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Geometry Instructional Supports for the Model Curriculum

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Geometry Instructional Supports for the Model Curriculum OHIO’S MODEL CURRICULUM WITH INSTRUCTIONAL SUPPORTS | Mathematics | 2018 1 Ohio’s Model Curriculum Mathematics with Instructional Supports Geometry Course High School Geometry Course OHIO’S MODEL CURRICULUM WITH INSTRUCTIONAL SUPPORTS | Mathematics | 2018 2 Mathematics Model Curriculum with Instructional Supports Geometry Course TABLE OF CONTENTS INTRODUCTION 8 STANDARDS FOR MATHEMATICAL PRACTICE—GEOMETRY 10 MODELING (★) 11 GEOMETRY (G) 13 CONGRUENCE (G.CO) 13 EXPERIMENT WITH TRANSFORMATIONS IN THE PLANE. (G.CO.1-5) 13 EXPECTATIONS FOR LEARNING 13 CONTENT ELABORATIONS 14 INSTRUCTIONAL STRATEGIES 15 • Van Hiele Connection 15 • Defining Geometric Terms 16 • Transformations as Functions 19 • Definitions of Rigid Motions 20 • Transformations That Preserve Distance and Those That Do Not 24 • Representing and Drawing Transformations 25 • Symmetry 27 INSTRUCTIONAL TOOLS/RESOURCES 29 UNDERSTAND CONGRUENCE IN TERMS OF RIGID MOTIONS. (G.CO.6-8) 32 EXPECTATIONS FOR LEARNING 32 CONTENT ELABORATIONS 33 INSTRUCTIONAL STRATEGIES 34 • Van Hiele Connection 34 • Congruence and Rigid Motions 34 • Corresponding Parts of Congruent Figures are Congruent 36 INSTRUCTIONAL TOOLS/RESOURCES 39 High School Geometry Course OHIO’S MODEL CURRICULUM WITH INSTRUCTIONAL SUPPORTS | Mathematics | 2018 3 CONGRUENCE, CONTINUED (G.CO) 42 PROVE GEOMETRIC THEOREMS BOTH FORMALLY AND INFORMALLY USING A VARIETY OF METHODS. (G.CO.9-11) 42 EXPECTATIONS FOR LEARNING 42 CONTENT ELABORATIONS 43 INSTRUCTIONAL STRATEGIES 44 • Van Hiele Connection 46 • Proofs 46 INSTRUCTIONAL TOOLS/RESOURCES 54 MAKE GEOMETRIC CONSTRUCTIONS. (G.CO.12-13) 57 EXPECTATIONS FOR LEARNING 57 CONTENT ELABORATIONS 58 INSTRUCTIONAL STRATEGIES 59 • Van Hiele Connection 59 • Making Formal Constructions 59 CLASSIFY AND ANALYZE GEOMETRIC FIGURES. (G.CO.14) 69 EXPECTATIONS FOR LEARNING 69 CONTENT ELABORATIONS 70 INSTRUCTIONAL STRATEGIES 71 • Van Hiele Connection 71 • Hierarchies 72 INSTRUCTIONAL TOOLS/RESOURCES 74 SIMILARITY, RIGHT TRIANGLES, AND TRIGONOMETRY (G.SRT) 76 UNDERSTAND SIMILARITY IN TERMS OF SIMILARITY TRANSFORMATIONS. (G.SRT.1-3) 76 EXPECTATIONS FOR LEARNING 76 CONTENT ELABORATIONS 77 INSTRUCTIONAL STRATEGIES 78 • Van Hiele Connection 78 • Verifying Properties of Dilations Experimentally 79 • Similarity Transformations 91 INSTRUCTIONAL TOOLS/RESOURCES 99 High School Geometry Course OHIO’S MODEL CURRICULUM WITH INSTRUCTIONAL SUPPORTS | Mathematics | 2018 4 SIMILARITY, RIGHT TRIANGLES, AND TRIGONOMETRY, CONTINUED (G.SRT) 102 PROVE AND APPLY THEOREMS BOTH FORMALLY AND INFORMALLY INVOLVING SIMILARITY USING A VARIETY OF METHODS. (G.SRT.4-5) 102 EXPECTATIONS FOR LEARNING 102 CONTENT ELABORATIONS 103 INSTRUCTIONAL STRATEGIES 104 • Van Hiele Connection 104 • Proving and Applying Theorems about Triangles 105 • Solving Problems Using Triangle Congruence and Similarity Criteria 117 INSTRUCTIONAL TOOLS/RESOURCES 123 DEFINE TRIGONOMETRIC RATIOS, AND SOLVE PROBLEMS INVOLVING RIGHT TRIANGLES. (G.SRT.6-8) 127 EXPECTATIONS FOR LEARNING 127 CONTENT ELABORATIONS 128 INSTRUCTIONAL STRATEGIES 129 • Van Hiele