A Correlation of. To the. New York State Next Generation Mathematics Learning Standards Geometry
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1 A Correlation of 2018 To the New York State Next Generation Mathematics Learning Standards
2 Table of Contents Standards for Mathematical Practice Copyright 2018 Pearson Education, Inc. or its affiliate(s). All rights reserved
3 Learning Standards Standards for Mathematical Practice MP.1 Make sense of problems and persevere in solving them. SE/TE: 5-13, 22-27, 36-43, 58-63, 71-77, , , , , , , , , , , , , , , , , , , , , , , 479, , , MP.2 Reason abstractly and quantitatively. SE/TE: 5-13, 22-27, 44-50, 51-57, 58-63, 78-84, 85-91, 92-98, , , , , , , , , , , , , , , , , , , , , , , 514, , , MP.3 Construct viable arguments & critique the reasoning of others. SE/TE: 14-21, 28-34, 51-57, 71-77, 78-84, 92-98, , , , , , , , , , , , , , , , , , , , MP.4 Model with mathematics. SE/TE: 35, 44-50, 85-91, 92-98, 99, , 142, 166, , 217, , , 252, , , , 332, , , , , 373, , 392, , 435, , , 479, , , 514, , , MP.5 Use appropriate tools strategically. SE/TE: 14-21, , , , , , , , , , MP.6 Attend to precision. SE/TE: 36-43, 99, , , , , , , , , , , , , , , , , ,
4 Learning Standards MP.7 Look for and make use of structure. SE/TE: 5-13, 14-21, 22-27, 28-34, 36-43, 44-50, 51-57, 58-63, 78-84, 85-91, , , , , , , , , , , , , , , , , , , , , , , , , , , , , MP.8 Look for and express regularity in repeated reasoning. SE/TE: 28-34, 71-77, , , , , , , 514 GEO-G.CO Congruence Experiment with transformations in the plane 1. Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane. 2. Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not. SE/TE: 6, 8-9, 10, 11, 12-13, 23-24, 24, 25, TE: 13A-13B, 27A-27B SE/TE: , , , , , , TE: 112A-112B, 120A-120B, 128A-128B, 135A- 135B, 156A-156B, 309A-309B, 316A-316B 3. Given a regular or irregular polygon, describe the rotations and reflections (symmetries) that map the polygon onto itself. SE/TE: 136, 137, 138, 139, , 142, 157 TE: 141A-141B, 165A-165B 4. Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments. SE/TE: 105, 106, 110, 113, 121, 122, 133 TE: 112A-112B, 120A-120B, 128A-128B, 135A- 135B 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another. SE/TE: 106, 107, 108, 109, 110, , 114, 115, , 121, 122, 123, 126, , 131, , 149 TE: 112A-112B, 120A-120B, 128A-128B, 135A- 135B, 156A-156B 2
5 Learning Standards Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. SE/TE: 105, 110, , 115, , 130, 132, 133, , 149, 151, 152, 154, , 183, 183, 184, 185, , 190, 191, 191, TE: 112A-112B, 120A-120B, 135A-135B, 156A- 156B, 187A-187B, 193A-193B 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. SE/TE: 149, 150, 154, , , 170, , 174, 175, 184, TE: 156A-156B, 173A-173B, 181A-181B, 187A- 187B 8. Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS and HL (Hypotenuse Leg)) follow from the definition of congruence in terms of rigid motions. SE/TE: 70, 149, 150, 150, 154, , , , 170, , 173, 174, 175, , 178, , 182, 183, 185, , 189, 190, 191, Prove geometric theorems 9. Prove and apply theorems about lines and angles. TE: 156A-156B, 173A-173B, 181A-181B, 187A- 187B, 193A-193B SE/TE: 51-52, 53, 54, 55, 56-57, 61, 62-63, 71, 72, 73, 74, 75, 76-77, 78, 79, 80, 81, 82, 83-84, 184, , 189, , , 229, 231, 271, 273, , 276, 277, 281, 287, 345, 452 TE: 57A-57B, 63A-63B, 77A-77B, 84A-84B, 187A- 187B, 193A-193B, 208A-208B, 232A-232B, 278A- 278B, 285A-285B, 293A-293B, 353A-353B, 458A- 458B 10. Prove theorems about triangles. SE/TE: 85, 86, 86, 87, 87, 87-88, 89, 90-91, 164, 173, 176, 182, , 219, 220, 224, 228, 287, 334, 337, , TE: 91A-91B, 165A-165B, 173A-173B, 181A-181B, 187A-187B, 225A-225B, 232A-232B, 293A-293B, 339A-339B, 399A-399B 3
