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CLASS VI UNIT-6 CBSE-i BASIC GEOMETRICAL CONCEPTS 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 may be taken are duly mentioned. In case any thing has been missed out, the Board will be pleased to rectify the error at the earliest possible opportunity. All Rights of these documents are reserved. No part of this publication may be reproduced, printed or transmitted in any form without the prior permission of the CBSE-i. This material is meant for the use of schools who are a part of the CBSE-International only. CBSE-i BASIC GEOMETRICAL CONCEPTS Class - VI UNIT-6 Shiksha Kendra, 2, Community Centre, Preet Vihar,Delhi-110 092 India 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 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 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 Groups: Classes IX-X Mathematics : Science : Dr. K.P. Chinda Ms. Charu Maini Mr. J.C. Nijhawan Ms. S. Anjum Ms. Rashmi Kathuria Ms. Meenambika Menon Ms. Reemu Verma Ms. Novita Chopra Ms. Neeta Rastogi Ms. Pooja Sareen Political Science: Ms. Sharmila Bakshi Ms. Archana Soni Ms. Srilekha Dr. Rajesh Hassija Ms. Rupa Chakravarty Ms. Sarita Manuja Ms. Himani Asija Dr. Uma Chaudhry History : Ms. Jayshree Srivastava Ms. M. Bose Ms. A. Venkatachalam Ms. Smita Bhattacharya Economics: Ms. Mridula Pant Mr. Pankaj Bhanwani Ms. Ambica Gulati Material Production Groups: Classes VI-VIII English : Ms. Rachna Pandit Ms. Neha Sharma Ms. Sonia Jain Ms. Dipinder Kaur Ms. Sarita Ahuja Dr. Indu Khetarpal Ms. Vandana Kumar Ms. Anju Chauhan Ms. Deepti Verma Ms. Ritu Batra Science : Dr. Meena Dhami Mr. Saroj Kumar Ms. Rashmi Ramsinghaney Ms. Seema kapoor Ms. Priyanka Sen Dr. Kavita Khanna Ms. Keya Gupta Mathematics : Ms. Seema Rawat Ms. N. Vidya Ms. Mamta Goyal Ms. Chhavi Raheja Political Science: Ms. Kanu Chopra Ms. Shilpi Anand 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 Geography: Ms. Suparna Sharma Ms. Leela Grewal History : Ms. Leeza Dutta Ms. Kalpana Pant Ms. Nandita Mathur Ms. Seema Chowdhary Ms. Ruba Chakarvarty Ms. Mahua Bhattacharya Coordinators: Dr. Sadhana Parashar, Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi, Head (I and R) E O (Com) E O (Maths) O (Science) Shri R. P. Sharma, Consultant Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO Ms. Seema Lakra, S O Ms. Preeti Hans, Proof Reader Con ten t Preface Acknowledgment 1. Syllabus ............................................................................................... 1 2. Scope Document ..................................................................................... 2 3. Teacher's Support Material ......................................................................... 6 å Teacher's note ................................................................................................................ 7 Activity skill matrix ......................................................................................................... 12 å Warm up Activity W1 .................................................................................................... 14 Puzzle Time .................................................................................................................... 14 å Warm up Activity W2 .................................................................................................. 15 Appreciate your knowledge .......................................................................................... 15 å Pre Content Worksheet P1 ........................................................................................... 16 Making Math A - Maze- ing .......................................................................................... 16 å Pre Content Worksheet P2 ........................................................................................... 16 Origin of Geometry ....................................................................................................... 16 å Content Worksheet CW1 ................................................................................................. 16 Point, Line, Line Segment and a Ray-1 ............................................................................. 16 å Content Worksheet CW2 .................................................................................................. 16 Point, Line, Line Segment and a Ray-2 ............................................................................. 16 å Content Worksheet CW3 .................................................................................................. 17 Skill Drill ....................................................................................................................... å Content Worksheet CW4 ................................................................................................. Constructing Line Segment .......................................................................................... Content Worksheet CW5 ................................................................................................. å Constructing Line Segments by Compass ..................................................................... 17 17 17 17 17 å Content Worksheet CW6 ................................................................................................... Practice Worksheet ........................................................................................................ å Content Worksheet CW7 ................................................................................................... Angles, Interior and Exterior of an Angle ........................................................................ å Content Worksheet CW8 .................................................................................................. Types of Angles ............................................................................................................. å Content Worksheet CW9 .................................................................................................. Angles through Paper Folding ....................................................................................... å Content Worksheet CW10 ................................................................................................ Angles and Applications ................................................................................................ å Content Worksheet CW11 ................................................................................................ Practice Worksheet ....................................................................................................... å Content Worksheet CW12 ................................................................................................ Perpendicular Lines ....................................................................................................... å Content Worksheet CW13 ................................................................................................ Perpendicular Bisector .................................................................................................. å Content Worksheet CW14 ................................................................................................ Constructing Perpendicular Bisector ............................................................................. å Content Worksheet CW15 ................................................................................................ Measuring and Constructing Angles using Protractor ................................................... 18 18 18 18 20 20 22 22 22 22 23 23 23 23 26 26 26 26 27 27 å Content Worksheet CW16 ................................................................................................ 27 Copying and Duplicating Angles using Compass and Ruler ............................................ 27 å Content Worksheet CW17 ................................................................................................ 27 Constructing and Bisecting Angles ................................................................................ å Content Worksheet CW18 ................................................................................................. Triangles ........................................................................................................................ 27 28 28 å Content Worksheet CW19 ................................................................................................. 28 Quadrilaterals ................................................................................................................ 28 å Content Worksheet CW20 ................................................................................................. 28 Practice worksheet: Triangles and Quadrilaterals ........................................................... 28 å Content Worksheet CW21 ................................................................................................ Introducing Cirles .......................................................................................................... å Content Worksheet CW22................................................................................................. Circles............................................................................................................................ å Content Worksheet CW23................................................................................................. Fun with Circles 1 ................................................................................................................ å Content Worksheet CW24................................................................................................. Practice Worksheet ....................................................................................................... å Content Worksheet CW25................................................................................................. Fun with Circles 2 ................................................................................................................ 29 29 29 29 30 30 30 30 30 30 å Post Content Worksheet PCW1 .................................................................................... 31 å Post Content Worksheet PCW2 ..................................................................................... 31 Assessment of the Chapter............................................................................................. 31 4. Assessment Guidance Plan ............................................................................................... 32 5. Study Material .................................................................................................................... 36 6. Student's Support Material (Student's Worksheets) .......................................................... 76 å SW 1 : Warm up Activity W1..................................................................................... 77 Puzzle Time.................................................................................................................... 77 å SW 2 :Warm up W2: Appreciate your knowledge .......................................................... 79 å SW 3 :Pre Content Worksheet P1 ................................................................................. 82 Making Math A - Maze- ing ........................................................................................... 82 å SW 4 :Pre Content Worksheet P2................................................................................... 83 Origin of Geometry ........................................................................................................ 83 å SW 5 :Content Worksheet CW1 ........................................................................................ 85 Point, Line, Line Segment and a Ray-1 ............................................................................. 85 å SW 6 :Content Worksheet CW2 ....................................................................................... 89 Point, Line, Line Segment and a Ray-2 ............................................................................ 89 å SW 7 :Content Worksheet CW3 ....................................................................................... 97 Skill Drill ....................................................................................................................... 97 å SW 8 :Content Worksheet CW4 ........................................................................................ Constructing Line Segment ........................................................................................... 98 98 å SW 9:Content Worksheet CW5 ......................................................................................... 100 Constructing Line Segment by Compass.......................................................................... 100 å SW 10 :Content Worksheet CW6 ...................................................................................... 100 Practice Worksheet ........................................................................................................ å SW 11:Content Worksheet CW7....................................................................................... Angles, Interior and Exterior of an Angle........................................................................ å SW 12 :Content Worksheet CW8 ..................................................................................... Types of Angles ............................................................................................................ å SW 13 :Content Worksheet WC9 ..................................................................................... Angles through Paper Folding ....................................................................................... å SW 14 :Content Worksheet CW10 ................................................................................... Angles and Applications ............................................................................................... å SW 15 :Content Worksheet CW11 ................................................................................... Practice Worksheet: Angles and Types of Angles .......................................................... å SW 16 :Content Worksheet CW12 ................................................................................... Perpendicular Lines........................................................................................................ 100 104 104 108 108 110 110 112 112 113 113 116 116 å SW 17 :Content Worksheet CW13.................................................................................... 117 Perpendicular Bisector.................................................................................................. 117 å SW 18 :Content Worksheet CW14.................................................................................... 119 Constructing Perpendicular Bisector............................................................................ 119 å SW 19 :Content Worksheet CW15 .................................................................................... 120 Measuring and Constructing Angles using Protractor ................................................... 120 å SW 20 :Content Worksheet CW16 .................................................................................... 121 Copying or Duplicating Angles using Compass and Ruler ............................................... 121 å SW 21 : Content Worksheet CW17 .................................................................................. 122 Constructing and Bisecting Angles ............................................................................... 122 å SW 22 :Content Worksheet CW18 ................................................................................... 122 Triangles ...................................................................................................................... 122 å SW 23 :Content Worksheet CW19 .................................................................................... Quadrilaterals ................................................................................................................ 125 125 å SW 24 :Content Worksheet CW20 .................................................................................... 127 Practice worksheet: Triangles and Quadrilaterals .......................................................... 