Connection 130 • Modeling (★) 130 • Definition of Trigonometric Ratios 130 • General Information about Trigonometric Ratios 141 • Solving Problems Using Right Triangles 144 INSTRUCTIONAL TOOLS/RESOURCES 149 CIRCLES (G.C) 153 UNDERSTAND AND APPLY THEOREMS ABOUT CIRCLES. (G.C.1-4) 153 EXPECTATIONS FOR LEARNING 153 CONTENT ELABORATIONS 155 INSTRUCTIONAL STRATEGIES 156 • Van Hiele Connection 156 • Similarity of Circles 156 • Relationships Among Parts of a Circle 159 • Central and Inscribed Angles 161 • Constructions with Circles 165 INSTRUCTIONAL TOOLS/RESOURCES 173 High School Geometry Course OHIO’S MODEL CURRICULUM WITH INSTRUCTIONAL SUPPORTS | Mathematics | 2018 5 CIRCLES, CONTINUED (G.C) 176 FIND ARC LENGTHS AND AREAS OF SECTORS OF CIRCLES. (G.C.5) 176 EXPECTATIONS FOR LEARNING 176 CONTENT ELABORATIONS 177 INSTRUCTIONAL STRATEGIES 178 • Arc Lengths 178 • Areas of Sectors of Circles 181 INSTRUCTIONAL TOOLS/RESOURCES 185 EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS (G.GPE) 187 TRANSLATE BETWEEN THE GEOMETRIC DESCRIPTION AND THE EQUATION FOR A CONIC SECTION. (G.GPE.1) 187 EXPECTATIONS FOR LEARNING 187 CONTENT ELABORATIONS 188 INSTRUCTIONAL STRATEGIES 189 • Van Hiele Connection 189 • Relating the Distance Formula and/or the Pythagorean Theorem to the Equation of a Circle 190 • Finding the Center and Radius of a Circle by Completing the Square 196 • Finding the Point(s) of Intersection of a Line and Circle 199 INSTRUCTIONAL TOOLS/RESOURCES 200 USE COORDINATES TO PROVE SIMPLE GEOMETRIC THEOREMS ALGEBRAICALLY AND TO VERIFY SPECIFIC GEOMETRIC STATEMENTS. (G.GPE.4-7) 202 EXPECTATIONS FOR LEARNING 202 CONTENT ELABORATIONS 203 INSTRUCTIONAL STRATEGIES 204 • Van Hiele Connection 205 • Geometric Theorems with Coordinates 205 • The Distance Formula 210 • Slope 211 • A Point on a Directed Line Segment between Two Given Points 214 • Modeling (★) 216 • Computing Perimeters and Areas of Polygons Using Coordinates 216 INSTRUCTIONAL TOOLS/RESOURCES 217 High School Geometry Course OHIO’S MODEL CURRICULUM WITH INSTRUCTIONAL SUPPORTS | Mathematics | 2018 6 GEOMETRIC MEASUREMENT AND DIMENSION (G.GMD) 221 EXPLAIN VOLUME FORMULAS, AND USE THEM TO SOLVE PROBLEMS. (G.GMD.1,3) 221 EXPECTATIONS FOR LEARNING 221 CONTENT ELABORATIONS 222 INSTRUCTIONAL STRATEGIES 223 • Van Hiele Connection 223 • Modeling (★) 223 • Informal Arguments for Formulas 224 • Using Formulas to Solve Problems 238 INSTRUCTIONAL TOOLS/RESOURCES 239 VISUALIZE RELATIONSHIPS BETWEEN TWO-DIMENSIONAL AND THREE-DIMENSIONAL OBJECTS. (G.GMD.4) 244 EXPECTATIONS FOR LEARNING 244 CONTENT ELABORATIONS 245 INSTRUCTIONAL STRATEGIES 246 • Van Hiele Connection 246 • Two-dimensional Cross Sections of Three-dimensional Objects 246 • Three-dimensional Objects Generated by Rotations of Two-Dimensional Objects 248 • Modeling Connection (★) 249 INSTRUCTIONAL TOOLS/RESOURCES 250 UNDERSTAND THE RELATIONSHIPS BETWEEN LENGTHS, AREAS, AND VOLUMES. (G.GMD.5-6) 253 EXPECTATIONS FOR LEARNING 253 CONTENT ELABORATIONS 254 INSTRUCTIONAL STRATEGIES 255 • Van Hiele Connection 255 • Scaling 255 • Comparing Similar and Non-Similar Figures 258 INSTRUCTIONAL TOOLS/RESOURCES 264 High School Geometry Course OHIO’S MODEL CURRICULUM WITH INSTRUCTIONAL SUPPORTS | Mathematics | 2018 7 MODELING WITH GEOMETRY (G.MG) 267 APPLY GEOMETRIC