While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
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= Concept Map =
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=== Concept Map ===
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<mm>[[circle.mm|flash]]</mm>
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[[File:circle.mm|flash]]
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== Introduction ==
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The following is a background literature for teachers. It summarises the things to be known to a teacher to teach this topic more effectively . This literature is meant to be a ready reference for the teacher to develop the concepts, inculcate necessary skills, and impart knowledge in Geometry - Circles from Class 6 to Class 10.
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The first step is to understand how to define circles and related terms using geometric vocabulary. The next step is to understand what is Pi. That it is a constant and that for any circle the ratio of the circumference by the diameter is always a constant value Pi. The interesting properties of Pi – an irrational number can also be discussed here in the basic form. Ability for the child to do simple area and perimeter calculations. Next the learner should understand that the circle is a 2 dimensional plane figure and how to visualise solid 3-dimensional figures. What are the solid shapes that have a circle as a part of them. Mensuration – more complex area measurements which include circular shapes. Surface Area and Volume measurement of sold shapes such as cylinder, sphere and cone. Understand the properties of the circles by proving theorems deductively. Also acquire the skills of deductive proofs, understand that all the properties can be deduced from the axioms. Understand the relationship between lines and circles – secant and tangent
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== Additional Resources[edit | edit source] ==
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=== Resource Title ===
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[http://www.mathopenref.com/tocs/circlestoc.html Circles and Arcs]
##[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.
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##[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.
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##[http://www.khanacademy.org Khan academy] Has good educative videos.
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##[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information.
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# Books and journals
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## [http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.
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# Textbooks:
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## [http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circle
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## [http://ncert.nic.in/NCERTS/textbook/textbook.htm?jemh1=10-14 CLASS 10]
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# Syllabus documents
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=== Non-OER[edit | edit source] ===
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# Web resources
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#*[http://www.mathsisfun.com/geometry/circle.html maths is fun]Here you get description of terms of circles
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#*[http://www.scribd.com/doc/11489830/Circles-Intresting-Facts Intersting facts] this web link is full of circle facts.
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#*[http://www.sparknotes.com/math/geometry1/circles/section4.rhtml sparknotes] Gives some more details about properties of circles
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#*[http://www.regentsprep.org/Regents/math/geometry/GP14/PracCircleSegments.htm www.regentsprep.com] conatins good objective problems on chords and secants
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#*[http://www.mathwarehouse.com/geometry/circle/tangents-secants-arcs-angles.php www.mathwarehouse.com] contains good content on circles for different classes
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#*[http://staff.argyll.epsb.ca/jreed/math20p/circles/tangent.htm staff.argyll] contains good simulations
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#*[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.
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#*[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.
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#*This is a video showing construction of tangent at any point on a circle
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{{#widget:YouTube|id=LLKFqv71i0s|left}}
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This is a resource file created by Suchetha, Mathematics teacher, GJC Thyamangondlu
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*This is a video showing construction of tangent from external point and theorem
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{{#widget:YouTube|id=xvXaxx1u-iA|left}}
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This is a resource file created by Suchetha, Mathematics teacher, GJC Thyamangondl
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*This is a video showing Transverse common tangent
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{{#widget:YouTube|id=LA7afvv4u-A}}
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This is a resource file created by Gireesh KS , Assistant Teacher, GHS jalige, Bangalore Rural District
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** you want see the kannada videos on theorems and construction of circle [http://karnatakaeducation.org.in/KOER/index.php/%E0%B3%A7%E0%B3%A6%E0%B2%A8%E0%B3%87_%E0%B2%A4%E0%B2%B0%E0%B2%97%E0%B2%A4%E0%B2%BF%E0%B2%AF_%E0%B2%B5%E0%B3%83%E0%B2%A4%E0%B3%8D%E0%B2%A4_-_%E0%B2%B8%E0%B3%8D%E0%B2%AA%E0%B2%B0%E0%B3%8D%E0%B2%B6%E0%B2%95%E0%B2%A6_%E0%B2%97%E0%B3%81%E0%B2%A3%E0%B2%B2%E0%B2%95%E0%B3%8D%E0%B2%B7%E0%B2%A3%E0%B2%97%E0%B2%B3%E0%B3%81 click here] this is shared by Yakub koyyur GHS Nada.
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# Books and journals
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# Textbooks
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##[http://ktbs.kar.nic.in/New/Textbooks/class-x/english/maths/class-x-english-maths-chapter14.pdf Karnataka text book for Class 10, Chapter 14 - Chord properties]
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##[http://ktbs.kar.nic.in/New/Textbooks/class-x/english/maths/class-x-english-maths-chapter15.pdf Karnataka text book for Class 10, Chapter 15 - Tangent Properties]
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# Syllabus documents (CBSE, ICSE, IGCSE etc)
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== Learning Objectives ==
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* Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.
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* To make students know that circle is a 2-dimensional plane circular figure.
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* All points on its edge are equidistant from the center.
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* The method of drawing a circle
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* The size of the circle is defined by its radius.
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* To elicit the difference between a bangle or a circular ring and circle as such.
