Value Chart of Decimals 29 2 Content Worksheet C10 30 CONTENT Comparison of Fractions Decimals 2 Content Worksheet C11 31 Addition and Subtraction of Fractions 2 Content Worksheet C12 32 Addition and Subtraction of Fractions 2 Content Worksheet C13 33 Multiplication and Division as Repeated Addition and Subtraction 2 Content Worksheet C14 33 Word Problems Involving Fractions and Decimals 2 Content Worksheet C15 34 Rounding off and Estimation 2 Content Worksheet C16 35 Estimation and Significance 2 Content Worksheet C17 35 Ratio and Proportion 2 Post Content Worksheet P1 2 Post Content Worksheet P2 2 Post Content Worksheet P3 2 Post Content Worksheet P4 4. Assessment Guidance Plan 37 5. Study Material 41 6. Student’s Support Material SW 1 : Warm up W1 : Importance of Numbers 2 126 SW 2 : Warm up W2 : Tangrams 2 128 SW 3 : Pre Content Worksheet P1 : A Quality as a Part of a Whole 2 131 SW 4 : Pre Content Worksheet P2 : What is it any ways 2 133 SW 5 : Content Worksheet C1 : Understanding and Representing 2 136 Fractions and its Components CONTENT 2 SW 6 : Content Worksheet C2 : Locating Fractions on the 137 Number Line 2 SW 7 : Content Worksheet C3 : Introducing Decimals 147 2 SW 8 : Content Worksheet C4 : Placing Decimals on a Number Line 151 2 SW 9 : Content Worksheet C5 : Equivalent Fractions 153 2 SW 10 : Content Worksheet C6 : Converting Fractions to Decimals 155 2 SW 11 : Content Worksheet C7 : Simplification of Fractions 158 2 SW 12 : Content Worksheet C8 : Independent Practice 161 2 SW 13 : Content Worksheet C9 : Place Value Chart of Decimals 166 2 SW 14 : Content Worksheet C10 : Comparison of 169 Fractions/Decimals 2 SW 15 : Content Worksheet C11 : Addition and Subtraction 174 of Fractions 2 SW 16 : Content Worksheet C12 : Addition and Subtraction 179 of Fractions SW 17 : Content Worksheet C13 : Multiplication and Division as 2 180 Repeated Addition and Subtraction SW 18 : Content Worksheet C14 : Word Problems Involving 2 184 Fractions and Decimals SW 19 : Content Worksheet C15 : Rounding Off and Estimation 2 188 SW 20 : Content Worksheet C16 : Estimation and Significance 2 191 SW 21 : Content Worksheet C17 : Ratio and Proportion 2 193 SW 22 : Post Content Worksheet P1 2 197 SW 23 : Post Content Worksheet P2 2 198 SW 24 : Post Content Worksheet P3 2 201 SW 25 : Post Content Worksheet P43 2 202 Acknowledgments 2 205 Suggested Videos/Links/PPT’s 2 205 SYLLABUS - Unit 1 FRACTIONS AND DECIMALS Fractions • Understanding and representing fractions • Components of a fraction • Representing fraction on number line • Types of fractions • Simplifying a fraction • Comparing and ordering fractions • Addition and subtraction of fractions and extension to word problems • The fraction of a quantity • Division and multiplication of a fraction by a whole number • Extension of BODMAS to fractions • Ratio and proportion • Applications to real life Decimals • Converting a fraction to decimal • Representing a decimal on a place value chart • Expanding a decimal number • Components of a decimal number • Representing decimal on a number line • Comparing and ordering decimal numbers (extension of fractions and place value), • Basic operations on decimals and extension to word problems, • Extending addition and subtraction of decimals multiplication and division by a whole number, • Extension of BODMAS to decimals, • Estimation and significance 1 to SCOPE DOCUMENT FRACTIONS AND DECIMALS Prerequisite: Knowledge of Factors and multiples, skill of multiplication and division of two numbers, HCF and LCM, BODMAS on integers, simple word problems extending the concept of the four operations, Estimation. Concepts: • Understanding and representing fractions • Placing fractions on a number line • Understanding decimals and converting fractions into decimals • Representing decimals on number line • Equivalent fractions • Comparing and ordering fractions and decimals • Adding and subtracting fractions and decimals • Multiplying and dividing fractions and decimals by a whole number. • BODMAS with reference to fractions and decimals • Estimation with fractions and decimals • Applications of fractions - Ratio and Proportion • Application of decimals - Scientific notations, concept of significant figures Learning objectives At the end of this lesson student will be able to • Define fractions, its components and types of fractions • Represent fractions on the number line • Define decimal and equivalent decimals. • Identify through given pictures the whole number and fractional parts of a decimal and write decimals in expanded form. • Represent decimal numbers on the number line 2 • Differentiate between more than 1 and less than 1 fractions. • Define equivalent fractions and convert fractions to decimals and vice-versa. • Know about the place values of a decimal number, and apply it in various situations. • Compare and order two or more fractions/ decimals. • Perform addition and subtraction on fractions and decimal numbers and extend it to solving word problems • Extend repeated addition and subtraction to multiplication and division of fractions and decimals. • Give estimations with the significance of number specified. • Apply the knowledge of ratio and proportion to real life problems. Extension activities: 1. Write an article for the school magazine on how decimal system is used for classification of books in the library. 2. Research on Golden ratio / number and its patterns. Where do you see the golden number used in real life? Project activity: Using Ms Excel, divide two natural numbers chosen randomly and find out when the quotient comes out to be a proper fraction, an improper fraction or a natural number. You may choose numbers between 1 and 20. Refer to the following images to generate the random numbers. 3 4 t numberr format in the t column n C is choseen as fractio on upto 2 diigits. Note that the Now in th he 4th column D, div vide column A by co olumn B. Now the number n forrmat is ‘general’ which w takess in decimall format. Compare C th he fractionss obtained in i column 3 to the decimals obtained o in column 4. Record you ur observattions. 5 Cross currricular liinks ENGLISH H: Students would w resea arch and write w about how decim mal system m is useful in i classificaation of books in a library and d write an article a for th he school magazine. m SCIENCE E: Measure th he height an nd weightss of 5 friend ds. Find ou ut which of them havee their heigh hts and weights as natural nu umbers and d which of th hem have it i as decimaals. ART: Mathematiics of goldeen ratio exissts and is used u extensiively in all types of arrt. The mov vie’ “Da Vinci Code”, the pain nting of Mona Lisa and a many architectura a al wonders all have reeference to some asspect of gollden ratio and a proporttions. http://ww ww.markwa ahl.com/index.php?id d=22 6 Use fractions to make mosaic designs and represent by different colours used in fraction form. Fraction Dialogue Read a dialogue on fractions at http://nrich.maths.org/1510 Technology help: Web links for reference, research and remedial Power point presentations You tube to deliver content 7 Teacher’s Support Material 8 TEACHER’S NOTE The teaching of Mathematics should enhance the child’s resources to think and reason, to visualise and handle abstractions, to formulate and solve problems. As per NCF 2005, the vision for school Mathematics include : 1. Children learn to enjoy mathematics rather than fear it. 2. Children see mathematics as something to talk about, to communicate through, to discuss among themselves, to work together on. 3. Children pose and solve meaningful problems. 4. Children use abstractions to perceive relation-ships, to see structures, to reason out things, to argue the truth or falsity of statements. 5. Children understand the basic structure of Mathematics: Arithmetic, algebra, geometry and trigonometry, the basic content areas of school Mathematics, all offer a methodology for abstraction, structuration and generalisation. 6. Teachers engage every child in class with the conviction that everyone can learn mathematics. Students should be encouraged to solve problems through different methods like abstraction, quantification, analogy, case analysis, reduction to simpler situations, even guess-and-verify exercises during different stages of school. This will enrich the students and help them to understand that a problem can be approached by a variety of methods for solving it. School mathematics should also play an important role in developing the useful skill of estimation of quantities and approximating solutions. Development of visualisation and representations skills should be integral to Mathematics teaching. There is also a need to make connections between Mathematics and other subjects of study. When children learn to draw a graph, they should be encouraged to perceive the importance of graph in the teaching of Science, Social Science and other areas of study. Mathematics should help in developing the reasoning skills of students. Proof is a process which encourages systematic way of argumentation. The aim should be to develop arguments, to evaluate arguments, to make conjunctures and understand that there are various methods of reasoning. Students should be made to understand that mathematical communication is precise, employs unambiguous use of language and rigour in formulation. Children should be encouraged to appreciate its significance. At the upper primary stage, students get the first taste of power of Mathematics through the application of powerful abstract concepts like Algebra, Number System, Geometry etc. 9 Revisiting of the previous knowledge and consolidating basic concepts and skills learnt at the Primary Stage helps the child to appreciate the abstract nature of Mathematics. Whether it is Number system or algebra or Geometry, these topics should be introduced by relating it to the child’s every day experience and taking it forward to abstraction so that the child can appreciate the importance of study of these topics. The mathematics curriculum during the preschool, elementary school and the middle school years has many components. But at the heart of mathematics in those years are concepts of number and operations with numbers. Proficiency with the numbers and numerical operations is an important foundation for further education in mathematics and in fields that use mathematics. Fractions, Ratio and Proportion and Decimals are important concepts in the middle school curriculum, but their development and understanding takes place at the elementary school. The students should be made to understand that the key to success in mathematics is their belief that they can make sense of the mathematics they study. Unfortunately, many children lose this belief when they begin to work with fractions and decimals. If they believe that Mathematics is a set of rules to memorize and apply, it becomes very difficult for them to learn. An informal introduction to fractions begins long before the concept is introduced formally as a part of mathematics. When a child wants to share half of his chocolate with his friend or distributes his set of candies equally amongst three of his friends, he is unknowingly talking about fractions. The comparision of fractions can also be informally introduced by paper cutting before introducing it formally. For eg. If we are to compare 5/9 and ½, to show that 5/9 is bigger, take 5 strips and divide each strip into half. Now if these 10 strips have to be given to 9 friends, 1 part of the strip so still left out, showing that 5/9 is greater than ½ Take another eg of 5/7 and 1/3. 10 Again since a couple of parts are left over after giving out one part to each of the 7 friends, 5/7 is greater than 1/3. The students may be introduced to equivalent fractions by paper folding methods or overlapping methods before they are formally exposed to the mathematical procedure. The study of fractions should build on the students’ prior knowledge of whole number concepts and skills and their applications in everyday life. For example, students may use fractions and decimals to report measurements, to indicate scale factors and compare responses from samples of unequal sizes. The teacher may help the students deepen their understanding of fractions by encouraging flexible thinking and justifying their answers. Eg. 1. Given 3/4th of a strip, draw its ½. If the figure below is 3/4th of a given strip, draw half of the given strip 2. Given a points 1 and 1 located on the number line, locate ,2 etc. The area model may be used to represent a part of a whole by shading a portion of whole area to find fractional parts and even equivalent fractions. It may further be extended to the introduction of decimals 11 For eg. Using a 10x10 grid, shading 3/4th of the grid or 0.75 of the grid mean the same. This gives them an idea how ¾, 75/100, .75, 15/20 etc. mean the same. The comaprsion of frations can be introduced pictorially. Eg. Shaded portion represents 7/8 and Shaded portion represents ¾ The comparison is clear through the figure above that 7/8 is a grater fraction than ¾ although both have one piece left out of the whole. This reasoning may further be extended by mathematical explanations. 12 Multiplicative reasoning as a transition from Additive reasoning The teacher should emphasize on the development of multiplicative reasoning as a transition from the additive reasoning. This may be clarified by an example. If an investment of $5 gives me $15 back after one year, and another investment of $20 gives me $30 back after one year, then which one is a better investment? An obvious answer the teacher might get is that both are same as both earn me $10. This was additive reasoning. This is where the concept of ratio needs to be developed. The teacher now may convert the investments into simple comparable numbers such as $1 giving $5 in the first investment and $8 giving me $18 in the second investment. Now which one is better? Obviously I would like to invest in the first one as it is earning me $5 on an investment of $1 each time. So if I invest $8 in the first investment, I can earn $40! This is the development of multiplicative reasoning. This may give the teacher a base line to develop the concept of ratio. Proportional reasoning Proportional reasoning is fundamental to the understanding of the mathematics of the secondary school curriculum. Thus, it is important that the students, at this stage develop the ability to use the proportional reasoning before they go into more advanced mathematics. If the students can make sense of fractions, ratios and proportions and can reason in a logical way, the transition to the secondary school curricula will be a smooth one. Understanding proportionality should not only mean setting two ratios equal and solving for a missing terms; it means recognizing quantities in tables , graphs etc. and find out their relationship. To be taken care of… Teachers need to be attentive to the conceptual obstacles that the students may face as they make a transition from operations with whole numbers. Multiplying and dividing fractions and decimals can be challenging for many because of the problems that are primarily conceptual rather than procedural. From their experience with whole numbers, many students develop a thinking that multiplication makes a number greater and division makes a number smaller. Teacher should check to see of his/her students has this misconception and should take steps to build their understanding. While teaching addition of fractions, the teacher may encourage the students to use bench marks and estimations. For example : A very common error of the type can be avoided if the students are1 taught the method of estimation using bench marks: 13 3 1 4 1 so the sum ou ught to be greater than 1! A solid conceptua al foundation > and > 5 2 5 2 using estimations leads to less numb ber of mistakes. Since The division of fractions may be intrroduced as repeated subtraction; just as it is introduced i in whole numbers. For eg. Divide 7 by 3/5 can be refram med as: What is the number of pieces of striings obtained when a length of 3/5m is cut from f a string of length 7 Now there are 11 pieces of length 3/ /5 m and 2/5 th part of the 3/5m is left out; which w is 2 3 2 2 /5 m ie m of string is left out. So o, 7 ÷ = 11 3 3 /5 5 3 The understanding of the propertiies of associatively and commutatively can n be used to 7 4 simplify some of the computations in fractions. For eg. computation of 3 × × can be 2 3 7 4 simplified by using the associativee property as × (3 × ) . Similar other pro operties used 2 3 intelligently can simplify the operations on fractions. The unit Fractions and Decimals ha as activities where the students may talk abo out the use of fractions and decimals in their life; they can find and speak about the insttances where d around us. fractions and decimals can be found 14 The students may be encouraged to use and find the use of fractions in their daily lives. They may be asked to • Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. • Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? • Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. • Find the use of fractions and ratio while cooking a dish. ( the amount of ingredients used) • Use of fractions and ratio while building something. 15 COMMON ERRORS Type of error Error made Conversion to and from the mixed fractions 1 8 4 6 Addition of fractions with different denominators 4 Correction 4 6 5 14 4 4 1 8 3 3 16 24 19 24 3 24 Thumb rule: MAD: multiply, add, denominators same etc 4 = Thumb rule: QRD 14 3 Addition Of mixed fractions 2 Multiplying and dividing the mixed fractions 3 8 3 4 5 2 5 =5 Multiplying and dividing proper fratcions 3 4 2 16 21 48 168 3 ) 6 =5 Cancel the common factors in the numerator and denominator Conver the mixed fractions into improper fractions, cancel the common factors if any and then multiply the numerators and denominators and convert the fraction into a mixed fraction. 