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Roanoke County Public Schools Geometry Readiness Curriculum Guide Revised 2012 Geometry Readiness Curriculum Guide 2011 Mathematics Curriculum Guide Revised 2011. Available at www.rcs.k12.va.us. Roanoke County Public Schools does not discriminate with regard to race, color, age, national origin, gender, or handicapping condition in an educational and/or employment policy or practice. Questions and/or complaints should be addressed to the Deputy Superintendent/Title IX Coordinator at (540) 562-3900 ext. 10121 or the Director of Pupil Personnel Services/504 Coordinator at (540) 562-3900 ext. 10181. Acknowledgements The following people have made tremendous contributions to the completion of this curriculum guide and all are appreciated. Theresa Hartley Cave Spring High Kevin Minnix William Byrd High Sherri Mays William Byrd High Barbara Smith Cave Spring High/Cave Spring Middle Christina Hall Northside Middle Brooke Haun Cave Spring High Roanoke County Public Schools Administration Dr. Lorraine Lange Superintendent Cecil Snead Director of Secondary Instruction Rebecca Eastwood Director of Elementary Instruction Linda Bowden Mathematics Coordinator Preface This curriculum guide is written for the teachers to assist them in using the textbooks/resources in a most effective way. This guide will assist the mathematics teacher in preparing students for the challenges of the twenty-first century. As established by the National Council of Teachers of Mathematics Principles and Standards for School Mathematics, educational goals for students are changing. Students should have many and varied experiences in their mathematical training to help them learn to value mathematics, become confident in their ability to do mathematics, become problem solvers , and learn to communicate and reason mathematically. This guide, along with the available textbook resources, other professional literature, alternative assessment methods, and varied instruction in-service activities will assist the mathematics teacher in continuing to integrate these student goals into the curriculum. Geometry Readiness Curriculum Guide 2011 Table of Contents Introduction/General Comments ............................................................................................................................................. i Textbook/Resources Overview ............................................................................................................................................... ii Sequence of Instruction and Pacing Suggestions .............................................................................Error! Bookmark not defined. Mapping for Instruction - First Nine Weeks ............................................................................................................................ 3 Mapping for Instruction - Second Nine Weeks ...................................................................................................................... 11 Mapping for Instruction - Third Nine Weeks ......................................................................................................................... 17 Mapping for Instruction - Fourth Nine Weeks ....................................................................................................................... 23 SOL Blueprints ....................................................................................................................................................................... 31 SOL Enhanced Scope and Sequence ...................................................................................................................................... 32 Supplemental Resources ....................................................................................................................................................... 32 SOL 2009 Framework ............................................................................................................................................................ 