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