2 3 3 6 2 16 1 4 3 1 3 9 4 10 3 15 2 7 1 2 Overview of the students’ worksheets The first warm up (W1) activity motivates the students to write or speak about the number system that they have already studied. They shall recall the division of symmetrical figures. In the second warm up (W2) activity, the student shall use the famous tangram puzzles and its parts to introduce a fraction. They may also be encouraged to speak about the use of fractions in their daily life. In the first pre content activity (P1), the students shall be talking about the fractions informally. They may be motivated to generalise the problem given to them for n friends. The focus on the warm up an pre content activities shall be to refresh the previous knowledge of the students so that they can comfortably build up the new topic. The pre content activities act as a bridge between the previously learnt concepts and the new concepts to be studied. In the 2nd pre content activity (P2), the students shall cut various shapes given to them, colour them, and answer the given set of questions leading to the development of the concept of fraction, numerator and denominator. The content worksheets from C1 through C17 aim at achieving the above stated learning objectives. Not only shall the students learn the basic concepts of fractions and decimals, they shall be encouraged to apply them to their daily lives. The teacher may give examples where the students can find fractions and decimals and may find the ratio of the like quantities. The teacher may encourage them to make projects where they find fractions while dealing with a merchant or a milkman or even in the library. Further the post content activities are designed to assess the students’ understanding of the concepts learnt in the unit. 17 ACTIVITY – SKILL MATRIX Activity Warm up (W1) Name of activity Importance of numbers Skills learnt Previous knowledge of numbers and division of a symmetrical object into equal parts Warm up (W2) Tangrams Relating mathematics with art puzzles Pre content (P1) Fraction as a part of a whole Memory, knowledge, understanding Pre content (P2) Identifying the parts of a Memory, fraction Content worksheet (C1) knowledge, understanding Understanding of a fraction Comprehends and its components multiple instructions, recalls and follows the given instruction Content worksheet (C2) Locating fractions on a Drawing skill, Thinking skill number line Content Worksheet (C3) Introducing decimals Synthesis skill Content Worksheet (C4) Placing decimals on the Drawing skill, Thinking skill number line Content worksheet (C5) Equivalent fractions Diagrammatic and mathematical understanding Content Worksheet (C6) Content Worksheet (C7) Converting fractions to Diagrammatic and mathematical decimals understanding Simplification of fractions Diagrammatic and mathematical understanding Content Worksheet (C8) Independent practice Content Worksheet (C9) Place value chart decimals Knowledge, Memory of Understanding and computing skills 18 Content Worksheet (C10) Comparison of fractions Perception and application skills and decimals Content Worksheet (C11) Addition and subtraction of Rational fractions comparison, Computational, language comprehension and application Content Worksheet (C12) Addition and subtraction of Understanding , thinking and decimals Content Worksheet (C13) Extend rational comparison multiplication Comparison and application and division as repeated skill, Higher order thinking skill addition and subtraction Content Worksheet (C14) Word problems involving Application, fractions and decimals comprehension language and problems solving Content Worksheet (C15) Rounding off and Application skill, estimation skill estimation Content Worksheet (C16) Content Worksheet (C17) Estimation and Application significance thinking. Ratio and Proportion Knowledge, skill, critical understanding, critical thinking Post Content Worksheet Knowledge and self learning (PC1) Post Content Worksheet Knowledge and self learning (PC2) Post Content Worksheet Computational (PC3) solving Post Content Worksheet Computational (PC4) solving 19 and problem and problem ACTIVITY 1- WARM UP (W1), IMPORTANCE OF NUMBERS Specific Objective: To discuss importance of natural numbers and whole numbers Description: The students shall see the video clip 1 to understand how the knowledge of fractions is useful in real life situations. Questionnaire will also allow the students to think about how and where such numbers are used. Execution: Teacher will make a questionnaire which talks about different types of numbers and their importance, numbers which are neither whole nor integers. Parameters for Assessment: Students will be able to: • Recall different type of numbers they have come across or they have used. • Divide a quantity in equal parts. ACTIVITY 2- WARM UP (W2), TANGRAMS Specific Objective: To review and recall the concept of natural numbers and whole numbers Description: The students shall be told about tangram puzzle set. They would then complete the table given in the task .They shall be helped to relate it to part of a whole. Execution: Teacher may take print outs of the sheet and ask the students to write in the given space. Teacher may encourage them to make their own tangram puzzle set and use it to make a shape of their own choice. Parameters for Assessment: Students will be able to: • • Compare quantities using concept of division Represent a quantity as a part of a whole 20 ACTIVITY 3- PRE CONTENT (P1), A QUANTITY AS A PARTS OF A WHOLE Specific Objective: To represent a quantity as a part of a whole (Through hands on activity) Material Required -. Pattern blocks handout and worksheet for each child. Pre preparation - Teacher introduces the topic by reviewing the use of terms such as ½ in daily life. • Students quote examples from daily life: o I walked ½ km today; o I ate ½ of my brother’s share of chocolate. Teacher encourages the students to provide more examples of ½ and other fractions familiar to them. Description: The students shall cut out the shapes and through colouring activity they will be able to understand the meaning of the word ‘whole’ and ‘parts of a whole’ as well the meanings of the words ‘numerator’ and ‘denominator’. Execution: The Teacher distributes the hand out to the students. Different shapes other than a circle are taken to install the idea of a fraction as a part of a whole. Parameters for Assessment: Students will be able to: • • Comprehend given problem situation Represent a quantity as a part of a whole 21 ACTIVITY 4- PRE CONTENT (P2), WHAT IS IT ANYWAY Specific Objective: To represent a quantity as a part of a whole Description: The students shall interpret the given problem and write their answers in the given space. Execution: Teacher may take print outs of the sheet and ask the students to write in the given space. Teacher may ask them about notion of ‘one-third’, ‘two-fifth’ etc. Parameters for Assessment: Students will be able to: • • • Comprehend given problem situation Represent quantities using concept of division Represent a quantity as a part of a whole ACTIVITY 5- CONTENT WORKSHEET (C1), UNDERSTANDING AND REPRESENTING FACTORS AND ITS COMPONENTS Specific Objective To understand a fraction and its components Description: The students shall interpret the given problem and write their answers in the worksheet. The students may be divided into groups to do this activity. Execution: Teacher checks for understanding of the meaning of the words numerator, denominator and fraction through a worksheet. Parameters for Assessment: Students will be able to: 22 • • • Hass knowledg ge of variou us types of fractions. f Relaate the giveen problem to fractionss. Can n speak crea atively abou ut fractionss and their need n and usage. u Extra Reaading: http://w www.youtube.com/waatch?v=gJgusNWTIkA A&feature= =related http://w www.amath hsdictionary yforkids.com/ ACTIVIT TY 6- CO ONTENT T WORK KSHEET T (C2), LO OCATIN NG FRAC CTIONS ON O TEH H NUMBE ER LINE E O e: Specific Objective To represent fractionss on a numb ber line Activity 1 & 3: Descriptiion: The concept c of parts p of fracctions is to be used to introduce representaation of fractions on a number line. The students sh hall be told d about how w to represeent the fracction on the numbeer line with h help of video clip 4 by b making as many markings m ass the denom minator of the given fraction. Activity 2: 2 Materialss needed:: Gems for each e child, Cloth’s pin ns, Clotheslline with 0 and a 1 mark ked. 23 Pre preparation: Gems packets, a clothesline with markings 0 to 1 and clothes pins. Description – After distributing a packet of gems to each child, teacher asks each child to randomly pick up 10 gems from the packet and then follow the directions given in the instruction sheet. Teacher helps students to model. Teacher will allow student to come and demonstrate his/her knowledge on the board. Since each child has a different favourite colour hence will get to represent different fraction on the number line. This activity will give a visual representation to a fractional number.] Note: At this point teacher may want to extend the knowledge of number line and ask students to arrange the fractions representing different colours in the ascending or the descending order. Execution: Students follow the directions and complete the activity as well as some more questions based on knowledge acquired. Parameters for Assessment: Students will be able to: • • • Exhibit knowledge of parts of fraction. Divide a unit distance into as many equal parts as the denominator. Represent fractions on a number line. 24 ACTIVITY 7- CONTENT WORKSHEET (C3), INTRODUCING DECIMALS Specific Objective: To introduce decimals and represent quantities using decimal Description: The concept of parts of fractions is to be used to introduce decimal as a special case of a fraction with denominator in powers of 10. Execution: Activities given can be given in form of worksheet or practical examples to the students. Teacher may ask the students to do conversion from centimetre into metre etc. Parameters for Assessment: Students will be able to: • • Exhibit knowledge of fraction and parts of fraction. Represent fraction with denominator in powers of 10 using concept of decimal. • Represent quantities using decimal. Extra Reading: http://home.avvanta.com/~math/FDU1.HTM ACTIVITY 8 - CONTENT WORKSHEET (C4), PLACING DECIMALS ON THE NUMBER LINE Specific Objective: To represent decimal on a number line Description: The concept of representation of fractions on number line is to be used to introduce representation of decimal on number line, as a special case of a fraction with denominator in powers of 10. Students shall write their answers in the given space and then represent decimal on number line accordingly. Execution: Activity given to be given in form of worksheet to the students. Teacher may ask the students to represent decimal on a number line after viewing video clip 5. 25 Parameters for Assessment: Students will be able to: • • • Represent fractions on a number line. Divide a unit distance into 10 equal parts and represent related decimal on it. Represent decimal on a number line. ACTIVITY 9- CONTENT WORKSHEET (C5), EQUIVALENT FRACTIONS Specific Objective: To demonstrate, communicate and explain equivalent fractions. Description: The students shall use the hands out (cut outs) of congruent rectangles to represent fractions diagrammatically. . This worksheet will help students to represent same fraction in many ways hence understand equivalent fractions. Execution: Teacher will show Video clip 7 to the students to explain the concept visually. The teacher prepares a comprehensive worksheet for students to practice the concepts learnt. Parameters for Assessment: Students will be able to: • • Represent fractions diagrammatically Understand and find equivalent fractions to a given fraction Extra Reading: www.helpingwithmath.com/by-subject/fractions/fractions.htm 26 ACTIVITY 10- CONTENT WORKSHEET (C6), CONVERTING FRACTIONS TO DECIMALS Specific Objective: To convert fractions to decimals Description: This worksheet is based on equivalent fractions. The fractions are converted to an equivalent fraction with denominator 10 or powers of 10. Teacher induces the topic by showing the graphics where a figure is divided into number of parts. Students shall answer the questions based on the graphics. The concept and discussion is extended to the same shapes cut into 10 equal parts and then 100 equal parts. Execution: Teacher shows the video clip 8 again to show conversion of equivalent fractions to reinforce learning. A worksheet may be given to the students in which they try to convert fractions to decimals based on what they have learnt. Parameters for Assessment: Students will be able to: • • • Represent fractions diagrammatically Understand equivalent fractions Convert fractions to decimals Extra Reading: www.resources/games/fraction-games4/equivalent01.html 27 ACTIVITY 11 - CONTENT WORKSHEET (C7), SIMPLIFICATION OF FRACTIONS Specific Objective: To demonstrate simplification of fractions Description: Students view video clips 9 to understand the simplification of fractions visually. After the explanation and video clip students are allowed to work on the situations independently. Execution: This worksheet also deals with equivalent fractions but instead of finding multiples, teacher guides the students to divide by a common factor and arrive at a fraction whose numerator and denominator have no factor other than 1 common. Parameters for Assessment: Students will be able to: • • • Represent fractions diagrammatically Knowledge of equivalent fractions Understanding of simplification of fractions Extra Reading: http://www.funbrain.com/cgi-bin/fob.cgi?A1=s&A2=0 http://www.math-aids.com/Fractions/ 28 ACTIVITY 12- CONTENT WORKSHEET (C8), INDEPENDENT PRACTICES Specific Objective: To find Equivalent fractions and simplify fractions Description: Students recall Equivalent fractions and simplification of fractions and are allowed to work on the situations independently. Execution: Teacher may prepare a comprehensive worksheet to test student’s understanding of classifying fraction, simplify fractions and converting fractions to decimals, using concept of Equivalent fractions and simplification of fractions. Parameters for Assessment: Students will be able to: • • • Has knowledge of various types of fraction Is able to relate the given number to different kind of fractions Can speak creatively about numbers and their need and usage ACTIVITY 13- CONTENT WORKSHEET (C9), PLACE VALUE CHART OF DECIMALS Specific Objective: To learn to read and use Place value chart of decimals Description: Students watch video clip 3 and after a brief recall of the place value chart of numbers, they then answer questions based on what they saw in the video. Execution: Teacher asks questions to test understanding of the place value of each digit and then asks students to give the expanded form / the place value of a particular digit/ number name etc. Parameters for Assessment: 29 Students will be able to: • • Understands the place value chart for decimals Read and write decimals in expanded form ACTIVITY 14- CONTENT WORKSHEET (C10), COMPARISON OF FRACTIONS/DECIMALS Specific Objective: To visualise and compare fractions/decimals using manipulative and to define like and unlike fractions Description: In activity 1, Students to cut out fractional parts 1, 1/2 , ¾ , 1/6 , 2/5 , 1/10 , 1/12.Now arrange them on your desk from the smallest to greatest to the smallest. In activity 2- Like and unlike fractions Material Required: Poster depicting the graphic. With the help of the questions based on the above graphics teacher explains the concept of like and unlike fractions. Just as we cannot say that tower T2 is taller than tower T7, because they are not at the same level, we cannot compare the fractions of the type 2 5 and 1 4 . The teacher sums up the argument and concludes that for anything to be compared they should be on the same levels. Similarly in order to compare fractions, we need to put them on the same level (same denominator) 30 In activity 2 & 3, students shall write their answers in the given space. Execution: In activity 1, teacher will distribute fraction strips to all the students. Teacher will give instructions to students which will help them to cut out fractional parts 1, ½, 1 6 , 2 5 , ¾, 1 1 10 , 12 and arrange them on their desk from smallest to greatest or greatest to the smallest. In activity 2 & 3, Through modelling and questioning, teacher will extend the knowledge of comparison of like fractions to unlike fractions as well as to decimal numbers. Parameters for Assessment: Students will be able to: • Is able to understand fraction strip and use it creatively • Is able to understand like and unlike fractions • Is able to compare decimals Extra Reading: http://www.funbrain.com/cgi-bin/fob.cgi?A1=s&A2=0 ACTIVITY 15- CONTENT WORKSHEET (C11), ADDITION AND SUBTRACTION OF FRACTIONS Specific Objective: To understand addition and subtraction of fractions Description: In activity 1 – DONINO ACTIVITY students shall arrange cutouts of dominoes so that the matching expressions are equivalent and form a continuous loop. The outcome shall be: 31 Execution n: This iss a guided d worksheet where students will w learn how to use u the manipulatiive and solve the worksheet at th he same tim me. Teacherr will show video clip 12 and video clip 13 to reinfo orce additio on and subttraction of fractions. f Parameteers for Assessmentt: Students will be ablle to: • • • Has knowledge off various ty ypes of fracttions Is able to t add and d subtract frractions Can sollve related problems on o fractionss Extra Reaading: www.help pingwithma ath.com/by y-subject/frractions/fraactions.htm m http://ww ww.mathsneet.net/gcsee/worksheeets.html ACTIV VITY 16- CONTE ENT WO ORKSHE EET (C12 2), ADDITIION AND D SUBT TRACTIO ON OF FRACTIO F ONS Specific Objective O e: To understtand addition and sub btraction of decimals Descriptiion: This worksheet deals withh the addiition and subtraction s n of decimaals and extended to t real life word prob blems. In acctivity 1, sttudents apply the con ncept learn nt using manipulatiive by solviing the worrksheet. 32 Students shall write their answers in the given space. Execution: Teacher will show video clip 14 to the students to reinforce the concept. This is followed by the worksheet which is done by the students independently. Parameters for Assessment: Students will be able to: • • • Has knowledge of decimal Add and subtract decimals Can solve related problems on decimals Extra Reading: http://www.sometests.com/tests/Math/MathsQuizFractionsFor6Class.html ACTIVITY 17- CONTENT WORKSHEET (C13), MULTIPLICATION AND DIVISION AS REPEATED ADDITION AND SUBTRACTION Specific Objective: To extend multiplication and division as repeated addition and subtraction Description: This worksheet helps students to understand the fact that multiplication is repeated addition and similarly division is repeated subtraction. Students shall write their answers in the given space. Execution: Teacher will prepare questions where students will understand the concept and have questions to practice the concept as well. Teacher ensures that students follow all instructions and complete the worksheet. Parameters for Assessment: Students will be able to: • • Understand multiplication as repeated addition Understand division as repeated subtraction 33 ACTIVITY 18- CONTENT WORKSHEET (C14), WORD PROBLEMS INVOLVING FRACTIONS AND DECIMALS Specific Objective: To test understanding of fractions and decimals as a concept and application Description: Students test their understanding and apply skills of addition and subtraction of fractions and decimals to a game The Match Maker. Material needed required - Clue cards Teacher prepares cards for each student of the class. Cards are distributed to the students. Teacher will guide them to find a match for their card. There are two clues one on the front side and one at the back side of the card. This is just played as a game in the class. Students will record their observations on the worksheet. Students get 10 minutes to complete the whole chain of clues. Execution: Teacher will prepare a comprehensive worksheet for the students to test their understanding of fractions and decimals as a concept and application. Parameters for Assessment: Students will be able to: • • Has knowledge of fractions and decimals Can solve related problems on decimals and fractions ACTIVITY 19- CONTENT WORKSHEET (C15), ROUNDING OFF AND ESTIMATION Specific Objective: To explain rounding off and estimation Material Needed: Measuring tape. 34 Description: Students shall discuss rounding off and its purpose and shall view video clip no 15 to understand the process of rounding off. After watching the video students complete the task given in the worksheet. Execution: - Teacher will prepare a set of questions through which the students will understand the need of estimation in real life. Teacher will ask students to pair up and answer the same set of questions for their partners. Teacher will guide students to analyse and discuss the possible reasons for the variations in the two answers. She/he will pose questions to stimulate thinking. Parameters for Assessment: Students will be able to: • • • Understand the need of estimation in real life Can analyse the possible reasons for the variations in the two answers Understand the process of rounding off ACTIVITY 20- CONTENT WORKSHEET (C16), ESTIMATION AND SIGNIFICANCE Specific Objective: To understand significant figures and learn the rules for the significant figures Description: A guided worksheet which has questions explaining the concept of significant figures is prepared and is given to the students. Student answers the question and understand the appropriate significant digit to which the answer must be given. As reinforcement teacher shows Watch video clip 16 and write the rules of giving answers to the correct accuracy. Execution: Teacher ensures that students understand and follow instructions and record all discussions and then test their knowledge by solving questions on significant figures. Parameters for Assessment: Students will be able to: 35 • • Understand the rules for the significant figures. Solving questions on significant figures. ACTIVITY 21- CONTENT WORKSHEET (C17), RATIO AND PROPORTION Specific Objective: To understand and apply the knowledge