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Mathematics Standard 1: Geometry Glencoe ISBN#: 0-07865106-9 ALGEBRA, FUNCTIONS, AND GRAPHS: Students will understand algebraic concepts and applications. Essential Question: How can we solve math using symbols and letters instead of numbers? Benchmark Performance Standards A. Represent and analyze 1. Translate from verbal mathematical situations and expression to algebraic structures using algebraic formulae. symbols. 2. Given a graph, construct a function that represents the graph (linear only). End Learning Mastery Students will write an equation of a line given information about its graph. Students will solve problems by writing equations. Vocabulary/concepts: • point-slope form • rate of change • slope-intercept form Assessment(s) Practice Quiz 2, Student Edition (SE), p. 150 Resources Glencoe Key Chapters: 3 Glencoe Lesson 3.4 Open-Ended Assessment, Teachers Wraparound Edition (TWE), p. 150 Chapter 3 Resource Masters pp. 145-6 Enrichment, Chapter Resource Masters (CRM), p.148 geometryonline.com/self_ check_quiz TestCheck and Worksheet Builder CD-ROM, Lesson 3-4 ACE Assessment: A certain cellular phone company charges a flat rate of $19.95 per month for service. All calls are charged $0.07 per minute of air time t. The total charge C for a month can be represented by the equation C = 0.07t + 19.95. How can the equation of a line describe cellular telephone service? 1 a line describe cellular telephone service? • Explain how the fee for air time affects the equation, and • Describe how you can use equations to compare various plans. Mathematics Standard 1: Geometry ALGEBRA, FUNCTIONS, AND GRAPHS: Students will understand algebraic concepts and applications. Essential Question: How will pictures and concepts help me solve math problems in the real world? Benchmark B. Understand patterns, relations, functions, and graphs. Performance Standards End Learning Assessment(s) Mastery 1. Describe the concept of Students will write an Practice Quiz 2, Student the graph of a function. equation of a line given Edition (SE), p. 150 information about its graph. Open-Ended Assessment, 2. Translate among tabular, Students will solve Teachers Wraparound symbolic, and graphical problems by writing Edition (TWE), pp. 150, representations of equations. 469 functions. Resources Glencoe Key Chapters: 3, 9 Glencoe Lessons 3-4, 9-1 Chapter 3 Resource Masters pp. 143-7 Chapter 9 Resource Masters pp. 479-483 3. Describe the concept of a Students will draw reflected Enrichment, Chapter images. Resource Masters (CRM), graph of an equation. pp.148, 484 Students will recognize and 4. Understand symmetry of draw lines of symmetry geometryonline.com/self_ graphs. and points of symmetry. check_quiz Vocabulary/concepts TestCheck and Worksheet Builder CD-ROM, Lessons 3.4, 9.1 2 • • • • • • • • • point-slope form rate of change slope-intercept form isometry line of reflection line of symmetry point of symmetry reflection transformation Builder CD-ROM, Lessons 3.4, 9.1 ACE Assessment: A certain cellular phone company charges a flat rate of $19.95 per month for service. All calls are charged $0.07 per minute of air time t. The total charge C for a month can be represented by the equation C = 0.07t + 19.95. How can the equation of a line describe cellular telephone service? • Explain how the fee for air time affects the equation, and • Describe how you can use equations to compare various plans. Mathematics Standard 1: Geometry ALGEBRA, FUNCTIONS, AND GRAPHS: Students will understand algebraic concepts and applications. Essential Question: What are quantitative relationships? Benchmark Performance Standards C. Use mathematical models to represent and understand quantitative relationships. 1. Use a variety of computational methods. End Learning Assessment(s) Mastery Students will write an Practice Quiz 2, Student equation of a line given Edition (SE), p. 150 information about its graph. Resources Glencoe Key Chapters: 3, 5, 7 3 relationships. 2. Graph a linear equation and linear inequality in two Students will solve variables. problems by writing equations. 3. Solve applications Vocabulary/concepts involving systems of • point-slope form equations. • rate of change • slope-intercept form 4. Create a linear equation from a table of values containing co-linear data. Open-Ended Assessment, Teachers Wraparound Edition (TWE), pp. 150, 245, 370 5. Generate an algebraic sentence to model real-life situations. Assessment, Quiz 2, CRM, p.407 6. Write an equation of a line that passes through two given points. 