QCM520: Lesson plan assignment
Submitted by: Soon Yin Jie (19)
Date: 25 October 2005
Tutor: Ms Ivy Chow
Topic: Geometrical Properties of Circles
Contents
INTRODUCTION .......................................................................................................................................1
TOPIC...........................................................................................................................................................1
ADMINISTRATIVE DETAILS ..........................................................................................................................1
UNIT DESCRIPTION ......................................................................................................................................1
LEARNING THEORIES ...................................................................................................................................2
Co-operative Learning and Social Constructivism ................................................................................2
Guided Discovery ...................................................................................................................................2
Stages of Cognitive Development ...........................................................................................................2
Van Hiele Theory ...................................................................................................................................2
TEACHING APPROACHES .............................................................................................................................2
LEARNING DIFFICULTIES .............................................................................................................................3
LESSON DEVELOPMENT .......................................................................................................................4
PREREQUISITE KNOWLEDGE ........................................................................................................................4
SPECIFIC INSTRUCTIONAL OBJECTIVES ........................................................................................................4
KEY CONCEPTS............................................................................................................................................4
RESOURCES .................................................................................................................................................4
DETAILED LESSON PLAN .............................................................................................................................5
AFTERWORD ...........................................................................................................................................10
REFERENCES ..............................................................................................................................................10
APPENDIX ..................................................................................................................................................10
Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Introduction
Topic
The unit covered is “geometrical properties of circles”. The specific topic is “symmetrical
properties of chords in a circle”.
Administrative Details
Level:
Time:
Location:
Secondary 3 Express pupils of low-average ability
Double-period (70 minutes),
Regular classroom equipped with a whiteboard, overhead-projector, computer
and computer display projector.
Unit Description
The 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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Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Learning Theories
Co-operative Learning and Social Constructivism
Vygotsky 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 Discovery
Bruner 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 Development
Piaget, 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 Theory
Pierre 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 Approaches
Some 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
Page 2
Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
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 Difficulties
This 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.
Page 3
Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Lesson Development
Level:
Time:
Location:
Secondary 3 Express pupils of low-average ability
Double-period (70 minutes),
Regular classroom equipped with a whiteboard, overhead-projector, computer
and computer display projector.
Prerequisite Knowledge
Before 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 Objectives
At 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 Concepts
Perpendicular bisection of chords, chords equidistant from centre of a circle.
Resources
1.
2.
3.
4.
5.
Computer
Slides
Java applet demonstration site
Circle cut-outs (four for every student) – prepare before class
Teaching aid: large circle cut-out, string tied to ruler, magnet for sticking everything
to whiteboard
6. OHTs,
7. Whiteboard, markers
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Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Detailed Lesson Plan
Key questions and expected student responses are in italics.
Time
10 min
Teaching / Learning Activities
Starter activity
Material
Rationale
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.
Circles
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.
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.)
Slides
(reflecting
what is being
drawn on
students’
circles).
Appendix
pg.9-10.
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-pairshare”.
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
Page 5
Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Time
5 min
Teaching / Learning Activities
History and Java demo
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”.
Slides (on
Euclid and
history).
Appendix
pg.9-10.
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.
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).
“Euclid’s
Elements”
website (see
appendix for
printout,
references for
site). Appendix
pg.8.
Use of IT to help
visualise concepts
being taught.
Summarise main learning point from first
activity: the centre of a circle lies on the
perpendicular bisector of a chord in the
circle.
15 min
Property 1: Perpendicular chords
Distribute Worksheet 1 to the students, and
lead the students through the section
“Investigating Property 1”.
Worksheet 1.
Appendix
pg.1-2.
