Welcome to Common Core High School Mathematics Leadership Summer Institute 2015 Session 1 • 15 June 2015 Getting the Big Picture & congruence in grade 8

Today’s Agenda Introductions, norms and administrative details Paperwork, surveys, and assessments Rigid motions and congruence (Grade 8) Dinner Applications of congruence (Grade 8) Reflecting on CCSSM standards aligned to Grade 8 congruence Break Considering the nature of proof Daily journal Homework and closing remarks Timing for our purposes: Introductions, norms, admin: 3:30 Meghan with paperwork, surveys, and MKT items: 3:45 Content session 1: 4:45 Dinner: 5:45 (work on first proof task while you eat?) Content session 2: 6:15; reflection 7:15 Break: 7:25 Three proof tasks: 7:35 Daily journal: 8:20 Homework: 8:30

Activity 1 introductions, norms and administrative details Where do you teach and what do you teach? What are your learning goals for the summer? About mathematics content About mathematics teaching

Activity 1 introductions, norms and administrative details Start on Time  End on Time Name Tents Attention signal Raise hand!! Silence cell phones. Technology use for learning only please! Restrooms No sidebar conversations . . . Food Administrative fee

Activity 2 MKT Assessment Geometry knowledge assessment

Activity 2 Geometry Knowledge Assessment MKT Assessment Please give brief explanations for your rationale on the multiple choice questions. Link for online survey: http://bit.ly/15Geom

by Conceptual Categories Standards for Mathematical Practice Standards for Mathematics Content Make sense of problems & persevere in solving them Reason abstractly & quantitatively Construct viable arguments & critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for & make use of structure Look for & express regularity in repeated reasoning K–8 Standards by Grade Level High School Standards by Conceptual Categories ________________________ Domains Clusters Standards

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 6. Attend to precision. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. Reasoning and Explaining 4. Model with mathematics. 5. Use appropriate tools strategically. Modeling and Using Tools Seeing Structure and Generalizing 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Which of these do you see as most relevant to geometry? William McCallum, The University of Arizona

K-8 Domains & HS Conceptual Categories William McCallum, The University of Arizona

AN overview of the high school conceptual categories Algebra Functions Modeling Number & Quantity Geometry Statistics & Probability

Why the Engageny/Common Core/ eureka math books? First curriculum designed from the ground up for Common Core (not an existing curriculum “aligned” to the standards) Features tasks of high-cognitive demand that require students to think, reason, explain, justify, and collaborate Contains substantial teacher implementation support resources: lesson plans, notes on student thinking, assessments and rubrics Developed by an exceptional group of educators Source material is available free and could be integrated with existing programs your district may have

Learning Intentions and Success Criteria We are learning … to recognize and describe basic rigid motions; the definition of congruence in terms of basic rigid motions; the CCSSM Grade 8 expectations for congruence

Learning Intentions and Success Criteria We will be successful when we can: use appropriate language to describe the rigid motion (or sequence of rigid motions) that take one figure onto another; decide whether or not two figures are congruent; explain the CCSSM Grade 8 congruence standards.

Activity 3 Rigid Motions and congruence (Grade 8) definitions and properties of basic rigid motions EngageNY/Common Core Grade 8, Lessons 1, 2, 4, 5, 10 & 11

Activity 3 rigid motions and congruence (grade 8) Describe, intuitively, what kind of transformation will be required to move the figure on the left to each of the figures (1)-(3) on the right. To help with this exercise, use a transparency to copy the figure on the left. Note: begin by moving the left figure to each of the positions (1), (2) and (3). © 2014 Common Core, Inc.

Activity 3 rigid motions and congruence (grade 8) Given two segments AB and CD, which could be very far apart, how can we tell if they have the same length without measuring them individually? Do you think they have the same length? How can you check? © 2014 Common Core, Inc.

Activity 3 rigid motions and congruence (grade 8) Vocabulary A transformation of the plane is a rule that assigns to each point of the plane exactly one (unique) point. If we denote the transformation by F, we use F(P) to denote the unique point assigned to P by F. (We will sometimes write P’ instead of F(P).) The point F(P) will be called the image of P by F. We also say F maps P to F(P). If given any two points P and Q, the distance between the images F(P) and F(P) is the same as the distance between P and Q, then the transformation F preserves distance, or is distance-preserving. A distance-preserving transformation is called a rigid motion (or an isometry); the name suggests that it moves the points of the plane around in a rigid fashion.

Activity 3 rigid motions and congruence (grade 8) Turn and talk: What is a translation? A reflection? A rotation?

Activity 3 rigid motions and congruence (grade 8) Vocabulary A vector is a line segment with a specified direction If the segment is AB, and the specified direction is from A to B, we denote the corresponding vector by . Vector Vector

Activity 3 rigid motions and congruence (grade 8) Let’s play! Use transparencies to model: A translation (along a given vector ) A reflection (across a given line) A rotation (of a given angle about a given point) Note: translations, reflections, and rotations, are collectively known as basic rigid motions.

Activity 3 rigid motions and congruence (grade 8) Properties of basic rigid motions Explain why any basic rigid motion has the following three properties: It takes lines to lines, segments to segments, rays to rays, and angles to angles; It preserves distances between points (and so also lengths of segments); It preserves the degree measure of angles.

