CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Accelerated Coordinate Algebra/Analytic Geometry A
Unit 7: Similarity, Congruence, and Proofs
October 25, 2012
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CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Accelerated Coordinate Algebra/Analytic Geometry A
Unit 7: Similarity, Congruence, and Proofs
October 25, 2012
James Pratt – jpratt@doe.k12.ga.us
Brooke Kline – bkline@doe.k12.ga.us
Secondary Mathematics Specialists
These materials are for nonprofit educational purposes
only. Any other use may constitute copyright infringement.
Expectations and clearing up confusion
• Intent and focus of Unit 7 webinar.
• Framework tasks.
• GPB sessions on Georgiastandards.org.
• Standards for Mathematical Practice.
• Resources.
• http://ccgpsmathematics9-10.wikispaces.com/
• CCGPS is taught and assessed from 2012-2013 and beyond.
Welcome!
• The big idea of Unit 7
• Understanding congruence/similarity in terms of
transformations.
 Why do SSS, ASA, & SAS work? Why does AA work?
• Resources
Feedback
http://ccgpsmathematics9-10.wikispaces.com/
James Pratt – jpratt@doe.k12.ga.us
Brooke Kline – bkline@doe.k12.ga.us
Secondary Mathematics Specialists
Mathematical Communication
•Developing effective
mathematical communication
•Categories of mathematical
communication
•Organizing students to think,
talk, and write
•Updating the three-part
problem-solving lesson
•Tips for getting started
Research - Communication
•The value of student interaction
•Challenges the teachers face in
engaging students
•The teacher’s role
•Five strategies for encouraging highquality student interaction
1. The use of rich math tasks
2. Justification of solutions
3. Students questioning one
another
4. Use of wait time
5. Use of guidelines for Math Talk
My Favorite No
https://www.teachingchannel.org/videos/class-warm-up-routine
Wiki/Email Questions
• Unit 6 Frameworks
CONCEPTS/SKILLS TO MAINTAIN
It is expected that students will have prior knowledge/experience related to
the concepts and skills identified below. It may be necessary to pre-assess
in order to determine if time needs to be spent on conceptual activities that
help students develop a deeper understanding of these ideas.
• simplifying radicals
• calculating slopes of lines
• graphing lines
• writing equations for lines
Unit 6: New York Learning Task
• Teacher Edition – page 20
Unit 6: Geometric Properties Task
• Teacher Edition – page 43
Wiki/Email Questions – Unit 3
• MCC9‐12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) +
k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and
negative); find the value of k given the graphs. Experiment with cases and
illustrate an explanation of the effects on the graph using technology.
Include recognizing even and odd functions from their graphs and
algebraic expressions for them. (Focus on vertical translations of graphs
of linear and exponential functions. Relate the vertical translation of a
linear function to its y‐intercept.)
Announcements
System Test Coordinators,
Please note that we have posted today a revised EOCT Coordinate Algebra Study Guide. You can find the
guide at the GaDOE webpage below:
http://www.gadoe.org/Curriculum-Instruction-and-Assessment/Assessment/Pages/EOCT-Guides.aspx
The purpose of this revised posting was to edit the information that appeared on pages 147 – 148 regarding
strategies to fitting a line to data. The GaDOE Curriculum Division has determined that strategies for fitting a
line to data may include estimation (“eye-balling”) and/or the use of technology. The previous version of the
Study Guide specified that median-median was a required method for this purpose. However, that is not the
case. As a result, the pages referenced above, and those that contained related problems, have been edited
to clarify this point.
Please share this with the appropriate content experts in your local systems as you determine is
appropriate. The GaDOE Curriculum Division’s math specialists will be sharing this information with their
contacts in local systems as well.
Thank you!
Tony Eitel
Director, Assessment Administration
Assessment & Accountability
Office of Curriculum, Instruction, and Assessment
In each of the following diagrams, two triangles
are shaded. Based on the information given
about each diagram, decide whether there is
enough information to prove that the two
triangles are congruent.
In circle O, AB is congruent to CD
ABCD is a parallelogram
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?
The two triangles are congruent by SAS:
ABCD is a parallelogram
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?
The two triangles are congruent by SAS:
We have AX ≅ CX and DX ≅ BX since
the diagonals of a parallelogram bisect
each other, and ∠AXD ≅ ∠CBX since
they are vertical angles.
ABCD is a parallelogram
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?
The two triangles are congruent by SAS:
We have AX ≅ CX and DX ≅ BX since
the diagonals of a parallelogram bisect
each other, and ∠AXD ≅ ∠CBX since
they are vertical angles.
Alternatively, the two triangles are
congruent by ASA:
ABCD is a parallelogram
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?
The two triangles are congruent by SAS:
ABCD is a parallelogram
We have AX ≅ CX and DX ≅ BX since
the diagonals of a parallelogram bisect
each other, and ∠AXD ≅ ∠CBX since
they are vertical angles.
Alternatively, the two triangles are
congruent by ASA:
∠DAX ≅ ∠BCX and ∠ADX ≅ ∠CBX since they are opposite
interior angles. AD ≅ BC since opposite sides of a
parallelogram are congruent.
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?
Triangles are congruent.
In circle O, AB is congruent to CD
Triangle BOA is the result of reflecting
triangle COD across the perpendicular
bisector of AD
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?
What’s the big idea?
•Deepen understanding of
transformations.
•Develop understanding of
congruence and similar figures.
•Develop understanding of geometric
proof.
• Standards for Mathematical Practice.
Coherence and Focus
• K-8th
 Identification of figures in different
orientations
 Ratios and proportions
 Drawing of geometric figures with
specific characteristics
Transformations
Basic congruence and similarity
• 10th-12th
 Transformations of functions
 Trigonometric Functions
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Show that there is a translation of the plane which maps A to D
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Show that there is a translation of the plane which maps A to D
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Show that there is a rotation of the plane which does not move
D and which maps B’ to E.