33 Geometry Readiness Curriculum Guide 2011 Introduction/General Comments This curriculum guide follows the 2009 Virginia Geometry SOLs and uses the 2012 edition of Prentice Hall – Geometry Virginia Edition textbook as the primary resource. It is extremely important and required that the Sequence of Instruction and Pacing be followed as presented in the curriculum guide. This will allow the formative assessment tests to be an effective instructional tool. Students will take a formative assessment test during the second, third, and fourth nine weeks which will contain materials from the previous nine weeks. Each teacher-designed test will assess current 9-weeks skill levels of the SOLs in the Sequence of Instruction and Pacing. This tool will help teachers to pinpoint their students' strengths and weaknesses so they can make more informed decisions for instruction. The Mapping for Instruction has been set up according to each SOL that is taught in the specific 9-weeks. Note that only part of some SOLs may be taught in a certain 9-weeks. Each student should have a graphing calculator for use throughout this course. The instructor should use the calculator for investigation and verification of results as well as a powerful tool in solving the real-life application problems that are provided in each section of the textbook. Cooperative learning is very beneficial to students in learning many of the topics in geometry, particularly with proofs and application problem solving. The Geometer's Sketchpad software, available in every school and on high school students' laptops, should be used to enhance instruction throughout the course. Please note the following, keeping in mind this curriculum guide is meant to encompass all the different versions of Geometry Roanoke County offers in their middle and high schools: 1. Within the "Mapping for Instruction" the suggested pacing outlined in the "Comments" section was designed for high school geometry classes that meet for 95 minutes every other day. It is suggested that the allotted time be doubled for those teaching middle school Algebra courses. 2. Within the "Mapping for Instruction", under "Supporting Materials" section, there are suggestions for remediation and/or enrichment to help teachers meet the needs of their students. i Geometry Readiness Curriculum Guide 2011 Textbook/Resources Overview Course Title: Geometry, Virginia Edition Course Text: Prenctice Hall Publisher: Pearson The 2012 edition of Prentice Hall Geometry is divided up into 12 chapters. Each chapter of the teacher’s guide begins with Get Ready, a review of previously learned skills that are relevant for the chapter; Chapter Overview that introduces the Vocabulary, Big Ideas, My Math Video, and Chapter Preview; Math Background that summarizes main concepts and common errors; and Pacing and Assignment Guide that suggests how to differentiate pacing for Basic, Average, and Advanced students. A Resources page lists the supplemental materials available for each section of the chapter. Each section within a chapter is divided into 1) Interactive Learning; 2) Guided Instruction; 3) Lesson Check; 4) Practice, with suggested assignments differentiated for different levels; and 5) Assess & Remediate, where differentiated resources for Intervention, On-Level, and Extension activities are pictured. The end of each chapter contains Pull It All Together, where the “Big Ideas” of the chapter are recapped and connected to associated real-world applications; Chapter Review, which links the “Big Ideas” to answering essential questions; Chapter Vocabulary; Quick Review; and Chapter Test. Teachers are provided with the following resources available online (at PowerGeometry.com), in digital format (most are editable) and/or print format. Interactive Learning & Guided Instruction Lesson Check & Practice Assess & Remediate My Math Video Student Companion ExamView CD-ROM Solve It! Practice and Problem-Solving Workbook Lesson Quiz Student Companion Practice (Forms G and K) Quizzes and Tests (Forms G and K) Dynamic Activity Extra Practice Reteaching Online problems Find the Errors Performance Tasks Additional problems Enrichment Cumulative Review ELL support Answers and Solutions CD-ROM Progress Monitoring Assessments Activities, Games and Puzzles Standardized Test Prep Teaching with TI Technology TI-Nspire Support CD-ROM ii Geometry Readiness Curriculum Guide 2011 Sequence of Instruction and Pacing Suggestions First Nine Weeks SOL G.4a,b,e,f G.3a G.1a, b, d G.1 d, G.2 a,b,c G.3 a, b, G.4 c, d, g Chapter/Sections/Topic *Time Frame Chapter 1 Tools of Geometry; Lessons 2-8 VA-3 6.00 blocks Chapter 2 Reasoning and Proof; Lessons 1 - 6 VA-1 5.50 blocks Chapter 3 Parallel and Perpendicular Lines; Lessons 1 - 8 5.50 blocks Reviews, quizzes, and tests 5.50 blocks *Time Frame is based on 95 minutes of instruction per block. Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45 days in the middle schools. First Nine Weeks Total 22.5 blocks Second Nine Weeks SOL Chapter/Sections/Topic *Time Frame G.4 a, G.6 Chapter 4 Congruent Triangles; Lessons 1 - 7 7.00 blocks G.1 d, G.5 a, b, c, d Chapter 5 Relationships Within Triangles; Lessons1,2,(3 optional),4, 6 5.50 blocks G.8 Chapter 8 Right Triangles and Trigonometry; Lessons 1 - 4 4.50 blocks Reviews, quizzes, and tests 5.50 blocks *Time Frame is based on 95 minutes of instruction per block. Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45 days in the middle schools. 1 Second Nine Weeks Total 22.5 blocks Geometry Readiness Curriculum Guide 2011 Sequence of Instruction and Pacing Suggestions Third Nine Weeks SOL Chapter/Sections/Topic G.7, G.14 d Chapter 7 Similarity; Lessons 1 - 5 *Time Frame 4.50 blocks G.2 b, G.9, G.10 Chapter 6 Polygons and Quadrilaterals; Lessons 1 - 8 G.11 b, c, G.14 a, c, d Chapter 10 Area; Lessons 1, 2, 4 - 7 Pre-AP include Lesson 10-3 8.00 blocks 4.50 blocks Reviews, quizzes, and tests Include a review on Constructions (Lessons 1-6 and 3-6) *Time Frame is based on 95 minutes of instruction per block. Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45 days in the middle schools. Third Nine Weeks Total 5.5 blocks 22.5 blocks Fourth Nine Weeks SOL Chapter/Sections/Topic *Time Frame G.4, G.11 a, b, G.12 Chapter 12 Circles; Lessons 1 - 5 5.50 blocks Chapter 1 Nets and Drawings for Visualizing Geometry; Lesson 1(Optional) 0.00 blocks G.13, G.14 a, b, c, d Chapter 11 Surface Area and Volume; Lessons 2 - 7(be sure to include the defintions: face, edge, and vertex) 3.50 blocks G.3 c, d, G.10 Chapter 9 Transformations; Lessons 1 - 7 2.00 blocks 1.50 blocks Reviews, quizzes, and tests 4.50 blocks Sol Review 5.50 blocks Exam Review *Time Frame is based on 95 minutes of instruction per block. Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45 days in the middle schools. 2 Fourth Nine Weeks Total 22.5 blocks Geometry Readiness Curriculum Guide 2011 Mapping for Instruction - First Nine Weeks Tools of Geometry Chapter: 1 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics 1-2 Points, Lines and Planes Supporting Materials Comments “Geometry Plane & Simple” pages 1, 3, 5, and 6 “Milliken Geometry Reproducibles” page 3 0 .5 block 1-3 Measuring Segments “Geometry Plane & Simple” pages 2 and 4 “if Geometry” pages 7 and 8 0.5 block 1-4 Measuring Angles 1 block www.mathisfun.com/geometry/protractor-using.html “Geometry Plane & Simple” pages 9 and 10 “Geometry Teacher’s Activities Kit” pages 29 – 31, 36 - 46 G.2. The student will use the relationships 1-5 Exploring Angle Pairs between angles formed by two lines cut by a transversal to 1 block c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. • Solve real-world problems involving intersecting and parallel lines in a plane www.worksheetworks.com/math/geometry/measuringangles.html http://education.ti.com/calculators/timath/US/Activities/?sa= 5024# “Geometry Plane & Simple” pages 12, 13, 14,18, 19, and 20 “Milliken Geometry Reproducibles” pages 4 – 6 “if Geometry” pages 9, 11 – 13 “Geometry Teacher’s Activities Kit” pages 32 – 35, and 61 -64 G.4 The student will construct and justify the 1-6 Basic Constructions constructions of a) a line segment congruent to a given line 1 block segment; b) the perpendicular bisector of a line segment e) the bisector of a given angle f) an angle congruent to a given angle Justifications for constructions VA-3 G.3 The student will use pictorial representations, including computer software, constructions, and coordinate 1.7 Midpoint and Distance in the Coordinate Plane http://mathisfun.com/geometry/constructions.html http://whistleralley.com/construction/reference.html http://www.mathopenref.com/constructions.com http://www.youtube.com/watch?v=s-atfsonr8w “if Geometry” pages 45, 46, 49, 50 DOE ESS Sample Lesson Plan G.4 Constructions “Milliken Geometry Reproducibles” pages 12 and 19 “if Geometry” page 37 DOE ESS Sample Lesson Plan G.3 Distance and Midpoint 3 Review how to use a protractor VA-3 is located in the textbook between chapters 10 and 11 Concept will appear