Transcript School Year Session 11: March 5, 2014 Similarity: Is it just “Same Shape, Different Size”? 1.1

School Year Session 11: March 5, 2014
Similarity: Is it just “Same
Shape, Different Size”?
1.1
Agenda
•
•
•
•
•
•
Similarity Transformations
Circle similarity
Break
Engage NY assessment redux
Planning time
Homework and closing remarks
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
The Big Picture
An approximate timeline
1.4
• Someone in your group has recent experience
• Do not “bonk with the big blocks”
1.5
Activity 1:
Introducing Similarity
Transformations
• With a partner, discuss your definition of a
dilation.
1.6
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
1.7
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.
1.8
Activity 1:
Introducing Similarity
Transformations
• Read G-SRT.1
• Discuss how might you have students meet this
standard in your classroom?
1.9
Activity 2:
Circle Similarity
• Consider G-C.1: Prove that all circles are
similar.
• Discuss how might you have students meet this
standard in your classroom?
1.10
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?
1.11
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?
1.12
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?
1.13
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?
1.14
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?
1.15
Break
1.12
Activity 3:
Engage NY Redux
Last time, we left unanswered the question:
“Is the parabola with focus point (1,1) and directrix
y = -3 similar to the parabola y = x2?”
Answer this question, using the CCSSM definition
of similarity.
1.17
Activity 3:
Engage NY Redux
Are any two parabolas similar?
What about ellipses? Hyperbolas?
1.18
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
1.19
Activity 4:
Planning Time
Find someone who is teaching similar content to you, and
work as a pair.
Think about the unit you are teaching, and identify one key
content idea that you are building, or will build, the unit
around.
Identify a candidate task that you might use to address
your key idea, and discuss how that task is aligned to the
frameworks (cognitive demand/SBAC claims) we have seen
in class.
We will ask you to share out at 7:45.
1.20
Activity 5:
Homework & Closing
Remarks
Homework:
• Prepare to hand in your assessment and task modification
homework on March 19. You should include both the original
and the modified versions of both tasks (the end-of-unit
assessment and the classroom task), your assessment rubric,
and your reflections on the process and the results.
• Begin planning your selected lessons. You will have time to
discuss your ideas with your colleagues in class on March 19.
• Bring your lesson and assessment materials to class on March
19.
1.21