127 å SW 25 :Content Worksheet CW21 .................................................................................... 129 Introducing Circles ........................................................................................................ 129 å SW 26 :Content Worksheet CW22 .................................................................................... 132 Circles .......................................................................................................................... 132 å SW 27 :Content Worksheet CW23 ................................................................................... Fun with Circles ............................................................................................................ å SW 28 :Content Worksheet CW24 ................................................................................... Practice Worksheet: Circles .......................................................................................... å SW 29 :Content Worksheet CW25 ................................................................................... Fun with Circles 2 ......................................................................................................... 133 133 134 134 136 136 å SW 30 :Post Content Worksheet PCW1 ........................................................................... 138 å SW 31 :Post Content Worksheet PCW2........................................................................... 143 å Acknowledgments ........................................................................................................ 146 å Suggested videos/ links/ PPT's .................................................................................... 147 SYLLABUS UNIT-6 Introduction to geometry, Its linkage with and reflection in everyday experience. Line, line segment, ray Pair of lines- intersecting and parallel lines Measure of Line segment Drawing of a line segment Angle — Vertex, arm, interior and exterior, Types of angles- acute, obtuse, right, straight, reflex, complete and zero angle Measure of angles, Perpendicular lines Drawing a line perpendicular to a given line from a point a) on the line b) outside the line. BASIC GEOMETRICAL CONCEPTS Perpendicular bisector Construction of angles equal to a given angle (using compass)) Angle bisector making angles of angle 60o, 120o,30o, 45o, 90o etc. (using compass) Triangle — vertices, sides, angles, interior and exterior, altitude and median Quadrilateral — Sides, vertices, angles, diagonals, adjacent sides and opposite sides (only convex quadrilateral) Interior and exterior of a quadrilateral Circle — Centre, radius, diameter, arc, sector, chord, segment, semicircle, circumference, exterior. Construction of circle 1 interior and Scope document Review and Recall: Geometrical representation of fractions Algebra involving solving linear equations in one variable Concepts: Introduction to geometry and its linkage with and reflection in everyday experience Line, line segment, ray Pair of lines Intersecting and parallel lines Measuring and drawing line segments Angle — Vertex, arm, interior and exterior Types of angles- acute, obtuse, right, straight, reflex, complete and zero angle Fractions of a revolution and compass directions Perpendicular lines and its construction from a point a) on the line b) outside the line. - Through paper folding - Using set squares - And using compass and ruler Perpendicular bisector Measuring and drawing of angles using protractor. Constructing copying of an angle using compass and ruler Angle bisector Construction of angles 60o, 120o using ruler and compass 2 Making angles of 30o, 45o, 90o by ruler and compass Triangle — vertices, sides, angles, interior and exterior, altitude and median Quadrilateral — Sides, vertices, angles, diagonals, adjacent sides and opposite sides (only convex quadrilateral) Circle — Centre, radius, diameter, arc, sector, chord, segment, semicircle, circumference, interior and exterior. Construction of circle Learning objectives Understands and explore linkage of geometry with and reflection in everyday experience Understands and differentiates between line, line segment and ray Identifies the pair of lines and differentiates between intersecting and parallel lines Measures and draw line segments using ruler Define and name angle vertex and its arm, identifies interior and exterior of an angle Differentiate between different types of angles Find fractions of a revolution and identify and explore compass directions Define perpendicular lines and construct it from a point a) on the line b) outside the line. - Through paper folding - Using set squares - And using compass and ruler 3 Define and construct perpendicular bisector Measure and draw angles using a protractor Construct duplicate of an angle using compass and ruler Define and construct angle bisector Construct angles 60o and 120o using ruler and compass Construct angles 30o, 45o, 90o by ruler and compass Identify and name triangle , its vertices, sides, angles, interior and exterior Define altitude and median of a triangle Identify and name quadrilateral, its sides, vertices, angles, diagonals, adjacent sides and opposite sides (only convex quadrilateral) Identify and define circle, its centre, radius, diameter, arc, sector, chord, segment, and semicircle, circumference, interior and exterior. Construct circles Activities/resources/projects: 1. Measure your angle of vision by working in teams of two. Your partner should stand behind you holding a pencil at your eye height. This pencil should be gradually moved forward until you can see it. 4 Cross curricular links: Art: Make a poster about angles from the following suggestions: Sports: Gymnastics- Marks different types of angles in the following figures. Farms: Angles between fencelines, the angles through which gates swing, the angles of a loading ramp, the angle between the sides and the bottom of a trough. Cars: Angles between windscreens and bonnets, angles of seat backs, angles through which the doors and bonnet and boot move as they are opened. Clothing: Angles between seams, angles of collar peaks, angles in patterns for a garment. Leisure: In snooker, investigate the angle at which a ball must be hit if it is to be sunk. You would have to consider the balls at different positions on the table. 5 6 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. 2. 3. 4. 5. 6. Children learn to enjoy mathematics rather than fear it. Children see mathematics as something to talk about, to communicate through, to discuss among themselves, to work together on. Children pose and solve meaningful problems. Children use abstractions to perceive relation-ships, to see structures, to reason out things, to argue the truth or falsity of statements. 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. 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. Revisiting of the previous knowledge and consolidating basic concepts 7 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 students in the middle grades have an informal knowledge about a point, line and plane. By this stage they are aware of various 2 dimensional and 3 dimensional shapes. They have an intuitive idea about different polygons, angles and triangles. They are now expected to define and draw all these terms and their components mathematically. The student should be able to describe and represent characteristics and relationships between two-dimensional shapes and three-dimensional objects in a variety of orientations and positions. They should be able to explore a variety of geometric shapes and examine their characteristics. The teacher should encourage here, the use of geoboards and dot papers and dynamic geometric softwares such as goegebra for 2 dimensional geometric shapes to explore the properties of the figures. Through this deductive approach of building up the lesson, the teacher can ensure that the students find out all the properties by guidance. The teacher should encourage the children to quote the examples of parallel, perpendicular and intersecting lines around them. They should also develop a clear idea about the difference between a line, line segment and a ray. They should be encouraged to find out different types of angles existing around them, in nature and also some approximate measures that they can make using their body arms and legs. One interesting and simple example to identify the different types of angles is by finding out some maps through google earth or bing maps which have some circles or angles, capture the snapshots of these places and insert them in geogebra worksheet. The teacher can then ask the students to find out or draw the angles or circles and then measure the angles or the radii formed. Questions like what should the OB (see figure below) and like represent? One example of geogebra worksheet is as shown below 8 B O Such an activity not only reinforces the learning that has taken place, but also gives the students an idea that geometrical figures can be found everywhere. To explore about the various types of angles, the students may be asked to work in pairs, use a digital camera to click one of the two students making various types of angles that they can make using their body parts. They can be as creative as possible. Now ask them to make a photostory or a photosynth on their computers. You may down load photosynth from http://photosynth.net/ and photostory from http://www.microsoft.com/download/en/details.aspx?id=11132. This pair can ask the other students to quickly find out and list down the angle that is been formed in the pictures. To be taken care of … At this stage the students face a transition from the informal knowledge of geometrical concepts to a formal mathematical definition of the geometrical terms. The teacher 9 should be careful in introducing the correct terminologies and not only telling them the common error areas but also the reason why those errors may occur and why they are wrong. The definition of a line and a point should be carefully explained. The need and difference between a line, line segment and a ray should be clarified and may be explained using google/bing maps. Common Errors Type of error Notations Ray Naming of angle Error made Correction a) Symbols of line segment and line b) Using small letters to represent a line Ray OA or OA ray AO or AO a) b) Write AB for line segment AB When line is represented using two points, use capital letters.eg. line AB or AB Line may also be represented by small letters as line l or l v/s. Ray OA or OA is different from ray AO or AO as they have different initial points and direction. Naming of two or more angles with the same The angles are angle BAC, angle vertex B CAD and angleBAD. C A D The names of the angles cannot be given as angle A 10 Overview of the students’ worksheets The first Warm up activity (W1) is a brain exerciser where the students are using their previous knowledge of informal geometry and shall generalize the results for the given figures. In Warm Up activity (W2), the students use their previous knowledge of mathematics to associate the concepts like fractions with geometrical figures. Through Precontent 1, (P1), the students are informally introduced to line segments through recreational activity. In Precontent 2 (P2), the students try to find out how and where geometry exists around us. The focus on the warm up and 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. The content worksheets from C1 through C29 aim at achieving the above stated learning objectives. Not only shall the students learn the basic concepts of geometry, they shall be encouraged to find how closely and beautifully mathematics is related to their daily lives. The teacher may encourage them to make projects where they can appreciate the applicability of geometry in real life. Further the post content activity is designed to assess the students’ understanding of the concepts learnt in the chapter. The post content worksheet 2, (p2) is an assessment test to test the concepts learnt in totality. It is not included in the students’ Worksheets. The teacher may use it as a timed test by giving print outs to students. 11 Activity – Skill Matrix Activity Name of the activity Skills learnt Warm up (W1) Puzzle time Visual and mathematical understanding Warm up (W2) Appreciate your knowledge Knowledge and understanding Pre content (P1) Making Math A – Visual understanding Maze- ing Pre content (P2) Origin of Geometry Comprehension Content worksheet (CW1) Point, line, line segment Reasoning and comparison and a ray-1 skills Content worksheet (CW2) Point, line, line segment Comparative and and ray-2 explorative skills Content Worksheet (CW3) Skill drill Knowledge and understanding Content Worksheet (CW4) Constructing line segments Measurement skills Content worksheet (CW5) Constructing line segments Geometrical skills by compass Content Worksheet (CW6) Practice Worksheet Knowledge and Understanding Content Worksheet (CW7) Angles, exterior interior Content Worksheet (CW8) Types of Angles and Representation skills Diagramatical understanding and appreciation of 12 mathematics Content Worksheet (CW9) Content Worksheet Angles through Paper Appreciation of Folding mathematics Angles and applications Understanding and (CW10) Appreciation of mathematics (CW 11) Practice worksheet : Angles Understanding and and types of angles geometrical skills Content Worksheet Perpendicular Lines Content Worksheet (CW12) Content Worksheet Understanding and geometrical skills Perpendicular Bisector Exploration skills (CW13) Content Worksheet Constructing Perpendicular geometrical skills (CW14) Bisector Content Worksheet Measuring and constructing angles using protractor (CW15) Content Worksheet (CW16) geometrical skills Copying and Duplicating geometrical skills angles using compass and ruler (CW17) Constructing and bisecting geometrical skills Angles Content Worksheet Triangles Content Worksheet Reasoning (CW18) Content Worksheet and thinking Quadrilaterals Reasoning skills Practice worksheet Quadrilaterals and Knowledge and (CW19) Content Worksheet (CW20) 13 understanding critical Triangles Content Worksheet Introducing circles (CW21) Knowledge and understanding Content Worksheet Circles Application skills Fun with Circles 1 Application skills Practice Worksheet: Circles Knowledge and (CW22) Content Worksheet (CW23) Content Worksheet (CW24) understanding Content Worksheet Fun with Circles 2 (CW25) Application and knowledge skills Post Content Worksheet Extended Practice (PCW1) Knowledge and self learning Post Content Worksheet Extended Practice (PCW2) Knowledge and Self Learning WARM UP WORKSHEET – W1 Puzzle time Description – This is taken as a class activity. Teacher divides the class into groups of 23 students. Each group is given a puzzle to solve. The groups interchange their puzzles after solving them. This activity is done to prepare the children for understanding the basic geometrical shapes in the chapter. Solutions of the puzzles: Puzzle 1 - Solution - 27 Puzzle 2 – Solution 14 Puzzle 3 – a) Solution b) Puzzle 4 a) b) c) Execution – Teacher asks the students to express their views one by one. WARM UP WORKSHEET – W2 APPRECIATE YOUR KNOWLEDGE Objective – Recapitulation of geometrical representation of fractions, Algebra involving solving linear equations in one variable Description – The teacher will hand out the worksheet for the recapitulations of the concepts done in the previous chapter. Through this worksheet the teacher will be able to identify the students who are weak in the concepts of the previous chapter and will be able to work with them. 15 PRE CONTENT WORKSHEET P1 Making Math A – Maze- ing Objective - Explore and know about the need and importance of lines. Description –Teacher will give the handout of Math maze. Teacher will encourage the students to use ruler and pencil to solve this puzzle. This will give them the idea of moving in straight line, intersecting lines etc. PRE CONTENT WORKSHEET P2 Origin of Geometry Objective - Explore and know about the origin and importance of geometry Description – A handout of a passage on origin of geometry is given to students. The passage talks about origin and application of geometry in daily life. The students read the passage answer the questions that follow. Students will be able to identify the geometrical shapes those can be used in art/architecture around them. Follow up – Teacher will discuss the passage and the answers to the questions in the class. CONTENT WORKSHEET C W1 Point, Line, Line Segment and a Ray-1 Activity 1 - Defining Point, line, line segment and a ray Description – Teacher will take up this class activity. Through this inductive activity teacher will introduce the formal definitions of a point, line segment and a ray. This activity will help students visualize and hence differentiate between a line, a line segment and a ray. CONTENT WORKSHEET CW2 Point, Line, Line Segment and a Ray-2 Activity 2 - Joining the points 16 Objective – To reinforce the idea of point, line and line segment Material Required - Dictionary/Picture math dictionary available online. Description - Student would refer to the dictionary and find the meaning of the word point, line and line segment. This activity also introduces the concept of the shortest distance between two points and its practical applications. CONTENT WORKSHEET CW 3 Skill drill Objective – Test the understanding of the meaning of point, line and line segment. Description – This worksheet will help students revising all the concepts learnt so far in a play way method. Two exercises are given in the game form. The first game is matching which will help students recall the knowledge. The second game is the innovation in the game of knot and cross game. While playing the game students will apply the concepts they have learnt. CONTENT WORKSHEET CW4 Constructing Line Segment Activity 4 – Measuring and Constructing Line Segments Objective - Make students measure and construct line segments using a ruler. Material Required – Pencil and ruler Description- Teacher introduces the use of ruler through this activity. While during this worksheet the students will learn how to measure and construct the straight lines appropriately. CONTENT WORKSHEET CW5 Constructing Line Segment using Compass Activity 5- To construct and copy a line segment using compass and ruler Objective: To build the skill of accurate construction using compass and ruler. 