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Transcript

Essentials of Geometry Piscataway High School Teacher: Your Name Email: [email protected] Course Title: Geometry Textbook: Geometry (2014), HMH (Kanold, Burger, et al.) Course Overview Full year course: 5.0 credits (Honors Geometry has a 5 point weight added to the final grade) Prerequisite: Algebra 1 Description: Geometry is a college prep course that uses constructions and transformations to explore geometric relationships and figures. Students will use formal logic and various forms of proof in order to explore topics such as congruence and similarity, properties of geometric shapes, measurement of plane and solid figures, and trigonometry. The Geometry course is structured as follows: Unit Topic Length Unit 1 Constructions and Introduction to Geometry 6 Days Unit 2 Isometry and Transformations 14 Days Unit 3 Introduction to Proof 12 Days Unit 4 Triangles and Triangle Congruence 17 Days Unit 5 Quadrilaterals and Coordinate Proofs 10 Days Unit 6 Similarity 12 Days Unit 7 Right Triangles and Trigonometry 17 Days Unit 8 Circles 20 Days Unit 9 Modeling in Three Dimensions 12 Days 1 Geometry Scope and Sequence Unit Timing Topic (1 day = 1 hour) Concepts and Skills Assessment for Units 1 and 2 administered by the end of Cycle 4 Unit 1 6 Days Constructions and Introduction to Geometry [1.1, 1.2, & 1.4] Unit 2 14 Days Mandatory Performance Tasks: 1.4 #28 (p.56) Isometry and Transformations [1.3, Modules 2 & 3] Identify basic geometry vocabulary (points, lines, planes, intersections, etc.) Use undefined terms such as points, lines and planes to create axiomatic system to build other geometric terms (i.e. line segment, ray, angle, circle, etc.) Draw geometric diagrams and name/label geometric figures using the proper notation and symbols Constructions (with compass and straightedge, string, reflective devices, paper folding, and technology): copy segment, bisect segment, copy angle, bisect angle – in conjunction w/ book Use the distance, midpoint, and slope formulas Use the Segment Addition Postulate and Angle Addition Postulate Introduce conditional statements and counterexamples Complete simple proofs and algebraic proofs using properties and postulates to justify statements Rigid Transformations o Translations (using vectors), reflections, and rotations o Perform transformations both on and off the coordinate plane o Represent transformations mapping points from preimage to image using coordinate notation o Use slope and midpoint formulas to find the equation of the line of reflection o Use tracing paper investigations with ruler and protractor – should we find a specific activity? Define congruence through rigid transformations on the coordinate plane and validate using distance formula and midpoint formula Define compositions of transformations to map congruent figures onto one another Write congruence statements for congruent figures and identify corresponding parts of congruent figures Identify lines of symmetry and angles of rotational symmetry for figures Assessment for Unit 3 administered by the end of Cycle 6 Unit 3 12 Days Parallel & Perpendicular Lines [Module 4] Mandatory Performance Tasks: Lesson 4.2 (p. 189) – Do after Lesson 4.3 Review properties of angles (complements, supplements, vertical angles, linear pairs) Prove vertical angles are congruent Review parallel line properties and angle relationships (corresponding, alternate interior, alternate exterior, consecutive interior) o Prove angle theorems related to parallel lines cut by a transversal Discuss converse statements o Prove lines are parallel Construct a line parallel to a given line o Prove the construction using the Converse of the Corresponding Angles Theorem Construct perpendicular bisectors o Prove all points on a perpendicular bisector are equidistant from the endpoints of the segment Graph, find the slope of, and write equations for perpendicular and parallel lines Geometry Scope and Sequence Unit Timing Topic (1 day = 1 hour) Concepts and Skills Assessment for Unit 4 administered by the end of Cycle 9 Unit 4 17 Days Triangles and Triangle Congruence [Module 7, 5, 6, 8] Mandatory Performance Tasks: 6.2 (p. 204) Triangle Theorems [Module 7] o Triangle Angle Sum Theorem (using auxiliary lines), Polygon Angle Sum Theorem (using auxiliary diagonals), Exterior Angle Theorem, Isosceles Triangle Theorem (Base Angle Theorem), Equilateral Triangle Theorem Triangle Congruence Theorems [Modules 5 & 6] o AAS, ASA, SSS, SAS, HL congruence theorems o Prove triangles congruent using two-column, paragraph, and coordinate proofs o Use rigid motions to map congruent figures onto one another o Corresponding Parts of Congruent Figures are Congruent o Justify constructions using triangle congruence Inscribed and Circumscribed Triangles [Module 8] o Construct circumscribed and inscribed circles of a triangle o Locate circumcenter and incenter of a triangle on the coordinate plane o Application of incenter and circumcenter o Theorem: Medians of a Triangle meet at a point (prove using construction) o Airport Problem Assessment for Unit 5 administered by the end of Cycle 11 Unit 5 10 Days Quadrilaterals and Coordinate Proofs [Module 9, 10] Review properties of quadrilaterals o Parallelogram, rectangle, rhombus, square, trapezoid, isosceles trapezoid, kite o Discuss sufficient criteria to classify a quadrilateral Coordinate Proofs o Use distance formula and midpoint formula o Review slopes of parallel and perpendicular lines o Use distance, midpoint, and/or slopes to classify quadrilaterals in the coordinate plane o Position figures appropriately in the plane and use general coordinates to prove properties o Calculate area and perimeter of triangles and quadrilaterals in the coordinate plane Assessment for Unit 6 administered by the end of Cycle 14 Unit 6 12 Days Similarity [Module 11, 12] Define similarity and create similarity statements Discuss similarity through the use of dilations: o Dilations completed from the origin and other points on coordinate plane o Scale factor and center of dilation to explain similarity o List corresponding parts (congruent or proportional) o Finding the center of dilation given a dilation on a coordinate plane o Prove all circles are similar o Subdivide a Segment in a Given Ratio (partition a line using different ratios) Establish Triangle Similarity Postulates o AA, SAS, SSS (prove using above discussions/activities) o Prove Triangle Proportionality Theorem (include midsegments & coordinate proofs) Modeling: Using triangle similarity to solve indirect measurement problems Similarity in right triangles using altitude (Lesson 12.4) Geometry Scope and Sequence Unit Timing Topic (1 day = 1 hour) Concepts and Skills Assessment for Unit 7 administered by the end of Cycle 17 Unit 7 17 Days Trigonometry [Module 13] Review & prove Pythagorean Theorem Prove the Converse to the Pythagorean Theorem Special right triangles and Pythagorean triples Use similar triangles to define trigonometric ratios and their inverses in right triangles o Demonstrate and apply relationship and sine and cosine of complementary angles o Discuss range of trig ratios (e.g., certain ratios cannot have a value over 1) Apply trigonometry and Pythagorean Theorem to solve word problems o Include angles of elevation and depression Calculate area and perimeter of triangles, using trig to find altitudes Assessment for Unit 8 administered by the end of Cycle 21 Unit 8 20 Days Circles [Lessons 17.1, 15.1-15.3, 15.4 (chords only), 15.5???, 16.2, 16.3] Review definition of circle and associated vocabulary Use definition of circle to derive equation o Complete the square to find radius and center Review Constructions of inscribed and circumscribed polygons o Prove properties of angles for a quadrilateral inscribed in a circle (opposite angles are supplementary) Identify Circle relationships o Relationship between inscribed, central, and circumscribed angles o Inscribed angles on diameter o Radius is perpendicular to tangent at point of intersection Find arc length and sector area o Include application problems o Define radians Include informal discussion of area and perimeter of circumscribed polygons (limits, as the number of sides increases what happens?) Assessment for Unit 9 administered by the end of Cycle 23 Unit 9 12 Days Modeling in Three Dimensions [Lesson 17.2, Module 18, 19, 20] Relationships between 2D and 3D Shapes o Identify nets and review surface area o Discuss cross sections of conic o Derive equation of parabola (with directrix & focus) o Identify 3-D objects created by rotating 2-D objects Discuss volume using Cavalieri’s Principle o Informal limit process to find formulas o Volume of prisms, pyramids, cones and spheres o Volume of compound figures Complete application problems with 3-D shapes regarding surface area and volume o Density; Designing packaging to maximize volume or minimize surface area