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Line 185: − ==Concept # 2. 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. === − {| 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: 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 − <ggb_applet width="1280" height="572" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /> − *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.=== − {| 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 : 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 − <ggb_applet width="1278" height="571" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> − *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 === − {| 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 :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 − <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: − # 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.=== − {| 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: 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 − <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: − # 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 ?
no edit summary
*Question Corner
*Question Corner
= Hints for difficult problems =
= Hints for difficult problems =