1. 2.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2015 SESSION 2 16 JUNE 2015 CONGRUENCE IN HIGH SCHOOL

2. 2.2 TODAY’S AGENDA  Homework Review and discussion  Rigid motions and congruence (High School), Part I  The Case of Charlie Sanders  Dinner  Rigid motions and congruence (High School), Part II  Reflecting on CCSSM standards aligned to Grade 8 congruence  Break  Student work analysis  Developing criteria for proof  Peer-teaching planning  Daily journal  Homework and closing remarks

3. 2.3 LEARNING INTENTIONS AND SUCCESS CRITERIA We are learning …  precise definitions of the basic rigid motions;  the CCSSM High School expectations for congruence

4. 2.4 LEARNING INTENTIONS AND SUCCESS CRITERIA We will be successful when we can:  use appropriate language to describe a basic rigid motion precisely;  explain the CCSSM High School congruence standards;  describe the progression in the CCSSM congruence standards from Grade 8 to High School.

5. 2.5 ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

6. 2.6 ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION Table discussion:  Compare your answers to last night’s “Extending the mathematics” prompt.  Identify common themes, as well as points of disagreement, in your responses to the “Reflection on teaching” prompt.

7. 2.7 ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL), PART I PRECISE DEFINITIONS OF ROTATIONS AND REFLECTIONS ENGAGE NY /COMMON CORE GRADE 10, LESSONS 12, 13 & 14

8. 2.8 ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Work with a partner to complete the task. You are allowed a protractor, compass, and straightedge.  Partner A: Without showing the card to your partner, describe how to draw the transformation indicated on the card. When you have finished, compare your partner’s drawing with the transformed image on your card. Did you describe the motion correctly?  Partner B: Your partner is going to describe a transformation to be performed on the figure on your card. Follow your partner’s instructions, and then compare the image of your transformation to the image on your partner’s card.

9. 2.9 ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) What did you discover as you worked on the previous task? Would your students reactions to the task be essentially the same as yours? Why or why not?

10. 2.10 ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Defining rotations  Read the definition of a rotation in Lesson 13 (page S.70)

11. 2.11 ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Finding angles and centers of rotations  Complete Exercises 1, 2 & 3 (pages S.72-73). If you wish to use Geogebra, you may find this file useful: https://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day02/G10_M1_Lesson13_Ex1-3.ggb

12. 2.12 ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Exploration  Complete the Lesson 14 Exploratory Challenge (Page S.77) If you wish to use Geogebra, you may find this file useful:https://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day 02/G10_M1_Lesson14_Explore.ggbhttps://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day 02/G10_M1_Lesson14_Explore.ggb  What do you notice about these perpendicular bisectors?

13. 2.13 ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Defining reflections  Read the definition of a reflection in Lesson 14 (page S.78)

14. 2.14 ACTIVITY 2 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Finding lines of reflection  Complete Examples 2, 3 & 4 (pages S.79 & S.80). If you wish to use Geogebra, you may find these files useful: https://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day02/G10_ M1_Lesson14_Examples2-3.ggb https://pantherfile.uwm.edu/kevinm/www/CCHSML/Summer_2015/Day02/G10_ M1_Lesson14_Example4.ggb

15. 2.15 ACTIVITY 3 DEVELOPING IN STUDENTS A NEED FOR PROOF THE CASE OF CHARLIE SANDERS CASES OF REASONING AND PROVING IN SECONDARY MATHEMATICS

16. 2.16 ACTIVITY 3 THE CASE OF CHARLIE SANDERS  With your small group, discuss what you noticed about the opportunities Charlie’s students had to engage in reasoning-and-proving.  In what ways were Charlie’s and Kathy’s implementations of the task sequence similar and in what ways were they different?  Specifically, which of the Principles to Actions Mathematics Teaching Practices did you notice in each classroom?  Discuss with your group, and record your most important similarity and difference on the whiteboards.

17. 2.17 ACTIVITY 3 THE CASE OF CHARLIE SANDERS What is The Case of Charlie Sanders a case of?

18. Dinner

19. 2.19 ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL), PART II PRECISE DEFINITION OF TRANSLATION ENGAGE NY /COMMON CORE GRADE 8, LESSON 16

20. 2.20 ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Constructing parallel lines  Use Geogebra to complete the Lesson 16 Exploratory Challenge (page S.89).  Explain why this construction works.

