use the following symmetry properties of circles:teachers.stupidchicken.com/files/active/0/QCM520...

QCM520: Lesson plan assignment

Submitted by: Soon Yin Jie (19)Date: 25 October 2005Tutor: Ms Ivy Chow

Topic: Geometrical Properties of Circles

Contents

INTRODUCTION........................................................................................................................................1

TOPIC............................................................................................................................................................1ADMINISTRATIVE DETAILS...........................................................................................................................1UNIT DESCRIPTION.......................................................................................................................................1LEARNING THEORIES....................................................................................................................................2

Co-operative Learning and Social Constructivism.................................................................................2Guided Discovery....................................................................................................................................2Stages of Cognitive Development...........................................................................................................2Van Hiele Theory....................................................................................................................................2

TEACHING APPROACHES..............................................................................................................................2LEARNING DIFFICULTIES..............................................................................................................................3

LESSON DEVELOPMENT........................................................................................................................4

PREREQUISITE KNOWLEDGE.........................................................................................................................4SPECIFIC INSTRUCTIONAL OBJECTIVES........................................................................................................4KEY CONCEPTS............................................................................................................................................4RESOURCES..................................................................................................................................................4DETAILED LESSON PLAN..............................................................................................................................5

AFTERWORD............................................................................................................................................10

REFERENCES...............................................................................................................................................10Appendix....................................................................................................................................................10

Geometrical Properties of Circles: Symmetrical Properties of ChordsSoon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005

Introduction

TopicThe unit covered is “geometrical properties of circles”. The specific topic is “symmetrical properties of chords in a circle”.

Administrative Details Level: Secondary 3 Express pupils of low-average abilityTime: Double-period (70 minutes),Location: Regular classroom equipped with a whiteboard, overhead-projector, computer

and computer display projector.

Unit DescriptionThe entire unit, “geometrical properties of circles”, can be broken down into two specific subtopics, namely “symmetrical properties of circles” and “angle properties of circles”. The corresponding syllabus requirements from each subtopic are listed below:

A. Symmetrical properties of circles (under section 22, Symmetry): Students should be able to use the following symmetry properties of circles: 1. Equal chords are equidistant from the centre; 2. The perpendicular bisector of a chord passes through the centre; 3. Tangents from an external point are equal in length.

B. Angle properties of circles (under section 23, Angle): Students should be able to calculate unknown angles and solve problems (including problems leading to some notion of proof) using the following geometrical properties: 1. Angle in a semi-circle; 2. Angle between tangent and radius of a circle; 3. Angle at the centre of a circle is twice the angle at the circumference; 4. Angles in the same segment are equal; 5. Angles in opposite segments are supplementary.

The unit will be covered in three double-period lessons. The first lesson, “symmetrical properties of chords”, will cover points 1 and 2 under subtopic A. This lesson will be covered in greater detail in this lesson plan.

The second lesson (double-period), “symmetry and angle properties of tangents”, will cover point 3 from subtopic A as well as points 1 and 2 from subtopic B. This is because point 3 from subtopic A deals with tangents to the circle, the angle properties of which are dealt with in point 2 of subtopic B. Points 1 and 2 from subtopic B join well together because both are properties relating right angles and circles.

The final lesson (double-period), “interior angle properties of circles”, of this unit will cover points 3, 4 and 5 from subtopic B. Students will be taught the various relations between angles inside the circle, such as angle at the centre of a circle vs. angle at circumference, angles in the same segment, and angles in opposite segment. This will wrap up the entire unit on “geometrical properties of circles”.

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Learning Theories

Co-operative Learning and Social ConstructivismVygotsky introduced the concept of a Zone of Proximal Development (ZPD) in learners, where scaffolding can make the difference between what a learner can do by himself (unassisted performance) or should be able to achieve with guidance (assisted performance). In this lesson, students are able to solve simple geometric problems without assistance, whereas some scaffolding from the teacher will increase the range of problems they can solve.