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Show that there is a rotation of the plane which does not move
D and which maps B’ to E.
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Show that there is a reflection of the plane which does not
move D or E and which maps C’’ to F.
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
AB ≅ DE, AC ≅ DF, BC ≅ EF. Show △ABC ≅ △DEF
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?
Examples & Explanations
The triangle in the upper left is reflected over a line to the
triangle in the lower right. Using a compass and straightedge,
determine the line of reflection.
Adapted from Illustrative Mathematics G.CO.5, G.CO.12 Reflected Triangles
Examples & Explanations
The triangle in the upper left is reflected over a line to the
triangle in the lower right. Using a compass and straightedge,
determine the line of reflection.
Adapted from Illustrative Mathematics G.CO.5, G.CO.12 Reflected Triangles
Examples & Explanations
The triangle in the upper left is reflected over a line to the
triangle in the lower right. Using a compass and straightedge,
determine the line of reflection.
Adapted from Illustrative Mathematics G.CO.5, G.CO.12 Reflected Triangles
Examples & Explanations
In the picture below AD and BC intersect at X. AB and CD are
drawn forming △AXB and △CXD.
𝐷𝑋
The lengths AX, XB, CX, and DX satisfy the equation 𝐴𝑋
=
𝐵𝑋 𝐶𝑋
A
C
X
D
B
Adapted from Illustrative Mathematics G.SRT.2 Are They Similar?
Examples & Explanations
In the picture below AD and BC intersect at X. AB and CD are
drawn forming △AXB and △CXD.
𝐷𝑋
The lengths AX, XB, CX, and DX satisfy the equation 𝐴𝑋
=
𝐵𝑋 𝐶𝑋
Are the two triangles similar, if so describe
the sequence of transformations.
Adapted from Illustrative Mathematics G.SRT.2 Are They Similar?
Examples & Explanations
𝐷𝑋
The lengths AX, XB, CX, and DX satisfy the equation 𝐴𝑋
=
𝐵𝑋 𝐶𝑋
Rotate △ABX 180 degrees about point X, so
∠AXB coincides with ∠DXC. Then dilate △ABX
𝐷𝑋
by a factor of 𝐷𝑋
.
This
moves
A
to
D,
since
𝐴𝑋( )=𝐷𝑋, and
𝐴𝑋
𝐴𝑋
likewise moves B to C. Therefore △AXB is similar to △DXC
Adapted from Illustrative Mathematics G.SRT.2 Are They Similar?
Resource List
The following list is provided as a
sample of available resources and
is for informational purposes only.
It is your responsibility to
investigate them to determine
their value and appropriateness
for your district. GaDOE does not
endorse or recommend the
purchase of or use of any
particular resource.
Resources
• Common Core Resources
 SEDL videos - http://bit.ly/RwWTdc
or http://bit.ly/yyhvtc
 Illustrative Mathematics - http://www.illustrativemathematics.org/
 Dana Center's CCSS Toolbox - http://www.ccsstoolbox.com/
Common Core Standards - http://www.corestandards.org/
 Tools for the Common Core Standards - http://commoncoretools.me/
Phil Daro talks about the Common Core Mathematics Standards - http://bit.ly/URwOFT
•Assessment Resources
MAP - http://www.map.mathshell.org.uk/materials/index.php
Illustrative Mathematics - http://illustrativemathematics.org/
 CCSS Toolbox: PARCC Prototyping Project - http://www.ccsstoolbox.org/
 PARCC - http://www.parcconline.org/
Online Assessment System - http://bit.ly/OoyaK5

Resources
•Professional Learning Resources
 Inside Mathematics- http://www.insidemathematics.org/
Annenberg Learner - http://www.learner.org/index.html
 Edutopia – http://www.edutopia.org
 Teaching Channel - http://www.teachingchannel.org
 Ontario Ministry of Education - http://bit.ly/cGZlce
 Capacity Building Series: Communication in the Mathematics Classroom - http://bit.ly/acoWR9
 What Works? Research into Practice - http://bit.ly/SRYTuM
•Blogs
Dan Meyer – http://blog.mrmeyer.com/
Timon Piccini – http://mrpiccmath.weebly.com/3-acts.html
Dan Anderson – http://blog.recursiveprocess.com/tag/wcydwt/
Resources
Learnzillion.com
•
•
•
•
•
•
Review
Common Mistakes
Core Lesson
Guided Practice
Extension Activities
Quick Quiz
Resources
Learnzillion.com
~Thank you! Thank you! Thank you! This webinar was great, and the
site has great resources that I can use tomorrow! I just shared it with
everyone at my school! It is like going to a Common Core Conference
and receiving all the materials for every session and having them in
one place! I love it!
~I watch so many math videos for our common core lessons and I am
speechless, how awesome all these small video clips are.
~Thanks for this. I attended the webinar last week and really like this
site. I'm planning on having a PL session at school on Thursday.
https://attendee.gotowebinar.com/recording/2385067565478552832
Thank You!
Please visit http://ccgpsmathematics6-8.wikispaces.com/ to share your feedback, ask
questions, and share your ideas and resources!
Please visit https://www.georgiastandards.org/Common-Core/Pages/Math.aspx
to join the 6-8 Mathematics email listserve.
Follow on Twitter!
Follow @GaDOEMath
Brooke Kline
Program Specialist (6‐12)
bkline@doe.k12.ga.us
James Pratt
Program Specialist (6-12)
jpratt@doe.k12.ga.us
These materials are for nonprofit educational purposes only.
Any other use may constitute copyright infringement.