1. Session will be begin at 8:00 am
CCGPS Mathematics Unit-by-Unit Grade Level Webinar Analytic Geometry Unit 1: Similarity, Congruence, and Proofs May 7, 2013
Session will be begin at 8:00 am
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2. CCGPS Mathematics Unit-by-Unit Grade Level Webinar Analytic Geometry Unit 1: Similarity, Congruence, and Proofs May 7, 2013
James Pratt –
Brooke Kline –
Secondary Mathematics Specialists
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.
James & Brooke

3. Expectations and clearing up confusion
Intent and focus of Unit 1 webinar.
Framework tasks.
GPB sessions on Georgiastandards.org.
Standards for Mathematical Practice.
Resources.
James

4. What is a Wiki?
James

5. CCGPS Mathematics Sequence for Implementation
Brooke

6. CCGPS Mathematics Resources for Implementation
Brooke

7. Welcome! The big idea of Unit 1
Understanding congruence/similarity in terms of transformations.
Why do SSS, ASA, & SAS work? Why does AA work?
Standards for Mathematical Practice
Resources
Brooke

8. Feedback http://ccgpsmathematics9-10.wikispaces.com/
James Pratt – Brooke Kline –
Secondary Mathematics Specialists
Brooke

9. Parent Communication
Explanation to parents of the need for change in mathematics
What children will be learning in high school mathematics
Parents partnering with teachers
Grade level examples
Parents helping children learn outside of school
Additional resources
Brooke

10. Parent Communication
An overview of what children will be learning in high school mathematics
Topics of discussion for parent-teacher communication regarding student academic progress
Tips for parents that will help their children plan for college and career
Brooke

11. Parent Communication
An overview of what children will be learning in high school mathematics
Topics of discussion for parent-teacher communication regarding student academic progress
Tips for parents that will help their children plan for college and career
Brooke

12. Wiki/Email Questions MCC9-12.G.CO.6
What’s the difference between 8th grade MCC8.G.2 and what we do in Analytic Geometry?
MCC9-12.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
MCC8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
James

13. Wiki/Email Questions MCC9-12.G.CO.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Can you provide more information about this standard:
Do we need the students to be able to prove the medians of a triangle meet at a point?
Do the students need to know centroid, incenter, circumcenter, and orthocenter?
Do the students need to know how to apply each proof listed in this standard to “skill based” problems?
James

14.   
Brooke

15. Cost: $50 for GCTM members and $60 for GCTM non-members   
Cost: $50 for GCTM members and $60 for GCTM non-members
Travel expenses will be reimbursed for all participants who complete the academy and are Georgia certified K-12 educators under contract with a Georgia school
Registration will opened on April 1, 2013
Registration closing dates:
Academy 1 – May 15, 2013
Academy 2 – May 22, 2013
Academy 3 – May 29, 2013
Academy 4 – June 5, 2013
Payments must be received prior to the closing date of registration
Brooke

16. Call 1-855-ASK-GCTM (ext 4) for questions about the academy  
Visit for more details concerning these quality professional development opportunities
Call ASK-GCTM (ext 4) for questions about the academy
Peggy Pool – GCTM Vice President for Regional Services and Director of Academies,
Brooke

17. Activate your Brain
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
Brooke
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?

18. The two triangles are congruent by SAS:
Activate your Brain
The two triangles are congruent by SAS:
ABCD is a parallelogram
Brooke
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?

19. Activate your Brain 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
Brooke
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?

20. Activate your Brain 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
Brooke
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?

21. Activate your Brain 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
Brooke
∠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?

22. Activate your Brain Triangles are congruent.
Triangle BOA is the result of reflecting triangle COD across the perpendicular bisector of AD
In circle O, AB is congruent to CD
Brooke
Adapted from Illustrative Mathematics G.CO Are the Triangles Congruent?

