MATH 675: High School Mathematics from an Advanced Viewpoint

1. MATH 675: High School Mathematicsfrom an Advanced Viewpoint Similarity: Is it just “Same Shape, Different Size”? 1.1

2. Adapted from:Common Core High School MathematicsMarch 5, 2014 Similarity: Is it just “Same Shape, Different Size”? 1.2

3. Learning Intentions & Success Criteria Learning Intentions: We are learning similarity transformations as described in the CCSSM Success Criteria: We will be successful when we can use the CCSSM definition of similarity, and the definition of a parabola, to prove that all parabolas are similar

4. Introducing Similarity Transformations Activity 1: • With a partner, discuss your definition of a dilation. 1.4

5. Introducing Similarity Transformations Activity 1: • (From the CCSSM glossary) A dilation is a transformation that moves each point along the ray through the point emanating from a common center, and multiplies distances from the center by a common scale factor. Figure source: http://www.regentsprep.org/Regents/math/geometry/GT3/Ldilate2.htm 1.5

6. Introducing Similarity Transformations Activity 1: (From the CCSSM Geometry overview) • Two geometric figures are defined to be congruent if there is a sequence of rigid motions (translations, rotations, reflections, and combinations of these) that carries one onto the other. • Two geometric figures are defined to be similar if there is a sequence of similarity transformations (rigid motions followed by dilations) that carries one onto the other. 1.6

7. Circle Similarity Activity 2: • Consider G-C.1: Prove that all circles are similar. • Discuss how you might have students meet this standard in your classroom. 1.7

8. Circle Similarity Activity 2: Begin with congruence • On patty paper, draw two circles that you believe to be congruent. • Find a rigid motion (or a sequence of rigid motions) that carries one of your circles onto the other. • How do you know your rigid motion works? • Can you find a second rigid motion that carries one circle onto the other? If so, how many can you find? 1.8

9. Circle Similarity Activity 2: Congruence with coordinates • On grid paper, draw coordinate axes and sketch the two circles x2 + (y – 3)2 = 4 (x – 2)2 + (y + 1)2 = 4 • Why are these the equations of circles? • Why should these circles be congruent? • How can you show algebraically that there is a translation that carries one of these circles onto the other? 1.9

10. Circle Similarity Activity 2: Turning to similarity • On a piece of paper, draw two circles that are not congruent. • How can you show that your circles are similar? 1.10

11. Circle Similarity Activity 2: Similarity with coordinates • On grid paper, draw coordinate axes and sketch the two circles x2 + y2 = 4 x2 + y2 = 16 • How can you show algebraically that there is a dilation that carries one of these circles onto the other? 1.11

12. Circle Similarity Activity 2: Similarity with a single dilation? • If two circles are congruent, this can be shown with a single translation. • If two circles are not congruent, we have seen we can show they are similar with a sequence of translations and a dilation. • Are the separate translations necessary, or can we always find a single dilation that will carry one circle onto the other? • If so, how would we locate the centre of the dilation? 1.12

13. Other Conic Sections Activity 3: • Are any two parabolas similar? • What about ellipses? Hyperbolas? 1.13

14. Learning Intentions & Success Criteria Learning Intentions: We are learning similarity transformations as described in the CCSSM Success Criteria: We will be successful when we can use the CCSSM definition of similarity, and the definition of a parabola, to prove that all parabolas are similar