Greenwich Public Schools Mathematics Curriculum Objectives. Geometry

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1 Mathematics Curriculum Objectives Geometry June 30, 2006

2 NUMERICAL AND PROPORTIONAL REASONING Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technology. Enduring Understandings: Geometric figures can be represented in the coordinate plane. The structure of the coordinate system can be used to calculate measures contained in geometric figures. The Distance Formula is an application of the Pythagorean Theorem. The measures of geometric figures can be calculated and analyzed using a variety of strategies, tools, and technologies. A change in one dimension of an object results in predictable changes in area and or volume. A constant ratio exists between corresponding lengths of sides of similar figures. Essential Questions: What is the relationship of the different measures in two and three dimensional objects? How does a change in one dimension of an object affect the other dimensions? What is the difference between congruence and similarity? What is the relationship between similar geometric figures and proportions? Cluster: Coordinate Geometry Geo.1. Geo.2. Geo.3. Geo.4. Geo.5. Geo.6. Define the Cartesian plane and its components. Plot points on a coordinate plane. Calculate the distance between two points on the number line. Calculate the distance between two points on a coordinate plane. Calculate the midpoint of a segment. Calculate the midpoint of two points on a coordinate plane. Cluster: Ratio and Proportion Geo.7. Geo.8. Geo.9. Geo.10. Calculate the geometric mean of two numbers. Set up and solve problem using proportions. Understand and apply the relationships between the scale factor and the perimeters and areas of similar figures. Use the areas of similar polygons to solve real-world problems. Cluster: Similarity Geo.11. Understand and apply the scale factor between two similar polygons. 1

3 Geo.12. Geo.13. Geo.14. Geo.15. Geo.16. Understand the definition of similar triangles and how it is used to prove triangles similar. Understand proportions in similar triangles. Prove triangles similar by AA, SSS, SAS Use the properties of similar polygons to deduce information about segments and angles. Use proportionality theorems. 2

4 WORKING WITH DATA: PROBABILITY AND STATISTICS Data can be analyzed to make informed decisions using a variety of strategies, tools and technology. Enduring Understandings: The basic concepts of probability can be applied to lengths and areas to determine the probability of hitting a point on the line or an area within a defined region. Essential Question: How can probability be used to predict the likelihood of hitting a particular point on a given segment or particular area within a defined region? Cluster: Geometric Probability Geo.17. Geo.18. Define probability. Use area and lengths to solve problems involving geometric probability. 3

5 GEOMETRY AND MEASUREMENT Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools, and technology. Enduring Understandings: Proof is a justification that is logically valid and based on definitions, postulates, and theorems. Logical arguments consist of a set of premises or hypotheses and a conclusion. Two intersecting lines form angles with specific relationships. Parallel lines cut by a transversal form angles with specific relationships. Problems involving two and three-dimensional measurements can be solved using geometric relationships and trigonometry. Essential Questions: What are the properties of two and three-dimensional figures? What is the relationship between lines and angles? What are the relationships between the angels formed by two intersecting lines? How do geometric relationships and the application of measurements help us to solve real-world problems? Cluster: Area and Perimeter. Geo.19. Geo.20. Geo.21. Geo.22. Compute the perimeter of a polygon. Know and use area formulas to compute the areas of rectangles, squares, parallelograms, triangles, trapezoids, and rhombuses. Know and use the formula to compute the area of a regular polygon. Use the areas of polygons to solve real-world problems. Cluster: Surface Area and Volume. Geo.23. Geo.24. Geo.25. Geo.26. Geo.27. Identify types of right prisms, pyramids, cylinders, and cones. Identify parts of prisms, pyramids, cylinders, and cones. Define a sphere. Use and apply the formulas for lateral area, surface area, and volume to right prisms, right pyramids, cylinders, and cones to solve problems. Use and apply the formula for surface area and volume to a sphere to solve problems. 4

6 Cluster: Points, Lines, and Planes. Greenwich Public Schools Geo.28. Geo.29. Geo.30. Geo.31. Geo.32. Geo.33. Geo.34. Geo.35. Geo.36. Geo.37. Geo.38. Geo.39. Geo.40. Geo.41. Geo.42. Geo.43. Define or describe what is meant by a point, line, plane, ray, and plane. Draw and label points, lines, planes, rays, and segments. Identify types of angles (obtuse, right, acute, and straight) State and use the Ruler Postulate and the Segment Addition Postulate to solve problems. State and use the Protractor Postulate and the Angle Addition Postulate to solve problems. Identify coplanar points and lines. Define and identify adjacent angles. Apply the definition of vertical angles. Apply the definition of complementary and supplementary angles. Apply the definition of a linear pair. Distinguish between intersecting lines, parallel lines, and skew lines. Identify angles when two lines are cut by a transversal Identify the congruent angles when two parallel lines are cut by a transversal. Use the point, line, and plane postulates. Construct angle bisectors. Construct segment bisectors. Cluster: Reasoning and proof Geo.44. Geo.45. Geo.46. Geo.47. State the hypothesis and the conclusion of a conditional statement. Write conditional statements. Understand bi-conditional statement. Use counterexamples to disprove conditional statements. Cluster: Properties of Triangles. Geo.48. Geo.49. Geo.50. Geo.51. Geo.52. Geo.53. Geo.54. Geo.55. Identify special segments in triangles. Apply inequalities for lengths of sides and angles measures of triangles. Apply inequalities in two triangles. Apply the Triangle Inequality Theorem. Use the properties of a mid-segment to solve problems. Construct the perpendicular bisector of the sides of a triangle. Construct the medians and altitudes of a triangle. Identify the incenter, circumcenter, centroid, orthocenter of a circle and know their properties. Cluster: Congruent Triangles Geo.56. Classify triangles by sides and angles. 5

7 Geo.57. Geo.58. Geo.59. Geo.60. Geo.61. Geo.62. Geo.63. Use the Angle Sum Theorem. Apply the Exterior Angle Theorem. Identify congruent triangles using the definition. Show that two triangles are congruent using SSS, SAS, AAS, ASA. Show that two right triangles are congruent using HL, HA, LL, LA. Use the Isosceles Triangle Theorem. Prove triangles congruent given information in the form of a figure or statement, using algebraic, coordinate or deductive proofs. Cluster: Quadrilateral. Geo.64. Geo.65. Geo.66. Geo.67. Geo.68. Geo.69. Geo.70. Geo.71. Investigate and identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and diagonals. State the definition and properties of parallelograms. State the definition and properties of rectangles. State the definition and properties of rhombi. State the definition and properties of trapezoids. State the definition and properties of isosceles trapezoids. Understand the median and midsegment of trapezoid. Prove what type of quadrilateral is formed by four ordered pairs. Cluster: Polygons Geo.72. Geo.73. Geo.74. Geo.75. Geo.76. Classify polygons by the number of sides. Identify convex and concave polygons and state the differences between the two. Identify regular polygons. Apply the Sum of Interior Angles Theorem to solve problems. Apply the Sum of Exterior Angles Theorem to solve problems Cluster: Right Triangles and Trigonometry Geo.77. Identify the types of right triangles Geo.78. State and apply the Pythagorean Theorem and its converse to solve problems. Geo.79. Apply rules of special right triangles ( ) and ( ) Geo.80. Identify common Pythagorean triples. Geo.81. Apply the geometric mean to find the missing lengths of different parts of a triangle Geo.82. Define and apply trigonometric ratios to real world situations. Geo.83. Use inverse trigonometric functions to find the measure of angles. Geo.84. Solve word problems using angles of elevation and angles of depression. 6

8 Cluster: Circles Geo.85. Geo.86. Geo.87. Geo.88. Geo.89. Geo.90. Geo.91. Geo.92. Geo.93. Geo.94. Geo.95. Understand the definition of a circle and write the equation of a circle whose center is on or off the origin. Define the terms radius, chord, and diameter and identify the segments in a circle. Understand and use inscribed and central angles. Define and calculate the measure and length of an arc of a circle. Find the area of a sector in a circle. Find the circumference and area of a circle Apply theorems about chords of circle. Understand tangent and secant lines Apply theorems that relate tangents and radii Derive the general equation of a circle and use it to write the equation of a circle whose center is at the origin or off the origin. Solve problems involving angles formed by chords, secants, or tangents. Find the area of a segment of a circle. 7

9 ALGEBRAIC REASONING: PATTERNS AND FUNCTIONS Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools, and technology. Enduring Understandings: The slope of a linear function represents a constant rate of change for f ( ) x when x changes by a fixed amount. The equation of a line defines the relationship between two variables. The slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals of each other. Essential Questions: What information is given in the different forms of a linear equation: slope-intercept form, point-slope form, and standard form? What is a system of equations and what are the different methods that can be used to solve a system of linear equations? Cluster: Lines Geo.96. Write the equation of a line in slope-intercept form, point-slope form and standard form. Geo.97. Calculate the slope of a line using algebraic methods or a graph. Geo.98. Write the equation of a line parallel to a given line. Geo.99. Write the equation of a line perpendicular to a given line. Geo.100. Solve linear equations with one unknown variable. Geo.101. Solve systems of linear equations. 8

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