# Aldine ISD Benchmark Targets /Geometry SUMMER 2004

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1 ASSURANCES: By the end of Geometry, the student will be able to: 1. Use properties of triangles and quadrilaterals to solve problems. 2. Identify, classify, and draw two and three-dimensional objects (prisms, pyramids, cylinders, cones, and spheres). 3. Use logical reasoning to justify steps in problem solving. 4. Use properties and attributes of parallel and perpendicular lines to solve problems. 5. Apply concepts of congruence and similarity. Problem solving applications must be included in daily instruction. The teacher needs to pace instruction so students are able to develop a conceptual understanding. Therefore, the timeline is flexible. It is critical that the objectives for each quarter are completed by the end of that quarter. The teacher should incorporate some of the following TEKS in every lesson: A Identify and apply mathematics to everyday experiences, to activities in and out of school, with other disciplines, and with other mathematical topics. (TEKS b8.14a) B Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness. (TEKS 8.14B) C Select or develop an appropriate problem-solving strategy from a variety of different types, including: work backwards guess and check make a table draw a picture look for a pattern work a simpler problem acting it out D Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic models. (TEKS 8.15A) E Use logical reasoning to make conjectures from patterns or sets of examples and nonexamples (TEKS 8.16 A) F Validate conclusions using mathematical properties and relationships. (TEKS 8.16 B) Sara Ptomey Geometry - Page 1 Summer 2004

2 Geometry A Unit 1: Introduction to Geometry acute angle adjacent angles angle betweeness bisector collinear complementary angles congruent coplanar construct definition degree draw endpoint exterior of an angle Euclidean Geometry geometry interior of an angle line line segment linear pair midpoint non-collinear obtuse angle opposite rays parallel perpendicular plane point postulate property ray right angle sketch skew space straight angle supplementary angles theorem The student will: 1. Recognize the historical development of geometry (i.e. Euclid) (TEKS b1b) 2. Study the undefined terms in geometry and the relationships among them. (TEKS a3) 3. Develop an understanding of the relationships among definitions, postulates, theorems, and properties. (TEKS b1a) 4. Study properties of lines and parts of lines and use to solve problems (i.e. segments, rays, parallel, perpendicular, skew, and segment addition postulate) (TEKS a3) 5. Study properties of angles and angle pairs and use to solve problems involving angles and angle pairs (i.e. acute, right, obtuse, linear vertical adjacent, complementary, supplementary, and angle addition postulate) (TEKS a3) 6. Use rulers and protractors to measure line segments to the nearest 1/16 of an inch and 1/10 of a centimeter and angles to the nearest degree. (TEKS a5) 7. Use constructions to copy lines, parts of lines, and angles. (TEKS b2a) 8. Use constructions to explore segment and angle bisectors. (TEKS b2a) Sara Ptomey Geometry - Page 2 Summer 2004

3 Unit 2: Coordinate Geometry angle of rotation Cartesian Plane center of rotation dilation dimensions distance image isometry line of symmetry mapping point of symmetry pre-image reflection rotation scale factor symmetry transformation translation 9. Use one and two-dimensional coordinate systems to represent points, lines, line segments, and figures. (TEKS d2a) 10. Develop the distance, midpoint, and slope formulas and apply to solve problems. (TEKS d2c) 11. Use slope to investigate parallel and perpendicular lines. (TEKS d2b) 12. Demonstrate an understanding of the four basic transformations (translation, rotation, reflection, and dilation). (TEKS b2b) 13. Use congruence transformations to make conjectures about and justify the properties of geometric figures. (TEKS e3a) 14. Make and verify conjectures about the effects of transformations on the signs of coordinates, slopes, and distances. (TEKS e3a, f1) Unit 3: Lines and Angle Pairs alternate exterior angles alternate interior angles consecutive angles corresponding angles distance equidistant transversal 15. Make and verify conjectures about angle pairs that occur when lines are cut by a transversal. (TEKS b2b) 16. Formulate and test conjectures about the angle pair relationships that occur when parallel lines are cut by a transversal. (TEKS e2a) 17. Recognize that the distance between two geometric figures must be measured along a perpendicular line. (TEKS a3) 18. Use constructions to explore perpendicular and parallel lines. (TEKS b2a) Sara Ptomey Geometry - Page 3 Summer 2004

4 Unit 4: Triangles Aldine ISD Benchmark Targets /Geometry acute triangle altitude angle bisector base base angles corresponding parts equiangular equilateral triangle exterior angles hypotenuse included side included angle isosceles triangle leg median obtuse triangle perpendicular bisector polygon remote interior angles right triangle scalene triangle triangle vertex vertex angle 19. Use constructions to explore and classify various types of triangles. (TEKS b2a) 20. Use constructions of perpendicular bisectors, medians, altitudes, and angle bisectors of a triangle. (TEKS b2a) 21. Use slopes and midpoints to identify special segments of triangles. (TEKS d2b) 22. Formulate and test properties of triangles and their parts. (TEKS e2b) 23. Formulate and test properties of isosceles and equilateral triangles. (TEKS e2b) 24. Analyze inequalities in relationships between sides and angles in a triangle. (TEKS e2) 25. Justify and apply triangle congruence relationships (SSS, SAS, ASA, AAS, HL, HA, LL, LA, and CPCTC) Unit 5: Quadrilaterals base base angle diagonal isosceles trapezoid kite median midsegment parallelogram polygon quadrilateral rectangle rhombus right trapezoid square trapezoid vertex angles 26. Use constructions to classify various types of quadrilaterals. (TEKS a3, b2a) 27. Use numeric and geometric patterns to make generalizations about geometric properties, such as angle relationships in quadrilaterals. (TEKS c1) 28. Make and verify conjectures about the properties of parallelograms (rectangles, rhombi, and squares). (TEKS b2b) 29. Make and verify conjectures about the properties of trapezoids (isosceles trapezoids and right trapezoids). (TEKS b2b) 30. Make and verify conjectures about the properties of kites. (TEKS b2b) Sara Ptomey Geometry - Page 4 Summer 2004

