The contents of this curriculum include

Transcription

1 This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. To view a copy of this license, visit 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 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. January 2008 Respectfully Submitted, Chris Alvarez Katie Cook Kathy Gilbert Dina Kushnir Nick Lore Kate Nowak

2 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 9 Logic 9 Beginning Geometry Proofs 18 Parallel and Perpendicular Lines, Angle Relationships in 12 Polygons Relationships Within Triangles 9 Quadrilaterals 15 Coordinate Geometry 10 Similar Triangles 11 Right Triangles and Trigonometry 8 Transformational Geometry 11 for Regents 13 for Honors Area, Surface Area, Volume 14 Circle Geometry 13 Locus 8 for Regents; 11 for Honors Constructions 6 Quadratics Unit (Optional for Regents, Mandatory for Honors) 10 Total Use remaining days for projects, quiz days, quarterly exams, and review. 153 days for Regents; 168 days for Honors

3 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 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.

4 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..

5 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.

6 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.

7 Sections and page numbers refer to Geometry (New York Version), Prentice Hall Mathematics, 2007

8 Unit 1: Intro to Geometry 1 Points, Lines, and Planes Section 1-3 G.G.1 G.G.11 2 Segments, Rays, Parallel Lines and Planes Definition of Midpoint Section 1-4 Section 1-5 None (Review) 3 Types of Angles and Angle Pairs Section 1-6 None (Review) 4 Classifying Triangles Triangle Congruence SSS and SAS Supplement classifications Section 4-1 Section 4-2 G.G.28, 29 5 ASA and AAS Section 4-3 G.G.28 6 More Problem Solving Review of solving quadratics and 2 equations with 2 unknowns 7 Properties of Equality and Algebraic Proofs 8 Review 9 Test p. 234 (systems of linear) Supplement quadratics Section 2-4 None (Review) None

9 Unit 2: Logic 1 Statements, Disjunction, Conjunction, Supplement G.G.25 Negation 2 Conditional Statements Section 2-1 G.G.25 Counterexamples 3 Inverse, Converse, Contrapositive, Section 2-1 G.G.26 Logical Equivalence Supplement 4 Biconditionals Section 2-2 G.G.25 5 Deductive Reasoning Section 2-3 G.G.27 Laws of Detachment and Syllogism 6 Proving Angles and Segments Section 2-4 G.G.27 Congruent (Logical Reasoning) Section Logic Proofs (Required for Honors, Supplement G.G.27 optional for Regents) 8 Review 9 Test

10 Unit 3: Beginning Geometry Proofs 1 Definitions and Mini-Proofs Postulates vs. Definitions vs. Section 2-2 (7 11, 27) Section 2-3 (3, 5, 7, 38, 39) GG.25 GG.27 Theorems Mini-Proofs Packet 2 Team Practice with mini-proofs More mini-proofs (packet) GG.27 3 Substitution, Transitive Property, Sec. 2-4 (5-23) GG.27 Beginning multi-step proofs 4 Complements and Supplements of Congruent Angles, Other Angle Theorems 5 Practice Proving Segments and Angles Congruent Using Definitions, Angle Theorems, and Properties of Equality 6 Definition of Congruent Triangles Proving Triangles Congruent Using SSS, SAS Supplemental Proofs Select from Sec 2-5 (1 32) Supplemental Proofs Supplemental Proofs Select Problems from sections 4-1 and Congruent Triangle Proofs Using ASA and AAS Select problems from Sec 4-3 Supplemental Proofs 9 Proofs involving CPCTC Select problems from Sec 4-4 Supplemental Proofs 10 Theorems regarding Isosceles and Select problems from Sec 4-5 Equilateral Triangles. Supplemental Proofs Using CPCTC to prove midpoint, segment bisector, angle bisector, isosceles triangle GG.27 GG.27 GG GG. 27 GG.28 GG. 27 GG. 29 GG , Congruent Triangle Proofs using HL Select problems from Sec 4-6 GG , 31 Supplemental Proofs 12 Double Triangle and Overlapping Select problems from Sec 4-7 GG , 31 Triangle Proofs Supplemental Proofs 13 Double Triangle and Overlapping Select problems from Sec 4-7 GG , 31 Triangle Proofs Supplemental Proofs 14 Using Addition and Subtraction Postulates to Prove Segments and Angles Congruent Supplement GG , Congruent Triangle Proofs Using Addition and Subtraction Postulates 16 More Congruent Triangle Proofs (Practice Day) 17 Review Review Sheet 18 Exam Supplement GG , 31 Chapter 4 Review (p. 249 ) Supplemental Proofs GG , 31