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== Teaching Outlines ==
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==== Concept #1 Introduction to Circle ====
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When I tell people that circles are the mother of all inventions, the first thing they ask is, “circles are inventions?”
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Yes, a circle isn’t something that exists in nature. It isn’t something that people discovered like gold or the new lands of America. It is a mental construct, a symbolic representation that was invented much the same as language and the alphabet.
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There is no way to be certain, but anthropologists generally agree that the circle was created long before recorded history. It is quite likely that it was drawn by a stick in the sand. With the sun being a constant in early man’s existence and the source of all life, it is quite likely that the first circle represented the sun.
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Through the years man’s understanding of the circle has evolved substantially with Euclidean geometry being its crowning point of technological understanding. (Having said that, I assure you this blog is not going to be about mathematics or boring scientific equations.)
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What we will say that is without the rudimentary understanding of circles, the world would not be anything like it is today. Without circles, there would be no wheel, which is man’s crowning achievement dating back to the Neolithic Age (circa 9500 BC).
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The three other great achievements are the ability to make fire, the agriculture of crops, and the domestication of animals. While the circle didn’t have any direct bearing on these advancements, the understanding of circles certainly contributed to their proliferation and expansion.
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Besides the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a Ferris wheel, or watching the moon landing on our television set.
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If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. They are an intrinsic component in the invention of almost everything that we see around us.
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I had a microbiologist challenge me that his field didn’t have much use for circles. Not knowing anything about microbiology, I asked him what was the shape of the lens in his microscope.
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= Textbook =
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The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind.
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==ncert books==
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[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.
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[http://ncert.nic.in/NCERTS/textbook/textbook.htm?jemh1=10-14Class 9 Mathematics] contain higher level description and theorems on circles.
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=Additional Information=
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And it doesn’t exist except in our mental construct. It is a symbol, not a thing. We talk about it in every language on earth. It is written about in millions of textbooks and all over the Internet, and yet we cannot put it in a wheel barrel. It doesn’t exist in a three dimensional world or even a two dimensional world. It is merely a representation.
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==Useful websites==
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#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A GOOD WEBSITE ON DEFINITIONS FOR CIRCLES
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#[http://www.coolmath.com/reference/circles-geometry.html COOL MATH] GIVES CLEAR AND EASY DEFINITIONS
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#[http://www.mathopenref.com/circle.html OPEN REFERENCE] CONTAINS FEW SIMULATIONS
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#[http://en.wikipedia.org/wiki/Math_circle#History WIKIPEDIA] CONTAINS EXPLANATIONS FOR CIRCLES
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#[http://www.khanacademy.org KHAN ACADEMY] CONTAIN GOOD VIDEOS
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#[http://www.arvindguptatoys.com ARVIND GUPTA TOYS] CONTAIN VERY GOOD BOOKS ON MATHEMATICS AND SCIENCE
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#[http://nrich.maths.org/2490 nrich.maths.org] Contain very good description of Pie
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==Reference Books==
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Immanuel Kant’s famous phrase “ding an sich” applies to the circle. A circle is not a “thing-in-itself.” It is a semantic fabrication that exists only in our imagination. As Alfred Korzybski, the father of General Semantics, would say, it is “the map, not the territory.”
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#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.
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= Teaching Outlines =
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But here we are getting off on a philosophical tangent that might be subject to a future blog entry. For now let’s just say that circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why I think the circle is so fantastic.
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br>
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br>
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.
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===Learning objectives===
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=== Circle Properties ===
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# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.
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* A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.
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# To make students know that circle is a 2-dimensional plane circular figure.
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* Equal chords of a circle (or of congruent circles)subtend equal angles at the centre.
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# All points on its edge are equidistant from the center.
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* If the angles subtended by two chords of a circle(or of congruent circles) at the centre(corresponding centres) are equal, the chords are equal.
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# The method of drawing a circle
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* The perpendicular from the centre of a circle to a chord bisects the chord.
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# The size of the circle is defined by its radius.
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* The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
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# To elicit the difference between a bangle or a circular ring and circle as such.
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* There is one and only one circle passing through three non-collinear points.
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* Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres).
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* Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.
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* If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.
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* Congruent arcs of a circle subtend equal angles at the centre.
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* The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
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* Angles in the same segment of a circle are equal.
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* Angle in a semicircle is a right angle.
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* If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.
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* The sum of either pair of opposite angles of a cyclic quadrilateral is 1800.
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* If sum of a pair of opposite angles of a quadrilateral is 1800, the quadrilateral is cyclic.
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===Activities===
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====[[A discussion on “Life without circular shaped figures.”|A discussion on “Life without circular shaped figures.”]]====
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Discussion based activity to relate and assimilate circular shapes seen in our surroundings.
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====== [[Circle as a shape|Circle as a shape]] ======
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A circle is the set of all points in the plane that are a fixed distance from a fixed point.
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[[File:circle.gif]]
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==== [[Is circle a Polygon ? - A debate|Is circle a Polygon ? - A debate]] ====
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A polygon when increased in number of sides tends to form a circle is shown with this interesting activity.
