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Secondary Course211 - MathematicsPractical ManualCourse CoordinatorNeeraj Pratap SinghNATIONAL INSTITUTE OF OPEN SCHOOLING(An autonomous organisation under MHRD, Govt. of India)A-24-25, Institutional Area, Sector-62, NOIDA-201309 (U.P.)Website: www.nios.ac.in, Toll Free No. 18001809393

National Institute of Open SchoolingAugust, 2012 (23,000 copies)Published by the Secretary, National Institute of Open Schooling, A-24/25, Institutional Area,Sector-62, Noida-201309 and Printed at M/s Aravali Printers & Publishers (P) Ltd., W-30, Okhla IndustrialArea, Phase-II, New Delhi - 110020

ADVISORY COMMITTEEDr. S.S. JenaChairmanNIOSDr. Kuldeep AgarwalDirector (Academic)NIOSMrs Gopa BiswasJt. Director (Academic)NIOSCURRICULUM COMMITTEEChairpersonSh. G.D. DhallReader (Retd.), NCERT,K-171, LIC Colony,Paschim Vihar,New Delhi, Pin-110087Prof. Mohan LalSecretary DAV College Managing Committee,E-182, New Rajendra Nagar,New Delhi, Pin-110060,Sh. J.C. NijhawanProf. RamavtarVice Principal (Retd.),Professor (Retd.), NCERT,Sarvoday Vidalay, C-Block,533, Sec-7, Urban Estate,Sarasawti Vihar,Gurgaon, Haryana, Pin-122001New Delhi, Pin-110087Sh. P.K. GargRetd. Principal- Ramjas School,169, Pundrik Vihar, Saraswati Vihar,New Delhi, Pin-110034Sh. Mahendra ShankarLecturer (Retd.), Selection Grade, NCERT,DP-203, Maurya Enclave, Pitampura,New Delhi, Pin-110088Sh. Suvendu Sekhar DasAsst. Director (Academic),National Institute of Open Schooling,A-24/25, Institutional Area, Sector-62,Noida, Pin-201309Sh. Neeraj Pratap SinghSenior Executive Officer (Mathematics),National Institute of Open Schooling,A-24/25, Institutional Area, Sector-62,Noida, Pin-201309Sh. Ishwar ChandraReader (Retd.), NCERT,H.No.- WZ 1427D, Nangal Raya,New Delhi, Pin-110046LESSON WRITERSSh. G.D. DhallReader (Retd.), NCERT,K-171, LIV Colony,Paschim Vihar, New Delhi-110087Sh. J.C. NijhawanVice Principal (Retd.),Sarvoday Vidalay, C-Block,Sarasawti Vihar, New Delhi, -110087COURSE EDITORSSh. P.K. GargRetd. Principal- Ramjas School,169, Pundrik Vihar, Saraswati Vihar,Delhi, Pin-110034Sh. Suvendu Sekhar DasAsst. Director (Academic),National Institute of Open Schooling,A-24/25, Institutional Area, Sector-62,Noida, Pin-201309Dr. Rajpal SinghLecturer Mathematics,Rajkiya Pratibha Vikas Vidyalay,218, Maitri Appt. IP Extention,Patparganj, Delhi, Pin-110092Sh. Neeraj Pratap SinghSenior Excutive Officer ( Mathematics),National Institute of Open Schooling,A-24/25, Institutional Area, Sector-62,Noida, Pin-201309Dr. I.K. BansalProfessor and Head (Retd.), NCERT,Department of Elementary Education,129, Pocket C-13, Sec-3, Rohini,Delhi, Pin-110085GRAPHIC ARTISTSh. Mahesh SharmaGraphic Artist,National Institute of Open Schooling, Noida

Chairman’s MessageDear learner,As the needs of the society in general, and some groups in particular, keep on changingwith time, the methods and techniques required for fulfilling those aspirations alsohave to be modified accordingly. Education is an instrument of change. The right typeof education at right time can bring about positivity in the outlook of society, attitudinalchanges to face the new/fresh challenges and the courage to face difficult situations.This can be very effectively achieved by regular periodic curriculum renewal. A staticcurriculum does not serve any purpose, as it does not cater to the current needs andaspirations of the individual and society.For this purpose only, educationists from all over the country come together at regularintervals to deliberate on the issues of changes needed and required. As an outcomeof such deliberations, the National Curriculum Framework (NCF 2005) came out,which spells out in detail the type of education desirable/needed at various levels ofeducation – primary, elementary, secondary or senior secondary.Keeping this framework and other national and societal concerns in mind, we havecurrently revised the curricula of all the subjects at the secondary level, making themcurrent and need based. Textual material production is an integral and essential partof all NIOS programmes offered through open and distance learning system. Therefore,we have taken special care to make the learning material user friendly, interestingand attractive for you.I would like to thank all the eminent persons involved in making this materialinteresting and relevant to your needs. I hope you find it appealing and absorbing.On behalf of National Institute of Open Schooling, I wish you all a bright andsuccessful future.(Dr. S. S. Jena)Chairman, NIOS