Furthermore, scaffolding also includes group and co-operative learning, a concept that can be drawn upon for better results in the Mathematics classroom. Students will be asked to work in pairs to come up with ideas for a problem they are posed with at the beginning of class, and the teacher will utilise a “think-pair-share” classroom model to encourage discussion and sharing of ideas.

Guided DiscoveryBruner introduced the guided discovery approach in the 1970s, where “pupils try to discover mathematics on their own by working through various activities.” In this lesson, students will be made to discover various properties on their own, albeit with guidance from the teacher, through a series of investigative enquiry tasks in the first worksheet. This way, students can discover things for themselves, and would end up more motivated to learn.

Stages of Cognitive DevelopmentPiaget, in his work in the 1960s, put forward his organisation of cognitive development into a series of stages. Many students in our secondary schools have been shown to be in the concrete operational stage, hence supporting the greater usage of concrete material for learning. As such, this lesson will begin with students getting to “play” with circles and geometrical instruments, before moving on to other equally-concrete investigative acts in the first worksheet. The use of concrete objects will help solidify the foundation of these abstract properties in the students’ minds.

Van Hiele TheoryPierre and Dina van Hiele postulated in the 1960s that students learnt geometrical concepts within a certain hierarchy of levels. The levels are recognition at level 1, analysis, ordering, deduction, and finally rigour at level 5. The van Hiele theory suggests that students can only progress from one level to the next, without skipping levels of understanding.

In this topic, the lesson is structured to accommodate for the van Hiele theory. To begin with, students are asked to recognise the general appearance of the properties, i.e., the circles, chords and perpendicular bisectors that they will be dealing with. After the initial recognition stage, students will be brought up to the analysis and ordering stages through the use of the investigative worksheet. Finally, as the students are not required to produce rigorous mathematical proofs, an informal proof of each property will be presented to satisfy students at the deductive stage.

Teaching ApproachesSome of the teaching approaches employed here are commonly used in teaching geometry (as seen in the “Teaching of Geometry” section in Teaching Secondary Mathematics). In this lesson, deductive reasoning will be utilised, asking students to “deduce, from previously

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known information” such as perpendicular bisectors and chords, the method to obtain the centre of a circle without folding.

The approach of inductive reasoning and conjecturing, wherein the teacher would have the student make “conscious guesses of generalisations by observing instantiations or analogies”, is reflected in an investigate-style worksheet that asks students to measure angles and induce relations and properties.

Finally, some problem-based teaching will be used to engage the students. When beginning the class, students will be challenged to find the centre of a circle with restrictions on what they can do (e.g. no folding of the circle), and with limited tools. This will drum up greater interest in the topic among the students.

Learning DifficultiesThis topic can be difficult for students to visualise. The very properties that the students have to learn seem very complex when spelt out in words, and this does not help the students visualise what they have to learn. This is addressed in many parts of the lesson – the lesson begins by letting the students use their geometrical instruments to find the centre of a cut-out circle, and the student continues to investigate the properties of these cut-out circles as the lesson moves on. Various visual aids are utilised to ensure the students understand what the geometrical terms are referring to, such as slides, OHTs, and one self-constructed Teacher’s Aid that shows why equal chords are equidistant from the centre of a circle.

A further misconception could arise on the definition of “equal chords”, and this will be addressed as necessary when the term is introduced.

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Lesson DevelopmentLevel: Secondary 3 Express pupils of low-average abilityTime: Double-period (70 minutes),Location: Regular classroom equipped with a whiteboard, overhead-projector, computer

and computer display projector.

Prerequisite KnowledgeBefore the lesson, students should be able to:

1. Define the meanings of the terms “perpendicular bisector” and “equidistant”.2. Construct a perpendicular bisector using a compass.3. State that a chord is a straight line joining two distinct points on the circumference of

a circle.4. Give examples of rotation and reflection.5. State that any diameter of a circle is a line of symmetry.6. State and apply Pythagoras’ Theorem.

