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MATHEMATICS

EXEMPLAR PROBLEMS

Class VI

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FOREWORD

The National Curriculum Framework (NCF) 2005 initiated a new phase ofdevelopemnt of syllabi and textbooks for all stages of school education. Consciouseffort has been made to discourage rote learning and to diffuse sharp boundaries

between different subject areas. This is well in tune with NPE 1986 and LearningWithout Burden 1993 that recommend child centred system of education. Thetextbooks for Classes VI, VII and VIII were released respectively in 2006, 2007 and

2008. Overall the books have been well received by students and teachers.

NCF 2005 notes that treating the prescribed textbooks as the sole basis ofexamination is one of the key reasons why other resources and sites of learning are

ignored. It further reiterates that the methods used for teaching and evaluationwill also determine how effective these textbooks prove in making childrens life atschool a happy experience, rather than a source of stress or boredom.

Learning Mathematics is not about remembering solutions or methods butknowing how to solve problems. We hope that teachers will give their students a lotof opportunities to create and formulate problems themselves. We believe it would

be a good idea to ask them to formulate as many new problems as they can. Thiswould help children in developing an understanding of the concepts and principlesof Mathematics. The nature of the problems set up by them becomes varied andmore complex as they become confident with the ideas they are dealing in.

Problem solving strategies give learners opportunities to think rationally,

enabling them to understand and create methods and processes; they becomeactive participants in the construction of new knowledge rather than being passivereceivers. Learners need to identify and define a problem, select or design possiblesolutions and revise or redesign the steps, if required. Thus the role of a teacher gets

modified to that of a guide and facilitator. On being presented a problem, children firstneed to decode it. They need to identify the knowledge required for attempting itand build model for it.

In order to address such issues, the Department of Education in Science andMathematics (DESM) has made an attempt to provide this additional learningmatherial at upper Primary Stage. This resource book contains different types ofquestions of varying difficulty level. These problems are not meant to serve merely

as question bank for examinations but are primarily meant to improve the qualityof teaching/learning process in schools. It is expected that these problems wouldencourage teachers to design quality questions on their own. Students and teachersshould always keep in mind that examination and assessment are meant to testcomprehension, information recall, analytical thinking and problem-solving ability,

creativity and speculative ability.

A team of experts and practicing teachers with an understanding of the subjectworked hard to accomplish this task. The material was thoroughly discussed andedited.

NCERT will welcome suggestions from students, teachers and parents whichwould help us to further imporve the quality of material in subsequent editions.

Professor Yash PalNew Delhi Chairperson

National Steering Committee

NCERT

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PREFACE

The Department of Education in Science and Mathematics (DESM), National Councilof Educational Research and Training (NCERT), initiated the development of

Exemplar Problems in science and mathematics for Upper Primary stage aftercompleting the preparation of textbooks based on National Curriculum Framework2005.

The main objective of the book on Exemplar Problems in Mathematics is toprovide the teachers and students a large number of quality problems with varying

cognitive levels to facilitate teaching learning of concepts in mathematics that arepresented through the textbook for Class VI. It is envisaged that the problemsincluded in this volume would help the teachers to design tasks to assess

effectiveness of their teaching and to know about the achievement of their studentsbesides facilitating preparation of balanced question papers for unit and terminal

tests. The feedback based on the analysis of students responses may help theteachers in further improving the quality of classroom instructions. In addition,

the problems given in this book are also expected to help the teachers to perceivethe basic characteristics of good quality questions and motivate them to framesimilar questions on their own. Students can benefit themselves by attempting the

exercises given in the book for self assessment and also in mastering the basictechniques of problem solving. Some of the questions given in the book are expectedto challenge the understanding of the concepts of mathematics of the students and

their ability in applying them to novel situations.

The problems included in this book were developed and refined through aseries of workshops organised by DESM, that involved practising teachers, subject

experts from universities and institutes of higher learning and the members of themathematics group of DESM. We gratefully acknowledge their efforts and thankthem for their valuable contribution in our endeavour to provide good qualityinstructional material for the school system.

