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CONGRUENCY & SIMILARITY

CONGRUENCY & SIMILARITY. 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.

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CONGRUENCY & SIMILARITY

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  1. CONGRUENCY & SIMILARITY

  2. Our Teaching Package

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

  4. Concept Map of Topic

  5. Learning Theories

  6. Teaching of Geometry Students’ perception of geometry: • Proving theorems, and • Applying theorems to artificial problems. Students tend to be uninterested

  7. Motivational Strategies • 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

  8. Teaching Geometric Thoughts Van Hiele’s theory Level 0 - Visual: • Classification tasks Level 1 – Analysis: • Investigate relationships Level 2 – Informal Deduction • Conclude based on logic

  9. Congruency

  10. Congruent Figures • Congruent figures have • Same size • Same shape

  11. Worksheets for Congruency Refer to worksheets : • Appendix 1 • Appendix 2

  12. B A E F D C H G 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

  13. Tests For Congruent Triangles For Upper Secondary / For Higher Ability Lower Secondary

  14. 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

  15. 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

  16. 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

  17. 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.

  18. Similarity

  19. Definition of Similarity Figures that have the same shape but not necessarily the same size are similar, i.e. different sizes

  20. Worksheets for Similarity Refer to worksheet : • Worksheet Appendix 3

  21. 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.

  22. Worksheets for Similarity Refer to worksheet : • Worksheet Appendix 4

  23. Similar Figures The conventional definition: For two figures to be similar, • Corresponding angles are equal • Corresponding sides are proportional • .

  24. Worksheets for Similarity Refer to worksheet : • Worksheet Appendix 5 & 6

  25. 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)

  26. Applications of Similarity

  27. Applications of Similarity • Indirect measurement • Finding areas and volumes of similar objects • Finding unknown sides and angles of similar triangles

  28. 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.

  29. The triangles are similar because corresponding angles are congruent. Write a proportion: Girl’s shadow 2.5 1.5Girl’s height Tree’s shadow 37.5 x Tree’s height x = 22.5 m =

  30. A 3 3 9 = 32 B 9 9 Areas of Similar figures B is similar to A Scale 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 For similar figures: Ratio of areas = scale factor2

  31. 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

  32. Extension • Shapes other than cubes? • Triangles? • Cuboids? • What about spheres?

  33. Summary A and B are similar Length of B /Length of A = k = scale factor

  34. Worksheets for Similarity Refer to worksheets : • Worksheet Appendix 7,8, 9 & 10

  35. Congruent & Similar Figures : Transformations

  36. Congruent & Similar Figures : Transformations

  37. Worksheets for Similarity and Congruency Refer to worksheets : • Worksheet Appendix 11

  38. Difficulties And Misconceptions In Learning Congruent And Similar Figures

  39. D E F B A C 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. By cutting these two Δs, one is placed on top of the other. They are “matched” and are identical.

  40. D A 4.5 m 7.5 m 3 m 10 m B C 4 m E F 11.25 m 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.

  41. A D 9 cm G 6 cm 4 cm E B H 450 450- 450 4 cm I 6 cm 9 cm F C 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.

  42. l1 P1 A1 l1 = = ( )2 A2 P2 l2 l2 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

  43. E-Lessons

  44. 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.html http://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

  45. Sample of website (1)

  46. Sample of website (2)

  47. CDROM • Through the Ages with Congruency & Similarity

  48. Screen Sample of CD-DROM (1)

  49. Screen Sample of CD-DROM (2)

  50. 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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