5 Unit 6: Logic and Reasoning conclusion conditional conjecture contrapositive converse counterexample deductive reasoning hypothesis inductive reasoning inverse negation proof truth value 31. Use inductive reasoning to formulate a conjecture. (TEKS b3d) 32. Understand the writing process of a conjecture as a conditional statement (If-Then). (TEKS b1a, b3) 33. Write a conditional statement as a converse, inverse, and a contrapositive and determine the truth values of each statement. (TEKS b3a) 34. Use numeric and geometric patterns to demonstrate deductive reasoning (sum of angles, triangles in polygons, angles formed by rays). (TEKS a6, c) 35. Understand the connection between algebraic and geometric properties. (TEKS a4) 36. Develop and demonstrate the use of logical reasoning using two-column, flowchart, and/or paragraph proofs (algebraic, parallel line, and triangle proofs). (TEKS b1a, b3c) Sara Ptomey Geometry - Page 5 Summer 2004

6 Geometry B Unit 7: Similarity corresponding parts extremes means proportional scale factor similar figures similarity 37. Use proportions to solve problems in real-world situations. (TEKS a4, b4) 38. Use ratios to solve problems involving similar polygons. (TEKS f2) 39. Use properties of similarity to explore and justify conjectures of geometric figures and their corresponding parts. (TEKS f1) 40. Develop and use triangle similarity relationships (AA, SSS, and SAS). (TEKS f2) 41. Use numeric and geometric patterns to make generalizations about geometric properties, such as ratios in similar figures (perimeter, area, and volume ratios). (TEKS c1) Unit 8: Right Triangles angle of depression angle of elevation cosine geometric mean Pythagorean triples sine tangent trigonometric ratios trigonometry 42. Develop, extend, and use the Pythagorean Theorem and Pythagorean triples to solve problems. (TEKS c3, f3, e1c) 43. Develop, apply, and justify triangle similarity relationships using right triangle ratios (geometric mean). (TEKS f3) 44. Identify and apply the properties of special right triangles to solve problems. (TEKS c3) 45. Develop and apply the trigonometric ratios of the acute angles of a right triangle. (TEKS f3) 46. Use a variety of representations to describe geometric relationships and solve problems involving angle of elevation and angle of depression. (TEKS a4, b1b, b4) Unit 9: Polygons apothem annulus circumscribed concave convex diameter geometric probability inscribed n-gons polygons radius regular sector segment of a circle Sara Ptomey Geometry - Page 6 Summer 2004

7 47. Identify and classify polygons and their relationships. (TEKS a3) 48. Formulate and test conjectures about properties and attributes of the angles of polygons. (TEKS e2b) 49. Find the perimeters and areas of polygons and composite figures. (TEKS e1, e1a) 50. Find areas of circles, sectors, segments, and annuli. (TEKS e1b) 51. Use a variety of representations to describe geometric relationships and solve problems involving probability. (TEKS b4) 52. Use constructions to explore regular polygons. (TEKS b2a) Unit 10: Solids base cone cross-section cylinder edge face great circle height hemisphere lateral area lateral edge lateral face net oblique perspective polyhedron prism pyramid right similar solids slant height sphere surface area vertex volume 53. Analyze the characteristics of three-dimensional figures and the component parts (i.e. face, base, height, slant height, edge, vertex, etc.). (TEKS e2d) 54. Use nets to represent and construct three-dimensional objects. (TEKS d1b) 55. Describe and draw the cross section (any slice) of three-dimensional objects. (TEKS d1a) 56. Use top, front, side, and corner views of three-dimensional objects to create accurate and complete representations and solve problems. (TEKS d1c) 57. Find surface areas and volumes of three-dimensional solids including composite solids. (TEKS e1d) 58. Describe the effect on perimeter, area, and volume when parameters (length, width, height of a three-dimensional solid are changed. (TEKS f4) 59. Use numeric and geometric patterns to make generalizations about geometric properties, such as ratios in similar figures and solids (perimeter, area, and volume ratios). (TEKS c1, f1) 60. Use ratios to solve problems involving similar solids. (TEKS f2) Sara Ptomey Geometry - Page 7 Summer 2004

8 Unit 11: Circles Aldine ISD Benchmark Targets /Geometry arc arc length arc measure center central angle chord major arc minor arc circle circumference concentric circles diameter inscribed angle intercepted arc pi point of tangency radius secant semicircle tangent 61. Identify parts of circles. (TEKS a1) 62. Formulate and test conjectures about properties of angles and arcs of circles. (TEKS e2c) 63. Formulate and test conjectures about properties of tangents, secants, and chords. (TEKS e1b, e2c) 64. Find segment and arc lengths of circles using proportional reasoning. (TEKS e1b) 65. Use the equation of a circle in the coordinate plane. (TEKS d2a) Unit 12: Geometry and Beyond arc fractal great circle iteration Non-Euclidean Geometry self-similarity spherical tessellation 66. Use properties of transformations and their compositions to make connections between mathematics and their real world in applications such as tessellations and fractals.(teks c2) 67. Compare and contrast the structures and implications of Euclidean and non-euclidean geometries. (TEKS b1c) Sara Ptomey Geometry - Page 8 Summer 2004

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