Similarity: Is it just “Same Shape, Different Size”? 1.1

Similarity: Is it just “Same Shape, Different Size”? 1.2

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 1.3

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

Introducing Similarity Transformations (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: ometry/GT3/Ldilate2.htm Activity 1: 1.5

Introducing Similarity Transformations 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. 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. Activity 1: (From the CCSSM Geometry overview) 1.6

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

Circle Similarity Activity 2: 1.8 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?

Circle Similarity Activity 2: 1.9 Congruence with coordinates On grid paper, draw coordinate axes and sketch the two circles x 2 + (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?

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

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

Circle Similarity Activity 2: 1.12 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?

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

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 1.14