6 Learning Standards 11. Prove theorems about parallelograms. SE/TE: 263, 264, 265, 266, 267, 268, 269, 271, 272, 273, , 276, , 281, 283, , 286, , 290, 291, 292 TE: 270A-270B, 278A-278B, 285A-285B, 293A- 293B Make geometric constructions 12. Make, justify, and apply formal geometric constructions. SE/TE: 14, 15, 16, 17, 18, 19, 20-21, 166, 181, 207 TE: 21A-21B, 181A-181B, 208A-208B 13. Make and justify the constructions for inscribing an equilateral triangle, a square and a regular hexagon in a circle. SE/TE: 439, 442, 450, 461 TE: 443A-443B, 451A-451B GEO-G.SRT Similarity, Right Triangles, and Trigonometry Understand similarity in terms of similarity transformations 1. Verify experimentally the properties of SE/TE: dilations given by a center and a scale factor. TE: 309A-309B a. Verify experimentally that dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. SE/TE: TE: 309A-309B b. Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. SE/TE: 149, , 303, 307, , 318 TE: 156A-156B, 309A-309B, 323A-323B 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Explain using similarity transformations that similar triangles have equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. SE/TE: 301, 314, , 317, 318, 319, 321, TE: 309A-309B, 316A-316B, 323A-323B 4
7 Learning Standards 3. Use the properties of similarity transformations to establish the AA~, SSS~, and SAS~ criterion for two triangles to be similar. SE/TE: 317, 318, 321 TE: 323A-323B Prove theorems involving similarity 4. Prove and apply similarity theorems about triangles. SE/TE: 333, 334, 335, 337, , 346 TE: 339A-339B, 353A-353B 5. Use congruence and similarity criteria for triangles to: a. Solve problems algebraically and geometrically. SE/TE: 178, , 191, , 262, 317, 319, 321, , 324, 325, 333 TE: 181A-181B, 193A-193B, 270A-270B, 323A- 323B, 331A-331B, 339A-339B b. Prove relationships in geometric figures. SE/TE: 178, , 191, , 262, 317, 319, 321, , 324, 325, 333 TE: 181A-181B, 193A-193B, 270A-270B, 323A- 323B, 331A-331B, 339A-339B Define trigonometric ratios and solve problems involving right triangles 6. Understand that by similarity, side ratios in SE/TE: 354, 355, 358, , 377 right triangles are properties of the angles in the triangle, leading to definitions of sine, cosine and TE: 360A-360B, 379A-379B tangent ratios for acute angles. 7. Explain and use the relationship between the sine and cosine of complementary angles. SE/TE: , 358 TE: 360A-360B 8. Use sine, cosine, tangent, the Pythagorean Theorem and properties of special right triangles to solve right triangles in applied problems. SE/TE: 282, 345, 346, 347, 349, 350, 351, , 358, , 367, 373, 377, , 386 TE: 285A-285B, 353A-353B, 360A-360B, 372A- 372B, 379A-379B, 391A-391B 5
