our teaching package contents teaching theories adopted & motivation strategies congruency &...
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Our Teaching Package
CONTENTS
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. Indicate a void in students’ knowledge.
2. Present a challenge.
3. Show a sequential achievement.
4. Indicate a usefulness of a topic.
5. Use recreational mathematics.
6. Tell a pertinent story.
7. Get students involved in justifying mathematical curiosity.
8. Use teacher-made or commercially prepared materials
Teaching Geometric Thoughts
Van Hiele’s theoryLevel 0 - Visual: Classification tasksLevel 1 – Analysis: Investigate relationshipsLevel 2 – Informal Deduction Conclude based on logic
Congruency
Congruent Figures
Congruent figures have Same size Same shape
Worksheets for Congruency
Refer to worksheets :
Appendix 1
Appendix 2
Congruent Figures
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
A B
CD
E F
GH
Tests For Congruent Triangles
For Upper Secondary /
For Higher Ability Lower Secondary
Tests of Congruency for triangles (1)
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)
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)
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)
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 :
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 :
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 :
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
Indirect measurement
Finding areas and volumes of similar objects
Finding unknown sides and angles of similar triangles
Using Similarity for Indirect Measurement
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 2.5 1.5 Girl’s height
Tree’s shadow 37.5 x Tree’s height
x = 22.5 m
=
Areas of Similar figures
B is similar to AScale factor = 9/3=3
Area of A = 3 x 3 = 9 cm2
Area of B = 9 x 9 = 81cm2
Area of B Area of A
Ratio of areas = scale factor2
For similar figures:
3
A3
9
B 99 = 32
Volumes of similar figures
2 cm
4 cm
A
B
Cube A and B are similarScale factor = 4/2 = 2
Volume of A = 2 x 2 x 2 = 8 cm2
Volume of B = 4 x 4 x 4 = 64 cm2
Volume of BVolume of A 64 / 8 = 8 =23
Ratio of volumes = scale factor3
For similar figures:
Extension
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 similarLength of B /Length of A = k = scale factor
Worksheets for Similarity
Refer to worksheets :
Worksheet Appendix 7,8, 9 & 10
Congruent & Similar Figures : Transformations
Congruent Figures
Similar Figures
Rotate
Translate
Reflect
Enlarge
Reduce
Congruent & Similar Figures : Transformations
Worksheets for Similarity and Congruency
Refer to worksheets :
Worksheet Appendix 11
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.
Given ΔABC and ΔDEF.
Difficulties And Misconceptions In Learning Congruent And Similar Figures
A
B
C
D
EFBy cutting these two Δs, one is placed on top of the other. They are “matched” and are identical.
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.
4 m
3 m 4.5 mA
B C11.25 m
10 m7.5 m
D
E F
ΔABC is similar to ΔDEF but their sides are not congruent.
Difficulties And Misconceptions In Learning Congruent And Similar Figures
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.
9 cm
9 cm
450-
A
B
C
450
6 cm
6 cm
D
E
F
450
4 cm
4 cm
G
H
I
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
A1A2
=l1l2
( )2
Difficulties And Misconceptions In Learning Congruent And Similar Figures
P1
P2=l1
l2
E-Lessons
Websites for Congruency & Similarity Introductory level:http://www.mathleague.com/help/geometry/coordinates.htm#congruentfigures Intermediate level:http://www.math.com/school/subject3/lessons/S3U3L1GL.htmlhttp://dev1.epsb.edmonton.ab.ca/math14_Jim/math9/strand3/3203.
htm Advanced level:http://matti.usu.edu/nlvm/nav/frames_asid_165_g_4_t_3.html?open=instructor
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
General Mathematics, VCE units 1& 2,R.Chalker J, Dolman, B.Hodgsan, J. Seymour
Navigating Through Geometry in grades 6-8
Twists & Turns and Tangles in Math and Physics : Instructional Material for developing scientific & Logical Thinking
http://www.cut-the-knot.com
Q & A Session
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