Sequenced Units for Arizona s College and Career Ready Standards MA32 Honors Geometry

Transcription

1 Sequenced Units for Arizona s College and Career Ready Standards MA32 Honors Geometry Year at a Glance Semester 1 Semester 2 Unit 1: Basics of Geometry (12 days) Unit 2: Reasoning and Proofs (13 days) Unit 3: Parallel and Perpendicular Lines (12 days) Unit 4: Transformations (13 days) Unit 5: Congruent Triangles (17 days) Unit 6: Relationships Within Triangles (15 days) Unit 7: Quadrilaterals and Other Polygons (13 days) Unit 8: Similarity (11 days) Unit 9: Right Triangles and Trigonometry (17 days) Unit 10: Circles (18 days) Unit 11: Circumference, Area, and Volume (19 days)

2 Students began their study of geometric concepts in middle school mathematics. They studied area, surface area, and volume and informally investigated lines, angles, and triangles. Students in middle school also explored transformations, including translations, reflections, rotations, and dilations. The Geometry course outlined in this document begins with developing the tools of geometry, including transformations, proof, and constructions. These tools are used throughout the course as students formalize geometric concepts studied in earlier courses and extend those ideas to new concepts presented in the high school standards. There is a focus on modeling, problem solving, and proof throughout the course. This document reflects our current thinking related to the intent of Arizona s College and Career Ready Standards for Mathematics and assumes 160 days for instruction, divided among 11 units. The number of days suggested for each unit assumes 45-minute class periods and is included to convey how instructional time should be balanced across the year. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the standards; however, the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Strikethroughs in the text of the standards are used in some cases in an attempt to convey that focus, and comments are included throughout the document to clarify and provide additional background for each unit. Throughout Geometry, students should continue to develop proficiency with Arizona s College and Career Ready Standards eight Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 4. Model with mathematics. 8. Look for and express regularity in repeated reasoning. These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. In a high school geometry course, reasoning and developing viable arguments are particularly important, as are use of strategic tools and precision of language. Opportunities for highlighting certain practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in those units. When using this document to help in planning your district's instructional program, you will also need to refer to the standards document, relevant progressions documents for the standards, and the appropriate assessment consortium framework. Mesa Public Schools 1 May 2015

3 Unit 1: Basics of Geometry Suggested number of days: 12 Students will be familiar with the basic geometric terms of point, line, segment, ray, and Plane from previous courses. They should also be familiar with how each is represented, meaning the notation used to identify these geometric objects. Sketching geometric relationships is challenging for many students, therefore specific steps are given throughout the unit to help students. Ensure you have lots of common objects around the room for students to reference when, for example, they are asked to describe the intersection of two lines, or the intersection of a line and a plane. A. Experiment with transformations in the plane 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. D. Make geometric constructions 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. Expressing Geometric Properties with Equations G-GPE B. Use coordinates to prove simple geometric theorems algebraically 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 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). Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Mesa Public Schools 2 May 2015

4 Unit 2: Reasoning and Proofs Suggested number of days: 13 In this unit, students are introduced to inductive and deductive reasoning as well as conditional statements written in if-then form. Students examine when conditional statements are true and false and connections are made to the reasoning students did in an algebra setting when solving equations, as there is always justification for the steps taken in solving an equation, although we do not always write them. Students are introduced to two-column, flowchart, and paragraph proofs. These different formats are practiced while proving statements about segments and angles. C. Prove geometric theorems 9. Prove theorems about lines and angles. 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. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; 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. 11. Prove theorems about parallelograms. 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. Similarity, Right Triangles, and Trigonometry G-SRT B. Prove theorems involving similarity 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. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Mesa Public Schools 3 May 2015

5 Unit 3: Parallel and Perpendicular Lines Suggested number of days: 12 Students have been introduced to the idea of an axiomatic system and have a growing list of theorems, postulates, definitions, and properties. In this unit, the list will grow some more, and students will continue to develop their deductive skills. This unit begins with line relationships (parallel, skew, coplanar, intersecting) and angle-pair relationships (corresponding, alternate interior, alternate exterior, and consecutive interior) and then statements about parallel lines cut by a transversal and the relationships between the pairs of angles formed are presented. The converse of statements is explored and there is a greater focus on proofs as the unit progresses. It is in this unit that students often use circular reasoning, using a theorem in the process of trying to prove the theorem. Skills from algebra are connected, allowing for a coordinate approach to justifying that two lines are parallel or two lines are perpendicular. Throughout the unit, students should not lose sight of the bigger picture with regard to writing proofs. The goal is to write a logical argument based on known information or information that can be deduced from given information. A. Experiment with transformations in the plane 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. C. Prove geometric theorems 9. Prove theorems about lines and angles. 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. D. Make geometric constructions 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. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Expressing Geometric Properties with Equations G-GPE B. Use coordinates to prove simple geometric theorems algebraically 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). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Mesa Public Schools 4 May 2015

