CONGRUENCY

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CONGRUENCY SIMILARITY
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Our Teaching Package
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CONTENTS

• Teaching theories adopted motivation strategies
• Congruency its proof
• Similarity
• Applications of similarity congruency
• Difficulties and misconceptions
• E-Lesson

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Concept Map of Topic
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Learning Theories
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Teaching of Geometry

• Students perception of geometry
• Proving theorems, and
• Applying theorems to artificial problems.

Students tend to be uninterested
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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

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Teaching Geometric Thoughts

• Van Hieles theory
• Level 0 - Visual
• Classification tasks
• Level 1 Analysis
• Investigate relationships
• Level 2 Informal Deduction
• Conclude based on logic

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Congruency
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Congruent Figures

• Congruent figures have
• Same size
• Same shape

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Worksheets for Congruency

• Refer to worksheets
• Appendix 1
• Appendix 2

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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, CDGH, DAHE

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Tests For Congruent Triangles

• For Upper Secondary /
• For Higher Ability Lower Secondary

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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

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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

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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

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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.

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Similarity
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Definition of Similarity

• Figures that have the same shape but not
  necessarily the same size are similar, i.e.
  different sizes

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Worksheets for Similarity

• Refer to worksheet
• Worksheet Appendix 3

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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.

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Worksheets for Similarity

• Refer to worksheet
• Worksheet Appendix 4

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Similar Figures

• The conventional definition
• For two figures to be similar,
• Corresponding angles are equal
• Corresponding sides are proportional

.
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Worksheets for Similarity

• Refer to worksheet
• Worksheet Appendix 5 6

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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)

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Applications of Similarity
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Applications of Similarity

• Indirect measurement
• Finding areas and volumes of similar objects
• Finding unknown sides and angles of similar
  triangles

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Using Similarity for Indirect Measurement

• At any one time, vertical objects, the suns ray
  and shadows produced a set of similar triangles
• Make an indirect measurement to find height of
  tree.

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• The triangles are similar because
• corresponding angles are congruent.
• Write a proportion
• Girls shadow 2.5 1.5
  Girls height
• Trees shadow 37.5 x
  Trees height
• x 22.5 m

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Areas of Similar figures
B is similar to A Scale factor 9/33 Area of A
3 x 3 9 cm2 Area of B 9 x 9 81cm2 Area
of B Area of A
For similar figures
Ratio of areas scale factor2
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Volumes of similar figures
Cube A and B are similar Scale 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 B Volume of A
A
2 cm
64 / 8 8 23
B
For similar figures
Ratio of volumes scale factor3
4 cm
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Extension

• Shapes other than cubes?
• Triangles?
• Cuboids?
• What about spheres?

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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
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Worksheets for Similarity

• Refer to worksheets
• Worksheet Appendix 7,8, 9 10

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Congruent Similar Figures Transformations
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Congruent Similar Figures Transformations
Congruent Figures Similar Figures
Rotate
Translate
Reflect
Enlarge
Reduce
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Worksheets for Similarity and Congruency

• Refer to worksheets
• Worksheet Appendix 11

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Difficulties And Misconceptions In Learning
Congruent And Similar Figures
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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 ?ABC and ?DEF.
By cutting these two ?s, one is placed on top of
the other. They are matched and are identical.
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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 ?ABC and
?DEF.
?ABC is similar to ?DEF but their sides are not
congruent.
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Difficulties And Misconceptions In Learning
Congruent And Similar Figures
Case 3 Similar shapes "does not match exactly
when magnified or shrunk". Given similar ?ABC,
?DEF and ?GHI.
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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

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E-Lessons
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Websites for Congruency Similarity

• Introductory level
• http//www.mathleague.com/help/geometry/coordinate
  s.htmcongruentfigures
• Intermediate level
• http//www.math.com/school/subject3/lessons/S3U3L1
  GL.html
• http//dev1.epsb.edmonton.ab.ca/math14_Jim/math9/s
  trand3/3203.htm
• Advanced level
• http//matti.usu.edu/nlvm/nav/frames_asid_165_g_4_
  t_3.html?openinstructor

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Sample of website (1)
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Sample of website (2)
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CDROM

• Through the Ages with Congruency Similarity

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Screen Sample of CD-DROM (1)
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Screen Sample of CD-DROM (2)
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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

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Q A Session