7. Verify that a point lies on a line, given an equation of the line, and be able to derive linear equations by using the point-slope formula. Enrichment, Chapter Resource Masters (CRM), pp.148, 250, 374 Glencoe Lessons 3.4,5.1, 7.4 Chapter 3 Resource Masters pp. 143-7 Chapter 5 Resource Masters pp. 245-249 Chapter 7 Resource Masters pp. 369-373 Assessment, Mid-Chapter Test, CRM, p.409 geometryonline.com/self_ check_quiz TestCheck and Worksheet Builder CD-ROM, Lessons 3.4, 5.1, 7.4 ACE Assessment: The old surveyor’s telescope shown on p. 364 is called a theodolite. It is an optical instrument used to measure angles in surveying, navigation, and meteorology. It consists of a telescope fitted with a level and mounted on a tripod so that it is free to rotate about its vertical and horizontal axes. After measuring angles, surveyors apply trigonometry to calculate distance and height. 4 surveyors apply trigonometry to calculate distance and height. How do surveyors determine angle measures? • Where are theodolites used? • What kind of information can one obtain from a theodolite? Mathematics Standard 1: Geometry ALGEBRA, FUNCTIONS, AND GRAPHS: Students will understand algebraic concepts and applications. Essential Question: What is it, and how is it changing? Benchmark Performance Standards D. Analyze changes in various contexts. 1. Analyze the effects of parameter changes on polynomial functions. End Learning Mastery Students will find the geometric mean between two numbers. Students will solve 2. Solve routine two- and three-step problems relating problems involving relationships between parts to ratios. of a right triangle and the altitude to its hypotenuse. 3. Estimate the rate of change of a function or Students will use the equation by finding the Pythagorean Theorem. slope between 2 points on the graph. Students will use the converse of the 4. Evaluate the estimated Pythagorean Theorem. rate of change in the context of the problem. Assessments Open-Ended Assessment, Teachers Wraparound Edition (TWE), pp. 144, 287, 331, 348, 356 Enrichment, Chapter Resource Masters (CRM), pp.142, 300, 330, 356, 362 Assessment, Mid-Chapter Test, CRM, p. 177 Resources Glencoe Key Chapters: 3, 6, & 7 Glencoe Lessons: 3-3, 6-1, 6-6, 7-1, 7-2 Chapter 3 Resource Masters pp. 137-141 Chapter 6 Resource Masters pp. 295-299, 325-329 Chapter 7 Resource Masters pp. 351-355, 357-361 Assessment, Quiz 2, CRM, p. 175; Quiz 4, p.346; Quiz Technology: 1, p. 407 Graphing Calculator and Computer Masters geometryonline.com/self_ pp. 22, 27, 28, 30 5 context of the problem. Students will write ratios. check_quiz 5. Know Pascal’s triangle and use it to expand binomial expressions that are raised to positive integer powers. Students will use properties of proportions. TestCheck and Worksheet Builder CD-ROM, Lessons 3.3, 6.1, 6.6, 7.1, 7.2 D. Analyze changes in various contexts. (cont’d.) Students will find slopes of lines. ACE Assessment: How are Students will use slope to right triangles used to build identify parallel and suspension bridges? perpendicular lines. Include he following in your answer: Students will recognize • locations of the right and describe characteristics triangles. of fractals. • An explanation of which parts of the Students will identify nonright triangle are geometric iteration. formed by the cables. Vocabulary/Concepts • geometric mean • Pythagorean triple • cross-product • extremes • means • proportion • ratio • rate of change • slope • slope-intercept form • fractal • iteration • self-similar Mathematics Standard 2: Geometry 6 GEOMETRY and TRIGONOMETRY: Students will understand geometric concepts and applications. Essential Question: How can I use math to describe two-dimensional shapes and three-dimensional solid shapes? A. Analyze characteristics 1. Interpret and draw twoStudents will find Practice Quizzes, Student Glencoe Key Chapters: and properties of two- and dimensional objects and perimeters and areas of Edition (SE): Quiz 2, p. 80; 2-9, 11-13 three-dimensional find the area and perimeter parallelograms. Quiz 1, p. 138; Quiz 1, p. Glencoe Lessons: 2.1, 2.3, geometric shapes and of basic figures (triangles, 198; Quiz 1, p. 306; Quiz 2.4, 2.8, 3.1, 3.2, 3.5, 4.1, develop mathematical rectangles, circles, rhombi, Students will determine 1, p. 423; Quiz 2, p. 497; 4.3-4.5, 5.1, 6.2, 6.3, 7.2, arguments about geometric parallelograms, trapezoids) whether points on a Quiz 1, p. 609; Quiz 2, p. 8.1, 8.3, 8.4, 9.1, 9.5, 11.1relationships. coordinate plane define a 621; Quiz 1, p. 659; Quiz 11.4, 12.2-12.6, 13.1 parallelogram. 