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This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. The purpose of this curriculum project was to align our local Geometry curriculum with the latest New York State Learning Standards. These standards go into effect September 2008, with the first Geometry Regents exam being administered in June 2009. Six high school teachers from worked together on this project. First, we combed through the new standards thoroughly to determine similarities and differences between these standards and our current local curriculum. We collectively decided on additional topics F-M would include in its local curriculum, even if those topics were not part of the state mandates. Next, we determined which standards were adequately covered in our Geometry textbooks and which topics needed to be supplemented with other resources or teacher-made materials. Calendars for the Regents and Honors level Geometry courses were created for each unit and chapter, outlining daily objectives and suggested homework assignments. The daily objectives outlined in the unit calendars encompass all of the NY State content standards, as well as the additional topics we decided to add to our local curriculum. After creating these unit calendars, the teachers decided which units/objectives would be included in our local “Topics in Geometry” course. A table is included which indicates the units/objectives to be taught, along with a suggested number of instructional days. Once the curriculum was mapped out, we discussed use of the graphing calculator in each unit of study. We came to consensus on when/how the calculators could be used on various assessments, and when they should be used as instructional tools. A summary of calculator guidelines is included on the pages entitled Graphing Calculator Guidelines Unit by Unit. The contents of this curriculum include… Overview of “Geometry Regents” and “Geometry Honors” courses Overview of “Topics in Geometry” course Graphing Calculator Guidelines, Unit by Unit Geometry Unit Calendars for Regents/Honors A copy of the Geometry performance indicators published by NYSED A copy of the Geometry Crosswalk, published by NYSED. Respectfully Submitted, Chris Alvarez Katie Cook Kathy Gilbert Dina Kushnir Nick Lore Kate Nowak January 2008 Course Overview for Geometry Regents and Geometry Honors The unit calendars that follow have been created for the Regents level of Geometry. Honors-level topics and optional enrichment topics are also included on the calendars. Below is a summary of the number of days dedicated to each unit of study: Unit of Study # Days Intro to Geometry Logic Beginning Geometry Proofs 9 9 18 Parallel and Perpendicular Lines, Angle Relationships in Polygons Relationships Within Triangles Quadrilaterals Coordinate Geometry Similar Triangles Right Triangles and Trigonometry Transformational Geometry 12 Area, Surface Area, Volume Circle Geometry Locus Constructions Quadratics Unit (Optional for Regents, Mandatory for Honors) Total Use remaining days for projects, quiz days, quarterly exams, and review. 9 15 10 11 8 11 for Regents 13 for Honors 14 13 8 for Regents; 11 for Honors 6 10 153 days for Regents; 168 days for Honors Overview of “Topics in Geometry” Course Topics in Geometry is a course designed for students who need an additional math credit but struggle to succeed in a regular Regents level course. Students in this course will take a local final exam, and will have the option of taking the Geometry regents exam based on teacher recommendation. Modifications to the Regents curriculum are indicated under each unit heading. The extra days afforded