Specific Instructional ObjectivesAt the end of the lesson, the students should be able to:

1. State that the centre of a circle lies on the perpendicular bisector of a chord in the circle.

2. State the following symmetrical property of circles (“Property 1: Perpendicular chords”): The perpendicular to a chord, drawn from the centre of a circle, bisects the chord.

3. Infer, from the above, that the converse is true: A straight line drawn from the centre of a circle to bisect a chord, which is not a diameter, is perpendicular to the chord.

4. State the following symmetrical property of circles (“Property 2: Equidistant chords”): In a circle, equal chords are equidistant from the centre.

5. Infer, from the above, that the converse is true: Chords, which are equidistant from the centre, are equal.

6. Solve problems involving the application of Property 1 and Property 2 together with Pythagoras’ Theorem to find the unknown length of chords or lines perpendicular to chords in a circle.

Key ConceptsPerpendicular bisection of chords, chords equidistant from centre of a circle.

Resources1. Computer 2. Slides3. Java applet demonstration site4. Circle cut-outs (four for every student) – prepare before class5. Teaching aid: large circle cut-out, string tied to ruler, magnet for sticking everything

to whiteboard6. OHTs, 7. Whiteboard, markers

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Detailed Lesson PlanKey questions and expected student responses are in italics.

Time Teaching / Learning Activities Material Rationale10 min Starter activity

Show a diagram of a circle and challenge students: I bet you can’t find the centre of a circle using only a ruler, compass and pencil only, without folding the circle! Distribute four blank pre-cut circles to each student. While distributing, ask students to think of solutions verbally and invite them to share with the class.

Let students try to work out a solution in pairs for one minute. While they are doing so, revise the concept of chords: What is a chord? Mention that this is a hint, and ask students to try drawing some chords on their circle. Next, ask: What is a perpendicular bisector? Again, this is a hint for them to bisect the chords with their compasses. By drawing out the perpendicular bisectors, students would then obtain the centre of their circle. Students can prove they have the right answer by folding the circle into quarters to find the centre. Each step of the drawing will be reflected in the on-screen slides, to aid those who are lost or inattentive.

Ask students to prove to themselves that any two chords will do in finding the centre of the circle, using the other circle. Do you get the same result using the same method with other other two chords? (Yes.) How many chords do you need to determine the centre? (Two.)

Briefly introduce the objectives of the lesson: Learn about bisection properties of

chords in a circle Learn about equal-length properties of

chords in a circle Apply these properties to solve for

unknown lengths and angles

Circles

Slides (reflecting what is being drawn on students’ circles). Appendix pg.9-10.

This section makes use of van Hiele’s theory to begin introducing the students to recognise (stage 1) geometrical concepts they are about to learn.

Providing a challenge to the student using material they can play with helps engage them with concrete examples, especially those learners at a concrete operational stage.

Furthermore, this provides opportunity for revision of previously learnt concepts of chords and perpendicular bisectors that are essential to this subtopic.

Students are allowed to work in pairs for collaborative learning and to “think-pair-share”.

5 min History and Java demo

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Time Teaching / Learning Activities Material Rationale

Using slides, introduce Euclid and his book, “The Elements” (in this case, Book III: Theory of Circles). Mention that Euclid had discovered many of these properties in 300 B.C., showing examples from the Java website on “Euclid’s Elements”.

Site has the proposition in full detail – go through one to demonstrate to students how Euclid formulated and proved these properties.

Specifically, show applet from Book III Proposition 1: “To find the center of a given circle” and its corollary, “If in a circle a straight line cuts a straight line into two equal parts and at right angles, then the center of the circle lies on the cutting straight line.” Ask: Does this property look familiar? (Yes, we did it just now to find the centre of the circle).

Summarise main learning point from first activity: the centre of a circle lies on the perpendicular bisector of a chord in the circle.