11 Unit 4: Parallel and Perpendicular Lines, Angle Relationships in Polygons 1 Alternate interior and p. 130 #4, p.131 #1-5 G.G.35 corresponding angles # Proving lines parallel p #1-8, G.G.9, G.G.35 p.143 # Slopes of parallel and p #1-11, G.G.62, G.G.63 perpendicular lines #16-19, Writing equations of parallel p.178 #12-15, G.G.64, G.G.65 and perpendicular lines 5 Congruent triangle proofs using parallel and perpendicular lines supplement 6 Mixed practice supplement 7 Sum of angles and exterior angles of triangles p #1-5,16, #18-20,30,32 8 Polygon angle sum p #11-14, theorem #16-18, 21, Interior and exterior angles p #22-25, of regular polygons # Mixed practice supplement 11 Review supplement 12 Test G.G.30, G.G.32 G.G.36 G.G.37

12 Unit 5: Relationships Within Triangles 1 Mid-segments P262 (1-21, 38,39) G.G.21, G.G.42 2 Angle and Segment Bisectors P (1-25,31) G.G.21 3 Concurrent Lines, Medians, and P G.G.21, G.G.43 Altitudes (1-5,8-12, 14-16,19-23) 4 Quiz P279 (1-10) 5 Indirect Proofs P282 has notes G.G.24 P283 (10-19) Need to Supplement with actual 2 column proofs 6 Triangle Inequalities P292 (1-28) G.G.33, G.G.34 7 Inequality Proofs (honors only) Needs to be supplemented 8 Review 9 Test

13 Unit 6: Quadrilaterals 1 Properties of a Parallelogram 6-2 G.G.38 2 More work on Parallelograms (Algebraic) 3 Parallelogram Proofs 4 Proving a Parallelogram 6-3, supplement proofs G.G.41 5 More work on Parallelograms 6 Properties of a Rectangle 6-4 G.G.39 7 Properties of a Rhombus 6-4 G.G.39 8 Proving a Rectangle 6-4, supplement proofs G.G.41 Proving a Rhombus 9 Review Day 10 Properties of a Square Proving a Square Supplement with algebraic problems and proofs G.G.39 G.G Properties of Trapezoids and 6-5, supplement G.G.40 Isosceles Trapezoids including mid-segments 12 Trapezoids Proofs 6-5, supplement proofs Review of all Parallelograms 13 Properties of Kites 6-5 Review of all Parallelograms 14 Review Day 15 Unit Test on Quadrilaterals

