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
Activity 1:
Introducing Similarity
Transformations
• With a partner, discuss your definition of a
dilation.
1.4
Activity 1:
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:
http://www.regentsprep.org/Regents/math/ge
ometry/GT3/Ldilate2.htm
1.5
Activity 1:
Introducing Similarity
Transformations
(From the CCSSM Geometry overview)
• 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.
1.6
Activity 2:
Circle Similarity
• Consider G-C.1: Prove that all circles are
similar.
• Discuss how you might have students meet this
standard in your classroom.
1.7
Activity 2:
Circle Similarity
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?
1.8
Activity 2:
Circle Similarity
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?
1.9
Activity 2:
Circle Similarity
Turning to similarity
• On a piece of paper, draw two circles that are
not congruent.
• How can you show that your circles are
similar?
1.10
Activity 2:
Circle Similarity
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?
1.11
Activity 2:
Circle Similarity
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?
1.12
Activity 3:
Other Conic Sections
Are any two parabolas similar?
What about ellipses? Hyperbolas?
1.13
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