exemplar module analysis

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Functions

Exemplar Module AnalysisGrade 10 – Module 1


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Functions

Session Objectives:

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• Understand the role of transformations under the CCSS



AGENDA

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• Transformations: Then and Now• Coherence from Grade 8• Examples



What is the major change in Geometry?

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• 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



What are the types of questions that come to mind when you think of transformations?

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• 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



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Consider this checklist of “rules” as you brainstorm:



Past assessment questions on transformations

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January 2013Geometry Regents



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January 2013Geometry Regents

Past assessment questions on transformations



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What do these questions have in common?

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• 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



Transformations: Then and Now

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Then NowBound to the coordinate plane Coordinate-free (in 10th 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



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AGENDA

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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.



Example: Understanding a Rotation in Grade 8

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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



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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.



Example: Understanding a Rotation in Grade 10

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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˚.

Instruction emphasizes precision in language



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Grade 8 vs. Grade 108th Grade 10th 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˚. ∠



AGENDA

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Triangle Congruence Criteria: Proving S-A-S

#1 Translate

#2 Rotate #3 Reflect



Example: Rotations and Potential Questions

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• 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.



Assessment question: G.CO.5, G.CO.12

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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.




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Assessment Question Rubric

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Key Points

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• 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

exemplar module analysis

Documents

role of transformations

textbook question

module 1agenda3transformations

state assessment question

ccss views geometry

existing curriculum

major change

notable change