Slides (on Euclid and history). Appendix pg.9-10.

“Euclid’s Elements” website (see appendix for printout, references for site). Appendix pg.8.

Provide some historical perspective on the subject, and let students have a chance to get to know their famous Mathematicians, while appreciating how long ago these properties were formulated.

Use of IT to help visualise concepts being taught.

15 min Property 1: Perpendicular chords

Distribute Worksheet 1 to the students, and lead the students through the section “Investigating Property 1”.

In this section, students will be asked to draw any chord through the circle. Fold the circle into quarters to find the centre of the circle; next, drop a perpendicular from the centre of the circle to the chord. This will be demonstrated on the OHP with an OHT “circle”. Ask the students: Measure the two lengths of the chord – are they equal? (Yes.) What can you say about the perpendicular? (It bisects the chord.)

State property 1 for the students: A straight line drawn from the centre of a circle to bisect a chord, which is not a diameter, is perpendicular to the chord. Students are to

Worksheet 1. Appendix pg.1-2.

Circles

OHT. Appendix pg.6.

In this section, students will be led through an investigation into the first property being taught, once again using physical materials that can provide measurable proof of the properties being investigated. By measuring the right angles and equivalent lengths of chords, students will be more convinced of the properties, results supported by Piaget

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Time Teaching / Learning Activities Material Rationalefill in the blank on the worksheet.

Next, create another chord on the circle. Measure half the length of the chord and draw a line to the centre. Demonstrate using OHT as well. Ask the students: From what you know, what do you think is the angle formed between the chord and the line drawn (90.) Confirm this by measuring with your protractor. What can you say about the bisector? (It is perpendicular to the chord.)

Introduce the idea of the converse of property 1: The perpendicular to a chord, drawn from the centre of a circle, bisects the chord. Students are to fill in the blank on the worksheet with this converse.

Provide an informal proof of this property using reflection – in a circle split down the middle, half the chord when reflected will give the other half. This can be shown on the OHT.

and van Hiele’s theory (second level: analysis, recognising properties; third level: ordering, forming logically ordered relationships).

These hands-on activities also allow students to better visualise the relations between the various elements (chords and perpendicular bisectors).

As students are not required to produce formal proofs, the teacher shows a simple informal proof, as in van Hiele’s theory (fourth level: deduction, offering proof).

15 min Property 2: Equidistant chords

Lead students through second section of Worksheet 1, “Investigating Property 2”. Students are to draw two chords of equal length (e.g. 5 cm) on another circle, and measure the respective distances of their perpendicular bisectors to the centre of the circle. Demonstrate using OHT. Ask: “What is the distance of each bisector?” (The same.) “If I drew two equal lines from the centre of the circle, then drew chords perpendicular to them, do you think these chords will be equal?” (Yes.) Let the students try out the converse in the second part of the investigation.

State the property and its converse: In a circle, equal chords are equidistant from the centre. Furthermore, chords, which are equidistant from the centre, are equal. Ask


Circles

OHT. Appendix pg.6.

Teaching Aid: Large circle, string and ruler, magnet. Appendix pg.7.

Once again, the hands-on activities will help to engage learners who have better kinaesthetic intelligences (Gardner – multiple intelligence theory) and learn better by “doing things”. Students will be led through the stages of van Hiele’s theory once again through these activities.

The activities culminate in an informal proof

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Time Teaching / Learning Activities Material Rationalefor clarification: “What are equal chords?” (Misconception: Equal chords means the same chord. Actual answer: Chords with equal length.) Students to fill in the blanks on Worksheet 1.

Provide an informal proof using large cut-out circle, attached to the whiteboard with a magnet, together with string and ruler (Teacher’s Aid 1). The string and ruler are “anchored” to the centre of the circle with a magnet, and rotated around, wherein the string is the bisector and the ruler, tied perpendicular to the string, is the chord. Ask: “What do you notice about the ruler during rotation?” (Remains as a chord to the circle.) “What does the string represent?” (The bisector of the chord.) “Do the two change over rotation?” (No.)

