1.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2015 SESSION 3 17 JUNE 2015 SIMILARITY IN MIDDLE SCHOOL

1.2 TODAY’S AGENDA  Homework Review and discussion  Discussion of CCSSM Congruence standards  Introduction to Similarity (Grade 8)  The Cases of Vicky Mansfield and Nancy Edwards  (Read these during extended break)  Set up teacher discourse moves reading  Lunch  Applications of similarity (Grade 8)  Reflecting on CCSSM standards aligned to Grade 8 similarity  Peer-teaching planning  Daily journal  Homework and closing remarks

1.3 LEARNING INTENTIONS AND SUCCESS CRITERIA We are learning …  to recognize and describe basic similarity transformations;  the definition of similarity in terms of basic similarity transformations;  the CCSSM Grade 8 expectations for similarity.

1.4 LEARNING INTENTIONS AND SUCCESS CRITERIA We will be successful when we can:  use appropriate language to describe the similarity transformation (or sequence of similarity transformations) that take one figure onto another;  decide whether or not two figures are similar;  explain the CCSSM Grade 8 similarity standards.

1.5 ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

1.6 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.

1.7 ACTIVITY 1.5 DISCUSSION OF CCSSM CONGRUENCE STANDARDS

1.8 Read the Grade 8 and High School Congruence standards from the CCSSM.  Which of these standards have been addressed in our last two sessions?  How do you see the expectations for Geometry content change between Grade 8 and High School? ACTIVITY 1.5 DISCUSSION OF CCSSM CONGRUENCE STANDARDS Reflecting on CCSSM standards alignment

1.9 ACTIVITY 2 INTRODUCTION TO SIMILARITY (GRADE 8) DEFINITIONS AND PROPERTIES OF SIMILARITY TRANSFORMATIONS ENGAGE NY /COMMON CORE GRADE 8, MODULE 3, LESSONS 1-8

1.10 ACTIVITY 2 INTRODUCTION TO SIMILARITY (GRADE 8)  At your tables, examine the 8 pairs of shapes in the Exploratory Challenge (pages S.1-S.2). For which pairs are the two shapes similar?  Why do we need a better definition of similarity than “same shape, not necessarily the same size”?

1.11 ACTIVITY 2 INTRODUCTION TO SIMILARITY (GRADE 8) Turn and talk:  What is a dilation?

1.12 ACTIVITY 2 INTRODUCTION TO SIMILARITY (GRADE 8) (From the CCSSM glossary) A dilation is a transformation that moves each point along the ray through the point emanating from a common center, and multiplies distances from the center by a common scale factor. Figure source:

1.13 ACTIVITY 2 INTRODUCTION TO SIMILARITY (GRADE 8) Understanding scale factors  Discuss Exercises 1 & 2 (page S.5)

1.14 ACTIVITY 2 INTRODUCTION TO SIMILARITY (MIDDLE SCHOOL) Properties of dilations  Fold a piece of paper into fourths.  At the top of each fourth, write one of line, segment, ray, and angle, along with a diagram of each.  What do you think will happen to each of these after a dilation? Explain.

1.15 ACTIVITY 2 INTRODUCTION TO SIMILARITY (MIDDLE SCHOOL) Properties of dilations Any dilation has the following properties: 1. It takes lines to lines, rays to rays, segments to segments, and angles to angles; In fact, any line is taken to a parallel line (or to itself, if the center of dilation is on the line); 2. It multiplies all distances by the same scale factor; 3. It preserves the degree measure of angles.

1.16 ACTIVITY 2 INTRODUCTION TO SIMILARITY (MIDDLE SCHOOL) Example  Complete Exercise 1 from the Lesson 2 problem set (page S.10).  Complete Exercises 1 and 2 from the Lesson 3 problem set (page S.14) (Feel free to use Geogebra, but not the dilation tool.).

1.17 ACTIVITY 2 INTRODUCTION TO SIMILARITY (MIDDLE SCHOOL) The Fundamental Theorem of Similarity (FTS) Given a dilation with center O and scale factor r, then for any two points P and Q in the plane so that O, P, and Q are not collinear, the lines PQ and P’Q’ are parallel (where P’ and Q’ are the images of P and Q respectively, under the dilation). Furthermore, |P’Q’| = r|PQ|.  How could your students verify this result experimentally?

1.18 ACTIVITY 2 INTRODUCTION TO SIMILARITY (MIDDLE SCHOOL) Verifying the FTS  Discuss Lesson 6, Exercise 7 (page S.28).

1.19 ACTIVITY 3 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Definitions A similarity transformation is a sequence of (one or more) congruence transformations and dilations. Two geometric figures are similar if there is a similarity transformation which takes one of them onto the other.

1.20 ACTIVITY 2 INTRODUCTION TO SIMILARITY (MIDDLE SCHOOL) Understanding similarity  Discuss Lesson 8, Exercise 3 (page S.37).

1.21 ACTIVITY 3 THE CASES OF VICKY MANSFIELD AND NANCY EDWARDS  Read the two cases during your extended break. Consider as you read:  In The Case of Vicky Mansfield, make note of what students were learning in the class  In The Case of Nancy Edwards, which aspects of the effective Mathematics Teaching Practices did Ms. Edwards use that supported student learning?

Break

1.23 ACTIVITY 3 THE CASES OF VICKY MANSFIELD AND NANCY EDWARDS  When you read the Vicky Mansfield case, you kept track of what students were learning in Vicky’s class.  With your small groups, consider: What did Vicky’s students learn, and how did Vicky support their learning?  Use evidence from the case to support your claims.

