1. 8.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2015 SESSION 8 1 JULY 2015 POINTS OF CONCURRENCIES; UNKNOWN ANGLE PROOFS; DIVIDING THE KING’S FOOT; AXIOMATICS

2. 8.2 TODAY’S AGENDA  Homework Review and discussion  G10 M1 L5: Points of Concurrencies  Discussion  Break  G10 M1 L9: Unknown Angle Proofs—Writing Proofs  Discussion  Break  G10 M2 L10: Dividing the King’s Foot into 12 Equal Pieces  Lunch  Possible axioms for transformational geometry  Daily journal  Homework and closing remarks

3. 8.3 ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

4. 8.4 ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION Table discussion:  Compare your answers to the “Extending the mathematics” prompt from our last session.  Identify common themes, as well as points of disagreement, in your responses to the “Reflection on teaching” prompt.

5. 8.5 ACTIVITY 2 POINTS OF CONCURRENCIES ENGAGE NY /COMMON CORE GRADE 10 MODULE 1, LESSON 5

6. 8.6 ACTIVITY 2 POINTS OF CONCURRENCIES Connecting the lesson goals with pedagogy  Which of the Effective Mathematics Teaching Practices did you see modeled in this lesson?  How did the use of those practices support progress towards the lesson’s stated goals (learning intentions and success criteria)? 1. Establish mathematics goals to focus learning 2. Implement tasks that promote reasoning and problem solving 3. Use and connect mathematical representations 4. Facilitate meaningful mathematical discourse 5. Pose purposeful questions 6. Build procedural fluency from conceptual understanding 7. Support productive struggle in learning mathematics 8. Elicit and use evidence of student thinking

7. Break

8. 8.8 ACTIVITY 3 UNKNOWN ANGLE PROOFS—WRITING PROOFS ENGAGE NY /COMMON CORE GRADE 10 MODULE 1, LESSON 9

9. 8.9 ACTIVITY 3 UNKNOWN ANGLE PROOFS—WRITING PROOFS Connecting the lesson goals with pedagogy  Which of the Effective Mathematics Teaching Practices did you see modeled in this lesson?  How did the use of those practices support progress towards the lesson’s stated goals (learning intentions and success criteria)? 1. Establish mathematics goals to focus learning 2. Implement tasks that promote reasoning and problem solving 3. Use and connect mathematical representations 4. Facilitate meaningful mathematical discourse 5. Pose purposeful questions 6. Build procedural fluency from conceptual understanding 7. Support productive struggle in learning mathematics 8. Elicit and use evidence of student thinking

10. Break

11. 8.11 ACTIVITY 4 DIVIDING THE KING’S FOOT INTO 12 EQUAL PIECES ENGAGE NY /COMMON CORE GRADE 10 MODULE 2, LESSON 11

12. 8.12 ACTIVITY 4 DIVIDING THE KING’S FOOT INTO 12 EQUAL PIECES Connecting the lesson goals with pedagogy  Which of the Effective Mathematics Teaching Practices did you see modeled in this lesson?  How did the use of those practices support progress towards the lesson’s stated goals (learning intentions and success criteria)? 1. Establish mathematics goals to focus learning 2. Implement tasks that promote reasoning and problem solving 3. Use and connect mathematical representations 4. Facilitate meaningful mathematical discourse 5. Pose purposeful questions 6. Build procedural fluency from conceptual understanding 7. Support productive struggle in learning mathematics 8. Elicit and use evidence of student thinking

13. Lunch

14. 8.14 ACTIVITY 5 POSSIBLE AXIOMS FOR TRANSFORMATIONAL GEOMETRY Session goals  To deepen understanding of the role of axioms in proofs  To understand one proposed set of axioms for transformational geometry

15. 8.15 ACTIVITY 5 POSSIBLE AXIOMS FOR TRANSFORMATIONAL GEOMETRY Reading Wu’s “Teaching Geometry in Grade 8 and High School”  Read the section, “Goals of High School Geometry” (pages 79-80)

16. 8.16 ACTIVITY 5 POSSIBLE AXIOMS FOR TRANSFORMATIONAL GEOMETRY Reading Wu’s “Teaching Geometry in Grade 8 and High School”  Read the eight “assumptions”, (A1) through (A8). They can be found on the following pages:  (A1), (A2): page 81.  (A3): page 85; (A4): page 87.  (A5): page 89.  (A6): page 91.  (A7): page 110.  (A8): page 119.

17. 8.17 ACTIVITY 5 POSSIBLE AXIOMS FOR TRANSFORMATIONAL GEOMETRY Reading Wu’s “Teaching Geometry in Grade 8 and High School”  Read Wu’s Lemma 2 (page 82). (Do not bother to read the proofs.)  Since Wu does give a proof of this lemma, we know that it follows from (some of) his other assumptions. What would be the consequences of ignoring the proof and simply adding the statement of this lemma to our list of assumptions?

18. 8.18 ACTIVITY 6 DAILY JOURNAL

19. 8.19 Take a few moments to reflect and write on today’s activities. ACTIVITY 6 DAILY JOURNAL

20. 8.20  G10 M1 L5 (Points of Concurrencies): last page of class handout; G10 M1 L9&10 (Unknown Angle Proofs): Problem set #2; G10 M2 L10: (Dividing the King’s Foot into 12 Equal Pieces): Homework problem from class handout.  Extending the mathematics: In class today, we saw a geometric construction for dividing a line segment into any desired number of equal parts. Try to think of constructions for adding, subtracting, multiplying, and dividing the lengths of 2 given line segments.  Reflecting on teaching: What principles would you follow in determining a reasonable balance between axioms and theorems in your high school geometry class? How would you decide whether to include a result in your list of assumptions, or to prove it as a theorem? And more broadly, how do you support students in distinguishing assumptions, axioms, and theorems in geometry? ACTIVITY 7 HOMEWORK AND CLOSING REMARKS