17 Material Required – Compass, ruler and pencil Description – Teacher will make the students practice using compass for 10-15 minutes. Before they start this activity, teacher will make sure that all the students are able to handle the pair of compasses with comfort. Teacher demonstrates the use of pair of compasses. Students are given some tasks where the students use compass to construct a line segment, copy a line segment and even to measure the line segment. Video clip 2 is also shown to the students. In this video clip students learn how to copy a given line segment with the help of pair of compasses. Follow up - Teacher checks and monitors the constructions done by each student. CONTENT WORKSHEET CW6 Practice Worksheet Activity 1- Independent Practice Objective - Recapitulation of the work done in content C 1.1. Description - This is a recapitulation worksheet to make students recall all the concepts learnt in content C 1.1. Execution - Students will solve the worksheet given to them and note down their answers. Teacher will discuss answers to check for accuracy and understanding. Follow up - Teacher checks the worksheet for check the student’s concept, understanding and accuracy. CONTENT WORKSHEET CW7 Angles, Interior and Exterior of an Angle Activity 1- Defining angles, interior and exterior of an angle Objective: To understand the concept of angles, interior and exterior of an angle. Description: Teacher takes this as a class activity. Few figures are given to the students. The students are asked to observe the figures carefully before teacher introduces the concept of the angle, interior and exterior of the angles. Once the students are 18 conversant with the concept a worksheet is given to them to reinforce these concepts. Through this worksheet the concept of naming the angles, vertex and arms of the angles are also introduced to the students. Sample figures shown to the students: - . 19 CONTENT WORKSHEET CW8 Types of Angles Activity 2- Types of angles Objective: to distinguish and name different types of angles. Materials required: Wall clock, paper, pair of scissors, protractor. Description: Teacher may introduce the different types of angles with the help of different methods as described below. After the explanation the teacher will distribute hand out of worksheet 12 to the students and asks them to solve them using the tester. Sample: 20 Teacher asks the students to make a right angle tester using paper to find right angles around them. Using this tester, make a list of angles that are less than 90o. Use math dictionary to name all the angles less than 90o. _________________ Make a list of angles that are more than 90o but less than the 1800. What are these angles known as?____________________ Which angles gives you a straight line?_________________. What would be the name of those angels?._______________. Make a list of angles that are greater than the straight angle but less than the 3600. Find out a name for these angles. _____________ Turn by four right angles. How much angle did you turn in total? Which direction are you facing? _______________. Name the angle you have turned in the above case. __________ 21 CONTENT WORKSHEET CW9 Angles through Paper Folding Activity 1 – Fun Activity Objective – Creating and recognizing angles through Paper folding Materials required: Origami sheet of 10 cm by 10 cm. Description- Teacher will take this fun activity of recognizing and making angles by paper folding. Students follow the instructions given by the teacher and make the penguin. This will be good and interesting way of learning and practicing angles. CONTENT WORKSHEET CW10 Angles and Application Activity 2- Application of Geometry in day to day life Objective: To understand and learn the application of geometrical concepts in daily life. Description: Through this worksheet the teacher will try to show the importance of geometry in day to day life. Teacher takes the example of wall clock as the students see it all the time and can visualize the concept easily. . 22 CONTENT WORKSHEET CW11 Practice Worksheet: Angles and Types of Angles Activity 3- Independent practice Objective - Recapitulation of the work done in content C 2.1and 2.2 Description - This is a recapitulation worksheet to make students recall all the concepts learnt in content C 2.1 and 2.2. CONTENT WORKSHEET C 12 Perpendicular lines Activity 1 – To construct perpendicular to a line through a point on it Objective: To learn to construct perpendicular to a line through a point on and outside a given line. Description – Teacher will demonstrate the activity by taking two pencils in the class to represent intersection of two lines at various angles as shown below: Figure 1 Figure 2 Teacher introduces the concept of construction of perpendicular lines using various methods: Sample of each is given below: Perpendicular lines through Paper Folding: 23 Use the right angle tester to check whether the lines are perpendicular to each other or not. To construct perpendicular to a line through a point on it by Paper Folding: To construct perpendicular to a line through a point on it by using Set Squares: A line l is given with a point P on it. Place the edge of the ruler along the line l and hold it firmly. 24 Now, place a set- square with right angled corner in contact with the ruler and its edge aligned with that of the ruler. Slide the set- square along the edge of ruler until its right angled corner coincides with P. Hold the set- square firmly and draw PQ along the edge of the set-square. To construct perpendicular to a line through a point on it by using ruler and compass Teacher is to show the video clip 3 to the students. Students simultaneously construct in their note books better learning. 25 Note: Before using any geometrical instrument teacher to make sure that the students are comfortable handling it. While the students are doing the constructions in their note books teacher to make sure that there is no conceptual error made. If required teacher to do the construction on the board and students to follow. Questions can then be given to the students for independent practice CONTENT WORKSHEET CW13 Perpendicular Bisector Activity 2 – Perpendicular Bisector Objective: To make the students learn and understand the meaning of the term bisector and perpendicular bisector. Material Required - Dictionary/Picture math dictionary available online. Description - Student would refer to the dictionary and find the meaning of the word bisector and perpendicular bisector. CONTENT WORKSHEET CW14 Constructing Perpendicular Bisector Activity 3- Video Watch Description – Teacher will show Video clip 5 to the students. This video talks about how to construct perpendicular bisector of a line segment using a ruler and a compass. After seeing video clip 5 a worksheet is given to the students to practice the concept. 26 CONTENT WORKSHEET CW15 Measuring and Constructing Angles using Protractor Activity 1 - Measuring and constructing angles using Protractor Objective: To learn to measure different types of angles using a protractor and construct 60o angle using compass. Description – Teacher will show Video clip 6 to the students. This video talks about how to measure angles using protractor. Teacher will simultaneously demonstrate how to measure angles using protractor on the board for better learning. Following this questions are given in the form of a worksheet for independent practice. CONTENT WORKSHEET CW16 Copying or Duplicating Angles using Compass and Ruler Activity 2- Copy of angle using ruler and compass. Objective: To make students learn to duplicate a given angle using ruler and compass. Description – Teacher will show Video clip 8 and 9 to the students. This video talks about how to duplicate angles using ruler and compass. Teacher will encourage students to copy angles from worksheet 19 in their notebooks. CONTENT WORKSHEET CW17 Constructing and Bisecting Angles Activity 3 - Constructing angles and angle bisectors using ruler and compass Objective – To make students learn to construct angles without protractor and bisect given angles. Description- Teacher will show Video clip 10 to the students. This video talks about how to construct 60o angle using compass and ruler. Teacher makes students practice construction of 60o angle on their own after seeing the video. The students learn how to construct angle bisector by looking and following the method shown in the video clip 10. 27 This worksheet also introduces the construction of angles which are multiples of 600. The students will be able appreciate the application of angle of 600 after doing this worksheet. CONTENT WORKSHEET CW18 Triangles Activity 1 - Identifying and naming three sided figure Description: This is an inductive activity in which teacher demonstrates and helps the students to know how to name the triangles. Through this worksheet the teacher will introduce different elements of a triangle like vertices, sides, angles, medians, altitudes, perpendicular bisectors, centroid, orthocenter and circumcentre. CONTENT WORKSHEET CW19 Quadrilaterals Activity 2- From three sides to four sides Description: The objective of the worksheet is to extend the discussion from three sided figure to a four sided figure. On the basis of activity 2 Content worksheet C4.1, the students are made to name different elements of a quadrilateral (four sided closed figure) CONTENT WORKSHEET CW20 Practice Worksheet: Triangles and Quadrilaterals Activity 3- Independent Practice Objective - Recapitulation of the work done in content C 4.1. Description - This is a recapitulation worksheet to make students recall all the concepts learnt in content C 4.1. Ask students to do the work as directed. Students will solve the worksheet given to them and note down their answers. 28 CONTENT WORKSHEET CW21 Introducing Circles Activity 1 - Introducing circles Description:- Teacher let students play on the web link 2 for a day Instructions to the students Explore the circle given on the website. Make an account of the all the terms you have come across while playing with the circle. Write down the term and the meaning what you understood from the figure. Discuss with your friend and then with your teacher. Based on what they see/learn from the weblink, students try to answer the questions in worksheet 25. Follow up: Teacher discusses the responses given by the students in the class CONTENT WORKSHEET CW22 Circles Activity 2 - Discovering a circle Description – Through this worksheet the students will be able to discover the circle and the elements of the circle on their own. The students read the worksheet follow the instruction and with the help of math dictionary keep building the concepts of the circles. This is a process approach and will be fun in learning the very obvious topic circles. 29 CONTENT WORKSHEET CW23 Fun with Circles 1 Activity 3 - Make your own designs Objective- Explore and make new patterns with circles. Execution- Teacher displays different patterns using circles and asks the students to make their own poster using circles. Students have fun and display their work. This can also be done as interclass/intersection activity. CONTENT WORKSHEET CW24 Practice Worksheet: Circles Activity 4- Independent Practice Objective - Recapitulation of the work done in content C 5. Description - This is a recapitulation worksheet to make students recall all the concepts learnt in content C 5. Ask students to do the work as directed. CONTENT WORKSHEET CW25 Fun with Circles 2 Activity 5 - Fun corner Description In this activity the teacher will prepare a crossword to sum up all the concepts learnt in the chapter. This will be good and interesting way of revising the geometrical terms learnt in the chapter. Jumble words will further enhance their knowledge of geometrical terms. Pre preparation: Teacher will prepare the cross word and hand out to the students to start the exercise. Follow up: Teacher will discuss the cross word and jumble words for concepts . 30 POST CONTENT WORKSHEET PCW1 Objective: To Practice the concepts in totality learnt in the chapter Pre Preparation: Teacher will prepare the comprehensive worksheets of the chapter. Description: Teacher will hands out the worksheets to the students Follow up: Teacher will assess level of her students on the basis of the post content worksheets PS1 and give remedial wherever required. Note for the teacher: 1. Students weak at the concepts must be given the enough practice through the basic worksheets and then post content worksheets may be given to them. 2. Students who have grasped the concepts very well and are able to solve regular problems quite easily may be advised to move to extension activities. POST CONTENT WORKSHEET PCW2 ASSESSMENT OF THE CHAPTER Objective: To test the concepts in totality learnt in the chapter Pre Preparation: Teacher will prepare the comprehensive worksheets of the chapter. Description: Teacher will hands out the following worksheets to the students This will be timed test given to the students. Follow up: Teacher will assess her students on the basis of the post content worksheet PS2 as per the rubrics given. 