by omitting topics can be used for review, practice, or projects. Unit 1: Intro to Geometry Omit solving 2 equations with 2 variables Omit Algebra Proofs (Day 7) Unit 2:Logic Omit Logic Proofs (Day 7) Unit 3: Beginning Geometry Proofs Omit Days 12 – 15 Unit 4: Parallel and Perpendicular Lines, Angle Relationships Include all content but add some extra practice days. Unit 5: Relationships Within Triangles Omit Indirect Proofs (Day 5) Omit Inequality Proofs (Day 7) Unit 6: Quadrilaterals: Students should know all the properties of the various quadrilaterals and use them to solve algebraic problems. No proofs will be done in this unit, unless time and the skill level of the students allows it. Time gained by omitting proofs can be dedicated to explorations (e.g. – Geometer Sketchpad) . Unit 7: Coordinate Geometry Spend extra time writing equations of lines. Only do coordinate geometry proofs involving triangles and parallelograms. Proofs should be broken down into smaller steps. When having students solve a linear-quadratic system of equations, emphasize use of the graphing calculator. Unit 8: Similar Triangles Prove triangles similar using AA Theorem only. (Omit the SSS and SAS similarity theorems). Unit 9: Right Triangles and Trigonometry Omit Vectors Unit 10: Transformations Omit Isometry and preservation of properties. Rotations should be about the origin only. Unit 11: Area, Surface Area, Volume Omit area and arc length of sectors. Omit surface area of a cone or pyramid. Unit 12: Circles Algebraic/numeric problems only (based on Circle Theorems). No proofs. Unit 13: Locus Include all content in this unit. Keep the coordinate geometry problems on the simple side. Unit 14: Constructions Include all content in this unit. . Graphing Calculator Guidelines Unit by Unit Unit 1: Intro to Geometry Students should be tested on their ability to solve quadratic equations AND systems of linear equations, both graphically AND algebraically WITHOUT use of a graphing calculator. Quadratic Equations and systems of equations will be solved in the context of a geometry problem. Unit 2:Logic No restrictions on calculator use. Unit 3: Beginning Geometry Proofs No restrictions on calculator use. Unit 4: Parallel and Perpendicular Lines, Angle Relationships No restrictions on calculator use. Unit 5: Relationships Within Triangles No restrictions on calculator use. Unit 6: Quadrilaterals: Students should be tested on their ability to solve quadratic equations AND systems of linear equations, both graphically AND algebraically WITHOUT use of a graphing calculator. Quadratic Equations and systems of equations will be solved in the context of a geometry problem. Unit 7: Coordinate Geometry Students should be tested on their ability to solve a quadratic-linear system of equations (including the equation of a circle), both graphically AND algebraically WITHOUT use of a graphing calculator. Unit 8: Similar Triangles No restrictions on calculator use. Unit 9: Right Triangles and Trigonometry No restrictions on calculator use. Unit 10: Transformations No restrictions on calculator use. Unit 11: Area, Surface Area, Volume No restrictions on calculator use. Unit 12: Circles Students should be tested on their ability to solve quadratic equations AND systems of linear equations, both graphically AND algebraically WITHOUT use of a graphing calculator. Quadratic Equations and systems of equations will be solved in the context of a geometry problem. Unit 13: Locus No restrictions on calculator use. Unit 14: Constructions No restrictions on calculator use. Unit 15: Quadratics Unit The graphing calculator should be used as a tool for discovering