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Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY Course Description: Honors Geometry follows the same course of study as Geometry however this course will have a quickened pace that allows for the mathematical concepts to be explored with greater depth including a heightened level of critical thinking. This course integrates the concepts of plane, solid, and coordinate geometry. The course investigates the properties of plane geometric figures. Using inductive and deductive methods of reasoning, students present proof of angle and line relationships. A scientific calculator is required. I. Content Standard: Number, Number Sense and Operations Standard Students demonstrate number sense, including an understanding of number systems and reasonable estimates using paper and pencil, technology-supported and mental methods. Benchmarks (Grade 8-10 Band) Geometry Indicators A. Connect physical, verbal and symbolic representations of integers, rational numbers and irrational numbers. 1. Connect physical, verbal and symbolic representation of irrational numbers; B. Estimate, compute and solve problems involving scientific notation, square roots and numbers with integer exponents. 1. e.g., construct 2 as a hypotenuse or on a number line. Estimate the solutions for problem situations involving square and cube roots. 1 Dublin City Schools February 2008 Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY II. Content Standard: Data Analysis and Probability Standard Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data. Benchmarks (Grade 8-10 Band) A. Compute probabilities of compound events, independent events, and simple dependent events. Geometry Indicators 1. Model problems dealing with uncertainty with area models (geometric probability). 2 Dublin City Schools February 2008 Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY III. Content Standard: Patterns, Functions and Algebra Standard Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs and equations. Benchmarks (Grade 8-10 Band) A. Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations. Geometry Indicators 1. Solve equations and formulas for a specified variable; e.g., express the base of a triangle in terms of the area and height. 2. Use algebraic representations and functions to describe and generalize geometric properties and relationships. 3. Dublin Indicator: Describe and apply inequalities in the context of angles and sides in triangles. B. Solve and graph linear equations and inequalities. 1. Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a specific point. C. Solve quadratic equations with real roots by graphing, formula and factoring. 1. Graph the quadratic relationship that defines circles. D. Describe and interpret rates of change from graphical and numerical data. 1. Compute and interpret slope, midpoint and distance given a set of ordered pairs. 2. Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals. 3 Dublin City Schools February 2008 Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY IV. Content Standard: Measurement Standard Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies. Benchmarks Indicators A. Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements and rates. 1. Use unit analysis to check computations involving measurement. 2. Use the ratio of lengths in similar two-dimensional figures or threedimensional objects to calculate the ratio of their areas or volumes respectively. 3. Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures. 4. Solve problems involving unit conversion for situations involving distances, areas, volumes and rates within the same measurement system. 5. Determine the measures of central and inscribed angles and their associated major and minor arcs. 6. Convert rates within the same measurement system; e.g., miles per hour to feet per second; kilometers per hour to meters per second. B. Estimate and compute areas and volume in increasingly complex problem situations.. 1. Derive a formula for the surface area of a cone as a function of its slant height and the circumference of its base. 2. Calculate distances, areas, surface areas and volumes of composite three-dimensional objects to a specified number of significant digits. 3. Dublin Indicator: Determine the area of the sector of a circle. 4 Dublin City Schools February 2008 Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY V. Content Standard: Geometry and Spatial Sense Standard Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems. Benchmarks (Grade 8-10 Band) A. Formally define geometric figures. Geometry Indicators 1. Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles Of polygons; b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter); d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). 2. Recognize and explain the necessity for certain terms to remain undefined, such as point, line and plane. 3. Identify the reflection and rotation symmetries of two- and threedimensional figures. 4. Solve problems involving chords, radii, and arcs within the same circle. 5. Dublin Indicator: Define and apply the concept of bisect to shapes. 6. Dublin Indicator: Identify and describe collinear points and coplanar points, lines and segments. 7. Dublin Indicator: Describe, draw and identify polygons and their properties. 8. Dublin Indicator: Classify triangles based on their side and angle measures. 5 Dublin City Schools February 2008 Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY B. Describe and apply the properties of similar and congruent figures; and justify conjectures involving similarity and congruence. 1. Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of twodimensional figures and threedimensional objects. 2. Use proportions in several forms to solve problems involving similar figures (part-to-part, part-to-whole, corresponding sides between figures). 3. Dublin Indicator: Understand and apply triangle congruence postulates and theorems. C. Recognize and apply angle relationships in situations involving intersecting lines, perpendicular lines, and parallel lines. 1. Recognize the angles formed and the relationship between the angles when two lines intersect and when parallel lines are cut by a transversal. 2. Solve problems involving chords, radii, and arcs within the same circle. 3. Dublin Indicator: Define, describe, and apply operations involving angles. D. Use coordinate geometry to represent and examine the properties of geometric figures. 1. Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of twodimensional figures and threedimensional objects. 2. Represent and analyze shapes using coordinate geometry; e.g., given three vertices and the type of quadrilateral, find the coordinates of the fourth vertex. E. Draw and construct representations of two- and three-dimensional geometric objects 1. Draw nets for a variety of prisms, pyramids, cylinders and cones. 2. Construct right triangles, equilateral triangles, parallelograms, trapezoids, rectangles, rhombuses, squares and kites. 3. Construct congruent or similar figures using tools. 4. Perform reflections and rotations. 6 Dublin City Schools February 2008 Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY F. Represent and model transformations in a coordinate plane and describe the results. 1. Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations. 2. Derive coordinate rules for translations, reflections and rotations of geometric figures in the coordinate plane. 3. Show and describe the results of combinations of translations, reflections and rotations (compositions); e.g., perform compositions and specify the result of a composition as the outcome of a single motion, when applicable. G. Prove or disprove conjectures and solve problems involving two- and three-dimensional objects represented within a coordinate system. 1. Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram. 7 Dublin City Schools February 2008 Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY H. Establish the validity of conjectures about geometric objects, their properties and relationships by counterexample, inductive and deductive reasoning, and critiquing arguments made by others. 1. Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of twodimensional figures and threedimensional objects. 2. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: a. prove the Pythagorean Theorem; b. prove theorems involving triangle similarity and congruence; c. prove theorems involving properties of lines, angles, triangles and quadrilaterals; d. test a conjecture using basic constructions made with a compass and straightedge or technology. 3. Solve problems involving chords, radii, and arcs within the same circle. 4. Dublin Indicator: Identify and describe conditional statements and determine the if-then, converse, inverse, and contrapositive of conditional statements. I. Use right triangle trigonometric relationships to determine lengths and angle measures. 1. Define the basic trigonometric ratios in right triangles: sine, cosine and tangent. 2. Apply proportions and right triangle trigonometric ratios to solve problems involving missing lengths and angle sizes in similar figures. 8 Dublin City Schools February 2008 Dublin City Schools Mathematics Graded Course of Study HONORS GEOMETRY VI. Content Standard: Mathematical Processes The benchmarks for mathematical processes articulate what students should demonstrate in problem solving, representation, communication, reasoning and connections at key points in their mathematics program. Specific grade-level indicators have not been included for the mathematical processes standard because content and processes should be interconnected at the indicator level. Therefore, mathematical processes have been embedded within the grade-level indicators for the five content standards. By the end of the 8-10 program: A. Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution. B. Apply mathematical knowledge and skills routinely in other content areas and practical situations. C. Recognize and use connections between equivalent representations and related procedures for a mathematical concept; e.g., zero of a function and the x-intercept of the graph of the function, apply proportional thinking when measuring, describing functions, and comparing probabilities. D. Apply reasoning processes and skills to construct logical verifications or counterexamples to test conjectures and to justify and defend algorithms and solutions. E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas. F. Use precise mathematical language and notations to represent problem situations and mathematical ideas. G. Write clearly and coherently about mathematical thinking and ideas. H. Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner. 9 Dublin City Schools February 2008 Mathematics Targets Geometry Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and transformations are used to analyze mathematical situations and solve problems. Essential Learning: Analysis and applications of two- and three-dimensional figures Target 1 I can… Problems and statements form, prove and disprove (using counterexamples) conjectures. involving logic and geometric figures can be use inductive and deductive reasoning to draw valid conclusions. analyzed and solved. identify and describe conditional statements and determine the if-then, converse, inverse and contrapositive of conditional statements. describe, compare/contrast and use geometric postulates, theorems, definitions and corollaries to form logical arguments (proofs). estimate solutions for problem situations involving square and cube roots. use algebraic representations and functions to describe and generalize geometric properties and relationships. solve equations and formulas for specified variables. model problems dealing with geometric probability. Target 2 I can … Basic geometric figures and their properties can be identified, represented, analyzed, and applied. use and apply the concepts of points, lines, planes, segments, rays, and angles. identify and describe collinear points and coplanar points, lines, and segments. draw, identify, and apply properties of perpendicular and parallel lines and segments. describe, identify, draw and utilize properties of adjacent, vertical, complementary, supplementary, linear, corresponding, alternate interior, alternate exterior, and consecutive interior pairs of angles. Dublin City Schools August 2007 Mathematics Targets Geometry Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and transformations are used to analyze mathematical situations and solve problems. Essential Learning: Analysis and applications of two- and three-dimensional figures Target 3 I can … Polygons and their properties can be identified, represented, analyzed, and applied. describe, draw and identify polygons and their properties. classify triangles based on their side and angle measures. generalize and apply relationships of interior and exterior angles of polygons, including regular polygons. understand and apply triangle congruence postulates and theorems. describe, identify, draw, and apply properties of medians, altitudes, angle bisectors and perpendicular bisectors in triangles. define and explain points of concurrency related to triangles. describe and apply inequalities in the context of angles and sides in triangles. identify, draw, and apply properties of the following parallelograms, rectangles, rhombi, squares and trapezoids. Dublin City Schools August 2007 Mathematics Targets Geometry Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and transformations are used to analyze mathematical situations and solve problems. Essential Learning: Analysis and applications of two- and three-dimensional figures Target 4 I can… Circles and their properties can be identified, represented, analyzed, and applied. define, draw, and apply the properties of tangents, secants, radii, diameters and chords. determine the measures of central and inscribed angles and their associated major and minor arcs. determine arc length. determine the area of the sector of a circle. Target 5 I can… Three-dimensional figures and their properties can be identified, represented analyzed and applied. identify, name, and draw representations of prisms, cylinders, cones, pyramids and spheres. draw nets for three-dimensional figures. Dublin City Schools August 2007 Mathematics Targets Geometry Big Idea: Spatial reasoning and properties of one-, two-, and three-dimensional objects and transformations are used to analyze mathematical situations and solve problems. Essential Learning: Relating geometric properties and figures to the coordinate plane Target 1 I can… Properties and measurements of lines and line segments can be determined find the length of a segment using the distance formula. find the length of a segment using the Pythagorean Theorem. define, describe and apply operations using a midpoint of a segment on the coordinate plane. apply the concept of slope to parallel and perpendicular lines Target 2 I can … Figures on the coordinate plane can be classified and analyzed. classify shapes when given the coordinates of its vertices. apply concepts of slope and distance formula to analyze shapes on the coordinate plane. graph a circle and write its equation. Target 3 I can… Geometric figures can be modified through transformations. define, describe, and perform rotations, translations, dilations, and reflections.. explain transformation effects on coordinates and shapes (including combinations of transformations). Dublin City Schools August 2007 Mathematics Targets Geometry Big Idea: Measurement involves using appropriate units, tools, and technology to a defined degree of estimation or accuracy. Essential Learning: Measurement of geometric figures Dublin City Schools August 2007 Target 1 I can… Measures of angles and segments can be represented, analyzed, and applied. define, describe and apply operations involving angles. Target 2 I can … Measures of twodimensional figures can be represented, analyzed and applied. apply the concept of perimeter and area to polygons, including regular, and circles. describe and apply the idea of congruence to segments and angles. describe and apply the idea of congruence to shapes. define and apply the concept of bisect to shapes. state, understand and apply the sine, cosine, and tangent ratios in right triangles. identify, draw and apply relationships of 30-60 and 45-45 right triangles. use the concepts of ratio and proportion to solve problems, including converting units between systems and checking computations with unit analysis. identify and solve problems involving similar figures, including working with perimeter, area, volume and corresponding parts within similar figures. Target 3 I can… Measures of threedimensional figures can be represented and analyzed. determine volume of prisms, cylinders, cones, pyramids, and spheres. determine surface area of prisms, cylinders, cones, pyramids, and spheres Dublin City Schools August 2007