of ratio and proportion to real life problems Description: Activity 1: Ratio Teacher to start with a mental starter where she/he asks questions related to daily life where student already has an idea about the terminology or may meaning of the word ratio. Teacher introduces ratio as a way of comparing amounts of something. For example: Use 3 parts blue paint to 1 part white This means , mixing paint in the ratio 3 :1 (3 parts blue paint to 1 part white paint) means 3 + 1 = 4 parts in all Teacher relate lesson to fraction by showing 3 parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white paint. Activity 2: Proportion In this activity teacher develops the lesson on proportion and to make students understand that how is ratio different from proportion. Activity3: Its Maggi, maggi maggi. Materials Required: Maggi packets and the home science lab to make the maggi. Teacher instructs students to pick a slip from the bowl kept on the table. The slip contains the number of maggi cakes the student is required to prepare. Teacher asks one student to read the instructions from the packet given to prepare maggi. Its a hands on activity on how proportion and ratio are useful in day to day life situations. Execution: Students follow the directions given by the teacher and do some work independently after the discussions are over. 36 Parameters for Assessment: Students will be able to: • • relate the given problem to ratio and solve them relate the given problem to proportion and solve them Extra Reading: www.helpingwithmath.com/by-subject/fractions/fractions.htm http://www.mathsnet.net/gcse/worksheets.html http://www.math-aids.com/Fractions/ 37 SAMPLE RUBRIC (ASSESSMENT) FRACTIONS AND DECIMALS 0 1 2 3 4 Ability to Can not With guidance can Can demonstrate With 90% Can recognize define follow instructions diagrammatically accuracy can demonstrate and define a fraction and represent the demonstrate understandin fraction and and fractions representation of understandin g of fraction a decimal decimal . diagrammatically a fraction and g of fraction and decimal using correct but fails to connect decimal and has and decimal and notations it to abstract as well clarity about the communicate communicate as extend it to a representation of with with decimal fraction and a appropriate appropriate representation decimal, but is notations notations unable to express with, as well with, as well it independently as without as without without a Manipulative. Manipulative, Parameter manipulative. in various contexts. Ability to Cannot Can Can Can Can represent relate, plot fractions and plot fractions -plot and -Plot and fractions and visualize decimals on a and decimals on a order number line, but number line, decimals and decimals and on a with a lot of independently fractions on fractions, in number guidance. but takes a lot of number line various time to follow with 90% contexts, on a instructions accuracy number line. decimals on a or work number line line. Locate is able extend it to order them and 38 explain the relationship of Decimal to division and fraction. Ability to Can Can classify Can classify Can classify Can classify classify recall fractions. Can fractions, identify fractions, fractions, fractions , definition simplify fractions equivalent identify identify find s of types and complete fractions through equivalent equivalent equivalent of patterns but is patterns only fractions fractions fractions and fractions unable to relate a .Can simplify if through through simplify but decimal to division guided ,can patterns as patterns as fractions. cannot of a fraction. understand link well as frame well as frame Can juggle classify between fraction a rule .Can a rule in between them. and decimal but simplify various lacks accuracy efficiently contexts. Can with understand simplify fractions and link between efficiently in decimals fraction and various decimal contexts. Can accurately, understand and is able to link between extend it to fraction and ordering with decimal guidance accurately, problems and is able to extend it to ordering . Ability to Cannot Knows the theory Can compare and Can order With a high compare and make any but can put the orders fractions fractions and degree of 39 orders sense of concepts into and decimals but decimals. Can effectiveness fractions and the order practice under lot is unable to work juggle can compare decimals , of of supervision. in an integrated between the and manner two and order and integrate fractions decimals learning . Is and decimals . able to and is able to communicate integrate his thoughts. learning fractions relate to various contexts. Ability to Cannot Can make and Can make and Can make Can make Model and work carry out carry out a plan and carry out and carry out perform the with a plan to pose and to pose and solve a plan to pose a plan to pose operations fractions solve problems problems with and and and with decimals and solve solve decimals decimals and fractions, using problems problems in any fractions applying some strategies with with context the skills , using a usually resulting decimals and decimals and limited range of in a partially fractions, fractions appropriate accurate using accurately, strategies; solution appropriate using which strategies of appropriate rarely results in an the skills strategies of accurate solution learnt the skills that usually may offer an resulting in innovative an accurate approach solution 40 Ability to Cannot Can perform Can perform Can perform Can perform extend skill relate through repeated multiplication multiplication multiplication of addition and has addition and and division, as and division and division in and not been subtractions only. repeated in various various subtraction able to additions but contexts with contexts to master makes errors. Has 90% accuracy efficiently and multiplicatio the not mastered the helps peers in n and addition process yet. understanding division of and fractions and subtracti decimals on skills Ability to Cannot Can apply With some With With 100% Apply to real apply knowledge of effectiveness can considerable accuracy can life knowled fraction to ratios apply knowledge effectiveness apply Ability to ge of any and solve real life of fraction to ratio can apply knowledge of strategies concept problems or gives though with a lot knowledge of fraction to and apply the answer to the of guidance fraction to ratio , comes the skill of correct significance contains lots of ratio coming up with ratio and very rarely. errors in up correct alternate proportion to estimation and answers to strategies to problems scientific the specified solve and coming notations degree of problems up with accuracy in posed and is solutions estimation always corrected to and scientific specific to the the specified notations degree of skills. accuracy degree of accuracy. 41 Study Material 41 FRACTIONS AND DECIMALS Introduction Earlier, we discussed counting numbers (also called natural numbers) 1, 2, 3,… and whole numbers 0,1,2,3,…. and operations (+,−, ×, ÷) on these numbers. We also discussed some properties of there numbers. In fact natural numbers help us in the process of quantification of objects. However, there are certain situations which can be quantified by natural numbers as such. For example, when an object is divided or broken into two equal parts, will natural numbers help to tell what one part represents? To deal with such cases, we introduce some other type of numbers called fractions. In this unit we shall discuss fractions and some other related concepts like decimals, ratio and proportion and their use in daily life. What is a fraction? Suppose we are to distribute equally 14 apples between 2 children. Clearly each child will get 2 apples. If we had 2 apples, then each child will get 1 apple. What will each child get it we have only 1 apple? In this situation, we have to cut the apple into two equal parts and give one part to each child. We say that each child is getting one part out of two equal parts. This is symbolically expressed as _ One apple Two equal parts ½ is known as a fraction Thus, we can say that a fraction is a part of a whole In this case, “Whole” is an apple. 42 Further, suppose the same apples to be distributed equally among 4 children. Again in this case, we have to cut the apple into 4 equal parts and each child will get part. This is symbolically expressed as One apple Four equal parts Here again, a whole (an apple) is divided into 4 equal parts and one part represents the fraction ¼ To understand more about fractions, look at the following Figures: In (i) A rectangle (whole) is divided into 2 equal parts. One part is shaded. The shaded part is represented by the fraction . Similarly unshaded part is also represented by 43 (why?) In (ii) A circle (whole) is divided into 4 equal parts here the shaded part is represented by and unshaded by In (iii) Shaded portion is represented by the fraction In (iv) Shaded portion is represented by and unshaded by and unshaded by In all the above cases, whole was a single object such as an apple a rectangle or a circle However, a whole can also be a collection of objects. For example, in case of distributing 4 apples between 2 children, the whole was a collection of 4 apples, and in this case each child got 2 apples. Divide the collection by a line into two equal parts. Obviously, each part is one half (1/2) of the whole collection (4 apples) It is written as Similarly, its 5 totters are distributed equally among 2 children, then each will get and so on. Thus we can also have fractions such as , , , Components of Fractions In the fraction , 1 is called the numerator and 2 is called the denominator of the fraction. 44 , , , etc. totters ator and deenominatorr are separaated by a baar (--------) Note that the numera Similarly, in , 2 is th he numerattor and 3 is the denom minator. and in 2 is the num merator an nd 4 is the denominato d or. is read as a 1 by 2 orr 1 upon 2 or o divided by b 2 Similaarly, is reaad as 2 by 3 or 2 upon n 3 or 2 divided by y 3 ad so on n. • In words, w is read r as onee half • iss read as tw wo thirds • iss read as thrree- fourthss and so on n. Examplee 1: Correesponding to the shaaded portioons write the t fraction n in each of the following if possible: Solution n (i) (as 2 paarts taken frrom 3 equaal parts) 45 1 4 (ii) (iii) (iv) (v) No fraction, as parts are not equal (vi) No fraction, as parts are not equal Example 2: Write fractions having (i) 3 as a numerator and 5 as the denominator (ii) 12 as the denominator and 7 as the numerator (iii) 1 as numerator and 8 as the denominator Solution: (i) 3 5 (ii) 12 7 (iii) 1 8 Example 3: Which of the following fractions which have numerator 2? , , , , , Solution: Fractions and have numerator 2 Example 4: Which of the following fractions have denominator 5? , , , , 46 Solution: Fractions and have numerator 5 Representation of fractions on the number line We have learnt how to represent whole numbers on a number line (fee figure below): We also know that a number to the right of a number is greater then the other. In a similar way, we can represent fractions on a nummular line. • Let us represent the fraction We know that on the number line is a part a whole in which out 2 equal parts, one part is taken So we divide one unit (the distance between 0 and 1) into two equal parts by the point A (see figure below) Point A represents the fraction • Let us now represent the fraction on the number line. is a part a whole is write 2 parts have been taken out of 3 equal parts. So, we divide unit distance, 0 to 1 into three equal parts by point A and B as shown in the figure below: Second point B represents the fraction . Cleary the first point A represents the fraction Let us now represent the fraction . on the number line. 47 Method 1 Divide the unit distance 0 to 1 in 4 equal parts by the points A, B and C (see figure below). Point A represents . Now, we move 5 steps (each equal to OA) towards right of O as shown in the figure and reach at the point D 5 The point D represents the fraction . 4 Method 2 Take 5 units from O i.e. , the distance 0 to 5 on the number line. Divide this distance into 4 equal parts by the points A, B and C as shown below. The point A will represent Example 5: Represent the following fractions on the number line: (i) 3 8 (ii) 5 2 Solution: (i) Divide unit distance 0 to 1 into 8 equal points. find point from 0 will represent 3 8 0 1 48 (ii) Method 1: Divide unit distance 0 to 1 into 2 equal parts. The fast point of division will represent . Move 5 steps to the right each step being . Point B will represent . Method 2: Divide the distance 0 to 5 into two equal parts. The point of division will represent The point B represents Types of fractions. Proper and improper fractions Look at two groups of fraction A and B given below Group A: , , , , Group B: , , , , , , , , , , Compare numerators and denominators of fraction in each group. What do you find? In Group A, the numerator of each fraction is less than the denominator. In Group B, the numerator of each fraction is greater or equal to the denominator. So, on the basis of values of numerators and denominators fractions can the classified as follows: Fractions with numerators smaller than denominators are called proper fractions 49 Fractions with numerator greater or equal to denominators are called improper fractions Thus, fractions in group A are all proper fractions and fractions in group B are all improper fractions Unit fractions Fractions like , , , , , , etc in which numerator is 1, are called unit fractions. , , , , etc not unit fractions (why?) Clearly, Like and Unlike fractions Look at the denominators of the following fractions: , , , , what do you observe? The denominators of all the fractions are the same i.e. 2 such fractions are called like fractions. , Thus, , , Similarly, , , , , , , , are also like fractions. , , are also like fractions. What about fractions ? Clearly, they are not like fractions as their denominators are not same (though their numerators may be same) Fractions which are not like fractions are called unlike fractions. For example, Similarly, , , , , , , , , , are unlike fractions are also unlike fraction because, all of them do not have the same denominator 50 Mixed fraactions (or mixed m num mbers.) gure there is a collecttion of 2 objects o (circcles). Each h object hass been In the folllowing fig divided in nto two equ ual parts. We choose 3 parts We see on ne circle and d circle sh haded Thus and an nd is briefly y expressed d as 1 Numbers of the typee 1 are calleed mixed frractions (orr mixed num mbers). Improper fractiion Some morre exampless of mixed fractions arre: 2 ccombinatio on of a whole n number (1) and a prop per 1 f fraction 2 3 2 11 5 etc. Examplee 6: Which of theses arre proper and a which are a impropeer fractionss? 51 Solution: , , Proper fractions , (numerator is less thus denominator) , , Improper Fractions: , , (numerator is greater thus or equal to denominator) Example 7: Which of the following are unit fractions? 2 1 3 8 1 1 3 , , , , , , 3 4 1 7 10 101 11 Solution Unit fractions are , , , (Numerator is 1) Example 8: Which of the following are mixed fractions? Which are improper fractions? , 3 , 2 , , 11 , , 5 Solution: Mixed fraction: 3 1 , 11 5 ,5 Improper fractions , , , 3 Note: 2 is neither a mixed fraction nor an improper fraction (why?) 2 Example 9: Which the following fractions are like and which are unlike? (i) 5 11 2 16 , , , 3 8 3 8 (ii) 2 2 2 2 , , , 5 11 9 61 (iii) 5 16 19 21 101 , , , , 11 11 11 11 11 (iv) 1 4 16 18 , , , 3 9 9 9 52 Solution (i) Not like fractions. Hence, they are unlike fraction because, denominators are not the same. (ii) They are unlike fractions, though their numerators are the same. (iii) They are unlike fractions (iv) They are unlike fractions. Expressing an improper fraction as a mixed fraction Consider improper fractions say (i) , 4 4 is an improper fractions i.e (ii) contains 4 whole and Similarly, i.e. 5 5 more =5 , i.e. 5 + 28 25 3 , is a mixed fraction Alternatively: And 4 is an improper fraction contains 5 whole and Now 5 more , i.e 4+ 17 16 1 , ,= , 16 ,+ 4 , 5 4 4 5 So, we can always express an improper fraction as a mixed fraction. Expressing a mixed fraction as an improper fraction Consider mixed fractions, say 2 (i) 2 , 7 29 3 3 3 is a mixed fraction and 2 = 2 + = 4 4 4 53 + 3 4 11 (improper fraction ) 4 Here, you can see that 2 So, we can directly write 2 (ii) 3 as 4 7 7 (improper fractions) Here again, 7 In general, mixed fractions: a , + can be easily writen as can be expressed as an improper fraction = Example 10: Express the following improper fractious into mixed fractions: (i) (ii) (iii) (iv) 2 Solutions: (i) 3 =2 5 (ii) =8 (iii) =5 (iv) =1 5 So an improper fraction of the type 4 8 13 10 3 , , 35 32 3 etc cannot be expressed as mixed fractions. Example 11: Express the following mixed fractions into improper fractions: (i) 7 1 4 (ii) 11 (iii) 27 54 9 10 (iv) Solution (i) 7 (ii) 11 (iii) 27 = = = = = Equivalent Fractions Look at the shaded portion of the following figure What do you observe? Shaded portion in (i) represents fraction Shaded portion in (ii) represents fraction Shaded portion in (iii) represents fraction Shaded portion in (iv) represents fraction 55 Note that the shaded portion of each figure represents the same equal part of the given rectangle so, we can say that correcting fractions , , , and are equal. Such fractions are called equivalent fractions. If you represent these fractions namely , , , and on the number line, you will find that all these fractions are represented by the same point. Check it!! Using the above process of shading suitable number of parts of congruent rectangles (triangles of same shape and size) to represent , , , , you can see that the fractions , , , , are also equivalent fractions Obtaining equivalent fraction We have seen that , , , , Are equivalent fractions Let us consider , and We see that Thus has been obtained from by multiplying its numerator and denominator by the same number (here it is 2) Similarly = Here also, equivalent fraction denominator of the fraction has been obtained by multiplying the numerator by the same number 3. Multiplying the numerator and denominator by 4 Thus, when the numerator and the denominator of a fraction are both multiplied by the same number (other than0), we obtain an equivalent fraction 56 Again consider and See that similarly, Thus, when the numerator and the denominator of a fraction are both divided by the same number (other them 0), we obtain an equivalent fraction. To check whether two fractions are equivalent are equivalent or not. We see that and are equivalent fractions i.e. We thus that 1 ( numerator of first ) x 4 (Numerator of second ) = 4 Also, 2(denominator of the first) x 2(numerator of the second)=4 thus 1 4 2 2 Similarly for equivalent fractions , And 3 8 6 ,2 4 6 and 24 4 3 12 and so an 57 Thus in a pair of equivalent fraction Numerator of the first x denominator of second = Denominator of first x numerator of second Example 12: Are the following two fraction equivalent? , (i) , (ii) (iii) , (iv) , Solutions: (i) since 1 5 2 So, (iii) 15 3 = 15. and So, (ii) 15 8 are equivalent 16, 7 3 21 16 21, and are not equivalent 25 40 20 5 1000 100 Since 25 40 20 5 Therefore and are not equivalent 58 (iv) 10 22=220 11 20=220 since 10 Therefore, Example 13: 22 11 20, and are equivalent fill in the blanks: (i) (ii) (iii) Solution: 135 (i) 120 (ii) 90 (iii) 45 therefore 2 4 5 45 30 therefore, 90 18 90 30 , 216 3 18 18 Example 14: find the equivalent fraction of whose (i) Denominator is 12 (ii) Numerator is 21. Solution : (i) Since denominator is 12 and 12 = 4 x 3, therefore required equivalent fraction = (ii) Since denominator is 21 and 21=3 7, therefore required equivalent fraction 59 = Fractions in the simplest form Look at the following fractions: 4 12 2 20 6 , , , , 6 18 3 30 9 Cheek that all these are equivalent fractions in the fraction , 2 and 3 are co prime that is, there is no common factor between 2 and 3 other than 1. Such a fraction is called a fraction is the simplest or lowest form or lowest terms. is not is the simplest form as 4 and 6 have a common factor 2 other than 1 Similarly is also not is the simplest form. Further is again a fraction is the simplest tent because, 9 and 16 have no common factor other then 1. Reducing a fraction in the simplest form Consider the fraction Since 12 and 18 have common factors 3 , 2 and 6 besides 1, so, is not in simplest form. To reduce it into its simplest form, we divide both numerator and denominator by their HCF. Here H C F (12, 18) =6 So, Similarly simplest form is also not is the simplest form as 36 and 162 have factors 2, 3, 6, 9, 18 (other than 1) So Note that H C F 36 ,162 Which is now is the simplest form. 60 18 Example 15: Reduce the following fraction is their simplest form: (ii) (i) (iii) Solution: (i) 20, 48 22,25 (ii) 1 68,86 (iii) 4 2 Example 16 : Which of the following fraction are in the simplest form? (i) (ii) (iii) (iv) Solution: (i) is in the simplest form as HCF (16, 39) =1 (ii) is not is the simplest form as HCF 218,242 (iii) is also not in the simplest form (why?) (iv) is in the simplest form (why?) 61 2 Comparison of fractions Comparing like fractions Look at the shaded portions of each of the figure given below: Obviously, the shaded portion which represents represents is larger than that of portion which . So, Similarly, it can the seen that Thus For two like fractions, the fraction having greater numerator is greater than the other. Comparing unlike fractions For comparing unlike fractions, we first convert them into like fractions and then compare them by comparing their numerators. For example to compare fractions Note that 30 is the L C M of the denominators 10 and 15 62 and As 16 Since 9 So, Another Method of comparing two fractions. , Consider: 3 15 45 10 8 80 And 80 10 45 80 So, 10 8. 