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Transcript

Geometry Piscataway High School Teacher: Mrs. Styles Email: [email protected] Course Title: Geometry/Essentials of Geometry/Honors Geometry/Geometry 9 Textbook: Geometry (2015), 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 7 Days Unit 2 Isometry and Transformations 14 Days Unit 3 Introduction to Reasoning and Proofs 13 Days Unit 4 Triangles and Triangle Congruence 15 Days Unit 5 Quadrilaterals and Coordinate Proofs 10 Days Unit 6 Similarity 15 Days Unit 7 Right Triangles and Trigonometry 13 Days Unit 8 Circles 20 Days Unit 9 Modeling in Three Dimensions 12 Days Unit 10 Analytic Geometry 8 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 5 Unit 1 7 Days Introduction to Geometry HSG.CO.A.1 HSG.CO.D.12 [1.1 & 1.2] Unit 2 14 Days HSG.CO.A.2 HSG.CO.A.3 HSG.CO.A.4 HSG.CO.A.5 HSG.CO.A.6 HSG.CO.B.6 HSG.CO.B.7 Isometry and Transformations [1.3, Modules 2 & 3] Identify basic geometry vocabulary (points, lines, planes) Use undefined terms such as points, lines and planes to create axiomatic system to build other defined geometric terms (i.e. line segment, ray, coplanar, parallel, collinear, postulate, midpoint, segment bisector) Construct a copy of a line segment (discuss radius of circles when doing construction) and find the length using the distance formula (Pythagorean Theorem). Use the construction of copying a line to discuss the Segment Addition postulate and solve problems. Construct a bisector of a segment (midpoint) and use the midpoint formula. Identify vocabulary related to angles (angles, vertex, classifying angles, angle bisector) Name angles based on proper notation (ex: 𝑚∠𝑀𝐾𝐺 is the same as 𝑚∠𝐺𝐾𝑀) Constructions - copy angle, bisect angle (options: with compass and straightedge, string, paper folding, and technology, tracing paper) – in conjunction w/book. Use Segment and Angle Addition Postulates Identify vocabulary related to transformations (image, preimage, rigid transformation, translation, reflection, rotation). Rigid Transformations o Translations (using vectors), reflections, and rotations o Perform transformations both on and off the coordinate plane o Use coordinate notation to map points from preimage to image and vice versa. o Use slope and midpoint formulas to find the equation of the line of reflection o Use tracing paper investigations for translation and reflection/ruler and protractor for rotation -(refer to lessons 2.1, 2.2, 2.3) Identify lines of symmetry and angles of rotational symmetry for figures Perform sequences of transformations to map congruent figures onto one another Given a sequence of transformations, identify the sequence of coordinates using the proper notation. Define congruence through rigid transformations on the coordinate plane and validate using distance formula and midpoint formula Use proper notation to write congruence statements for congruent figures and identify corresponding parts of congruent figures (ex: ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹) Discuss and apply congruence properties and use congruent corresponding parts in a proof. Assessment for Unit 3 administered by the end of Cycle 7 Unit 3 13 Days HSG.CO.C.9 HSG.CO.D.12 HSG.GPE.B.5 Introduction to Reasoning and Proofs [1.4, Module 4] Mandatory Performance Tasks: Lesson 4.2 (p. 184) – Do after Lesson 4.3 Review properties of angles (complements, supplements, vertical angles, linear pairs) Identify vocabulary related to reasoning (inductive reasoning, deductive reasoning, conjecture, theorem, conditional statements and counterexamples) Discuss postulates about Lines and Planes Complete simple proofs and algebraic proofs using properties and postulates to justify statements (refer to examples in book and dropbox) Prove vertical angles are congruent using the proof method of your choice (for all proofs). Review parallel line properties and angle relationships (corresponding, alternate interior, alternate exterior, consecutive/same-side interior) o Prove angle theorems related to parallel lines cut by a transversal Discuss converse statements o Prove that two lines are parallel for angle relationships. 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 (Perpendicular Bisector Theorem) 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 10 Unit 4 15 Days HSG.CO.C.10 HSG.CO.B.7 HSG.CO.B.8 HSG.CO.D.12 Triangles and Triangle Congruence [Module 7(intro), 5, 6, 7] Mandatory Performance Tasks: 6.2 (p. 294) Vocabulary embedded within each module – begin with defining and constructing Isosceles and Equilateral Triangles to use in Module 5 (Module 7.2). 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 Use corresponding Parts of Congruent Triangles are Congruent in proofs o Justify angle bisector and perpendicular bisector constructions using triangle congruence Triangle Theorems [Module 7.1 & 7.2] – use the following theorems in proofs as well as algebraic examples o Triangle Sum Theorem (using auxiliary lines), Polygon Angle Sum Theorem (using auxiliary diagonals), Exterior Angle Theorem, Isosceles Triangle Theorem (Base Angle Theorem), Equilateral Triangle Theorem Assessment for Unit 5 administered by the end of Cycle 12 Unit 5 10 Days HSG.CO.C.11 HSG.GPE.B.4 HSG.GPE.B.5 HSG.GPE.B.7 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 Review 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 Perform coordinate proofs to identify the type of quadrilaterals on the coordinate plane o Calculate area and perimeter of triangles and quadrilaterals on and off of the coordinate plane Assessment for Unit 6 administered by the end of Cycle 15 Unit 6 15 Days Similarity HSG.SRT.A.1 HSG.SRT.A.1.A HSG.SRT.A.1.B HSG.SRT.A.2 HSG.SRT.A.3 HSG.SRT.B.4 HSG.SRT.B.5 [Module 11, 12, 8.4] Define similarity and create similarity statements using proper notation (ex: ∆𝐴𝐵𝐶~∆𝑋𝑌𝑍). 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 Discuss similarity through the definition of midsegments of a triangle and prove the triangle midsegment theorem using similarity (http://mathbitsnotebook.com/Geometry/SegmentsAnglesTriangles/SATMidSegments.html) 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 18 Unit 7 13 Days HSG.SRT.C.6 HSG.SRT.C.7 HSG.SRT.C.8 Trigonometry [Module 13] Review & prove Pythagorean Theorem Prove the Converse to the Pythagorean Theorem Discuss Special Right trianngles. 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 (examples in Dropbox) Calculate area and perimeter of triangles, using trig to find altitudes Assessment for Unit 8 administered by the end of Cycle 22 Unit 8 20 Days HSG.CO.A.1 HSG.GPE.A.1 HSG.C.A.1 HSG.C.A.2 HSG.C.A.4 HSG.C.B.5 HSG.CO.D.13 Circles [Lessons 17.1, 15.115.3, 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 inscribed polygons (limits, as the number of sides increases what happens?) Construct a square in a circle [15.2] https://www.khanacademy.org/math/geometry/geometric-constructions/polygonsinscribed-in-circles/v/constructing-square-inscribed-in-circle Assessment for Units 9 & 10 administered by the end of Cycle 25 (time permitting) Unit 9 12 Days HSG.GMD.A.1 HSG.GMD.A.3 HSG.GMD.B.4 HSG.MG.A.1 HSG.MG.A.2 HSG.MG.A.3 HSG.GPE.A.2 Modeling in Three Dimensions [Lesson 17.2, Module 18, 19, 20] Unit 10 8 days Analytical Geometry HSG.C.A.3 [Module 8] 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 Inscribed and Circumscribed Triangles [Module 8] o Construct circumscribed and inscribed circles of a triangle and justify using the theorems o Locate circumcenter and incenter of a triangle on the coordinate plane o Theorem: Medians of a Triangle meet at a point (prove using construction) o Application of incenter, circumcenter, and median