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Line 18: − =Notes for teachers =+ − Introduction: − 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. Line 186:
Line 183: − + − *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 # 4. 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.=== − {| 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://www.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: − # The children should have prior knowledge of circle, radius, diameter and circumference. − # They should have measuring and computational skills. − *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 # 5 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://www.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 − *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://www.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 − <ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> − *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 ?
Measurements in circles (view source)
Revision as of 10:51, 6 December 2013
, 10:51, 6 December 2013→Notes for teachers
#[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.
==Concept # Measurements in circles ==
==Concept # Measurements in circles ==
*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 =