31 ASSESSMENT GUIDANCE PLAN Parameter 0 1 2 3 (LOWEST) 4 (HIGHES T) 1 Can Can Can Can Can differentiate differentiat differentiate differentiat between e between between e between line, line line, line line, line line, line segment, segment, segment, segment, ray, ray, ray, ray, Intersecting Intersectin Intersecting Intersectin and parallel and parallel g and and parallel g and lines lines parallel lines parallel with 30% accuracy lines with 90% lines with 50% accuracy accuracy with 100% Is able to measure and draw line segments, perpendicu lar from a point on and outside the line using either compass or set squares. and Is able to measure and draw line segments, perpendicul ar from a point on and outside the line using either compass or set squares. and perpendicul ar bisector of a line differentiate between line, line segment, ray, Intersecting 2 Does not possess any knowledge and cannot differentiat e between variables and constants. Is able to measure and draw line segments, perpendicul ar from a point on and outside the line using either compass or set squares. and perpendicul ar bisector of a line Is able to possess any measure and draw knowledge line of the segments, perpendicul concepts. ar from a point on and outside the line using either compass or set squares. and perpendicul ar bisector Does not 32 accuracy Is able to measure and draw line segments, perpendicu lar from a point on and outside the line using either compass or set squares. and segment 3 of a line segment with 30% accuracy perpendicu segment perpendicu lar bisector with 90% lar bisector of a line accuracy of a line segment segment with 50% with 100% accuracy accuracy Is able to Is able to Is able to Is able to Is able to Does not define and possess any define and define and define and define and differentiate knowledge differentiate differentiat differentiate differentiat between of the between e between between e between different concepts. different different different different types of types of types of types of types of angles and is angles but is angles and angles and is angles and able unable to is able able is able measure and measure measure measure but measure draw angles and draw but is able is able to but is able using angles using to draw draw angles to draw protractor. protractor. angles using angles using protractor using protractor with 90% protractor with 50% accuracy with 100% accuracy 4. Is able to construct angle 60o, can copy a given angle can bisect an angle and hence can construct Does not accuracy Is able to Is able to Is able to Is able to possess any construct construct construct construct knowledge angle 60o, angle 60o, angle 60o, can angle 60o, of the can copy a can copy a copy a given can copy a concepts given angle given angle but is given but is angle but able to bisect angle but 33 angles 30o, 45o, 90o using compass and ruler 5. Is able to identify and name triangle and quadrilateral s , its vertices, sides, angles, interior and exterior, is able to define altitude and median of a triangle unable to is able to an angle and is able to bisect an bisect an construct bisect an angle and angle and angles 30o, angle and construct construct 45o, 90o using construct angles 30o, angles 30o, compass and angles 30o, 45o, 90o 45o, 90o ruler with 45o, 90o using using 90% using compass compass accuracy compass and ruler and ruler and ruler with 50% with 100% accuracy. accuracy. Is able to possess any identify and name knowledge triangle and of the quadrilatera ls but is concepts unable to define its vertices, sides, angles, interior and exterior, is unable to define altitude and median of a triangle Does not 34 Is able to identify and name triangle and quadrilater als, to define its vertices, sides, angles, interior and exterior but is unable to define altitude and median of a triangle Is able to identify and name triangle and quadrilaterals , to define its vertices, sides, angles, interior and exterior but is able to define altitude and median of a triangle with 90% accuracy. Is able to identify and name triangle and quadrilate rals, to define its vertices, sides, angles, interior and exterior but is able to define altitude and median of a triangle with 100% accuracy. 6. Identify and define circle, its centre, radius, diameter, arc, sector, chord, segment, semicircle, circumferenc e, interior and exterior, is able to construct circle Identify and possess any define circle, its knowledge centre, of the radius, diameter, concepts arc, sector, chord, segment, semicircle, circumferen ce, interior and exterior, is able to construct circle with 30% accuracy. Does not Identify and define circle, its centre, radius, diameter, arc, sector, chord, segment, semicircle, circumfere nce, interior and exterior, is able to construct circle with 50% accuracy. Identify and define circle, its centre, radius, diameter, arc, sector, chord, segment, semicircle, circumferenc e, interior and exterior, is able to construct circle with 90% accuracy. Identify and define circle, its centre, radius, diameter, arc, sector, chord, segment, semicircle, circumfer ence, interior and exterior, is able to construct circle with 100% accuracy. 35 Unit 6 Basic Geometry 1. Introduction The word `geometry’ is said to be derived from the Greek word `geo’ meaning earth (or land) and metron meaning `measurement’. Thus, the origin of geometry can be traced back to the times when human beings first felt the need of measuring lands. Ancient Egyptians were perhaps the first people to study geometry in the process of demarcating and restoring the landmarks affected by the annual flood of the river Nile. They were mainly concerned with finding the perimeters and areas of some rectilinear figures such as rectangular, squares, etc. Thus, it can be said that the study of geometry began with the present day `mensuration’. With the passage of time, this study took the form of the study of figures or shapes formed by points, lines and planes. In this chapter, we shall begin the study of geometry with some basic geometrical concepts and shapes such as point, line, plane, line segment, ray, angle, triangle, etc. 2. Geometry in Our Environment If we look around in our immediate environment, we observe a number of objects with different shapes and sizes. Some of these objects are books, ball, ruler, different containers, wheels, and so on (see fig. 1) 36 Fig. 1 In addition to the above, we may find some beautiful and symmetrical buildings, leaves of trees, different paintings depicting various types of shapes. This shows that geometry has very close links with nature, environment and our daily life. Example 1: Give some situations where you observe the use of geometry. Solution: Some of the usages are: Construction of buildings, construction of roads, construction of bridges and dams and flyovers, laying of various types of cables and pipelines, etc. 3. Basic Geometrical Concepts: Point, line, plane, line segment etc. are called the building blocks of geometry. Let us take them one by one. Point When you take a sharp tip pencil and mark a dot on paper, you have represented a point. This almost invisible looking dot, known as point, determines the location if put on a map. Thinner the dot, better the representation of a point. The point shows the exact position of an objects or a place and has no length, breadth or thickness. That is 37 why we say: thinner the dot better representation of the point it is. We name a point with the capital letters A, B, C etc. and read it as point `A’, point `B’, etc. (Fig. 2) Fig.2 A star in the sky looks like a dot and gives an idea of a point. Line We observe many examples of lines around us such as straight electric wires, tightly stretched strings, edges in a room, edges of a paper etc. all give us a feel of the part of a line. Fig.3 Basic idea of a line is in its straightness and that it has no breadth. That is why when you draw a line, it should have very little thickness. A line extends in both the directions endlessly. This is represented by drawing the arrows on both the sides of the line. Thus, by a line, we mean a complete line and not a portion or part of it. Two arrow heads at both the ends represent this fact. Let us mark two points say C and D on a paper and try to draw a line through them. We observe that we can draw only one line passing through them (Fig. 4) So, we can name a line Fig.4 by taking any two points on it. Thus, line of Fig. 3 can be named as lines CD. 38 We can also represent a line by a small letter l (or a, b, x) (Fig. 5) Fig.5 If we take any two points say A and B on the line, then we can also represent the line by AB (Fig. 6) Fig.6 Note that line AB is the same as line BA. Plane Observe the surface of a blackboard, walls of your room, top of your book, etc. All these surfaces are flat. These flat surfaces give us an idea of (a part or portion of) a plane. To get an idea of a plane, we shall have to imagine the extension of such a flat surface on all sides indefinitely. Thus we can think of a plane as a flat surface extending endlessly in all the directions. Fig. 7 A plane has only length and breadth but no thickness. Line Segment When we study geometry, we work with only a part of a line which is known as line segment. Let us try to understand its meaning. If we take two points A and B and join them, there are many ways to join them (Fig. 7). Observe and see for ourselves which one is the shortest. 39 (i) (ii) (iii) (iv) Fig. 8 In other words we may say that there are many ways to reach B from A. We observe that Fig 8 (iv) above, depicts the shortest way of reaching B from A. Only this way or path is considered as a point or portion of the line AB and is called the line segment AB. What are the end points of this line segment? Clearly, point A and B The distance between these two points A and B is called length of the line segment AB. Further, there will be only one line segment joining the points A and B because only one line can pass through any two given points on a plane. Also, note that line segment AB is the same as line segment BA. It is denoted as AB or BA. Ray When we hear the word ray, rays of light or sun’s rays come to our mind (Fig. 8) Fig. 9 40 Here, the source of light such as a bulb or sun can be equated to a point and sun’s rays or light rays can be considered as a part of a line extending in one direction indefinitely from the source of light. We have also studied that a line extends unendingly or infinitely in both the directions. A part of a line which extends only in one direction unendingly or infinitely from a given point is known as ray. Fig.10 This part of a line is extending only in one direction starting from point O is an example of a ray. By taking a point say A on the ray, we can name it ray OA. It is also written as OA (Fig. 10). It means that the ray starts from its initial point O and passes through A and extends in the direction of OA. Infinitely many rays can be drawn from a point O as an initial point. But there will be only one ray passing through a given point `A’ (Fig. 11) Fig. 11 Note that ray AB is different from ray BA (Fig. 11) 41 Fig. 12 Example 2: (i) (ii) (iii) (iv) (v) State which of the following statements are true and which are false: A line has two end points A ray has one end point Only one line can pass through two given points Only one ray can pass through a given point A line segment has two end points Solution: (i) (ii) (iii) (iv) (v) False, it has no end points True, it is also called its initial point True False, infinitely many rays can pass through a given point True Example 3: Give two examples each for the following from your environment. (i) Point (ii) Line (iii) Plane Solution: (i) (ii) (iii) Corners of a tea box, corner of a Joker’s cap Edges of a ruler, a stretched wire Paper sheet, table top Example 4: Identify lines, line segments and rays in the following figure: 42 Fig.13 Also, write the name of each: Solution: Lines: (i) Line AM (iv) Line AB (viii) Line SK Line segment: (iii) Line segment PQ (iv) Line segment CD (ix) Line segment TP Ray: (ii) Ray MN (vi) Ray PR (vii) Ray NM 4. Pair of Lines When we draw two lines on a plane or a paper, there can be many ways, the lines can be drawn. Some of these are shown below. Fig. 14 43 Intersecting Lines In (i), (ii), (iv) and (v), the two lines meet each other (note that a line extends indefinitely in both the directions. These two lines are said to be intersecting lines. Observe that these lines intersect at only one point. Some examples of such lines in our real life are: Fig. 15 Parallel Lines In Fig. (iii) and (vi) of Fig. 13, two lines do not meet each other. When two lines drawn on a plane do not meet each other or intersect with each other, then we call them as parallel lines. (Fig. 16) There are many examples around us in which we see edges / lines being parallel to each other. Fig. 16 When two lines AB and CD are parallel to each other, we write line AB || line CD or AB || CD or simply AB||CD. (Fig. 16) 44 Fig. 17 If these two lines are named as l1 and l2 respectively, then we write l1 || l2 Can you think of more examples of parallel lines? Example 5: Give two examples each of the following up from your environment. (i) Intersecting lines (ii) Parallel lines Solution: (i) (ii) Edges of a cuboidal tea packet, adjacent edges of a floor Opposite edges of floor, opposite edges of a book Example 6: In Fig. 18, identify, which are intersecting lines and which are parallel lines. In case of parallel lines, write them in symbolic form. In case of (i) intersecting lines, write their point of intersection Fig 18 45 Solution: Intersecting lines: (i) Point of intersection is P (iii) Point of intersection is C (v) Point of intersection is A. Parallel lines: (ii) AB || MN (iii) PQ || ST (vi) DP || EQ 5. Open and Closed Curves Take a pencil and doodle something on a piece of paper. Some of the curves or figures may be as shown below: Fig 19 Now try tracing with the pencil these doodles or figures starting from one end to the other end. You will find that in some curves or figures while tracing the path you reach the point from where you started, without retracing the path and lifting the pencil, while in some other figures starting and ending points are different. For example, in (ii), (iii), (v), (vi) and (vii), starting from one point, we can reach the same point without retracing the path. Such curves or figures are called closed curves. Thus, curves in (ii), (iii), (v), (vi) and (vii) are closed curves. The other curves, i.e. (i), (iv) and (viii) are called open curves. Simple Curves The curve that does intersect itself is called a simple curve. 46 For example, in (i), (ii), (iv), (v) and (vi) of Fig. 18, the curves are simple, while in (iii), (vii) and (viii) the curves are not simple. Do you observe that curves (ii), (v) and (vi) are closed as well as simple. Such curves are called simple closed curves. Note that curves (iii) and (vii) are closed curves but they are not simple closed curves clearly, curve (viii) is open but not simple. Now, let us consider a simple closed curve in a plane and try to mark a point in the plane. There can be only three possible positions of the point as shown below (Fig. 19). Fig. 20 The point can be placed inside the curve or outside the curve or on the curve. The part of the plane which is enclosed by the curve is called the interior of the curve. Fig 21 The part of the plane which is outside the boundary i.e. the curve itself is known exterior of the figure. Thus, every simple closed curve divides the plane into three parts, namely (i) its interior (ii) its exterior and (iii) the curve itself. The curve and its interior together is called a region corresponding to that curve. 47 Polygons A simple closed curve made up of only line segments is called a polygon. Look at the following figures: Fig. 22 Clearly, (i), (iii) and (vi) are polygons. Curve (ii) is a closed curve made up of line segments only, but it is not a polygon, because it is not a simple closed curve. Convex and Non-convex polygons Observe the polygon (i) & (ii) in Fig. 23. Fig. 23 48 Take any two points A and B in their interiors and join them. In (i) line segment AB lies wholly is the exterior of the polygon. What about line segment AB in polygon (ii)? It does not lie wholly in its interior. Polygons of the type given in (i) are called convex polygons whereas polygon of the type given in (ii) are called non-convex (or concave) polygons. Some examples of convex polygons are: Fig.24 Some examples of non-convex polygons are: Fig 25 Unless slated otherwise, by a polygon we shall always mean a convex polygon. 49 6. Polygons: Sides, Vertices and Diagonals Consider the polygons: Fig 26 The polygon (i) is made up of line segments AB, BC, CD and DA (i) (ii) (iii) (iv) (v) These line segments are called sides of the polygon ABCD. A, B, C and D are called vertices of the polygon ABCD. Sides AB and BC, BC and CD are called adjacent sides. Similarly, CD and DA are adjacent sides and AD and AB are adjacent sides. Sides AB and DC are called opposite sides of the polygon ABCD. Similarly AD and BC are opposite sides of the polygon. In the same way, A and B, B and C, C and D etc. are called adjacent vertices. A and C, B and D are called opposite vertices. If we join A to C and D to B (as shown with dotted line, AC and BD are called its diagonals. Thus, the line segments joining the vertices other than the adjacent vertices are called diagonals of the polygon. What are sides, vertices and diagonals of the polygon in Fig. 25 (ii)? Clearly sides are: AB, BC, CD, DE and EA Vertices are: A, B, C, D and E 50 Diagonals are: AC, AD, BD, BE and CE Polygon in Fig. 25 (i) has four sides. It is called a quadrilateral. Polygon in Fig. 25 (ii) has five sides. It is called a pentagon. Similarly, a polygon with six sides is called a hexagon, a polygon with seven sides is called a septagon (or heptagon) and a polygon with eight sides is called a octagon. Example 7: Which of the following are open, closed and simple closed curves? Fig. 27 Open curves: (i), (iv) and (vi) Closed curves: (ii), (iii) and (v) Simple closed: (iii) and (v) Example 8: Which of the following are polygons? 