To demonstrate the value of Pi move the slider named a from minimum to maxiumum value and observe the circumference
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====Evaluation====
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Use the GeoGebra file [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Constant_Pi.html]] and illustrate and verify that the ratio is true for different radius by moving the radius slider and using the table below to compute the values.
# Ask the children to make a list of all circular objects that they can think of :
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# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)
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# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.
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*Multimedia resources
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*Website interactives/ links/ / Geogebra Applets
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*Process:(How to do the activity)
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==== Perimeter of a circle ====
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Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.
To apply the use of calculating the perimeter of a circle in a real life example .
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# Can you imagine bicycles and your other vehicles without circular wheels ?
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====Material and Resources Required====
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# How different life would have been if wheel was not disovered ?
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Pencil, Paper
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# How could we explain centrifugal force ?
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====Pre-requisites/Instructions====
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# What about potter's wheel and stone mill?
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Draw the following sketch and do the calculations for the evaluation questions. The sketch shows the two main dimensions of a standard 400 metres running track.
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# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify
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====Evaluation====
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*Question Corner:
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#Calculate the inside perimeter of this shape.
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# Are shapes important ? How?
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##'''Why do you think that it is not equal to 400 metres? '''The inside runner cannot run at the very edge of the lane (there is normally an inside kerb) but let us assume that the athlete runs at a constant distance of, say, x cm from the inside edge.
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# Is bangle a circle ?
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#
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# Would it be possible to understand pi without understanding circles ?
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#What is the radius of the two circular parts run by the athlete in the inside lane?
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#
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#Show that the total distance travelled, in centimetres, is 2 π (3650 + x ) + 16878 and equate this to 40 000 cm to find a value for x.
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##Is it realistic? For 200 m and 400 m races, the runners run in specified lanes. Clearly, the further out you are the further you have to run, unless the starting positions are staggered.
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#
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#The width of each lane is 1.22 m, and it is assumed that all runners (except the inside one) run about 20 cm from the inside of their lanes.
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##With these assumptions, what distance does the athlete in Lane 2 cover when running one complete lap? Hence deduce the required stagger for a 400 m race.
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##What should be the stagger for someone running in Lane 3 ?
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#
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#If there are 8 runners in the 400 m race, what is the stagger of the athlete in Lane 8
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compare with that in Lane 1 ? Is there any advantage in being in Lane 1?
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===== Further Explorations =====
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1. This link gives an overview of what Pi is. [[http://en.wikipedia.org/wiki/Pi]]
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=== Concept #2 Terms associated with circles ===
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===== Activities =====
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====== [[Centre of a circle|Centre of a circle]] ======
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All points on a circle are at fixed distance from a point, which is the center of a circle.
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====== [[Radius and diameter of a circle|Radius and diameter of a circle]] ======
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Marking radius and diameter of a circle and understand their relation.
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====== [[Circumference of a circle|Circumference of a circle]] ======
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Measuring circumference to understand it as the perimeter of the shape.
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====== [[Semicircle|Semicircle]] ======
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Partitioning a circle into two halves to form semicircles by drawing diameter.
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====== [[Interior and exterior of a circle|Interior and exterior of a circle]] ======
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Points on the planar surface of the circle within its circumference are said to be interior points and points on the outside of circumference are said to be its exterior points.
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====== [[Basic elements of a circle]] ======
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Investigation to understand basic parameters associated with circles.
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====== [[Chord of a circle|Chord of a circle]] ======
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Chords of a circle are of different sizes.The length of the chord increases as it moves closer to the centre and decreases as it moves away from the center.
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====== [[Arc of a circle|Arc of a circle]] ======
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The part of the circumference within the two points in either directions are called its arcs.
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====== [[Sector of a circle|Arcs and Sector of a circle]] ======
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Slice of a circle enclosed between any two radii is called a sector.Semicircle and quadrant are special types of sectors.
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=== Concept #3: Circles and Lines ===
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===== Activities =====
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====== [[Introduction to chords]] ======
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A chord is the interval joining two distinct points on a circle. This activity investigates formation of chord and compares with the diameter of the circle.
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====== [[Activity1 Angles in the same segment are equal]] ======
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====== [[Angle subtended by an arc]] ======
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====== [[Secant and tangent of a circle]] ======
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A tangent is a line touching a circle in one point. A secant is the line through two distinct points on a circle.
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=== Concept #4: Theorems and properties ===
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A chord is a straight line joining 2 points on the circumference of a circle.Chords within a circle can be related in many ways.
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The theorems that involve chords of a circle are :
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* Perpendicular bisector of a chord passes through the centre of a circle.
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* Congruent chords are equidistant from the centre of a circle.
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* If two chords in a circle are congruent, then their intercepted arcs are congruent.
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* If two chords in a circle are congruent, then they determine two central angles that are congruent.
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===== Activities =====
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====== [[Chord length and distance for centre of the circle]] ======
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For a chord the distance from the centre is the perpendicular distance of the chord such that it passes through the centre.
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====== [[The longest chord passes through the centre of the circle]] ======
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Investigating the diameter is the longest chord of a circle.