6 Unit 4: Transformations Suggested number of days: 13 Students should have a conceptual understanding of transformations from middle school, where they studied translations, reflections, rotations and have been introduced to glide reflections. The focus on plane transformations in the Arizona Common Core State Standards makes sense in terms of the continuity from middle school to high school. In middle school, the conditions for triangle congruence were informally explored. In high school, once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures. The criteria for triangle similarity are established through similarity transformations. In this unit, key postulates and theorems relating to rigid motions are presented. Translations, reflections, glide reflections, and rotations are all postulated to be rigid motions. While all could be proven and established as theorems, the authors of the resource have chosen to treat them as postulates. Dilations are introduced as non-rigid transformations where the scale factor k results in an enlargement (k > 1) or a reduction (0 < k < 1). The composition of a dilation with rigid motions results in a similarity transformation. Similar figures are defined in terms of similarity transformations and the use of dynamic geometry software for the explorations and formal lessons is highly encouraged. This tool provides students the opportunity to explore and make conjectures, mathematical practices we want to develop in all students. A. Experiment with transformations in the plane 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). 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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. B. 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. Similarity, Right Triangles, and Trigonometry G-SRT A. Understand similarity in terms of similarity transformations 1. Verify experimentally the properties of dilations given by a center and a scale factor: 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. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 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. continued on next page Mesa Public Schools 5 May 2015

7 Unit 4: Transformations Suggested number of days: 13 constraints or minimize cost; working with typographic grid systems based on ratios). Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Mesa Public Schools 6 May 2015

8 Unit 5: Congruent Triangles Suggested number of days: 17 In this unit, students will work with a variety of proof formats as they investigate triangle congruence. Methods for establishing triangle congruence (SAS, SSS, ASA, and AAS) are established using rigid motions. The proof of each congruence criteria for triangles is done by composing transformations. This means a sequence of rigid motions maps one triangle onto another triangle. Other proof styles presented in this unit include the two-column proof, the paragraph or narrative proof, and finally, the coordinate proof. In addition to working with proofs, there are properties of equilateral and isosceles triangles that are proven. The use of dynamic geometry software for the explorations is highly encouraged. This tool provides students the opportunity to explore and make conjectures, mathematical practices we want to develop in all students. B. Understand congruence in terms of rigid motions 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. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. C. Prove geometric theorems 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; 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. D. Make geometric constructions 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Similarity, Right Triangles, and Trigonometry G-SRT B. Prove theorems involving similarity 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Expressing Geometric Properties with Equations G-GPE B. Use coordinates to prove simple geometric theorems algebraically 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). 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). constraints or minimize cost; working with typographic grid systems based on ratios). Mesa Public Schools 7 May 2015

9 Unit 6: Relationships Within Triangles Suggested number of days: 15 This unit uses the deductive skills developed in the last unit, Unit 5, to explore special segments in a triangle. These segments include perpendicular bisectors, angle bisectors, medians, altitudes, and mid-segments. Students are able to discover special properties of these segments by using dynamic geometry software. To prove these relationships, a variety of proof formats and approaches are used: transformational, synthetic, analytic, and paragraph. This unit includes inequalities within one triangle and in two triangles. The indirect proof is introduced and used to prove several of the theorems and triangles have been the geometric structure used to help students develop their deductive reasoning skills. In the next unit, students will study quadrilaterals and other polygons. C. Prove geometric theorems 9. Prove theorems about lines and angles. 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. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; 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. D. Make geometric constructions 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. Circles G-C A. Understand and apply theorems about circles 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 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). constraints or minimize cost; working with typographic grid systems based on ratios). Mesa Public Schools 8 May 2015

10 Unit 7: Quadrilaterals and Other Polygons Suggested number of days: 13 Students have studied the special quadrilaterals in middle school and should be familiar with their definitions and some of their properties. Special quadrilaterals include parallelograms, rectangles, rhombuses, squares, trapezoids, isosceles trapezoids, and kites. Students have investigated the properties of these special quadrilaterals using inductive reasoning. In this unit, students are able to investigate the properties of special quadrilaterals by using dynamic geometry software. The properties are then proven using a variety of proof formats: transformational, synthetic, analytic, and paragraph. The unit begins with the angle measures in a polygon where students first derive a formula for the sum of the interior angles and then a formula for the sum of the exterior angles. The unit continues by studying special quadrilaterals. In Unit 5, students proved triangles congruent given different hypotheses. The reasoning used in that unit is applied to proofs involving quadrilaterals in this unit. C. Prove geometric theorems 11. Prove theorems about parallelograms. 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. Similarity, Right Triangles, and Trigonometry G-SRT B. Prove theorems involving similarity 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 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). constraints or minimize cost; working with typographic grid systems based on ratios). Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Mesa Public Schools 9 May 2015