In this section,
students will be led
through an
investigation into the
In this section, students will be asked to
Circles
first property being
draw any chord through the circle. Fold the
taught, once again
circle into quarters to find the centre of the
OHT.
using physical
circle; next, drop a perpendicular from the
Appendix pg.6. materials that can
centre of the circle to the chord. This will be
provide measurable
demonstrated on the OHP with an OHT
proof of the
“circle”. Ask the students: Measure the two
properties being
lengths of the chord – are they equal? (Yes.)
investigated. By
What can you say about the perpendicular?
measuring the right
(It bisects the chord.)
angles and equivalent
lengths of chords,
State property 1 for the students: A straight
students will be more
line drawn from the centre of a circle to
convinced of the
bisect a chord, which is not a diameter, is
properties, results
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Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Time
Teaching / Learning Activities
perpendicular to the chord. Students are to
fill in the blank on the worksheet.
Material
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.)
These hands-on
activities also allow
students to better
visualise the relations
between the various
elements (chords and
perpendicular
bisectors).
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.
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).
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.
15 min
Rationale
supported by Piaget
and van Hiele’s
theory (second level:
analysis, recognising
properties; third
level: ordering,
forming logically
ordered
relationships).
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
Worksheet 1.
Appendix
pg.1-2.
Once again, the
hands-on activities
will help to engage
learners who have
Circles
better kinaesthetic
intelligences
OHT.
(Gardner – multiple
Appendix pg.6. intelligence theory)
and learn better by
Teaching Aid: “doing things”.
Large circle,
Students will be led
string and
through the stages of
ruler, magnet.
van Hiele’s theory
Appendix pg.7. once again through
these activities.
The activities
culminate in an
Page 7
Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Time
Teaching / Learning Activities
equidistant from the centre, are equal. Ask
for 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.
Material
Rationale
informal proof
(deduction stage of
van Hiele’s theory),
allowing students to
deepen their
understanding of the
matter and remember
it better.
Worksheet 2.
Appendix
pg.2-4.
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.
Provide an informal proof using large cutout 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.)
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.)
Whiteboard,
markers.
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.
Page 8
Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Time
Teaching / Learning Activities
Questions 3 and 4 are trickier questions,
requiring 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.
Material
Rationale
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.
The lesson is
summarised for
closure, and students
are asked to recall the
Homework 1.
key principles.
Appendix pg.5. 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.
Page 9
Geometrical Properties of Circles: Symmetrical Properties of Chords
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Afterword
References
1. 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.
Appendix
1.
2.
3.
4.
5.
6.
7.
Worksheet 1: Investigations (2 pages)
Worksheet 2: In-class problems (2 pages)
Homework 1
OHT
Teaching Aid
“Euclid’s Elements” printout
Slides (2 pages)
Page 10
Geometrical Properties of Circles: Symmetrical Properties of Chords
Worksheet 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.
2.
3.
4.
Draw any chord through the circle. Label the two sides A and B.
Fold the circle in four, to find the centre of the circle. Label this O.
Drop a perpendicular to the chord. Label the point of intersection C.
Find the lengths of AC and BC:
a. AD =
cm
b. BD =
cm
5. What can you conclude about AD and BD?
6.
7.
8.
9.
Now, draw another chord on the circle. Label the two sides D and E.
Measure half the length of DE, and mark that point F.
Draw a line connecting O and F.
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.
Property 1:
Appendix Page 1
Geometrical Properties of Circles: Symmetrical Properties of Chords
Worksheet 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.
Property 2:
Appendix Page 2
Geometrical Properties of Circles: Symmetrical Properties of Chords
Worksheet 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 Chords
Worksheet 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 Chords
Homework 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
OHT
Soon 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 Aid
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Teaching Aid for Property 2
The teaching aid can be constructed as follows:
1.
2.
3.
4.
5.
Cut a large circle out of thick construction paper.
Tie a piece of string to a ruler.
Use the magnet to stick the circle to the whiteboard
Attaching the string and ruler together with it.
Rotate!
Ruler – “chord”
String – “bisector”
Circular cut-out
Magnet – “centre”
Appendix Page 7
“Euclid’s Elements” printout
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
These controls are
draggable for
demonstration purposes.
Appendix Page 8
Presentation Slides
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Appendix Page 9
Presentation Slides
Soon Yin Jie / QCM 520 / NIE PGDE (Sec) July 2005
Appendix Page 10