“ “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 Billy Sparks used to ask (as we were rewriting Wisconsin Model Academic Standards), “Does that rise to the level of a standard?” That is the question that first came to my mind as I read this Common Core standard in the high school Geometry conceptual category. What are your thoughts? Note: in this presentation, we will be doing something I often argue should not be done with the CCSSM: we are going to be focusin on a singel standard, rather than on a cluster. Hopefully the justification for this will become apparent as we proceed.

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?

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. Source: CCSSM High School Geometry overview.

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. Source: CCSSM glossary. Figure source: http://www.regentsprep.org/Regents/math/geometry/GT3/Ldilate2.htm

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? Ask: when should two circles be congruent (and not merely similar)? Answer: two circles are congruent if and only if they have equal radii. (Justify this statement, using the definitions of circle, and of rigid motion.)

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? (This is a diversion from the main thread of the session, but could be a valuable classroom activity.) Ask: how easy would it be to find the other types of congruence transformations algebraically?

Turning to Similarity On a piece of paper, draw two circles that are not congruent. How can you show that your circles are similar? (This is a diversion from the main thread of the session, but could be a valuable classroom activity.) Ask: how easy would it be to find the other types of congruence transformations algebraically?

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? (This is a diversion from the main thread of the session, but could be a valuable classroom activity.) Ask: why is it sufficient to consider circles centred at the origin?

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? Have participants discuss this last question, with two circles drawn on paper. Turn to Geogebra worksheets.

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.

Questions? kevinm@uwm.edu