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====== [[Perpendicular bisector of a chord passes through the center of a circle|Perpendicular bisector of a chord passes through the centre of a circle]] ======
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Since every perpendicular bisector passes through the centre, the centre must lie on every one of them, so the centre must be their single common point.
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====== [[Perpendicular from centre bisect the chord]] ======
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====== [[Congruent chords are equidistant from the centre of a circle|Congruent chords are equidistant from the centre of a circle]] ======
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In the same circle or in circles of equal radius:
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• Equal chords are equidistant from the centre.
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• Conversely, chords that are equidistant from the centre are equal.
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====== [[Angles in a circle subtended by a chord]] ======
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The angle made at the centre of a circle by the radii at the end points of a chord is called the central angle or angle subtended by a chord at the centre.
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=== Concept #5: Cyclic Quadrilateral ===
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In Euclidean geometry, a '''cyclic quadrilateral''' or inscribed '''quadrilateral''' is a '''quadrilateral''' whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
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=====Activities=====
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===Activity No #2. Circle of varying radius using Geogebra ===
A quadrilateral ABCD is called cyclic if all four vertices of it lie on a circle.In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees.If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic.In a cyclic quadrilateral the exterior angle is equal to interior opposite angle.
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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======[[Properties of cyclic quadrilateral]]======
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|}
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Relation between the angles of a cyclic quadrilateral are explored with this hand on activity.
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*Estimated Time: 20 mins
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=== Concept #6 Constructions in circles ===
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The students should know that tangent is a straight line touching the circle at one and only point.They should understand that a tangent is perpendicular to the radius of the circle.The construction protocol of a tangent.Constructing a tangent to a point on the circle.Constructing tangents to a circle from external point at a given distance.A tangent that is common to two circles is called a common tangent.A common tangent with both centres on the same side of the tangent is called a direct common tangent.A common tangent with both centres on either side of the tangent is called a transverse common tangent.
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*Materials/ Resources needed: Laptop, geogebra,projector and a pointer
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=== [[Circles_Constructions]] ===
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*Prerequisites/Instructions, if any:
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=== Concept #7 Tangents ===
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#The students should know a circle
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A line which touches a circle at exactly one point is called a tangent line and the point where it touches the circle is called the point of contact.
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#They should know the meaning of radius .
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#They should know to measure radius using compass
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#They should know to draw a circle of given radius using compass.
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*Multimedia resources: Laptop
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==== Properties of tangent ====
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The tangent at any point of a circle is perpendicular to the radius through the point of contact. We can also conclude that at any point on a circle there can be one and only one tangent.
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*Explain the theorem that if two segments from the same exterior point are tangent to a circle, then they are congruent.
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*Solve for an unknown in a problem involving tangents.
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*Apply properties of tangents to solve problems involving triangles circumscribed about a circle.
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*Website interactives/ links/ / Geogebra Applets:
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==== Tangents from a point outside the circle ====
#As you increase the number of sides ask them to count the number of sides.
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#Ask them the name of the shape formed.
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#As the number of sides increase,let them speak about the length of the sides ?
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'''*Developmental Questions'''
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#How many sides does this figure have ?
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#Name the figure formed.
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#What is hapenning to the length of the sides as the number of sides is increased ?
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#What shape is this ?
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#So, can circle be considered a polygon ? Justify
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*Evaluation:
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#Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?
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#Are the students able to appreciate the application of polygon anology to circles.
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*Question Corner
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Debate between two groups with these two perspectives.<br>
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#Circle seems to have derived from polygons . Circle can be considered a polygon.
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Vs
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#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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|}
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*Estimated Time
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*Materials/ Resources needed
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*Prerequisites/Instructions, if any
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*Multimedia resources
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*Website interactives/ links/ / Geogebra Applets
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==== Common tangents ====
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Common tangents are lines or segments that are tangent to moret han one circle at the same time.
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*Process/ Developmental Questions
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==== Direct common tangents ====
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The centres of the circles lie on the same side of the common tangent.(dct)
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*Evaluation
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[[Image:KOER%20Circles_html_m202ccc14.gif|link=]]
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*Question Corner
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==Concept # 2. Measurements in circles ==
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[[Image:KOER%20Circles_html_m244a7f98.png|link=]]
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===Learning objectives===
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# The students should learn to measure radius, diameter, circumference, chord length and angles subtended at the centre and on the circumference of the circle.
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# The students should understand that radius, diameter and chord lengths are linear measurements.
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# They should learn to relate the size of the circle with radius.
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# They realise that to draw a circle knowing the measure of radius or diameter is essential.
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# There can be infinite radii in a circle.
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# Diameter is twice the radius.
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# The students should understand what a chord is.
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# Chords of different lengths can be drawn in a circle.
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# Chord length can be measured using a scale and its units is cm.
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# They should know that the length of the chord increases as it moves closer to the diameter.
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# The longest chord in the circle is its diameter.
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# Distance of chord from the centre is its perpendicular distance from the centre.
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# A chord divides the circle into two segments.
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# Angle at the centre of the circle is 360º.