the properties of a quadratic function in various forms (standard form, vertex form, factored form). Students should then be able to achieve all the objectives in the unit without use of a graphing calculator. Sections and page numbers refer to Geometry (New York Version), Prentice Hall Mathematics, 2007 Integrated Geometry Unit 1: Intro to Geometry Day Topic(s)/Objectives Covered Suggested Assignment 1 2 Points, Lines, and Planes Segments, Rays, Parallel Lines and Planes Definition of Midpoint Types of Angles and Angle Pairs Classifying Triangles Triangle Congruence SSS and SAS Section 1-3 Section 1-4 Section 1-5 Section 1-6 Supplement classifications Section 4-1 Section 4-2 None (Review) G.G.28, 29 5 ASA and AAS Section 4-3 G.G.28 6 More Problem Solving p. 234 (systems of linear) Review of solving quadratics and Supplement quadratics 2 equations with 2 unknowns Properties of Equality and Algebraic Section 2-4 Proofs Review Test 3 4 7 8 9 Standards Addressed G.G.1 – G.G.11 None (Review) None (Review) None Integrated Geometry Unit 2: Logic Day Topic(s)/Objectives Covered Suggested Assignment 1 Supplement Standards Addressed G.G.25 Section 2-1 G.G.25 Section 2-1 Supplement Section 2-2 Section 2-3 G.G.26 Section 2-4 Section 2-5 Supplement G.G.27 2 3 4 5 6 7 8 9 Statements, Disjunction, Conjunction, Negation Conditional Statements Counterexamples Inverse, Converse, Contrapositive, Logical Equivalence Biconditionals Deductive Reasoning Laws of Detachment and Syllogism Proving Angles and Segments Congruent (Logical Reasoning) Logic Proofs (Required for Honors, optional for Regents) Review Test G.G.25 G.G.27 G.G.27 Integrated Geometry Unit 3: Beginning Geometry Proofs Day Topic(s)/Objectives Covered Suggested Assignment 1 Section 2-2 (7 – 11, 27) Section 2-3 (3, 5, 7, 38, 39) Mini-Proofs Packet More mini-proofs (packet) Sec. 2-4 (5-23) Supplemental Proofs Select from Sec 2-5 (1 – 32) Supplemental Proofs 2 3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 Definitions and Mini-Proofs Postulates vs. Definitions vs. Theorems Team Practice with mini-proofs Substitution, Transitive Property, Beginning multi-step proofs Complements and Supplements of Congruent Angles, Other Angle Theorems Practice Proving Segments and Angles Congruent Using Definitions, Angle Theorems, and Properties of Equality Definition of Congruent Triangles Proving Triangles Congruent Using SSS, SAS Congruent Triangle Proofs Using ASA and AAS Proofs involving CPCTC Theorems regarding Isosceles and Equilateral Triangles. Using CPCTC to prove midpoint, segment bisector, angle bisector, isosceles triangle Congruent Triangle Proofs using HL Double Triangle and Overlapping Triangle Proofs Double Triangle and Overlapping Triangle Proofs Using Addition and Subtraction Postulates to Prove Segments and Angles Congruent Congruent Triangle Proofs Using Addition and Subtraction Postulates More Congruent Triangle Proofs (Practice Day) Review Exam Standards Addressed GG.25 GG.27 GG.27 GG.27 GG.27 Supplemental Proofs GG.27 Select Problems from sections 4-1 and 4-2 GG.27 - 29 Select problems from Sec 4-3 Supplemental Proofs Select problems from Sec 4-4 Supplemental Proofs Select problems from Sec 4-5 Supplemental Proofs GG. 27 GG.28 GG. 27 GG. 29 GG. 27 – 29, 31 Select problems from Sec 4-6 Supplemental Proofs Select problems from Sec 4-7 Supplemental Proofs Select problems from Sec 4-7 Supplemental Proofs Supplement GG. 27 – 29, 31 Supplement GG. 27 – 29, 31 Chapter 4 Review (p. 249 ) Supplemental Proofs Review Sheet GG. 27 – 29, 31 GG. 27 – 29, 31 GG. 27 – 29, 31 GG. 27 – 29, 31 Integrated Geometry Unit 4: Parallel and Perpendicular Lines, Angle Relationships