14 Unit 7: Coordinate Geometry 1 Review Pythagorean Theorem & Its converse Section 8-1 p Section 1-8 G.G.48 G.G.67 Distance Formula p.56 #1-17,32-40,43,44-52 Review simplifying radicals (and cube roots in HONORS) p.58 #65,67 p.73 #34-36, 38 3D distance formula optional for HONORS supplement for honors 2 Midpoint and Slope formulas Section 1-8 Section 3-7 G.G Equations of Lines Parallel, Perp. Or neither supplement for honors 3D midpoint formula optional for HONORS 3 Writing the equation of the p.178 #23,46 G.G.68 perpendicular bisector of a segment given the endpoints of the line segment needs to be supplemented 4 Coordinate Geometry Proofs Section 6-6 Section 6-7 Proofs in text are too easy, needs to be supplemented G.G.69 G.G Coordinate Geometry Proofs For Regents practice more coordinate geometry proofs For HONORS do coordinate geometry proofs with variables 6 Equations of Circles and Appropriate Problem Solving 7 FOR HONORS ONLY write the eq. of a circle given any three points that lie on the circle For Regents Practice Day 8 Review solving linear-quadratic system graphically and algebraically (including those that involve circles) 9 Review 10 TEST Section 6-6 Section 6-7 Proofs in text are too easy, needs to be supplemented Section 12-5 Needs to be supplemented p.355 #13-20 p.699 #53-58 Review from Algebra 1 G.G.69 G.G G.G G.G.70

15 Unit 8: Similar Triangles 1 Ratios and Proportions P (2-28 evens, G.CN.4 40,41,42 Review of solving quadratics p372 2 Similar Polygons P (1-35 odd) G.RP.2 3 Proving Triangles Similar P (1-19) G.G.44, G.G.45 4 Proving Triangles Similar using Need to be supplemented G.G.44, G.G.45 2-column proofs 5 Quiz 6 Similarity in Right Triangles P394 (1-22) G.G.45, G.G.47 7 More Similarity in Right Triangles Supplement more practice Use Practice Proportions in Triangles P (1-26) G.G.45, G.G.46 9 Product of Means and Extremes Need to be supplemented G.G.45, G.G Review 11 Test

16 Unit 9: Right Triangles and Trigonometry 1 Special Right Triangles P (1-22) G.G.48 2 More Special Right Triangles Supplement Use Practice The Tangent Ratio P (1-20,31,32) 4 Sine and Cosine Ratios P (1-24) Option: Activity Lab p444 5 Angles of Elevation and P (1-18,28,29) Depression 6 Vectors P (1-28) 7 Review 8 Test

17 Unit 10: Transformation Geometry 1 Introduction to Transformations and Isometries Translations Preserving Segment and Angle Congruence 9-1 supplement G.G.54 G.G.55 G.G.61 2 Line Reflections Preserving Parallelism and Perpendicularity 9-2 supplement G.G.54 G.G.55 G.G.57 G.G.61 3 Rotations 9-3 G.G.54 G.G.55 G.G.61 Mandatory for HONORS ONLY Rotation of any degree measure about any point. Include special angles. (1 or 2 days) 4 Mixed Practice 5 Line Symmetry (Reflectional) Rotational Symmetry Supplement 6 Dilations Similarity vs Congruence 9-5 G.G.58 G.G.59 G.G.60 G.G.61 7 Compositions 9-6 G.G.60 8 Glide Reflections Direct Isometries Opposite Isometries 9 More Compositions Mixed Practice 10 Review 11 Unit Test G.G.55 G.G.56 G.G.57 supplement

18 Unit 11: Area, Surface Area, Volume Day Topic Assignment Standards Covered 1 Areas of Triangles, Quadrilaterals, Circles, and Composite Figures (see examples below) Select problems from sections 10-1, 10-2, and 10-7 (#25 27, 36 40) 2 Find Area of a Polygon Given the Coordinates of the Vertices; Find Area of a Polygon Bound by 3 or 4 Linear Equations 3 Relationship Between Areas and Perimeters of Similar Figures 4 Finding Areas Using Trig. and Special Right Triangles 5 Mandatory for Honors Only: Areas of Regular Polygons 6 Circles and Sectors (Arc Length and Area) 7 Surface Area of Prisms and Cylinders Honors: Bases are Regular Polygons 8 Surface Area of Pyramids and Cones Honors: Bases are Regular Polygons 9 Volumes of Prisms and Cylinders Honors: Bases are Regular Polygons 10 Volumes of Pyramids and Cones 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 12 Mandatory for Honors Only: Relationship Between Volumes of Similar Solids 13 Review 14 Exam Examples for Day 1: Supplement composite figures Section 10 1 (24 33) Section 10 2 (25 27) Select problems from Section 10-4 Select problems from Section 10-5 Select problems from Sections 10 3 and 10-5 Select problems from Sections 10 6 and 10-7 Select problems from Section 11-2 Select problems from Section 11-3 Select problems from Section 11-4 Select problems from Section 11-5 Select problems from Section 11 6 Supplement Select problems from Section 11-7 GG.10 GG. 14 GG.13 GG.15 GG.11 GG.12 GG.14 GG.13 GG.15 GG.16 6 x Express the area of the shaded space in terms of x. 8

