Objectives Geometry (Building a system of geometric knowledge and its application to the real world)

Transcription

1 Applied Geometry High School Math Grades Recommended: 10 th or 11 th grade Course Code: MA Prerequisites: Algebra II, MA311/321 Length of course: one year Description In this course, students will learn basics of Euclidean and non-euclidean geometry and algebraic concepts through practical applications using computer modeling (computer assisted design), computer calculations, and hands-on projects. Students will learn relationships between angles, parallel lines, perpendicular lines, triangles trigonometry, quadrilaterals, and circles. Students will design objects using geometric shapes and algebraic concepts, make basic constructions, and demonstrate competence in inductive and deductive proofs. Using computer assisted drawing as a basis for computer training and visualization of geometry in both 2D and 3D space, students will learn to read and analyze problems, construct visual images illustrating the problems, and solve those problems by applying the technology of computers. Objectives Geometry (Building a system of geometric knowledge and its application to the real world) 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 and the real world application of 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 and their application to the real world develop an understanding of the relationships among parts of circles and between circles and various angles, segments, and arcs, generalizing their findings to the real world. demonstrate the use of real and irrational numbers to solve geometric problems involving perimeter, circumference, area (regular, lateral, and surface), and volume and their application to the real world

2 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 the real world. analyze and use reasoning in geometry: proofs, hypothesis, logic, deductive reasoning (using definitions, theorems, postulates) and inductive reasoning and apply this reasoning to the real world. develop and explore the 1) trigonometric ratios from both right triangles and unit circles, 2) trigonometric identities and apply this knowledge to solving problems in the real world. Algebraic concepts demonstrate understanding of lines, their equations, their slopes and the relationships between lines Technology and Other Resources This course will be taught in a computer lab as each unit will require Auto CAD (computer assisted design), Solid Works, and/or Microsoft Office. Each quarter the students will be asked to design and build a project. Software: Auto CAD Solid Works Microsoft Office Grading System Projects 30 %, Tests 30 %, Class Portfolio 30%, Class Participation 10 %

3 Syllabus Applied Geometry Unit 1: Basics of Computers, Design, and the Building Blocks of Geometry (9 weeks) All students will use computer aided design software to draw 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,5) polygons and 6)planes and explain there interconnection. Main Topics A. Introduction to Computers, Computer Design, and Measurement. 1. Basic computer functions, Windows, Microsoft Office. 2. Beginning Computer Assisted Design. 3. Introduction to drawing, drawing tools and drawing aids. B. Undefined Terms and Basic Definitions and their relationship to drawings and the real world 1. Points, lines, and planes. 2. Segments, rays, and distance. 3. Angles, their measurement, types of, pairs of, and angle addition C. Perpendicular/Parallel Lines and Planes and their relationship to drawings and the real world. 1. Algebra (slope of line, equation of line solving equations) 2. Lines that are perpendicular and the angles formed 3. Lines that are parallel and the angles formed 4. Applying parallel lines to polygons. Learning Outcomes: ASW Apply measurement units and calculations to computer assisted design problems Explain how measurement errors affect computer assisted design; carry units through computer assisted designs so that results include correct units. Determine what is a reasonable degree of accuracy for measurement in a given computer assisted design situation Understand and use concepts of accuracy, error, tolerance and accumulated error in applied situations (drawings of actual parts) Know and interpret the relationship between the coefficients of a linear equation and the slope and x- and y- intercepts of a graph as applied to computer assisted drawings Understand and identify the undefined terms of geometry Investigate postulates about points lines and planes and apply to design Apply properties of vertical angles, linear pairs of angles, supplementary angles,

4 complementary angles, and right angles to solve multi-step design problems Apply properties of corresponding angles, alternate interior angles, alternate exterior angles and same-side (consecutive) interior angles to solve design problems Identify n-gons from n=3 to n=12 and apply to computer assisted drawings Products/Projects Through an understanding of mathematics concepts and computer assisted design the students will design and build a model which will incorporate at least three of the following topics: angles, perpendicular lines, parallel lines, polygons, slopes, and equations of lines. (This project will have a written proposal and a materials list submitted three weeks before the end of this unit.) Major Assessments The project described above and three section tests.

5 Syllabus Applied Geometry Unit 2: Mathematical Reasoning, Polygons (9 weeks) All students will use mathematical reasoning and logic to analyze and devise strategies to solve problems and apply this reasoning to polygon and circles. Main Topics D. Introduction to Proof analyzing and applying to drawings and to the real world. 1. Properties from Algebra. 2. Proving Theorems. 3. Special Pairs of Angles. 4. Perpendicular Lines. 5. Planning a proof. 6. Postulates relating points, lines, planes. E. Polygons, recognizing similarities and general patterns to make predictions about them, drawings and their applications to the real world. Triangles. 1. Angle measure 2. Classification of 3. Area and perimeter of 4. Geometric inequalities 5. Congruent triangles a. Corresponding parts in congruence. b. Theorems based on congruent triangles. c. Proving triangles congruent Polygons with more than three sides 1. Regular 2. Interior and exterior angles of polygons 3. Area and perimeter of 4. Symmetry of Circles 1. Parts of 2. Area and Circumference of Learning Outcomes: ASW 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, contra positive to reason about mathematical concepts.

6 Solve multi-step problems involving angle measure, length, perimeter, or area of scalene, isosceles, equilateral, acute, right or obtuse triangles and use computer design to draw them Know, justify, and use the properties of median, altitudes, perpendicular bisectors of sides, and angles bisectors of a triangle and use computer design to draw each. 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 and draw and example of Prove theorems about the angle sum of a triangle and the exterior angles of a triangle. Prove that triangles are congruent by suing the side-side-side (SSS), side-angleside (AAS), angle-side-angle (ASA) and hypotenuse-leg (HL) theorem Justify that the hypotenuse-angle, leg-leg and leg-angle theorems in right triangles are special cases of the general triangle congruence theorems (SAS, ASA, and SSS). Solve multi-step problems involving angle measure, length, perimeter, or area involving squares, rectangles, parallelograms, kites, and trapezoids Use the properties of squares, rectangles, parallelograms, kites, and trapezoids and construct them with CAD Describe the hierarchies among quadrilaterals Derive the formulas for the interior and exterior angles of quadrilaterals and draw representations of each 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 polygon and make drawings showing these dissections Know, justify, and use properties for squares, rectangles, rhombi, parallelograms, kites and trapezoids, draw these quadrilaterals 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 Use drawing programs to describe and derive a rule for the symmetries of polygons Derive, use and justify by drawing the formulas for the perimeter and area of regular n-gons Derive, use and justify by drawing the formulas to find the interior and exterior angles of a regular n-gon Draw the tessellations of polygons and analyze and explain why only certain polygons will tile the plane Describe the symmetries of regular polygons Describe the relationship between multisided polygons and circles 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.

7 Products/Projects Applying the knowledge the student has acquired about polygons the students will design the floor plan of a building with the following requirements: minimal 2,000 square feet of space, at least eight rooms, at least 2 rooms that are trapezoidal in shape and at least one room that is triangular in shape, the use of polygons with more than 4 sides will count extra points. This floor plan will be enlarged to 3 x3 piece of foam core and will be used 3 rd marking period for the next project. Remember to include the fact that this is an energy efficient home with walls made of 2 x 6 studs Major Assessments The project described above and four section tests.

8 Syllabus Applied Geometry Unit 3: 2 and 3 Dimensional Shapes (9 weeks) All students will demonstrate knowledge of 2-dimensional and 3-dimensional shapes and their application to the world of today. Main Topics F. Introduction to Solid Works as a tool to discover mathematical concepts G. Right Triangles and their use in drawings and the real world.. 1. The Pythagorean Theorem. 2. Special Right Triangles. 3. Symmetry H. Shapes in 3-dimensional space and their relationship/applications to drawings and the real world 1. Drawings from 3 views (front, right, top) 2. Pyramids, Prisms, Polyhedra a. Parts of b. Surface Area, Volume, Relationships c. Symmetry of 3. Cylinders, Cones, Hemispheres, and Spheres a. Parts of b. Surface Area, Volume, Relationships c. Symmetry of I. Similar Shapes and their relationship/applications to drawings and the real world 1. Dilations and Scale factors 2. Similar polygons a.. Relationships of parts b. Relationships of perimeter and area 3. Similar 3 dimensional shapes a.. Relationships of Parts b. Relationships of Perimeter, Surface Area, and Volume Learning Outcomes: ASW 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, and find these four special points using computer design software. Describe symmetries of triangles. 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

9 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 fie 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 Given a 2-dimensional view, draw the 3 dimensional object Given a 3-dimensional figure, draw a 2-dimensional representation Identify and/or draw cross-sections of 3 dimensional figures. Prove that triangles are similar by using SSS, SAS, and AA conditions for similarity Know and use the Fundamental Theorem of Similarity: Products/Projects Students will demonstrate knowledge of polygons and 3-dimensional shapes by constructing a 3-dimensional model on their 3 x3 floor plan from the last unit project. Included in this project will be front, back, left, right, top, and 3-D view of their home, which now has windows (one which will be circular and one which will be hexagonal), doors, any porches, roof and chimney. Major Assessments The project described above and three section tests

10 Syllabus Applied Geometry Unit 4: Circles, Right Triangles, Coordinate Geometry, and Transformations (9 weeks). All students will demonstrate knowledge of circles, right triangles, coordinate geometry and transformations and apply this knowledge to the real world. Main Topics J. Circles and their relationship/applications to drawings and the real world 1. Tangents, Arcs, and Chords. 2. Angles and Segments 3. Inscribed and circumscribed K. Right Triangles and their relationship/applications to drawings and the real world 1. Trigonometry L. Coordinate Geometry and their relationship/applications to drawings and the real world 1. Using the Distance Formula. 2. Lines. 3. Coordinate Geometry Proofs. M. Transformations and their relationship/applications to drawings and the real world 1. Basic Mappings. 2. Products and Symmetry. Learning Outcomes: ASW 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 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. 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 haw 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.

11 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. 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, and analyze tessellations of the plane (using computer assisted design. 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 using computer design 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 Students will demonstrate knowledge of Solid Works and basic geometry skills by using their 3 dimensional model from last 9 weeks and drawing the outside siding and yard landscaping on all 4 sides of the house (making any modifications necessary from last marking period s project). The landscaping must contain 3 unique tessellations; a chord, a tangent, and a secant illustrated in a circle; a circle inscribed in a polygon; and a polygon inscribed in a circle. Upon completion of this project each student will have a complete model with removable roof and a set of 10 (6 from the previous marking periods project and 4 new landscape drawings) design prints for their house. Major Assessments The above project and 4 section tests.