on FA#3 Geometry Readiness Curriculum Guide 2011 Tools of Geometry Chapter: 1 SOL with Essential Knowledge and Skill methods, to solve problems involving symmetry and transformation. This will include a) investigating and using formulas for finding distance, midpoint, and slope; Textbook Chapters/Sections/Topics Supporting Materials 1 block Formulas 1.8 Perimeter, Circumference and Area “Milliken Geometry Reproducibles” pages 27, 28, 29, 32, and 33 “if Geometry” page 72 1 block 4 Comments Geometry Readiness Curriculum Guide 2011 Reasoning and Proof Chapter: 2 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials G.1. The student will construct and judge the validity of a logical 2-1 Patterns and Inductive argument consisting of a set of premises and a conclusion. This Reasoning will include • Select and use various types of reasoning and methods of 0.5 block proof, as appropriate. DOE ESS Sample Lesson Plan G.1 Inductive and Deductive Reasoning G.1 The student will construct and judge the validity of a logical 2-2 Conditional Statements argument consisting of a set of premises and a conclusion. This VA-1: Review Venn Diagrams will include a) identifying the converse, inverse, and contrapositive of a 1 block conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; and • Identify the converse, inverse, and contrapositive of a conditional statement. “Geometry Plane & Simple” pages 37 and 38 “if Geometry” pages 91 and 92 “Geometry Teacher’s Activities Kit” pages 65 - 67 Translate verbal arguments into symbolic form, such as (p q) and (~p ~q). • Determine the validity of a logical argument. • Use valid forms of deductive reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples. • Select and use various types of reasoning and methods of proof, as appropriate. • Use Venn diagrams to represent set relationships, such as intersection and union. • Interpret Venn diagrams. • Recognize and use the symbols of formal logic, which include →, ↔, ~, , , and . . G.1 The student will construct and judge the validity of a logical 2-3 Biconditionals and Definitions argument consisting of a set of premises and a conclusion. This will include 1 block a) identifying the converse, inverse, and contrapositive of a conditional statement; 5 DOE ESS Sample Lesson Plan G.1 Logic and Conditional Statements Comments Need to supplement logic notation including therefore Geometry Readiness Curriculum Guide 2011 Reasoning and Proof Chapter: 2 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials b) translating a short verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; and • Identify the converse, inverse, and contrapositive of a conditional statement. Translate verbal arguments into symbolic form, such as (p q) and (~p ~q). • Determine the validity of a logical argument. • Use valid forms of deductive reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples. • Select and use various types of reasoning and methods of proof, as appropriate. • Use Venn diagrams to represent set relationships, such as intersection and union. • Interpret Venn diagrams. • Recognize and use the symbols of formal logic, which include →, ↔, ~, , , and . . G.1 The student will construct and judge the validity of a logical 2-4 Deductive Reasoning argument consisting of a set of premises and a conclusion. This will include 0 .5 block a) identifying the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; and • Identify the converse, inverse, and contrapositive of a conditional statement. Translate verbal arguments into symbolic form, such as (p q) and (~p ~q). • Determine the validity of a logical argument. • Use valid forms of deductive reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples. • Select and use various types of reasoning and methods of proof, as appropriate. • Use Venn diagrams to represent set relationships, such as 6 “Geometry Plane & Simple” pages 39 and 40 DOE ESS Sample Lesson Plan G.1 Inductive and Deductive Reasoning Comments Geometry Readiness Curriculum Guide 2011 Reasoning and Proof Chapter: 2 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials intersection and union. • Interpret Venn diagrams. • Recognize and use the symbols of formal logic, which include →, ↔, ~, , , and . . G.1 The student will construct and judge the validity of a logical 2-5 Reasoning in Algebra and argument consisting of a set of premises and a conclusion. This Geometry will include d) using deductive reasoning 1.5 block • Use valid forms of deductive reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples. • Select and use various types of reasoning and methods of proof, as appropriate G.1 The student will construct and judge the validity of a logical 2-6 Proving Angles Congruent argument consisting of a set of premises and a conclusion. This will include 1 block d) using deductive reasoning • Use valid forms of deductive reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples. . 