Bureaucrats, oversight, Administrators
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From Karnataka Open Educational Resources
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Line 30: − + − # The teacher should have the necessary skill of using geogebra tool. − + Line 42:
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Line 69: − # The students should have prior knowledge of circle, radius , diameter and circumference.. − # The teacher should have knowledge of using geogebra. Line 78:
Line 74: − + − # 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. Line 94:
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Line 95: − + − + − + − + − # 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. − + − # 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. Line 153:
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Text replacement - "</mm>" to ""
[[File:Measurements in circles.mm|flash]]
==Concept # Measurements in circles ==
==Concept # Measurements in circles: 1. Radius and Diameter ==
===Learning objectives===
===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.
# Ability 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.
# Radius, diameter and chord lengths are linear measurements.
# They should learn to relate the size of the circle with radius.
# Relate the size of the circle with radius.
# They realise that to draw a circle knowing the measure of radius or diameter is essential.
# They realise that to draw a circle knowing the measure of radius or diameter is essential.
# There can be infinite radii in a circle.
# There can be infinite radii in a circle.
# Chords of different lengths can be drawn in a circle.
# Chords of different lengths can be drawn in a circle.
# Chord length can be measured using a scale and its units is cm.
# 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 length of the chord increases as it moves closer to the diameter.
# The longest chord in the circle is its diameter.
# The longest chord in the circle is its diameter.
# Distance of chord from the centre is its perpendicular distance from the centre.
# Distance of chord from the centre is its perpendicular distance from the centre.
{| style="height:10px; float:right; align:center;"
{| style="height:10px; float:right; align:center;"
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
|}
|}
*Estimated Time: 15 mins
*Estimated Time: 15 mins
#students' geometry box
#students' geometry box
*Prerequisites/Instructions, if any:
*Prerequisites/Instructions, if any:
# Children should have the knowledge of circle, centre, radius, diameter and circumference.
# Circle and its basic parts should have been done.
*Multimedia resources: Laptop
*Multimedia resources: Laptop
*Website interactives/ links/ / Geogebra Applets
*Website interactives/ links/ / Geogebra Applets : This file was done by ITfC-Edu-Team.
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" />
*Process:
*Process:
# Given diameter, radius = D/2.
# 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.
# 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.
# To measure 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.
Then she can project the digital tool 'geogebra.' and further clarify concepts.
*Developmental Questions:
*Developmental Questions:
# Name the centre of the circle.
# Name the centre of the circle.
{| style="height:10px; float:right; align:center;"
{| style="height:10px; float:right; align:center;"
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
|}
|}
*Estimated Time : 10 minutes
*Estimated Time : 10 minutes
Laptop, geogebra file, projector and a pointer.
Laptop, geogebra file, projector and a pointer.
*Prerequisites/Instructions, if any:
*Prerequisites/Instructions, if any:
*Multimedia resources:
*Multimedia resources:
Laptop, geogebra file, projector and a pointer.
Laptop, geogebra file, projector and a pointer.
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
*Process:
*Process:
# The teacher can review the concept of a circle , radius , diameter and circumference .
# Show the geogebra file and ask the following questions.
*Developmental Questions:
*Developmental Questions:
# The teacher can point to centre of circle and ask the students as to what it is.
# The teacher can point to centre of circle and ask the students as to what it is.
# How do you measure a chord and in what units ?
# How do you measure a chord and in what units ?
*Evaluation:
*Evaluation:
Were the students able to distinguish between radius, diameter and chord ?
# Were the students able to distinguish between radius, diameter and chord ?
*Question Corner:
*Question Corner:
After drawing a chord,what are the two segregated parts of the circle called ?
3 After drawing a chord,what are the two segregated parts of the circle called ?
==Concept # 3 Angles in circles==
==Concept # 2. Angles in circles==
===Learning objectives===
===Learning objectives===
# students should understand that the angle at the centre of the circle is 360 degrees.
# students should understand that the angle at the centre of the circle is 360 degrees.
{| style="height:10px; float:right; align:center;"
{| style="height:10px; float:right; align:center;"
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
|}
|}
*Estimated Time :40 minutes
*Estimated Time : 40 minutes
*Materials/ Resources needed :Laptop, geogebra file, projector and a pointer.