51 Fig. 28 Solution: Polygons are: (i), (ii), (iv), (vi) and (vii) Example 9: Write the sides, vertices and diagonals of the following polygons. Fig.29 52 Solution: Example 10: Write the opposite sides and opposite vertices and adjacent side of the quadrilateral LOVE. Solutions: Opposite sides: LO and EV; LE and OV Opposite vertices: L and V; O and E Adjacent sides: LO, OV; OV, EV; EV, EL; EL, LO 7. Angles You must have come across same physical objects as shown below, which have essentially two arms say OA and OB joined together say at O. Fig 30 53 The two arms OA and OB are inclined towards each other and have an opening between them. These give a basic idea of an angle in geometry. An angle is formed by two rays with the same initial point. Let us look at the angle shown below: This is formed by two rays OA and OB, with a common initial point O. Fig 31 This point O is called the vertex of the angle and rays OA and OB, are called arms of the angle. Symbol ` ’ is used to denote the angle. This angle can be denoted as AOB. Note that the common initial point O or the vertex is in the middle. It can also be named as BOA. Sometimes, we denote an angle by its vertex as O, or angle O. Another notation for an angle is to place a number 1, 2, 3 etc. or a small letter a, b, c etc. near the circular arc. Interior & Exterior of an angle Let PQR be an angle as shown in the figure. Fig. 31 54 Observe that (i) (ii) (iii) PQR divides the plane in three parts: Part of the plane between two arms or OR and OP of the angle. This part is called the interior of the angle. Point A is in the interior of PQR. Part of the plane beyond the arms of the angle. This part is called the exterior of the angle. Point B is in the exterior of the angle. Part which is angle itself. Points D and C are on the angle itself. The angle PQR along with its interior is called angular region PQR. Example 11: Write arms and vertices of the angles given below: Fig 33 Solution: For angle (i): arms are ML and MN. Vertex is M. For angle (ii): arms are OP and OQ. Vertex is O. For angle (iii): arms are YX and YZ. Vertex is Y. 55 For angle (iv): arms are IH and IG. Vertex is I. Example 12: Write all possible names of the angle as shown in the adjoining figure: Fig 34 Solution: Possible names are: 2, A, PAQ, PAB, CAB, CAQ, QAP, Example 13: List the points which are in the (i) exterior of A (see figure) (ii) interior of A and also the points which lie on A. Fig 35 Solution: The points in the exterior of A: K, E 56 BAP, BAC, QAC The points in the interior of The points which lie on 8. A: L, D and M A: A, Q, B, P and C Degree Measure of an Angle (i) An angle can also be thought of as rotation of a ray from its initial position OA to OB as shown in the figure below: Fig 36 The amount of rotation from OA to OB is called the measure or magnitude of the angle. (ii) When the ray OA makes one complete resolution around O, we say that it has completed one turn and angle so formed is called a complete angle. Fig 37 (iii) If we divide this one turn into 360 equal parts, each part is called one degree. This degree is our basic unit of measurement of an angle. The unit `degree’ is denoted by a small circle. So, 1 degree is written as 1o 20 degree is written as 20o 90 degree as 90o Thus, 1 turn (complete rotation) = 360o 57 (iv) If ray OA takes a half turn as shown in the following figure: Fig 38 Then we say that the ray has moved through an angle of 180o. or 180o is the measure of this angle. An angle of measure 180o is called a straight angle. Straight in the sense that arms OA and OB are two opposite rays making a line BOA. (v) If ray OA takes a quarter turn ( turn) as shown in the figure below, then we say that the ray has moved through an angle of = 90o An angle of measure 90o is called a right angle. When the two lines meet each other at a right angle, they are called perpendicular lines. If the two lines meet at an angle which is not a right angle, then they are not perpendicular lines. 58 (vi) If ray OA does not move at all, we say that the ray has moved through an angle at 0o. An angle of measure 0o is called a zero angle. Types of Angles (i) Acute angle: An angle whose measure is less than a right angle (90o) but greater than 0o is called an acute angle (see fig.). Fig 39 (ii) Obtuse angle: An angle whose measure is greater than a right angle (90o) but less than a straight angle (180o) is called an obtuse angle (see fig.) 59 Fig 40 (iii) Reflex angle An angle whose measure is greater than a straight angle (180o) but less than a complete angle is called an reflex angle. (see fig.) Fig 41 Example 14: Classify the angles whose measures are given below: (i) 75o (ii) 120o (iii) 16o (iv) 179o (v) 182o (vi) 210o (vii) 360o (viii) 89o (ix) 12o (x) 0o Solution: Acute angle: (i), (iii) (viii) and (ix) Zero angle: (x) Obtuse angle: (ii) and (iv) Reflex angle: (v) and (vi) Complete angle: (vii) Example 15: Classify the following angles as acute, right, obtuse, straight or reflex, without measurement. 60 Fig 42 Solution: Acute angle: (i) Straight angle: (ii) and (vii) Right angle: (iv) Obtuse angle: (v) Reflex angle: (iii) and (vi) Measuring an Angle Protractor A protractor is an instrument (Fig. 43) designed to measure angles. It is marked in degrees, from 0o to 180o marked on it in multiples of 10 and has two scales, clockwise and anti-clockwise. So, you can measure angles on both directions. To measure an angle, place the centre of the protractor over the vertex of the angle in such a manner that the baseline of the protractor lies exactly on the base arm of the angle. (As shown in the figure) Read the marking on the outer reading where the arm cuts the protractor. This gives the measure of the angle. 61 Fig. 43 The measure of given angle BAC is 57o. Now let us measure PQR (See fig. 43) P Q R Fig 44 Fig. 45 The measure of PQR = 350 (See Fig. 45) 62 Which is also written as m or simply 8. PQR=350 PQR=350 Triangles Recall that a simple closed curve made up of only line segments is called a polygon. A polygon made up of three line segments is called a triangle (See Fig. 46). Fig. 46 The line segments forming a triangle are called sides of the triangle. In Fig. 45, AB, AC and BC are side of triangle ABC, symbolically written as ABC. Points A, B and C are called vertices of the triangle. Observe that at each vertex there is an angle formed by two line segments. For example, at vertex A, angle BAC or A is formed by line segments AB and AC (determined by rays AB and AC respectively) and so on. Thus, a triangle has three angles. For example, in ABC, three angles are C. Side BC is opposite to vertex A or A, Side CA is opposite to vertex B or B and Side AB is opposite to vertex C or C. A, Interior and Exterior of a Triangle A triangle (being a simply closed curve) divides a plane into three parts: (i) (ii) Interior: Part of the plane enclosed by three sides of the triangle. Exterior: Part of the plane beyond the sides of the triangle. 63 B and (iii) Triangle itself. Fig 47 Points K, D, E lie in the interior of ABC (Fig. 47) Points P, H lie in the exterior of ABC (Fig. 47) Points N, S and O lie on the triangle itself. Triangle ABC along with its interior is called a triangular region ABC. Example 16: In BIT, write the following: (i) (ii) (iii) (iv) (v) (vi) Vertices Angles Sides Side opposite B Side opposite vertex I Angle opposite side BI Fig 48 64 Solutions: (i) (ii) (iii) (iv) (v) (vi) Vertices are: B, I, T Angles are: B, I, T Sides are: BI, BT, TI Side opposite B is IT Side opposite vertex I is BT Angle opposite side BI is T Example 17: In the figure, identify the points which are: (i) (ii) (iii) (iv) in the interior of the triangle VAN in the exterior of the triangle VAN on the triangle VAN itself on the triangular region VAN Fig 49 Solution: (i) (ii) (iii) (iv) Interior: Points L, R and Q Exterior: Points Z, M, T, P and D On the triangle: Points S, N, X, C, V, A and N Triangular region: Points L, R, Q, S, N, X, C, V, A and N Median of a triangle In ABC, if A is joined to the mid-point M of opposite side BC, then the line segment AM is called median of ABC. 65 Clearly, a triangle has three medians (See Fig. 50) and these three medians meet at a point called centroid of the triangle. In Fig. 51, O is the centroid. Fig 50 Fig 51 Note that the centroid of a triangle always lies in its interior. Altitudes of a Triangle Consider ABC Let AL be perpendicular from A an opposite side BC. Then line segment AL is called an altitude of ABC. (Fig. 52) 66 Fig. 52 Clearly a triangle has three altitudes. In ABC, three altitudes are AL, BM and CN. These altitudes meet at a point. This point is called orthocenter of the triangle in the ABC, O is the orthocentre. Example 18: In CAR, M and K are the midpoints of sides AR and CA respectively. AHR=90o. (Fig. 53) Fig. 53 Write the altitude and median of the triangle. Solution: RK and CM will be median (why?) and AH is an altitude (why?). 9. Angles of a Quadrilateral You have seen that a triangle has three angles formed at three vertices. Recall that a quadrilateral has four vertices. So, a quadrilateral has four angles formed at its vertices (Fig. 54). In quadrilateral ABCD, the angles are A, B, C and D. 67 Fig. 54 10. Circles A circle is most familiar geometric figure which you came across in daily life such as a wheel, a round clock, a coin etc. It is a simple closed curve whose every point is at a constant distance from a fixed point in the plane. (See fig. 55) Fig. 55 The fixed point is called the centre of the circle. In the figure O is the centre of the circle. The constant distance is called the radius of the circle. In the figure, OP = OQ = OT = OR = OM = = radius of the circle Word `radius’ of a circle is used in two senses: (i) (ii) Constant distance of a point on the circle to its centre. The line segment joining any point on the circle to its centre. 68 In the sense of a distance, a circle has a unique radius while in the sense of a line segment a circle has infinitely many radii (plural of `radius’) Chord and Diameter Consider a circle with centre O and radius r. Take any two points, say A and B on the circle. The line segment AB is called a chord of the circle. PQ, RS, TS are also chords of the circle (See fig. 56). Fig. 56 Note that the chord TS passes through the centre of the circle. Chord TS is called a diameter of the circle. Thus, a chord passing through the centre of the circle is called a diameter of the circle. Note that a diameter is the longest chord of a circle. Also diameter TS = OT + OS = OT + OT =2OT = 2r Thus, diameter = 2 x radius Arc of a circle Consider a circle with Centre O. Take any two points P and Q on the circle (See fig. 57). 69 Fig. 57 These two points divide the circle into two parts. Each part is called an arc of the circle. In general, one part is smaller than the other. Smaller part is called minor arc and the other (larger) part is called major arc. In the figure, PSQ is minor arc, also denoted as QPS, and PTQ is major arc, denoted as QPT. In case, points P and Q divide the circle into two equal parts, then each part is called a semi circle (See fig. 58) Fig. 58 In the figure arc PSQ and arc PTQ are semi-circles. 70 Circumference Length or measure of a circle is called its perimeter or circumference of a circle. Interior and Exterior of a Circle As a circle is also a simple closed curve, so, it divides a plane into three parts: (i) (ii) (iii) Interior: part of the plane enclosed by the circle. Exterior: part of the plane beyond the circle. Circle itself (See fig. 59) Fig. 59 In the figure, points P, O, N, S and Z are in interior of the circle, points X, M, Q are in the exterior of the circle and the points K, T, R and Y are on the circle. A circle along with its interior is called the circular region 71 Sector of a Circle Take a circle with centre O and its two radii OA and OB (Fig. 60). Fig. 60 Radii OA and OB have divided the circular region with two parts. Each part is called a sector of the circle. In general, one part is smaller than the other. The smaller part is called the minor sector (shaded) and the greater part is called the major sector (unshaded) of the circle. Segment of a Circle Take a circle with centre O and draw a chord AB (Fig. 61) Fig. 61 Chord AB has divided the circular region into two parts. Each part is called a segment of the circular region or simply the segment of a Circle. 72 In the figure, shade and region shows the minor segment and unshaded portion shows the major segment. If these two segments are equal, then each is called semi-circular region (Fig. 62) Fig. 62 What happens when the two sector of the figure become equal? Example 19: Fill in the blanks: (i) (ii) (iii) (iv) (v) A chord of a circle is a line segment with its end points ............ Radius of a circle is a line segment with one end at .............. and the other end at ......... A diameter of a circle is a chord and that .............. centre. The end points of a chord of a circle divides the circle in two parts, where each part is called ................... of the circle. The end points of a diameter of a circle divides it into two equal parts. Each part is called a ............... Solution: (i) (ii) (iii) (iv) (v) on the circle the centre, the circle passes through the an arc semi-circle 73 Example 20: Identify the points (Fig. 62) which are: (i) (ii) (iii) in the interior in the exterior in the circle itself Fig. 63 Solution: (i) (ii) (iii) O, P and S Q, M and N T and R Example 21: Write the points (Fig 64) which are: (i) (ii) (iii) (iv) (v) in the minor sector OAPB minor segment ATB major sector OAQB major arc AQB minor arc APB 74 Fig. 64 Solution: (i) (ii) (iii) (iv) (v) O, A, P, T, B, U and N A, P, T, B and U O, A, Q, S, B and D A, Q, S and B A, P, T and B 75 STUDENT’S SUPPORT MATERIAL 76 STUDENT’S WORKSHEET – 1 Puzzle Time WARM UP ACTIVITY W1 Name of the student ______________________ Date ____________ Activity 1- Puzzle time Solve your puzzles keep the answers ready. Discuss the answers with your friends and teacher. Puzzle 1 - Count the total number of triangles in the given figure Puzzle 2 – Thinking out of the box Connect all 9 dots with four straight lines without lifting the pencil off the paper and without going over the line already drawn. Puzzle 3 - Slice the rectangle a) Slice the square into 4 identical sections, so that in each section there is 1 caterpillar with its leaf. One caterpillar will not have a leaf, she is taking a diet. 77 b) Slice the picture into two sections such that it becomes a square with 8 rows and 8 columns. Puzzle 4 - Let’s Play with Matchsticks a) Move two matchsticks to make 11 squares. b) This cow has the following parts: head, body, horns, legs and tail. It is looking to the left. Move two matches so that it is looking to the right. c) Collect six pencils and try to place them in such a way that each pencil is in touch with all the other five pencil. 78 STUDENT’S WORKSHEET – 2 APPRECIATE YOUR KNOWLEDGE WARM UP ACTIVITY W2 Name of the student ______________________ Date ___________ th 1. Which figure shaded is 2. Which two figures 2 of the whole? 5 have shaded fractions? 79 parts that represent equivalent 3. Use fractions to write the part of the whole shape that is shaded? 4. What is half of half equal to? 5. What fraction of each of these are coloured? 80 6. Simplify the following expressions: a) 3x – 4 + 2x – 3 7. Solve these equations: a) x + 9 = 32 8. b) – 2y – y + 3 – y b) 3y – 8 = 10 c) y/ 9 = 3 How many squares are there in the figures shown below? 