8 , in the same way, for fraction 5 9 And 45 , 45 , 6 4=24 24 , , Example 17: Find the larger (greater) fraction in each pair (i) , (ii) ; , (iii) (iv) , Solution (i) Because denominator is the same and numerator of numerator of . (ii) (iii) 35 is LCM of 7 and 5 63 is greater than So, , Alternatively: since 21 4 5 20, 7 3 21 20, So, (iii) 11 15 165 18 13 234 As 234 165, Therefore 17 (iv) 49 833 As 945 > 833, Therefore, Example 18: Compare the fraction 4 17 4 25 Solution 4 25 100 17 4=68 Since 100 > 68, therefore, Note that amongst the fractions that have the same numerator; the fraction with smaller denominator is greater. Example 19: Arrange , , , is ascending order. 64 Solution: These are unlike fractions (Why) They can be expressed as like fractions as following. 1 LCM = 2 x 3 = 6 1 3 1 3 2 2 2 6 1 2 1 3 2 2 2 6 1 0 1 6 Now So given fractions in the ascending order are , , Note: In this case, numerator is the same So, we can compare the fractions by comparing their denominators. Example 20: Arrange the following is descending order: 4 3 5 11 , , , 7 8 14 18 65 LCM = 2 2 2 3 3 7 =504 Solution: = hence, the given fractions in descending order are , , , Addition of fractions Adding like fractions The sum of two or more like fraction = For example, = 17 Similarly 66 Adding unlike fraction To add two or more unlike fractions convert them in to like fractions and then add For example, = = (45 is the lcm of 9 and 15) 25 6 45 31 45 Example 21: Find the following sums: (i) (ii) (iii) 1 (iv) 5 3 Solution (i) 1 Unlike fractions are converted into equivalent like fractions and then addition is carried out 67 (i) = 1 = (ii) 1 (iii) 5 3 4 (LCM of 7,9,14) Subtraction of fractions o Subtracting like fractions The difference between two like fractions = For example, 68 Subtracting Unlike Fractions To find the difference between two unlike fractions, convert them to like fractions and the subtract. For example, 6 = 4 12 Example 22: Find (i) (ii) 4 2 21 3 2 7 (iii) Solutions: (i) (ii) 4 3 (iii) Unlike fraction are converted into equivalent like fractions and then subtraction is carried out 69 Example 23: Simplify 3 (i) (ii) 3 (iii) 1 1 9 20 4 37 15 1 2 3 1 Solutions : 3 (i) (ii) 1 3 3 4 = 6 Alternatively, we can also simplify in the following way: Step 1: Calculating two whole number part It is : 3 1 4 3 4 1 6 Step 2: Calculating two fractional part It is: Combing the two parts we get 3 (iii) 1 1 2 3 1 1 5 4 1 2 6 29 30 29 30 1 = (1) 6 1 Multiplication of a fraction by whole number. Observe the following figures: 70 If Represents 1 Then Represents And Represents . Please make equal divisions It also represent two times = x2 2 Thus, 3 Similarly, 4 4 2 3 4 2 3 8 3 And so on We may say that : The product of a fraction and a whole number is also a fraction whose numerator is the product of the numerator of the given fraction and whole number and the denominator is the same as that of the given fraction. 71 Fraction of a quantity Look at collection given below It has 6 balls. The dotted lines have divided the collection of 6 balls into three equal parts and each small collection is one third of the whole collection. There are 2 balls in each small collection. It means that one third of 6 balls is 2 balls, i.e, 6 6 Similarly and 2 6 6 4 or 6 2 or, 6 4 or 6 6 2 5 50 2 5 In the same way, 50 = =Rs 20 72 2 5 50 Similarly, 1 3 and 147 540 1 3 540 540 3 180 147 = km = 84 km Thus, When we are to find a fraction of a quantity we multiply the fraction with the given quantity, In other words, the word ‘of ‘means ‘multiply’ Division of a fraction by a whole number Let us find 4 number, Fraction Observe the following figure: In the figure (i) shaded portion represents . To divided by 4, we divide the shaded portion into 4 equal parts by drawing two diameters as shown in figure (i). Each of these 4 parts represents 1/2 (ii) represents 4 and also each part in figure . 73 4 So, = Similarly, it can be seen that 6 2 And and so on. Thus A fraction whole number (other then 0) = Example 24 : 10 (i) 20 (ii) (iii) 4 2 12. (i) (ii) 13 (iii) 0 17 26 Solution (i) 10 (ii) 12 8 2 74 20 17 26 (iii) (iv) 13 13 17 26 221 26 17 2 13 1 8 2 We can also do as: = 13 (v) 4 (vi) 0 2 1 4 9 9 4 8 4 9 4 1 = 9 0 Example 25: Calculate (i) (ii) 50 280 (ii) 242 132 (vi) Solution (i) 50 50 (ii) 280 (iii) 242 33 280 5 40 200 4 x 22 1 242 88 (iv) 132 Example 26 : Find: (i) (iii) 14 1 6 132 5 1 3 1 x 22 1 18 99 75 22 Solutions : (i) 5 (ii) 18 (iii) 14 (iv) 99 Simplification using BODMAS You know bow to simplify expressions involving whole numbers using BODMAS. This is now extended in simplifying fractions including whole numbers. Example 27 : simplify 2 (i) (iii) 1 2 (ii) 2 4 7 6 18 3 7 4 35 Solution: 2 (i) 6 = = = 4 76 = 4 = = (i) 2 48 12 6 1 = - = = = 3 Another way to simplify: 2 = 2 13 1 1 5 7 4 7 2 21 = = = = = 3 77 (ii) 2 4 4 3 7 18 = 4 35 18 = = 44 3 7 4 35 = 44 15 35 4 35 = 44 11 35 = 44 15 4 35 44 11 35 44 44 44 Example 28 : simplify (i) 3 1 2 (ii) 9 7 15 2 3 16 71 2 3 2 3 16 2 1 2 3 x9 27 4x2 Solution:(i) 3 1 2 2 = 16 1 2 x9 3 x9 = = 15 = 15 = 15 78 = = (ii) 9 7 35 18 2 3 71 = 71 = 71 = + = x2 = x2 = 27 4x2 27 18 4x2 4x2 x2 9 9 Word Problems on fractions Example 29: Amit travelled a distance of 13 km in the morning on his bike and a distance of 8 km in the evening find the distance travelled by him altogether? Solution: Total distance travelled =13 km+8 km = 13 8 km km = = km km 79 = km =22 14 km Note: we can also proceed as 13 1 2 8 = 13 8 =21 5 4 =21 1 =22 3 4 1 2 13 1 2 3 4 3 4 8 2 4 21 3 4 1 4 1 4 1 Rita buys a ribbon of length 5 , meters. Sheela buys a ribbon of length 3 1 8 meters. Who has the lager ribbon and by how much? What is the total length of 7 ribbons? Example 30: Solution:Clearly the length 8 1 7 5 1 3 ? So, Mena’s ribbon is lager than Rita’s Now we find: 8 1 7 5 = = = = So, Mena’s ribbon is longer by of2 17 meters 2 80 5 Total longer of two ribbons = = = meters = meters = meters =13 You can also find 8 1 7 5 1 3 10 21 1 7 by writing as 8 1 3 5 13 10 21 Example 31: 3 1 m of cloth is needed for preparing a shirt. How much cloth will be required to prepare 4 6 such 6 shirts ? 3 Solution: Cloth for one shirt = 1 4 m Cloth for 6 shirts = = 7 m 4 x6 m m = m 10 m 3 A box contains 1 kg. of sweats. It is equally distributed among 40 5 children of a class. How many sweets will each child get? Example 32: 3 Solution: Weight of sweets in the box = 1 5kg = kg No. of children =40 So, amount of sweets received by each child= 81 40 . = 1 25 kg Example 33: Geoff spent of his salary on food, 16 on children’s education and other items. If his monthly salary is $5000, what is his monthly savings? Solution: Monthly Salary of Geoff = $ 5000 $ 5000 Expenditure on food = 5000 =$ =$ = $ 2000 $ 5000 Expenditure on children’s education = 5000 =$ = $ 500 Expenditure on other items $ 5000 = =$ 1 8 5000 = $ 625 Total expenditure = $ 2000 $ 500 $ 625 Total monthly saving of Geoff = $ 5000 $ 3125 $ 3125 Alternative method Total expenditure (in fraction) = 82 $ 1875 on = = = Savings = 1 5 8 8 8 Thus monthly savings 2 8 5 8 = 8 5 8 3 8 $ 5000 5000 =$ =$ =$ 1875 Ratio And Proportion Ratio A car costs Rs 200000 and a motorbike costs Rs 50000. We may say that price of car is Rs 200000 - Rs 50000 i.e. Rs 150000 more than that of motorbike. This is called comparison by subtraction. Another way is that the car’s price is times that of the motorbike i.e. 4 times of the price of motorbike. This way of comparison is known as comparison by division and is also known as a ratio. Thus the ratio of price of car and price of motorbike = It is also written as 4:1 and read as 4 to 1 or 4 ratio 1. the sign “:” is read as “is to” Example 34: In a class there are 30 girls and 24 boys. What is the ratio of number of girls to that of the boys? And what is the ratio of number of boys to that girls ? Solution: Number of girls = 30 Number of boys = 24 So, the ratio of number of girls to number of boys 83 =30: 24 (i) Next, the ratio of number of boys to the number of girls = 24: 30 (ii) You can see that 30: 24 is different from 24:30 Thus, ratio a : b different from b : a, if a 0. Ratio 30: 24, can also be written as . It means 30: 24 is the same as 5:4 In ratio 30:24, 30 is called the first term of antecedent of the ratio and 24 is called the second term of consequent of the ratio. Example 35: (i) Length of a room is 15 m and its breadth is 10 m. What is the ratio of length to breadth of the room? (ii) Cost of a toffee is 75 cents and cost of a chocolate is $ 3 find the ratio of their prices. (iii) Length of a cylinder is 25 cm and its diameter is 30 mm what is the ratio of the length and diameter of the cylinder? (iv) Weight of a box is 20 kg and its length is 60 cm. What is the ratio of the weight of the box to the length of the box? Solution: (i) Ratio of length to breadth = 15 : 10 or 5 :2 (ii) Can we say that the ratio of the cost of toffee to the cost of chocolate is 75 : 3 i.e. 25:1 ? Which means cost of toffee is more then is more cost of chocolate. But is it so? No! Here cost of toffee is in cents and that of chocolate in $. 84 To get true comparison, of two costs, they must be in the same unit either in cents or dollars. It the cost is in cents, then cost of chocolate is 300 cents and required ratio = 75 : 300 3 = 12 In the cost are in dollars, then cost of toffee in Rs :3 Then required ratio = x 3 12 Two quantities can be companied only if they are in the same units. (iii) Length of cylinder 25 Diameter 30 So, required ratio = 250:30 = 25:3 (iv) Weight of box in kg = 20 Height in cm 250 = 60 Note that the two quantities ‘kg’ and ‘cm’ are not in the same unit. Can you convert kg into cm or cm to kg? It is not possible!! So, in this case, we can not find the ratio. 85 Ratio and fraction You have a aleady seen that a ratio can be written as a fraction and a fraction can also be written as a ratio. For example, in the above case, the ratio 5:2 = etc. Similarly, 2 3, 4 As ,75:300 = 9 etc. While discussing fractions, you have seen that fractions like Equivalent Fractions Equivalent Ratios 25:1 = , , are In the same very, we can say that the ratio 2:3, 4:6, 6:9 etc are is the simplest form of the fraction , etc in the same way, 2:3 is the simplest form of the ratio 4:6, 6:9 etc Example 36: Write the fraction Solution : 24: 36 in the form of a ratio in the simplest form. 2: 3 Example 37: Express the ratio 15:35 in the form of a fraction in the simplest form Solution: 15:35 = • Proportion Consider the following situation: Cost of 1 liter of petrol =Rs. 60 Cost of 1 liter of petrol will be Rs 60 5 300 Cost of 12 liters of petrol will be Rs 60 x 12 = Rs 720 What is the ratio of quantity of petrol? It is 5: 12 or What in the ratio of their costs? It is 300: 720 = Thus, we obese that: 86 The ratio of quantities of petrol (5:12) = The ratio of their costs (300: 720) i.e 5:12=300:720 Such an equality of two ratios is called proportion. Sometimes, the symbol “::” is used to denote the equality of two ratio. Thus, for the present situation, we may write 5:12:: 300:720 Here four numbers are involved which are called the respective terms of the proportion. Extreme terms or Extremes Middle terms or Means Further note that in the proportion 5: 12 : : 300 : 720 Product of first and fourth terms = 5 Product of second and third terms 720 12 3600 300 3600 So, product of first and fourth terms = Product of second and third terms. Thus, Four numbers a, b c and d are said to form a proportion or are in proportion if product of extremes = product of means (middle terms) i.e. a x d = b x c 87 Example 38 : Find whither the numbers are in proportion (i) (ii) (iii) 3, 6, 21, 42, 2,5,10,20 7,9,56,81 Solution (i) Product of extremes terms = 3 x 42 =126 Product of middle terms = 6 x 21 = 126 Since product of extreme terms = product of middle terms therefore, the numbers 3, 6, 21, 42 are in proportion. (ii) Product of extreme terms = 2 x 20 = 40 Product of middle terms = 5 x 10 = 50 Since 40 50, so the number 2, 5, 10, 20are not in proportion. (iii) Product of extreme terms = 7 x 81 = 567 Product of middle terms = 9 x 56 = 506 Since 567 506, So, the numbers 7, 9, 56, 81 are not is proportion. Example 39: Determine if the following ratios form a proportion (i) 440m:2km and 55cm:3m (ii) 250 ml:100 ml and Rs.75 : Rs. 30 Solution (i) 440 : 2 km = 440 m : 2 x 1000 m 11 = 50 and 55 cm : 3 m = 55 cm : 3 x 100 cm =55 = 3 55 3 100 11: 60 Since 11:50 11 : 60, therefore, the given ratios do not form a proportion. 88 (i) 250 ml : 100 ml = Rs 75: Rs 30 = 5: 2 5 2 Here, the ratio 250 ml: 100 ml and Rs 75: Rs 30 form a proportion. Note. Unit used in the first ratio is ml while unit used in second ratio is Rs. So, we may have different unit is different ratios but we can not have different unit in the same ratio. Example 40: A train covers a distance of 200 km speed of 50 km/ hr in 4 hours and covers the same distance at 40 km /hr. in 5 hours. Do the number 50, 40, 4, 5 make a proportion? Solution: The numbers will be in proportion if 50 x 5 = 40 x 4 or 250 = 160, which is not true So, the numbers 50, 40, 4, 5 are not is a proportion. Example 41: If two places are 8 cm apart on a map with a scale of 1: 40000. What is the actual distance between them? Solution: 1 cm on the map represents 4000 cm in reality So, 8 cm in the map represents 8 2 cm on the map represents 5 = 40000 cm in reality = 4000 = = 336000 = = 3 89 Example 42: A bullock cart travels 36 km in 4 hours and a car travels 120 km in 2 hours. Using the relation , find the ratio of their speeds. speed = Solution: Speed of bullock cart = Speed of car = 60 9 / / So, the ratio of speed of bullock cart and car = 9 :60 = 3 :20 Example 43: In example 42 find the ratio of distance travelled by the bullock cart and car and also the ratio or the time taken by them. Are the two ratios forming a proportion? Solution: Ratio of distance travelled by the bullock cart and car = 36:120 = 3:10 Ratio of time taken by them (in hrs.) by them =4:2 = 2:1 Since 3:10 2:1, So, the two ratio do not form a proportion Example 44: Nelson buys 6 note books for Rs. 90 and Jacob buys 25 notebooks for Rs 375. Do the ratio of number of note looks and their costs form a proportion? Solution: Ratio of the number of note books = 6: 25 = Ratio of cost of note books = 90: 375 = So, the two ratios are in proportion. 90 Example 45: To make 12 buns, Sarita needs: (i) 240g flour (ii) 60g margarine (iii) 24g sugar (iv) 75 ml milk (v) 12g salt To make 18 buns, Nitu needs (i) 360g flour (ii) 90g margarine (iii) 30g sugar (iv) 100 ml milk (v) 16g salt Find the ratio of: (a) Number of buns made by Sarita and Nitu (b) Quantity of flour used by both (c) Quantity of margarine used by both (d) Quantity of sugar used by both (e) Quantity of milk used by both (f) Quantity of salt used by both Which two ratios form a proportion? Solution: (a) 12:18 = 2:3 (b) 240:360 = 2:3 91 (c) 60:90 = 2:3 (d) 24:30 = 4:5 (e) 75:100 = 3:4 (f) 12:16 = 3:4 We find that (a) and (b), (b) and (c), (a) and (c) form a proportion. Also, (e) and (f) form a proportion. But (a) and (d), (b) and (d), (c) and (d), (d) and (e), (d) and (f) do not form a proportion. Example 46: Check whether the numbers 3, 9, 9, 27 in proportion? Solution: Product of extremes = 3 × 27 = 81 Product of means = 9 x 9 = 81 Clearly 81 = 81 Hence, the numbers 3, 9, 9, 27 are in proportion. Note that in the proportion, 9 is repeated at: 2nd and 3rd places or 2nd and 3rd terms are the same. 3, 9, 9, 27 9 is repeated In such cases, we say that 3, 9, 27 are in continued proportion or simply that 3, 9, 27 are in proportion. In such cases, it is understood that middle terms are the same. Thus, by saying that a, b, c are in continued proportion, we mean a:b = b:c or or b2=ac 92 DECIMALS Concept of a Decimal Consider the fractions: , , , , , , , , , , Out of these fractions, the fractions , , , , Are such whose denominators are 10, 100, 1000 etc. Such fractions whose denominators are 10, 100, 1000,........ are called decimal fractions. Some more examples of decimal fractions are , , , , etc. Recall that means 7 parts taken out of 10 equal parts of a whole. We may call each part as one tenth So, means 7 tenths and 2 . or 2 93 7 10 2 and 7 tenths represents 25 parts out of 100 equal parts of a whole (See fig. above) We may call each part as one hundredth 1 Now Similarly, 1 and 19 hundredths represents 21 parts out of 1000 equal parts of a whole We may call each part as one thousand th So, . means 21 thousandths. We write the fraction as 0.7 (read as zero point seven) as 2.7 (read as two point seven) as 0.25 (read as zero point two five or decimal two five) as 1.19 (read as one point one nine) and so on... The point (.) in each of the numbers 0.7, 2.7, 0.25, 1.19 etc. is called a decimal point. 94 And the number itself is called a decimal number or simply a decimal. Like whole numbers we can also write decimal numbers on a place value chart as given below. Representing a decimal on a place value chart as given below Decimal Decimal Thousands Hundreds Tens Ones Decimal Tenths Number (x1000) (x100) (x10) (x1) • x 2351.84 219.135 17.09 0.7 2.7 1.19 0.124 2 3 2 5 1 1 1 9 7 0 2 1 0 • • • • • • • Hundred x 8 1 0 7 7 1 1 4 3 9 9 2 In the above table: Value at the hundreds (100) place is 1 i.e. 100 1000 10 Similarly value at the tens (10) place is 1 i.e. 10 x 1000 10 Proceeding in this way, Value at the ones (1) place is 1 i.e. 1 x 10 10 Value at the tenths i.e. of the value at hundred (100) place of the value at tens (10) place of the value at ones (1) place 1 Value at hundredths i.e. place is of the value at thousand (1000) place place is of the value at tenths and so on. 95 place Thousand 1 1000 5 4 Thus, the value at each place is of the value of place just left to it as was in the case of place value chart for whole numbers. Expanded from of a decimal number Look at the face value and place value of each digit in the number 2351.84 give in the table above Face value 2 3 5 1 8 Place Place value 2 Thousands Hundreds Tens Ones Tenths 1 10 8 4 hundredths 4x 1000 3 100 5 10 1 1 2000 300 50 1 8 or 0.8 10 1 100 4 or 0.04 100 Adding all the place values, we get 2000 + 300 + 50 + 1 + 0.8 + 0.04 = 2351.84 Left hand side is called the expanded form of the decimal number 2351.84 on the right. In the decimal number 2351.84, 2351 is called the integral part or whole number part. and 84 is called the decimal part. Two parts are separated by a dot (.) called decimal point. Similarly expanded form of 219.135 is 200 + 10 + 9 + 0.1 + 0.03 +0.005 and expanded form of the decimal number 2059.101 0 is 2 × 1000 + 0 × 100 + 5 × 10 + 9 × 1 + 1 × = 2000 + 50 + 9 + 0.1 + 0.001 96 1 Example 47: Write the following numbers in the place value chart. (i) 20.3 (ii) 3.4 (iii) 89.1 Solution, Tens Ones . Tenths 2 0 . 