# Florida Mathematics Standards for Geometry Honors (CPalms # )

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1 A Correlation of Florida Geometry Honors 2011 to the for Geometry Honors (CPalms # )

2 Geometry Honors (# ) Course Standards MAFS.912.G-CO.1.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. SE/TE: 11-13, 20-22, 27-29, 44, 52-53, , , TE: 19A-19B, 26A-26B, 33A-33B, 56A-56B, 146A-146B, 169A-169B, 657A-657B MAFS.912.G-CO.1.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). MAFS.912.G-CO.1.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. MAFS.912.G-CO.1.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. MAFS.912.G-CO.1.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. MAFS.912.G-CO.2.6: 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. MAFS.912.G-CO.2.7: 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: , 552 (Concept Byte), , , (Concept Byte) TE: 551A-551B, 558A-558B, 565A-565B, 581A- 581B, 592A-592B SE/TE: 549 (#17), 554, TE: 565A-565B SE/TE: , 552 (Concept Byte), , TE: 551A-551B, 558A-558B, 565A-565B SE/TE: 549 (#13-15), 552 (Concept Byte), 555 (#3), 556 (#20-24), 562 (#9-16), 563 (#24-29), 564 (#38, 54), (Concept Byte) TE: 551A-551B, 558A-558B, 565A-565B For related content, please see: SE/TE: , 548 (#4) TE: 551A-551B For related content, please see: SE/TE: 544, 549 (#7-9) TE: 551A-551B 2

3 MAFS.912.G-CO.2.8: Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. MAFS.912.G-CO.3.9: Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. MAFS.912.G-CO.3.10: Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180 ; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MAFS.912.G-CO.3.11: Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MAFS.912.G-CO.4.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. MAFS.912.G-CO.4.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. SE/TE: 225 (Concept Byte), , , TE: 233A-233B, 241A-241B, 264A-264B SE/TE: , , 147 (Concept Byte), , , , , 300 (Concept Byte), 363 TE: 127A-127B, 146A-146B, 155A-155B, 163A- 163B, 169A-169B, 299A-299B SE/TE: , , 284 (Concept Byte), , 308 (Concept Byte), , TE: 178A-178B, 256A-256B, 291A-291B, 315A- 315B, 331A-331B SE/TE: , , 375, 378 TE: 366A-366B, 374A-374B, 382A-382B SE/TE: 43-46, TE: 48A-48B, 188A-188B SE/TE: 667 (Concept Byte) 3

4 MAFS.912.G-SRT.1.1: Verify experimentally the properties of dilations given by a center and a scale factor: MAFS.912.G-SRT.1.1.a A 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. MAFS.912.G-SRT.1.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MAFS.912.G-SRT.1.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. MAFS.912.G-SRT.1.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. MAFS.912.G-SRT.2.4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. SE/TE: TE: 581A-581B SE/TE: TE: 581A-581B SE/TE: TE: 581A-581B SE/TE: , , 470 (Concept Byte), TE: 431A-431B, 447A-447B, 458A-458B, 478A- 478B SE/TE: , TE: 458A-458B SE/TE: , 490 (Concept Byte), 497 (#49) TE: 498A-498B SE/TE: , , , , , , , , , 741 (Concept Byte), TE: 366A-366B, 374A-374B, 382A-382B, 388A- 388BB, 447A-447B, 458A-458B, 467A-467B, 641A-641B, 749A-749B MAFS.912.G-SRT.3.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. MAFS.912.G-SRT.3.7: Explain and use the relationship between the sine and cosine of complementary angles. SE/TE: , , 506 (Concept Byte), TE: 467A-467B, 505A-505B, 513A-513B SE/TE: , 512 (#37) TE: 513A-513B 4

5 MAFS.912.G-SRT.3.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MAFS.912.G-SRT.4.10: Prove the Laws of Sines and Cosines and use them to solve problems. MAFS.912.G-SRT.4.11: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). MAFS.912.G-C.1.1: Prove that all circles are similar. MAFS.912.G-C.1.2: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MAFS.912.G-C.1.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MAFS.912.G-C.1.4: Construct a tangent line from a point outside a given circle to the circle. SE/TE: , , TE: 498A-498B, 505A-505B, 513A-513B SE/TE: (Concept Byte) SE/TE: (Concept Byte) This standard is outside the scope of Prentice Hall Florida Geometry Honors SE/TE: , 789 (Concept Byte), TE: 787A-787B, 797A-797B SE/TE: 301, 306 (#20-21), 766, 768 (#28) TE: 307A-307B, 769A-769B SE/TE: 768 (#28) MAFS.912.G-C.2.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. MAFS.912.G-GPE.1.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. SE/TE: TE: 787A-787B SE/TE: TE: 803A-803B 5