81 STUDENT’S WORKSHEET – 3 Making Math A – Maze- ing PRE CONTENT WORKSHEET P 1 Name of Student____________________ Date_____________ Activity Solve this maze to help Ted reach his tricycle. (Use straight line to reach to the bicycle) 82 STUDENT’S WORKSHEET – 4 Origin of Geometry PRE CONTENT WORKSHEET P 2 Name of the student ______________________ Date ___________ Read the passage and answer the questions following it. Geometry has been used by us since a very long time. ‘Geometry’ is the English version of the Greek word’ Geometron’ where ‘Geo’ means Earth and ‘metron’ means Measurement. The geometrical ideas shaped up in ancient times, due to the need in art, architecture and measurement. They might be traced back to the times when the boundaries of cultivated lands had to be marked for farmers. Further, it found its need in the construction of magnificent palaces, temples, lakes, dams and cities, art and architecture for its aesthetics and stability. Even today geometrical ideas are reflected in all forms of art, measurements, architecture, engineering, cloth designing etc. Most manmade structures today are in a form of Geometric figures. Well, some examples would be, say a CD, which is a circle and the case could be a rectangular box. Buildings, cars, rockets, planes, maps are all great examples. Here are few examples on how the world uses Geometry in buildings and structure:This is a building at MIT. This building is made up of cubes, squares and a sphere. The cube is the main building and the squares are the windows. The doorways are rectangle, like always. On this building, there is a structure on the room that is made up of a sphere. 83 These are the Pyramids, in Indianapolis. There are many 3D geometric shapes in this pyramid building. The windows are made up of tinted squares and the borders of the outside walls and windows are made up of 3D geometric shapes. This is a Chevrolet SSR Roadster Pickup. This car is built with geometry. The wheels and lights are circles, the doors are rectangular prisms, the main area for a person to drive and sit in it is a half sphere with the sides chopped off which makes it 1/4 of a sphere. If a person would look very closely he would see a lot more shapes in the car. You observe and use different objects like boxes, tables, books and so on. All such objects have different shapes. Some are straight while some others are round. The ruler which you use, the pencil with which you write with are straight. The pictures of a bangle, the one rupee coin or a ball appear round. Answer the following questions: 1. Make a list of things around you and identify various shapes in them. 2. Identify and enlist few items from your surroundings that comprises of more than 5 geometrical figures. 3. Geometry plays a dominant role in our life.’ Justify this statement. 84 4. Find out more from art/architecture where geometry has been used to make it beautiful. STUDENT’S WORKSHEET – 5 Point, Line, Line Segment and a Ray-1 CONTENT WORKSHEET CW1 Name of the student ______________________ Date ____________ Activity 1 – Defining Point, line, line segment and a ray Given above is a street plan of a Town used by a Taxi driver to travel along the streets. Imagine yourself to be a taxi driver and you need to position your Taxi at various location. Answer the following questions 1. How will you represent the position of your Taxi at the intersection of Elm street and Third Avenue on the Street Plan? ________________________________________________________________________ ________________________________________________________________________ 85 2. Will you draw a big car at that position or will you use something really small may be a dot, a star or a cross to represent? Explain. ________________________________________________________________________ ________________________________________________________________________ 3. Which among these -dot, cross and star will be the best representation of the exact location of your taxi? Explain _______________________________________________________________________ _______________________________________________________________________ 4. Let’s draw a dot, star and a cross on the plain sheet. Ask your friends to mark the three locations using different colour pen. A dot or a point indicates the exact location. In mathematics, a dot is referred to as a point and is a basic unit of geometry. 86 5. Does the figure given in a rectangular box represent a point? Discuss with your friend. What do you infer from this? ___________________________________________________________________ ___________________________________________________________________ 6. Following are some model of points. Draw one more model of point from day to day life. 7. Mark 5 different location of your taxi on the Street plan. How will you distinguish them? What according to you will make those points different from each other? How will it be convenient for you to talk about those points? ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 8. If points are named by single alphabet in capital letters, Mark a point A at the intersection of Ash Street and Fifth Avenue and a point B at the intersection of Dogwood Street and Fourth Avenue. 87 Extension: Observe the night sky. Does the star in the sky give an idea of a point? Mark the points of stars in any three constellations. 88 STUDENT’S WORKSHEET – 6 Point, Line, Line Segment and Ray-2 CONTENT WORKSHEET CW2 Name of the student ______________________ Date ___________ Activity 2 – Joining the points 1. Mark the shortest possible way of your taxi from a) Birch Street First Avenue to the Sixth Avenue. b) Chestnut Street Third Avenue to Fir Street Third Avenue. Explain by drawing it in the map given below. Two points can be connected to each other in shortest possible way by drawing a straight line between them. This shortest possible route joining two points in Mathematics is termed as __________________. Discuss with your friends and your teacher about the notations AB and BA . ______________________________________________________________________ ______________________________________________________________________ Represent the distance between the two points, you marked in the map, by the notation discussed above. 89 2. How many streets pass from point A as well as point B? 3. How many line segments can be drawn through two given points? Draw and explain in the box given below. 3. How many line segments can you draw through three points at different locations as given in the two figures below? Draw them in the figure given below. 4. What is the difference between the points in figure 1 and those in figure 2? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 90 Refer to the math dictionary and name the points given in figure 1 and figure 2. The points D, E and F in figure 2 are ___________________ as __________________________________________________________________________ The points A, B and C in figure 1 are ___________________ as __________________________________________________________________________ 5. Mark 5 sets of three points each on the Street plan that are collinear and two sets of three points that are non collinear. Conclusion: The straight line distance between two points in Mathematics is known as ____________________. 6. Observe the following cartoon carefully. 91 State three differences between line and a line segment. ____________________________________________________________________ ____________________________________________________________________ Give two examples of line and two of a line segment from your surroundings. ______________________________________________________________________ ______________________________________________________________________ Complete the statements: 7. (i) _____________________ is the extension of ___________________. (ii) _____________________ is the part of ________________________. Draw as many lines as possible through a single point. How many lines can you draw passing through a single point? _____________________ What is a special name given to these lines? (hint: starts from C and end at T) _____________________________________________________________________ . Three or more lines passing through a single point are said to be _______________. 92 8. Observe the following figures and state the difference(s). Record them in the space given below. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Explore and find out, what is the name given to the new geometrical figure you see in the above figure. _____________________________________________________________ How can you represent it mathematically? ______________________________________ State the differences between line, line segment and a ray. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 93 9. Name the rays given in the picture. Is S the starting point of each of these rays? Can we say that R is the starting point of ray SR ? Explain. ______________________________________________________________________________ ______________________________________________________________________________ 10. Imagine me, Dash and Yeoh driving our vehicles along the path shown in the figure. Write down whether Yeoh will _______________________ 94 meet me and Dash or not? Can we say, path of Yeoh intersects mine and Dash’s path? _____________________ If yes, write down the intersections where Yeoh will meet me and Dash. ____________ Will Dash ever meet me if both of us keep moving on our own straight paths? _______ If yes, where will Dash meet me? _____________________________________________ If no, why don’t we meet? __________________________________________________ The lines which do not meet or intersect each other are called ______________. Give two examples from day to day life which reflects the same path as that of mine and Dash’s i.e. just like railway tracks that never meet. ________________________________________________________________________ ________________________________________________________________________ Observe the above rail track and think about the mathematical symbol What does What does means? _________________. mean? _________________. 95 . Fill in the blanks with appropriate symbol: l _________ m, m _________ n and l _________ n Extension: Which lines in these pictures are parallel and which lines are intersecting? Mark in the diagrams or Make a list of them. Walk around your classroom, school building or grounds. Make a list of the parallel lines that you see. Choose an activity such as playing football, going shopping, dancing, taking a photograph etc. Discuss what parallel and intersecting lines you would be likely to find. 96 STUDENT’S WORKSHEET – 7 Skill Drill CONTENT WORKSHEET CW3 Name of the student ______________________ Date ______ Matching Terms with their Meanings Cut out the terms, meanings and diagrams. Fit them in a table as given below in your notebooks. Term 2. Meaning Diagram CROSS AND KNOTS Batsu and Maru played the game of Knots and crosses. Batsu answered yes while Maru answered No to solve the geometrical problem shown in the grid. Batsu played first and marked O, and Maru played next and marked X. Who was the winner? At the end of the game, which boxes had X's and which had O's? 97 STUDENT’S WORKSHEET – 8 Constructing Line Segments CONTENT WORKSHEET CW4 Name of the student ______________________ Date ______ Activity 4 - Measuring and constructing lines What do the points marked on the ruler represent? What do you understand by A = 5.6? ________________________________________________________________________ ________________________________________________________________________ Any point on the ruler determines the distance between _______ and ________. What type of scale has been used? If 0 is represented by the point O the OA = 5.6 _________. 98 Draw OA on the paper. Do we always have to start measuring from O? Explain. _________________________________________________________________________ _________________________________________________________________________ Draw OA with the help of the following ruler Mark clearly the points O and A, and then the line segment OA in the above case. 1. Measure the length of following line segments: Caution: While using the ruler you should look straight down else you may get incorrect measure. 2. Draw a line segments AB = 8.2 cm and PQ = 4.8 cm using ruler. 99 STUDENT’S WORKSHEET – 9 Constructing Line Segments by Compass CONTENT WORKSHEET CW5 Name of the student ______________________ Date __________ Activity 5 – To construct and copy a line segment using compass and ruler Answer the following questions after the video watch. 1. Draw any line segment PQ . Without measuring PQ , construct a copy of PQ . 2. Given some line segment AB , whose length you do not know, construct PQ such that the length of PQ is twice that of AB . 3. Construct AB of length 8.6 cm. From this, cut off AC of length 4.7 cm. Measure BC . STUDENT’S WORKSHEET – 10 Practice Worksheet CONTENT WORKSHEET CW6 Name of the student ______________________ Date ______ Activity 1: Independent Practice 1. Given AB of length 4.7 cm, construct PQ such that the length of PQ is twice that of AB . 2. Given AB of length 7.7 cm and CD of length 4.3 cm, construct a line segment PQ such that the length of PQ is equal to the difference between the lengths of AB and CD . Verify by measurement. 3. The length of XY = x + 3, and YZ = 2x + 7. If Y is between points X and Z, and XZ = 31, what is the length of YZ? Note, X, Y, Z are collinear. Draw a diagram to help solve the problem. 4. In the Figure, name 100 5. a) The line, in as many ways as possible. b) Three points on the line. c) Two line segments. Take any three points A, T, E in your notebook. Join them in pairs. How many lines do you get if a) A, T, E are collinear 3. 4. b) A, T, E are not collinear In the given figure name: a) Four non- collinear points. b) Point of intersection of the lines t and s. c) Point on intersection of the lines m and n. d) Point of intersection of the lines l and n. e) Point of intersection of the lines m and l. f) Four line segments. g) Two points on the line l. h) How many lines are drawn in the figure? In the given figure name: a) Four collinear points. b) Three non- collinear points. c) Three pairs of intersecting lines. d) Lines which are concurrent at O. e) Lines which are concurrent at T. f) Lines which contain the point D. 101 5. 6. State true or false and correct the false statement. a) A line has a definite length. b) Two distinct lines always intersect at a point. c) One and only one line can be drawn through two given points. d) Only one line can be drawn through one point. e) The maximum number of points of intersection of four distinct lines is six. f) A point has no length, no breadth but has thickness. g) A line contains infinite number of points. Lines k, l, m are concurrent. Also, lines l, m, n are concurrent. Draw a figure and find if the lines k, l, m, n are all concurrent. 7. Identify and name the line segments and rays in each of the following figures: 8. Count the number of line segments drawn in each of the following figures and name them. 102 9. Take any five points C, R, A, Z, Y in your notebook in such a way that no three of them are collinear. Join them in pairs as shown in the figure. a) How many such lines are there? b) How many lines pass through A? c) How many lines pass through B? d) Name all lines. e) Name any three line segments. f) Name the shaded figure you get enclosed within the line segments joining C, R, A, Z, Y points. 10. In the given figure, name a) all pairs of intersecting lines. b) all pairs of intersecting lines. c) all sets of collinear points. d) lines which intersect at the point e) point of intersection of lines f) points of intersection of lines g) lines which contains the point h) Mark points O and D at the two intersection points left unmarked in the figure above. i) Using the points of intersection in the figure, fill in the blanks to find out a hidden message Me told Yeoh 103 j) Find out five more alphabets hidden in the figure, one is done for you. STUDENT’S WORKSHEET – 11 Angles, Interior and Exterior of an Angle CONTENT WORKSHEET CW7 Name of the student ______________________ Date ______ Activity 1 – Defining angles, interior and exterior of an angle, Types of angles 1. Name and shade all the angles in each of the following clocks: 104 2. Name the shaded angles in each of the following : 105 3. Take any angle ABC. a) Shade the portion bordering BC and where BA lies. b) Shade the portion bordering BA and where BC lies. c) Name the common blue shaded area of the angle CBA. d) What does the yellow region represent? e) Mark two points F and K in the interior (inside) of CBA . f) Mark two points L and M in the exterior (outside) of CBA . g) Mark points P and Q in the interior of ABC . h) Which four points are in the same region? 