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# Angles in circles are measured using protractor.
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# Circumference and area are calculated using formula.
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===Notes for teachers===
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===Activity No # 1. Measuring radius and diameter. ===
# Initially the teacher can explain the terms: circle, its centre, radius, diameter and circumference.
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# Ask the children “What parameter is needed to draw a circle of required size ?”
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# Show them how to measure radius on the scale accurately using compass.
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# Show them to draw a circle.
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# Given diameter, radius = D/2.
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# Also the other way i.e. If a circle is given, then its radius can be measured by using scale which is the linear distance between centre of the circle and any point on the circumference.
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#To measur diameter, measure the length of that chord which passes through the centre of the circle.
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Then she can project the digital tool 'geogebra.' and further clarify concepts.
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*Developmental Questions:
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[[Image:KOER%20Circles_html_m6e667170.png|link=]]
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# Name the centre of the circle.
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# Name the point on the circumference of the circle.
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# What is the linesegment AB called ?
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# Name the line passing through the centre of the circle.
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# Using what can you measure the radius and diameter.
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# Name the units of radius/diameter.
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*Evaluation:
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# How do you measure exact radius on the compass?
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# Are the children able to corelate the radius/diameter of a circle with its size ?
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*Question Corner:
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# If the centre of the circle is not marked , then how do you get the radius for a given circle.
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# How many radii/diameter can be drawn in a circle?
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# Are all radii for a given circle equal ?
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# Is a circle unique for a given radius/diameter ?
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# In how many parts does a diameter divide the circle ? What is each part called ?
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===Activity No # 2 Measuring a chord in a circle.===
The correct use of reasoning is at the core of mathematics, especially in constructing proofs. Many statements, especially in geometry. Recall that a proof is made up of several mathematical statements, each of which is logically deduced from a previous statement in the proof, or from a theorem proved earlier, or an axiom, or the hypotheses. The main tool, we use in constructing a proof, is the process of deductive reasoning.
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*Estimated Time
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*Materials/ Resources needed
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*Prerequisites/Instructions, if any
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*Multimedia resources
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*Website interactives/ links/ / Geogebra Applets
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*Process/ Developmental Questions
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*Evaluation
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*Question Corner
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===Activity No # 4.Equal chords subtend equal angles at the centre. ===
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We start the study of this chapter in deductive reasoning using several examples.
*Tangent: line that intersects a circle in exactly one point, called the point of tangency
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*Radius from centre of circle to the point of tangency is always perpendicular to the tangent line. If
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*The radius is not perpendicular to the line, the line is not tangent to the circle.
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*Estimated Time
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*Recall the Pythagorean Theorem:
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*Materials/ Resources needed
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*Use the fact that a tangent line and the radius through that point of tangency are perpendicular to solve for a third value. Show how you can also use this fact to deduce whether or not a line is tangent to a specific circle.
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*Prerequisites/Instructions, if any
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*Tangents from an external point are equal in length.
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*Estimated Time
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*Materials/ Resources needed
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*Prerequisites/Instructions, if any
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*Multimedia resources
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*Website interactives/ links/ / Geogebra Applets
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*Process/ Developmental Questions
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*Evaluation
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*Question Corner
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==Concept # 4. Finding the Circumference of a circle==
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==Types of tangents==
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===Learning objectives===
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*Recognise the difference between a secant and a tangent of a circle.
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# The children understand that the distance around the edge of a circle is known as circumference.
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*Construct a tangent to a circle at a given point on it.
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# The children learn to measure the circumference of the circle.
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*Construct and verify that, the radius drawn at the point of contact is perpendicular to the tangent.
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# Derivation of formula for circumference.
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*Construct tangents to a circle from an external point.
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# They understand what is pi.
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*Recognise the properties of direct common tangents and the transverse common tangents.
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===Notes for teachers===
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==Touching circles==
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The circumference of a circle relates to one of the most important mathematical constants in all of mathematics. This constant pi, is represented by the Greek letter П. The numerical value of π is 3.14159 26535 89793 , and is defined by the ratio of a circle's circumference to its diameter.
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Common tangents of two circles How many common tangents do two circles have. Informally draw all different cases, with 0, 1, 2, 3, 4 common tangents.
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C = п. D or C = 2пr.
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===Activity No # 1 Derivation of formula for circumference and the value for pi.===
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For any two different circles, there are five possibilities regarding their common tangents:
*One circle touches the other from inside. There is one common tangent, located at this touching point.
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*The two circles intersect in two points. They have two common tangents, which lie symmetrically to the axis connecting the two centres.
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*The two circles touch each other from outside. They have three common tangents.
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*Estimated Time : 45 mins
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*The two circles lie outside of each other. They have four common tangents.
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*Materials/ Resources needed:
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Note books, compass, pencil, mender, scale.
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*Prerequisites/Instructions, if any:
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# The children should have prior knowledge of circle, radius, diameter and circumference.
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# They should have measuring and computational skills.
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*Multimedia resources:
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*Website interactives/ links/ / Geogebra Applets
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*Process:
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#Ask the children to draw five circles with different radii.
