similarity: is it just “same shape, different size”? 1.1
TRANSCRIPT
MATH 675: High School Mathematicsfrom an Advanced Viewpoint
Similarity: Is it just “Same Shape, Different Size”?
1.1
Adapted from:Common Core High School Mathematics
March 5, 2014
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
Introducing Similarity Transformations
•With a partner, discuss your definition of a dilation.
Activity 1:
1.4
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/geometry/GT3/Ldilate2.htm
Activity 1:
1.5
Introducing Similarity Transformations
• 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.
Activity 1:
(From the CCSSM Geometry overview)
1.6
Circle Similarity
• Consider G-C.1: Prove that all circles are similar.
• Discuss how you might have students meet this standard in your classroom.
Activity 2:
1.7
Circle SimilarityActivity 2:
1.8
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?
Circle SimilarityActivity 2:
1.9
Congruence with coordinates• On grid paper, draw coordinate axes and sketch the two
circlesx2 + (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?
Circle SimilarityActivity 2:
1.10
Turning to similarity
• On a piece of paper, draw two circles that are not congruent.
• How can you show that your circles are similar?
Circle SimilarityActivity 2:
1.11
Similarity with coordinates• On grid paper, draw coordinate axes and
sketch the two circlesx2 + y2 = 4x2 + y2 = 16
• How can you show algebraically that there is a dilation that carries one of these circles onto the other?
Circle SimilarityActivity 2:
1.12
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?
1.13
Other Conic SectionsActivity 3:
Are any two parabolas similar?
What about ellipses? Hyperbolas?
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