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Common Core High School Mathematics:Transforming Instructional Practice for a New Era
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
1.4
An approximate timeline
Select a focus unit
Specify a set of
learning intentions
Design or modify an
assessment
Design a task for a
focus lesson(s)
within the unit
Teach the unit &
lesson and collect
evidence
Plan, Teach, Reflect
project with lesson &
assessment evidenceJanuary 22
February 19March 5
March 5-19By April 5
May 7
The Big Picture
• Someone in your group has recent experience• Do not “bonk with the big blocks”
1.5
Introducing Similarity Transformations
•With a partner, discuss your definition of a dilation.
Activity 1:
1.6
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:
1.7
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)
1.8
Introducing Similarity Transformations
• Read G-SRT.1
• Discuss how might you have students meet this standard in your classroom?
Activity 1:
1.9
Circle Similarity
• Consider G-C.1: Prove that all circles are similar.
• Discuss how might you have students meet this standard in your classroom?
Activity 2:
1.10
Circle SimilarityActivity 2:
1.11
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:
1.12
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:
1.13
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:
1.14
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:
1.15
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?
Break
1.12
1.17
Engage NY ReduxActivity 3:
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.
1.18
Engage NY ReduxActivity 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
1.19
1.20
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.
Planning TimeActivity 4:
1.21
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.
Activity 5:
