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SENECA VALLEY SCHOOL DISTRICT CURRICULUM Course Title: Course Number: Grade Level(s): Periods per Week: Length of Period: Length of Course: Credits: Faculty Author(s): Date: Geometry 0324 9 5 42 Minutes Full Year 1.0 Mary McConaughy, Debbie Skinger May 17, 2016 COURSE DESCRIPTION: Geometry is an academically challenging course which includes an in-depth analysis of plane, solid, and coordinate geometry as they relate to both abstract mathematical concepts, as well as real-world problem situations. Significant emphasis is placed on algebra which is integrated throughout all units. The course is designed to focus on the key topics of geometry including inductive reasoning, deductive reasoning and proofs, perpendicular and parallel lines, triangles, quadrilaterals, proportions, right triangles and trigonometry, circles, surface area, and volume. PREREQUITE(S): 70% or better in Algebra 1 OR 90% or better and teacher recommendation in Algebra 1B. ESSENTIAL QUESTIONS: 1. How can patterns be used to describe relationships in mathematical situations? 2. How can recognizing repetition or regularity assist in solving problems more efficiently? 3. How are spatial relationships, including shape and dimension, used to draw, construct, model, and represent real situations or solve problems? 4. How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving? 5. How can geometric properties and theorems be used to describe, model, and analyze situations? Geometry –1– COURSE OUTLINE I. OBJECTIVES (PA standard) Tools of Geometry A. Inductive Reasoning Define and apply inductive reasoning to real-life situations B. Number Patterns Find the next term in number patterns C. Picture Patterns Draw the next picture in a sequence of picture patterns D. Points, Lines & Planes Use terminology and notation of points, lines and planes Apply the postulates of Geometry Use terminology and notation of segments, rays, parallel and skew lines, and parallel planes E. Measuring Segments & Angles Determine measures of segments and angles Define and illustrate line and angle relationships F. Defining Line and Angle Relationships Define and classify polygons and related terms G. Defining Polygons Define and classify triangles and their related parts H. Defining Triangles Define and classify special quadrilaterals and their related parts I. J. Defining Special Quadrilaterals Solve statement problems by first translating each into a labeled diagram Translating Word Problems into Diagrams Geometry –2– COURSE OUTLINE OBJECTIVES (PA standard) II. Geometric Construction A. Constructing Perpendicular Bisectors Construct perpendicular bisectors given a line segment B. Construct Perpendiculars Construct perpendicular lines given a point on a line and a point not on a line C. Construct Angle Bisectors Construct angle bisectors D. Construct Medians, Altitudes, and Angle Bisectors in Triangles Construct medians, altitudes, and angle bisectors in triangles Geometry –3– COURSE OUTLINE OBJECTIVES (PA standard) III. Exploring Geometry A. Discovering Angle Relationships B. Triangle Sum Conjecture C. Polygon Sum Conjecture D. Exterior Angles of a Polygon E. Triangle Inequalities F. Properties of Isosceles Triangles G. Properties of Parallel Lines Use properties of angles formed by intersecting lines to find the measures of missing angles. (G.2.2.1.1) CC.2.3.HS.A.3 To find missing angle measures by using linear pair and vertical angles conjectures. Identify and/or use properties of triangles. (G.1.2.1.1) CC.2.3.HS.A.3, CC.2.3.HS.A.13 Solve problems by applying the triangle sum conjecture. Identify and/or use properties of regular polygons. (G.1.2.1.4) CC.2.3.HS.A.3, CC.2.3.HS.A.13 Solve problems by applying the polygon exterior sum conjecture. Identify and/or use properties of triangles. (G.1.2.1.1) CC.2.3.HS.A.3, CC.2.3.HS.A.13 Determine and apply the triangle inequality conjecture. Identify and/or use properties of isosceles and equilateral triangles. (G.1.2.1.3) CC.2.3.HS.A.3, CC.2.3.HS.A.13 Use properties of angles formed when two parallel lines are cut by a transversal to find the measures of missing angles. (G.2.2.1.2) CC.2.3.HS.A.3 Geometry –4– COURSE OUTLINE IV. Special Quadrilaterals OBJECTIVES (PA standard) Identify and/or use properties of quadrilaterals. (G.1.2.1.2) CC.2.3.HS.A.3, CC.2.3.HS.A.13 A. Properties of Trapezoids Apply properties of trapezoids. B. Properties of Mid-segments Apply properties of mid-segments in triangles and trapezoids. C. Properties of Parallelograms D. Properties of Special Parallelograms Apply properties of parallelograms. Apply properties of