11 Unit 8: Similarity Suggested number of days: 11 This is a short unit that revisits similarity, a concept first introduced in Unit 4. This unit begins with what it means for two polygons to be similar: corresponding sides are proportional and corresponding angles are congruent. Then methods for proving two triangles are similar are presented. One of the methods involves only angles (AA), with the other two methods involving only sides (SSS) or sides and the included angle (SAS). Unlike congruency methods, the corresponding sides must be proportional versus congruent. Several proportionality theorems, mostly connected to triangles, are presented and properties of similar triangles will be needed in Unit 9 when trigonometric ratios are defined. Similarity, Right Triangles, and Trigonometry G-SRT A. Understand similarity in terms of similarity transformations 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. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. B. Prove theorems involving similarity 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. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Expressing Geometric Properties with Equations G-GPE B. Use coordinates to prove simple geometric theorems algebraically 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric probl ems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 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). constraints or minimize cost; working with typographic grid systems based on ratios). Arizona s College and Career Ready Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Mesa Public Schools 10 May 2015

12 Unit 9: Right Triangles and Trigonometry Suggested number of days: 17 This unit introduces students to right triangle trigonometry. Students will encounter a more in-depth study of trigonometry in Algebra 2. The Pythagorean Theorem will not be completely new to students who will have familiarity with this theorem from middle school. Students will use knowledge of similar triangles to investigate relationships in special right triangles ( and ) as well as similar triangles that are formed when the altitude to the hypotenuse is drawn in a right triangle. Being familiar with these relationships and solving for segment lengths in triangles will be helpful in subsequent lessons. Presented are the tangent, sine, and cosine ratios with the focus being to solve for parts of a right triangle. Many real-life applications are presented. The Law of Sines and the Law of Cosines are learned so that non-right triangles can be solved. Similarity, Right Triangles, and Trigonometry G-SRT B. Prove theorems involving similarity 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. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. C. Define trigonometric ratios and solve problems involving right triangles 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. 7. Explain and use the relationship between the sine and cosine of complementary angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. D. Apply trigonometry to general triangles 9. Derive the formula A = ½ ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10. Prove the Laws of Sines and Cosines and use them to solve problems. 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). 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). constraints or minimize cost; working with typographic grid systems based on ratios). Mesa Public Schools 11 May 2015

13 Unit 10: Circles Suggested number of days: 18 This unit is all about circles. Many of the theorems will be investigated using dynamic geometry software. Vocabulary and symbols related to circles are introduced first, followed by looking at circular arcs that are intercepted by chords. All of the angle relationships that occur when two chords, secants, or tangents intersect a circle are investigated, as well as the investigation of segment relationships that occur when two chords, secants, or tangents intersect a circle. Circles are then presented in the coordinate plane where the standard form of the equation is derived. Much of the earlier units has been devoted to lines, triangles, and polygons. A. Experiment with transformations in the plane 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. D. Make geometric constructions 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Circles G-C A. Understand and apply theorems about circles 1. Prove that all circles are similar. 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. 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 4. Construct a tangent line from a point outside a given circle to the circle. Expressing Geometric Properties with Equations G-GPE A. Translate between the geometric description and the equation for a conic section 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. 2. Derive the equation of a parabola given a focus and directrix. B. Use coordinates to prove simple geometric theorems algebraically 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). 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). constraints or minimize cost; working with typographic grid systems based on ratios). Mesa Public Schools 12 May 2015

14 Unit 11: Circumference, Area, and Volume Suggested number of days: 19 This unit on circumference, area, and volume finishes the study of measurement of solids that began in middle school. Students will come to this unit with knowledge of many formulas for surface area and volume. These will be reviewed and a few new formulas added to the list. Different from junior high school is that students now have a greater ability to solve equations. Now that students also know the Pythagorean Theorem and trigonometry, so they are able to solve for measures that previously had to be told to them. In this unit, students will do additional work with circles involving arc length and area of Sectors, as well as find the area of regular polygons. A. Experiment with transformations in the plane 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. Circles G-C B. Find arc lengths and areas of sectors of circles 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. Geometric measurement and dimension G-GMD A. Explain volume formulas and use them to solve problems 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. 2. Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. B. Visualize relationships between two-dimensional and three- dimensional objects 4. Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 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). 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). constraints or minimize cost; working with typographic grid systems based on ratios). Mesa Public Schools 13 May 2015