3 (i) 20.3 (ii) 3.4 3 . 4 (iii) 89.1 8 9 . 1 Example 48: Write the following as decimals (i) (ii) Two ones and four tenths Twenty and six tenths Solution: (i) (ii) Two ones and four tenths = 2.4 Twenty and six tenths = 20.6 Example 49: Write as decimals 3 8 6 23 34 , , , , 10 10 10 100 1000 Solution: 0.3 0.8 0.6 0.23 0.034 97 Example 50: Write as fractions 0.2, 0.5, 0.9, 0.67 Solution: 0.2 0.5 0.9 0.67 Example 51: Read and write the number name, first one is done for you. (i) (ii) (iii) (iv) (v) 7.5 9.4 18.2 225.71 1087.31 Solution: (i) seven and five tenths (ii) Nine and 4 tenths (iii) Eighteen and two tenths (iv) Two hundred twenty five and seventy one hundreds (iii) One thousand eighty seven and thirty one hundreds Example 52: Write in decimal form (i) Eight tenths (ii) Eighty two and three tenths (iii) Two hundred twenty seven and seven tenths (iv) One hundred and twenty two hundredths 98 Solution: (i) 0.8 (ii) 82.3 (iii) 227.7 (iv) 100.22 Example 53: Write the following decimal numbers in expanded form. (i) 25.197 (ii) 0.401 Solution: 9 (i) 25.197 = 2 × 10 + 5 × 1 + 1 × (ii) = 20 + 5 + 0.1 + 0.09 + 0.007 1 1 0.401 = 4 0 1 10 100 = 0.4 + 0.001 7 1 1000 Example 54: Write the decimal number of each of the following expansion: (i) 3 1000 (ii) 1 10 2 7 100 1 0 5 1 10 10 1 1 1 1 8 10 1 1 5 100 1000 5 1 100 Solution: (i) Required decimal number = 3000 + 200 + 50 + 1 + 0.8 + 0.05 = 3251.85 (ii) Required decimal number = 10 + 7 + 0 + 0.01 + 0.005 = 17.015 Representing decimals on the number line We already discussed representation of whole numbers and fractions on a number line. Recall that for representing a fraction say on a number line we divided the unit distance 0 to 1 into 5 equal parts and the second point of division represented the fraction . As decimals are also fractions (with denominators 10, 100, 1000 etc.), we can also represent decimals on the number line. For example to represent a decimal say 0.4 on the number line, divide unit 0 to 1 into 10 equal parts and the 4th point of division represents 99 i.e. 0.4 as shown below. To represeent say 1.2 on the num mber line, divide d each h unit into 10 1 equal paarts and maark the twelfth po oint of divission startin ng from 0. This T point will w represen nt i.e. 1.2. vely, since 1.2 1 = 1 + 0.2, 0 therefore, divide the t unit disstance betw ween 1 and 2 into Alternativ ten equal parts and mark m the seecond pointt of division n. This pointt will represent 1 + 0.22 i.e. 1.2 on the number line. Fraction ns as Decim mals We have already a seeen how a frraction with h denominaator 10, 1000, 1000 etc. can be con nverted to decimaal. Let us con nvert the fra action say into decim mal. Thu us, Sim milarly, and d an nd so on 100 Decimals as Fractions We can also convert decimals to fractions. For example, 1.2 1 2 10 10 10 2 10 12 10 6 5 We can directly write 1.2 (Remove the decimal point. In the denominator place one zero after 1 as the number of digits after the decimal point in 1.2 is one) = Similarly 6.35 6 Here, also we can directly write 6.35 + Here also we can directly write (Remove the decimal point. In the denominator place one zero after 1 as the number of digits after the decimal point in 1.2 is one) 6.35 = . In the same way . 11.309 = Example 55: Convert the following fractions to decimals (i) (ii) 101 Solution: 0.6 (i) 0.95 (ii) Note: To convert a fraction into a decimal, convert the denominator as 10, 100, 1000 etc. Example 56: Convert the following decimals to fractions (i) 9.25 (ii) 628.375 Solution (i) 9.25 (ii) 628.375 . Comparing Decimal Numbers Like whole numbers, we can also compare decimal numbers using the place value concept. For example to compare 0.05 and 0.1, we first look at the whole number parts of the two numbers and compare them as whole number. In 0.05, whole number part = 0 In 0.1, whole number part = 0 So, we then compare, the tenths digit of the two numbers. In 0.05, the tenths digit is 0 In 0.1 the tenths digit is 1 As 1 > 0, so, 0.1 > 0.05 Consider one more example; let us compare 12.959 and 12.961 In these two numbers, again compare whole number parts. Here is each case it is 12. So, we compare tenths digit of the two numbers. Again, they are the same each being 9. 102 Then, we compare hundredths digit in the two numbers. In 12.959, hundredths digit is 5 In 12.961, hundredths digit is 6 Since 6 > 5, so 12.961 > 12.959 In the same way, it the hundredths digits are the same, then compare the thousandths digits and so on. Example 57: Compare the numbers 0.4 and 0.09. Solution: Step 1: Compare whole number parts. They are equal as each in 0. Step 2: Compare tenths digit of both the number. In 0.4, tenths digit is 4 In 0.09, tenths digit is 0 Since 0 < 4, therefore, 0.09 < 0.4 Example 58: Compare 18.191 and 9.999 Solution: Step 1: Compare whole number parts In 18.191, it is 18 In 9.999, it is 9 As 18 > 9, therefore 18.191 > 9.999 Example 59: Compare the numbers 4.8307 and 4.8316. Solution: Step 1: Compare whole number parts. It is 4 in each case. Step 2: Compare the tenths digits. It is 8 in each case. 103 Step 3: Compare the hundredths digits. It is 3 in each case. Step 4: Compare the thousandths digits. In 4.8307, it is 0 In 4.8316, it is 1 As 0 < 1, so 4.8307 < 4.8316 Example 60: Arrange in ascending order. 6.78, 7.77, 9.43, 3.78, 7.79 Solution: Compare who number parts 3<6<7<9 So, 3.78 < 6.78 < 7.77 or 7.79 < 9.43 Now let us compare 7.77 and 7.79 Here again, whole number parts are same, tenths digit of both numbers are also same, so, we compare the hundredths digits. In 7.79, hundredths digit is 9 In 7.77, hundredths digit is 7 Since, 9 > 7, so 7.79 > 7.77 i.e. 7.77 < 7.79 Hence, required ascending order is 3.78, 6.78, 7.77, 7.79, 9.43 104 Operations on Decimals We have already learnt about decimals as fraction with denominators 10, 100, 1000 and so on. We have also represented decimals using a place value chart using this place value chart. We can perform different basic operation on decimals as we do for whole numbers. We shall explain it through examples. Addition: Example 61: Find the following sum (i) 25.38 + 31.51 (ii) 31.47 + 5.21 Solution: (i) We write the decimals as follows: 25 . 38 (Decimal points in the same column, + 31 . 51 tenth digits in the same column, etc.) Sum 56 . 89 (ii) We write the decimals as follows: 31.47 (Decimal points in the same column, + 05.21 tenth digits in the same column, Sum 36.68 there is no tens digit in the number 5.21) Here, the two decimals have been added without carrying over as it is done in whole numbers. Decimal can be added with carrying over in the same way as in the case of whole numbers. Example 62: Find the following sum: (i) 47.38 + 35.24 (ii) 659.45 + 42.35 Solution: (i) We again write the decimals such that decimal points are in the same column and digits of different places (tens, units, tenths, etc.) are in the same column as follows. 105 1 1 ← Carry over 47.38 +35.24 Sum 8 2 . 6 2 (ii) We write the decimals as follows: 11 659. +0 4 2 . Sum 7 0 1 . 1 ← Carry over 45 (There is no digit in the hundreds 35 place in the second number.) 80 Example 63: Find the following sum. (i) 27.824 + 261.16 (ii) 733.5 + 24.369 Solution: (i) In the first decimal, there are three digits 8, 2 and 4 after the decimal point and two digits 1 and 6 after the decimal point. To make the number of digits equal in both the decimal, we put a `0’ in the end in the second decimal. Now, we write these decimals as follows: 27.824 +261.160 ← (0 added) Sum 288.984 (ii) Here also, we add two `0’ in the first decimal to make the number of digits in the two decimals after the decimal point the same. Now, we write as follows: 733.500 ← (Two `0’ have been added) + 24.369 Sum 757.869 Example 64: Find the following sum. (i) 246.59 + 29.347 (ii) 578.348 + 28.7 106 Solution: (i) We write as follows: 1 1 ← Carry over 2 4 6 . 5 9 0 ← (One `0’ as added) + 29.347 Sum 2 7 5 . 9 3 7 (ii) We write as follows: 1 1 1 ← Carry 578.348 + 2 8 . 7 0 0 ← (Two `0’ are added) Sum 6 0 7 . 0 4 8 We can add three or more decimals in the same manner. Example 65: Find 22.17 + 242.2 + 1.289 + 87.4315 Solution: We write the decimals as follows: 111 11 022.1700 +242.2000 +001.2890 +087.4315 Sum 353.0905 ← Carry ← (Two `0’ have been added) ← (Three `0’ have been added) ← (One `0’ has been added) Subtraction: Example 66: Find the following difference: (i) (ii) 56.89 – 25.38 36.68 – 5.21 Solution: (i) We write the decimals as follows: 56.89 (Decimal points are in the same column) – 25.38 Difference→31.51 (ii) We write the decimals as follows: 36.68 – 05.21 Difference→31.47 107 Comparing (i) and (ii) of example 61 with (i) and (ii) of example 66, can you see that subtraction is the inverse operation of addition in the case of decimals also? In the above example, we were able to subtract one decimal from the other without any borrowing. But due to the place value chart, we can perform the subtraction with borrowing, if necessary, as shown in the following examples. Example: Find the following difference. (i) (ii) 93.73 – 53.34 601.85 – 45.36 Solution: (i) We write as follows: 6 9 3 . 7 13 –53.3 4 ← Borrow (1 has been borrowed from 7 to make it 6 and 3 as 13) (Note: 1 tenth = 10 hundredths) Difference→ 4 0 . 3 9 (ii) We write as follows: 5 9 7 ← Borrow 1 1 6 0 1 . 8 5 (1 hundred = 10 tens, 1 ten = 10 units, – 0 4 5 . 3 6 1 tenth = 10 hundredths) Difference→ 5 5 6 . 4 9 Example 67: Find the following difference: (i) (ii) 388.96 – 29.824 659.359 – 9.78 Solution: (i) There are two digits after the decimal point in the first decimal and there are three digits in the decimal part of the second decimal. So, we add one 0 to the first decimal and write as follows: (i) 7 5 ← Borrow 1 1 (One 0 added) 3 8 8.9 6 0 –0 2 9.8 2 4 (One hundredth = 10 thousandth, one ten = 10 units) Difference→ 3 5 9 . 1 3 6 108 (ii) 4 18 12 ← Borrow 1 659.3 5 9 –009.7 8 0 (One 0 added) Difference→ 6 4 9 . 5 7 9 Mixed Operations If we have an expression involving both the operations addition and subtraction, we combine the decimals involving + sign and – sign separately and then simplify. Example 68: Simplify: (i) (ii) 24.327 – 4.003 – 12.389 + 112.309 228.37 – 29.516 + 12.059 – 1.09 Solution: (i) 24.327 – 4.003 – 12.389 + 112.309 = 24.327 + 112.309 – 4.003 – 12.389 =(24.327 + 112.309) – (4.003 + 12.389) I II Note that we have used brackets in the same way as we did in the case of whole numbers. Now, we simplify the two brackets as follows. I II 1 1 24. 3 2 7 14.003 +12.389 +112. 3 0 9 136. 6 3 6 26.392 Now, we subtract II from I (why)? 5 1 3 6 . 6 13 6 –0 2 6 . 3 9 2 110.2 44 (ii) 228.37 – 29.516 + 12.059 – 1.09 = 228.37 + 12.059 – 29.516 – 1.09 = (228.37 + 12.059) – (29.516 – 1.09) I II I II 1 1 1 1 109 228.370 (One 0 added) 29.516 +012.059 +01.090 240.429 30.606 Now, we subtract II from I as follows: 39 2 4 0 . 14 2 9 –0 3 0 . 6 0 6 209. 823 (One 0 added) Word Problems We shall explain the process of solving word problems on addition and subtraction of decimals through some examples. Example 69: A contractor collected 143.75 liters of milk from his first milk vendor, 138.08 liters from the second vendor and 165.5 liters of milk from the third vendor. How much milk was collected by him? Solution: Total collection = (143.75 + 138.08 + 165.5) liters = (143.75 + 138.08 + 165.50) liters Now, we add as follows: 111 1 143. 75 + 138.08 + 165.50 447.33 Thus, total milk collected = 447.33 liters, which in terms of liters and milliliters can be written as 447 liters 330 milliliters. (Recall that 1 liter = 1000 milliliters) Example 70: Sum of two numbers is 27.57. If one number is 14.38, find the other number. Solution: Two other number = 27.57 – 14.38 Thus, we have: 4 110 27. 5 17 – 14. 3 8 13. 1 9 So, the required number is 13.19. Example 71: In a school fete, the collections at four counters were £585.75, £410, £375.50 and £333.25. Find the total collection of the four counters. Also find the difference between the highest and lowest collection. Solution: Total collection = £585.75 + £410 + £ 375.50 + £333.25 We add them as follows: 211 1 585.75 410.00 (Two 0 are added) 375.50 333.25 1704.50 Thus, total collection was £1704.50 Now, highest collection = £585.75 Lowest collection = £333.25 We find their difference as follows: 585.75 -333.25 252.50 Thus, required difference is £252.50 Example 72: From a barrel containing 453.76 kg of rice, a cook used 28.985 kg of rice in the first three days of the week. How much rice left in the barrel? Solution: Rice left in barrel = 453.76 kg – 28.985 kg 111 We find it as follows: 12 16 15 4 4 5 3 . 7 6 10 kg (One 0 is added) 0 2 8 . 9 8 5 kg 4 2 4 . 7 7 5 kg Thus, rice left in the barrel is 424.775 kg. Example 73: Kanta purchased 3m25cm cloth for her shirt and 2m5cm for her trouser. How much cloth was purchased by her in all? Solution: Cloth for shirt = 3m25cm, which can be expressed as 3.25 m (Recall 1m =100 cm) Cloth for trouser = 2m5cm = 2.05 m (Recall 1m = 100 cm) So, total cloth purchased = 3.25m + 2.05m Now, we add as follows: 1 3.25 m + 2.05 m 5.30 m Thus, total cloth purchased by Kanta is 5.30m or 5m30cm. Multiplication and Division by a Whole Number We know that decimal fractions are fractions with denominator 10, 100, 1000 etc. We have also discussed the multiplication and division of fraction by whole numbers in this unit. Therefore, we can multiply and divide decimals by whole numbers by treating them as decimal fractions. We explain the process through examples. Multiplication Example 74: Find the following products. (i) 3.75 × 10 (ii) 22.109 × 100 (iii) 325.1 × 1000 112 Solution: (i) 3.75 × 10 10 [3.75 is written as a decimal fraction = = ] [Recall how we multiply a fraction by a whole number] = = 37.50 = 37.5 [Note that 37.50 and 37.5 are the same (why?). Also, note that decimal point has been shifted to the right by one place] (ii) 22.109 × 100 100 [22.109 is written as a decimal fraction] = = = 2210.900 = 2210.9 [Note that decimal point has been shifted to the right by two places] (iii) 325.1 × 100 1000 = = 3251000 = 325100.0 [Note that decimal point has been shifted to the right by three places] = 325100 Observe that on multiplying a decimal by 10, 100, 1000 etc. the decimal point is respectively shifted to the right by one place, two places, three places, etc. in the product. Example 75: Find the following products: (i) 23.6 × 3 (ii) 4.375 × 7 (iii) 147.35 × 14 113 Solution: (i) 23.6 × 3 = 3 [23.6 is written as a decimal fraction] = [Recall how a fraction is multiplied by a whole number] = 708 = 70.8 [Decimal fraction is written as a decimal] (ii) 4.375 × 7 7 = = = 1000 [4.375 is written as decimal fraction] 30625 = 30.625 [Decimal fraction written as a decimal] (iii) 147.35 × 14 14 [Decimal number is written as a decimal fraction] = = = 100 206290 = 2062.90 = 2062.9 [Decimal fraction written as a decimal] Observe that in the multiplication of a decimal by a whole number, the decimal is written in decimal fraction and finally in the product the decimal fraction is converted in a decimal. 114 Division Example 76: Find the following quotients: (i) 37.59 ÷ 10 (ii) 442.921 ÷ 100 (iii) 3895.23 ÷ 1000 Solution: (i) 37.59 ÷ 10 = 10 [Decimal is written as a decimal fraction] = [Recall how a fraction is divided by a whole number] = = 3.759 [Note that decimal print has been shifted to the left by one place] (ii) 442.921 ÷ 100 100 [Decimal is written as a decimal fraction] = = = 442921 = 4.42921 [Note that decimal point has been shifted to the left by two places] (iii) 3895.23 ÷ 100 1000 [Decimal is written as a decimal fraction] = = = 100 100000 = 3.89523 [Note that decimal point has been shifted to the left by three places] Observe that on dividing a decimal by 10, 100, 1000, etc. the decimal point is shifted to the left respectively by one place, two places, three places, etc. in the quotient. 115 Example 77: Find the following quotients. (i) 46.5 ÷ 4 (ii) 112.56 ÷ 3 (iii) 28.305 ÷ 15 Solution: (i) 46.5 ÷ 4 4 [Decimal is written as a decimal fraction] = = [Recall how a fraction is divided by a whole number] = 40 [465 is divided by 40 as whole numbers, using long division] = 11.625 Alternative solution: After obtaining = 465 , we proceed as follows: 40 = [Numerator and denominator are multiplied and divided by 25 to obtain 1000in denominator] = .625 = 11.625 (ii) 112.56 ÷ 3 3 = = = = = 100 11256 [ or divide 11256 by 300 by long division] [Using by division] = 37.52 116 Alternative solution: After obtaining 11256 , we can proceed as follows: = [Numerator and denominator have been divided by 3 to obtain 100 in the denominator] = = 100 = 37.52 (iii) 28.305 ÷ 15 = 28305 15 = = 28305 Alternative solution: After obtaining = = , we can proceed as follows: 15000 28305 [Numerator and denominator have been divided by 15 to obtain 1000 in the denominators] = = 1.887 Word Problems Example 78: Weight of one watermelon is 5 kg 215g. Find the weight of five such watermelons. Solution: Weight of one watermelon = 5 kg 215g = 5.215 kg So, weight of 5 such watermelons: = 5.215 kg × 5 117 = 5.215 × 5 kg 5 kg = = kg = 26.075 kg = 26 kg 75g Example 79: A train travels a distance of 375 km 15m in 3 hours. Find the distance travelled by the train in one hour. Solution: Distance travelled in 3 hours: = 375 km 15m = 375.015 km So, distance travelled by train in 1 hour = 375.015 km ÷ 3 = 375.015 ÷ 3 km = = = = = 1000 ÷ 3 km 375015 375015 km km km 3000 1000 km = 125.005 km = 125 km 5m Example 80: Cost of one book is $29.57. Find the cost of 3 such books. Solution: Cost of 1 book = $29.57 118 So, cost of 3 books = $29.57 × 3 3 =$ =$ = $88.71 Example 81: 3 kg 275g of sweets were distributed among 25 students of class. Find the sweets received by each student. Solution: Total sweets = 3 kg 275g = 3.275 kg Number of students = 25 So, sweets received by one student: = 3.275 kg ÷ 25 = 3.275 ÷ 25 kg = 25 kg = kg kg = = = 25000 1000 kg kg = 0.131 kg or 131g 119 BODMAS in Decimals Simplification of expressions involving decimals using BODMAS can be done in the same way as was done in whole numbers Example 82: Simplify the following expressions: (i) 2.5 × 7 – 3.5 ÷ 10 × 6 – (3.75 + 1.15) (ii) 3.75 × 1000 + 2.85 × 8 + (2.6 ÷ 2 × 10 – 5.2) Solution: (i) 2.5 × 7 – 3.5 ÷ 10 × 6 – (3.75 + 1.15) = 0.35 6 4.90 4.90 = = 17.5 4.90 = 17.5 – 2.10 – 4.90 = 17.5 – (2.10 + 4.90) = 17.5 – (7.00) = 17.5 – 7.0 = 10.5 (ii) 3.75 × 1000 + 2.85 × 8 + (2.6 ÷ 2 × 10 – 5.2) = 3750 285 8 100 = 3750 2280 100 = 3750 22.80 = 3750 22.80 26 10 2 26 10 2 5.2 5.2 26 10 5.2 20 260 20 10 5.2 = 3750 + 22.80 + (13 – 5.2) 120 = 3750 + 22.80 + (13.0 – 5.2) = 3772.80 + 7.8 = 3772.8 + 7.8 = 3780.6 Estimation Recall that there are a number of situations in which we do not require to know the exact quantity but need to know only reasonable guess or estimate about it. For example, while presenting a budget the Finance Minister gives some budget estimates of amount of money to be spent on the Education Programmes of the country. We have already discussed some methods of estimation for whole numbers in earlier units. There we have discussed the process of estimating the outcomes of the operations on whole numbers, using the technique of rounding off numbers. The same process can be applied to estimate the outcomes on operations on decimals. The main purpose of such estimation is to check the reasonableness of the answers obtained. The rules of rounding off a decimal to a certain place are the same as for whole numbers. For example, we want to round off 465.275 to the nearest tenth place (say), then we look for the digit in the `hundredth place’. It is 7 here (it is more than 5). So, we shall round off 465.275 to 465.3. (Note that tenth place digit 2 has been increased to 3). Suppose, we want to round off 465.275 to the nearest hundredth place, then we shall look for the next place, to thousandth place. Here the digit is 5. So, we round off 465.275 to the nearest hundredth place as 465.28 (Note that digit 7 has been increased to 8). Suppose we want to round off 26.895472 say fifth decimal place. For this, we look for the sixth decimal place. Here the digit is 2. So, 26.895472 shall be rounded off to the fifth decimal place as 26.89547 (Note that the digit 7 has not been changed because 2 < 5) Suppose we want to round off 26.895472 to the nearest fourth decimal place, than we shall look for the next place, namely the fifth decimal place. Here the digit is 7. So, 26.895472 is rounded off to the nearest fourth place (i.e. ten thousandths place) as 26.8955. (Note that 4 has been increased to 5 as next digit 7 > 5). Suppose, we want to round off 26.895472 to the nearest thousandth place. Now, we should look for the ten thousandth place. Here, the digit is 4, which is less than 5. 