4. Shade Interior of a) FED yellow and exterior of GED red. Interior of which angle will now appear to be orange? b) Mark three points in the exterior of FEG but interior of GED . How do we measure the length of a table? _____________________ How do we measure the weight of an object? ___________________ How do we measure time? _______________ 106 How do we differentiate between the angles shown in two figures below? ___________________________________________________________________________ How do we say that which angle is bigger and by how much? __________________________________________________________________________ INVESTIGATE: You wish to divide a complete turn into an exact number of parts. Is 360 a good number? Try dividing 360 by each of the whole numbers from 1 to 12. Now do the same for 100. Which is the better number of degrees to have in one complete turn, 100 or 360? Give reasons for your choice. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 107 STUDENT’S WORKSHEET – 12 Types of Angles CONTENT WORKSHEET CW8 Name of the student ______________________ Date ______ Activity 2 – Types of angles TRY THESE: 1. Mark different types of angles in the following figures. First one is done for you. 108 How many right angles can you find in the tennis court? Hint: There are more than 30 but less than 50 2. In the figures colour the right angles in green, obtuse angles in pink and acute angles in yellow. (some are done for you) 109 3. The pictures below are a signaling system called ‘semaphore’ which was used by the navy. Name the type of angle each signal forms. 4. Picture shows some of the angles a striker made on a snooker table. Name all the ten angles shown in the picture. STUDENT’S WORKSHEET – 13 Angles through Paper Folding CONTENT WORKSHEET CW9 Name of the student ______________________ Activity 1 – Fun activity Creating and recognizing angles through Paper folding 110 Date ______ Questions: 1. How many different types of angles can you see in the penguin? 2. Open up the sheet and note all creases. Mark and name right angle and obtuse angles you can observe. 3. Make an airplane using paper folding and compare different types of angles involved in it. 4. Observe if changing the angles in paper folding affect the speed and time of fight of your airplane. 111 STUDENT’S WORKSHEET – 14 Angles and Applications CONTENT WORKSHEET CW10 Name of the student ______________________ Date ______ Activity 2 – Application of Geometry in day to day life Fractions of Revolution- In a clock When the hands of a clock starts at say 12, and moves round until it stops at 12 again; we say that it has gone through one complete turn. Measure of 1 complete turn or revolution is ___. Draw the clocks and answer the following questions. Second hand of a clock starts at 12 and stops at 3. What fraction of the revolution has the hand moved? _____ What degree of the revolution has the hand moved? _____ How many right angles has the hand of the clock moved? ____ Second hand of a clock starts at 12 and stops at 7. What fraction of the revolution has the hand moved? _____ What degree of the revolution has the hand moved? _____ Where will the second hand stop if it starts now at 7 and turns through half a revolution? _______________________________________ 112 Fractions of Revolution- In Compass Direction There are four main directions in Geography, i.e. North, South, East and West. If you stand facing north and turn clockwise to turn east, by what angle would you move?_______________________ If you stand facing east and move clockwise through two right angles, what direction you will be facing now?___________________ How many right angles do you turn through if you face north and turn clockwise to turn west? ________________________ If you stand facing west and turn ½ a revolution, in which direction are you facing? ______________________ Does it matter if you turn clockwise or anticlockwise? ___________________ If you stand facing west and turn anticlockwise to turn north what part of a revolution have you turned?_______________________ STUDENT’S WORKSHEET – 15 Practice Worksheet : Angles and Types of Angles CONTENT WORKSHEET CW11 Name of the student ______________________ Activity 3- Independent practice 1. Name the vertex and the arms of the following angles: 113 Date ______ 2. a) How many angles are shown in the figure below? Name them. b) Name a right angle, an acute angle and an obtuse angle in the figure. 3. Name the angles shown in the figure. a) How many of these angles can be named using a single letter or the vertex letter only? b) Name the points to the interior of of DCA but exterior ACB . 4. Write another name for a) 1 b) 2 c) 4 d) What is the sum of angle 1 and angle 2? 114 e) Is the sum of angle 4 and angle 2 equal to sum of angle 1 and angle 2? Justify your answer. 5. Draw angles to represent FOR , STR and AND . 6. In the figure, a) Which of the following statements are true? (i) Point A is the interior of AOD . (ii) Point B is the interior of AOB . (iii) Point B is the interior of AOC . (iv) Point C is the exterior of AOB . (v) AOC . Point D is the exterior of b) Which points are in the interior of angle BOD? c) Which points are to the exterior of angle AOD? d) Which points are on the angle AOD? e) Name a right angle and a reflex angle. f) Name two angles whose sum is a right angle. 7. State which of the following statements is true: (i) The vertex of an angle lies in its interior. (ii) The vertex of an angle lies on its exterior. (iii) The vertex of an angle lies on it. 8. Where does the second hand of the clock stop if a) it starts at 6 and turns through ¼ of a turn. b) it starts at 12 and turns through 2/3 of a revolution. 9. If you stand facing west and turn anticlockwise through ¾ of a revolution, in which direction are you facing? 10. If you stand facing north and turn, in either direction, through a complete revolution, in which direction are you facing? 115 11. If you stand facing south and turn through one and a half revolution, in which direction are you facing? 12. How many right angles does the seconds hand of a clock turn through when it a) starts at 9 and stops at 6, b) face north and turn anticlockwise to face east, c) face north and turn to face north again. 13. How many degrees are there in three right angles? 14. How many degrees has the seconds hand of a clock turned through when it moves from a) 8 to 5 b) 10 to halfway between 11 and 12 c) 7 to 9 d) 3 to halfway between 4 and 5 15. The second’s hand of a clock starts at 12. Which number is it pointing to when it has turned through an angle of a) 90o b) 150o c) 540o d) 360o e) 210o f) 720o 16. In the following figures, write down the size of the angle marked with a letter: Note: Diagrams are not to scale. 116 17. Each of the equal angles marked p is 35o. Find the reflex angle q. 18. Angle s is twice angle p. Find angle t. 19. The angle marked d is 75o. Find angle g. STUDENT’S WORKSHEET – 16 Perpendicular Lines CONTENT WORKSHEET CW12 Name of the student ______________________ Date ___________ Activity 1 : To construct perpendicular to a line through a point on it 1. Draw any line segment AB . Mark any point M on it. Through M, draw a perpendicular to AB . (by paper folding) 2. Draw any line segment PQ . Take any point R not on it. Through R, draw a perpendicular to PQ . (Use ruler and set-square) 117 3. Draw a line l and a point X on it. Through X, draw a line segment XY perpendicular to l. Now draw a perpendicular to XY at Y. (Use ruler and set square) 4. Draw any line segment XY. Take a point Q not on it. Through Q, draw a perpendicular to XY using ruler and compass. 5. Draw a line AB. Take a point P outside it. Draw a line passing through P and perpendicular to AB. STUDENT’S WORKSHEET – 17 Perpendicular Bisector CONTENT WORKSHEET CW13 Name of the student ______________________ Date ____________ Activity 2: Perpendicular Bisector Look at the Math dictionary to find the meaning of the following 1. Bisector 2. Perpendicular bisector Define these terms in your own words: ______________________________________________________________________ ______________________________________________________________________ Mark perpendicular bisectors in the figure of a Tennis ground. 118 STUDENT’S WORKSHEET – 18 Constructing Perpendicular Bisector CONTENT WORKSHEET CW14 Name of the student ______________________ Date ____________ Activity 3 : Video Watch (Video clip 5) Answer the following questions based on it:1. Draw AB of length 6.3 cm and find its bisector. 2. Draw a line segment of length 8.5 cm and construct its perpendicular bisector. 3. Draw the perpendicular bisector of XY whose length is 11.8 cm. (a) Take any point P on the bisector drawn. Examine whether PX = PY. (b) If M is the mid point of XY , what can you say about the lengths MX and XY? 4. Draw a line segment of length 12.4 cm. Using compass; divide it into four equal parts. 5. With PQ of length 5.4 cm as diameter draw a circle. Draw any chord AB . Construct the perpendicular bisector of AB and examine if it passes through C. 7. Draw a circle of radius 6 cm. Draw any two of its chords. Construct the perpendicular bisectors of these chords. Where do they meet? 8. Draw any angle with vertex O. Take a point A on one of its arms and B on another such that OA = OB. Draw the perpendicular bisectors of OA and OB . Let them meet at P. Is PA = PB? STUDENT’S WORKSHEET – 19 Measuring and Constructing Angles using Protractor CONTENT WORKSHEET CW15 Name of the student ______________________ Date _________ Activity 1 - Measuring and constructing angles using Protractor Watch video 6 and 7 and try these questions:Measure the following angles using a protractor. 119 STUDENT’S WORKSHEET – 20 Copying and Duplicating Angles using Compass and Ruler CONTENT WORKSHEET CW16 Name of the student ______________________ Date __________ Activity 2: - Copy of angle using ruler and compass. Watch video 8 and while watching video 9 and practice copying angle. Copy the angles given in worksheet 19 in your notebooks using ruler and compass. 120 STUDENT’S WORKSHEET – 21 Constructing and Bisecting Angles CONTENT WORKSHEET CW17 Name of the student ______________________ Date ___________ Activity 3: - Constructing angles and angle bisectors using ruler and compass. Watch video 10 and 11 and try the following questions: 1. Construct with ruler and compasses, angles of following measures: (a) 60° (b) 30° (c) 90° (d) 120° (e) 45° (f) 135° 2. Draw an angle of measure 45° and bisect it. 3. Draw a right angle and construct its bisector. 4. Draw an angle of measure 157°using a protractor and construct its bisector. 5. Draw an angle of measure 148° using a protractor and divide it into four equal parts. 6. Draw an angle of 120° and bisect the angle. Measure the bisected angle using a protractor. STUDENT’S WORKSHEET – 22 Triangles CONTENT WORKSHEET CW18 Name of the student ______________________ Date ______ Activity 1: - Identifying and naming three sided figure Refer to math dictionary to fill in the gaps as you move along. Take three points on a paper such that they are not in the same straight line. Name them as A, R and C, 121 Join the points pair wise using straight lines. How many line segments have you drawn? _____________________________________ How many angles are enclosed in the figure? ____________________________________ How many vertices (corners) are there? ________________________________________ Name the line segments drawn in the figure. Name the line segment which is opposite to vertex A. ________________________________________________________________ Name the angle formed by joining each pair of line segments. ______________________ The three sided figure is known as ____________. Name this three sided figure as a triangle. ACR__ or ________ or ________. Yellow shaded portion is the ______________of the triangle while the rest is the _____________of the triangle. Points M and N are the _______ points of the triangle. In the given triangle, position your pencil at vertex A. In how many ways can you move away from A? _________ 122 These line segments or arms of angle A are said to be ________________ of A. Which line segment is left out? ________________ The left out line segment CR is said to be ______________ of angle A. Name the adjacent and opposite sides of angle R. Fold the triangle through vertex A, draw perpendicular to the opposite side CR. (acute angled triangles only) Open the folding and draw the crease. Name the point where crease meet CR as M. AM is line segment which is known as _________________ from A to CR. Make altitude from R and C on the respective opposite sides. What do you observe?______________________________________________________ Do all the altitudes meet at same point? ________________________________________ What can you say about the three altitudes of a triangle? ___________________________ Define orthocenter. ___________________________________________________________________________ ___________________________________________________________________________ ______________________________________________________ 123 Take another triangle ABC (acute angled) . Fold side BC to find the mid point of side BC. Open the folding and name the mid point as M. Fold the triangle again and try to join vertex A to mid point M of BC. Open the fold and mark the crease. The line segment joining the mid point of BC to the opposite vertex A is called ______________________. Mark all the three medians. Record you conclusions. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ Define centroid. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 1. Draw a rough sketch of a triangle ABC. Mark a point M in its interior and a point N in its exterior. Is the point A in its exterior or in its interior? 2. a) Identify three triangles in the figure. b) Write the names of seven angles. c) Write the names of six line segments. d) Which two triangles have B as common? 124 STUDENT’S WORKSHEET – 23 Quadrilaterals CONTENT WORKSHEET CW19 Name of the student ______________________ Date ______ Activity2: - From three sides to four sides Take four points A, B, C and D such that no three points are collinear. Join all the points in such a way that no triangle is formed. What do you get? Name the figure so obtained?(hint: ) 125 Justin and Kevin went skating to their Quadri park. Kevin reached the park early and went inside the Quadric park while Justin was looking and calling him out. Read the conversation between them to learn more about the Quadri park. Read the above role play and answer the following questions. 1. Draw a rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them. Is the meeting point of the diagonals in the interior or exterior of the quadrilateral? 2. Draw a rough sketch of a quadrilateral KLMN. State, a) two pairs of opposite sides b) two pairs of opposite angles c) two pairs of adjacent sides d) two pairs of adjacent angles. INVESTIGATE: Why is it that structures like electric towers make use of triangular shapes and not quadrilaterals? Take hints from the following and discuss. Use strips and fasteners to make a triangle and a quadrilateral. 126 Try to push inward at any one vertex of the triangle. Do the same to the quadrilateral. Did the triangle get distorted? Did the quadrilateral get distorted? Which is stronger, triangle or a quadrilateral? What can you do to make a quadrilateral stronger? STUDENT’S WORKSHEET – 24 Practice Worksheet : Quadrilaterals and Triangles CONTENT WORKSHEET CW20 Name of the student ______________________ Activity3: - Independent practice 1) In the given figure name (a) The triangle in at least three different ways (b) The vertex opposite to the side PQ (c) The side opposite to the vertex Q 2) In the given figure (a) name a point in the interior of ABC 127 Date ______ (b) name a point in the exterior of (c) name a point on the ABC ABC (d) name a segment in the triangular region ABC 3) In the given figure, name (a) all triangles (b) all angles (c) all segments (d) Triangles having LE as common angle. 4) Make any four points (no three collinear) in your copy book. Try to make as many triangles as possible. How many maximum numbers of triangles can you make? 5) In the figure, a) Write down the names of all possible triangles. b) Which triangle has, (i) A as a vertex? (iii) C as a vertex? (ii) B as a vertex? (iv) D as a vertex? (v) E as a vertex? (vi) F as a vertex? (vii) J as a vertex? (viii) H as a vertex? c) Which triangle has C on at least one of their sides? d) Which of the triangles have the point H in the exterior? 6) Which of the following are quadrilaterals? 