NYS COMMON CORE MATHEMATICS CURRICULUM A Story of

NYS COMMON CORE MATHEMATICS CURRICULUM Participant Poll • • • Classroom teacher Math trainer

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Session Objectives • Examination of:

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda • Orientation to Materials

NYS COMMON CORE MATHEMATICS CURRICULUM What’s In a Module? Teacher Materials • Module Overview

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Types of Lessons 1. Problem

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What’s In a Lesson? Teacher

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Changes in CCSS that

NYS COMMON CORE MATHEMATICS CURRICULUM January 2014 Geometry Regents © 2012 Common Core, Inc.

NYS COMMON CORE MATHEMATICS CURRICULUM A Checklist of “rules”: © 2012 Common Core, Inc.

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 1: Congruence, Proof, and

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A: Basic Constructions •

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 1: Construct an Equilateral

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1, Example 1 © 2012 Common Core, Inc.

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1, Example 2 © 2012 Common Core, Inc.

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B: Unknown Angles •

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B: Unknown Angles Lesson 8, Exercise 2: Lesson

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B: Unknown Angles What can be established about

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 10, Unknown Angle Proofs-

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C: Transformations/Rigid Motions •

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Rigid Motions in Grade 8

NYS COMMON CORE MATHEMATICS CURRICULUM Grade 8, Module 2, Lesson 4 • Grade 8

NYS COMMON CORE MATHEMATICS CURRICULUM (Grade 10) Lesson 14: Reflections • Exploratory Challenge ©

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Reflections and the Perpendicular Bisector

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14: Reflections • Example 1 © 2012 Common

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14: Reflections Definition of Reflection: © 2012 Common

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14: Reflections Application of Reflection: © 2012 Common

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 19: Construct and Apply

NYS COMMON CORE MATHEMATICS CURRICULUM Topic D: Congruence • • • Standards G-CO. 7,

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 22: Congruence Criteria for

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 26: Triangle Congruency Proofs

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic G: Axiomatic Systems •

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Example from Mid-Module Assessment ©

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Themes of Module 1:

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 2: Similarity, Proof, and

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A: Scale Drawings •

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2: Ratio Method © 2012 Common Core, Inc.

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2: Ratio Method, Example 1 © 2012 Common

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2, Example 2 © 2012 Common Core, Inc.

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3: Parallel Method, Example 2 © 2012 Common

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3, Example 2 © 2012 Common Core, Inc.

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consequences of the Ratio and

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B: Dilations • •

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: How Do Dilations

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C: Similarity and Dilations

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: What are Similarity

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 15: Angle-Angle Criterion for

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 17: SAS and SSS

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 19 and 20: Modeling

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic D: Applying Similarity to

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 21: Special Relationships with

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 22: Proving the Pythagorean

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 23 and 24: Multiplying

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic E: Trigonometry • •

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25: Incredibly Useful Ratios Quote from Lesson 25:

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 27: Sine and Cosine

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 28 -29: Solving missing

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions L 28, Exercise 6 ©

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions L 29, Example 2 ©

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Themes of Module 2:

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Biggest Takeaway • • What

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NYS COMMON CORE MATHEMATICS CURRICULUM Participant Poll • • • Classroom teacher Math trainer Principal or school leader District representative / leader Other © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Session Objectives • Examination of: • The development of mathematical understanding across the module using a focus on concept development within the lessons. • Examples that highlight themes and changes according to the Common Core State Standards. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Agenda • Orientation to Materials (if needed) • Review of Module 1: Congruence, Proof, and Constructions - Themes and Examples • Module 2: Similarity, Proof, and Trigonometry - Themes and Examples © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM What’s In a Module? Teacher Materials • Module Overview • Topic Overviews • Daily Lessons • Assessments Student Materials • Daily Lessons with Problem Sets Copy Ready Materials • Exit Tickets • Assessments © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Types of Lessons 1. Problem Set Students and teachers work through examples and complete exercises to develop or reinforce a concept. 2. Socratic Teacher leads students in a conversation to develop a specific concept or proof. 3. Exploration Independent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge. 4. Modeling Students practice all or part of the modeling cycle with real-world or mathematical problems that are ill-defined. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What’s In a Lesson? Teacher Materials Lessons • Student Outcomes and Lesson Notes (in select lessons) • Classwork • General directions and guidance, including timing guidance • Bulleted discussion points with expected student responses • Student classwork with solutions (boxed) • Exit Ticket with Solutions • Problem Set with Solutions Student Materials • Classwork • Problem Set © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Changes in CCSS that Impact M 1 & M 2 “The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally)… In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries onto the other. ” Common Core State Standards for Mathematics, p 74 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Changes in CCSS that Impact M 1 & M 2 Transformations • How they are first introduced • The manner in which they are described and studied • Their use in the definition of congruence and similarity • Other uses to which transformations are put, e. g. , reasoning and steps in proofs © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 1: Congruence, Proof, and Constructions Topic A: Basic Constructions Topic B: Unknown Angles Topic C: Transformations/Rigid Motions Topic D: Congruence Topic E: Proving Properties of Geometric Figures Topic F: Advance Constructions Topic G: Axiomatic Systems © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A: Basic Constructions • Standards G-CO. 1, G-CO. 12, G-CO. 13 • • Constructions How to: • • • construct an equilateral triangle copy an angle bisect an angle construct a perpendicular bisector determine the incenter and circumcenter of a triangle © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 1: Construct an Equilateral Triangle Flow of the lesson: • Opening Exercise: (1) A question that jogs students ideas on distance (2) key vocabulary review • Example 1: Guide students to discovering how to construct an equilateral triangle (with the use of a compass and straightedge). • Example 2: Read an excerpt from Elements that describes Euclid’s approach to constructing an equilateral triangle; have students develop their own set of steps on how to construct an equilateral. • Geometry Assumptions*: Introduce the notion that geometry is an axiomatic system- everything in the subject can be traced back to a short list of basic assumptions we take for granted. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B: Unknown Angles • • • Standards G-CO. 9 Breakdown of Lessons 6 -11: Lessons 6 -8: Unknown angle problems involving angles and lines at a point, transversals, and angles in a triangle. Students use simple justification in non-algebraic steps. Lessons 9 -11: Unknown angle proofs, non-numeric problems emphasizing justification of a relationship, sometimes a known existing relationship. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Topic B: Unknown Angles Lesson 8, Exercise 2: Lesson 9, Discussion: © 2012 Common Core, Inc. All rights reserved. commoncore. org What can be established about x, y, and z? A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 10, Unknown Angle Proofs- Proofs with Constructions Problem Set, 1 AB DE and BC EF. Prove that m ABC = m DEF. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C: Transformations/Rigid Motions • • • Standards G-CO. 2, G-CO. 3, G-CO. 4, G-CO. 5, G-CO. 6, G-CO. 7 G-CO. 12 Key concepts in Lessons 12 -21: Formal definition of each rigid motion (rotation, reflection, translation) and how each rigid motion is related to constructions Compositions of rigid motions Applications of rigid motions: discussion regarding correspondence, how congruence is defined © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Rigid Motions in Grade 8 • • • What knowledge do students arrive with? Rigid motions first introduced in Grade 8, Module 2 G 8 focus: • Understanding of what a transformation is • Intuitive understanding and experimental verification of the properties of the rigid motions (translation, reflection, and rotation). • Descriptive definitions • Hands-on exploration with the use of transparencies © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 19: Construct and Apply a Sequence of Rigid Motions We define two figures in the plane as congruent if there exists a finite composition of basic rigid motions that maps one figure onto the other. • • The idea of “Same size, same shape” only paints a mental picture; not specific enough. It is also not enough to say that two figures are alike in all respects except position in the plane. A congruence gives rise to a correspondence. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic G: Axiomatic Systems • • • Standards All A mathematician’s story: Many years ago I taught calculus for business majors. I started my class with an offer: "We have to cover several chapters from the textbook and there approximately forty formulas. I may offer you a deal: you will learn just four formulas and I will teach you how to get the rest out of these formulas. " The students gladly agreed. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Themes of Module 1: Congruence, Proof, and Constructions • Module is anchored by the definition of congruence • Emphasis is placed on extending the meaning and use of vocabulary in constructions • There is an explicit recall and application of facts learned over the last few years in unknown angle problems and proofs • Triangle congruence criteria are indicators that a rigid motion exists that maps one triangle to another; each criterion can be proven to be true with the use of rigid motions. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Module 2: Similarity, Proof, and Trigonometry Topic A: Scale Drawings Topic B: Dilations Topic C: Similarity and Dilations Topic D: Applying Similarity to Right Triangles Topic E: Trigonometry © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic A: Scale Drawings • • • Standards G-SRT. 1, G-SRT. 4 Key Concepts Two ways two construct scale drawings: Ratio Method and Parallel Method Triangle Side Splitter Theorem Dilation Theorem © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consequences of the Ratio and Parallel Methods • Students prove that the Ratio and Parallel Methods produce the same scale drawing for a give figure • Resulting theorems used frequently in Topic B: • Triangle Side Splitter Theorem. A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side. • Dilation Theorem. If a dilation with center O and scale factor r sends point P to P’ and Q to Q’, then P’Q’ = r PQ. Furthermore, if r≠ 1, and O, P, and Q are the vertices of a triangle then PQ P’Q’. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic B: Dilations • • Standards G-SRT. 1, G-SRT. 4 Key Concepts Proving properties of dilations, including how it is that dilations map segments to segments, lines to lines, rays to rays, and angles to angles. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 7: How Do Dilations Map Segments? Example 3 • To show that dilations map segments to segments, call on: • The definition of a dilation • The Triangle Side Splitter Theorem • Construction of a ray © 2012 Common Core, Inc. All rights reserved. commoncore. org For r < 1

