Geometry - Eureka Math Scope and Sequence€¦ · Geometry - Eureka Math Scope and Sequence Module...

Geometry - Eureka Math Scope and Sequence

Module 1: Congruence, Proof, and Constructions Module 2: Similarity, Proof, and Trigonometry Module 3: Extending to Three Dimensions

Module 4: Connecting Algebra and Geometry through Coordinates

Module 5: Circles with and Without Coordinates

Topic A Topic B Topic C Topic D Topic E Topic F Topic G Topic A Topic B Topic C Topic D Topic E Topic A Topic B Topic A Topic B Topic C Topic D Topic A Topic B Topic C Topic D Topic E

Basic Constructions

Unknown Angles

Transformations/Rigid Motions

Congruence Proving Properties of Geometric Figures

Advanced Constructions

Axiomatic Systems

Scale Drawings

Dilations Similarity and Dilations

Applying Similarity to Right Triangles

Trigonometry Area Volume Rectangular and Triangular Regions Defined by Inequalities

Perpendicular and Parallel Lines in the Cartesian Plane

Perimeters and Areas of Polygonal Regions in the Cartesian Plane

Partitioning and Extending Segments and Parameterization of Lines

Central and Inscribed Angles

Arcs and Sectors

Secants and Tangents

Equations for Circles and Their Tangents

Cyclic Quadrilaterals and Ptolemy’s Theorem

G-CO.A.1 G-CO.C.9 G-CO.A.2 G-CO.B.7 G-CO.C.9 G-CO.D.13 G-CO.A.1 G-SRT.A.1 G-SRT.A.1 G-SRT.A.2 G-SRT.B.4 G-SRT.C.6 G-GMD.A.1 G-GMD.A.1 G-GPE.B.7 G-GPE.B.4 G-GPE.B.7 G-GPE.B.4 G-C.A.2 G-C.A.1 G-C.A.2 G-GPE.A.1 G-C.A.3

G-CO.D.12 G-CO.A.3 G-CO.B.8 G-CO.C.10 G-CO.A.2 G-SRT.B.4 G-SRT.B.4 G-SRT.A.3 G-SRT.C.7 G-GMD.A.3 G-GPE.B.5 G-GPE.B.6 G-C.A.3 G-C.A.2 G-C.A.3 G-GPE.B.4

G-CO.D.13 G-CO.A.4 G-CO.C.11 G-CO.A.3 G-MG.A.3 G-SRT.B.5 G-SRT.C.8 G-GMD.B.4 G-C.B.5

G-CO.A.5 G-CO.A.4 G-MG.A.1 G.MG.A.1

G-CO.B.6 G-CO.A.5 G.MG.A.2

G-CO.B.7 G-CO.B.6 G.MG.A.3

G-CO.D.12 G-CO.B.7

G-CO.B.8

G-CO.C.9

G-CO.C.10

G-CO.C.11

G-CO.D.12

Priority Content G-CO.D.13

1. Module 1, Topic B (all Lessons)2. Module 1, Topic C (all Lessons)3. Module 1, Topic D (all Lessons)4. Module 2, Topic B (all Lessons)5. Module 2, Topic C (omit Lessons 19, 20)6. Module 2, Topic D (all Lessons)7. Module 2, Topic E (omit Lesson 33)8. Module 4, Topic B (all Lessons)9. Module 4, Topic C (all Lessons)10. Module 4, Topic D (all Lessons)

�1

Mathematics Curriculum GEOMETRY • MODULE 1

Table of Contents1

Congruence, Proof, and Constructions Module Overview .................................................................................................................................................. 3

Topic A: Basic Constructions (G-CO.A.1, G-CO.D.12, G-CO.D.13) ......................................................................... 7

Lessons 1–2: Construct an Equilateral Triangle ........................................................................................ 8

Lesson 3: Copy and Bisect an Angle ........................................................................................................ 21

Lesson 4: Construct a Perpendicular Bisector ........................................................................................ 30

Lesson 5: Points of Concurrencies .......................................................................................................... 37

Topic B: Unknown Angles (G-CO.C.9) .................................................................................................................. 42

Lesson 6: Solve for Unknown Angles—Angles and Lines at a Point ....................................................... 43

Lesson 7: Solve for Unknown Angles—Transversals .............................................................................. 55

Lesson 8: Solve for Unknown Angles—Angles in a Triangle ................................................................... 63

Lesson 9: Unknown Angle Proofs—Writing Proofs ................................................................................ 69

Lesson 10: Unknown Angle Proofs—Proofs with Constructions ............................................................ 76

Lesson 11: Unknown Angle Proofs—Proofs of Known Facts .................................................................. 83

Topic C: Transformations/Rigid Motions (G-CO.A.2, G-CO.A.3, G-CO.A.4, G-CO.A.5, G-CO.B.6, G-CO.B.7, G-CO.D.12) ............................................................................................................................................. 91

Lesson 12: Transformations—The Next Level ........................................................................................ 93

Lesson 13: Rotations ............................................................................................................................. 102

Lesson 14: Reflections .......................................................................................................................... 111

Lesson 15: Rotations, Reflections, and Symmetry ............................................................................... 118

Lesson 16: Translations ........................................................................................................................ 125

Lesson 17: Characterize Points on a Perpendicular Bisector................................................................ 133

Lesson 18: Looking More Carefully at Parallel Lines ............................................................................ 139

Lesson 19: Construct and Apply a Sequence of Rigid Motions ............................................................ 149

Lesson 20: Applications of Congruence in Terms of Rigid Motions ..................................................... 153

1 Each lesson is ONE day, and ONE day is considered a 45-minute period.

Module 1: Congruence, Proof, and Constructions

1

A STOR O UNCTIONS

© 2014 Common Core, Inc. All rights reserved. commoncore.org

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US



M1 Module Overview

GEOMETRY

Lesson 21: Correspondence and Transformations ............................................................................... 159

Mid-Module Assessment and Rubric ................................................................................................................ 164

Topics A through C (assessment 1 day, return 2 days, remediation or further applications 3 days)

Topic D: Congruence (G-CO.B.7, G-CO.B.8) ....................................................................................................... 178

Lesson 22: Congruence Criteria for Triangles—SAS ............................................................................. 179

Lesson 23: Base Angles of Isosceles Triangles ...................................................................................... 189

Lesson 24: Congruence Criteria for Triangles—ASA and SSS ............................................................... 197

Lesson 25: Congruence Criteria for Triangles—AAS and HL ................................................................. 204

Lessons 26–27: Triangle Congruency Proofs ........................................................................................ 211

Topic E: Proving Properties of Geometric Figures (G-CO.C.9, G-CO.C.10, G-CO.C.11) ..................................... 221

Lesson 28: Properties of Parallelograms .............................................................................................. 222

Lessons 29–30: Special Lines in Triangles ............................................................................................. 231

Topic F: Advanced Constructions (G-CO.D.13) .................................................................................................. 241

Lesson 31: Construct a Square and a Nine-Point Circle ........................................................................ 242

Lesson 32: Construct a Nine-Point Circle .............................................................................................. 247

Topic G: Axiomatic Systems (G-CO.A.1, G-CO.A.2, G-CO.A.3, G-CO.A.4, G-CO.A.5, G-CO.B.6, G-CO.B.7, G-CO.B.8, G-CO.C.9, G-CO.C.10, G-CO.C.11, G-CO.C.12, G-CO.C.13) ................................................. 251

Lessons 33–34: Review of the Assumptions ......................................................................................... 253

End-of-Module Assessment and Rubric ............................................................................................................ 264

Topics A through G (assessment 1 day, return 2 days, remediation or further applications 3 days)

Module 1: Congruence, Proof, and Constructions

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A STORY OF FUNCTIONS





GEOMETRY • MODULE 2

Mathematics Curriculum

Module 2: Similarity, Proof, and Trigonometry 1

Table of Contents1

Similarity, Proof, and Trigonometry Module Overview .................................................................................................................................................. 3

Topic A: Scale Drawings (G-SRT.A.1, G-SRT.B.4, G-MG.A.3) ................................................................................. 9

Lesson 1: Scale Drawings ........................................................................................................................ 11

Lesson 2: Making Scale Drawings Using the Ratio Method .................................................................... 27

Lesson 3: Making Scale Drawings Using the Parallel Method ................................................................ 44

Lesson 4: Comparing the Ratio Method with the Parallel Method ........................................................ 59

Lesson 5: Scale Factors ........................................................................................................................... 72

Topic B: Dilations (G-SRT.A.1, G-SRT.B.4) ........................................................................................................... 88

Lesson 6: Dilations as Transformations of the Plane .............................................................................. 90

Lesson 7: How Do Dilations Map Segments? ....................................................................................... 104

Lesson 8: How Do Dilations Map Lines, Rays, and Circles? .................................................................. 120

Lesson 9: How Do Dilations Map Angles? ............................................................................................ 135