7 “Geometry Plane & Simple” pages 15 - 17 Comments Geometry Readiness Curriculum Guide 2011 Parallel and Perpendicular Lines Chapter: 3 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials 3-1 Lines and Angles G.2 The student will use the relationship between angles formed by two lines cut by a 1 block transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. http://www.curriki.org/xwiki/bin/view/MyCurriki/Collecti ons?user=IsaacNewton “Milliken Geometry Reproducibles” page 6 3-2 Properties of Parallel Lines G.2 The student will use the relationship between angles formed by two lines cut by a 1 block transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal including corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles. “Geometry Plane & Simple” page 60 “Geometry Teacher’s Activities Kit” pages 47 - 52 and 61 – 64 G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include d) using deductive reasoning. G.2 The student will use the relationships 3-3 Proving Lines Parallel “Geometry Plane & Simple” page 59 1 block DOE ESS Sample Lesson Plan G.2 Lines and Angles 8 DOE ESS Sample Lesson Plan G.2 Lines and Angles Comments Geometry Readiness Curriculum Guide 2011 Parallel and Perpendicular Lines Chapter: 3 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Comments between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversa l• Use algebraic and coordinate methods as well as deductive proofs to verify whether two lines are parallel. “Geometry Teacher’s Activities Kit” pages 53 -57 G.2 The student will use the relationships between angles formed by two lines cut by a transversal to c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. 3-4 Parallel and Perpendicular Lines G.2 The student will use the relationships between angles formed by two lines cut by a transversal to c) solve real-world problems involving angles formed when parallel lines are cut by a transversalThe student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Solve problems by using the relationhips between pairs of angles formed bye the intersection of two parallel lines and a transversal including correspondings anles, alternate intreior angles, alternate exterior angles, and same-side interior angles • Solve real-world problems involving intersecting and parallel lines in a plane. 3-5 Parallel Lines and Triangles “Milliken Geometry Reproducibles” page 8 “Geometry Teacher’s Activities Kit” pages 91 - 95 0.5 block G.4 The student will construct and justify the constructions of c) a perpendicular to a given line from a point not on the line; 3-6 Constructing Parallel and Perpendicular Lines 0.5 block http://www.mathopenref.com/constructions.html “if Geometry” pages 46, 47, 48,51 VA-3: Justifying a 9 VA-3 is located in the textbook between chapters 10 and 11. Geometry Readiness Curriculum Guide 2011 Parallel and Perpendicular Lines Chapter: 3 SOL with Essential Knowledge and Skill d) a perpendicular to a given line at a given point on the line; f) an angle congruent to a given angle; and g) a line parallel to a given line through a point not on the given line. G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include a) investigating and using formulas for finding distance, midpoint, and slope; G.2 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and • Use algebraic and coordinate methods as well as deductive proofs to verify whether two lines are parallel.. • Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal including corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles. • Solve real-world problems involving intersecting and parallel lines in a plane. Textbook Chapters/Sections/Topics Supporting Materials Construction) 0.5 block “if Geometry” pages 32 - 36 3.7 Equations of Lines in the Coordinate Plane 0.5 block 3-8 Slopes of Parallel