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Geometry August 19, 2014 Geometry Common Core Test Guide Sample Items Old or New????? Old or New? Old or New? • Trees that are cut down and stripped of their branches for timber are approximately cylindrical. A timber company specializes in a certain type of tree that has a typical diameter of 50 cm and a typical height of about 10 meters. The density of the wood is 380 kilograms per cubic meter, and the wood can be sold by mass at a rate of $4.75 per kilogram. Determine and state the minimum number of whole trees that must be sold to raise at least $50,000. • The diameter of a sphere is 5 inches. Determine and state the surface area of the sphere, to the nearest hundredth of a square inch. Agenda Congruence • Geometry Regents Test Guide • Grade 8 CCSS-Geometry • Grade 8 Geometry Tasks • HS CCSS-Geometry • High School Geometry Tasks Similarity • Grade 8 CCSS-Geometry • Grade 8 Geometry Tasks • HS CCSS-Geometry • High School Geometry Tasks NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Key Changes in CCSS “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 assumed to preserve distance and angles… Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of "same shape" and "scale factor" developed in the middle grades.” Common Core State Standards for Mathematics, p74 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Regents example? © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM A Checklist of “rules”: © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions ? Congruence Building From Grade 8 Definition of Translation Definition of Reflection • The reflection R across a given line L, where L is called the line of reflection, assigns each point on L to itself, and to any point P not on L, R assigns the point R(P) which is symmetric to it with respect to L, in the sense that L is the perpendicular bisector of segment joining P to R(P) Definition of Rotation • • • • • • Rotations require information about the center of rotation and the degree in which to rotate. Positive degrees of rotation move the figure in a counterclockwise direction. Negative degrees of rotation move the figure in a clockwise direction. Basic Properties of Rotations: (R1) A rotation maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle. (R2) A rotation preserves lengths of segments. (R3) A rotation preserves degrees of angles. When parallel lines are rotated, their images are also parallel. A line is only parallel to itself when rotated exactly 180˚. • The rotation Ro of t degrees (-360 < t < 360) around a given point O, called the center of the rotation, is a transformation of the plane defined as follows. Given a point P, the point Ro(P) is defined as follows. The rotation is counterclockwise or clockwise depending on whether the degree is positive or negative, respectively. If P is distinct from O, then by definition, Ro(P) is the point Q on the circle with center O and radius |OP| so that <QOP = t and so that Q is in the counterclockwise direction of the point P. Geometry Course Overview Similarity NYS COMMON CORE MATHEMATICS CURRICULUM Grade 8 – Module 3 – Lesson 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM *Lesson 2, Example 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM Triangle Side Splitter Theorem: © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 8 – Module 3 •Lesson 4: Lined Paper Activity leads to an understanding of similarity. •Similarity in terms of Dilation •Given a dilation with center O and scale factor r, then for any two points P, Q in the plane (when points O, P, Q are not collinear), the lines PQ and P’Q’ are parallel, where P’ = Dilation(P) and Q’ = Dilation(Q), and furthermore, |P’Q’| = r |PQ|. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions *Lesson 5: Scale Factors Facts known: • Ratio and Parallel Methods produce the same scale drawing. • Triangle Side Splitter Theorem. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Grade 8 •Lesson 2: Properties of Dilations • Students informally verify the properties of dilations: • • • • Lines map to lines, Rays map to rays, Segments map to segments, and Angles map to angles of the same degree. Lesson 7: Informal Proofs of Properties of Dilations – Optional Lesson • *Informal proof that angles map to angles • Informal proofs that lines map to lines, rays to rays, segments to segments. © 2012 Common Core, Inc. All rights reserved. commoncore.org 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: Similarity transformations, the AA criterion, prove theorems about triangles, use congruence and similarity to prove relationships in geometric figures. 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 mapping 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 15: Angle-Angle Criterion for Two Triangles to be Similar Theorem. Two triangles with two pairs of equal corresponding angles are similar. Use your handout to outline a proof of the theorem. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM *Lesson 15, Discussion: © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions *Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM *Lesson 17: Discussion © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM *Lesson 17: Discussion © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions NYS COMMON CORE MATHEMATICS CURRICULUM © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Functions