“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)
• 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 = 4
x2 + 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