Activity 3 rigid motions and congruence (grade 8) Prove it! Let P’ be the image of P by a reflection in line L. Use the properties of basic rigid motions to explain why L must be the perpendicular bisector of the segment PP’. L P’ . . P

Activity 3 rigid motions and congruence (grade 8) Definitions A congruence transformation (or just congruence, for short) is a sequence of (one or more) basic rigid motions. Two geometric figures are congruent if there is a congruence which takes one of them onto the other.

Activity 3 rigid motions and congruence (grade 8) Properties of congruence transformations Explain why any congruence transformation has the same three properties as the basic rigid motions: It takes lines to lines, segments to segments, rays to rays, and angles to angles; It preserves distances between points (and so also lengths of segments); It preserves the degree measure of angles.

Activity 3 applications of congruence (grade 8) Explore Complete Lesson 10, Exercises 1 through 4. (Pages S.51 & S.52) Discuss Lesson 10 Problem Set 1, Problem 1 (Page S.54)

Activity 3 rigid motions and congruence (grade 8) Vocabulary A transformation of the plane is a rule that assigns to each point of the plane exactly one (unique) point. If we denote the transformation by F, we use F(P) to denote the unique point assigned to P by F. (We will sometimes write P’ instead of F(P).) The point F(P) will be called the image of P by F. We also say F maps P to F(P). If given any two points P and Q, the distance between the images F(P) and F(P) is the same as the distance between P and Q, then the transformation F preserves distance, or is distance-preserving. A distance-preserving transformation is called a rigid motion (or an isometry); the name suggests that it moves the points of the plane around in a rigid fashion.

Dinner

Activity 4 What is Proof? Please complete the “What is Proof?” questions as you eat.

Activity 5 applications of congruence (Grade 8) Angles associated with parallel lines and triangles EngageNY/Common Core Grade 8, Lessons 12 & 13

Activity 5 applications of congruence (grade 8) Prove it! Vertical angles, such as angle 1 and angle 3 in the diagram, are congruent, and so have equal degree measure. 1 2 4 3

Activity 5 applications of congruence (grade 8) Prove it! If lines L1 and L2 are parallel, then corresponding angles, such as angles 1 and 5, are congruent, and so have equal degree measure. Moreover, alternate interior angles, such as angles 4 and 6, are congruent. 1 2 4 3 5 6 8 7

Activity 5 applications of congruence (grade 8) Prove it! The sum of the degree measures of the three angles in any triangle is 180O. A B C D Hint. Start by drawing line CE parallel to line AB.

Activity 5 applications of congruence (grade 8) Reflecting on CCSSM standards aligned to lesson 1 Read MP3, the third 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 MP3? Are there other standards for mathematical practice that were prominent as you and your groups worked on the tasks?

Activity 5 applications of congruence (grade 8) CCSSM MP.2 EngageNY MP.2 MP.2 Reason abstractly and quantitatively Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MP.2 Reason abstractly and quantitatively. This module is rich with notation that requires students to decontextualize and contextualize throughout. Students work with figures and their transformed images using symbolic representations and need to attend to the meaning of the symbolic notation to contextualize problems. Students use facts learned about rigid motions in order to makes sense of problems involving congruence.

Activity 5 applications of congruence (grade 8) CCSSM MP.3 EngageNY MP.3 MP.3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. MP.3 Construct viable arguments and critique the reasoning of others. Throughout this module, students construct arguments around the properties of rigid motions. Students make assumptions about parallel and perpendicular lines and use properties of rigid motions to directly or indirectly prove their assumptions. Students use definitions to describe a sequence of rigid motions to prove or disprove congruence. Students build a logical progression of statements to show relationships between angles of parallel lines cut by a transversal, the angle sum of triangles, and properties of polygons like rectangles and parallelograms.

Break

Activity 6 The Nature of Proof

Activity 6 Three Proof Tasks How many different 3-by-3 squares are in the 4-by-4 square shown? How many different 3-by-3 squares are there in a 5-by-5 square? How many different 3-by-3 squares are there in a 60-by-60 square? Are you sure that your answer is correct? Why?

Activity 6 Three Proof Tasks Place different numbers of spots around a circle and join each pair of spots by straight lines. Explore a possible relation between the number of spots and the greatest number of non-overlapping regions into which the circle can be divided by this means. When there are 15 spots around the circle, is there an easy way to tell for sure what is the greatest number of non-overlapping regions into which the circle can be divided? (Geogebra or Geometer’s Sketchpad may be helpful here.)

Activity 6 Three Proof Tasks Discuss the student statement:
 This problem teaches us that checking 5 cases is not enough to trust a pattern in a problem. Next time I work with a pattern problem, I’ll check 20 cases to be sure.

Activity 6 Three Proof Tasks Does the expression 1 + 1141n2 (where n is a natural number) ever give a square number? No… until you get to 30,693,385,322,765,657,197,397,207

Activity 7 daily journal

Activity 7 daily journal Take a few moments to reflect and write on today’s activities.

Activity 8 Homework and Closing Remarks During Week 2 (June 29-July 2), together with a small group of your peers, you will teach one of the following lessons (or some other one of your choice, with the approval of the course facilitators): 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. You will be asked to select your preferred lesson tomorrow…

Activity 8 Homework and Closing Remarks Complete Problems 1, 2 and 3 from the Grade 8 Module 2 Lesson 13 Problem Set in your notebook (pages S.69-S.70) Extending the mathematics: Can you find an example of a congruence that is not just a single basic rigid motion? Can you prove that your example is not one of the basic types? Reflecting on teaching: Consider a typical class of 8th grade students in your district. How would their understanding of and experiences with congruence compare with what you have see today?