1. 6.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2015 SESSION 6 29 JUNE 2015 SEQUENCING BASIC RIGID MOTIONS; THE KOU-KU THEOREM

2. 6.2 TODAY’S AGENDA  Homework Review and discussion  G8 M2 L8: Sequencing Reflections and Translations  Discussion  Break  G8 M2 L9: Sequencing Rotations  Discussion  Break  Kou-Ku and the Peacock’s Tail (Part I)  Lunch  Kou-Ku and the Peacock’s Tail (Part II)  Daily journal  Homework and closing remarks

3. 6.3 ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

4. 6.4 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. 6.5 ACTIVITY 2 SEQUENCING REFLECTIONS AND TRANSLATIONS ENGAGE NY /COMMON CORE GRADE 8 MODULE 2, LESSON 8

6. 6.6 ACTIVITY 2 SEQUENCING REFLECTIONS AND TRANSLATIONS 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. 6.8 ACTIVITY 3 SEQUENCING ROTATIONS ENGAGE NY /COMMON CORE GRADE 8 MODULE 2, LESSON 9

9. 6.9 ACTIVITY 3 SEQUENCING ROTATIONS 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. 6.11 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Session goals  To explore the concept of area  To discover an important theorem about area  To explain the session title

12. 6.12 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Turn and talk:  What is “area”?

13. 6.13 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Selected CCSSM standards related to area: Grade 3 Geometric measurement: understand concepts of area and relate area to multiplication and to addition.  3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement  a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.  b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.  3.MD.6 Measure areas by counting unit squares.

14. 6.14 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL What is the area of this figure?  Find the area of the parallelogram in as many ways as you can. (At least two.)  Explain your reasoning to your group.

15. 6.15 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL What is the area of this figure?

16. 6.16 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL What is the area of this figure?

17. 6.17 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL What is the area of this figure?

18. 6.18 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Properties of area  “Moving property”: the area of a shape is not changed under a rigid motion. (Congruent shapes have equal areas.)  “Combining property”: the total area of two (or more) non-overlapping shapes is equal to the sum of their areas.

19. 6.19 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Selected CCSSM standards related to area: Grade 6 Solve real-world and mathematical problems involving area, surface area, and volume.  6.G.1 Find the area of right triangles, other triangles, special quadrilaterals and polygons, by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

20. 6.20 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Developing a conjecture about areas  Draw a right-angled triangle near the center of a sheet of grid paper.  You should draw the triangle with two of its sides lying along grid lines.  Draw a square on each side of the triangle.

21. 6.21 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Developing a conjecture about areas  Draw a right-angled triangle near the center of a sheet of grid paper.  You should draw the triangle with two of its sides lying along grid lines.  Draw a square on each side of the triangle.

22. 6.22 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Developing a conjecture about areas  Use the area techniques we have developed to find the areas of each of your 3 squares.  Write your answers on the whiteboard.  What pattern do you see?

23. 6.23 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Proving your area conjecture  Examine the picture below. How does this picture prove your conjecture? (Or is it not yet a proof?) Picture credit: http://www.cut- theknot.org/pythagoras/index.s html

24. 6.24 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL 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.

25. 6.25 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Proofs of the Pythagorean Theorem  You can find 111 proofs of the Pythagorean Theorem at the Cut the Knot website, http://www.cut-theknot.org/pythagoras/index.shtml.http://www.cut-theknot.org/pythagoras/index.shtml  The picture proof we have just seen is Proof #9 in its entirety.  Which proof was discovered by a U.S. president?

26. 6.26 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Selected CCSSM standards related to area: Grade 8 Understand and apply the Pythagorean Theorem.  8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

27. Lunch

28. 6.28  There are three similar right triangles in the picture. Use your knowledge of similarity to show that a 2 + b 2 = c 2.  Why does this constitute a proof of the Pythagorean theorem? (If it does.) ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL A similarity proof of the Pythagorean Theorem Picture credit: Engage NY /Common Core

29. 6.29 ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL Selected CCSSM standards related to the Pythagorean Theorem: Grade 8 Understand and apply the Pythagorean Theorem.  8.G.6 Explain a proof of the Pythagorean Theorem and its converse. High School Geometry Prove theorems involving similarity.  G-SRT.4 Prove theorems about triangles. Theorems include:... The Pythagorean Theorem proved using triangle similarity.

30. 6.30 ACTIVITY 5 DAILY JOURNAL

31. 6.31 Take a few moments to reflect and write on today’s activities. ACTIVITY 5 DAILY JOURNAL

32. 6.32  Engage NY G8 M2 L8, Problem Set #1: G8 M2 L9: Problem Set #2.  Extending the mathematics: Each of the proofs of the Pythagorean theorem we looked at today used the Triangle Angle Sum Theorem. Identify where that theorem was used in each proof. Find at least one more proof of the Pythagorean theorem (from your current curriculum, or from the Cut-the-Knot website). Does that proof also use the Triangle Sum Theorem?  Reflecting on teaching: Give a few (2≤x≤4) reasons why learning the Pythagorean Theorem is important for all students. How might we foster stronger connections between the Pythagorean Theorem and related mathematical ideas? ACTIVITY 6 HOMEWORK AND CLOSING REMARKS