and Perpendicular Lines DOE ESS Sample Lesson Plan G.3 Slope 0.5 block 10 Comments Geometry Readiness Curriculum Guide 2011 Mapping for Instruction - Second Nine Weeks Congruent Triangles Chapter: 4 SOL with Essential Knowledge and Skill G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Textbook Chapters/Sections/Topics 4-1 Congruent Figures 1 block Supporting Materials Comments http://www.mathsisfun.com/geometry/trianglescongruent.html http://www.cliffsnotes.com/study_guide/CongruentTriangles.topicArticleId-18851,articleId-18788.html http://www.mathopenref.com/congruenttriangles.html All of these websites review the triangle congruences and offer some activities. “Geometry Plane & Simple” page 21 G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. • Use definitions, postulates, and theorems to prove triangles congruent. • Use coordinate methods, such as the distance formula and the slope formula, to prove two triangles are congruent. G.4a The student will construct and justify the construction of a line segment congruent to a given line segment. 4-2 Triangle Congruence by SSS and SAS VA - 4 Congruent and Similar Triangles G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. • Use definitions, postulates, and theorems to prove triangles congruent. • Use coordinate methods, such as the distance formula and the slope formula, to prove two triangles are congruent. 4-3 Triangle Congruence by ASA and AAS G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. • Use definitions, postulates, and theorems to prove triangles congruent. 4-4 Using Corresponding Parts of Congruent Triangles 1 block http://www.youtube.com/watch?v=TPL12Tk7L6U This youtube video www.uen.org/.../downloadFile.cgi?...Congruent_Trian describes an activity gles_Activity. you can use with your students to Section VA - 4 is found in the Virginia Section of the prove triangle textbook between Chapters 10 and 11. congruences. “Geometry Plane & Simple” page 41, 42,44 “if Geometry” page 25 1 block 1 block 11 http://illuminations.nctm.org/ActivityDetail.aspx?id=4 “Geometry Plane & Simple” page 43 “Milliken Geometry” page 14, 15, 16 “if Geometry” pages 26, 27, 29 and 30 “Geometry Teacher’s Activities Kit” pages 160 - 163 “Geometry Plane & Simple” page 45, 46, 47,48 Geometry Readiness Curriculum Guide 2011 Congruent Triangles Chapter: 4 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials • Use coordinate methods, such as the distance formula and the slope formula, to prove two triangles are congruent. • Use algebraic methods to prove two triangles are congruent. G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. • Use definitions, postulates, and theorems to prove triangles congruent. • Use algebraic methods to prove two triangles are congruent. 4-5 Isosceles and Equilateral Triangles G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. • Use definitions, postulates, and theorems to prove triangles congruent. • Use algebraic methods to prove two triangles are congruent. 4-6 Congruence in Right Triangles G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. • Use definitions, postulates, and theorems to prove triangles congruent. 4-7 Congruence in Overlapping Triangles 1 block 1 block “Geometry Plane & Simple” pages 41, 42, 43, 44,46, 47 1 block 12 Comments Geometry Readiness Curriculum Guide 2011 Relationships Within Triangles Chapter: 5 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a.) order the sides by length, given the angle measures; b.) order the angles by degree measure, given the side lengths; c.) determine whether a triangle exists; and d.) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. 5-1 Midsegments of Triangles G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a.) order the sides by length, given the angle measures; b.) order the angles by degree measure, given the side lengths; c.) determine whether a triangle exists; and d.) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. 5-2 Perpendicular and Angle Bisectors G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a.) order the sides by length, given the angle measures; b.) order the angles by degree measure, given the side lengths; c.) determine whether a triangle exists; and d.) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. 