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

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

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

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

5. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What are the types of questions that come to mind when you think of transformations? 5 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

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

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

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

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

10. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions What do these questions have in common? 10 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

11. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Transformations: Then and Now 11 ThenNow Bound to the coordinate planeCoordinate-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 size and same shape Congruent: 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

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

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

14. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions 14 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.

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

16. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions 16 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.

17. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Example: Understanding a Rotation in Grade 10 17 The rotation of θ degrees around C (or the center C) is the transformation R C,θ defined as follows: 1.For the center point C, R C,θ (C) = C, and 1.For any other point P, R C,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˚. Instruction emphasizes precision in language

18. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions 18 Grade 8 vs. Grade 10 8 th Grade10 th Grade Rotation: Intuitive understanding Rotation: Precise definition The rotation of θ degrees around C (or the center C) is the transformation R C,θ defined as follows: 1. For the center point C, R C,θ (C) = C, and 2. For any other point P, R C,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˚.

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

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

21. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Example: Rotations and Potential Questions 21 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.

22. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessment question: G.CO.5, G.CO.12 22 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, G.CO.12 23 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, G.CO.12 24 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, G.CO.12 25 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: G.CO.5, G.CO.12 26 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.

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

28. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Points 28 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