Difference between revisions of "Measurements in circles"
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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. | #[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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==Concept # Measurements in circles == | ==Concept # Measurements in circles == | ||
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*Multimedia resources : Laptop, Projector. | *Multimedia resources : Laptop, Projector. | ||
*Website interactives/ links/ / Geogebra Applets | *Website interactives/ links/ / Geogebra Applets | ||
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= Hints for difficult problems = | = Hints for difficult problems = | ||
Revision as of 10:51, 6 December 2013
Error: Mind Map file Measurements in circles.mm not found
Textbook
ncert books
Class 9 Mathematics contain simple description and theorems on circles. 9 Mathematics contain higher level description and theorems on circles.
Additional Information
Useful websites
- maths is fun A GOOD WEBSITE ON DEFINITIONS FOR CIRCLES
- COOL MATH GIVES CLEAR AND EASY DEFINITIONS
- OPEN REFERENCE CONTAINS FEW SIMULATIONS
- WIKIPEDIA CONTAINS EXPLANATIONS FOR CIRCLES
- KHAN ACADEMY CONTAIN GOOD VIDEOS
- ARVIND GUPTA TOYS CONTAIN VERY GOOD BOOKS ON MATHEMATICS AND SCIENCE
- nrich.maths.org Contain very good description of Pie
Reference Books
- School Geometry By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.
Concept # Measurements in circles
Learning objectives
- The students should learn to measure radius, diameter, circumference, chord length and angles subtended at the centre and on the circumference of the circle.
- The students should understand that radius, diameter and chord lengths are linear measurements.
- They should learn to relate the size of the circle with radius.
- They realise that to draw a circle knowing the measure of radius or diameter is essential.
- There can be infinite radii in a circle.
- Diameter is twice the radius.
- The students should understand what a chord is.
- Chords of different lengths can be drawn in a circle.
- Chord length can be measured using a scale and its units is cm.
- They should know that the length of the chord increases as it moves closer to the diameter.
- The longest chord in the circle is its diameter.
- Distance of chord from the centre is its perpendicular distance from the centre.
- A chord divides the circle into two segments.
- Angle at the centre of the circle is 360º.
- Angles in circles are measured using protractor.
- Circumference and area are calculated using formula.
Notes for teachers
Activity No # 1. Measuring radius and diameter.
- Estimated Time: 15 mins
- Materials/ Resources needed:
- Laptop, goegebra tool, projector and a pointer.
- students' geometry box
- Prerequisites/Instructions, if any:
- Children should have the knowledge of circle, centre, radius, diameter and circumference.
- The teacher should have the necessary skill of using geogebra tool.
- Multimedia resources: Laptop
- Website interactives/ links/ / Geogebra Applets
- Process:
- Initially the teacher can explain the terms: circle, its centre, radius, diameter and circumference.
- Ask the children “What parameter is needed to draw a circle of required size ?”
- Show them how to measure radius on the scale accurately using compass.
- Show them to draw a circle.
- Given diameter, radius = D/2.
- 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.
- To measur diameter, measure the length of that chord which passes through the centre of the circle.
Then she can project the digital tool 'geogebra.' and further clarify concepts.
- Developmental Questions:
- Name the centre of the circle.
- Name the point on the circumference of the circle.
- What is the linesegment AB called ?
- Name the line passing through the centre of the circle.
- Using what can you measure the radius and diameter.
- Name the units of radius/diameter.
- Evaluation:
- How do you measure exact radius on the compass?
- Are the children able to corelate the radius/diameter of a circle with its size ?
- Question Corner:
- If the centre of the circle is not marked , then how do you get the radius for a given circle.
- How many radii/diameter can be drawn in a circle?
- Are all radii for a given circle equal ?
- Is a circle unique for a given radius/diameter ?
- In how many parts does a diameter divide the circle ? What is each part called ?
Activity No # 2 Measuring a chord in a circle.
- Estimated Time : 10 minutes
- Materials/ Resources needed:
Laptop, geogebra file, projector and a pointer.