Lesson 10: Dividing the King’s Foot into 12 Equal Pieces ..................................................................... 148

Lesson 11: Dilations from Different Centers ........................................................................................ 162

Topic C: Similarity and Dilations (G-SRT.A.2, G-SRT.A.3, G-SRT.B.5, G-MG.A.1) .............................................. 179

Lesson 12: What Are Similarity Transformations, and Why Do We Need Them? ............................... 181

Lesson 13: Properties of Similarity Transformations ............................................................................ 195

Lesson 14: Similarity ............................................................................................................................. 217

Lesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be Similar ........................................ 229

Lesson 16: Between-Figure and Within-Figure Ratios.......................................................................... 242

Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar ........................................................................................................................ 255

Lesson 18: Similarity and the Angle Bisector Theorem ........................................................................ 271

Lesson 19: Families of Parallel Lines and the Circumference of the Earth ........................................... 283







M2 Module Overview GEOMETRY

Module 2: Similarity, Proof, and Trigonometry 2

Lesson 20: How Far Away Is the Moon? ............................................................................................... 297

Mid-Module Assessment and Rubric ................................................................................................................ 306 Topics A through C (assessment 1 day, return 1 day, remediation or further applications 4 days)

Topic D: Applying Similarity to Right Triangles (G-SRT.B.4) ............................................................................. 333

Lesson 21: Special Relationships Within Right Triangles—Dividing into Two Similar Sub-Triangles ...................................................................................................................... 334

Lesson 22: Multiplying and Dividing Expressions with Radicals ........................................................... 348

Lesson 23: Adding and Subtracting Expressions with Radicals ............................................................ 363

Lesson 24: Prove the Pythagorean Theorem Using Similarity .............................................................. 373

Topic E: Trigonometry (G-SRT.C.6, G-SRT.C.7, G-SRT.C.8) ................................................................................ 385

Lesson 25: Incredibly Useful Ratios ...................................................................................................... 387

Lesson 26: The Definition of Sine, Cosine, and Tangent....................................................................... 401

Lesson 27: Sine and Cosine of Complementary Angles and Special Angles ......................................... 414

Lesson 28: Solving Problems Using Sine and Cosine ............................................................................ 424

Lesson 29: Applying Tangents .............................................................................................................. 437

Lesson 30: Trigonometry and the Pythagorean Theorem .................................................................... 450

Lesson 31: Using Trigonometry to Determine Area ............................................................................. 462

Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle ......................................... 473

Lesson 33: Applying the Laws of Sines and Cosines ............................................................................. 485

Lesson 34: Unknown Angles ................................................................................................................. 498

End-of-Module Assessment and Rubric ............................................................................................................ 511 Topics A through E (assessment 1 day, return 1 day, remediation or further applications 4 days)








Module 3: Extending to Three Dimensions

1

Table of Contents1

Extending to Three Dimensions Module Overview .................................................................................................................................................. 2

Topic A: Area (G-GMD.A.1) ................................................................................................................................... 6

Lesson 1: What Is Area? ............................................................................................................................ 8

Lesson 2: Properties of Area ................................................................................................................... 25

Lesson 3: The Scaling Principle for Area ................................................................................................. 38

Lesson 4: Proving the Area of a Disk ....................................................................................................... 51

Topic B: Volume (G-GMD.A.1, G-GMD.A.3, G-GMD.B.4, G.MG.A.1, G.MG.A.2, G.MG.A.3) ............................. 67

Lesson 5: Three-Dimensional Space ....................................................................................................... 69

Lesson 6: General Prisms and Cylinders and Their Cross-Sections......................................................... 85

Lesson 7: General Pyramids and Cones and Their Cross-Sections ....................................................... 102

Lesson 8: Definition and Properties of Volume .................................................................................... 118

Lesson 9: Scaling Principle for Volumes ............................................................................................... 137

Lesson 10: The Volume of Prisms and Cylinders and Cavalieri’s Principle ........................................... 150

Lesson 11: The Volume Formula of a Pyramid and Cone ..................................................................... 162

Lesson 12: The Volume Formula of a Sphere ....................................................................................... 177

Lesson 13: How Do 3D Printers Work? ................................................................................................. 195

End-of-Module Assessment and Rubric ............................................................................................................ 206

Topics A through B (assessment 1 day, return 1 day)








Table of Contents1

Connecting Algebra and Geometry Through Coordinates Module Overview .................................................................................................................................................. 3

Topic A: Rectangular and Triangular Regions Defined by Inequalities (G-GPE.B.7) .............................................. 7