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# Let them carefully measure their circumferences using wool.
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# Mark the distance around the circle on the wool with a sketch pen.
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# Measure the length of the measured wool using a scale.
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# Make a table with columns radius, diameter and circumference
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# For every circle find Circumference / diameter.
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# Round C/d to two decimal places.
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# Observe the answers in each case. It would be aprroximately 3.14 .
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# The value 3.14 is the value of pi which is constant.
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C/d = п or C = п d or C = 2п r.
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*Developmental Questions:
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# Have you noted down radius, diameter and their respective circumferences.
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# Check if your calculations are correct.
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# What do you infer from the observed results ?
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*Evaluation:
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# Are the children taking correct measurements.
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# Are they comparing the results of C/d with all circles.
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# Are they noticing that it is constant .
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# Are they questioning why it is constant?
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*Question Corner:
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# How do you derive the formula for circumference of a circle ?
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# What is the name of that constant ?
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# Try to collect more information on Pi.
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==Concept # 5 Finding the area of a circle.==
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==== '''Construction of tangents''' ====
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===Learning objectives===
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*[[Image:KOER%20Circles_html_50027288.png|link=]]
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# The child should understand that the area of a circle is the entire planar surface.
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*<u>To draw a tangent to a circle from an external point </u> [[Image:KOER%20Circles_html_m520802ec.png|link=]]
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# Derivation of the formula for area of the circle.
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*<u>To draw direct common tangents to two given circles of equal radii, with centres ‘d’ units apart. </u> [[Image:KOER%20Circles_html_4b7743eb.png|link=]]
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# Area of the circle is dependent on its radius.
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*<u>To draw a direct common tangent to two circles of different radii. </u> [[Image:KOER%20Circles_html_3b9c6f9.png|link=]]
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# The formula for area of a circle is derived by converting the circle into an equally sized parallelogram.
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To construct Transverse common tangents to two circles.
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===Notes for teachers===
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1.Proof for area of a circle refer to them following link.
# Show the students the two figures circle and square.
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# Tell them that the radius and side of square are of same measure as it would help us in deducing the formula for area of circle.
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# Formulas are easy ways of calculating area .
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# If formulas are not known then the entire area in question can be divided into small squares of 1 unit measure and can deduce the formula of the whole.
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# First the number of full squares is counted.
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# Then two half squares would add up to 1 full square.
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# Ignore less than quarter . Take 3/4 as full.
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# Finally adding up the whole number would give us the full area of the figure in question.
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# Divide area of circle with that of square and deduce formula for square with known formula for square.
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*Developmental Questions:
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# Which are these two figures?
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# What inputs do you need to draw a circle ? And for a square ?
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# What do you observe as constant in the two figures ?
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# Do you think the size of both the figures are same ?
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# How do we find it ?
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# What is the formula to find the area of a square ?
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# When we do not know the formula for area, how do we deduce it ?
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# Count the number of squares in the entire area of circle ?
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# How to add half and quarter squares ?
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# Approximately how many total 1 unit squares cover the circle ?
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# So, what is the area of the circle ?
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# What are we trying to deduce (get) through this activity ?
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# Fine lets try dividing the area of circle with area of square and observe the proceedings while we try to deduce the formula for area of circle.
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*Evaluation;
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# Has the student understood the concept of area.
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# Was the student aligned with the assignment and was he able to follow the sequence of steps ?
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# Is the student able to appreciate the analogy ?
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*Question Corner;
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# What is Pi ?
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# What do you understand by area ?
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# What is the formula to find the area of square and that of a circle ?
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= Hints for difficult problems =
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[[Circles Tangents Problems]]
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====== [[Construction of direct common tangent]] ======
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The direct common tangents to two circles meet on the line of centres and divide it externally in the ratio of the radii.
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= Project Ideas =
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====== [[Construction of transverse common tangent]] ======
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The transverse common tangents also meet on the line of centres and divide it internally in the ratio of the radii.
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==Further Explorations==
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1. This link gives an overview of what tangents are, [[http://en.wikipedia.org/wiki/Pi]]
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=See Also=
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Click [http://www.youtube.com/watch?v=BPTJ9P4vQ78 here] for some interesting videos on constructions of circles.
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=Teachers Corner=
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The major portion of the contributions for this topic are from '''Radha N, GHS Begur''' and '''Roopa N GHS Nelavagilu''' .
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==GeoGebra Contributions==
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#The GeoGebra file below verifies the theorem
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##The angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle.
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##Arcs and Angles http://karnatakaeducation.org.in/KOER/Maths/Arc_angle.html
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##Download ggb file here http://karnatakaeducation.org.in/KOER/Maths/Arc_angle.ggb
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##Arcs and Angles Part 2 http://karnatakaeducation.org.in/KOER/Maths/Same_segment_angle.html
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##Download ggb file here http://karnatakaeducation.org.in/KOER/Maths/Same_segment_angle.ggb
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##See a video to understand this theorem http://www.youtube.com/watch?v=0B0v0NCHZx0
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#This GeoGebra file shows how a cone can be constructed from a sector of a circle
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##Cone Construction http://karnatakaeducation.org.in/KOER/Maths/conesurfacearea.html
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##Download ggb file here http://karnatakaeducation.org.in/KOER/Maths/conesurfacearea.ggb
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====== Solved problems/ key questions (earlier was hints for problems). ======
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= Math Fun =
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===Projects (can include math lab/ science lab/ language lab) ===
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#Collect different types of circular objects
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#Collect different '''Pie Charts'''.
