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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. 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: Classification tasks Level 1 – Analysis: Investigate relationships Level 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 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) 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 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 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 : 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 : 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 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 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