Transcript Common Core High School Mathematics: Transforming Instructional Practice for a New Era 8.1

Common Core High School Mathematics:
Transforming Instructional Practice for a New Era
8.1
Learning Intentions & Success Criteria
Learning Intentions:
We are learning to deepen our understanding of the
Common Core State Standards and the implications for
teaching and learning mathematics.
Success Criteria:
We will be successful when we can describe how the
content standards and math practice standards are
evident in the implementation of a mathematical task.
8.2
Agenda
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Homework review and discussion
Proving triangle and circle theorems
Reading G-CO.9-13
Introducing similarity transformations
Break
“Prove that all circles are similar”
Reading G-SRT.1 and G-C.1
Homework and closing remarks
8.3
Activity 1:
Homework Review and
Discussion
Table Discussion:
Discuss your write up for the day 7 math task
Patty Paper Geometry Open Investigations:
• Compare your strategies with others at your
table.
• Reflect on how you might revise your own
solution and/or presentation.
8.4
Activity 2:
Proving Triangle and Circle
Theorems
(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.
Use this definition to prove that two line segments are
congruent if and only if they have the same length.
8.5
Activity 2:
Proving triangle and circle
theorems
Find proofs of each of the following theorems,
using either transformation or traditional
methods.
Theorem 1 (Isosceles triangle symmetry)
Let ΔABC be isosceles, with AB  AC . Then the
angle bisector at A is the perpendicular bisector
of BC .
8.6
Activity 2:
Proving triangle and circle
theorems
Theorem 2 (SAS triangle congruence criterion)
Let ΔABC and ΔDEF be two triangles, AB  DE ,
AC  DF ,A D. Then ΔABC and ΔDEF are
congruent.
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8.7
Activity 2:
Proving triangle and circle
theorems
Theorem 3 (Three points determine a circle)
Let A, B and C be three non-collinear points. Then
there is one and only one circle which contains A, B and
C.
Theorem 4 (Circle intersection theorem)
Two distinct circles can intersect in at most 2 points. If
there is exactly one point of intersection, it lies on the
line through the centres of the circles; if there are two
intersection points, they are reflections of each other in
this line.
8.8
Activity 2:
Proving triangle and circle
theorems
Theorem 5 (SSS triangle congruence criterion)
Let ΔABC and ΔDEF be two triangles, with
AB  DE , AC  DF , and BC  EF . Then ΔABC
and ΔDEF are congruent.
8.9
Activity 2:
Proving triangle and circle
theorems
Theorem 6
Any angle inscribed in a semicircle is a right
angle
Theorem 7
A tangent line to a circle is perpendicular to the
radius through the point of tangency.
8.10
Activity 3:
Reading G-CO.9-13
• Read these standards from the high school
Geometry conceptual category
• Turn and talk:
• How do you see these standards in the
activity you have just completed?
• How might you prove some of the theorems
mentioned in these standards that we have
not covered today?
8.11
Activity 4:
Introducing Similarity
Transformations
• With a partner, discuss your definition of a
dilation.
8.12
Activity 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/ge
ometry/GT3/Ldilate2.htm
8.13
Activity 4:
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.
8.14
Activity 4:
Introducing Similarity
Transformations
• Read G-SRT.1
• Discuss how might you have students meet this
standard in your classroom?
8.15
Break
8.16
Activity 5:
“Prove that all circles are similar”
What Kind of a Standard is That?
•Read G-C.1
•What, precisely, is the student expectation in
this standard?
8.17
Activity 5:
“Prove that all circles are similar”
What Kind of a Standard is That?
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?
8.18
Activity 5:
“Prove that all circles are similar”
What Kind of a Standard is That?
Turning to similarity
•On a piece of paper, draw two circles that are
not congruent.
•How can you show that your circles are similar?
8.19
Activity 5:
“Prove that all circles are similar”
What Kind of a Standard is That?
• 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?
8.20
Activity 6: Reading G-SRT.1 and G-C.1
• Read these standards from the high school
Geometry conceptual category
• Turn and talk:
• How do you see these standards in the
activity you have just completed?
• How might you prove some of the theorems
mentioned in these standards that we have
not covered today?
8.21
Learning Intentions & Success Criteria
Learning Intentions:
We are learning to deepen our understanding of the
Common Core State Standards and the implications for
teaching and learning mathematics.
Success Criteria:
We will be successful when we can describe how the
content standards and math practice standards are
evident in the implementation of a mathematical task.
8.22
Activity 7:
Homework and Closing
Remarks
Homework (to be included in journal):
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Day 8 Math Task: Tangent Line and Radius
Day 8 Class Reflection
Reading
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Geometry, Big Idea 2: Geometry is about working
with variance and invariance, despite appearing to
be about theorems
8.23