121 So, 26.895472 is rounded off to the nearest thousandth place as 26.895. (Note that 5 remains the same here as 4 < 5). Suppose, we want to round off 26.895472 to the nearest hundredth place. We shall look for the next place, i.e. thousandth place. Here, the digit is 5. So, the digit at the hundredth place (which is 9 here) is to be increased by 1. It will become 9+1=10. Thus, in 26.895472, digit 9 will become 0 and digit 8 will become 8+1=9. Thus, 26.895472 is round off to the nearest hundredth place as 26.90. Clearly, if we want to round off 26.895472 to the nearest tenth place, it will be 26.9. Now, we shall explain the estimation of outcomes of the operations on decimals through examples. Example 83: Estimate the outcomes of the following. (i) 12.27 + 3.42 (ii) 583.21 + 2.534 (iii) 9.005 + 25.655 + 2.509 Solution: (i) (ii) (iii) In this case, we may round off 12.27 and 3.42 each to the nearest tenth place and than estimate the operation (+). We have: 12.27 = 12.3 and 3.42 = 3.4 So, estimation of (12.27 + 3.42) = 12.3 + 3.4 = 15.7 Here, we may round off 583.21 and 2.534 each to the nearest hundredth place. So, we have: 583.21 + 2.534 = 583.21 + 2.53 = 585.74 Here, we may round off 9.005, 25.655 and 2.509 each to the nearest hundredth place. So, we have: 9.005 + 25.655 + 2.509 = 9.01 + 25.66 + 2.51 = 37.18 Recall that there is no hard and fast rule of rounding off different numbers involved in the operations, regarding to the nearest places, they must be rounded off. For example, we may round off each of the decimals 9.005, 25.655 and 2.509 to the nearest tenth place. Thus, we have the following estimation: 9.005 + 25.655 + 2.509 = 9.0 + 25.7 + 2.5 = 37.2 122 Note that we may also round off 9.005 and 2.509 to the nearest hundredth place and 25.655 to nearest tenth place and obtain the following estimation: 9.005 + 25.655 + 2.509 = 9.01 + 25.7 + 2.51 = 9.01 + 25.70 + 2.51 = 37.22 Example 84: Estimate each of the following difference. (i) 14.372 – 4.005 (ii) 36.38 – 17.846 (iii) 122.50 – 31.2395 Solution: (i) We may round off 14.372 and 4.005 each to the nearest hundredth place and obtain the following estimation: = 14.372 – 4.005 = 14.37 – 4.01 = 10.36 We may also round off each of these decimals to the nearest tenth place and obtain the estimation as follows: = 14.372 – 4.005 = 14.4 – 4.0 = 10.4 (ii) We may round off 17.846 to the nearest hundredth place, without any change in first decimal 36.38. Thus, we obtain the estimation as: = 36.38 – 17.846 = 36.38 – 17.85 = 18.53 We may also round off each decimal to the tenth place and obtain the estimation: = 36.38 – 17.846 = 36.4 – 17.8 = 18.6 (iii) We may round off 31.2395 to the nearest hundredth place and keep other decimal 122.50 as it is. Thus, we have the following estimation: = 122.50 – 31.2395 = 122.50 – 31.24 = 91.26 In this case, we may also round off each decimal to the tenth place and obtain the estimation as: = 122.50 – 31.2395 = 122.5 – 31.2 = 91.3 Significant Figures We can write any number using only ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 under the base ten system of numeration. For example, we have numbers like: 435860709 273201008 123 509381092, and so on, Using each of the digits of the above numbers, we can read the numbers with some meaning in each case. Now, let us consider the following numbers: 008507219 092168308 00070958621, and so on, If we look at the number 008507219, we note that there is no difference between number 8507219 and 008507219. Thus, it can be said the two zeros (00) preceding 8507219 have no meaning. The other way of expressing the zero fact is that the two digits `00’ before 8507219 have no significant, i.e. these are not significant digits for two numbers. Similarly, in 092168308, the digit `0’ preceding 92168308 is not significant and in 0007095862, the digits `000’ preceding 7095862 are not significant. In general, it can be said that all the digits 0 written before a number (be a natural number) are not significant. It shall not be mistaken that all the digits 0 used in a number are not significant. In fact, in any number, all 0s (except those written before a number) are significant. Thus, in 921128009300, all the four `0s’ are significant. Thus, in a number: • • • Digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 are always significant. Digits 0 occurring between two significant digits (1 to 9) and also on the right of the number are all significant. Digits 0 on the left of the number are not significant. What can thought of about significant digits in a decimal number? As regards the digits 1 to 9 is concerned, there is no change in the situation. However, in the case of digit 0, the situation is a little bit reverse. Note that in decimals, 2.73, 2.730 and 2.7300 are considered as the same decimal. Thus, 0 used at the end of the decimal have no meaning, i.e. they are not significant. Thus, digits 0 used on the right of a decimal number are not significant, while the digits 0 used on the left of a number (natural number) are not significant. 124 Thus, for a decimal number, we have the following rules for significant digits: • Digits 1 to 9 are always significant. • Digits 0 between two significant digits are always significant. For example, in decimal number 2.31008095, all `0s’ are significant. • Digits 0s to the right of the decimal part and digits 0s to the left of the whole number part of a decimal are not significant. • Digits 0s used only for spacing the decimal point (i.e. as place holders) is not significant. For example, in the decimal 0.02507, one 0 to the left of the decimal point is not significant. However, the remaining five digits 0, 2, 5, 0 and 7 after the decimal point are all significant in this decimal. If we consider the decimal 1.02507, then all the six digits 1, 0, 2, 5, 0 and 7 are significant. Example 85: Write all the significant digits in each of the following decimals: (i) 345.098070 (ii) 450.108007 (iii) 0.001503 Solution: (i) The significant digits are: 3, 4, 5, 0, 9, 8, 0, 7 (ii) The significant digits are: 4, 5, 0, 1, 0, 8, 0, 0, 7 (iii) The significant digits are: 9, 0, 0, 1, 5, 0, 3 (iv) The significant digits are: 1, 5, 0, 3 125 09.001503 (iv) STUDENT’S WORKSHEET – 1 IMPORTANCE OF NUMBERS WARM UP W1 Name of the student _____________________ 1. Date ______ What are natural numbers. How did our ancestors use them ? ____________________________________________________________________________ ____________________________________________________________________________ 2. What are whole numbers? How are whole numbers different from natural numbers. Give some examples of whole numbers? ____________________________________________________________________________ ____________________________________________________________________________ 3. What are directed numbers? What was the need to introduce directed numbers. Write a few lines to explain . ____________________________________________________________________________ ____________________________________________________________________________ 4. What are the part of the whole numbers known as? Give some situations in life where you use such numbers? ____________________________________________________________________________ ____________________________________________________________________________ 5. What do the numbers like 32.9, $4.50, 162.5cm mean ? What are they known as? ____________________________________________________________________________ ____________________________________________________________________________ 126 6. Divide the figure into equal number of parts (number of parts mentioned alongside) a. 8 parts b. c. 3 parts 2 parts d. 7 parts SELF ASSESSMENT RUBRIC – WARM UP (W1) Parameter Recall different type of numbers they have come across or they have used. Divide a quantity in equal parts 127 e. 9 part STUDENT WORKSHEET 2 TANGRAMS WARM UP W2 Name of the student ______________________ Date ______ The Tangram puzzle set given above ( literally "seven boards of skill") is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. The objective of the puzzle is to form a specific shape using all seven pieces, which may not overlap. Observe and answer the following questions: 1. Total number of pieces = ___________ 2. Number of 3-sided pieces = ___________ 3. Number of 4-sided pieces = ___________ 4. There are ___________ number of 3-sided pieces out of total ________pieces. 5. There are ___________ number of 4-sided pieces out of total ________pieces. 128 Thus we can say that ________ out of total 7 pieces are 3-sided pieces. In the same manner ________ out of total 7 pieces are 4-sided pieces. We can say that are 4-sided. Recall how the number ½ , ¾ , ¼ etc are used by you and your family in day to day life. Write five such incidences and discuss it with the class. __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ Recall how the number 0.5, 6.25, $1.5 etc are used by you and your family in day to day life. Write five such incidences and discuss it with the class. __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ 129 SELF ASSESSMENT RUBRIC – WARM UP (W2) Parameter Compare quantities using concept of division Represent a quantity as a part of a whole 130 STUDENT’S WORKSHEET – 3 PRECONTENT P1 A QUANTITY AS A PART OF A WHOLE Name of the student ______________________ Date ______ A student is given a chocolate to share it equally among two friends 1. If the chocolate is to be divided into two friends equally, then how much does each friend get? 2. Represent the number of chocolates each friend gets as a part of the total number of pieces in the given box. 3. Now complete the statement: Each friend gets ________ of the chocolate. 4. Now repeat the same activity by dividing the chocolate equally among three friends 131 5. Now complete the statement: Each friend gets ________ of the chocolate. 6. Can you find the part of chocolate that each friend gets, if there were 5 friends? SELF ASSESSMENT RUBRIC PRE CONTENT (P1) Parameter Comprehend given problem situation Represent quantities using concept of division Represent a quantity as a part of a whole 132 STUD DENT’S WORKS SHEET – 3 W WHAT IS S IT ANY YWAYS PRECO ONTENT T P2 Name of the studentt ________________________ 133 Date ______ Do as directed (complete and discuss) A) Instructions for students: (i) Cut out figure 1 (ii) Total number of triangles = _______________ (iii) Color the figure as given below: (vi) How many green triangles are there in your figure? ______________ (v) Green portion is _____________ of the total triangles. Conclusion: The green portion is half of the total triangles. B) Instructions: Now cut out figure 2. C) (i) Color the triangles in figure 2 in green. (ii) How many parts of the whole have you colored? _____________ (iii) Remove one green triangle. (iv) What part of the figure has been removed? ____________ (v) Numerator is __________ and Denominator is __________. Instruction: (i) Cut out and color figure 3. (ii) Remove one green triangle. (iii) What part has been removed? __________ (iv) State the numerator and denominator. _________________. 134 D) E) F) Instruction: (i) Color figure 5 with Blue triangles. (ii) Remove one triangle. (iii) What part of the blue triangles are left? _______________ (iv) Specify the numerator and the denominator. _________________ Instruction: (i) Color figure 6 with blue blocks. (ii) Remove one blue block. (iii) What part of the figure is covered by blue? ___________ Instruction: (i) Color half of the figure 7 with red blocks. Extension: (i) Color 2 (ii) Color 3 3 5 of figure 2 with blue triangles. of figure 5 with brown triangles. SELF ASSESSMENT RUBRIC – PRE CONTENT (P2) 135 Parameter Is able to follow multiple instructions for a question Represent quantities using concept of division Represent a quantity as a part of a whole STUDENT’S WORKSHEET 5 CONTENT WORKSHEET C1 UNDERSTANDING OF REPRESENTING A FRACTION AND ITS COMPONENTS Name of the student ____________________ 1. Date ______ Complete the following table(for shaded parts): Fraction Numerator 136 Denominator 2. Circle the denominator and box the numerator in each of the following: 4 4 a) 1 4 1 5 1 2 1 8 1 7 Do you observe something special in all these fractions? Is there any exception? _______________________________________________________________ b) Which of the above numbers is odd one out? Why? _______________________________________________________________ 3. Read the clues and give an example for each of the following: (a) A fraction whose denominator is three times the numerator.______ (b) A fraction whose numerator is four less than the denominator.________ (c) A fraction which has no common factor in the numerator and denominator. (d) A fraction whose denominator is 2 times the numerator. 4. A pizza is sliced into 8 parts. One part was given to the child and 2 parts was taken by the father. Draw a figure to show how much of the pizza was left. 5. Approximately, what fraction of chocolate has been eaten? 6. Observe the fractions and answer the following questions- 1/4 , 10/25, 5/7 , 7/2 , 3/11, ½, 4/7 , 5/3, 8/14, 137 4 , 6 , 2/3 , 3/9, 7/14, 12/34, 8/9, Write (a) Unit fractions __________________________________________________________________________ (b) Proper fraction __________________________________________________________________________ (c) Improper fraction __________________________________________________________________________ (d) Write like fractions __________________________________________________________________________ 7. I bought ten chocolates. Each chocolate is divided into eight equal parts. After distributing ½ of the chocolate to each of the 16 friends, find out: (a) How many chocolates am I left with? ___________________________________________________________________________ (b) Write the number of chocolates left as a fraction of the total number of the chocolates? __________________________________________________________________________ 8. An egg crate contains 12 egg in all. Two eggs have been removed. (a) Write the number of eggs left as a fraction of the total number of eggs. (b) Explain why the fraction is written as 1/6 in the figure and not 2/12. (c) 9. If an apple represents the number 1 then, 138 represents_________________ Now draw & show how can you represent 4 ½ apples? 10. Observe the figure & write the fraction for the shaded part. 139 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET 1 Parameter Has knowledge of various types of numbers. Is able to relate the given number to different kind of numbers Can speak creatively about numbers and their need and usage 140 STUD DENT’S WORKS SHEET – 6 LOCA ATING FRACTIO F ONS ON N A NUM MBER LIN NE CON NTENT WORKS W SHEET C2 C Name of th he student ________________________ Date ______ _ Activity 1 – Locating fractions on o a numbeer line Look at a your ruleer and take a guess to the t location n of the fracction ½ , 1 ½ , 2 ½ and d so on…… ……………… … Write and expreess what thought t prrocess/ straategy will you apply y to locatee these numbeers? _________________ _____________________________________________________________________ _________________ _____________________________________________________________________ Answer th he followin ng question ns on the baasis of videeo clip 4. 1 What did d you lea arn from vid deo clip? ________________ ______________________________________________________________________ ________________ ________________________________________________________________ 2 For rep presenting the t fraction n 4 a) 10 on th he number line l Th he portion between b 0 and a 1 is div vided into _______ _ equ ual parts. b) Eaach part of the t numberr representss _______________ fraction. c) 3 Drraw a numb ber line and d mark the position off 4 10 on it. Now lo ook at the number n linee given belo ow and ansswer the folllowing queestions. 141 If the whole line is divided into 10 equal parts, then explain: 1. Why does ‘A’ represent 1 10 (the first division out of 10)? ________________________________________________________________________ 2. Why does the point ‘B’ represent 5 10 ? _______________________________________________________________________ 3. What fractions do C, D and E represent? __________________________________________________________________________ 4. Represent 4 7 on the following number line. 5. Write another fraction of your choice. (a) ____________ To represent it on number line the portion between 0 and 1 is divided into _______ parts. (b) (c) Draw a number line and mark the fraction on it. Explain the procedure in detail. (Refer to the previous question for some ideas.) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ (d) Now represent the following fractions on the number line: 2 5 , 3 8 , 1 7 , 1 9 , 5 9 142 143 ACTIVITY 2 –‘GEM POWER’FRACTIONS ARE NUMBERS 1 Do as directed You will need a pack of gems to answer the questions. But before you eat them!! 1. Count the number of gems of each colour in your pile . Red: ___ Blue: ____ Orange:___ Yellow: ____Green: ____ Any other ____ 2. Write down each color gems as a fraction of the total number of gems you had. Red: ___ Blue: ____ Orange:___ Yellow: ____Green: ____ Any other ____ 3. Draw a number line in the space provided: 4. Using a ruler, divide your number line into as many equal parts as the total number of gems you have. 5. Starting from left to right, mark the first point as 0 and the last point as 1. 6. Explain why is the last point was marked as 1? Is there any other way you could mark it as? ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 144 Mark the fractions representing different color gems on the number line using that colour pen/pencil. ACTIVITY 3 – LOCATING MORE FRACTIONS ON A NUMBER LINE 1. Observe 11/4 on number line. What do we observe? 11 = 2 ¾ (remember! Mixed fraction) 4 11 4 = 4 4 + 4 4 +¾=1+1+¾ The numbers on the number line can be regrouped in groups of 4 every time. Portion between each pair of consecutive integer on number line is divided into four equal parts. • We see that 11 • Portion between 3 and 4 is divided into 4 equal parts and 3rd part is 11 . 4 lies between 2 and 3. 4 2. Now use number line to mark following fractions: 15 , 7 24 , 5 16 , 9 145 20 . 3 SELF ASSESSMENT RUBRIC – CONTENT WORKSHEET C2 Parameter 146 Knowledge of parts of fraction. Divide a unit distance into as many equal parts as the denominator. Represent fractions on a number line. STUDENT’S WORKSHEET – 7 INTRODUCING DECIMALS CONTENT WORKSHEET C3 Name of the student ______________________ Date ______ 1. Due to over use, the markings are faded and I can see markings only till 10. 2. How much length is visible here? 3. What portion of the visible length is the arrowed length? _______________ 147 _______________ a) Represent it as a fraction of visible length. _______________ b) Is there any other way in which this information can be represented? __ c) What can you say about this form of representation? Note: A decimal is a special case of a fraction with denominator in powers of 10. Examples: • The fraction 3 • The fraction 54 • The fraction 456 10 means 3 out of 10 which is also represented as 0.3 100 means 54 out of (10 x 10) or 5.4 out of 10 or 0.54 1000 means 456 out of (10 x 10 x 10) or 45.6 out of (10 x 10) or 4.56 out of 10 or 0.456. 4. Now, observe this image and answer the following; The measuring tape shown above is a meter long. a) Write down the length of the measuring tape in centimeters. b) A toy was measured by the tape and found to be 32 cm long. Express the length of the toy as a fraction of the total length of the tape._____ c) 5. Express your answer to part (b) as a decimal. ______ Refer to online math dictionary and write what is meant by a decimal, decimal notation and its component. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 6. Complete the following table: 148 Fraction 1 10 0.1 5 0.5 10 7 100 9 4 1000 10 3 100 39 100 435 7. Decimal 1000 If each shaded box is worth 1, then represent the shaded part using decimal. Decimals 8. Observe the figure of the decimal blocks below and complete the table given below 149 150 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C3 Parameter Exhibit knowledge of fraction and parts of fraction. Represent fraction with denominator in powers of 10 using concept of decimal. Represent quantities using decimal. STUDENT’S WORKSHEET – 8 151 PLACING DECIMALS ON A NUMBER LINE CONTENT WORKSHEET C4 Name of the student ______________________ Date ______ Activity 1 –‘Gem power’ - Only 10 gems 1. Based on video clip 5, describe what did you understand? ____________________________________________________________________________ ____________________________________________________________________________ 2. Now the gems activity (take any 10 gems for this activity) I. Count gems of different colors and write your answer here: Red_______, green ______, yellow______, pink______, other colors_____. II. Represent all color gem fractions and decimals. Red_______, green ______, yellow______, pink______, other colors_____. III. Now draw a number line and represent all your different color gems as fractions on the number line. IV. Also write their corresponding decimal representation below the fraction number now on a number line. What is 3 in other words known as? 3. Represent 3 4. Draw a number line and represent 0.7 on the number line. 5. If the line below represents a unit length, which is divided into 10 equal parts. The 10 10 line is marked with points labeled A - G. Write the decimal numbers for each of these points. A=_____, B=______, G=_____. 152 C=_____, D=______, E=_____, F=______, SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C4 Parameter Has knowledge of various types of numbers. Is able to relate the given number to different kind of numbers Can speak creatively about numbers and their need and usage 153 STUDENT’S WORKSHEET – 9 EQUIVALENT FRACTIONS CONTENT WORKSHEET C 5 Name of the student ______________________ 1. The fractions 2 3 and 4 6 are represented diagrammatically. 2 What do you observe? Is 2 2. Check for 1 Date ______ 3 4 3 =4 6 6 ? _______________________________ 3 5 and 15 They are also called______________________________________. 3. Write a general procedure for converting fractions to equivalent fractions. ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 4. Find the missing terms in each of the following: (a) 4/7 = __/21 (d) 2/9 = 18/__ (b) 40/13=80/__ (e) 24/18= 3/__ (c) 25/30= 5/__ 154 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C5 Parameter Represent fractions diagrammatically Understand and find equivalent fractions to a given fraction 155 STUDENT’S WORKSHEET 10 CONVERTING FRACTIONS TO DECIMALS CONTENT WORKSHEET C6 Name of the student ______________________ Date ______ 1. Represent the shaded portion as a fraction: 2. Now divide the other two circles in equal portions and shade to represent equivalent fractions 3. Divide each bar into equal portions and shade in the bars to represent three equivalent fractions: If the rectangle is cut into 10 equal parts instead, shade and represent an equivalent fraction for the above fractions. 156 4. Why do you think we want to cut the shape into ten equal parts? ____________________________________________________________________________ ____________________________________________________________________ 5. Is there another way to represent this fraction? (Hint: 3 6. 5 = 10 = 100) Discuss the procedure for converting a fraction to decimal and summarize with your partner Hint: 7. Based on the video clip 8, write the procedure for the conversion of a fraction to a decimal by an algorithm and summarize: Hint: 157 8. Now Try to convert following fractions to decimals: a) 6 b) 7 25 10 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C6 Parameter Represent fractions diagrammatically Understand equivalent fractions Convert fractions to decimals 158 STUDENT’S WORKSHEET 11 SIMPLIFICATION OF FRACTIONS CONTENT WORKSHEET C7 Name of the student ______________________ Date ______ Observe the pattern and explain what you see/? 24 28 , 18 21 , 12 14 , 6 7 ____________________________________________________________________________ ____________________________________________________________________________ Now watch video clips 9 and then answer the following questions: 1. Observe the picture and write down the simplification process. Also mark how many slices were joined or merged together and show the arrows and the division process as in the example a) ______slices merged together b) ______slices merged together c) ______slices merged together 4. Find the HCF of the numerator and the denominator of each of the original fractions given above. Explain how the knowledge of HCF would help in simplification. 159 ________________________________________________________________________________ ________________________________________________________________________________ _______________________________________________________________________________ 5. Some of the following fractions are such that you CANNOT simplify them. Cross them out. Why is that? Discuss. Simplify the ones you can 4 , 15 , 15 , 13 , 13 , 12 , 12 18 22 51 52 21 17 7 6. Write each of the following decimal as a fraction (denominator as a power of 10) Decimal Fraction 0.2 2 10 0.3 0.4 0.33 5.7 7. Write each decimal in the fraction form (simplify wherever possible) 0.27 0.01 0.55 0.95 0.82 0.17 0.48 0.25 0.02 1.05 160 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C7 Parameter Represent fractions diagrammatically Knowledge of equivalent fractions Understanding of simplification of fractions 161 STUDENT’S WORKSHEET – 12 INDEPENDENT PRACTICE CONTENT WORKSHEET C8 Name of the student ______________________ Date ______ Activity 1: Independent Practice 1. a) Put a tick in the appropriate column and complete the table: Proper fraction 1 4 Mixed fraction 3 3 11 7 2 1 6 Improper fraction 7 11 b) Explain what do you understand by fractions 4 3 , 1 1 and 2 1 ? 7 7 c) Represent each one of them diagrammatically 2. From the number line given below, identify n and express your answer as a fraction. 3. Now, Place the following improper fractions on the number line (guided practice) 4. Observe the pattern in part a) and b) and choose the appropriate answers for all questions in part c). a. 162 b. c. Circle the correct improper fraction for the parts that are shaded. (i) 5 , 6 5 2 , 6 1 3 (ii) 13 , 14 1 1 , 6 1 2 (iii) 9 , 4 5 1 , 4 2 1 1 , 3 (iv) 4 , 6 1 4 4 3 5. Complete the following table: Improper fraction to Mixed number Mixed fraction 12 16 11 10 7 2 1 7 13 9 5 7 2½ 5 32 3 3 4¼ 4 163 number to Improper 6. Fill in the missing numbers to make equivalent fractions: 1 2 = 3 2 4 = 9 3 9 = 7 1 10 = 10 2 8 ? ? = = = 7 ? 69 ? 10 = 2 5 3 3000 = 8 ? ? 13 39 = = = ? 8 104 ? 7. Write each of the following fractions as an equivalent fraction with: Denominator 24 1 2 1 3 3 4 5 12 3 4 6 7 4 5 Numerator 12 1 6 8. Some of the following equivalent fractions are correct but two of them are wrong. Find the incorrect ones and explain why they are incorrect. 2 6 = 5 15 9. 2 4 = 3 9 3 6 = 7 14 4 12 = 9 27 7 77 = 10 100 Observe and complete the pattern ½, 2 2 4 3 5 , 3 4 , 4 , 8 6 10 6 ,_____,_______,_____,______ , 6 , 12 9 , _____,______,______,_____ 15 ,_____,______,_______,______ Discuss with your partner how you would decide on the pattern. 10. Draw a rectangle and cut it into 4 equal parts, a) Represent 3 4 by shading on the rectangle. 164 9 18 = 13 26 Now divide the rectangle into 16 equal parts. b) Represent by shading a fraction equivalent to the previous fraction. Place the two rectangles one below the other. c) Explain, how you know by observation, that the two fractions are equal. d) How will you be able to show and explain fractions smaller and bigger than this fraction? 11. Convert 0.8 into a fraction. Represent 4 5 and 0.8 on the same number line. Explain what you observe. 12. Convert ( if required ) and then reduce the given fractions to lowest terms in the space provided.( try to work with prime factors) 1) 0.5 2) 0.6 3) 32 4) 99 5) 96 6) 600 7) 525 8) 1.25 40 126 1125 132 800 165 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C8 Parameter Has knowledge of various types of fraction. Is able to relate the given number to different kind of fractions. Can speak creatively about numbers and their need and usage 166 STUDENT’S WORKSHEET 13 PLACE VALUE CHART OF DECIMALS CONTENT WORKSHEET C9 Name of the student ______________________ 1. 2. Date ______ Use the above place value table and write the following numbers in the expanded form. a) 6402307 b) 703297 c) 109990 Based on the Place value chart above and the knowledge of expanded form, write the expanded form of following (teacher to explain if required) a) 54.2 b) 231.42 167 c) 5910.016 9. Fill in the missing words and numbers on this chart 10. a) Place the digits of the number mentioned in the place value chart appropriately. b) Write its number name c) Write each number in the expanded form. Is there a special name for the last number in the series? (Hint: You researched on this in unit1) 168 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C9 Parameter 169 Understands the place value chart for decimals Read and write decimals in expanded form STUDENT’S WORKSHEET 14 COMPARISON OF FRACTIONS / DECIMALS CONTENT WORKSHEET C10 Activity1: Cut out one fractional part of each type Students to cut out fractional parts 1, 1/2 , ¾ , 1/6 , 2/5 , 1/10 , 1/12 Now arrange them on your desk from the smallest to greatest to the smallest 170 Activity 1- Like and unlike fractions 171 Image1 Image 2 1. Observe the above images carefully and discuss: In image 1: The heights of the two people were measured and we came to the conclusion that the man is taller than the women. What do you think? In image 2 it was stated that Tower T 4 is smaller than T 7 and Tower T 2 is taller than T 5. What can you say about this analysis? Give an argument to justify your answer. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ 2. Convert all the fractions given above with the same denominator. (Hint - 5, 4, 8 all have common multiple 40. Remember!) _______________________________________________________________________ _______________________________________________________________________ Verify 2/5=16/40 ¼=10/40 3/8=15/40 3. Convert the following fractions to like fractions and arrange them in the ascending order. 1/5, 1/2, 4/5, 3/2, 12/10 4. Which of the following pairs of fraction are like fractions and which are unlike fractions. (2/5, 3/5), (12/15, 3/5), (½, 6/14), ACTIVIT 3 - DECIMALS Observe the following examples carefully: a) Write 1.4 as a fraction in two parts? If 1.4 can be shown as 172 (4/5, ¾) OR 1.4 = 1 and 0.4 = 10 10 + 4 10 = 14 10 true! What kind of fraction is this? b) 2.4 is made of 2 and .4 = 1 and 1 and .4 = 10 10 = 24 1 c) + 10 10 10 100 10 + 10 10 10 1 23.4 = 20 and 3 and .4 = + 4 100 4 10 + 30 3 10 10 + 4 = 0.4 10 = 234 10 (Why?) .4 1. Using the algebra tiles express diagrammatically the decimals 14.3, 21.7, 20.8. 2. The decimal number 23.4 is read as twenty three point four. Now complete the following sentences 1.3 is read as one point three four and so on…… 1.4 is read as_______________________________________ 109.09 is read as_________________________________________ 173 25.002 is read as _______________________________________ 3. Put the correct sign in the blank: (<,>or=) a) 9.16________9.166 b) 2.034__________2.03 c) 2.16__________54/25 d) 12.61_________12.16 e) 2.54________2.45 f) 1.045_________1.405 g) 0.402__________0.420 h) 1/2_________0.5 i) 63.24_________163.24 j) 16.78_______1.678 k) 41.56__________14.56 l) 0.0045_______0.0405 m) $4.85________$417/50 n) $6and 5 cents_________$6 and 50 cents o) 0.22__________ 12/48 p) 18.05_________1.805 q) 5.95___________5.59 r) 13/51__________0.3 s) 2.54_________2.6 t) 5/7__________7/10 u) 4/10__________0.34 Parameter SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C10 174 Is able to understand fraction strip and use it creatively Is able to understand like and unlike fractions Is able to compare decimals STUDENT’S WORKSHEET 15 ADDITION AND SUBTRACTION OF FRACTIONS CONTENT WORKSHEET C11 Name of the student ______________________ 175 Date ______ ACTIVIT TY 1 – DO OMINO ACTIVITY A Y Begin with h the 'Start' domino an nd then add d dominoes so that the matching expressions e s are equivalentt to form a continuous c loop. (a) 1. ACT TIVITY 2 – SOME MO ORE PROB BLEMS Add d and then verify yourr answer ussing fraction n strips. (a) ½+½ (b) (c) (d) (e) 1/6 + 1/ /6 +1/6 1/3 + ¼ 1/3 + 1/ /5 1/5 + ¼. ¼ 2. Now w try ½ +½ 3. ve the follow wing : Solv using fraction strips and then n without using u fractio on strips. a) 1/4 + 2/4 4 c) 4/15 + 6//15 - 3/15 b b) d) ½+ 1/2 /2 1/9 + 2/3 +2/9 176 e) 4/5 + 7/10 + - ½ f) 7/15 + 1/3 – 1/5 4. If Ali spends 3/5 of his allowance to buy candies, then spends 1/5 of the money for a sandwich , how much portion of his allowance is spent in total? 5. During Easter, my aunt put leftover cakes into the fridge. She noticed that the fridge had 5/11 of a cake filled with cherries, 8/11 of a cake filled with blueberries, and 1/11 of a cake filled with peaches. How many left over cake did my aunt have in all? Simplify your answer and write it as a proper fraction or as a whole or mixed number. ( assuming that all the cakes were of same size ) 6. Do as directed :- Add the fractions. Add the whole numbers. 2 a) e) 7. 4 1 5 2 b) 3 1 8 f)2 1 2 2 7 8 c)1 5 6 3 2 5 g)5 2 5 2 3 4 Fill in the missing number. 1 6 ______ 4 7 ________ 2 3 1/5 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C11 177 d) 3 2 5 h) 5 1 1 4 7 10 3 1 6 STUDENT’S WORKSHEET 16 ADDITION AND SUBTRACTION OF DECIMALS CONTENT WORKSHEET C12 Name of the student ______________________ Date ______ ACTIVITY1 : STAR BLOCK Solve the given problems and use the given table for coloring Parameter Has knowledge of various types of fractions Is able to add and subtract fractions Can solve related problems on fractions 178 If the answer lies between 0 and 10 11 and 99 100 and 999 Color the shape red Blue yellow ACTIVITY -2 : Do as directed. 1. Louis has $4.50 and Ronald $3 ¼. How much money do they have together? 179 2. What is the combined thickness of these five strips of ribbon 0.008, 0.125, 0.15,0.185, 0.005 cm? 3. The ice cream parlor has been charging $3.70 for its jumbo banana split 4. Because of inflation the prices were increased by 75 percent. 5. a) Write the increased price as decimal b) What is the new price. Jhon is making a telescope for his science project. He joined two metal tubing, one 30.45 inches long and other 12.75 inches long. How long would the telescope be when the two tubes are joined? 6. For winter’s snowstorm, Mrs. Brown bought two coats for her children. One coat cost $55.75 and the other cost $9.30.What was the difference in the cost of the two coats? 7. In the school café the cost of items are as followed. Ryan ordered one of the each item a) How much money does Ryan have to pay b) He paid with a $10 note. How much money will he get back? 8. A piece of webbing is 7.6 m long. If two pieces each 2.3 m and 1.5 m long respectively cut off, how much is left? Pizza (small) $5.50 Sandwich $1.20 Juice 75 cents Veg burger $1 .60 180 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C12 Parameter Has knowledge of decimal Add and subtract decimals Can solve related problems on decimals 181 STUDENT’S WORKSHEET 17 EXTEND MULTIPLICATION AND DIVISION AS REPEATED ADDITION AND SUBTRACTION. CONTENT WORKSHEET C13 Name of the student ______________________ Date ______ Activity 1: Repeated addition and subtraction 1. Do as directed : Solve ______ a) b) Rewrite this addition problem as a multiplication problem.__________ Verify your answer by using fraction strips. _____________ c) Now frame this information as a multiplication fact and solve : 1/12 + 1/12 +1/12 +1/12+1/12+ 1/12 =1/12 X _____=________ 2. What can you say about the multiplication of a decimal by whole number : (a) What is 0.4 + 0.4=_________? (b) Write this as a multiplication fact : _______ (c) Now complete the statement- 0.4 X ______=______ (d) Now model the solution using decimal blocks and paste the tiles/draw a diagram if possible 3. Do as directed: (a) Explain what do you understand by 1 divided by (b) Represent this diagrammatically. (c) What does ½ divided into two mean? (d) Confirm this by superimposing the result on the ¼ fraction strip. Paste your result. (e) Write the steps of the process you have just followed. 4. Now try to write the algorithm for the fraction division. 182 5. Can you do the same for division of decimal with a whole number. Explore with decimal blocks and draw a diagram to show your findings. Take help from this graphic: (a) 1 ÷ 2 , in a pile of 10 slabs each represents 0.1 so (b) 4 slabs divided into two groups = 2 slabs each. So 0.4 ÷ 2 = 0.2 Try to demonstrate the same for the following cases (you may use a manipulative if you need) ½x5 ¾ x4 ½÷3 ¼ ÷3 0.9 ÷ 3 0.9 x 3 2.4 ÷ 6 2.4 ÷ 4 Activity 2: Fraction of a whole Fraction of a quantity Give a scenario: A student is given the chocolate to share it equally among two friends 1. Count the number of pieces the chocolate has? (1 chocolate =_____ pieces) 2. If the chocolate is to divided into two friends equally, how much does each friend get? 183 3. Represent the number of chocolates each friend gets as a fraction of the total number of pieces. 4. Simplify your answer. _______________________________________________________ ____________________________________________________ 5. Now complete the statement: Each friend gets __________________ of the chocolate ______________________________________________________________________ Now repeat the same activity by dividing the chocolate equally among three friends 1 3 of 24 = _____________ Now let us find out 2/3 of a chocolate Write and explain how you would solve such a problem ________________________________________________________________________ ________________________________________________________________________ 11. If I have a stack of 60 coins arranged like this plan a strategy and verify that 1 3 of 60 = 20 Now Try these: 2 5 of 100 1 184 3 of 90 4 7 of 49 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C13 Parameter Understand multiplication as repeated addition Understand division as repeated subtraction 185 STUDENT’S WORKSHEET 18 WORD PROBLEMS INVOLVING FRACTIONS AND DECIMALS CONTENT WORKSHEET C14 Activity 1 – Try them all!!! 1. Annabelle cut a piece of rope and labeled it 1 4 5 . Ryan cut another piece of ropeand labels it as 1.8. They measured the rope lengths and found them to be the same. Explain how are they equal? 2. Ravi and Karan were both given some math questions to do. Ravi completed the whole work in 1 2 5 hours and karan the same task in 1.5 hours. Who completed the work late? 3. The thickness of four sheets of metal is 35/1000 inches, 13/100 inches, 97/1000 inches and 1/10 inches. Arrange these sheets in ascending order of thickness. 4. Four swimmers entered into a competition, the timings of the four of the swimmers were 9.8o seconds, 9.75 seconds, 9.79 seconds, 9.801 seconds. In what minimum time the fifth swimmer should swim in order to win the competition? 5. To make a miniature ice-cream truck we need tyres with a diameter between 1.465 cm and 1.42 cm. Will a tyre with 1.469 cm in diameter work? Explain why or why not? 6. ”I SCREAM ICE-CREAM” All right you bums, I got your at bats and hit right here, and if any of you have less than 0.250 average you don’t get any ice cream. 186 Check for all the players getting ice-cream Players / Mickey: bats averages 15/30 Roger: 10/25 Joe: 19/35 Yogi: 9/42 Boby: 6/15 Activity 2 - The Match Maker In this card game, there are two clues one on the front side and one at the back side of the card. This is just played as a game in the class. Students will record their observations on the worksheet. Students get 10 minutes to complete the whole chain of clues. For e.g. In the following cards Card 1 follows card 12 and card 7 follows card 1 and so on. Card No. 12 Front Back 6 I have 6. Who has 2 .5 less? 1 3 .5 7 3 I have 3 1 . Who has its double? 