(deduction stage of van Hiele’s theory), allowing students to deepen their understanding of the matter and remember it better.

20 min In-class worksheet practice

Distribute Worksheet 2. Tell students to attempt questions 1 and 2 first – these are simple application questions on the two properties above.

After five minutes, ask two students to give their answers for questions 1 and 2, and explain their solutions. If necessary, prompt the students by asking them, Do you remember what the two properties are? How can we apply them to this situation?

For example, question 1 requires application of property 1. Ask: What do the indicators of equal length on the chord tell us? (The chord is bisected.) Hence, what is the angle, based on property 1? On the application of property 2, ask: Are the lengths of the bisectors equal? (Yes.) What does this imply about the lengths of the chords? (The chords are of equal length) Similarly, for question 2: What does the right angle between the line from the centre and the chord tell us about the chord, based on property 1? (It is bisected.)

Questions 3 and 4 are trickier questions,


Whiteboard, markers.

The in-class practice will serve to reinforce what the students have already learnt. The simple application questions let the students have a taste of why they were taught these properties.

For the trickier questions, the teacher begins by modelling how to solve a problem, before leading the students into guided practice. If time allows, students could be given time for independent practice, but that component would be captured in the homework assignment.

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Time Teaching / Learning Activities Material Rationalerequiring some application of other past knowledge, namely Pythagoras’ Theorem. For these questions, let the students try doing themselves, but after that, go through each step of the question and ask for students’ ideas on how to proceed. Demonstrate each solution on the board carefully.

For questions 3 and 4, go through how to obtain the radius (these are not immediately obvious because it is not drawn in). Emphasise that any line from the centre to the edge of a chord is the length of a radius. Emphasise on drawing more lines on the diagrams for clarity.

5 min Conclusion

Recap the two properties taught, and the general solution of problems involving chords and symmetry. Ask the students to recall the two properties and the special case involving the centre of the circle using choral response. Show the conclusion slide with those three properties, and ask students to infer the converses from the properties shown.

Assign homework for the students to do – distribute Homework 1.

Leave the students with some food for thought for the next lesson – how can we find the centre of a circle with only a ruler and compass, but by drawing only two lines in the circle itself? (This will be answered when they learn about tangents to circles – tangents give right angles to the circle, which can be intersected to give the centre.)

Slides. Appendix pg.9-10.

Homework 1. Appendix pg.5.

The lesson is summarised for closure, and students are asked to recall the key principles. Homework is assigned for independent practice, and the conclusion is related to the trigger activity at the beginning of the lesson, while setting the stage for the next lesson.

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Afterword

References1. Euclid’s Elements, by D.E. Joyce, Clark University. Applets and explanations

available at http://aleph0.clarku.edu/~djoyce/java/elements/toc.html. 2. GCE Mathematics Ordinary Level (Syllabus 4017). Ministry of Education, Singapore.3. New Elementary Mathematics Syllabus D 3B (Teacher’s Guide), by Sin Kwai Meng.

Pan Pacific Publications, Singapore.4. Teaching Secondary School Mathematics: A Resource Book, by Lee Peng Yee.

McGraw Hill, Singapore.

Appendix1. Worksheet 1: Investigations (2 pages)2. Worksheet 2: In-class problems (2 pages)3. Homework 14. OHT5. Teaching Aid6. “Euclid’s Elements” printout7. Slides (2 pages)

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http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

Geometrical Properties of Circles: Symmetrical Properties of ChordsWorksheet 1: Investigations

Name: ( ) Date:

Class:

INVESTIGATION

Use one of the pre-cut circles each for your investigation. Diagrams provided are for your reference and are not drawn to scale.