CONCEPTS IN MODELING SITUATIONS. (G.MG.1-3) 267 EXPECTATIONS FOR LEARNING 267 CONTENT ELABORATIONS 268 INSTRUCTIONAL STRATEGIES 269 • Van Hiele Connection 269 • Modeling (★) 269 • Describing Objects with Geometric Models 270 • Concepts of Density 272 INSTRUCTIONAL TOOLS/RESOURCES 275 STATISTICS AND PROBABILITY (S) ERROR! BOOKMARK NOT DEFINED. CONDITIONAL PROBABILITY AND RULES OF PROBABILITY (S.CP) ERROR! BOOKMARK NOT DEFINED. UNDERSTAND INDEPENDENCE AND CONDITIONAL PROBABILITY, AND USE THEM TO INTERPRET DATA. (S.CP.1-5) ERROR! BOOKMARK NOT DEFINED. EXPECTATIONS FOR LEARNING ERROR! BOOKMARK NOT DEFINED. CONTENT ELABORATIONS ERROR! BOOKMARK NOT DEFINED. INSTRUCTIONAL STRATEGIES 281 • GAISE Model 281 • Modeling (★) 282 • Characteristics of Outcomes 282 • Two-way Tables 287 • Relationships Between Sets and Probability 288 • Conditional Probability 292 • Independence 294 INSTRUCTIONAL TOOLS/RESOURCES 297 USE THE RULES OF PROBABILITY TO COMPUTE PROBABILITIES OF COMPOUND EVENTS IN A UNIFORM PROBABILITY MODEL. (S.CP.6-9) ERROR! BOOKMARK NOT DEFINED. EXPECTATIONS FOR LEARNING ERROR! BOOKMARK NOT DEFINED. CONTENT ELABORATIONS ERROR! BOOKMARK NOT DEFINED. INSTRUCTIONAL STRATEGIES 304 • Modeling (★) 304 • Conditional Probability 304 • Addition Rule 308 • Multiplication Rule (+) 309 • Permutations and Combinations (+) 311 INSTRUCTIONAL TOOLS/RESOURCES 312 High School Geometry Course OHIO’S MODEL CURRICULUM WITH INSTRUCTIONAL SUPPORTS | Mathematics | 2018 8 Introduction PURPOSE OF THE MODEL CURRICULUM Just as the standards are required by Ohio Revised Code, so is the development of the model curriculum for those standards. Throughout the development of the standards (2016-17) and the model curriculum (2017-18), the Ohio Department of Education (ODE) has involved educators from around the state at all levels, Pre-K–16. The model curriculum reflects best practices and the expertise of Ohio educators, but it is not a complete a curriculum nor is it mandated for use. The purpose of Ohio’s model curriculum is to provide clarity to the standards, a foundation for aligned assessments, and guidelines to assist educators in implementing the standards. The model curriculum is not a collection of lessons nor a full curriculum; it does not suggest pace, sequence, or amount of time spent on topics. It provides information about a topic related to the standards including ideas for examples, strategies for teaching, and possible connections between topics in addition to highlighting some misconceptions. COMPONENTS OF THE MODEL CURRICULUM The model curriculum contains two sections: Expectations for Learning and Content Elaborations. Expectations for Learning: This section begins with an introductory paragraph describing the cluster’s position in the respective learning progression, including previous learning and future learning. Following are three subsections: Essential Understandings, Mathematical Thinking, and Instructional Focus. • Essential Understandings are the important concepts students should develop. When students have internalized these conceptual understandings, application and transfer of learning results. • Mathematical Thinking statements describe the mental processes and practices important
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