2 I have 3 8 8 . Who has 1 8 more? Now try it out yourself. Card No. Front Back 1 3 .5 I have 3 1 . Who has its double? 2 ½ I have ½. Who has 1/4 more? 3 33 4 2 8 2 I have 3 3 . Who has 2 less? 8 I have 2. Who has 7 187 8 more? 5 43 6 11 7 3 8 67 9 3 4 4 I have 4 3 8 less? 4 I have 1.25. Who has 3 .5 more? I have 3 8 . Who has 3 4 . Who has 1 8 I have 6 7 more? 8 . Who has 6 1 less? 12 8 I have 3. Who has 3 more? 10 13 11 4 I have 4. Who has 5/8 less? 12 6 I have 6. Who has 2 .5 less? 13 7 I have 7. Who has 1/8 less? 14 1 15 27 16 0.75 I have 1 3 . Who has 5 8 8 I have 1 8 8 8 I have 2 7 8 . Who has 1 1 more? 8 8 . Who has 1 I have 0.75. Who has 5 188 5/8 more? 8 8 more? less? SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C14 Parameter Has knowledge of fractions and decimals Can solve related problems on decimals and fractions 189 STUDENT’S WORKSHEET – 19 ROUNDING OFF AND ESTIMATION CONTENT WORKSHEET C15 Name of the student ______________________ Date ______ Activity 1 – 1. Estimate your waist meaurement and write it down. __________________________ Now pair up and measure up each others waist and record_____________________ Write what do you observe? ______________________________________________________________________________ ______________________________________________________________________________ Discuss the possible reasons for these variations ______________________________________________________________________________ ______________________________________________________________________________ What specifications do you need to give to the shop keeper, if you wish to buy a dress for yourself. ______________________________________________________________________________ ______________________________________________________________________________ If a person’s waist size is measured as 22.6 inches, what waist size would the shop owner give? 23 or 22 ______________________________________________________________________________ Why does this happen? ______________________________________________________________________________ What are the consequences of buying a size 23? ______________________________________________________________________________ What are the consequences of buying a size 22? 190 After watching video clip no 15: to understand the process of rounding off. Recall: What did we learn? __________________________________________________________________________________ __________________________________________________________________________________ What are the rules of rounding off? __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ What are the exceptions? __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ How do we take care of these? __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ Based on what you have learnt: Test yourself: 1. Complete the following sentences and round the number as directed: (a) 1.248 is between 1.24 and 1.25 and when rounded to (nearest hundredth) is 1.25 (b) 4. 534is between ______and _____ and when rounded to (nearest tenths) is ______. (c) 47.203 is between _____and _____and when rounded to ( nearest ones) is _____. (d) 125.598 is between _____and _____ and when rounded to (nearest tenths) is ______ 2. Round each to the nearest tenths 6.25 4.23 17.65 122. 456 3. Round the following to the nearest 100: 101+ 34 295+ 67 345- 32 798 191 54.37 90.099 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C15 Parameter Understand the need of estimation in real life Can analyse the possible reasons for the variations in the two answers Understand the process of rounding off 192 STUDENT’S WORKSHEET – 20 ESTIMATION AND SIGNIFICANCE CONTENT WORKSHEET C16 Name of the student ______________________ 1. Date ______ Measure and write your height to the nearest 10 mm, to the nearest cm, correct to 2 decimal places. 2. In the number 1.2304 Write which is the most significant figure. ______________________ Write the second most significant digit___________________ Write the third most significant digit__________________________ 4. Observe the examples and answer the questions that follow: (a) 170.6, 1(the non zero figure) is the first, 7 is the second and 0 is the third significant figure. (b) In 0.02509, 2 is the first (the non zero figure), 5 the second and 0 is third significant figure. For figure 0.001503, write down (a) the first significant figure( 1 the first non zero figure) (b) The third significant figure (0 ) (c) Watch video clip and write down the rules for significant digits: ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 5. Rewrite the answer in 2 significant figures: 1) 7803 ________________ 2) 1009 ________________ 3) 918.010_______________ 4) 0.0001 _______________ 5) 0.00390_______________ 6) 8120 ________________ 193 7) 7.991 _______________ 8) 729 ________________ SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C16 Parameter Understand the rules for the significant figures. Solving questions on significant figures. 194 STUDENT’S WORKSHEET – 21 RATIO AND PROPORTION CONTENT WORKSHEET C 17 Name of the student ______________________ Date ______ Activity 1: Ratio 1. Discuss and write where in your daily life situations do you use ratio and proportion. ______________________________________________________________________________ ______________________________________________________________________________ 2. Based on the discussions in class: Write how would you define a ratio? ______________________________________________________________________________ ______________________________________________________________________________ 3. Now try the following questions: Look at the pattern below and answer the following questions: (a) Ratio of white beads to total beads ? ________________ (b) Ratio of white beads to total beads ? _______________________ (c) Ratio of black to white ?________________________ (d) Ratio of white to black ? ___________________ (Remember to simplify a ratio, wherever possible) 4. If length of the string is now increased (a) Express the number of black beads of the total number of beads as a fraction. ________________ (b) Observe and describe what is the change in the fraction of black beads to the total beads in the both the cases. ___________________________________________________________________ 195 5. Write the following ratio as instructed 6. Colour 4 of these tins red and 3 tins blue. What is the ratio of red to blue? Activity 2: Proportion Complete the following information, as the discussion takes place in class. 1. To paint a picture I need to make color , mixing 2 scoops of white and 8 scoops of blue.How much color do I need to mix to get the same shade but for three similar paintings?________________________________________________________________ ________________________________________________________________________ 2. In a group there are 15 males and 12 females. (a) What is the ratio of males to females? Give your answer in its simplest form.____________________________________________________________________ (b) What is the ratio of males to the total number of people in the group. _________________________________________________________________________ (c) What does it mean to you? ______________________________________________ (d) I observe that for every 27 people, there are 15 males. If the number of people is trebled, what would the number of males be? 196 3. What is the difference between a ratio and a proportion: ______________________________________________________________________________ ______________________________________________________________________________ 4. Which of the following are in proportion? 35 7 = 20 4 3 12 = 8 16 54 3 = 81 4 64 8 = 40 5 Activity3: Its Maggi, maggi maggi Read the following instructions carefully For each packet of maggi add two cups of water, and allow it to boil. Once the water comes to boil add one maggi cake, and a packet of the spices given along with maggi. Simmer for 2 minutes and presto, your maggi is ready. Now tabulate the ingredients used by you: No. of cakes Cups Maggi Water 1 2 of water used Ratio ½ 2 3 No. of cakes Time taken to prepare Ratio Maggi maggi 1 2 min ½ 2 3 197 SELF ASSESSMENT RUBRIC CONTENT WORKSHEET C17 Parameter Is able to relate the given problem to ratio and solve them Is able to relate the given problem to proportion and solve them 198 STUDENT’S WORKSHEET – 22 POST CONTENT WORKSHEET PC 1 Name of the student ______________________ Date ______ Activity : Test yourself 1. Observe the colored stircase and answer the following; How can you compare the length of the orange staircase to the red one? What is the ratio of the length of the orange to the red staircase? What is the ratio of the length of the green to the blue staircase? Is there any other pair you could choose with same result ? 2. In a bag there are 14 apples and 16 oranges. Write this as a ratio of Oranges: apples Apples: Oranges Apples: total fruit Oranges: total fruit 3. Find the missing number n if 48 n = 72 6 35 7 = 60 n 4. Find the missing number n if 2, 5, 32 and n respectively are in proportion? ( 2 4, 7, n and 84 respectively are in proportion? ( 4 199 5 7 = 32 ) n = n 84 ) 5. The ratio of girls to boys in a class is 2: 3. If there are 12 boys, how many girls are there? 6. The ratio of the marks received by Ali and Amir are in the ratio 6:7. If Ali received 42 out of 50 in the test how many marks did Amir get out of 50? 7. The length and the breadth of a rectangle are in the ratio 1:3. If the breadth of the rectangle is 21cm what is the length of this rectangle? 8. An investment of $5200 is divided in the ratio 6:7between two brothers Ram and Lakshman. (a) How much money does each brother get? (b) Represent the money received by Ram as a fraction of the total money and simplify it. (c) Represent the money received by Lakshman as a fraction of the total money and simplify it. (d) Is there an easier method of writing these fractions (observe the ratio carefully) STUDENT’S WORKSHEET – 23 POST CONTENT WORKSHEET PC 2 Name of the student ______________________ Date ______ 1. Choose the correct answer: (a) Which is the biggest? 4.013 (b) 4.0014 Calculate 2.15 + 0.37? 2.87 (c) Write 16 1 (d) 4.004 3 2.62 20 2.52 in its simplest form. 2 6 4 5 Three friends compare their weights. Who is the heaviest? 200 Ann is 41.58kg (e) Christine is 41.62kg Sabah calculates that she spends 2 3 Drew is 41.47kg of her pocket money on video games, and 1 5 on magazines. What fraction of her pocket money does Sarah spend on games and magazines? 5 11 15 6 (f) What is 42 minutes as a fraction of 1 hour? 3 (g) 13 15 2 5 7 3 10 What is the missing number from this ratio? 11:7 = [ ]:35 55 (h) 10 7 A map has a scale of 1:25000. How much does 1cm on the map represent in real life? 250cm I. 2m 25000cm Bruce, Alice and Sally enter a 'guess the weight of the cake' competition. The actual weight of the cake is 800g. Who is closest to the correct answer? Bruce guesses 788g II. Sally guesses 784g Round 57.53m to the nearest ten meters 54 III. Alice guesses 856g 60 50 Round 3.643 to 2 decimal places. 3.6 3.64 3.7 2. Evaluate the following; 6 2 3 + 11 3 4 5 7 3 −1 12 4 1 + 1 − 1 4 3 2 1 +2 − 1 +2 2 3 6 9 1 7 10 2 −1 11 7 201 10 +41 5 1 − 1 2 8 6 3 4 - 4 3 3. Arrange the following in ascending order: 7 10 , 13 20 ,2 3 0.875, 0.833, 13 16 4. Jane used ½ a piece of ribbon and her sister used 2 3 of the piece. Who used more ribbon and by how much? 5. Write the natural numbers from 102 to 113. What fraction of them are prime numbers? 6. Mrs. Bell made 40 cookies. Her son ate 1 7. Harry was given $15 allowance each week. He spent 3 5 of them .How many cookies did he eat? 5 of it. What fraction did he save? How much did he save in dollars? 8. At a sale some shirts are sold at ½ of their original price. If the original price of these shirts is $30, what is the sale price? 9. Represent the value 4.56 on the number line. 10. Read the value indicated by the pointer; 202 STUD DENT’S S WORK KSHEET T – 24 Posst CONT TENT Wo orksheett PC.3 1. These notices n werre seen on two t markett stalls. At which w stalll was the prrice of one orange cheaperr and by ho ow much? H father gave g him a choice c of getting g it on n a weekly or on a 2. Rob waanted an allowance. His daily basis. He sa aid he woulld either paay him $1.225 a week or pay him m in the folllowing mannerr for a weeek: On Monday M he would giv ve him $0.01; On Tu uesday $0.02; On Wednesday $0.04 and so on through Sunday. Wh hat would you y tell Ro ob to do so he can get morre allowancce? 3. A Drug g Store park king lot hass space for 1000 cars. 2 5 of the spaces aree for compaact cars. On Tueesday, theree were 200 compact caars and som me standard d size cars in the park king lot. The parrking lot was 3 4 full.. How man ny standard d size cars were w in the parking p lott? 4. Shane the t Snail sttarted at thee dot. Whaat side will he be on when w he haas crawled 13 20 of the disttance aroun nd the regular pentago on of equal sides? 5. Jenny bought b 7 t--shirts, onee for each of o her seveen brothers,, for $9.95 each. The cashier charged d her an ad dditional $13.07 in salees tax. She left l the storre with a measely m $7.28. How much money m did Jenny J start with? 203 6. On an average day, Canadians spend $958904.00 buying video games. Of this total $767123.00 is spent on Nintendo games. In one week how much do Canadians spend on Nintendo Games. How much do they spend on other video games in a week? STUDENT’S WORKSHEET – 25 POST CONTENT WORKSHEET PC 4 Name of the student ______________________ Date ______ Activity 3 - Test yourself 1. Solve the following: a) 4 + 1 5 2 b) 1 + 2 3 5 c) 2 − 1 3 4 d) 3 − 3 7 14 e) 1 + 2 + 2 9 3 9 f) 4 + 7 − 1 5 10 2 g) 1 + 2 + 1 6 3 4 h) 7 2. From what number should 4 15 + 1 − 1 3 5 be subtracted to get 1? Give your answer in the simplest 7 form wherever possible. 3. Of the students in the musical, 1 12 play the flute and another 1 dance. What fraction of 3 the students in the band play either the flute or dance? Simplify your answer and write it as a proper fraction or as a whole or mixed number. 4. In the morning, Aman drove to school and used 7 he drove to the movie theater and used 1 3 15 of a gallon of gas. In the afternoon, of a gallon of gas. How much gas did Aman use in all? Simplify your answer and write it as a proper fraction or as a whole or mixed number. 5. Louisa initially filled a measuring cup with 3 she poured 1 3 5 of a cup of syrup from a large jug. Then of a cup back into the jug. How much syrup remains in the measuring cup? Simplify your answer. 204 6. Mr. Higgins cut two pieces of wood for making a toy. One piece of wood measured 5/8 of a meter and the other piece of measured 1 4 of a meter. How much more is the first piece of wood more than the second piece? Simplify your answer. 7. Ben watched a cockroach and an ant on the floor. The cockroach walked 5 6 of a yard and the ant moved ½ of a yard in the same time. How much farther did the cockroach crawl than the ant? 8. What is the combined thickness of these five strips of ribbon 0.008, 0.125, 0.15, 0.185, and 0.005 cm? 9. John is making a telescope for his science project. He joined two metal tubing, one 30.45 inches long and the other 12.75 inches long. How long would the telescope be when the two tubes are joined? 10. For winter’s snowstorm, Mrs Brown bought two coats for her children. One coat cost $55.75 and the other cost $9.30.What was the difference in the costs of the two coats? 11. Two boys were riding their bicycles to school. One boy was traveling at the speed of 15 1 2 km per hour and the other was traveling at a speed of 11.2 km per hour. How much faster is the first boy traveling than the other boy? 12. I took my car for servicing .The usual price for a service is $30.50. However, I got a coupon in a lucky draw in which I got a $15.75 off on the car servicing. How much do I still have to I pay for the service? 13. The planet Mars takes 1 4 5 of our year to go around sun. The planet Jupiter which is farther away from Mars takes 11.9 of our years to go around the sun. How much longer does Jupiter take then Mars to go around the sun? 14. George has three pieces of wood cut from a 22ft long log. One is 10 1 ft long, another is 8 6 1 ft long and the third is 5 7 ft long. 4 8 How far do they stretch if they are laid end to end? How much wood is left over after the three pieces are cut? 15. In the school café the cost of the items are as follows: 205 Ryan ordered one of each item a) How much money does Ryan have to pay? b) He paid with a $10 note. How much money will he get back? 16. A piece of webbing is 7.6m long. If two pieces each 2.3 m and 1.5 m long respectively are cut off, how much is left? 17. Answer the following: (a) Which row shows the highest prices? (b) Which row shows the lowest prices? (c) Which is the cheapest kind of fuel you can buy? The difference between the highest and the lowest rates is called the range. The higher the range the less consistent the fuel is. (d) Which is the most consistent fuel price in the table given above? (e) Which is the least consistent fuel price in the table given above? 206 ACKNOWLEDGMENTS: www.aaamath.com/fra.html www.helpingwithmath.com› BySubject › Fractions www.homeschoolmath.net/worksheets/fraction-decimal.php www.mathsisfun.com/worksheets/decimals.php www.teach-nology.comMathWorksheets www.teachingideas.co.uk/.../contents08fracdecperratprop.htm ww.math-drills.com/decimal.shtml www.aplusmath.com/Worksheets/index.html www.theteachersguide.com/mathlessonplans1.html ww.superteacherworksheets.com/fraction-cont.html www.newtonanddescartes.com/protected/content/ipe_na/.../g6_03_05.pdf http://www.jamit.com.au/worksheets/ http://www.mystfx.ca/special/mathproblems/grade6.html http://www.lessonplanspage.com/Math45.htm http://staff.argyll.epsb.ca/jreed/math7/strand1/1201.htm http://www.dr-mikes-math-games-for-kids.com/fractions-worksheets.html http://www.learningwave.com/lwonline/workingfront/decimalsfront.html TILTLE/LINK Video clip 1 Introduction to fractions http://www.youtube.com/watch?v=cy-8lPVKLIo http://www.youtube.com/watch?v=gJgusNWTIkA&feature=related Video clip 2 Introduction to decimals http://www.youtube.com/watch?v=M_xskCkyAmo&feature=relatedi mproper 207 Video clip 3 placing fractions on a number line http://www.youtube.com/watch?v=7X3sn2Bj-AM&feature=related Video clip 5 placing decimals on a number line http://www.youtube.com/watch?v=uCAXYAu1y7g&feature=related Video Clip 6 fraction converted to mixed fraction http://il.youtube.com/watch?v=cLYa05dOy8E&feature=channel Video clip7 introduces concept of equivalent fractions Video clip 8 Converting fraction to a decimal http://www.youtube.com/watch?v=SGiSW2fvKdw&feature=relatedc onvertin Video clip 9 Process of simplifying fractions http://www.youtube.com/watch?v=7P7zT-iuH80&feature=related Video clip 11 Comparing decimals http://il.youtube.com/watch?v=HCC96awA-FM&feature=related Video clip 12 Adding unlike fractions http://www.youtube.com/watch?v=MEo0rSZ4O3w&NR=1 Video clip 13 how to add and subtract mixed fractions http://www.youtube.com/watch?v=KW4XN0fs1K8&feature=related Video clip 15 rounding of decimals http://www.youtube.com/watch?v=SfdFD1bZatU&feature=channel Video clip 16 Significant figures http://www.youtube.com/watch?v=QXKMj-kaXcE Web link 1 review of four operations on fractions ppt http://www.slideshare.net/kamstrak/fractions-power-point-1261979 Web link 2 check fraction and related words http://www.amathsdictionaryforkids.com/ 208 Web link 3 Ordering of decimals http://www.mathsisfun.com/numbers/ordering-game.php?m=DecTricky Web link 4 Presentation on fractions and important concepts ( To be referred continuously) http://www.kidsolr.com/math/fractions.html Web link 5 Game on Simplification of fractions http://www.funbrain.com/cgi-bin/fob.cgi?A1=s&A2=0 Web link 6 more drill on fraction and decimals http://home.avvanta.com/~math/FDU1.HTM Web link 8 more work for remedial / practice http://www.math-drills.com/fractions.shtml Web link 10 more work for remedial / practice http://www.ixl.com/math/grade/sixth/ 209