Circles

ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ

The Story of Mathematics

Philosophy of Mathematics

Teaching of Mathematics

Curriculum and Syllabus

Topics in School Mathematics

Textbooks

Question Bank

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Concept Map

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Additional Resources[edit | edit source]

OER[edit | edit source]

1. Web resources :
   1. Cool math For clear and easy definitions.
   2. Wikipedia Has good explanations on circles.
   3. Khan academy Has good educative videos.
   4. Arvind gupta toys Contains good information.
2. Books and journals
   1. School Geometry By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.
3. Textbooks:
   1. Class 9 Mathematics contain simple description and theorems on circle
   2. CLASS 10
4. Syllabus documents

Non-OER[edit | edit source]

1. Web resources
   • maths is funHere you get description of terms of circles
   • Intersting facts this web link is full of circle facts.
   • sparknotes Gives some more details about properties of circles
   • www.regentsprep.com conatins good objective problems on chords and secants
   • www.mathwarehouse.com contains good content on circles for different classes
   • staff.argyll contains good simulations
   • Open reference Contains good simulations.
   • nrich.maths.org Refer for understanding Pi.
   • This is a video showing construction of tangent at any point on a circle

 : This is a resource file created by Suchetha, Mathematics teacher, GJC Thyamangondlu

1. • This is a video showing construction of tangent from external point and theorem

 : This is a resource file created by Suchetha, Mathematics teacher, GJC Thyamangondlu

• • • you want see the kannada videos on theorems and construction of circle click here this is shared by Yakub koyyur GHS Nada.

1. Books and journals
2. Textbooks
   1. Karnataka text book for Class 10, Chapter 14 - Chord properties
   2. Karnataka text book for Class 10, Chapter 15 - Tangent Properties
3. Syllabus documents (CBSE, ICSE, IGCSE etc)

Learning Objectives

• 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

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.

Activities

A discussion on “Life without circular shaped figures.”

Discussion based activity to relate and assimilate circular shapes seen in our surroundings.

Circle as a shape

A circle is the set of all points in the plane that are a fixed distance from a fixed point. 

Is circle a Polygon ? - A debate

A polygon when increased in number of sides tends to form a circle is shown with this interesting activity.

Concentric circles

Drawing concentric circles, with this hands on activity circle as a shape and variations in it is explored.

Congruent circles

Equal circles are circles with same radius is a concepts that is introduced in this activity.

Equal parts in a circle

Dividing a circle in to parts and exploring to divide it into equal parts is show in this activity.

Concept #2 Terms associated with circles

Activities

Centre of a circle

All points on a circle are at fixed distance from a point, which is the center of a circle.

Radius and diameter of a circle

Marking radius and diameter of a circle and understand their relation.

Circumference of a circle

Measuring circumference to understand it as the perimeter of the shape.

Semicircle

Partitioning a circle into two halves to form semicircles by drawing diameter.

Interior and exterior of a circle

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.

Basic elements of a circle

Investigation to understand basic parameters associated with circles.

Chord of a circle

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.

Arc of a circle

The part of the circumference within the two points in either directions are called its arcs.

Arcs and Sector of a circle

Slice of a circle enclosed between any two radii is called a sector.Semicircle and quadrant are special types of sectors.

Concept #3: Circles and Lines

Activities

Introduction to chords

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.

Secant and tangent of a circle

A tangent is a line touching a circle in one point. A secant is the line through two distinct points on a circle.

Concept #4: Theorems and properties

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.

The theorems that involve chords of a circle are :

• Perpendicular bisector of a chord passes through the center of a circle.
• Congruent chords are equidistant from the center of a circle.
• If two chords in a circle are congruent, then their intercepted arcs are congruent.
• If two chords in a circle are congruent, then they determine two central angles that are congruent.

Activities

Chord length and distance for centre of the circle

For a chord the distance from the center is the perpendicular distance of the chord such that it passes through the center.

The longest chord passes through the centre of the circle

Investigating the diameter is the longest chord of a circle.

Perpendicular bisector of a chord passes through the center of a 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.

Congruent chords are equidistant from the center of a circle

In the same circle or in circles of equal radius:

• Equal chords are equidistant from the centre.

• Conversely, chords that are equidistant from the centre are equal.

Angles in a circle subtended by a chord

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.

Activities

Cyclic quadrilateral

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.

Properties of cyclic quadrilateral

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.

Activities

Construction of direct common tangent

The direct common tangents to two circles meet on the line of centres and divide it externally in the ratio of the radii. 

Construction of transverse common tangent

The transverse common tangents also meet on the line of centres and divide it internally in the ratio of the radii.

Projects (can include math lab/ science lab/ language lab)

1. Collect different types of circular objects
2. Collect different Pie Charts.
3. Collect different photographs of tools of cutting circles
4. Collect different coins of circular shape
5. Collect different images of medals