19 Unit 12: Circles 1 Tangents angle formed by Section 12-1 G.G.50 tangent and radius 2 Tangents two segments Section 12-1 G.G.50 tangent to a circle from a point outside the circle are congruent 3 Chords and Arcs congruent Section 12-2 G.G.49 chords have congruent arcs 4 Chords perpendicular bisector Section 12-2 G.G.49 of a chord contains the center of the circle 5 Inscribed Angles and their relationship to intercepted arcs Section 12-3 p.682 #25 (needs to be G.G.51 G.G.52 6 Angle Measures and Segment Lengths secant lines and their relationships with intercepted arcs 7 FOR HONORS ONLY Honors Level Questions 8 Review Day 9 Circle Proofs w/ Similar Triangles 10 More Circle Proofs: Similar Triangles FOR HONORS ONLY 11 Circle Proofs: Congruent Triangles 12 Review 13 TEST supplemented) Section 12-4 Needs to be supplemented Needs to be supplemented Needs to be supplemented Needs to be supplemented G.G.51 G.G.53 G.G.27 G.G.27 G.G.27

20 Unit 13: Locus 1 Locus Packet pp. 2 3 G.G.22 (#1-7, 10-12,14,17-19, 21) 2 Compound Locus Packet pp.4 5 (#1-19,22) 3 More compound locus worksheet 4 Locus with coordinate geometry Packet p. 7 (#2-8even, G.G odd, 26-30) 5 More locus with coordinate geometry Packet p.7 (#31-41,44-46) p.8 (#1-6, 10-12) 6 Locus review review packet * Honors Only: 2 3 days of supplement 3-D locus and 3-D compound loci 7 More review supplement 8 Test

21 Unit 14: Constructions NOTE: This unit may be taught as a stand alone unit or incorporated into other units as appropriate 1 Introduction to Constructions Constructions and Justifications: -Congruent Line Segments -Congruent Angles -Perpendicular Bisector of a Line Segment -Median of a Triangle -Bisecting an Angle 1-7 supplement (especially justifications) G.G.17 G.G.18 2 Constructions and Justifications: - Perpendicular at a Point on a Line - Perpendicular From a Point to a Line -Altitude of a Triangle 3 Constructions and Justifications: - A line parallel to a given line - A Parallelogram - An equilateral Triangle - Trisecting a line segment (optional) 4 Investigate the Concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of sides of triangles 5 Review and Practice 6 Unit Quest 3-8 supplement (especially justifications) 3-8 supplement 5-2 (HW # 32) 5-2 (HW # 33) 5-3 Geometer s Sketchpad Lab Activity G.G.19 G.G.19 G.G.20 G.G.21

22 Unit 15: Quadratic Functions Mandatory for HONORS only Day Topic(s)/Objectives Covered Vertex form of a parabola Pages A-C Vertex form of a parabola Pages D - E Writing a Quadratic Equation given the Roots Solving Quadratic Equations by Completing the Square Writing the Equation of a Parabola 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 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 9 Review Study! 10 TEST Packet, pg Due Tomorrow: June 2005 Honors Final Part 3 ( #1-4) Standards