A Note From the DirectorDear Learner,The Academic Department at the National Institute of Open Schooling tries tobring you new programmes every now and then in accordance with your needsand requirements. We are now revising the curriculum in all the subjects at thesecondary level. In order to bring to you a curriculum which is at par with otherboards in the country, we consulted the curriculum in different subjects at theCBSE and Several State Boards of Secondary Education. The National CurriculumFramework developed by the National Council for Educational Research andTraining was kept as a reference point. After making a comprehensive comparativestudy, we developed the curriculum that was functional, related to life situationsand simple. Leading educationists of the country were involved and under theirguidance, we have been able to revise and update the curriculum.At the same time, we also had a look at the learning material. We have removedold, outdated information and added new, relevant things and have tried to makeit attractive and appealing for you.I hope you will find the new material that is now in your hands interesting andexciting. Any suggestions for further improvement are welcome.Let me wish you all a happy and successful future.(Dr. Kuldeep Agarwal)Director (Academic), NIOS

A Word With YouDear Learner,You might be enjoying Mathematics Book I and Book II provided to you by theNational Institute of Open Schooling. Some of the concepts in Mathematics areof abstract nature and in learning such concepts becomes easier when learntthrough activities performed in mathematics laboratory. Mathematics activitiescan be carried out by facilitator and learners to explore, to learn and to createinterest of learners in the subject and develop positive attitude towards thesubject.Keeping the above in view, National Institute of Open Schooling has developeda laboratory manual, which is in your hands. This is in addition to the two booksof your theory part of mathematics.In the beginning, this Laboratory Manual has few pages under introduction,given an idea about the importance and meaning of practical work in mathematics.There are 30 activities given in this manual. Each activity, has detailedinstructions about how to perform the experiment and how to take observationto reach at the conclusion.Though the manual has the scope of recording your observations in tabularform, you are required to maintain a record book, as it carries weightage inpractical examination.In case of any doubt or problem while doing the activity, do not hesitate to writeto us.We hope, you will enjoy performing these activities.Wishing you all the success.Yours,(Neeraj Pratap Singh)Senior Executive Officer (Maths)

IntroductionIt is a general saying that mathematics is by doing only. The concepts for which theproof/verification is done by experimentation or activities are better understood bylearners are retained in their brains for longer period of time. Jen Piaget, anpsychologists, while writing his thesis on concept formation in children, has broughtout that all abstract concepts, can be brought down to the concrete operational stage,and can be understood and retained in a better way. For example, if the abstractconcept of number two, is illustrated by showing two apples, two oranges or anyother two similar objects, which the learners can touch and handle, the learnerunderstands in better way.The human brain is capable of storing only limited amount of information. Theinformation (concepts) which are repeatedly learnt and practiced, are permanentlystored in the brains of children which help in learning of concepts. The activitieswhich are repeatedly done for understanding concepts help the understanding forthose concepts.Mathematics laboratory is a place where learners can learn and explore difficultmathematical concepts and verify mathematical facts, formulae and theorems/resultsthrough a variety of activities and handling related projects using non-costly materialsavailable in their environment. A mathematics laboratory can create mathematicalawareness, skill building, positive attitudes towards the subject and above all ideasof learning by doing.It is the place where learners can learn certain concepts using concrete objects andverify mathematical facts. Formulae and properties by using modes, measurementsand the other activities. Here the learners handle the concrete materials, make modelsof their own by their own imagination and verify the facts/formulae.Design and Lay out of the Mathematics LaboratoryIn mathematics laboratory there should be sufficient space to accommodate atleast30 learners for doing activities/experiments at a time.