21. 2.21 ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Defining translations  Read the definition of a translation in Lesson 16 (page S.90)  Complete Examples 2 & 3 (page S.91) You may use Geogebra to complete either of these examples, but you may not use the translation tool: mimic the straightedge and compass constructions.

22. 2.22 Read MP6, the sixth CCSSM standard for mathematical practice.  Recalling that the standards for mathematical practice describe student behaviors, how did you engage in this practice as you completed the lesson?  What instructional moves or decisions did you see occurring during the lesson that encouraged greater engagement in MP6?  Are there other standards for mathematical practice that were prominent as you and your groups worked on the tasks? ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE Reflecting on CCSSM standards alignment

23. 2.23  How do you see the expectations for the standards for mathematical practice change between Grade 8 and High School? ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE Reflecting on CCSSM standards alignment

24. 2.24 Read the High School Congruence standards from the CCSSM.  How do you see the expectations for Geometry content change between Grade 8 and High School? ACTIVITY 4 RIGID MOTIONS AND CONGRUENCE Reflecting on CCSSM standards alignment

25. Break

26. 2.26 ACTIVITY 5 STUDENT WORK ANALYSIS

27. 2.27 ACTIVITY 5 STUDENT WORK ANALYSIS Consider the following: Prove the conjecture that when you add any two odd numbers, your answer is always even.

28. 2.28 ACTIVITY 5 STUDENT WORK ANALYSIS  Imagine that the students in your class produced responses A-J to the “odd + odd = even” task.  Review the ten student responses individually and record whether or not each response qualifies as a proof & provide the rationale that led you to that conclusion.  Discuss your ratings and rationale with members of your group, come to a group consensus on which responses are and are not proofs and why, and record you group’s decision on the Proof Evaluation Chart on the whiteboard.  As you work, be able to describe the criteria you’re using to judge whether or not an argument is a proof.

29. 2.29 ACTIVITY 6 DEVELOPING CRITERIA FOR PROOF

30. 2.30 ACTIVITY 6 DEVELOPING A CRITERIA FOR PROOF Some ideas from you:  Justify a given statement using facts  Uses universally agreed-upon truths  Contains a series of logical steps  You have finished when you get to the statement you set out to prove  Simplifies complex ideas  There can be no counterexamples Think about the criteria for judging a mathematical argument as a proof.

31. 2.31 ACTIVITY 6 DEVELOPING A CRITERIA FOR PROOF An argument that counts as proof must meet the following criteria:  The argument must show that the conjecture or claim is (or is not) true for all cases.  The statements and definitions that are used in the argument must be ones that are true and accepted by the community because they have been previously justified.  The conclusion that is reached from the set of statements must follow logically from the argument made. In addition, a valid proof may vary along the following dimensions:  type of proof (e.g., demonstration, induction, counterexample)  form of the proof (e.g., two-column, paragraph, flow chart)  representation used (e.g., symbols, pictures, words)  explanatory power (e.g., how well the proof explains why the claim is true) Variance on these dimensions, however, does not matter as long as the preceding three criteria for proof are met.

32. 2.32 ACTIVITY 7 PEER TEACHING PLANNING

33. 2.33 Write your name on three separate Post-it notes, label each Post-it note as your first, second or third choice, and place the notes underneath the appropriate lesson number. Remember, you choices are:  Grade 8, Module 2: Lesson 8 or Lesson 9.  Grade 8, Module 3: Lesson 11 or Lesson 12.  Grade 10, Module 1: Lessons 1 & 2, Lesson 3; Lesson 5; Lesson 9; or Lesson 10.  Grade 10: Module 2: Lesson 7; Lesson 8; Lesson 9; Lesson 10; or Lesson 11. ACTIVITY 7 PEER TEACHING PLANNING

34. 2.34 ACTIVITY 8 DAILY JOURNAL

35. 2.35 Take a few moments to reflect and write on today’s activities. ACTIVITY 8 DAILY JOURNAL

36. 2.36  Complete Problems 5 and 6 from the Grade 10 Module 1 Lesson 16 Problem Set in your notebook (page S.95)  Extending the mathematics: We have made implicit use of parallel and perpendicular lines throughout our discussion of rigid motions. Identify as many of these uses as you can (from both the Grade 8 and Grade 10 material).  Reflecting on teaching: Consider a typical class of high school Geometry students in your district. How would their understanding of and experiences with congruence compare with what you have see today? ACTIVITY 8 HOMEWORK AND CLOSING REMARKS