2, p. 670 2. Find the area and Glencoe Chapter Resource perimeter of a geometric Students will find areas of Open-Ended Assessment, Masters: figure composed of a triangles, trapezoids, and Teachers Wraparound pp. 57-61, 69-73, 75-79, combination of two or Edition (TWE), pp. 66, 80, 99-103, 125-129, 131-135, more rectangles, triangles, rhombi. 87, 114, 131, 138, 157, 149-153, 183-187, 195and/or semicircles with Students will find areas of 183, 198, 206, 213, 245, 199, 201-205, 207-211, just edges in common. regular polygons and 297, 306, 356, 409, 423, 245-249, 301-305, 307circles. 430, 469, 497, 600, 609, 311, 357-361, 417-421, 3. Find and use measures 616, 621, 648, 654, 659, 429-433, 435-439, 479of sides and interior and 483, 503-507, 611-615, exterior angles of triangles Students will find areas of 665, 670, 694 irregular figures, and of 617-621, 623-627, 629and polygons to classify Enrichment, Chapter 633, 667-671, 673-677, figures (scalene, isosceles, irregular figures on the Resource Masters (CRM), 679-683, 685-689, 691and equilateral, rectangles, coordinate plane. pp. 62, 74, 80, 104, 130, 695, 723-727 and other convex Students will identify and 136, 154, 188, 200, 206, polygons.) classify triangles by 212, 250, 306, 312, 362, Technology: 422, 434, 440, 484, 508, Graphing Calculator and 4. Interpret and draw three- angles, and by sides. 616, 622, 628, 634, 672, Computer Masters dimensional objects and Students will find the sum 678, 684, 690, 696, 728 pp. 19, 30-32, 37-38 find the surface area and of the measures of the volume of basic figures interior angles of a Assessment, Mid-Chapter (spheres, rectangular polygon, and the sum of Test, CRM, p. 121, 241, solids, prisms, polygonal the exterior angles of a 347, 475, 657, 719 cones), and calculate the surface areas and volumes polygon. of these figures as well as figures constructed from unions of rectangular solids and prisms with faces in common, given 7 A. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. (Cont’d) figures constructed from unions of rectangular solids and prisms with faces in common, given the formulas for these figure. 5. Demonstrate an understanding of simple aspects of a logical argument: • • identify the hypothesis and conclusion in logical deduction use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. 6. Demonstrate an understanding of inductive and deductive reasoning, explain the difference between inductive and deductive reasoning, and identify and provide examples of each: • for inductive reasoning, demonstrate Students will recognize the conditions that ensure a quadrilateral is a parallelogram. Students will prove that a set of points forms a parallelogram in the coordinate plane. Students will draw 2dimensional models for 3dimensional figures. Students will find surface area. Students will find lateral areas and surface areas of prisms. Students will find lateral areas and surface areas of cylinders. Students will find lateral areas and surface areas of regular pyramids. Students will find lateral areas and surface areas of cones. Assessment, CRM, p.119, Quiz 2; p. 120, Quiz 4; p. 175, Quiz 1; p. 176, Quiz 3; p. 239, Quiz 2; p. 345, Quiz 1; p. 345, Quiz 2; p. 407, Quiz 1; p. 473, Quiz 2; p. 655, Quiz 1 & Quiz 2; p. 656, Quiz 3; p. 717, Quiz 1 & Quiz 2; p. 718, Quiz 3 geometryonline.com/self_ check_quiz TestCheck and Worksheet Builder CD-ROM, Lessons 2.1, 2.3, 2.4, 2.8, 3.1, 3.2, 3.5, 4.1, 4.3-4.5, 5.1, 6.2, 6.3, 7.2, 8.1, 8.3, 8.4, 9.1, 9.5, 11.1-11.4, 12.2-12.6, 13.1 ACE Assessment: Explain the similarities and differences between finding the lateral area and surface area of a cone versus the lateral area and surface area of a regular pyramid. Students will find volumes of prisms and cylinders. 8 A. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. (Cont’d) demonstrate • understanding that showing a statement is true for a finite number of examples does not show it is true for all cases unless the cases verified are all cases. • For deductive reasoning, prove simple theorems. Students will analyze statement in if-then form. Students will write the converse, inverse, and contrapositive of if-then statements. 