NYS COMMON CORE MATHEMATICS CURRICULUM A Story of

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NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Exemplar Module Analysis Grade 10

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Exemplar Module Analysis Grade 10 – Module 1 © 2012 Common Core, Inc. All rights reserved. commoncore. org

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

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Session Objectives: • Understand the role of transformations under the CCSS 2 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • Transformations: Then and

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • Transformations: Then and Now • Coherence from Grade 8 • Examples 3 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What is the major change

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What is the major change in Geometry? • 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 4 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What are the types of

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What are the types of questions that come to mind when you think of transformations? • Recall a state assessment question, or a textbook question • Share the question with your neighbor • Discuss the skills students need to successfully complete the question • Discuss how you delivered the content 5 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consider this checklist of “rules”

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consider this checklist of “rules” as you brainstorm: 6 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Past assessment questions on transformations

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Past assessment questions on transformations January 2013 Geometry Regents 7 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Past assessment questions on transformations

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Past assessment questions on transformations January 2013 Geometry Regents 8 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consider this checklist of “rules”

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consider this checklist of “rules” as you brainstorm: 9 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What do these questions have

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What do these questions have in common? • The transformations are anchored in the coordinate plane • A set number of transformations exist • Transformations are performed relative to the origin or an axis • There are several seemingly isolated rules to memorize • The “answer”, the full purpose, is to locate where the figure is after the transformation 10 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Transformations: Then and Now Then

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Transformations: Then and Now Then Now Bound to the coordinate plane Coordinate-free (in 10 th grade) Transformations existed as a special topic, and only select transformations are examined Transformations are understood to be abundant - Predict the effect of a transformation - Identify the transformation that yields a particular result ‘Then’ + We use transformations to define congruence and similarity, and we use them as tools for reasoning and proof Congruent: Two figures are congruent if they have the same Two figures in a plane are congruent if there size and same shape exists a finite composition of basic rigid motions that maps one figure onto the other figure 11 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consider this checklist of “rules”

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Consider this checklist of “rules” as you brainstorm: 12 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • Transformations: Then and

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • Transformations: Then and Now • Coherence from Grade 8 • Examples 13 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 8 Geometry: Foundations for

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 8 Geometry: Foundations for Grade 10 • Build an intuitive idea of how each rigid motion behaves with the help of manipulatives (8. G. 1). • For example, transparencies easily illustrate what makes rigid motions “rigid. ” • Learn to pay attention to specific aspects of these experiences and to describe them in precise ways (8. G. 1). • For example, rigid motions “preserve lengths of segments and measure of angles. ” • Differentiate between the mathematical concept of transformation and closely-related common-sense concepts. • For example, a transformation in the mathematical sense operates on all points of the plane; the motions that we apply to a model cannot fully capture this. 14 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Example: Understanding a Rotation in

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Example: Understanding a Rotation in Grade 8 Instruction emphasizes observation and an intuitive understanding A rotation around a given point C of a fixed degree Teaching Geometry According to the Common Core Standards http: //math. berkeley. edu/~wu/ 15 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 10 Geometry: Module 1

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 10 Geometry: Module 1 • The intuitive understanding of rigid motions and congruence and the relationship between are made fully explicit and precise through mathematical definitions (G. CO. 4). • Students learn each rigid motion in exact terms, manipulate the rigid motions individually and in sequence, and culminate in the definition of congruence (G. CO. 6). • Two figures in a plane are congruent if there exists a finite composition of basic rigid motions that maps one figure onto the other figure. • The journey leading to congruence in Module 1 is supported by the Mathematical Practice 6—Attend to precision. 16 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Example: Understanding a Rotation in

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Example: Understanding a Rotation in Grade 10 Instruction emphasizes precision in language The rotation of θ degrees around C (or the center C) is the transformation RC, θ defined as follows: 1. For the center point C, RC, θ(C) = C, and 1. For any other point P, RC, θ(P) is the point Q on the circle with center C and radius CP found by turning in a counterclockwise direction along the circle from P to Q such that ∠QCP = θ˚. 17 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 8 vs. Grade 10

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 8 vs. Grade 10 8 th Grade 10 th Grade Rotation: Intuitive understanding Rotation: Precise definition The rotation of θ degrees around C (or the center C) is the transformation RC, θ defined as follows: 1. For the center point C, RC, θ (C) = C, and 2. For any other point P, RC, θ (P) is the point Q on the circle with center C and radius CP found by turning in a counterclockwise direction along the circle from P to Q such that ∠QCP = θ˚. 18 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • Transformations: Then and

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions AGENDA • Transformations: Then and Now • Coherence from Grade 8 • Examples 19 © 2012 Common Core, Inc. All rights reserved. commoncore. org

A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM Triangle Congruence Criteria: Proving S-A-S

A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM Triangle Congruence Criteria: Proving S-A-S #1 Translate #2 Rotate #3 Reflect 20 © 2012 Common Core, Inc. All rights reserved. commoncore. org

A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM Example: Rotations and Potential Questions

A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM Example: Rotations and Potential Questions • Determine the center of rotation using the necessary constructions. • Determine the angle of rotation. • Given a center of rotation, name one of the angles that measures the angle of rotation. • Given the original figure, apply a rotation of 32˚ about a center of your choice. 21 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5,

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5, G. CO. 12 In the figure below, there is a reflection that transforms △ABC to triangle △A'B'C'. Use a straightedge and compass to construct the line of reflection and list the steps of the construction. 22 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5,

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5, G. CO. 12 In the figure below, there is a reflection that transforms △ABC to triangle △A'B'C'. Use a straightedge and compass to construct the line of reflection and list the steps of the construction. 23 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5,

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5, G. CO. 12 In the figure below, there is a reflection that transforms △ABC to triangle △A'B'C'. Use a straightedge and compass to construct the line of reflection and list the steps of the construction. 24 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5,

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5, G. CO. 12 In the figure below, there is a reflection that transforms △ABC to triangle △A'B'C'. Use a straightedge and compass to construct the line of reflection and list the steps of the construction. 25 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5,

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G. CO. 5, G. CO. 12 In the figure below, there is a reflection that transforms △ABC to triangle △A'B'C'. Use a straightedge and compass to construct the line of reflection and list the steps of the construction. 26 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment Question Rubric 27 ©

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment Question Rubric 27 © 2012 Common Core, Inc. All rights reserved. commoncore. org

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Points • The major

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Points • The major change in geometry under the CCSS is the role of transformations, and the expectations of their presentation in 8 th and 10 th grade • Transformations: • Serve as the foundation for the concept of congruence • Are not limited to a “list of rules” • Are not based in the coordinate plane 28 © 2012 Common Core, Inc. All rights reserved. commoncore. org