23. What’s the big idea? Understand congruence in terms of rigid motions.
Understand similarity in terms of similarity transformations.
Prove theorems involving similarity.
Prove geometric theorems.
Make geometric constructions.
Brooke

24. Standards for Mathematical Practice
What’s the big idea?
Standards for Mathematical Practice
Brooke

25. What’s the big idea?
SMP 3 – Construct viable arguments and critique the reasoning of others
Student Sample Work
Feedback/Critique and Revision
Brooke
Expeditionary Learning

26. Coherence and Focus K-8th 10th-12th
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
James

27. Examples & Explanations
𝐴𝐵 ≅ 𝐷𝐸 , 𝐴𝐶 ≅ 𝐷𝐹 , 𝐵𝐶 ≅ 𝐸𝐹 . Show △ABC ≅ △DEF
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

28. 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
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

29. 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
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

30. Examples & Explanations
𝐴𝐵 ≅ 𝐷𝐸 , 𝐴𝐶 ≅ 𝐷𝐹 , 𝐵𝐶 ≅ 𝐸𝐹 . Show △ABC ≅ △DEF
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

31. 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.
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

32. 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.
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

33. Examples & Explanations
𝐴𝐵 ≅ 𝐷𝐸 , 𝐴𝐶 ≅ 𝐷𝐹 , 𝐵𝐶 ≅ 𝐸𝐹 . Show △ABC ≅ △DEF
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

34. 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.
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

35. Examples & Explanations
𝐴𝐵 ≅ 𝐷𝐸 , 𝐴𝐶 ≅ 𝐷𝐹 , 𝐵𝐶 ≅ 𝐸𝐹 . Show △ABC ≅ △DEF
James
Adapted from Illustrative Mathematics G.CO.8 Why does SSS Work?

36. 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.
James
Adapted from Illustrative Mathematics G.CO.5, G.CO.12 Reflected Triangles

37. 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.
James
Adapted from Illustrative Mathematics G.CO.5, G.CO.12 Reflected Triangles

38. 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.
James
Adapted from Illustrative Mathematics G.CO.5, G.CO.12 Reflected Triangles

39. Proofs in CCGPS
Encourage multiple ways of writing proofs, such as in narrative
paragraphs, using flow diagrams, in two‐column format, and using diagrams without words.
Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.
James

40. Examples & Explanations
In the picture below 𝐴𝐷 and 𝐵𝐶 intersect at X. 𝐴𝐵 and 𝐶𝐷 are drawn forming △AXB and △CXD.
The lengths AX, XB, CX, and DX satisfy the equation 𝐴𝑋 𝐵𝑋 = 𝐷𝑋 𝐶𝑋 A B X C D
James
Adapted from Illustrative Mathematics G.SRT.2 Are They Similar?

41. Examples & Explanations
In the picture below AD and BC intersect at X. AB and CD are drawn forming △AXB and △CXD.
The lengths AX, XB, CX, and DX satisfy the equation 𝐴𝑋 𝐵𝑋 = 𝐷𝑋 𝐶𝑋
Are the two triangles similar, if so describe
the sequence of transformations.
James
Adapted from Illustrative Mathematics G.SRT.2 Are They Similar?

42. Examples & Explanations
The lengths AX, XB, CX, and DX satisfy the equation 𝐴𝑋 𝐵𝑋 = 𝐷𝑋 𝐶𝑋
Rotate △ABX 180 degrees about point X, so
∠AXB coincides with ∠DXC. Then dilate △ABX
by a factor of 𝐷𝑋 𝐴𝑋 . This moves A to D, since 𝐴𝑋( 𝐷𝑋 𝐴𝑋 )=𝐷𝑋 , and
likewise moves B to C. Therefore △AXB is similar to △DXC
James
Adapted from Illustrative Mathematics G.SRT.2 Are They Similar?

43. 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.
James

44. Resources
James

45. Resources
James

46. Resources
Brooke

47. Resources Common Core Resources Assessment Resources
SEDL videos - or
Illustrative Mathematics -
Dana Center's CCSS Toolbox -
Common Core Standards -
Tools for the Common Core Standards -
Phil Daro talks about the Common Core Mathematics Standards -
Assessment Resources
MAP -
Illustrative Mathematics -
CCSS Toolbox: PARCC Prototyping Project -
PARCC -
Online Assessment System -
Brooke

48. Resources Professional Learning Resources Blogs
Inside Mathematics-
Annenberg Learner -
Edutopia –
Teaching Channel -
Ontario Ministry of Education -
Expeditionary Learning: Center for Student Work -
Blogs
Dan Meyer –
Timon Piccini –
Dan Anderson –
Brooke

49. James Pratt Program Specialist (6-12) jpratt@doe.k12.ga.us
Thank You! Please visit to share your feedback, ask questions, and share your ideas and resources! Please visit to join the 9-12 Mathematics listserve. Follow on Twitter!
Brooke Kline
Program Specialist (6‐12)
James Pratt Program Specialist (6-12)
James & Brooke
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.