5-4 Medians and Altitudes Supporting Materials http://www.youtube.com/watch?v=cSZYvU_dsAw 1 block 2 blocks 0.5 block 13 “Geometry Teacher’s Activities Kit” pages 144 -,145 “Geometry Plane & Simple” page 73 Comments Geometry Readiness Curriculum Guide 2011 Relationships Within Triangles Chapter: 5 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics G.1d The student will construct and judge the 5-6 Inequalities in One validity of a logical argument consisting of a set of Triangle premises and a conclusion. This will include a.) identifying the converse, inverse, and 2 blocks contrapositive of a conditional statement; b.) translating a short verbal argument into symbolic form; c.) using Venn diagrams to represent set relationships; and d.) using deductive reasoning. • Use valid forms of deductive reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples. G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a.) order the sides by length, given the angle measures; b.) order the angles by degree measure, given the side lengths; c.) determine whether a triangle exists; and d.) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. • Order the sides of a triangle by their lengths when given the measures of the angles. • Order the angles of a triangle by their measures when given the lengths of the sides. • Given the lengths of three segments, determine whether a triangle could be formed. • Given the lengths of two sides of a triangle, determine the range in which the length of the third side must lie. • Solve real-world problems given information about the lengths of sides and/or measures of angles in triangles. Supporting Materials http://illuminations.nctm.org/lessondetail.aspx?ID=L68 1 http://www.mathwarehouse.com/geometry/triangles/tri angle-inequality-theorem-rule-explained.php http://www.mathwarehouse.com/geometry/triangles/ 14 Comments Geometry Readiness Curriculum Guide G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. • Determine whether a triangle formed with three given lengths is a right triangle. • Solve real-world problems, using right triangle trigonometry and properties of right triangles. 2011 8-1 The Pythagorean Theorem and its Converse 1 block http://regentsprep.org/Regents/math/geometry/GP13/TRe Chapter VA in the sourcePyth.htm textbook provides additional lessons and http://teachers.henrico.k12.va.us/math/igo/WU_Algebra/ standards review to WU7_4Alg.htm get students ready for the end of the year http://teachers.henrico.k12.va.us/math/igo/07RightTriangl SOL test. VA es/7-2PythagoreanThm/CW7-2.pdf Standards Review Pythagorean http://teachers.henrico.k12.va.us/math/igo/07RightTriangl Theorem (page VA es/7-2PythagoreanThm/HW7-2.pdf 14) could be used after Chapter 8 is http://www.math-videos-online.com/easy-pythagoreancompleted for theorem-proofs.html additional review for the SOL test. http://www.math-play.com/Geometry-Math-Games.html http://regentsprep.org/Regents/math/geometry/GP13/Pra cPyth.htm DOE ESS Sample Lesson Plan G.8 The Pythagorean Relationship SOL with Essential Knowledge and Skill G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. • Solve for missing lengths in geometric figures, using properties of 45-45-90 triangles. • Solve for missing lengths in geometric figures, using properties of 30-60-90 triangles. • Solve real-world problems, using right triangle trigonometry and properties of right triangles. Textbook Chapters/Sections/Topics 8-2 Special Right Triangles Supporting Materials http://teachers.henrico.k12.va.us/math/igo/07RightTriangl es/7-3SpecialRtTriangles/CW7-3.pdf 1 block http://teachers.henrico.k12.va.us/math/igo/07RightTriangl es/7-3SpecialRtTriangles/HW7-3.pdf “Geometry Plane & Simple” pages 80, 81, and 82 “Milliken Geoemtry” pages 17, 18 “if Geometry” page 62 “Geometry Teacher’s Activities Kit” pages 154 – 157 DOE ESS Sample Lesson Plan G.8 Special Right Triangles and Right Triangles G.8 The student will solve real-world 8-3 Trigonometry http://teachers.henrico.k12.va.us/math/igo/07RightTriangl 15 Comments Geometry Readiness Curriculum Guide 2011 problems involving right triangles by using the Pythagorean Theorem and its converse, 1.5 blocks properties of special right triangles, and right triangle trigonometry. • Solve problems involving right triangles, using sine, cosine, and tangent ratios. • Solve real-world problems, using right triangle trigonometry and properties of right triangles. • Explain and use the relationship between the sine