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic C: Similarity and Dilations • • • Standards G-SRT. 2, G-SRT. 3, G-SRT. 4, G-SRT. 5 Key Concepts Similarity Transformation. The composition of a finite number of dilations and/or rigid motions. Two figures are said to be similar if a similarity transformation exists maps one figure onto another. A congruence is a similarity with scale factor 1. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 12: What are Similarity Transformations and Why Do We Need Them? Example 1 • Describe a similarity transformation that maps figure Z onto figure Z’. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar Discussion Theorem. Two triangles with two pairs of equal corresponding angles are similar. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar Theorem. Two triangles with two pairs of equal corresponding angles are similar. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 19 and 20: Modeling Lessons using Similarity Eratosthenes’ calculation of the Earth’s circumference Aristarchus’ calculation of the distance from the Earth to the Moon © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic D: Applying Similarity to Right Triangles • • • Standards G-SRT. 6, G-SRT. 7, G-SRT. 8 Key Concepts The special relationships that result when an altitude is drawn in a right triangle from the vertex of the right angle to the hypotenuse. Proving the Pythagorean Theorem using similarity Simplifying radical expressions © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 21: Special Relationships with Right Triangles. Dividing into Two Similar Sub-Triangles Discussion • We can use the AA criterion to show that each of the smaller two right triangles is similar to the large right triangle; similarity is transitive, so all three triangles are similar. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 21: Special Relationships with Right Triangles. Dividing into Two Similar Sub-Triangles © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 22: Proving the Pythagorean Theorem by Similarity Discussion • How can the ratios in right triangles (e. g. shorter leg: hypotenuse) be used to prove the Pythagorean Theorem? © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 22: Proving the Pythagorean Theorem by Similarity • Which right triangle ratios can be used to express a in terms of other variables? © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 22: Proving the Pythagorean Theorem by Similarity longer leg: hypotenuse © 2012 Common Core, Inc. All rights reserved. commoncore. org shorter leg: hypotenuse

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 23 and 24: Multiplying and Dividing Expressions with Radicals © 2012 Common Core, Inc. All rights reserved. commoncore. org Adding and Subtracting Expressions with Radicals

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Topic E: Trigonometry • • • Standards G-SRT. 6, G-SRT. 7, G-SRT. 8 Key Concepts Definition of sine, cosine, tangent Relationship between sine and cosine Application of the trig ratios to solve missing value, real world, and area problems. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25: Incredibly Useful Ratios Quote from Lesson 25: “In fact if, since we know the values of these ratios are constant for a given angle measure, we could compile a table of every acute angle between 0˚ and 90˚ and list the values of the adj/hyp and opp/hyp ratios with respect to each acute angle measure. ” © 2012 Common Core, Inc. All rights reserved. commoncore. org A Story of Functions

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lesson 27: Sine and Cosine of Complementary Angles and Special Angles • What conclusions can be drawn based on each of the ratios? © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Lessons 28 -29: Solving missing value problems • • • Selected examples to try: Lesson 28: Problems using Sine and Cosine, Exercise 6 Lesson 29: Applying Tangents, Example 2 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Themes of Module 2: Similarity, Proof, and Trigonometry • Just as rigid motions are used to define congruence, so dilations are added to define similarity. • A close look at the way dilations are constructed yields information about how dilations behave. • Right triangle similarity is built on the concepts of withintriangle ratios and between-triangle ratios leading to an understanding of trigonmetric ratios. © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Biggest Takeaway • • What is your biggest takeaway with respect to Modules 1 and 2? How can you support successful implementation at your school/s given your role? © 2012 Common Core, Inc. All rights reserved. commoncore. org