Exemplar Module Analysis

1. Exemplar Module Analysis Grade 10 – Module 1

2. Session Objectives: • Understand the role of transformations under the CCSS

3. AGENDA • Transformations: Then and Now • Coherence from Grade 8 • Examples

4. 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

5. 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

6. Consider this checklist of “rules” as you brainstorm:

7. Past assessment questions on transformations January 2013Geometry Regents

8. Past assessment questions on transformations January 2013Geometry Regents

9. Consider this checklist of “rules” as you brainstorm:

10. 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

11. Transformations: Then and Now

12. Consider this checklist of “rules” as you brainstorm:

13. AGENDA • Transformations: Then and Now • Coherence from Grade 8 • Examples

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. 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/

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. Example: Understanding a Rotation in Grade 10 Instruction emphasizes precision in language The rotationof θ degrees around C (or the center C) is the transformation RC,θ defined as follows: For the center point C, RC,θ(C) = C, and 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. Grade 8 vs. Grade 10 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˚.

19. AGENDA • Transformations: Then and Now • Coherence from Grade 8 • Examples

20. Triangle Congruence Criteria: Proving S-A-S #1 Translate #3 Reflect #2 Rotate

21. 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.

22. 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. 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. 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. 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. 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.

27. Assessment Question Rubric

28. Key Points • The major change in geometry under the CCSS is the role of transformations, and the expectations of their presentation in 8th and 10th 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