rectangles, rhombi and squares. Geometry –5– COURSE OUTLINE V. Congruent Triangles A. Congruent Figures B. Triangle Congruence by SSS and AAA? C. Triangle Congruence by SAS and SSA? D. Triangle Congruence by ASA and AAS? E. Using Congruent Triangles: CPCTC F. OBJECTIVES (PA standard) Identify and/or use properties of congruent and similar polygons or solids. (G.1.3.1.1) CC.2.3.HS.A.2, CC.2.3.HS.A.5, CC.2.3.HS.A.6 Define and apply congruent polygons, segments and angles. Determine congruent triangles by Side-Side-Side Congruence Conjecture. Determine congruent triangles by Side-Angle-Side Congruence Conjecture. Determine congruent triangles by Angle-Side-Angle and AngleAngle-Side Congruence Conjectures. Overlapping Triangles Determine whether segments or angles are congruent by using CPCTC. Determine congruent triangles by SSS, SAS, ASA and AAS Congruence Conjectures in overlapping triangles. Geometry –6– COURSE OUTLINE VI. Deductive Reasoning & Geometric Proof A. Deductive Reasoning B. Premises of Geometry OBJECTIVES (PA standard) Identify and/or use properties of triangles. (G.1.2.1.1) CC.2.3.HS.A.3, CC.2.3.HS.A.13 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction). (G.1.3.2.1) CC.2.3.HS.A.3, CC.2.3.HS.A.6, CC.2.3.HS.A.8, 2.2.HS.C.9 C. Algebraic Proofs D. Proving Angle Pair Relationships E. Proving Properties of Parallel Lines F. Proving Parallel Lines G. Proving Congruent Triangles H. Using CPCTC in Proofs I. Using Overlapping Triangles in Proofs J. Indirect Proofs Define and apply deductive reasoning to real-life situations. Draw conclusions from given information and justify these using premises (definitions and undefined terms, properties of algebra, equality, and congruence, postulates of geometry, and previously accepted/proven geometric theorems for geometric arguments Write simple algebraic proofs Prove and apply theorems about angles and their corollaries. (Vertical Angles, Congruent Supplements, and Congruent Complements theorems, and their corollaries) Prove and apply properties of parallel lines intersected by a transversal. (Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, Same-Side Interior Angles Theorem, Same-Side Exterior Angles Theorem) Prove lines parallel. (Converses of the Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, Same-Side Interior Angles Theorem, Same-Side Exterior Angles Theorem) Prove two triangles congruent using SSS, SAS, ASA, and AAS Congruence Postulates. Prove that parts of two triangles are congruent using triangle congruence and CPCTC Prove two triangles or parts congruent in overlapping triangles. Write indirect proofs Geometry –7– COURSE OUTLINE VII. Circles A. Circle Parts B. Properties of Chords, Arcs & Central Angles OBJECTIVES (PA standard) Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle. (G.1.1.1.1) CC.2.3.HS.A.8, CC.2.3.HS.A.9, CC.2.3.HS.A.13 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle. (G.1.1.1.2) CC.2.3.HS.A.8, CC.2.3.HS.A.9, CC.2.3.HS.A.13 C. Tangent Properties D. Inscribed Angles & their Intercepted Arcs Use chords, tangents, and secants to find missing arc measures or missing segment measures. (G.1.1.1.3) CC.2.3.HS.A.8, CC.2.3.HS.A.9, CC.2.3.HS.A.13 E. Circumference/Diameter Ratio Define, identify and use proper notation for circles and their parts F. Determine and apply properties of chords, arcs and central angles in circles Arc Length G. Other Angle Relationships in Circles H. Segment Lengths in Circles Determine and apply properties of tangents in circles (Tangent Theorem and Tangent Segments Theorem) Apply properties of inscribed angles and their intercepted arcs in circles Find the area and circumference of a circle. Solve problems involving area and circumference of a circle(s). (M07.CG.2.2.1) CC.2.3.7.A.1 Define and apply arc length theorem Find the measures of angles formed by chords, secants and tangents Find the lengths of segments associated with circles Geometry –8– COURSE OUTLINE VIII. Area and Surface Area A. Areas of Rectangles and Parallelograms OBJECTIVES (PA standard) Determine the area of triangles and special quadrilaterals (i.e., square, rectangle, parallelogram, rhombus, and trapezoid. (M06.C-G.1.1.1) CC.2.3.6.A.1 B. Areas of Triangles and Trapezoids Find the measurement of a missing length, given the perimeter, circumference, or area. (G.2.2.2.2) CC.2.3.HS.A.3, CC.2.3.HS.A.9, CC.2.3.HS.C.1 C. Areas of Regular Polygons Solve multi-step problems involving area and perimeter of rectangles and parallelograms D. Areas of Circles Solve multi-step problems involving area and perimeter of triangles and trapezoids E. Areas of a Sector, a Segment and an Annulus F. Irregular