7. Write geometric proofs (including proofs by contradiction), including: Students will name and label corresponding parts of congruent triangles. a. Theorems involving the properties of parallel lines cut by a transversal line and the properties of quadrilaterals. Students will identify congruence transformations. b. Theorems involving complementary, supplementary, and congruent angles. c. Theorems involving congruence and similarity. d. The Pythagorean Theorem. Students will make conjectures based on inductive reasoning. Students will find counterexamples. Students will identify and use perpendicular bisectors and angle bisectors for triangles. Students will identify and use medians and altitudes in triangles. Students will recognize and apply properties of rectangles. Students will determine whether parallelograms are rectangles. 9 whether parallelograms are rectangles. A. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. (Cont’d) Students will draw reflected images. Students will recognize and draw lines of symmetry and points of symmetry. Students will determine whether a dilation is an enlargement, a reduction, or a congruence transformation. Students will determine the scale factor for a given dilation. Students will use the Law of Detachment, and the Law of Syllogism. Students will identify the relationships between two lines and two points. Students will name angles formed by a pair of lines and a transversal. Students will use the properties of parallel lines to determine congruent angles. 10 to determine congruent angles. Students will use algebra to find angle measures. A. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. (Cont’d) Students will recognize angle conditions that occur with parallel lines. Students will prove that two lines are parallel based on given angle relationships. Students will write proofs involving supplementary and complementary angles. Students will write proofs involving congruent and right angles. Students will use the SSS and SAS postulates to test for triangle congruence. Students will use the ASA Postulate and AAS Theorem to test for triangle congruence. Students will identify similar figures. Students will solve 2problems involving scale factors. 11 2problems involving scale factors. Students will identify similar triangles. A. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. (Cont’d) Students will use similar triangles to solve problems Students will use the Pythagorean Theorem. Students will use the converse of the Pythagorean Theorem. Vocabulary/Concepts • apothem • irregular figure • irregular polygon • acute triangle • equiangular triangle • equilateral triangle • isosceles triangle • obtuse triangle • right triangle • scalene triangle • diagonal • axis • circular cone • lateral area • lateral edges • lateral faces • net • oblique cone • oblique cylinder 12 A. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. (Cont’d) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • oblique prism regular pyramid right cone right cylinder right prism slant height surface area volume conclusion conditional statement contrapositive converse hypothesis if-then statement inverse logically equivalent related conditionals conjecture counterexample inductive reasoning congruence transformations altitude centroid circumcenter concurrent lines incenter median orthocenter perpendicular bisector point of concurrency rectangle isometry 13 • • • • • • • • A. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. (Cont’d) • • • • • • • • • • • • • • • line of reflection line of symmetry point of symmetry reflection transformation dilation similarity transformation deductive reasoning Law of Detachment Law of Syllogism alternate exterior angles alternate interior angles consecutive interior angles corresponding angles parallel lines parallel planes skew lines transversal included angle included side scale factor similar polygons Pythagorean triple 14 Mathematics Standard 2: Geometry GEOMETRY and TRIGONOMETRY: Students will understand geometric concepts and applications. Essential Question: How can a point be located on a plane (flat) surface? Benchmark B. Specify locations and describe spatial relationships using coordinate geometry and other representational systems. Performance Standards 1. Demonstrate understanding of the construction of the coordinate plane, know the names of the origin, coordinate axes and four quadrants, draw and label them correctly, find the coordinates of an indicated point, and plot a point with given coordinates. 