*Materials/ Resources needed : Laptop, geogebra file, projector and a pointer.
*Prerequisites/Instructions, if any
*Prerequisites/Instructions, if any
# The students should have prior knowledge of a circle and circumference.
# Circles and its parts should have been done.
*Multimedia resources: Laptop and a projector.
*Multimedia resources: Laptop and a projector.
*Website interactives/ links/ / Geogebra Applets
*Website interactives/ links/ / Geogebra Applets
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
*Process:
*Process:
# The teacher should initially discuss about the circle , radius, centre and circumference.
# Project the geogebra file and ask the questions listed below.
*Developmental Questions:
*Developmental Questions:
# Name the centre of the circle?
# Name the centre of the circle?
{| style="height:10px; float:right; align:center;"
{| style="height:10px; float:right; align:center;"
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
|}
|}
*Estimated Time: 40 minutes
*Estimated Time: 40 minutes
*Materials/ Resources needed:Laptop, projector, geogebra file and a pointer.
*Materials/ Resources needed:Laptop, projector, geogebra file and a pointer.
*Prerequisites/Instructions, if any
*Prerequisites/Instructions, if any
#The students should have prior knowledge of a circle, angles, arcs and segments.
# Knowledge of a circle, angles, arcs and segments.
#The students should have a thorough knowledge about the types of angles.
# About the types of angles.
#They should have the skill of drawing a circle , angles and measuring them.
# Skill of drawing a circle , angles and measuring them.
*Multimedia resources : Laptop, Projector.
*Multimedia resources : Laptop, Projector.
*Website interactives/ links/ / Geogebra Applets: This file has been done by Mallikarjun Sudi of Yadgir.
<ggb_applet width="1280" height="600" version="4.0" 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*Process:
# The teacher can recall the concept of circle, arc segment.
# She can then project the geogebra file , change slider and make clear the theorems about angles in a circle.
Developmental Questions:
# Name the minor and major segments.
# Name the angles formed by them.
# Where are the two angles subtended ?
# What is the relation between the two angles.
# Name the major and minor arcs.
# What is an acute angle?
# What is an obtuse angle?
# What type of angles are formed by minor arc ?
# What type of angles are formed by major arc ?
# What are your conclusions ?
*Evaluation:
# How many angles can a segment subtend on the circumference ?
# What can you say about these angles ?
*Question Corner:
# Recall the theorems related to angles in a circle.
*Process:
# The teacher can recall the concept of circle, arc segment.
# She can then project the geogebra file , change slider and make clear the theorems about angles in a circle.
Developmental Questions:
# Name the minor and major segments.
# Name the angles formed by them.
# Where are the two angles subtended ?
# What is the relation between the two angles.
# Name the major and minor arcs.
# What is an acute angle?
# What is an obtuse angle?
# What type of angles are formed by minor arc ?
# What type of angles are formed by major arc ?
# What are your conclusions ?
*Evaluation:
# How many angles can a segment subtend on the circumference ?
# What can you say about these angles ?
*Question Corner:
# Recall the theorems related to angles in a circle.
==Concept # 3. Finding the Circumference of a circle==
===Learning objectives===
# The children understand that the distance around the edge of a circle is known as circumference.
# The children learn to measure the circumference of the circle.
# Derivation of formula for circumference.
# They understand what is pi.
===Notes for teachers===
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.
C = п. D or C = 2пr.
===Activity No # 1 Derivation of formula for circumference and the value for pi.===
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|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
|}
*Estimated Time : 45 mins
*Materials/ Resources needed:
Note books, compass, pencil, mender, scale.
*Prerequisites/Instructions, if any:
# Circles basics should have been done.
*Multimedia resources:
*Website interactives/ links/ / Geogebra Applets
*Process:
#Ask the children to draw five circles with different radii.
# Let them carefully measure their circumferences using wool.
# Mark the distance around the circle on the wool with a sketch pen.
# Measure the length of the measured wool using a scale.
# Make a table with columns radius, diameter and circumference
# For every circle find Circumference / diameter.
# Round C/d to two decimal places.
# Observe the answers in each case. It would be aprroximately 3.14 .
# The value 3.14 is the value of pi which is constant.
C/d = п or C = п d or C = 2п r.