6 MAFS.912.G-GPE.1.2: Derive the equation of a parabola given a focus and directrix. This standard is addressed in Florida Honors Algebra See, for example, the following pages: SE/TE: TE: 629A-629B MAFS.912.G-GPE.1.3: Derive the equations of ellipses and hyperbolas given the foci and directrices. This standard is addressed in Florida Honors Algebra See, for example, the following pages: SE/TE: , TE: 644A-644B, 652A-652B MAFS.912.G-GPE.2.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). MAFS.912.G-GPE.2.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). MAFS.912.G-GPE.2.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. MAFS.912.G-GPE.2.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. MAFS.912.G-GMD.1.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. SE/TE: , TE: 412A-412B, 418A-418B SE/TE: , 205 (Pull It All Together) TE: 204A-204B SE/TE: 51-52, 402, 408, 411 (#22) TE: 56A-56B, 412A-412B For related content, please see: SE/TE: 401, 403 (#8-13), 404 (#17-30) TE: 405A-405B SE/TE: , 659 (Concept Byte), , , 725 (Concept Byte), TE: 657A-657B, 666A-666B, 724A-724B, 732A- 732B 6

7 MAFS.912.G-GMD.1.2: Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures. MAFS.912.G-GMD.1.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. MAFS.912.G-GMD.2.4: Identify the shapes of two-dimensional cross-sections of threedimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects. MAFS.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). MAFS.912.G-MG.1.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). MAFS.912.G-MG.1.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios) LAFS.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9 10 topics, texts, and issues, building on others ideas and expressing their own clearly and persuasively. SE/TE: , TE: 724A-724B, 732A-732B SE/TE: , 725 (Concept Byte), , 735, 737 (#17-22) TE: 724A-724B, 732A-732B, 740A-740B SE/TE: TE: 565B (Enrichment), 695A-695B For related content, please see: SE/TE: 354, 356 (#12-14), 365 (#31), 372 (#12), 390, 395 (#37-38), 693 (#34), 705 (#23), 710, 714 (#30), 728, 731 (#28) For related content, please see: SE/TE: 620 (#17), 633 (#30), 720, 722 (#21, 28), 723 (#30, 35), 730 (#15-16) This standard is outside the scope of Prentice Hall Florida Geometry Honors Students have an opportunity to participate in collaborative discussions as they complete Solve It! / Getting Ready! exercises at the beginning of each lesson, and Think About a Plan, Reasoning, Compare and Contrast, and Open- Ended problems in the Practice and Problem- Solving Exercises. For example, please see: SE/TE: 4, 82, 94, 106, 249, 250, 285, 301, 389, 395, 460, 491, 499, 516, 660 7

10 LAFS.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. LAFS.910.WHST.1.1: Write arguments focused on discipline-specific content. LAFS.910.WHST.1.1.a Introduce precise claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that establishes clear relationships among the claim(s), counterclaims, reasons, and evidence. LAFS.910.WHST.1.1.b Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form and in a manner that anticipates the audience s knowledge level and concerns. LAFS.910.WHST.1.1.c Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. SE/TE: 18 (#50), 25 (#35-36), 65 (#38), 255 (#27), 412 (#31), 446 (#32), 456 (#15), 475 (#23), 476 (#31-32), 520 (#29, 33), 556 (#19), 738 (#28), 768 (#20) Students have an opportunity to write arguments focused on discipline-specific content as they solve Open-Ended, Think About a Plan, Compare and Contrast, Writing, and Reasoning problems in the exercise sets. Sample references: SE/TE: 94, 112, 152, 154, 254, 357, 404, 465, 496, 520, 620, 640, 664, 714 Students have an opportunity to write arguments focused on discipline-specific content as they solve Open-Ended, Think About a Plan, Compare and Contrast, Writing, and Reasoning problems in the exercise sets. Sample references: SE/TE: 94, 112, 152, 154, 254, 357, 404, 465, 496, 520, 620, 640, 664, 714 Students have an opportunity to write arguments focused on discipline-specific content as they solve Open-Ended, Think About a Plan, Compare and Contrast, Writing, and Reasoning problems in the exercise sets. Sample references: SE/TE: 94, 112, 152, 154, 254, 357, 404, 465, 496, 520, 620, 640, 664, 714 Students have an opportunity to write arguments focused on discipline-specific content as they solve Open-Ended, Think About a Plan, Compare and Contrast, Writing, and Reasoning problems in the exercise sets. Sample references: SE/TE: 94, 112, 152, 154, 254, 357, 404, 465, 496, 520, 620, 640, 664,

