Accelerated Geometry: Course Level: 10th or 11th grade (students who have not had Geomtery I) Course Code: MA?? Course Length: ( Pre-requisite

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1 Accelerated Geometry: Course Level: 10 th or 11 th grade (students who have not had Geomtery I) Course Code: MA?? Course Length: (1 semester) Pre-requisite: Algebra I (MA113/123) Description: This accelerated course covers the topic usually included in one year of geometry, including Euclidean and non-euclidean geometry with an emphasis on reasoning and practical application. This course integrates coordinate and transformational geometry. Students will learn relationships between angles, parallel lines, perpendicular lines, triangles trigonometry, quadrilaterals, and circles. Students will find areas and volumes of designated geometric shapes, make basic constructions and demonstrate competence in inductive and deductive proofs. Graphing calculators will be used in this course. Objectives demonstrate understanding of points, lines and planes and their relationships to drawings demonstrate an understanding of angle measurement and its application to parallel lines and transversals in both 2 dimensional and 3 dimensional space demonstrate understanding of and application of the distance formula. apply and demonstrate geometric principals through constructions investigate relationships of geometric shapes using symmetry analyze and generalize mathematical models of patterns to make inferences, and predictions using transformations : rotations, translations, reflections, dilations recognize similarities and generalize patterns to make predictions about polygons (2 dimensional shapes) and polyhedra (3 dimensional shapes), their angle measures and derive formulas that apply to the interior angles, exterior angles, and number of diagonals. understand, derive, and apply the Pythagorean Theorem to solve problems and extend it to the special right triangles and their ratios, indirect measurement and proof of the Pythagorean Theorem.

2 develop an understanding of the relationships among parts of circles and between circles and various angles, segments, and arcs, generalizing their findings to theorems and apply them to geometric proofs. demonstrate the use of real and irrational numbers to solve geometric problems involving perimeter, circumference, area (regular, lateral, and surface), and volume explore similar and congruent geometric figures and apply their findings to angle measures, length of sides, area and volume of these figures, generalizing their findings to triangle similarity and congruence theorems and apply them to geometric proofs. analyze and use reasoning in geometry: proofs, hypothesis, logic, deductive reasoning (using definitions, theorems, postulates) and inductive reasoning. develop and explore the 1)trigonometric ratios from both right triangles and unit circles, 2) trigonometric identities and apply this knowledge to solving problems Technology and Other Resources Holt, Rinehart and Winston Geometry Textbook Graphing Calculator Compass and Protractor Grading System Tests/Projects 50%, Quizes 20% Homework 20% Class Participation 10%

3 Syllabus Unit 1: Basics of Geometry and Geometric Reasoning(9 weeks) All students will understand, draw, and apply to real life the representation of 1) the building blocks of geometry, 2)lines, there slopes and equations, 3)perpendicular lines and their angle relationships, 4)parallel lines and their angle relationships, and 5)planes and explain there interconnection. Main Topics: Chapters 1-5 A. The building blocks of Geometry and their relationship to the real world B. Reasoning in Geometry C. Lines and distance in Geometry D. Polygons 1. Triangles. 2. Polygons with more than three sides E. Transformations and their relationship/applications to drawings and the real world 1. Basic Mappings. 2. Products and Symmetry. Learning Outcomes: ASW Understand and identify the undefined terms of geometry. Apply basic terminology and notation (points, lines, planes, rays, angles, segments, vertex, union, intersection and triangle) to everyday life Find the lengths of segments and the measures of angles Classify angles. Investigate postulates about points lines and planes and apply to drawings Apply properties of vertical angles, linear pairs of angles, supplementary angles, complementary angles, and right angles to solve multi-step problems Understand the meaning of the term proof Create logical chains of information Write and explain two-column proofs and paragraph proofs Draw diagrams of statements to assist in logical explanations Evaluate inductive and deductive arguments Use terms and concepts from formal logic (e.g., axiom, proposition, negation, truth and falsity, implication, if and only if, converse, inverse, contrapositive) to reason about mathematical concepts. Solve multi-step problems involving angle measure, length, perimeter, or area of scalene, isosceles, equilateral, acute, right or obtuse triangles Given a line on the coordinate plane and a point not on the line find the slope of the line, write its equation, write the equation of the line through the point parallel to the given line and perpendicular to the given line.