in Polygons Day Topic(s)/Objectives Covered 1 Alternate interior and corresponding angles Proving lines parallel Suggested Assignment p. 130 #4, p.131 #1-5 #11-13 2 p.137-138 #1-8, 10-21 p.143 #4-10 3 Slopes of parallel and p.177-178 #1-11, perpendicular lines #16-19, 25-28 4 Writing equations of parallel p.178 #12-15, 20-23 and perpendicular lines 5 Congruent triangle proofs supplement using parallel and perpendicular lines 6 Mixed practice supplement 7 Sum of angles and exterior p.150-151 #1-5,16, angles of triangles #18-20,30,32 8 Polygon angle sum p.161-162 #11-14, theorem #16-18, 21, 33-35 9 Interior and exterior angles p.161-162 #22-25, of regular polygons #40-43 10 Mixed practice supplement 11 Review supplement 12 Test Standards Addressed G.G.35 G.G.9, G.G.35 G.G.62, G.G.63 G.G.64, G.G.65 G.G.30, G.G.32 G.G.36 G.G.37 Integrated Geometry Unit 5: Relationships Within Triangles Day Topic(s)/Objectives Covered Suggested Assignment 1 2 3 P262 (1-21, 38,39) P267-268 (1-25,31) P275-276 (1-5,8-12, 14-16,19-23) P279 (1-10) P282 has notes P283 (10-19) Need to Supplement with actual 2 column proofs P292 (1-28) Needs to be supplemented 4 5 Mid-segments Angle and Segment Bisectors Concurrent Lines, Medians, and Altitudes Quiz Indirect Proofs 6 7 8 9 Triangle Inequalities Inequality Proofs (honors only) Review Test Standards Addressed G.G.21, G.G.42 G.G.21 G.G.21, G.G.43 G.G.24 G.G.33, G.G.34 Integrated Geometry Unit 6: Quadrilaterals Day Topic(s)/Objectives Covered Suggested Assignment 1 2 6-2 Standards Addressed G.G.38 6-3, supplement proofs G.G.41 6-4 6-4 6-4, supplement proofs G.G.39 G.G.39 G.G.41 Supplement with algebraic problems and proofs 6-5, supplement G.G.39 G.G.41 G.G.40 3 4 5 6 7 8 9 10 11 12 13 14 15 Properties of a Parallelogram More work on Parallelograms (Algebraic) Parallelogram Proofs Proving a Parallelogram More work on Parallelograms Properties of a Rectangle Properties of a Rhombus Proving a Rectangle Proving a Rhombus Review Day Properties of a Square Proving a Square Properties of Trapezoids and Isosceles Trapezoids including mid-segments Trapezoids Proofs Review of all Parallelograms Properties of Kites Review of all Parallelograms Review Day Unit Test on Quadrilaterals 6-5, supplement proofs 6-5 Integrated Geometry Unit 7: Coordinate Geometry Day Topic(s)/Objectives Covered 1 2 3 4 5 6 7 8 9 10 Review Pythagorean Theorem & Its converse Distance Formula Review simplifying radicals (and cube roots in HONORS) 3D distance formula – optional for HONORS Midpoint and Slope formulas Equations of Lines Parallel, Perp. Or neither 3D midpoint formula – optional for HONORS Writing the equation of the perpendicular bisector of a segment given the endpoints of the line segment Coordinate Geometry Proofs Coordinate Geometry Proofs For Regents – practice more coordinate geometry proofs For HONORS – do coordinate geometry proofs with variables Equations of Circles and Appropriate Problem Solving FOR HONORS ONLY – write the eq. of a circle given any three points that lie on the circle For Regents – Practice Day Review solving linear-quadratic system graphically and algebraically (including those that involve circles) Review TEST Suggested Assignment Section 8-1 p.417-423 Section 1-8 p.56 #1-17,32-40,43,44-52 p.58 #65,67 Standards Addressed G.G.48 G.G.67 p.73 #34-36, 38 supplement for honors Section 1-8 Section 3-7 G.G.62-66 supplement for honors p.178 #23,46 G.G.68 needs to be supplemented Section 6-6 Section 6-7 Proofs in text are too easy, needs to be supplemented Section 6-6 Section 6-7 Proofs in text are too easy, needs to be supplemented Section 12-5 G.G.69 G.G.38-41 G.G.69 G.G.38-41 G.G.71-74 Needs