2. Determine the midpoint and distance between two points within a coordinate system and relate these ideas to geometric figures in the plane (e.g., find the center of a circle given two endpoints of a diameter of the circle). 3. Given two linear equations, determine whether the lines are parallel, perpendicular, or coincide. End Learning Mastery Students will find the distance between two points. Students will find the midpoint of a segment. Students will identify and name polygons. Students will find the perimeters of polygons. Students will write an equation of a line given information. Assessment(s) Practice Quiz 2, Student Glencoe Key Chapters: 1, Edition (SE), p. 150, p. 306, 3-6, & 11 p.609 Glencoe Lessons 1.3, 1.6, Open-Ended Assessment, 3.4, 4.1, 4.4, 5.1, 6.1-6.3, Teachers Wraparound 11.2 Edition (TWE), pp. 27, 50, 150, 183, 206, 245, 287, Glencoe Chapter Resource 297, 306, 609 Masters pp. 13-17, 31-35, 145-6, Enrichment, Chapter 183-187, 201-205, 245-249, Resource Masters (CRM), 295-299, 301-305, 307-311, pp. 18, 36, 148, 188, 206, 617-621 250, 300, 306, 312, 622 Assessment, Mid-Chapter Test, CRM, p. 53, 241, 347 Students will solve problems by writing equations. Assessment, CRM, p. 51, Quiz 2; p. 52, Quiz 4; p. 239, Quiz 2; p. 345, Quiz 1 Students will identify and & Quiz 2; p.655, Quiz 1 classify triangles by angles, geometryonline.com/self_ and by sides. check_quiz Students will use the SSS and SAS Postulates to test for triangle congruence. Resources Technology: Glencoe Graphing Calculator and Computer Masters Pp. 18, 27 TestCheck and Worksheet Builder CD-ROM, Lessons 1.3, 1.6, 3.4, 4.1, 4.4, 5.1, 6.1-6.3, 11.2 15 4. Use basic geometric ideas (e.g., the Pythagorean theorem, area, and perimeter of objects) in the context of the Euclidean Plane, calculate the perimeter of a rectangle with integer coordinates and sides parallel to the coordinate axes and with sides not parallel. 6.1-6.3, 11.2 Students will identify and use perpendicular bisectors, medians, altitudes, and ACE Assessment: How do angle bisectors. engineers use geometry? • Why do engineers Students will identify and use triangles in use medians and altitudes in construction? triangles. • Why do you think Students will write ratios. the pressure applied Students will use to the ground from properties of proportions. the Eiffel Tower is so small? Vocabulary: Division Property of Equality Multiplication Property of Equality inequality variable unknown numerical expression algebraic expression evaluate Students will model and solve contextualized problems using various representations, such as graphs, tables, formulas, and equations. 16 and equations. Vocabulary: Venn diagram sets models graphs charts Mathematics Standard 2: Geometry GEOMETRY and TRIGONOMETRY: Students will understand geometric concepts and applications. Benchmark B. Specify locations and describe spatial relationships using coordinate geometry and other representational systems. (cont’d.) Performance Standards End Learning Mastery Students will identify similar figures. Assessment(s) Resources Students will solve problems involving scale factors. Students will identify similar triangles. Students will use similar triangles to solve problems. Students will find areas of triangles, trapezoids, and rhombi. 17 B. Specify locations and describe spatial relationships using coordinate geometry and other representational systems. (cont’d.) Vocabulary/Concepts • midpoint • segment bisector • concave • convex • n-gon • perimeter • polygon • regular polygon • point-slope form • slope-intercept form • acute triangle • equiangular triangle • equilateral triangle • isosceles triangle • obtuse triangle • right triangle • scalene triangle • included angle • altitude • centroid • circumcenter • concurrent lines • incenter • median • orthocenter • perpendicular bisector • point of concurrency • cross-product • extremes • means • proportion • ratio • scale factor 18 • similar polygons Mathematics Standard 2: Geometry GEOMETRY: Students will understand geometric concepts and applications. Essential Question: How does it move? Benchmark C. Apply transformations and use symmetry to analyze mathematical situations. Performance Standards End Learning Mastery Students will draw reflected images. Assessment(s) 1. Describe the effect of rigid motions on figures in the