Areas G. Surface Area of Prisms, Pyramids, Cones, Cylinders and Spheres H. Maximizing Area I. J. Solve multi-step problems involving area and perimeter of regular polygons Changing Values in Linear Dimension Geometric Probability Find the area and circumference of a circle. Solve problems involving area and circumference of a circle(s). (M07.CG.2.2.1) CC.2.3.7.A.1 Find the area of the sector of a circle. (G.2.2.2.5) CC.2.3.HS.A.3, CC.2.3.HS.A.9, CC.2.3.HS.C.1 Find the area of circle portions (sector, segment and annulus) Develop and/or use strategies to estimate the area of a compound/composite figure. (G.2.2.2.4) CC.2.3.HS.A.3, CC.2.3.HS.A.9, CC.2.3.HS.C.1 Estimate area, perimeter, or circumference of an irregular figure. (G.2.2.2.1) CC.2.3.HS.A.3, CC.2.3.HS.A.9, CC.2.3.HS.C.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet. (G.2.3.1.1) CC.2.3.HS.A.14, CC.2.3.HS.A.12 Find the measurement of a missing length, given the surface area or volume. (G.2.3.1.3) CC.2.3.HS.A.14, CC.2.3.HS.A.12 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon. (G.2.2.2.3) CC.2.3.HS.A.3, Geometry –9– COURSE OUTLINE OBJECTIVES (PA standard) CC.2.3.HS.A.9, CC.2.3.HS.C.1 Describe how a change in the linear dimension of a figure affects its perimeter, circumference, and area. (G.2.2.3.1) CC.2.3.HS.A.9, CC.2.3.HS.A.8 Describe how a change in the linear dimension of a figure affects its surface area or volume. (G.2.3.2.1) CC.2.3.HS.A.13 Use area models to find probabilities. (G.2.2.4.1) CC.2.3.HS.A.14 Geometry –10– OBJECTIVES (PA standard) COURSE OUTLINE IX. Right Triangles A. Simplifying, Multiplying & Rationalizing Radicals Simplify radicals Multiply radicals Rationalize radicals B. The Pythagorean Theorem and Its Converse C. Pythagorean Primitives and their Multiples D. Isosceles Right and 30-60-90 triangles E. Right Triangle Application Use the Pythagorean theorem to write and/or solve problems involving right triangles. (G.2.1.1.1) CC.2.3.HS.A.7, CC.2.2.HS.C.9 Use the Pythagorean primitives and their multiples to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions Use isosceles right triangles and 30-60-90 triangles to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (M08.C-G.2.1.2) CC.2.3.8.A.3 Geometry –11– COURSE OUTLINE OBJECTIVES (PA standard) X. Volume A. Classifying Polyhedrons Classify polyhedrons by their number of faces Determine and use Euler’s Formula B. Prisms & Pyramids Identify and/or use properties of pyramids and prisms. (G.1.2.1.5) CC.2.3.HS.A.3, CC.2.3.HS.A.13 C. Solids with Curved Surfaces Identify and/or use the properties of a sphere or cylinder. (G.1.1.1.4) CC.2.3.HS.A.8, CC.2.3.HS.A.9, CC.2.3.HS.A.13 D. Volumes of Prisms and Cylinders D-F: Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. (G.2.3.1.2) CC.2.3.HS.A.14, CC.2.3.HS.A.12 E. Volumes of Pyramids and Cones F. Volumes of Spheres G. Changing Values in Linear Dimension D-F: Find the measurement of a missing length, given the surface area or volume. (G.2.3.1.3) CC.2.3.HS.A.14, CC.2.3.HS.A.12 Describe how a change in the linear dimension of a figure affects its surface area or volume. (G.2.3.2.1) CC.2.3.HS.A.13 Geometry –12– COURSE OUTLINE OBJECTIVES (PA standard) XI. Similarity A. Using Ratios and Proportions Use ratios and proportions to solve real-life applications B. Similar Polygons B-C: Identify and/or use properties of congruent and similar polygons or solids. (G.1.3.1.1) CC.2.3.HS.A.2, CC.2.3.HS.A.5, CC.2.3.HS.A.6 C. Proving Triangles Similar D. Indirect Measurement E. Corresponding Parts of Similar Triangles F. Perimeter, Area and Volume of Similar Figures D-E: Identify and/or use proportional relationships in similar figures. (G.1.3.1.2) CC.2.3.HS.A.2, CC.2.3.HS.A.5, CC.2.3.HS.A.6 Identify and/or use properties of congruent and similar polygons or solids. (G.1.3.1.1) CC.2.3.HS.A.2, CC.2.3.HS.A.5, CC.2.3.HS.A.6 Geometry –13– COURSE OUTLINE OBJECTIVES (PA standard) XII. Trigonometry A. Trigonometric Ratios Use trigonometric ratios to write and/or solve problems involving right triangles. (G.2.1.1.2) CC.2.3.HS.A.7, CC.2.2.HS.C.9 Geometry –14– COURSE OUTLINE OBJECTIVES (PA standard) XIII. Coordinate Geometry A. Distance Formula A-B: Calculate distance and a midpoint algebraically. (G.2.1.2.1) CC.2.3.HS.A.11 B. Midpoint Formula C. The Slope of a Line C-D: Relate slope to perpendicularity and/or parallelism. (G.2.1.2.2) CC.2.3.HS.A.11 D. Slope of Parallel and Perpendicular Lines E. Classifying Quadrilaterals by Coordinate Methods Use slope, distance, and/or midpoint between two points on a coordinate plane to establish properties of a 2-dimensional shape. (G.2.1.2.3) CC.2.3.HS.A.11 Geometry –15–