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#Collect different photographs of tools of cutting circles
The following is a background literature for teachers. It summarises the things to be known to a teacher to teach this topic more effectively . This literature is meant to be a ready reference for the teacher to develop the concepts, inculcate necessary skills, and impart knowledge in Geometry - Circles from Class 6 to Class 10.
The first step is to understand how to define circles and related terms using geometric vocabulary. The next step is to understand what is Pi. That it is a constant and that for any circle the ratio of the circumference by the diameter is always a constant value Pi. The interesting properties of Pi – an irrational number can also be discussed here in the basic form. Ability for the child to do simple area and perimeter calculations. Next the learner should understand that the circle is a 2 dimensional plane figure and how to visualise solid 3-dimensional figures. What are the solid shapes that have a circle as a part of them. Mensuration – more complex area measurements which include circular shapes. Surface Area and Volume measurement of sold shapes such as cylinder, sphere and cone. Understand the properties of the circles by proving theorems deductively. Also acquire the skills of deductive proofs, understand that all the properties can be deduced from the axioms. Understand the relationship between lines and circles – secant and tangent
Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.
To make students know that circle is a 2-dimensional plane circular figure.
All points on its edge are equidistant from the center.
The method of drawing a circle
The size of the circle is defined by its radius.
To elicit the difference between a bangle or a circular ring and circle as such.
Teaching Outlines
Concept #1 Introduction to Circle
When I tell people that circles are the mother of all inventions, the first thing they ask is, “circles are inventions?”
Yes, a circle isn’t something that exists in nature. It isn’t something that people discovered like gold or the new lands of America. It is a mental construct, a symbolic representation that was invented much the same as language and the alphabet.
There is no way to be certain, but anthropologists generally agree that the circle was created long before recorded history. It is quite likely that it was drawn by a stick in the sand. With the sun being a constant in early man’s existence and the source of all life, it is quite likely that the first circle represented the sun.
Through the years man’s understanding of the circle has evolved substantially with Euclidean geometry being its crowning point of technological understanding. (Having said that, I assure you this blog is not going to be about mathematics or boring scientific equations.)
What we will say that is without the rudimentary understanding of circles, the world would not be anything like it is today. Without circles, there would be no wheel, which is man’s crowning achievement dating back to the Neolithic Age (circa 9500 BC).
The three other great achievements are the ability to make fire, the agriculture of crops, and the domestication of animals. While the circle didn’t have any direct bearing on these advancements, the understanding of circles certainly contributed to their proliferation and expansion.
Besides the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a Ferris wheel, or watching the moon landing on our television set.
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. They are an intrinsic component in the invention of almost everything that we see around us.
I had a microbiologist challenge me that his field didn’t have much use for circles. Not knowing anything about microbiology, I asked him what was the shape of the lens in his microscope.
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind.
And it doesn’t exist except in our mental construct. It is a symbol, not a thing. We talk about it in every language on earth. It is written about in millions of textbooks and all over the Internet, and yet we cannot put it in a wheel barrel. It doesn’t exist in a three dimensional world or even a two dimensional world. It is merely a representation.
Immanuel Kant’s famous phrase “ding an sich” applies to the circle. A circle is not a “thing-in-itself.” It is a semantic fabrication that exists only in our imagination. As Alfred Korzybski, the father of General Semantics, would say, it is “the map, not the territory.”
But here we are getting off on a philosophical tangent that might be subject to a future blog entry. For now let’s just say that circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why I think the circle is so fantastic.
Source: http://circlesonly.wordpress.com/tag/inventions/
Summary :
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.
Circle Properties
A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.
Equal chords of a circle (or of congruent circles)subtend equal angles at the centre.
If the angles subtended by two chords of a circle(or of congruent circles) at the centre(corresponding centres) are equal, the chords are equal.
The perpendicular from the centre of a circle to a chord bisects the chord.
The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
There is one and only one circle passing through three non-collinear points.
Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres).
Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.
If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.
Congruent arcs of a circle subtend equal angles at the centre.
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Angles in the same segment of a circle are equal.
Angle in a semicircle is a right angle.
If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.
The sum of either pair of opposite angles of a cyclic quadrilateral is 1800.
If sum of a pair of opposite angles of a quadrilateral is 1800, the quadrilateral is cyclic.
To demonstrate the value of Pi move the slider named a from minimum to maxiumum value and observe the circumference
Evaluation
Use the GeoGebra file [[2]] and illustrate and verify that the ratio is true for different radius by moving the radius slider and using the table below to compute the values.
Radius of circle r
Circumference of Circle C
C/2r
6
18.85
-
2.5
15.71
-
.........