8 Learning Standards Apply trigonometry to general triangles 9. Justify and apply the formula = 1/2 sin( ) to find the area of any triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. SE/TE: 374 (Additional Example 4), TE: 379A-379B GEO-G.C Circles Understand and apply theorems about circles 1. Prove that all circles are similar. SE/TE: 313, 314, TE: 316A-316B 2a. Identify, describe and apply relationships between the angles and their intercepted arcs of a circle. SE/TE: 428, 429, 432, , 436, 437, 438, 439, 440, 441, , 444, 445, 446, 448 TE: 434A-434B, 443A-443B, 451A-451B 2b. Identify, describe and apply relationships among radii, chords, tangents, and secants of a circle. SE/TE: 428, 429, 432, , 436, 437, 438, 439, 440, 441, , 444, 445, 446, 448 TE: 434A-434B, 443A-443B, 451A-451B Find arc lengths and areas of sectors of circles 5. Using proportionality, find one of the following given two others; the central angle, arc length, radius or area of sector. SE/TE: TE: 426A-426B GEO-G.GPE Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section 1a. Derive the equation of a circle of given center SE/TE: 400, 401 and radius using the Pythagorean Theorem. Find the center and radius of a circle, given the TE: 405A-405B equation of the circle. 1b. Graph circles given their equation. SE/TE: 400, 401 TE: 405A-405B 6
9 Learning Standards Use coordinates to prove simple geometric theorems algebraically 4. On the coordinate plane, algebraically prove SE/TE: 386, geometric theorems and properties. TE: 391A-391B 5. On the coordinate plane: a. Explore the proof for the relationship between slopes of parallel and perpendicular lines; SE/TE: 92-93, 94, 95, 96, 97-98, 99, 222, , 385, 389, 393, 397, TE: 98A-98B, 225A-225B, 391A-391B, 399A-399B b. Determine if lines are parallel, perpendicular, or neither, based on their slopes; and SE/TE: 92-93, 94, 95, 96, 97-98, 99, 222, , 385, 389, 393, 397, TE: 98A-98B, 225A-225B, 391A-391B, 399A-399B c. Apply properties of parallel and perpendicular lines to solve geometric problems. SE/TE: 92-93, 94, 95, 96, 97-98, 99, 222, , 385, 389, 393, 397, TE: 98A-98B, 225A-225B, 391A-391B, 399A-399B 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. SE/TE: 22, 23, 25, 26-27, 385, 389, 394 TE: 27A-27B, 391A-391B, 399A-399B 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. SE/TE: 24, 25, 26-27, 385, 386, 388, 389, TE: 27A-27B, 391A-391B GEO-G.GMD Geometric Measurement and Dimension Explain volume formulas and use them to solve problems 1. Provide informal arguments for the formulas SE/TE: 471, 472, 476, 480 for the circumference of a circle, area of a circle, volume of a cylinder, TE: 478A-478B, 486A-486B pyramid, and cone. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. SE/TE: 471, 472, 474, 475, 476, , 480, , , 484, , , 488, 489, 490, TE: 478A-478B, 486A-486B, 492A-492B 7
10 Learning Standards Visualize relationships between two-dimensional and three-dimensional objects 4. Identify the shapes of plane sections of threedimensional objects, and identify three- SE/TE: 465, 466, 467, 468, dimensional objects generated by rotations of TE: 470A-470B two-dimensional objects. GEO-G.MG Modeling with Apply geometric concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects. SE/TE: 70, 246, 252, 255, , 258, 259, 261, 263, 267, 270, 275, 278, 285, 288, 290, 293, 385 TE: 251A-251B, 261A-261B, 270A-270B, 278A- 278B, 285A-285B, 293A-293B, 391A-391B 2. Apply concepts of density based on area and volume of geometric figures in modeling situations. SE/TE: 474, 478, 489 TE: 478A-478B492A-492B 3. Apply geometric methods to solve design problems. SE/TE: 4, 70, 104, 148, 200, 244, 300, 344, 384, 418, 464, 498 8
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