11 LAFS.910.WHST.1.1.d Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing. LAFS.910.WHST.1.1.e Provide a concluding statement or section that follows from or supports the argument presented. LAFS.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research. Students have an opportunity to write arguments focused on discipline-specific content as they solve Open-Ended, Think About a Plan, Compare and Contrast, Writing, and Reasoning problems in the exercise sets. Sample references: SE/TE: 94, 112, 152, 154, 254, 357, 404, 465, 496, 520, 620, 640, 664, 714 Students have an opportunity to write arguments focused on discipline-specific content as they solve Open-Ended, Think About a Plan, Compare and Contrast, Writing, and Reasoning problems in the exercise sets. Sample references: SE/TE: 94, 112, 152, 154, 254, 357, 404, 465, 496, 520, 620, 640, 664, 714 Students are provided with opportunities to produce clear and coherent writing in mathematics on a daily basis in Open-Ended, Think About a Plan, Compare and Contrast, Writing, and Reasoning problems in the exercise sets. Sample references: SE/TE: 104, 145, 168, 238, 261, 357, 386, 403, 477, 503, 638, 664, 672, 714 Students analyze and correct errors in mathematical reasoning presented in the textbook. Performance tasks (Pull It All Together) at the end of every chapter require students to analyze, reflect, and do research. For example, one performance task requires students to create and analyze graphs and charts. Sample references: SE/TE: 128, 147, , 242, 272, 308, 352, , 479, 490, 506, 515, 531,

12 MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. This standard is addressed throughout the text. See, for example, the following pages: SE/TE: 65(#38, 40), 162 (#30), 202 (#27), 203 (#41), 240 (#21), 254 (#20), 255 (#27), 289 (#28), 297 (#23), 300 (Concept Byte), 314 (#34), 357 (#39), 410 (#14), 413 (Concept Byte), 655 (#36) 12

13 MAFS.K12.MP.2.1: Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. This standard is addressed throughout the text. See, for example, the following pages: SE/TE: 117 (#5-8), 125 (#13), 145 (#37-42), 168 (#10), 175 (#7), 187 (#21), 233 (#32), 264 (#26), 298 (#32, 36), 299 (#38), 306 (#26-29), 438 (#55-57) 13

14 MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. This standard is addressed throughout the text. See, for example, the following pages: SE/TE: 85 (#25-30), 86 (#33-37), 87 (#53), 93 (#16-19), 94 (#28), 143 (#10), 145 (#45), 170 (Concept Byte), 177 (#42), (Concept Byte), 187 (#26, 28), 240 (#23), 242 (Concept Byte), 284 (Concept Byte), 293 (#24-25) 14

15 MAFS.K12.MP.4.1: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. This standard is addressed throughout the text. See, for example, the following pages: SE/TE: 68 (Concept Byte), 87(#54), 146 (#48), 170 (Concept Byte), 242 (Concept Byte), 255 (#29), 264 (#26), 270 (#23-24), 293 (#24-25), 300 (Concept Byte), 306 (#20-21),412 (#31), (Concept Byte) 15

16 MAFS.K12.MP.5.1: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. MAFS.K12.MP.6.1: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. This standard is addressed throughout the text. See, for example, the following pages: SE/TE: 68 (Concept Byte), 146 (#48), 147 (Concept Byte), 170 (Concept Byte), (Concept Byte), 186 (#7-10), 187 (#16-18, 26, 28), 225 (Concept Byte), 240 (#22), 314 (#29-30), 352 (Concept Byte), 413 (Concept Byte), 454 (#30), 512 (#46), (Concept Byte), 641 (#47) This standard is addressed throughout the text. See, for example, the following pages: SE/TE: 65 (#31-33), 153 (#15-17), 155 (#27), 222 (#22-23, 26-27), 284 (Concept Byte), 290 (#46), 352 (Concept Byte), 404 (#33-34), 413 (Concept Byte), 456 (#15-16), 511 (#29), 515 (Concept Byte), 619 (#8-10), 626 (#17-19), 641 (#47) 16

17 MAFS.K12.MP.7.1: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MAFS.K12.MP.8.1: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x² + x + 1), and (x 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. This standard is addressed throughout the text. See, for example, the following pages: SE/TE: 85 (#21-24), 86 (#46), 87 (#47-48), 195 (#44-47), 201 (#11-14), 203 (#36-38), 231 (#18-20), 232 (#22), 289 (#30), 305 (#17-18), 437 (#27-32), 593 (Concept Byte) This standard is addressed throughout the text. See, for example, the following pages: SE/TE: 85 (#6-11), 86 (#38-43, 44-45), 88 (#58), (Concept Byte), 741 (Concept Byte), 748 (#34-37) 17

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