*Developmental Questions:
# Have you noted down radius, diameter and their respective circumferences.
# Check if your calculations are correct.
# What do you infer from the observed results ?
*Evaluation:
# Are the children taking correct measurements.
# Are they comparing the results of C/d with all circles.
# Are they noticing that it is constant .
# Are they questioning why it is constant?
*Question Corner:
# How do you derive the formula for circumference of a circle ?
# What is the name of that constant ?
# Try to collect more information on Pi.
==Concept # 4. Finding the area of a circle.==
===Learning objectives===
# The child should understand that the area of a circle is the entire planar surface.
# Derivation of the formula for area of the circle.
# Area of the circle is dependent on its radius.
# The formula for area of a circle is derived by converting the circle into an equally sized parallelogram.
===Notes for teachers===
1.Proof for area of a circle refer to them following link.
http://www.basic-mathematics.com/proof-of-the-area-of-a-circle.html
===Activity No # 1. To discover a formula for the area of a circle. ===
This activity has been taken from website :
http://www.mathsteacher.com.au/year8/ch12_area/07_circle/circle.htm
{| style="height:10px; float:right; align:center;"
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
|}
*Estimated Time:90 mins
*Materials/ Resources needed:A compass, pair of scissors, ruler and protractor , pencil and chart papers.
*Prerequisites/Instructions, if any
# Prior knowledge of circle, radius and parallelogram area.
# Skill to measure and draw accurately.
*Multimedia resources
*Website interactives/ links/ / Geogebra Applets
*Website interactives/ links/ / Geogebra Applets
<ggb_applet width="1280" height="600" version="4.0" 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*Process:
Refer this website :
http://www.mathsteacher.com.au/year8/ch12_area/07_circle/circle.htm
*Developmental Questions:
# Calculate the area of the figure in Step 6 by using the formula: Area = base x height
# What is the area of the circle drawn in Step 1?
# It appears that there is a formula for calculating the area of a circle. Can you discover it?
*Evaluation:
# Is the student able to comprehend the idea of area.
# Is the student able to corelate that the base of the parallelogram formed is half of the circle's circumference.
*Question Corner:
# What is the area of a parallelogram ?
# Is there any other way by which you can deduce the formula for area of a circle ?
===Activity No # 2. Proving area of the circle = п r² using geogebra applet.===
{| style="height:10px; float:right; align:center;"
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
|}
*Estimated Time: 45mins
*Materials/ Resources needed;
Laptop, geogebra file, projector and a pointer.
*Prerequisites/Instructions, if any:
Prior knowledge of circle, radius, square and area of square.
*Multimedia resources: Laptop.
*Website interactives/ links/ / Geogebra Applets: This file was done by Bindu.
<ggb_applet width="1280" height="600" version="4.0" 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*Process:
# Show the students the two figures circle and square.
# 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.
# Formulas are easy ways of calculating area .
# 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.
# First the number of full squares is counted.
# Then two half squares would add up to 1 full square.
# Ignore less than quarter . Take 3/4 as full.
# Finally adding up the whole number would give us the full area of the figure in question.
# Divide area of circle with that of square and deduce formula for square with known formula for square.
*Developmental Questions:
# Which are these two figures?
# What inputs do you need to draw a circle ? And for a square ?
# What do you observe as constant in the two figures ?
# Do you think the size of both the figures are same ?
# How do we find it ?
# What is the formula to find the area of a square ?
# When we do not know the formula for area, how do we deduce it ?
# Count the number of squares in the entire area of circle ?
# How to add half and quarter squares ?
# Approximately how many total 1 unit squares cover the circle ?
# So, what is the area of the circle ?
# What are we trying to deduce (get) through this activity ?
# 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.
*Evaluation;
# Has the student understood the concept of area.
# Was the student aligned with the assignment and was he able to follow the sequence of steps ?
# Is the student able to appreciate the analogy ?
*Question Corner;
# What is Pi ?
# What do you understand by area ?
# What is the formula to find the area of square and that of a circle ?
= Hints for difficult problems =
= Hints for difficult problems =