I express my gratitude to Professor Krishna Kumar, Director, NCERT andProfessor G.Ravindra, Joint Director, NCERT for their valuable motivation and

guidance from time to time. Special thanks are also due to Dr. A.K. Wazalwar,Reader in Mathematics, DESM for coordinating the programme, taking pains inediting and refinement of problems and for making the manuscript pressworthy.

We look forward for the feedback from students, teachers and parents for the

further improvement of the contents of this book.

Hukum SinghProfessor and Head

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DEVELOPMENT TEAM

EXEMPLAR PROBLEMS MATHEMATICS

MEMBERS

Amit Bajaj, PGT, CRPF Public School, Rohini, Delhi.

Avantika Dam, TGT, CIE Experimental Basic School, Department of Delhi, Delhi

Hridaykant Dewan, Vidya Bhawan Society, Udaipur, Rajasthan

Hukum Singh, Professorand Head, DESM, NCERT, New Delhi

Jyoti Tyagi, TGT, Sharda Sen RSKV, Trilok Puri, Delhi

K.A.S.S.V. Rao, Lecturer, DESM, R.I.E Bhopal (M.P)

Mahendra Shankar, Lecturer(S.G.) (Retd.), DESM, NCERT, New Delhi

Nagesh Mone, Principal, Kantilal Purushottam Das Shah Prashala, Sangli

(Maharashtra)

P.K. Chaurasia, Lecturer, DESM, NCERT, New Delhi

Priyadarshan Garg, PGT, Kendriya Vidyalaya Beawar, Beawar, (Rajasthan)

Ram Avtar, Professor(Retd.), DESM, NCERT, New Delhi

Sanjay Mudgil, Lecturer, DESM, NCERT, New Delhi

T.P. Sarma, Lecturer, DESM, NCERT, New Delhi

MEMBER - COORDINATOR

Ashutosh K. Wazalwar, Reader, DESM, NCERT, New Delhi

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ACKNOWLEDGEMENT

The Council gratefully acknowledges the valuable contribution of the followingparticipants in the Review Workshop:

Vedala Simhadri,TGT, Kendriya Vidyalaya Srikakulam, (A.P.); K. BalajiTGT,Kendriya Vidyalaya No. 1, Tirupati, (A.P.); Kapil Dev Shukla, TGT, Army School,

Daulat Singh Marg, Jhansi (U.P.); C.F. Mary Bennette, Maths Teacher, T.I MatricHigher Secondary School, Chennai (Tamil Nadu); Asha Gauri Shankar, Reader,Department of Mathematics, Lakshmibai College, Ashok Vihar III, Delhi;

Minakshi Verma, TGT, Delhi Public School, Sector-19, Indira Nagar, Lucknow (U.P);P.S. Khattri, TGTJawahar Navodaya Vidyalaya, Mungeshpur, Delhi; Ra jeevKumar, TGT, Kendriya Vidyalaya, New Mehrauli Road, New Delhi; Shri Anil

Bhaskar Joshi, Teacher, Manutai Kanya Shala, Tilak Road, Akola (Maharashtra);Omlata Singh, TGT, Presentation Convent Senior Secondary School, Delhi; S.K.S.

Gautam, Professor(Retd.), NCERT, New Delhi; Sutapa Mitra, TGT,K.V.No. 2, SaltLake, Kolkata (West Bengal).

Special thanks are due to Professor Hukum Singh, Head, DESM, NCERT forhis support during the development of this book.

The Council also acknowledges the efforts of Deepak Kapoor, Incharge,Computer Station; Ajeet Kumar Dabodiya, Vishan Devi and Disha Dhawan, DTPOperators; Abhimanu Mohanty, Proof Reader.

The contribution of APC Office, Administration of DESM, Publication

Department and Secretariat of NCERT is also duly acknowledged.

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CONTENTS

Foreword iiiPreface iv

Unit 1 Number System 1

Unit 2 Geometry 21

Unit 3 Integers 41

Unit 4 Fractions and Decimals 53

Unit 5 Data Handling 69

Unit 6 Mensuration 89

Unit 7 Algebra 105

Unit 8 Ratio and Proportion 117

Unit 9 Symmetry and Practical Geometry 133Answers 149

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