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Geometry Detailed Objectives Throughout the course as appropriate LG.1.G.6 Give justification for conclusions reached by deductive reasoning State and prove key basic theorems in geometry (i.e., the Pythagorean theorem, the sum of the measures of the angles of a triangle is 180°, and the line joining the midpoints of two sides of a triangle is parallel to the third side and half it’s length) DMEC Category Logic and Reasoning Points, Lines, Planes, etc. Objectives 1. Compare and contrast inductive and deductive reasoning. 2. Use Venn Diagrams to solve problems. 3. Use Matrix Logic to solve problems. 4. Identify the inverse, converse and contrapositive of a statement. 5. Recognize and describe relationships derived from figural patterns. 1. Represent points pictorially with proper identification. 2. Represent lines pictorially with proper identification. 3. Represent rays pictorially with proper identification. 4. Represent line segments pictorially with proper identification. 5. Represent angles pictorially with proper identification. 6. Represent planes pictorially with proper identification. 2009-2010 SLEs LG.1.G.1 Define, compare and contrast inductive reasoning and deductive reasoning for making predictions based on real world situations • Venn diagrams • matrix logic • conditional statements (statement, inverse, converse, and contrapositive) • figural patterns LG.1.G.3 Describe relationships derived from geometric figures or figural patterns. LG.1.G.2 Represent points, lines, and planes pictorially with proper identification, as well as basic concepts derived from these undefined terms, such as segments, rays, and angles Category Angle relationships Non-Euclidean Geometries Objectives 1. Solve problems involving complementary angles. 2. Solve problems involving supplementary angles. 3. Solve problems involving vertical angles. 4. Solve problems involving linear pairs of angles. 5. Solve problems involving angles formed by perpendicular lines. 6. Solve problems involving parallel lines cut by a transversal. SLEs LG.1.G.4 Apply, with and without appropriate technology, definitions, theorems, properties, and postulates related to such topics as complementary, supplementary, vertical angles, linear pairs, and angles formed by perpendicular lines LG.1.G.5 Explore, with and without appropriate technology, the relationship between angles formed by two lines cut by a transversal to justify when lines are parallel 7. Perform basic Euclidean constructions. M.3.G.5 Identify and apply properties of and theorems about parallel and perpendicular lines to prove other theorems and perform basic Euclidean constructions R.4.G.9 Explore non-Euclidean geometries, such as spherical geometry and identify its unique properties which result from a change in the parallel postulate 1. Identify unique properties which result when the parallel postulate is not included as one of the postulates for a geometry. Category Slope and Equations of Lines Objective 1. Identify horizontal and vertical lines by calculating the slope. 2. Determine if sets of lines are parallel or perpendicular by calculating their slope. 3. Write the equation of a line parallel to a line through a point not on the line. 4. Write the equation of a line perpendicular to a line through a given point. 5. Write the equation of the perpendicular bisector of a line segment. 1. Prove the Pythagorean Theorem. 2. Use the Pythagorean Theorem and its converse to solve problems. Pythagorean Theorem and Related Topics 3. Find the distance between two points in a rectangular coordinate system. 4. Find the coordinates of the midpoint of a line segment in the coordinate plane. SLEs CGT.5.G.1 Use coordinate geometry to find the slopes of parallel, perpendicular, horizontal, and vertical lines CGT.5.G.2 Write the equation of a line parallel to a line through a given point not on the line CGT.5.G.3 Write the equation of a line perpendicular to a line through a given point CGT.5.G.4 Write the equation of the perpendicular bisector of a line segment T.2.G.4 Apply the Pythagorean Theorem and its converse in solving practical problems LG.1.G.6 State and prove key basic theorems in geometry CGT.5.G.1 Use coordinate geometry to find the distance between two points, the midpoint of segments. 5. Apply the triangle inequality to solve problems. . T.2.G.2 Investigate the measures of segments to determine the existence of triangles (triangle inequality theorem) Category Special Segments of Triangles Properties of Polygons Objective 1. Identify altitudes of triangles and use them to solve problems. 2. Identify medians of triangles and use them to solve problems. 3. Identify angle bisectors of triangles and use them to solve problems. 4. Identify perpendicular bisectors of triangles and use them to solve problems. 5. Identify midsegments of triangles and use them to solve problems 1. Use properties of quadrilaterals to solve problems. 2. Solve problems involving the sum of the measures of the interior angles of a polygon. 3. Solve problems involving the interior and exterior angle measures of regular and irregular polygons. 4. Solve problems involving the number of sides and angles of polygons. 5. Classify figures given the set of points that form the vertices of the figure. 