-
-
Perimeter of a circle
Learning Objectives
To apply the use of calculating the perimeter of a circle in a real life example .
Material and Resources Required
Pencil, Paper
Pre-requisites/Instructions
Draw the following sketch and do the calculations for the evaluation questions. The sketch shows the two main dimensions of a standard 400 metres running track.
Evaluation
Calculate the inside perimeter of this shape.
Why do you think that it is not equal to 400 metres? The inside runner cannot run at the very edge of the lane (there is normally an inside kerb) but let us assume that the athlete runs at a constant distance of, say, x cm from the inside edge.
What is the radius of the two circular parts run by the athlete in the inside lane?
Show that the total distance travelled, in centimetres, is 2 π (3650 + x ) + 16878 and equate this to 40 000 cm to find a value for x.
Is it realistic? For 200 m and 400 m races, the runners run in specified lanes. Clearly, the further out you are the further you have to run, unless the starting positions are staggered.
The width of each lane is 1.22 m, and it is assumed that all runners (except the inside one) run about 20 cm from the inside of their lanes.
With these assumptions, what distance does the athlete in Lane 2 cover when running one complete lap? Hence deduce the required stagger for a 400 m race.
What should be the stagger for someone running in Lane 3 ?
If there are 8 runners in the 400 m race, what is the stagger of the athlete in Lane 8
compare with that in Lane 1 ? Is there any advantage in being in Lane 1?
Further Explorations
1. This link gives an overview of what Pi is. [[3]]
Points on the planar surface of the circle within its circumference are said to be interior points and points on the outside of circumference are said to be its exterior points.
Chords of a circle are of different sizes.The length of the chord increases as it moves closer to the centre and decreases as it moves away from the center.
A chord is the interval joining two distinct points on a circle. This activity investigates formation of chord and compares with the diameter of the circle.
Since every perpendicular bisector passes through the centre, the centre must lie on every one of them, so the centre must be their single common point.
The angle made at the centre of a circle by the radii at the end points of a chord is called the central angle or angle subtended by a chord at the centre.
Concept #5: Cyclic Quadrilateral
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
A quadrilateral ABCD is called cyclic if all four vertices of it lie on a circle.In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees.If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic.In a cyclic quadrilateral the exterior angle is equal to interior opposite angle.
Relation between the angles of a cyclic quadrilateral are explored with this hand on activity.
Concept #6 Constructions in circles
The students should know that tangent is a straight line touching the circle at one and only point.They should understand that a tangent is perpendicular to the radius of the circle.The construction protocol of a tangent.Constructing a tangent to a point on the circle.Constructing tangents to a circle from external point at a given distance.A tangent that is common to two circles is called a common tangent.A common tangent with both centres on the same side of the tangent is called a direct common tangent.A common tangent with both centres on either side of the tangent is called a transverse common tangent.
A line which touches a circle at exactly one point is called a tangent line and the point where it touches the circle is called the point of contact.
Properties of tangent
The tangent at any point of a circle is perpendicular to the radius through the point of contact. We can also conclude that at any point on a circle there can be one and only one tangent.
Explain the theorem that if two segments from the same exterior point are tangent to a circle, then they are congruent.
Solve for an unknown in a problem involving tangents.
Apply properties of tangents to solve problems involving triangles circumscribed about a circle.
Tangents from a point outside the circle
The lengths of two tangents from an external point are equal.
The tangents drawn from an external point to a circle are equally inclined to the line joining the point to the centre of the circle.
Secant
A line which intersects the circle in two distinct points is called a secant line (usually referred to as a secant).
Touching circles
Common tangents
Common tangents are lines or segments that are tangent to moret han one circle at the same time.
Direct common tangents
The centres of the circles lie on the same side of the common tangent.(dct)
Transverse common tangents
The centres of the circles lie on either side of the common tangent(tct)
Evaluation
1.How many direct common tangents can be drawn to 2 intersecting circles and 2 separate circles?
2.Can you draw tct to 2 intersecting circles?
3. How many umber of tangents to a circle which are parallel to a secant ?
4. How many number of tangents that can be drawn through a point which is inside the circle ?
Proofs and verification of properties of tangents
The correct use of reasoning is at the core of mathematics, especially in constructing proofs. Many statements, especially in geometry. Recall that a proof is made up of several mathematical statements, each of which is logically deduced from a previous statement in the proof, or from a theorem proved earlier, or an axiom, or the hypotheses. The main tool, we use in constructing a proof, is the process of deductive reasoning.
We start the study of this chapter in deductive reasoning using several examples.
we can verify the theorems by practical construction. And also by using GeoGebra tool.
Tangents to a circles:
Tangent: line that intersects a circle in exactly one point, called the point of tangency
Radius from centre of circle to the point of tangency is always perpendicular to the tangent line. If
The radius is not perpendicular to the line, the line is not tangent to the circle.
Recall the Pythagorean Theorem:
Use the fact that a tangent line and the radius through that point of tangency are perpendicular to solve for a third value. Show how you can also use this fact to deduce whether or not a line is tangent to a specific circle.
Tangents from an external point are equal in length.
