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“ “ Prove that All Circles are Similar” -- What Kind of Standard is that?

“ “ Prove that All Circles are Similar” -- What Kind of Standard is that?. Kevin McLeod (UW-Milwaukee Department of Mathematical Sciences) WMC Annual Conference Green Lake, May 2, 2013. CCSSM Definitions: Congruence and Similarity. Read the handout (CCSSM high school geometry overview)

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“ “ Prove that All Circles are Similar” -- What Kind of Standard is that?

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  1. ““Prove that All Circlesare Similar” -- What Kind of Standard is that? Kevin McLeod (UW-Milwaukee Department of Mathematical Sciences) WMC Annual Conference Green Lake, May 2, 2013

  2. CCSSM Definitions:Congruence and Similarity • Read the handout (CCSSM high school geometry overview) • How does the Common Core define congruence? Similarity? • How (if at all) do these definitions differ from those you use in your geometry classes?

  3. CCSSM Definitions:Congruence and Similarity • 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.

  4. CCSSM Definition: Dilation • 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

  5. 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?

  6. 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?

  7. Turning to Similarity • On a piece of paper, draw two circles that are not congruent. • How can you show that your circles are similar?

  8. 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?

  9. 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?

  10. What Kind of Standard is “Prove that all circles are Similar”? • A very good one! • Multiple entry points. • Multiple exit points. • Multiple connections to other content standards (not only in the Geometry conceptual category). • Multiple connections to practice standards.

  11. Questions? kevinm@uwm.edu

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