4 Know, justify, and use the properties of median, altitudes, perpendicular bisectors of sides, and angles bisectors of a triangle. Recognize and explain how the rigidity and stability of triangles is utilized in real world applications such as trusses, tripods, tricycles, etc. Use the Hinge Theorem to explain how a triangular linkage (a triangle with a side that may vary in length) allows for controlled length and/or angle change, and yet preserves rigidity Prove theorems about the angle sum of a triangle and the exterior angles of a triangle. Solve multi-step problems involving angle measure, length, perimeter, or area involving squares, rectangles, parallelograms, kites, and trapezoids Know, justify, and use properties of squares, rectangles, rhombi, parallelograms, kites an trapezoids Describe the hierarchies among quadrilaterals Derive the formulas for the interior and exterior angles of quadrilaterals Prove theorems about the interior and exterior angle sums of quadrilaterals Dissect any polygon into non-overlapping triangles and use properties of triangles to analyze the polygons and make drawings showing these dissections Know, justify, and use properties for squares, rectangles, rhombi, parallelograms, kites and trapezoids, and find their use in every day life Use drawing to describe and derive a rule for the symmetries of regular polygons and describe these symmetries Explain the relation between the area formula of a triangle and the area formula of a parallelogram Explain relations between the area formulas of various quadrilaterals Derive, use and justify the formulas for the perimeter and area of regular n-gons Derive, use and justify the formulas to find the interior and exterior angles of a regular n-gon Explain why only certain polygons will tile the plane and create tessellations using one or more polygons. Describe the symmetries of regular polygons Know and use the Pythagorean Theorem, its converse, and Pythagorean Triples Know, justify and use the properties of 30 o -60 o -90 o triangles and 45 o -45 o -90 o triangles. Know and use the properties of the centroid, circumcenter, orthocenter, and the incenter of triangles Describe symmetries of triangles. Prove that triangles are similar by using SSS, SAS, and AA conditions for similarity Find the image of a figure under a given isometry Given two figures that are images of each other under an isometry, describe completely the isometry Give a rule or mapping to describe a given isometry. Find the image of a figure under composition of two or more isometries, and tell whether the image is a reflection, rotation, translation, or glide reflection image of the preimage Use transformations to create original figures that will tessellate (tile) the plane,

5 and analyze tessellations of the plane. Given two congruent figures, tell which single transformation can be used to map one to the other Use the Two Reflection Theorem to solve problems Find the image of a figure under a dilation, given a center and size change factor Give a rule or mapping to describe a dilation with center at the origin and magnitude k. Find the image of a figure under a composite of a dilation and an isometry, with or without using coordinates. Products/Projects Working in nine groups of four, students are challenged to design and build one of the greens in a 9-hole miniature golf course. Each green needs to be constructed so that a "hole-in-one" can be achieved through three unique paths-none of which involved a straight shot at the hole. Instead, students need to construct their greens so that the "hole-in-one" shots would result from a trajectory that involved a ricochet, or a series of ricochets. Major Assessments The above project and 5 section tests

6 Syllabus Unit 2:Two and Three Dimensional Shapes:their relationships to life (9 weeks) All students will demonstrate knowledge of 2-dimensional and 3-dimensional shapes and their application to the world of today. Main Topics: Chapters 6-10 F. 3-diminesional shapes G. Similar Shapes and their relationship/applications to the real world H. Circles and their relationship/applications to the real world I. Geometric Constructions J. Trigonometry: Right Triangles and their relationship/applications to drawings and the real world K. Coordinate Geometry and their relationship/applications to the real world Learning Outcomes: ASW Solve multi-step problems involving surface area and volume of pyramids and prisms Describe symmetries of pyramids and prisms Know and use Euler s formula relating the number of vertices, faces, and edges in polyhedra Identify and describe five Platonic solids (regular polyhedra): tetrahedron, cube, octahedron, dodecahedron, icosahedron, and recognize any of the five regular polyhedra given its net. Explain why there are exactly five regular polyhedra Explain the relation between the volume formulas for pyramids and prisms Solve multi-step problems involving surface area and volume of cones, cylinders, hemispheres, and spheres Describe symmetries of cones, cylinders, hemispheres, and the spheres Explain the relations between the vole formulas fore cones and cylinders. Analyze the efficiency of the various 3-D forms by calculating the ratio of the surface area to volume Describe the relationship between multisided polygons and circles Identify and/or draw cross-sections of 3 dimensional figures Solve multi-step problems involving circumference and area of circles Explain how the area of a circle and the area of a parallelogram are related. Know and use properties of chords, tangent lines and tangent segments, and secant lines and secant segments of circles Know and use properties of central angles, inscribed angles, and angles formed by intersecting chords in circles. Know and use properties of arcs and sectors, and find lengths of arcs and areas of

7 sectors. Know and use relations between circles and inscribed or circumscribed polygons Recognize and explain how the rotational symmetry of a circle is key to real world applications. Apply concepts and properties of circles to analyzing gear ratios, pulleys. Given a line segment or an angle construct another one congruent to the first and construct their bisectors Given specific information about triangles (SAS, ASA, SSS, HL,) construct the triangle Given a line and a point not on the line, construct a line through the point that is parallel to the given line and one that is perpendicular to the given line and justify all steps. Express the sine, cosine, and tangent of angles in a right triangle Use the sine, cosine, and tangent of an angle in a right triangle to solve problems Describe how various strategies of measuring with triangles e.g. similar triangles, Pythagorean Theorem, trigonometry) are used to find unknown distances and angles, and determine which strategy(ies) might be useful in a given situation. Solve problems using right triangle trigonometry, the Law of Sines, and the Law of Cosines Know and use the distance formula and the midpoint formula in 2 and 3 dimensions Know and use the standard form of an equation for a plane Use coordinates to describe points, lines, polygons, and circles in a plane and using computer assisted design Use coordinates to describe 3-dimensional objects. Use coordinate representations to justify properties of polygons and circles. Products/Projects Students use their mini golf course greens created at the end of the first 9 weeks and now add three dimensional objects that are functional parts of the golf green. Each green must contain at least one rectangular prism, one triangular prism, one pyramid, one cylinder, one cone and one sphere or a hemisphere. This project will include the constructed golf green and a four page paper on the mathematics of their golf green. Included in this paper will be at least 20 terms from our geometry study and an explanation of how to get a hole in one. Major Assessments The above project and 6 unit tests.