- Prerequisites/Instructions, if any:
- The students should have prior knowledge of circle, radius , diameter and circumference..
- The teacher should have knowledge of using geogebra.
- Multimedia resources:
Laptop, geogebra file, projector and a pointer.
- Website interactives/ links/ / Geogebra Applets
- Process:
- The teacher can review the concept of a circle , radius , diameter and circumference .
- Any two points on the circumference can be joined.
- The joining line segment is called the chord.
- Let the students name the chord .
- Move the chord on the geogebra and let them observe its changing size.
- Let them observe the chord becoming a diameter while passing through the centre of the circle.
- The length of the chord is measued using a scale with its unit being cm.
- Developmental Questions:
- The teacher can point to centre of circle and ask the students as to what it is.
- She can point to radius and ask the students to name it.
- Then ask them if any two points on the circumference are joined by a line segment what is it called ?
- How many chords can be drawn in a circle ?
- Are lengths of all chords the same ?
- Name the biggest chord in a circle.
- How do you measure a chord and in what units ?
- Evaluation:
Were the students able to distinguish between radius, diameter and chord ?
- Question Corner:
After drawing a chord,what are the two segregated parts of the circle called ?
Concept # 3 Angles in circles
Learning objectives
- students should understand that the angle at the centre of the circle is 360 degrees.
Notes for teachers
Activity No # 1.The angle at the centre is double the angle at the circumference
- Estimated Time :40 minutes
- Materials/ Resources needed :Laptop, geogebra file, projector and a pointer.
- Prerequisites/Instructions, if any
- The students should have prior knowledge of a circle and circumference.
- They should know that an arc is a curved line along the circumference of a circle.
- If the end points of an arc are joined to a third point on the circumference of a circle, then an angle on the circumference is formed.
- If the end points of an arc are joined to the centre of a circle, then an angle at the centre of the circle is formed.
- They should know to measure the angles.
- Multimedia resources: Laptop and a projector.
- Website interactives/ links/ / Geogebra Applets
- Process:
- The teacher should initially discuss about the circle , radius, centre and circumference.
- Projecting geogebra file she can show the major and the minor arcs.
- Name the arc in discussion.
- Let students find out and name the angle subtended by the arc at the centre and angle subtended by the same arc on the circumference.
- Observe that the end points of the arc lie on the angle.
- Each side of the angle contains at least one end -point of the arc.
- Project different angles subtended by the same arc on the circumference. What is the inference ?
- Compare angle formed at the centre and angle formed on the circumference by the same arc.
- Change the angles/arc using slider. Note down the two angles in each case.
- Ask students what they observed ? Let them infer.
- Developmental Questions:
- Name the centre of the circle?
- Name the minor arc.
- Name the point on the circumference of the circle at which the arc subtends an angle.
- Name all radii from figure.
- What type of triangle is triangel APO ?
- Name the two equal sides of the triangle APO.
- Recall the theorem related to isosceles triangle.
- Name the two equal angles.
- Name the exterioe angle for the triangle APO
- Recall the exterior angle theorem.
- What relation do you observe between <p and <x.
- Similarly try to explain the same with triangle PBO.
- If <APO is half of <AOQ and <BPO is half of <BOQ what can you conclude about angles <AOB and <APB.
- What relation do you observe between the angle at the centre and that on the circumference formed by the same arc ?
- Evaluation:
- In a circle, how many angles are subtended by an arc at its centre?
- In a circle, how many angles are subtended by an arc at its circumference?
- Question Corner:
- What are the applications of this theorem.
Activity No # 2. Angles in a circle.
- Estimated Time: 40 minutes
- Materials/ Resources needed:Laptop, projector, geogebra file and a pointer.
- Prerequisites/Instructions, if any
- The students should have prior knowledge of a circle, angles, arcs and segments.
- The students should have a thorough knowledge about the types of angles.
- They should have the skill of drawing a circle , angles and measuring them.
- Multimedia resources : Laptop, Projector.
- Website interactives/ links/ / Geogebra Applets
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Hints for difficult problems
Project Ideas
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