The rough design is given here:

Materials Required: Sheets of paper of different Colours, Glazed paper scales, Woodenboards, Nails, Threads, Thermo cole piece, Cardboard square and Triangular grids, Pinsand Clips, Wooden and paper strip, Cutter (paper), Scissor, Adhesive/fevicol, sketchpen, Gun geometry box (Bigger – Wooden), Graph paper (inches/cm both, Pencils ofdifferent colours, Colour box, Knobs, Tracing paperHuman Resoure: It is desirable to have a laboratory assistant (with mathematicsbackground), in charge of the mathematical lab. He expected to have special skillsrequired to handle different instruments, needed for practical work. He should be able torepair things, if they are not in order and keep the materials ready for carrying out activitiesin the following days.Time – Desired: 15% to 20% of total time for mathematics syllabus to be devoted tomathematics laboratory.Scheme of Evaluation: 15 marksThe division of marks in the examination can be done as follows:i)ActivityMarkAssessment of Activity Performed/Recordsof activities prepared10Viva – voce5Total15The proposed practical test is suggested to be held at least 15 days before the theoryexamination.ii) Every students may be given two activities out of which he has to select one andperform it these (in case, the students not comfortable with the given activities, hemay be allowed to select one activity of his choice)iii) Viva-voce can be done at the examination centre by asking questions related to theactivity/project he/she has done.

ContentsS.No.List of ActivitiesPagesVerification of the identities (1 to 4)1.(a b)2 a2 2ab b212.(a-b) 2 a2 - 2ab b233.(a2-b2) (a b) (a-b)3332524.(a b) a b 3a b 3ab75.To find the HCF of two given numbers by division method.96.Equivalent fractions117.To verify that a linear equation in two variables has infinite number of solutions.138.To find the condition for consistency of a system of linear equations.159.To verify the relation between roots and coefficients of a quadratic equation.1910.To verify graphically that a quadratic polynomial can have at most two zeroes.2111.To verify that a given sequence is an A.P.2312.To find the sum of first n odd natural numbers.2513.To find the sum of first n natural numbers.2714.To find the sum of first n terms of an arithmetic progression.o3915.To verify that the sum of the angles of a triangle is 1803116.To verify that the angles opposite to equal sides of a triangle are equal.3317.To verify mid point theorem.3518.To verify basic proportionality theorem.3719.To verify Pythagoras theorem.3920.To verify the relation between the ratio of areas of two similar triangles and their sides.4121.To find the area of a circle.4322.To demonstrate that the opposite angles of a cyclic quadrilateral are supplementary.4523.To verify that the equal chords of congruent circles subtend equal angles at the centre.4724.To find the area of a trapezium.4925.To find the total surface area of a cube.5126.To find the formula for the curved surface area of a cone.5327.To find the relationship among the volumes of right circular cone, right circularcylinder and a hemisphere of same radius and same height.()5528.3322To verify the identity a b ( a b ) a ab b29.To draw a triangle equal in area to a parallelogram.5930.To find the incentre of different types of triangles.6157

SECONDARYLEVELLaboratory ManualACTIVITY 1Title: Verification of the identity (a b)2 a2 2ab b2Expected background knowledge:- Area of a square and a rectangle.Objectives:NotesAfter performing this activity the learner will be able to verify and demonstratethe identity (a b)2 a2 2ab b2Materials required:(i)Cardboard(ii)White chart paper(iii)Two glazed papers of different colours, say red and green(iv)Pair of scissors(v)Gum(vi)Coloured ball point pens.(vii)Pencil and geometrical instruments.1Mathematics Secondary Course1

SECONDARYLEVELNotesLaboratory ManualPreparation for the activity:(i)On a white chart paper, draw a square ABCD of side (a b) units (say a 7cm, b 4cm)Cut it out and paste it on the cardboard.(ii)Cut off two rectangles of dimensions a b (7cm 4 cm) from red colour paper and asquare of side b (4 cm) from green colour paper.(iii)Paste these cutouts on the square ABCD as shown in the figure and name them asrectangle EFGD, square FHCG and rectangle KBHFDemonstration and Use2In the figure, area of square ABCD (AB) 2 (a b) 2 unitArea of square AKFE (AK) 2 a2 unit2.Area of rectangle KBHF (KF FH) a b ab unit2Area of square FHCG (HC)2 b2 unit2Area of rectangle EFGD ED GD a b ab unit2From the figure we may observe that:Area of square ABCD Area of square AKFE area of rectangle KBHF area of squareFHCG area of rectangle EFGD.