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Class VI

MATHEMATICS EXEMPLAR PROBLEMS

F OREWORD
The National Curriculum Framework (NCF) 2005 initiated a new phase of developemnt of syllabi and textbooks for all stages of school education. Conscious effort 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 Learning Without Burden 1993 that recommend child centred system of education. The textbooks 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 of examination 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 evaluation will also determine how effective these textbooks prove in making childrens life at school a happy experience, rather than a source of stress or boredom. Learning Mathematics is not about remembering solutions or methods but knowing how to solve problems. We hope that teachers will give their students a lot of 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. This would help children in developing an understanding of the concepts and principles of Mathematics. The nature of the problems set up by them becomes varied and more 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 become active participants in the construction of new knowledge rather than being passive receivers. Learners need to identify and define a problem, select or design possible solutions 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 first need to decode it. They need to identify the knowledge required for attempting it and build model for it. In order to address such issues, the Department of Education in Science and Mathematics (DESM) has made an attempt to provide this additional learning matherial at upper Primary Stage. This resource book contains different types of questions of varying difficulty level. These problems are not meant to serve merely as question bank for examinations but are primarily meant to improve the quality of teaching/learning process in schools. It is expected that these problems would encourage teachers to design quality questions on their own. Students and teachers should always keep in mind that examination and assessment are meant to test comprehension, information recall, analytical thinking and problem-solving ability, creativity and speculative ability. A team of experts and practicing teachers with an understanding of the subject worked hard to accomplish this task. The material was thoroughly discussed and edited. NCERT will welcome suggestions from students, teachers and parents which would help us to further imporve the quality of material in subsequent editions. Professor Yash Pal Chairperson National Steering Committee NCERT

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PREFACE
The Department of Education in Science and Mathematics (DESM), National Council of Educational Research and Training (NCERT), initiated the development of Exemplar Problems in science and mathematics for Upper Primary stage after completing the preparation of textbooks based on National Curriculum Framework 2005. The main objective of the book on Exemplar Problems in Mathematics is to provide the teachers and students a large number of quality problems with varying cognitive levels to facilitate teaching learning of concepts in mathematics that are presented through the textbook for Class VI. It is envisaged that the problems included in this volume would help the teachers to design tasks to assess effectiveness of their teaching and to know about the achievement of their students besides facilitating preparation of balanced question papers for unit and terminal tests. The feedback based on the analysis of students responses may help the teachers in further improving the quality of classroom instructions. In addition, the problems given in this book are also expected to help the teachers to perceive the basic characteristics of good quality questions and motivate them to frame similar questions on their own. Students can benefit themselves by attempting the exercises given in the book for self assessment and also in mastering the basic techniques of problem solving. Some of the questions given in the book are expected to 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 a series of workshops organised by DESM, that involved practising teachers, subject experts from universities and institutes of higher learning and the members of the mathematics group of DESM. We gratefully acknowledge their efforts and thank them for their valuable contribution in our endeavour to provide good quality instructional material for the school system. I express my gratitude to Professor Krishna Kumar, Director, NCERT and Professor 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 in editing 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 Singh Professor 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, Professor and 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 following participants in the Review Workshop: Vedala Simhadri, TGT , Kendriya Vidyalaya Srikakulam, (A.P.); K. Balaji TGT , 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 Matric Higher 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, TGT Jawahar Navodaya Vidyalaya, Mungeshpur, Delhi; Rajeev Kumar, 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, Salt Lake, Kolkata (West Bengal). Special thanks are due to Professor Hukum Singh, Head, DESM, NCERT for his support during the development of this book.

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The Council also acknowledges the efforts of Deepak Kapoor, Incharge, Computer Station; Ajeet Kumar Dabodiya, Vishan Devi and Disha Dhawan, DTP Operators; Abhimanu Mohanty, Proof Reader . The contribution of APC Office, Administration of DESM, Publication Department and Secretariat of NCERT is also duly acknowledged.

CONTENTS
Foreword Preface Unit Unit Unit Unit Unit Unit Unit Unit Unit 1 2 3 4 5 6 7 8 9 Number System Geometry Integers Fractions and Decimals Data Handling Mensuration iii iv 1 21 41 53 69 89

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Algebra Ratio and Proportion 105 117 Symmetry and Practical Geometry Answers 133 149

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