Our Teaching
Package
CONTENTS
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Teaching theories adopted & motivation
strategies
Congruency & its proof
Similarity
Applications of similarity & congruency
Difficulties and misconceptions
E-Lesson
Concept Map of Topic
Learning Theories
Teaching of Geometry
Students’ perception of geometry:
 Proving theorems, and
 Applying theorems to artificial problems.
Motivational Strategies
1.
2.
3.
4.
5.
6.
7.
8.
Indicate a void in students’ knowledge.
Present a challenge.
Show a sequential achievement.
Indicate a usefulness of a topic.
Use recreational mathematics.
Tell a pertinent story.
Get students involved in justifying mathematical
curiosity.
Use teacher-made or commercially prepared materials
Teaching Geometric Thoughts
Van Hiele’s theory
Level 0 - Visual:
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Classification tasks
Level 1 – Analysis:
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Investigate relationships
Level 2 – Informal Deduction
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Conclude based on logic
Congruency
Congruent Figures
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Congruent figures have
 Same size
 Same shape
Worksheets for Congruency
Refer to worksheets :
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Appendix 1
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Appendix 2
Congruent Figures
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When 2 figures are congruent, all corresponding parts of
the 2 figures are congruent.
Ratio of length of corresponding sides will be 1: 1
ABCD  EFGH
AB = EF, BC =FG, CD=GH, DA=HE
B
A
D
C
E
H
F
G
Tests For Congruent Triangles
For Upper Secondary /
For Higher Ability Lower
Secondary
Tests of Congruency for triangles (1)
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SSS
If each of the three
sides of one triangle
is congruent to the
corresponding side of
another triangle, then
the triangles are
congruent
Tests of Congruency for triangles (2)
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AAS
If two angles and the
side opposite one of
them in one triangle
are congruent to the
corresponding parts
of another triangle,
then triangle are
congruent
Tests of Congruency for triangles (3)
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SAS
If two sides and the
included angle of one
triangle are congruent
to two sides and the
included angle of
another triangle,then
the triangles are
congruent
Tests of Congruency for triangles (4)
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ASA
If two angles and the
included side of one
triangle are congruent
to two angles and the
included side of
another triangle, then
the triangles are
congruent.
Similarity
Definition of Similarity
Figures that have the same shape but not
necessarily the same size are similar, i.e.
different sizes
Worksheets for Similarity
Refer to worksheet :
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Worksheet Appendix 3
Similar Figures
Similar figures have same shapes and different
sizes.
Two figures are similar if you can rotate,
translate and/or reflect one of them so that it
can be enlarged or reduced onto another.
Worksheets for Similarity
Refer to worksheet :
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Worksheet Appendix 4
Similar Figures
The conventional definition:
.
For two figures to be similar,
1. Corresponding angles are equal
2. Corresponding sides are proportional
Worksheets for Similarity
Refer to worksheet :
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Worksheet Appendix 5 & 6
Definition of Similarity
Figures that have the same shape but not
necessarily the same size are similar.
(congruent figures are special case of
similar figures)
Applications of Similarity
Applications of Similarity
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Indirect measurement
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Finding areas and volumes of similar
objects
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Finding unknown sides and angles of
similar triangles
Using Similarity for Indirect Measurement
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At any one time,
vertical objects, the
sun’s ray and
shadows produced a
set of similar triangles
Make an indirect
measurement to find
height of tree.
The triangles are similar because
corresponding angles are congruent.
Write a proportion:
Girl’s shadow
Tree’s shadow
2.5
37.5
1.5
=
x
x = 22.5 m
Girl’s height
Tree’s height
Areas of Similar figures
3
A
B is similar to A
Scale factor = 9/3=3
3
Area of A = 3 x 3 = 9 cm2
Area of B = 9 x 9 = 81cm2
B
9
Area of B
Area of A
9 = 32
For similar figures:
Ratio of areas = scale factor2
9
Volumes of similar figures
Cube A and B are similar
Scale factor = 4/2 = 2
A
2 cm
Volume of A = 2 x 2 x 2 = 8 cm2
Volume of B = 4 x 4 x 4 = 64 cm2
Volume of B
Volume of A
64 / 8 = 8 =23
B
For similar figures:
4 cm
Ratio of volumes = scale factor3
Extension
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Shapes other than cubes?
Triangles?
Cuboids?
What about spheres?
Summary
Length
Area
Volume
A
L1
A1
V1
B
L1 x k
A1 x k2
V1 x k3
A and B are similar
Length of B /Length of A = k = scale factor
Worksheets for Similarity
Refer to worksheets :
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Worksheet Appendix 7,8, 9 & 10
Congruent & Similar Figures :
Transformations
Congruent & Similar Figures : Transformations
Congruent
Figures
Rotate
Translate
Reflect
Enlarge
Reduce
Similar
Figures
Worksheets for Similarity and Congruency
Refer to worksheets :
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Worksheet Appendix 11
Difficulties And
Misconceptions In Learning
Congruent And Similar Figures
Difficulties And Misconceptions In Learning
Congruent And Similar Figures
Case 1 : Students do not realise that congruent shapes can be
"matched" by placing one atop the other.
D
Given ΔABC and ΔDEF.
By cutting these two Δs, one is
placed on top of the other. They
are “matched” and are identical.
E
F
B
A
C
Difficulties And Misconceptions In Learning
Congruent And Similar Figures
Case 2: Students think that similar shapes must have congruent
angles and congruent sides.
This needs not be so as similar shapes need not necessarily have
congruent sides.
Given ΔABC and ΔDEF.
ΔABC is similar to ΔDEF but
their sides are not congruent.
A
D
4.5 m 7.5 m
3m
10 m
B 4m C
E 11.25 m F
Difficulties And Misconceptions In Learning
Congruent And Similar Figures
Case 3 : Similar shapes "does not match exactly when magnified
or shrunk".
Given similar ΔABC, ΔDEF and ΔGHI.
A
D
9 cm
G
6 cm
4 cm
450
H 45 E
0
4 cm
I
6 cm
9 cm
F
C
450-
B
Difficulties And Misconceptions In Learning
Congruent And Similar Figures
Case 4 : Students might not realize that:
• the ratio of the perimeters is the same as the scale factor relating the
lengths
• the ratio of the areas is the square of that scale factor.
For figure 1 : length l1, perimeter P1 area A1.
For figure 2 : length l2, perimeter P2 area be A2
P1
P2
=
l1
l2
A1
l1 2
A2 = ( l2 )
E-Lessons
Websites for Congruency & Similarity
Introductory level:
http://www.mathleague.com/help/geometry/coordin
ates.htm#congruentfigures
 Intermediate level:
http://www.math.com/school/subject3/lessons/S3U
3L1GL.html
http://dev1.epsb.edmonton.ab.ca/math14_Jim/mat
h9/strand3/3203.htm
 Advanced level:
http://matti.usu.edu/nlvm/nav/frames_asid_165_g_
4_t_3.html?open=instructor
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Sample of website (1)
Sample of website (2)
CDROM

Through the Ages with Congruency &
Similarity
Screen Sample of CD-DROM (1)
Screen Sample of CD-DROM (2)
Acknowledgements
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General Mathematics, VCE units 1& 2,
R.Chalker J, Dolman, B.Hodgsan, J. Seymour
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Navigating Through Geometry in grades 68
Twists & Turns and Tangles in Math and
Physics : Instructional Material for
developing scientific & Logical Thinking
http://www.cut-the-knot.com
Q & A Session