to be supplemented p.355 #13-20 p.699 #53-58 Review from Algebra 1 G.G.70 Integrated Geometry Unit 8: Similar Triangles Day Topic(s)/Objectives Covered Suggested Assignment 1 Ratios and Proportions 2 3 Similar Polygons Proving Triangles Similar P368-370 (2-28 evens, 40,41,42 Review of solving quadratics p372 P375-377 (1-35 odd) P385-387 (1-19) 4 Proving Triangles Similar using 2-column proofs Quiz Similarity in Right Triangles More Similarity in Right Triangles Proportions in Triangles Product of Means and Extremes Review Test 5 6 7 8 9 10 11 Standards Addressed G.CN.4 G.RP.2 G.G.44, G.G.45 Need to be supplemented G.G.44, G.G.45 P394 (1-22) Supplement more practice Use Practice 7-4 P400-401 (1-26) Need to be supplemented G.G.45, G.G.47 G.G.45, G.G.46 G.G.45, G.G.46 Integrated Geometry Unit 9: Right Triangles and Trigonometry Day Topic(s)/Objectives Covered Suggested Assignment 1 2 Special Right Triangles More Special Right Triangles 3 4 The Tangent Ratio Sine and Cosine Ratios 5 Angles of Elevation and Depression Vectors Review Test P428-429 (1-22) Supplement Use Practice 8-2 P434-436 (1-20,31,32) P441-442 (1-24) Option: Activity Lab p444 P447-449 (1-18,28,29) 6 7 8 P455-459 (1-28) Standards Addressed G.G.48 Integrated Geometry Unit 10: Transformation Geometry Day Topic(s)/Objectives Covered Suggested Assignment 1 Introduction to Transformations and Isometries Translations Preserving Segment and Angle Congruence Line Reflections Preserving Parallelism and Perpendicularity 9-1 supplement Rotations 9-3 Mandatory for HONORS ONLY – Rotation of any degree measure about any point. Include special angles. (1 or 2 days) Mixed Practice Line Symmetry (Reflectional) Rotational Symmetry Dilations Similarity vs Congruence Supplement Compositions Glide Reflections Direct Isometries Opposite Isometries More Compositions Mixed Practice Review Unit Test 9-6 9-6 2 3 4 5 6 7 8 9 10 11 9-2 supplement Standards Addressed G.G.54 G.G.55 G.G.61 G.G.54 G.G.55 G.G.57 G.G.61 G.G.54 G.G.55 G.G.61 9-4 9-5 supplement G.G.58 G.G.59 G.G.60 G.G.61 G.G.60 G.G.55 G.G.56 G.G.57 Integrated Geometry Unit 11: Area, Surface Area, Volume Day 1 Topic Assignment Areas of Triangles, Quadrilaterals, Select problems from sections Circles, and Composite Figures 10-1, 10-2, and 10-7 (#25 – 27, 36 – (see examples below) 40) Supplement composite figures Find Area of a Polygon Given the Section 10 – 1 (24 – 33) Section 10 – 2 (25 – 27) Coordinates of the Vertices; Find Area of a Polygon Bound by 3 or 4 Linear Equations Relationship Between Areas and Select problems from Section 10 - 4 Perimeters of Similar Figures Finding Areas Using Trig. and Select problems from Section 10 - 5 Special Right Triangles Mandatory for Honors Only: Select problems from Sections Areas of Regular Polygons 10 – 3 and 10 - 5 Circles and Sectors Select problems from Sections (Arc Length and Area) 10 – 6 and 10 - 7 Surface Area of Prisms and Select problems from Section 11 - 2 Cylinders 2 3 4 5 6 7 Standards Covered GG.10 GG. 14 Honors: Bases are Regular Polygons 8 Surface Area of Pyramids and Cones Select problems from Section 11 - 3 GG.13 GG.15 Select problems from Section 11 - 4 GG.11 GG.12 GG.14 GG.13 GG.15 GG.16 Honors: Bases are Regular Polygons 9 Volumes of Prisms and Cylinders Honors: Bases are Regular Polygons 10 Volumes of Pyramids and Cones Select problems from Section 11 - 5 Honors: Bases are Regular Polygons 11 Volume and Surface Area of A Sphere; Great Circle of a Sphere; Planes Equidistant from Center of a Sphere Intersect Sphere in 2 Congruent Circles Mandatory for Honors Only: Relationship Between Volumes of Similar Solids Review Exam 12 13 14 Select problems from