6. Prove theorems about polygons. SLEs T.2.G.3 Identify and use the special segments of triangles (altitude, median, angle bisector, perpendicular bisector, and midsegment) to solve problems R.4.G.1 Explore and verify the properties of quadrilaterals R.4.G.2 Solve problems using properties of polygons: • sum of the measures of the interior angles of a polygon • interior and exterior angle measure of a regular polygon or irregular polygon • number of sides or angles of a polygon CGT.5.G.5 Determine, given a set of points, the type of figure based on its properties (parallelogram, isosceles triangle, trapezoid) LG.1.G.6 State and prove key basic theorems in geometry Category Proportional Reasoning Objective SLEs 1. Find missing measures of figures based on congruence T.2.G.1 Apply congruence (SSS …) and similarity (AA ...) relationships. correspondences and properties of figures to find missing parts of geometric figures and provide logical justification 2. Find missing measures of figures based on similarity relationships. Trigonometry 3. Use properties of parallel lines and proportional reasoning to find lengths of segments. 1. Use trigonometric ratios to find the measures of sides of right triangles to solve problems. 2. Use trigonometric ratios to find the measures of angles in right triangles to solve problems. 3. Use trigonometric ratios to solve problems involving angles of elevation. M.3.G.4 Use (given similar geometric objects) proportional reasoning to solve practical problems (including scale drawings) M.3.G.5 Use properties of parallel lines and proportional reasoning to find the lengths of segments T.2.G.6 Use trigonometric ratios (sine, cosine, tangent) to determine lengths of sides and measures of angles in right triangles, including angles of elevation and angles of depression 4. Use trigonometric ratios of solve problems involving angles of depression. T.2.G.7 Use similarity of right triangles to express the sine, cosine, and tangent of an angle in a right triangle as a ratio of given including angles of elevation and angles of depression 5. Use the special right triangle relationships to solve problems. T.2.G.5 Use the special right triangle relationships (30˚-60˚-90˚ and 45˚45˚-90˚) to solve problems Category Transformations 2-D Representations of 3-D Objects Objectives 1. Draw and interpret the results of transformations and successive transformations on figures in the coordinate plane. SLEs CGT.5.G.7 Draw and interpret the results of transformations and successive transformations on figures in the coordinate plane • translations • reflections • rotations (90˚, 180˚, clockwise and counterclockwise about the origin) • dilations (scale factor) 2. Identify and explain why figures tessellate. R.4.G.3 Identify and explain why figures tessellate 3. Identify the attributes of the five Platonic Solids. 1. Use orthographic drawings to represent threedimensional objects. 2. Use isometric drawings to represent three-dimensional objects. R.4.G.4 Identify the attributes of the five Platonic Solids R.4.G.7 Use orthographic drawings (top, front, side) and isometric drawings (corner) to represent threedimensional objects 3. Draw, examine and classify cross-sections of threedimensional objects. R.4.G.8 Draw, examine, and classify cross-sections of three-dimensional objects Category Perimeter, Area , Volume Objectives 1. Solve problems involving perimeter of polygons. 2. Solve problems involving the circumference of circles. 3. Solve problems involving perimeter of composite figures. 4. Solve problems involving area of polygons. 5. Solve problems involving area of circles. 6. Solve problems involving area of composite figures. 7. Solve problems involving surface area of prisms. 8. Solve problems involving surface area of pyramids. 9. Solve problems involving surface area of cones. 10. Solve problems involving surface area of cylinders. 11. Solve problems involving surface area of spheres. 12. Solve problems involving the surface area of composite objects. 13. Solve problems involving volume of prisms. 14. Solve problems involving volume of pyramids. 15. Solve problems involving volume of cones. 16. Solve problems involving volume of cylinders. 17. Solve problems involving volume of spheres. 18. Solve problems involving volume of composite objects. SLEs M.3.G.2 Apply, using appropriate units, appropriate formulas (area, perimeter, surface area, volume) to solve application problems involving polygons, prisms, pyramids, cones, cylinders, spheres as well as composite figures, expressing solutions in both exact and approximate forms 19. Relate changes in the measurement of one attribute of an object to changes in the measurement of other attributes. M.3.G.3 Relate changes in the measurement of one attribute of an object to changes in other attributes (Ex. How does changing the radius or height of a cylinder affect its surface area or volume?) 20. Use area models to calculate probabilities. M.3.G.1 Calculate probabilities arising in geometric contexts (Ex. Find the probability of hitting a particular ring on a dartboard.) Category Circles Objectives 1. Write the equation of a circle in standard form given a coordinate graph of the circle. 2. Write the equation of a circle in standard form given the coordinates of the center of the circle and the radius of the circle. SLEs CGT.5.G.6 Write, in standard form, the equation of a circle given a graph on a coordinate plane or the center and radius of a circle 3. Use properties of central angles o a circle to solve problems. 4. Use properties of central angles of circles to solve problems. 5. Use properties of angles inscribed in circles to solve problems. 6. Use properties of arc of circles to solve problems. 7. Use properties of chords of circles to solve problems. 8. Use properties of lines tangent to circles to solve problems. 9. Use properties of lines secant to circles to solve problems. R.4.G.5 Investigate and use the properties of angles (central and inscribed) arcs, chords, tangents, and secants to solve problems involving circles 10. Solve problems using inscribed figures. R.4.G.6 Solve problems using inscribed and circumscribed figures 11. Solve problems using circumscribed figures. Notes: Highlighting indicates areas not included in SLE’s from previous grade levels.