Lesson 1: Searching a Region in the Plane ................................................................................................ 9

Lesson 2: Finding Systems of Inequalities That Describe Triangular and Rectangular Regions ............. 18

Lesson 3: Lines That Pass Through Regions ............................................................................................ 29

Lesson 4: Designing a Search Robot to Find a Beacon ........................................................................... 41

Topic B: Perpendicular and Parallel Lines in the Cartesian Plane (G-GPE.B.4, G-GPE.B.5) ................................. 52

Lesson 5: Criterion for Perpendicularity ................................................................................................. 54

Lesson 6: Segments That Meet at Right Angles...................................................................................... 63

Lesson 7: Equations for Lines Using Normal Segments .......................................................................... 72

Lesson 8: Parallel and Perpendicular Lines ............................................................................................. 81

Mid-Module Assessment and Rubric .................................................................................................................. 92 Topics A through B (assessment 1 day, return and remediation or further applications 1 day)

Topic C: Perimeters and Areas of Polygonal Regions in the Cartesian Plane (G-GPE.B.7) ................................ 111

Lesson 9: Perimeter and Area of Triangles in the Cartesian Plane ....................................................... 112

Lesson 10: Perimeter and Area of Polygonal Regions in the Cartesian Plane ...................................... 123

Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities .............. 134

Topic D: Partitioning and Extending Segments and Parameterization of Lines (G-GPE.B.4, G-GPE.B.6) ......... 143

Lesson 12: Dividing Segments Proportionately .................................................................................... 145

Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means ................................ 156

Lesson 14: Motion Along a Line—Search Robots Again (Optional) ...................................................... 166

Lesson 15: The Distance from a Point to a Line .................................................................................... 175


Module 4: Connecting Algebra and Geometry Through Coordinates

1








End-of-Module Assessment and Rubric ............................................................................................................ 185 Topics A through D (assessment 1 day, return, and remediation or further applications 2 days)

Module 4: Connecting Algebra and Geometry Through Coordinates

2







Table of Contents1

Circles With and Without Coordinates Module Overview ............................................................................................................................................ 3

Topic A: Central and Inscribed Angles (G-C.A.2, G-C.A.3) ................................................................................. 8

Lesson 1: Thales’ Theorem .................................................................................................................10

Lesson 2: Circles, Chords, Diameters, and Their Relationships ............................................................21

Lesson 3: Rectangles Inscribed in Circles ............................................................................................34

Lesson 4: Experiments with Inscribed Angles ......................................................................................42

Lesson 5: Inscribed Angle Theorem and Its Applications .....................................................................52

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles ..................................................65

Topic B: Arcs and Sectors (G-C.A.1, G-C.A.2, G-C.B.5)......................................................................................77

Lesson 7: The Angle Measure of an Arc ..............................................................................................79

Lesson 8: Arcs and Chords ..................................................................................................................92

Lesson 9: Arc Length and Areas of Sectors ........................................................................................103

Lesson 10: Unknown Length and Area Problems ..............................................................................116

Mid-Module Assessment and Rubric ............................................................................................................127 Topics A through B (assessment 1 day, return, remediation, or further applications 1 day)

Topic C: Secants and Tangents (G-C.A.2, G-C.A.3) .........................................................................................143

Lesson 11: Properties of Tangents ....................................................................................................144

Lesson 12: Tangent Segments ..........................................................................................................155

Lesson 13: The Inscribed Angle Alternate—A Tangent Angle ............................................................168

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle ........................................................178

Lesson 15: Secant Angle Theorem, Exterior Case ..............................................................................190

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams ............................201

Topic D: Equations for Circles and Their Tangents (G-GPE.A.1, G-GPE.B.4) ....................................................212

Lesson 17: Writing the Equation for a Circle .....................................................................................213


Module 5: Circles With and Without Coordinates

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Lesson 18: Recognizing Equations of Circles .....................................................................................225

Lesson 19: Equations for Tangent Lines to Circles .............................................................................236

Topic E: Cyclic Quadrilaterals and Ptolemy’s Theorem (G-C.A.3) ...................................................................246

Lesson 20: Cyclic Quadrilaterals .......................................................................................................247

Lesson 21: Ptolemy’s Theorem .........................................................................................................261

End-of-Module Assessment and Rubric ........................................................................................................270 Topics C through E (assessment 1 day, return, remediation or further applications 1 day)

Module 5: Circles With and Without Coordinates

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Geometry - Eureka Math Scope and Sequence€¦ · Geometry - Eureka Math Scope and Sequence Module...

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