Investigation 1

1. Draw any chord through the circle. Label the two sides A and B.2. Fold the circle in four, to find the centre of the circle. Label this O.3. Drop a perpendicular to the chord. Label the point of intersection C.4. Find the lengths of AC and BC:

a. AD = cm

b. BD = cm

5. What can you conclude about AD and BD?

6. Now, draw another chord on the circle. Label the two sides D and E.7. Measure half the length of DE, and mark that point F.8. Draw a line connecting O and F.9. Find the values of the angles ∠OFD and ∠OFE:

a. ∠OFD =

b. ∠OFE =

10. What can you conclude about AD and BD?

11. This investigation leads us to Property 1. Write in the property in the blank box below.

Appendix Page 1

Property 1:

Geometrical Properties of Circles: Symmetrical Properties of ChordsWorksheet 1: Investigations

Investigation 2

1. Using a new circle, draw two 5 cm long chords on the circle. Label these chords AC and XZ.

2. Find the mid-point of each chord. Label these points B and Y respectively.3. Fold the circle in four, to find the centre of the circle. Label this O.4. Draw lines from B to O and Y to O. Verify that these lines are perpendicular to AC

and XZ respectively.5. Find the lengths of OB and OY:

a. OB = cm

b. OY = cm

6. What can you conclude about OB and OY?

7. Using a new circle, find the centre of the circle by folding in four. Label this O.8. Draw two lines from O to the circumference of the circle.9. Measure 6 cm from O on each line, and mark these points B and Y. 10. Draw perpendicular lines across B and Y to form chords. Label the ends AC and XZ

respectively.11. Find the lengths of AC and XZ:

c. AC = cm

d. XZ = cm

12. What can you conclude about AC and XZ?

13. This investigation leads us to Property 2. Write in the property in the blank box below.

Appendix Page 2

Property 2:

Geometrical Properties of Circles: Symmetrical Properties of ChordsWorksheet 2: In-class problems

Name: ( ) Date:

Class:

PROBLEM-SOLVING

1. Given that O is the centre of the circle below, find the value of the length x and the angle q.

2. Given that O is the centre of the circle below, find the value of the lengths x and y. All lengths given are in cm.

Appendix Page 3

Geometrical Properties of Circles: Symmetrical Properties of ChordsWorksheet 2: In-class problems

3. Given that O is the centre of the circle below (all lengths are in cm):a. Find the radius of the circle. Hint: draw a dotted line to the end of a chord to

make the radius clearer.b. Hence find the value of the length x by using Pythagoras’ Theorem.

4. O is the centre of a circle of radius 10 cm. XY is a chord of length 16 cm. Find the area of triangle OXY.

Appendix Page 4

Geometrical Properties of Circles: Symmetrical Properties of ChordsHomework 1

Name: ( ) Date:

Class:

HOMEWORK

1. Given that O is the centre of the circle, find the value of x and y in each case.

2. Given that O is the centre of the circle and AOD is a straight line, find the value of x and y in each case.

3. Given that O is the centre of the circle, find the value of x and y in each case.

Appendix Page 5

OHTSoon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005

Diagrams to clearly show investigations of properties

Extra circles to sketch on for demonstration purposes

Appendix Page 6

Teaching AidSoon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005

Teaching Aid for Property 2

The teaching aid can be constructed as follows:

1. Cut a large circle out of thick construction paper.2. Tie a piece of string to a ruler.3. Use the magnet to stick the circle to the whiteboard4. Attaching the string and ruler together with it.5. Rotate!

Appendix Page 7

Ruler – “chord”

String – “bisector”

Circular cut-out

Magnet – “centre”

“Euclid’s Elements” printout Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005

Appendix Page 8

These controls are draggable for demonstration purposes.

Presentation SlidesSoon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005

Appendix Page 9

Presentation SlidesSoon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005

Appendix Page 10

use the following symmetry properties of circles:teachers.stupidchicken.com/files/active/0/QCM520...

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