128 7) Name the following quadrilaterals: 8) In the given Figure, name (a) The quadrilateral (b) Two points in its interior (c) One point in the exterior (d) All points on the quadrilateral STUDENT’S WORKSHEET – 25 Introducing Circles CONTENT WORKSHEET CW21 Name of the student ______________________ Date ______ Activity 1: - Introducing circles Explore circles on the web link 2 Based on what you saw/ learn from the weblink try to answer the following 129 1. Look at the pictures given below. What do you observe common in them? They are all __________ in shape. Observe the movement of clock needles and the movement of a cow roped to a pole in a field. A circular shape has the same distance from a fixed point. 130 What is the fixed point known as? _______________ Take two pencils and a piece of thread. Loop the pencils at each end of the thread. Fix one pencil on the paper and move the other pencil such that there is no slag in the thread. What figure will you get with the traced pencil’s path?________________ What would the length of the thread be known as?________________ Consider a circle with center A. Centre is a fixed point. Blue shaded region is the region inside the circle and is termed as the _______ of the circle. Yellow shaded region is the ______________ of the circle. The distance across a circle through the center is called the ____________ (The Image shown is as seen in the mirror) AC = AD = radius Therefore, Diameter = 2 x __________ Explain chord in our own words. ________________________________________________________________________ Draw a circle with the help of any object. 131 Draw five concurrent line segments which can also represent the chords of a circle. Diameter also a chord of the circle. How is the diameter different from the chord of the circle? ________________________________________________________________________ ________________________________________________________________________ STUDENT’S WORKSHEET – 26 Circles CONTENT WORKSHEET CW22 Name of the student ______________________ Date ______ Activity 2: - Discovering a circle Cut out the following shapes according to the instructions given and discover more about circles. I. Cut from inner and out boundary. Cut at A and C. Observe the smaller portion. Try to find its name. Define it. __________________________________________________________________________ __________________________________________________________________________ Now observe the larger portion. Can it be also given the same name? What is the difference in that case? (Hint: Why are you not allowed driving?) II. 132 Cut out the adjacent circle. Cut the circle from E to E. Solve the jumble words to know what it is. Try to define it. __________________________________________________________________________ ______________________________________________________________ What about the other portion? Hope you can differentiate between them now. III. Cut out the adjacent figure from E through O and obtain the portion which has S and T in its interior. Rearrange the jumble words to discover its name. Name the other portion also. STUDENT’S WORKSHEET – 27 Fun with Circles 1 CONTENT WORKSHEET CW23 Name of the student ______________________ Date ______ Activity 3: - Make your own designs Draw a circle of any radius. Divide the circumference of the circle into 8 equal parts. 133 Use these diagrams to help you draw the interlocking square design. Explore and make more new patterns. STUDENT’S WORKSHEET – 28 Practice Worksheet : Circles CONTENT WORKSHEET CW24 Name of the student ______________________ Date __________ Activity 4: - Independent Practice: 1. Name the parts of the circle shown in red. The ‘ ’ shows the centre of the circle 2. Name the regions shaded in green. 134 3. Match these definitions with the correct name from the box. Circle, circumference, radius, arc, diameter, tangent, chord, segment, semicircle, sector a) A line that touches the circle at just one point. b) A line that joins two points on the circumference and passes through the centre of the circle. c) A region enclosed by a chord and an arc. d) A set of points equidistant from another fixed point. e) The distance round the circle. f) The distance from the center to the circumference. g) A region enclosed by an arc and two radii. h) The two regions the circle is divided into by the diameter. 4. A chord, PQ, of a circle is drawn. A point R is taken outside the circle. R is then joined to P and Q as shown in the figure. a) Imagine PQ moves to the left. What happens to R? b) Imagine PQ moves to the right. What happens to R? 5. From the figure, identify: (a) the centre of circle (b) three radii (c) a diameter 135 (d) a chord (e) two points in the interior (f) a point in the exterior (g) a sector (h) a segment 6. a) Is every diameter of a circle also a chord? Explain b) Is every chord of a circle also a diameter? Expalin 7. Draw circles with following radii (use compass and ruler to draw circles): a) 4 cm b) 2.5 cm c) 4.8 cm 8. Draw circles with diameters given below: a) 7cm b) 8 cm c) 5 cm 9. Draw three concentric circles with radii 2.5 cm, 3 cm and 3.5 cm. 10. Concentric circles are those which share the same center. Draw a circle with centre O and radius 4 cm. Draw a chord AB of the circle. Indicate by marking points X and Y, the minor arc AXB and major arc AYB of the circle. STUDENT’S WORKSHEET – 29 Fun with Circles2 CONTENT WORKSHEET CW25 Name of the student ______________________ Activity 5: - Fun corner Crossword Puzzle 136 Date ___________ Across 1. An instrument used to draw circles. 2. The shape of a circle. 4. The ratio of the circumference and diameter of any circle. 6. The number of square units occupied by space inside the circle. 8. A part of a circle named by its endpoints. 10. A location in space that has no thickness. 11. The distance from the center of a circle to any point on the circle. 12. The distance around a circle. Down 1. A line joining two points on the circle. 3. The distance across the circle through its center. 4. A circle divides a ____ into two parts. 5. Plural for half a diameter. 7. A circle has 360 of these units. 9. A shape with all the points on it at the same distance from its center. 137 13. All points in a circle are at the same distance from this point. Jumble words Unscramble the words given below. GOOD LUCK!! 1. earclinlo ______________ 11. alenp _____________ 2. renrtcuocn ______________ 12. otrbsice _____________ 3. yra ______________ 13. diclrpepneuar _____________ 4. pnito ______________ 14. eaplalrl _____________ 5. einl ______________ 15. sruaid _____________ 6. car ______________ 16. tiaaraelldruq _____________ 7. mreitaed ______________ 17. reevtx _____________ 8. rcdoh ______________ 18. tensgme _____________ 9. ieadnm ______________ 19. eregde _____________ 10. utaidlet ______________ 20. tsorec _____________ STUDENT’S WORKSHEET - 30 POST CONTENT WORKSHEET PCW1 Name of the Student ______________ Date. ______________ 1. In the given figure, state the total number of distinct lines and name them. Also, answer the following questions: a) What is the total number of points of intersection of all pairs of lines? Name them. b) Name each pair of lines intersecting at a point. c) Name any three points which are collinear. d) Name any three points which are non-collinear. e) Name any three lines which are non- concurrent. f) Do you find any three lines which are concurrent? 138 2. Answer each of the following questions: a) How many single lines can be drawn each passing through four non-collinear points? b) How many maximum number of lines can be drawn each passing through three collinear points? c) In how many maximum number of points can two distinct lines intersect? d) What is the least number of distinct points which determines a unique line? 3. Take six points A, B, C, D, E and F in a plane such that no three of them are collinear. Join each point with the remaining points. Now, answer the following question: a) What is the total number of distinct lines in the figure? Name them. b) What is the total number of distinct lines passing through A, B, C, D, E and F? Name them. c) Name any three lines which are non- concurrent. d) Name any five distinct lines which are concurrent. 4. In the given figure, a) Name all the line segments. b) Name all the planes. c) Name pair of lines which are parallel. 5. State whether the following statements are true or false: a) There are only four points in a plane. b) There are only two points on a line. c) We can draw only one line through two distinct points. d) We can draw only one line through only one point. 139 e) We can draw hundred lines through hundred collinear points. f) Three concurrent lines intersect each other at three distinct points. g) Four points are always collinear when they lie on a plane. h) When two lines intersect at a point, this point is called the point of intersection of the two lines and when three or more lines pass through a common point, then this point is called the point of concurrence of the lines. i) Three distinct lines, taken in pairs, can intersect at most three points. j) Two lines in a plane always intersect each other at a single point. k) A point has no dimensions, it has only the position. 6. In the given figure, verify by measurement that a) AC + BE = AE + BC b) AD + CE = AE + CD c) AB + CE = AE – BC d) AC + DE = AE – CD e) AD + BE = AE + BC + CD 7. Construct a line segment PQ of length 10.5 cm with a help of a ruler. Then, cut a line segment PR of length 4.8 cm from PQ with a help of a compass and a ruler. Measure the length of the remaining line segment. 8. Draw a line segment PQ = 8.4 cm. By measurement, divide the line segment into a) two equal parts b) three equal parts. Measure each part. 9. In the given figure, find a) any seven distinct rays. b) any three pairs of intersecting rays. c) any three pairs of non intersecting rays. d) a point where a line segment, a ray and a line are concurrent. 140 e) a point where three line segments and a ray are concurrent. 10. In the given figure, write another name for a. 1 b. AEB c. 7 d. 2 e. DBE f. 9 g. 8 h. 4 i. 3 f. 5 11. State the types of angles which are formed between the following directions shown in the figure: a) north and south b) north west and south east c) south and west d) north and east f) north east and east (as marked in the figure) g) south west and south h) south west and east 12. How many degrees are there in the angle between the minute hand and the hour hand of a clock when it is a) 6 o’clock b) 12 o’clock c) 9 o’clock d) 3 o’clock 13. A boy is walking towards north- west and a girl is walking towards south-east. What is the angle between them? 14. Construct angles which are multiples of 15 using compass and the rest using a protractor: a) 190 b) 75 c) 90 d) 85 e) 120 f) 82 g) 80 h) 47 i.) 50 j) 100 15. Find the value of x and hence, all the angles indicated in each of the figures given below where AOB 90 . 141 16. Fill in the blanks with correct words or symbols to make the following statements true. i) If r cm is the radius of a circle with centre at O and if P is the point in the circular region, then OP _____ r cm. ii) If d cm is the diameter of a circle and r cm is its radius, then d ____ 2r. iii) The line segment with its endpoints on a circle is called the __________ of the circle. iv) A point which is equidistant from all points on a circle is called the ________ of the circle. v) The circle along with its _______ is called a circular region. vi) Radius of the circle is equal to _________ of its diameter. vii) The distance between the centre of a circle and a point on the circle is called the ________ of the circle. viii) The diameter is the ________ chord of a circle. ix) The length of the perimeter of a circle is called its ________. x) A diameter of a circle divides the circular region into two parts. Each part is called the ________ region. xi) The circular bounded by an arc and the two radii joining the endpoints of the arc and the centre of a circle is called the __________ of the circle. 17. Draw a square, each of whose sides is 5cm. Use a pair of compasses and ruler in your construction. 18. Draw a line segment AB = 5.6cm. Draw the right bisector of AB. 142 19. Draw a line segment AB = 6cm. Take a point C on AB such that AC = 2.5cm. Draw CD perpendicular to AB. 20. Draw a circle of diameter 10.2 cm and name the following: a) Centre b) Three radii c) Diameter d) Two minor arcs (e) Two Chords f) Two sectors 21. Draw two concentric circles with radii 3 cm and 6 cm and centre at o. Take a point P such that 3 cm < OP < 6 cm. Does the point P lie in the interior of the circle with radius 6 cm? Student’s worksheet – 31 POST CONTENT WORKSHEET PCW2 Name of the student ______________________ Date ____________ 1. a. In the given figure, name a. any four lines which are concurrent. b. any three lines which are non- concurrent. c. any three points which are collinear. d.any three points which are non-collinear. 2. Fill in the blanks: a. A cuboidal box has _____ corner points. b. A cuboidal box has ______ edges representing the line segments. c. An edge of a ruler gives us an idea of the portion of a ______. d. Three or more points are ________, if they lie on a line. e. Three or more lines are _______, if they pass through a common point. f. The surface of a football is a ______ surface. g. Each surface of a box is the portion of a ______ surface. 143 3. In each of the figures given below, verify by measurement that a. PR + RQ = PQ b. PQ – QR = PR c. PQ – PR = RQ 4. If PQ = 4.9 cm and RS = 2.5 cm, construct the following line segments with the help of a ruler and compass. (a) 2 PQ (b) PQ + RS (c) 2PQ – RS (d) PQ – RS (e) 2PQ – 2RS 5. Answer the following question: a. Name all the rays with initial points C, E, F and D in the given figure. b. Name all the rays which are different from each other. c. Name pairs of opposite rays. d. Is the ray AC different from the ray DB? e. Are the rays EA and DB different? f. Are the rays EF and FE same? g. Are the rays CE and CB same? h. Are the ray EB and EC same? i. Are the rays CA and EF opposite rays? j. Are the rays AC and AB the same rays? 6. In the given figure, a. name the vertices of b. name the common arm of 2 and 5. c. name the common arm of 1 and 4. d. name the arms which are not common for e. 2 and 3 and 6. 5 , and name the arms which are not common for 1 and 4. 144 7. How many degrees are there in (a) 4 right angles? (b) 2 complete angles? (c) 1/3rd of a right angle? (d) 2/5th of a straight angle e. ½ of a complete angle? (f) 8 right angles? 8. A man is walking towards west. After sometime, he turns to his right through a right angle. In which direction is he walking now? If he again turns to his left through a right angle, in which direction will he be walking? 9. Find the value of x and hence all the angles indicated in each of the following figures, where the points A, O and B are collinear. 10. Find the value of x and hence, all the angles indicated in each of the following figures : 11. State whether the following statements are true or false. i. Any two semi-circles of the same circle are equal in length. ii. Any two arcs of the same circle are equal in length. iii. A diameter is the greatest chord of the circle. iv. A secant of a circle is the same as the corresponding chord. 145 v. All diameters of a circle are equal in length. vi. Each radius of a circle is also a chord of the circle. vii. Each diameter of a circle is also a chord of the circle. viii. Only one radius can be drawn to a circle. ix. Concentric circles have different radii, but the same centre. x. A radius of a circle is twice its diameter. xi. A diameter of a circle is twice its radius. xii. From a given point on a circle, infinite number of chords can be drawn. 12. Draw a rectangle whose two adjacent sides are 5cm and 3.5cm. Make use of a pair of compasses and ruler only. 13. Use a pair of compasses and construct the following angles: (i) 15o (ii) 135o (iii) 22 ½ o (iv) 67 ½ o (v) 45o ACKNOWLEDGMENTS : Websites Referred to: 3. www.hotmaths.com.au 4. http://www.creativeteensclub.org/ctc/node/21 5. http://www.mathrealm.com/CD_ROMS/GeometryWorld.php#Try 6. http://www.col.org/stamp/JSMath3.pdf Reference Books: 1. Ncert Mathematics for class 7 2. National Framework 8+ Mathematics By Tipler and Vickers 3. Composite maths for class 6 4. STP Mathematics 7A By Nelson and Thornes 5. Math plus for middle school by S.C Das 146 Video Links : http://www.mathsisfun.com/geometry/plane-geometry.html http://www.amathsdictionaryforkids.com/dictionary.html http://www.ixl.com/math/grade/sixth/ http://www.basic-mathematics.com/basic-geometry.html http://www.onlinemathlearning.com/basic-geometry.html 147