1.24 ACTIVITY 3 THE CASES OF VICKY MANSFIELD AND NANCY EDWARDS  When you read the Nancy Edwards case, you made note of the Effective Mathematics Teaching Practices that Ms. Edwards used that supported student learning.  With your small groups, identify two of the effective Mathematics Teaching Practices that you thought were the most critical for student learning and say why.  Use evidence from the case to support your claims.

1.25 ACTIVITY 3 THE CASES OF VICKY MANSFIELD AND NANCY EDWARDS  What can we learn from each of these teachers?  In what ways do these two cases illustrate the challenges of teaching reasoning-and-proving?

1.26 ACTIVITY 3 THE CASES OF VICKY MANSFIELD AND NANCY EDWARDS Reasoning and Proving Framework Mathematical Component Making Mathematical Generalizations Providing Support for Mathematical Claims Identifying a Pattern Making a Conjecture Providing a Proof Providing a Non-Proof Argument Plausible Pattern Definite Pattern Conjecture Generic Example Demonstration Empirical Argument Rationale Learner Component What are students’ perceptions of the mathematical nature of a pattern / conjecture / proof / non-proof argument? Pedagogical Component How does the mathematical nature of a pattern / conjecture / proof / non-proof argument compare with students’ perception of this nature? How can teachers help their students reconsider and change (if necessary) their perceptions to better approximate the mathematical nature of a pattern/conjecture/proof/non-proof argument?

1.27 ACTIVITY 4 TEACHER DISCOURSE MOVES

1.28 ACTIVITY 4 TEACHER DISCOURSE MOVES Productive discourse: provides access to mathematical content and discourse practices Powerful discourse: supports students’ identities as knowers and doers of mathematics Waiting providing students with time to process teacher questions and think about their responses, including waiting after a student responds Inviting student participation soliciting multiple strategies for the same answer, determining how many students arrived at their answers, making diverse solutions available for public consideration Revoicing restating or rephrasing a student’s contribution, including checking back with the original speaker to see if the devoicing captured the student’s meaning Asking students to revoicerequiring students to listen to one another and revoice other student’s ideas in their own words Probing a student’s thinking following up on an individual student’s solution, strategy, or question to have the student elaborate on his or her ideas Creating opportunities to engage with another’s reasoning asking students explicitly to engage with another student’s idea, such as using another student’s strategy to solve a similar problem or to agree/disagree

Lunch

1.30 ACTIVITY 5 APPLICATIONS OF SIMILARITY (MIDDLE SCHOOL) PROPERTIES OF SIMILARITY TRANSFORMATIONS; TRIANGLE SIMILARITY ENGAGE NY /COMMON CORE GRADE 8, MODULE 3, LESSONS 9 & 10

1.31 ACTIVITY 5 APPLICATIONS OF SIMILARITY (MIDDLE SCHOOL) Symmetric property of similarity  Complete the Lesson 9 Exploratory Challenge 1 (page S.43).  Explain why, for any two plane figures S and T, if S is similar to T, then also T is similar to S.

1.32 ACTIVITY 5 APPLICATIONS OF SIMILARITY (MIDDLE SCHOOL) Transitive property of similarity  Complete the Lesson 9 Exploratory Challenge 2 (page S.45).  Explain why, for any three plane figures S, T, and U, if S is similar to T, and T is similar to U, then also S is similar to U.

1.33 ACTIVITY 5 APPLICATIONS OF SIMILARITY (MIDDLE SCHOOL) Reflexive property of similarity  Explain why any plane figures S is similar to itself.

1.34 ACTIVITY 5 APPLICATIONS OF SIMILARITY (MIDDLE SCHOOL) Triangle similarity  Our goal is to show that any two triangles with equal angles are similar.  Do we need the triangles to have three equal angles? Would they have to be similar if they only had one equal angle? Two equal angles?

1.35  Use a straightedge and protractor to draw two triangles with two equal angles. Do your triangles appear to be similar?  Read the “more formal” proof of the special case of the AA similarity criterion in the middle of page 139 (in the teacher materials). Do you find it more convincing than your experimental evidence? Would your students? ACTIVITY 5 APPLICATIONS OF SIMILARITY (MIDDLE SCHOOL) Angle-angle (AA) criterion for triangle similarity

1.36 Read the Grade 8 Similarity standards from the CCSSM.  Which of these standards have we addressed in today’s content sessions? ACTIVITY 5 APPLICATIONS OF SIMILARITY (MIDDLE SCHOOL) Reflecting on CCSSM standards alignment

1.37 ACTIVITY 6 PEER TEACHING PLANNING

1.38 Work with your teaching partners to plan your lesson. ACTIVITY 7 PEER TEACHING PLANNING

1.39 ACTIVITY 8 DAILY JOURNAL

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

1.41  Complete Problems 1 and 2 from the Grade 8 Module 3 Lesson 9 Problem Set in your notebook (page S.47-S.49)  Extending the mathematics: We identified 3 properties of the relation of similarity: symmetry, transitivity, and reflexivity. Does the relation of congruence have these same properties? What other relations (not necessarily geometric relations) can you think of that have these same three properties? Can you think of any relations that have some, but not all, of the properties?  Reflecting on teaching: Think back to definitions of congruence and similarity you have learned or taught in the past. How do they differ from the CCSSM definition? What advantages and disadvantages do you see in each? ACTIVITY 9 HOMEWORK AND CLOSING REMARKS