coordinate plane and space that include rotations, Students will recognize translations, and reflections: and draw lines of symmetry and points of symmetry a. determine whether a Students will draw given pair of figures on a coordinate plane represents translated images using coordinates. the effect of a translation, reflection, rotation,and/or Students will draw dilation. translated images by using b. sketch the planar figure repeated reflections. that is the result of a given transformation of this type. Students will draw rotated images using the angle of rotation. Practice, Student Edition (SE), Quiz 2, p. 482; Quiz 2, p. 497 Students will identify 2. Deduce properties of figures with rotational figures using transformations that include symmetry. translations, rotations, reflections, and dilations in a coordinate system: Geometryonline.com/self_ check_quiz Open-Ended Assessment, Teachers Wraparound Edition (TWE), pp. 469, 475, 482, 497 Enrichment, Chapter Resource Masters (CRM), pp. 484, 490, 496, 508 Assessment, Mid-Chapter Test, CRM, p.409 Resources Glencoe Key Chapters 9 Glencoe Lessons 9.1-9.3, 9.5 Glencoe Chapter Resource Masters pp. 479-483, 485-489, 491495, 503-507 Technology: Glencoe Graphing Calculator and Computer Masters pp. 33-34 Assessment, CRM, Quiz 1, p. 535 19 translations, rotations, reflections, and dilations in a coordinate system: TestCheck and Worksheet Students will determine Builder CD-ROM, Lessons whether a dilation is an 9.1-9.3, 9.5 enlargement, a reduction, or a. identify congruency and a congruent transformation. similarity in terms of Students will determine the ACE Assessment: transformations. scale factor for a given Explain how amusement rides exemplify rotations. b. determine the effects of dilation. Include the following in the above transformations Vocabulary/Concepts: your answer: on linear and area • angle of rotation • a description of how measurements of the • center of rotation the Tilt-A-Whirl original planar figure. • composition actually rotates two • dilation ways, and • direct isometry • other amusement • glide reflection rides that use • indirect isometry rotation. • invariant points • isometry • line of reflection • line of symmetry • point of symmetry • reflection • rotation • rotational symmetry • similarity transformation • translation Mathematics Standard 2: Geometry GEOMETRY and TRIGONOMETRY: Students will understand geometric concepts and applications. Essential Question: Can problems be solved without using numbers? 20 Benchmark End Learning Assessment(s) Mastery D. Use visualization, spatial 1. Solve real-world Students will name and Practice Quiz, Student reasoning, and geometric problems using congruence label corresponding parts of Edition (SE), Quiz 1, p. modeling to solve and similarity relationships congruent triangles. 198; Quiz 1, p. 306; Quiz 2, problems. of triangles. p. 323; Quiz 1, p. 363; Quiz Students will identify con- 2, p. 393; Quiz 1, p. 609 2. Solve problems gruence transformations. Open-Ended Assessment, involving complementary, supplementary, and conStudents will identify Teachers Wraparound similar triangles. Edition (TWE), pp. 114, gruent angles. 198, 306, 315, 323, 356, 363, 370, 376, 383, 390, 3.Solve problems involving Students will use similar triangles to solve problems. 600, 609, 616 the perimeter, circumference, area, volume, and Students will use proporEnrichment, Chapter surface area of common tional parts of triangles. Resource Masters (CRM), geometric figures. pp. 104, 200, 312, 318, 324, 362, 368, 374, 380, 386, 4. Solve problems using the Students will divide a segment into parts. 392, 616, 622, 628 Pythagorean Theorem. D. Use visualization, spatial reasoning, and geometric modeling to solve problems. (Cont’d) Performance Standards 5. Understand and use elementary relationships of basic trigonometric functions defined by the angles of a right triangle. Students will recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles. 6. Use trigonometric functions to solve for the length of the second leg of a right triangle given the angles and the length of the first leg. Students will recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles. Assessment, Mid-Chapter Test, CRM, pp. 347, 409, 657 Resources Glencoe Key Chapters: 2, 4, 6, 7, 11 Glencoe Lessons: 2.8, 4.3, 6.3-6.5, 7.2-7.7, 11.1-11.3 Glencoe Chapter Resource Masters pp. 99-103, 195-199, 307311, 313-317, 319-323, 357-361, 363-367, 