1. 4.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2015 SESSION 4 18 JUNE 2015 SIMILARITY IN HIGH SCHOOL

2. 4.2 TODAY’S AGENDA  Homework Review and discussion  Similarity transformations (High School), Part I  Sticky gum  Lunch  Sticky gum student work analysis  Break  Properties of similarity transformations  Reflecting on CCSSM standards aligned to similarity  Peer-teaching planning  Daily journal  Homework and closing remarks

3. 4.3 LEARNING INTENTIONS AND SUCCESS CRITERIA We are learning …  precise definitions and properties of similarity transformations;  the CCSSM High School expectations for similarity

4. 4.4 LEARNING INTENTIONS AND SUCCESS CRITERIA We will be successful when we can:  use appropriate language to describe a similarity transformation precisely;  explain the CCSSM High School similarity standards;  describe the progression in the CCSSM similarity standards from Grade 8 to High School.

5. 4.5 ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

6. 4.6 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. 4.7 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL), PART I SCALE DRAWINGS AND SIMILARITY TRANSFORMATIONS ENGAGE NY /COMMON CORE GRADE 10 MODULE 2, LESSONS 1, 2, 3 & 4

8. 4.8 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Scale drawings  Turn and talk: What is a “scale drawing”?  How would you make a scale drawing of some geometric object?

9. 4.9 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Two methods for creating scale drawings  Ratio method  Construct the image of each point by scaling the point directly away (or towards) a fixed center by a fixed scale factor  Parallel method  Construct the image of one point by scaling; construct other image points by constructing lines parallel to corresponding lines in the original figure.

10. 4.10 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Figure source: http://www.regentsprep.org/Regents/math/geometry/GT3/Ldilate2.htm

11. 4.11 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL)  Complete Exercise 1 (page S.9). (If you use Geogebra to complete this exercise, first create rough copies of the arrow and the center O.)

12. 4.12 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL)  Complete Exercise 1 (page S.19). (If you use Geogebra to complete this exercise, first create rough copies of the quadrilateral XYZW and the center O.)

13. 4.13 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL)  We have now seen two methods of constructing scale drawings…but how do we know they give the same results?  We are going to show that the two methods are indeed consistent, in effect by proving a well-known and important theorem about certain lines in triangles.  We will need two lemmas, which you will be asked to prove.

14. 4.14 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Lemma 1 If triangles ΔABC and ΔABD share a base, and have vertices C and D on a line parallel to, then the two triangles have the same area. A B C D

15. 4.15 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Lemma 2 If two triangles have bases and that lie on the same line, and a common third vertex C, then the ratio of their areas is equal to the ratio of the length of their bases. A B B’ C

16. 4.16 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Equivalence of the parallel and ratio methods  Read the proof of equivalence of the two methods (pages S.23 and S.24).

17. 4.17 ACTIVITY 2 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) The Triangle Side Splitter Theorem  Read the statement of this theorem (page S.25, or the Lesson Summary box, page S.26).  Explain to a partner why this theorem states the fact that the two ratio and parallel methods are equivalent.

18. 4.18 ACTIVITY 3 STICKY GUM  Over lunch, discuss the Sticky Gum Problem with some colleagues. You don’t need to solve it fully.  Consider the following:  What approaches might you take to solving the task?  What would you anticipate that students would do in solving the task?  The task asks you to “Generalize the problem as much as you can.” What aspects of the solution can be generalized?

19. Lunch

20. 4.20 ACTIVITY 4 STICKY GUM STUDENT WORK ANALYSIS  Imagine that the students in your class produced responses A-H to A Sticky Gum Problem. Review the eight student responses and determine which of the students actually produced a proof. (Use the Criteria for Judging the Validity of Proof from Day 2 to justify your selections.)  Come to a group consensus on which responses are proofs and why.  As a group, select one response that you think is “close” to being a proof and determine what is missing and what questions you could ask to help the student make progress.

21. Break

22. 4.22 ACTIVITY 5 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL), PART II PROPERTIES OF SIMILARITY TRANSFORMATIONS ENGAGE NY /COMMON CORE GRADE 10 MODULE 2, LESSONS 5, 13 & 14

23. 4.23 ACTIVITY 5 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) The Dilation Theorem If a dilation with center O and scale factor r sends point P to P’ and point Q to Q’, then |P’Q’| = r|PQ|. Moreover, if r ≠ 1 and O, P and Q are the vertices of a triangle, then is parallel to.  Discuss this theorem with a partner until you are confident you know what it says, then try to prove it. (Hint: you will need to use the Triangle Side Splitter Theorem.)

24. 4.24 ACTIVITY 5 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Properties of similarity transformations  Take a few moments with a partner to list as many properties as you can that the basic rigid motions (rotations, reflections, and translations) and dilations all have in common.  Which of these properties will be inherited by compositions of these special transformations (i.e. by all similarity transformations)?

25. 4.25 ACTIVITY 5 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Properties of similarity transformations  Distinct points are mapped to distinct points;  Each point in the plane has a pre-image;  There is a scale factor r so that, for any pair of points P and Q,  |P’Q’| = r|PQ|;  A similarity transformation sends lines to lines, rays to rays, segments to segments, and parallel lines to parallel lines;  A similarity transformation sends angles to angles of equal degree measure;  A similarity transformation sends a circle of radius R to a circle of radius rR.

26. 4.26 ACTIVITY 5 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL)  Complete Example 2 (page S.84).

27. 4.27 Read the High School Similarity standards from the CCSSM.  How do you see the expectations for Geometry content change between Grade 8 and High School? ACTIVITY 5 SIMILARITY TRANSFORMATIONS (HIGH SCHOOL) Reflecting on CCSSM standards alignment

28. 4.28 ACTIVITY 6 PEER TEACHING PLANNING

29. 4.29  Work with your group to plan your lesson. ACTIVITY 6 PEER TEACHING PLANNING

30. 4.30 ACTIVITY 7 DAILY JOURNAL

31. 4.31 Take a few moments to reflect and write on today’s activities. ACTIVITY 7 DAILY JOURNAL

32. 4.32  Complete Problems 4, 5 & 6 from the Grade 10 Module 2 Lesson 4 Problem Set in your notebook (page S.29).  Extending the mathematics: Suppose L and L’ are (distinct) parallel lines. What is the result of a reflection across L followed by a reflection across L’? (Be as specific as you can in your description.) What happens if L and L’ are not parallel?  Reflecting on teaching: What challenges do you see in taking a transformation approach to teaching Geometry in high school? What advantages? ACTIVITY 8 HOMEWORK AND CLOSING REMARKS