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Geometry A Syllabus School Year: Summer School 2015 Certificated Teacher: Luke Leifer Desired Results Course Title: Geometry A Credit: __x__ one semester (.5) Prerequisites and/or recommended preparation: Completion of Algebra 1 Estimate of hours per week engaged in learning activities: 8 hours of class work per week for 6 weeks of summer school Instructional Materials: All learning activity resources and folders are contained within the student online course. Online course is accessed via login and password assigned by student’s school (web account) or emailed directly to student upon enrollment, with the login website. Other resources required/Resource Costs: This course requires a MathXL for Schools account which will be provided by your course instructor. Holt McDougal Geometry 2011 – online videos, examples, and activities. Course Description: Students will explore the basic concepts and methods of Euclidean Geometry while deepening their understanding about plane and solid geometry. Course topics include reasoning and proof, line and angle relationships, two and three dimensional figures, coordinate plane geometry, geometric transformations, surface area and volume. Core processes include reasoning, problem solving and communication. The Algebra End of Course (EOC) assessment given during the spring in this class is a graduation requirement. Enduring Understandings for Course (Performance Objectives): We can use a variety of mathematical tools to describe our world and help solve daily problems. We can use logic to reason out solutions to problems and prove or disprove conjectures about the world around us. Course Learning Goals (including WA State Standards, Common Core Standards, National Standards): Geometry Unit 1 CCSS Standards Unit 1: Geometric Transformations In high school, students formalize much of the geometric exploration from middle school. In this unit, students develop rigorous definitions of three familiar congruence transformations: reflections, translations, and rotations and use these transformations to discover and prove geometric properties. Throughout the course, students will use transformations as a tool to analyze and describe relationships between geometric figures. Common Core State Standards Congruence — G--‐CO A. Experiment with transformations in the plane 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. B. Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Common Core State Standards for Mathematical Practice 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. Geometry Unit 2 CCSS Standards Unit 2: Angles and Lines This unit gives students the foundational tools for developing viable geometric arguments using relationships students studied in middle school related to lines, transversals, and special angles associated with them. Students learn how to combine true statements within a mathematical system to deductively prove other statements. Students should begin to see the structure of a mathematical system as they make conjectures and then prove statements involving lines and angles. Congruence — G--‐CO A. Experiment with transformations in the plane 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. C. Prove geometric theorems 9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Expressing Geometric Properties with Equations — G--‐GPE B. Use coordinates to prove simple geometric theorems algebraically 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. 7. Look for and make use of structure. Geometry Unit 3 CCSS Standards Unit 3: Triangles This unit explores basic theorems and conjectures about triangles, including the triangle inequality conjecture, the Triangle Sum Theorem, and theorems regarding centers of a triangle. Students explored some of these relationships in middle school but will build on their work in unit 2 with deductive reasoning and proof related to triangles in this unit. Students make and verify conjectures related to isosceles triangles and explore physical properties of the centroid of a triangle. In this unit, students also learn basic construction techniques and use these as they explore triangle properties. Throughout this unit, students will use the precise definitions developed in G--‐ CO.A.1. C. Prove geometric theorems. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. D. Make geometric constructions 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Circles — G--‐C A. Understand and apply theorems about circles 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Modeling with Geometry — G--‐MG A. Apply geometric concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★ Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategically. 8. Look for and express regularity in repeated reasoning. Geometry Unit 4 CCSS Standards Unit 4: Triangles Congruence This unit builds on students’ work with transformations in unit 1 and properties of triangles in unit 3 to formalize the definition of congruent triangles. Students reason to identify criteria for triangle congruence and use precise notation to describe the correspondence in congruent triangles. Congruence — G--‐CO B. Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. 7. Look for and make use of structure. Geometry Unit 5 CCSS Standards Unit 5: Similarity Transformations This unit moves away from rigid motion and focuses on dilations and similarity. Students prove theorems involving similarity and apply dilations and similarity to model situations in the real world. C. Prove geometric theorems 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Similarity, Right Triangles, and Trigonometry — G--‐SRT A. Understand similarity in terms of similarity transformations 1. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. B. Prove theorems involving similarity 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Expressing Geometric Properties with Equations — G--‐GPE B. Use coordinates to prove simple geometric theorems algebraically 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Modeling with Geometry — G--‐MG A. Apply geometric concepts in modeling situations 3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★ Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 8. Look for and express regularity in repeated reasoning. Evidence of Assessment Performance Tasks: Units are arranged so that students work through MathXL to master many of the objectives. Additional exercises are provided to develop application of newly learned concepts. Each unit concludes with both a skills-based assessment and one with applications. Other Evidence (self-assessments, observations, work samples, quizzes, tests and so on): Work in Math XL for school Quizzes and Unit tests in Math XL for school Types of Learning Activities Each unit follows the same basic pattern: 1. Work in MathXL for school 2. Target Activities (including additional work if needed to meet standard) 3. Unit Assessments Direct Instruction Indirect Instruction _x___Structured Overview ____Mini presentation __x__Drill & Practice ____Demonstrations __x__Other (List) __x__Problembased ___Case Studies ____Inquiry ____Reflective Practice ____Project ____Paper ____Concept Mapping ____Other (List) MathXL Experiential Learning ____ Virt. Field Trip ____Experiments ____Simulations ____Games ____Field Observ. ___Role-playing ____Model Bldg. ____Surveys ____Other (List) Independent Study ____Essays _x__Self-paced computer __x__Journals ____Learning Logs ____Reports ____Directed Study ____Research Projects ____Other (List) Interactive Instruction _x___Discussion ____Debates ___Role Playing ____Panels ____Peer Partner Learning ____Project team ____Laboratory Groups ____Think, Pair, Share ____Cooperative Learning ____Tutorial Groups ____Interviewing ____Conferencing ____Other (List) Other: Learning Activities These learning activities are aligned with the successful completion of the course learning goals and progress towards these learning activities will be reported monthly on a progress report. 1st Semester Geometry Unit 1 Geometric Transformations In high school, students formalize much of the geometric exploration from middle school. In this unit, students develop rigorous definitions of three familiar congruence transformations: reflections, translations, and rotations and use these transformations to discover and prove geometric properties. Throughout the course, students will use transformations as a tool to analyze and describe relationships between geometric figures. Lessons 1. Points, Lines, and Planes 2. Angles 3. Parallel and Perpendicular Lines 4. Reflections 5. Translations 6. 7. 8. 9. Rotations Compositions of Transformations Congruence and Transformations Dilations Unit 2 Angles and Lines This unit gives students the foundational tools for developing viable geometric arguments using relationships students studied in middle school related to lines, transversals, and special angles associated with them. Students learn how to combine true statements within a mathematical system to deductively prove other statements. Students should begin to see the structure of a mathematical system as they make conjectures and then prove statements involving lines and angles. Lessons 1. Pairs of Angles 2. Lines and Angles 3. Parallel Lines and Transversals 4. Proving Lines Parallel 5. Perpendicular Lines 6. Slopes of Lines 7. Lines in the Coordinate Plane Unit 3 Triangles This unit explores basic theorems and conjectures about triangles, including the triangle inequality conjecture, the Triangle Sum Theorem, and theorems regarding centers of a triangle. Students explored some of these relationships in middle school but will build on their work in unit 2 with deductive reasoning and proof related to triangles in this unit. Students make and verify conjectures related to isosceles triangles and explore physical properties of the centroid of a triangle. In this unit, students also learn basic construction techniques and use these as they explore triangle properties. Throughout this unit, students will use the precise definitions developed in G--‐CO.A.1. Lessons 1. Classifying Triangles 2. Angle Relationships in Triangles 3. Isosceles & Equilateral Triangles 4. Perpendicular and Angle Bisectors 5. Bisectors of Triangles 6. Median of Triangles Unit 4 Triangle Congruence This unit builds on students’ work with transformations in unit 1 and properties of triangles in unit 3 to formalize the definition of congruent triangles. Students reason to identify criteria for triangle congruence and use precise notation to describe the correspondence in congruent triangles Lessons 1. Transformations and Congruence 2. Congruence and Triangles 3. Triangle Congruence Postulates 4. Proving Triangle Congruence Through Transformations 5. Using Congruence Criteria Unit 5 Similarity Transformations This unit moves away from rigid motion and focuses on dilations and similarity. Students prove theorems involving similarity and apply dilations and similarity to model situations in the real world. Lessons 1. Ratios and Proportions 2. Similarity and Transformations 3. Ratios in Similar Polygons 4. Triangle Similarity 5. 6. 7. 8. Pythagorean Theorem and Similarity Triangle Similarity AA,SSS, and SAS Properties of Similar Triangles Using Proportional Relationships