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“PROVE THAT ALL CIRCLESARE SIMILAR” --

WHAT KIND OF STANDARD IS THAT?

Kevin McLeod(UW-Milwaukee Department of Mathematical Sciences)

WMC Annual ConferenceGreen Lake, May 2, 2013

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.

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

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?

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?

Turning to Similarity

• On a piece of paper, draw two circles that are not congruent.

• How can you show that your circles are similar?

Similarity with Coordinates

• On grid paper, draw coordinate axes and sketch the two circles

x2 + y2 = 4x2 + y2 = 16

• How can you show algebraically that there is a dilation that carries one of these circles onto the other?

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?

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