369-373, 375-379, 381-385, 387-391, 611-615, 617-621, 623-627 Technology: Glencoe Graphing Calculator and Computer Masters pp. 28-30, 37-38 Assessment, CRM, Quiz 4, p. 120; Quiz 2, p. 345; Quiz 3, p. 346; Quiz 1, p. 407; Quiz 2, p.407; Quiz 3, p. 408; Quiz 4, p. 408; Quiz 1, p. 655; Quiz 2, p. 655 geometryonline.com/self_ check_quiz TestCheck and Worksheet Builder CD-ROM, Lessons 2.8, 4.3, 6.3-6.5, 7.2-7.7, 11.1-11.3 21 first leg. triangles. Builder CD-ROM, Lessons 2.8, 4.3, 6.3-6.5, 7.2-7.7, 11.1-11.3 7. Know and use angle and side relationships in problems with special right triangles. Students will write proofs involving supplementary and complementary angles. ACE Assessment: The Chicago Metropolitan Students will write proofs Correctional Center is a 27involving congruent and story federal detention right angles. center. The cells are arranged around a loungeStudents will find like common area. The perimeters and areas of architect found that a parallelograms. triangular floor plan allowed for the maximum Students will determine number of cells to be most whether points on a efficiently centered around coordinate plane define a the lounge. parallelogram. How are triangles used in building design? Students will find areas of • Why was the triangles, trapezoids, and building triangular rhombi. instead of rectangular? Students will find areas of Why couldn’t the Law of regular polygons and Sines be used to solve the circles. triangle? Students will use the Pythagorean Theorem. D. Use visualization, spatial reasoning, and geometric modeling to solve problems. (Cont’d) Students will use the converse of the Pythagorean Theorem. Students will use properties of 45-45-90 triangles. 22 problems. (Cont’d) Students will use properties of 30-60-90 triangles. Students will find trigonometric ratios using right triangles. Students will solve problems using trigonometric ratios. Students will solve problems involving angles of elevation, and angles of depression. Students will use the Law of Sines to solve triangles. Students will solve problems by using the Law of Sines. Students will use the Law of Cosines to solve triangles. Students will solve problems by using the Law of Cosines. D. Use visualization, spatial reasoning, and geometric modeling to solve problems. (Cont’d) Vocabulary/Concepts: 23 reasoning, and geometric modeling to solve problems. (Cont’d) • • • • • • • • • • • • • • • • • • • congruence transformations congruent triangles midsegment apothem ambiguous case angle of depression angle of elevation cosine Law of Cosines Law of Sines Pythagorean identity Pythagorean triple Reciprocal identities sine solving a triangle tangent trigonometric identity trigonometric ratio trigonometry Mathematics Standard 3: Geometry DATA ANALYSIS AND PROBABILITY: Students will understand how to formulate questions, analyze data, and determine probabilities. Essential Question: What is a statistical method? reasoning, and geometric modeling to solve A. Select and use 1. For univariate data, be problems. (Cont’d) appropriate statistical able to display the methods to analyze data. distribution and describe its shape using appropriate summary statistics, and understand the distinction between a statistic and a parameter: congruence transformations Students will identify and • congruent triangles use •perpendicular bisectors, midsegment medians, altitudes, and • apothem angle bisectors. • ambiguous case • angle of depression Students willofidentify and • angle elevation use •medians cosineand altitudes in triangles. • Law of Cosines • Law of Sines • Pythagorean identity • Pythagorean triple • Open-Ended Assessment, Teachers Wraparound Edition (TWE), p. 245 Glencoe Key Chapter: 5 Enrichment, Chapter Resource Masters (CRM), pp. 250 Glencoe Chapter 5 Resource Masters pp. 245-249 Glencoe Lessons 5.1 24 parameter: Vocabulary/Concepts: • altitude Calculate and apply • centroid measures of central • circumcenter tendency (mean, median, • concurrent lines mode) and measures of • incenter variability (range, quartiles, • median standard deviation) • orthocenter • perpendicular bisector • point of concurrency geometryonline.com/self_ check_quiz TestCheck and Worksheet Builder CD-ROM, Lesson 5.1 ACE Assessment: Explain how you can balance a paper triangle on a pencil point. Include the following in your answer: • which special point is the center of gravity, and • a construction showing how to find this point. 25