Section 11 – 6 Supplement Select problems from Section 11 - 7 Examples for Day 1: x 6 8 Express the area of the shaded space in terms of x. Integrated Geometry Unit 12: Circles Day Topic(s)/Objectives Covered Suggested Assignment 1 Section 12-1 Standards Addressed G.G.50 Section 12-1 G.G.50 Section 12-2 G.G.49 Section 12-2 G.G.49 Section 12-3 p.682 #25 (needs to be supplemented) Section 12-4 G.G.51 G.G.52 2 3 4 5 6 7 8 9 10 11 12 13 Tangents – angle formed by tangent and radius Tangents – two segments tangent to a circle from a point outside the circle are congruent Chords and Arcs – congruent chords have congruent arcs Chords – perpendicular bisector of a chord contains the center of the circle Inscribed Angles and their relationship to intercepted arcs Angle Measures and Segment Lengths – secant lines and their relationships with intercepted arcs FOR HONORS ONLY Honors Level Questions Review Day Circle Proofs w/ Similar Triangles More Circle Proofs: Similar Triangles FOR HONORS ONLY Circle Proofs: Congruent Triangles Review TEST G.G.51 G.G.53 Needs to be supplemented Needs to be supplemented G.G.27 Needs to be supplemented G.G.27 Needs to be supplemented G.G.27 Integrated Geometry Unit 13: Locus Day Topic(s)/Objectives Covered 1 Locus 2 3 4 Compound Locus More compound locus Locus with coordinate geometry 5 More locus with coordinate geometry 6 * Locus review Honors Only: 2 – 3 days of 3-D locus and 3-D compound loci More review Test 7 8 Suggested Assignment Packet pp. 2 – 3 (#1-7, 10-12,14,17-19, 21) Packet pp.4 – 5 (#1-19,22) worksheet Packet p. 7 (#2-8even, 11-25 odd, 26-30) Packet p.7 (#31-41,44-46) p.8 (#1-6, 10-12) review packet supplement supplement Standards Addressed G.G.22 G.G.23 Integrated Geometry Unit 14: Constructions NOTE: This unit may be taught as a stand alone unit or incorporated into other units as appropriate Day Topic(s)/Objectives Covered Suggested Assignment 1 1-7 supplement (especially justifications) 2 3 4 5 6 Introduction to Constructions Constructions and Justifications: -Congruent Line Segments -Congruent Angles -Perpendicular Bisector of a Line Segment -Median of a Triangle -Bisecting an Angle Constructions and Justifications: - Perpendicular at a Point on a Line - Perpendicular From a Point to a Line -Altitude of a Triangle Constructions and Justifications: - A line parallel to a given line - A Parallelogram - An equilateral Triangle - Trisecting a line segment (optional) Investigate the Concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of sides of triangles Review and Practice Unit Quest Standards Addressed G.G.17 G.G.18 3-8 supplement (especially justifications) G.G.19 3-8 supplement G.G.19 G.G.20 5-2 (HW # 32) 5-2 (HW # 33) 5-3 Geometer’s Sketchpad Lab Activity G.G.21 Integrated Geometry Unit 15: Quadratic Functions Mandatory for HONORS only Day Topic(s)/Objectives Covered Vertex form of a parabola 1 Pages A-C Vertex form of a parabola 2 Pages D - E Writing a Quadratic Equation 3 given the Roots Solving Quadratic Equations 4 by Completing the Square 5 Writing the Equation of a Parabola 9 Writing the Equation of a Parabola Practice Days 1-6 p.5 QUIZ Constructing a Circle that Circumscribes a Triangle Writing the Equation of the Circle that passes through 3 given points Review 10 TEST 6 7 8 Suggested Assignment Pages F-G #1-6 Pages G-H #7-15 Packet, pg. 1 Packet, pg. 2 Packet, pg. 3 – 4 You can do #10 for bonus, your answer has to be IN FRACTION FORM Bring a compass & straight edge to class on Day 7 Packet, pg. 6 – 7 Bring a compass & straight edge to class tomorrow Packet, pg 8 Packet, pg. 9-10 Study! Due Tomorrow: June 2005 Honors Final Part 3 ( #1-4) Standards Addressed