and cosine of complementary angles. es/7-4Trigonometry/CW7-4.pdf G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. • Solve real-world problems, using right triangle trigonometry and properties of right triangles. http://e-zgeometry.com/ 8-4 Angles of Elevation and Depression http://teachers.henrico.k12.va.us/math/igo/07RightTriangl es/7-4Trigonometry/HW7-4.pdf http://teachers.henrico.k12.va.us/math/igo/07RightTriangl es/7-4Trigonometry/7-4WU.htm http://www.youtheducationservices.ca/trigonometry.html# 1 block 16 Geometry Readiness Curriculum Guide 2011 Mapping for Instruction - Third Nine Weeks Similarity Chapter: 7 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials 7-1 Ratios and Proportions http://e-zgeometry.com/ 1 block http://www.kutasoftware.com/freeige.html http://teachers.henrico.k12.va.us/math/igo/05Similarity/51UsingProportions/CW5-1.pdf http://teachers.henrico.k12.va.us/math/igo/05Similarity/51UsingProportions/HW5-1.pdf G.14.d The student will use similar geometric objects in two- or three-dimentions to solve real-world problems about similar geometric objects. • Solve real-world problems involving measured attributes of similar objects. 7-2 Similar Polygons http://teachers.henrico.k12.va.us/math/igo/05Similarity/52SimilarPolygons/CW5-2.pdf 1 block G.7 The student, given information in the form 7-3 Proving Triangles Similar of a figure or statement, will prove two triangles are similar, using algebraic and 1 block coordinate methods as well as deductive proofs. • Use definitions, postulates, and theorems to prove triangles similar. • Use algebraic methods to prove that triangles are similar. • Use coordinate methods, such as the distance formula, to prove two triangles are similar. http://teachers.henrico.k12.va.us/math/igo/05Similarity/52SimilarPolygons/HW5-2.pdf http://teachers.henrico.k12.va.us/math/igo/05Similarity/53SimilarTriangles/CW5-3.pdf http://teachers.henrico.k12.va.us/math/igo/05Similarity/53SimilarTriangles/HW5-3.pdf G.7 The student, given information in the form 7-4 Similarity in Right of a figure or statement, will prove two Triangles triangles are similar, using algebraic and coordinate methods as well as deductive 0.5 block proofs. 17 Comments Geometry Readiness Curriculum Guide 2011 Similarity Chapter: 7 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Comments • Use definitions, postulates, and theorems to prove triangles similar. • Use algebraic methods to prove that triangles are similar. • Use coordinate methods, such as the distance formula, to prove two triangles are similar. 7-5 Proportions in Triangles http://teachers.henrico.k12.va.us/math/igo/05Similarity/54ProportionalParts/CW5-4.pdf 0.5 block http://teachers.henrico.k12.va.us/math/igo/05Similarity/54ProportionalParts/HW5-4.pdf G.7 The student, given information in the form VA-4 Congruent and Similar of a figure or statement, will prove two Triangles triangles are similar, using algebraic and coordinate methods as well as deductive 0.5 block proofs. • Use coordinate methods, such as the distance formula, to prove two triangles are similar. Click on the Jeopardy Review on the left side of this webpage http://teachers.henrico.k12.va.us/math/igo/05Similarity/5_4 .html 18 Chapter VA in the textbook provides additional lessons and standards review to get students ready for the end of the year SOL test. VA-4 Congruent and Similar Triangles (page VA 10) can be used at the end of Chapter 7 to teach using coordinate methods, such as the distance formula, to prove two triangles are similar. Geometry Readiness Curriculum Guide 2011 Polygons and Quadrilaterals Chapter: 6 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials G.10 The student will solve real-world problems 6-1 The Polygon Angle-Sum involving angles of polygons. Theorems • Find the sum of the measures of the interior and exterior angles of a convex polygon. 1 block • Find the measure of each interior and exterior angle of a regular polygon. • Find the number of sides of a regular polygon, given the measures of interior or exterior angles of the polygon. http://homepage.mac.com/efithian/Geometry/Activit y-04.html “Milliken Geometry” page 23 “Geometry Teacher’s Activities Kit” pages 139 141 G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • Solve problems, including real-world problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids. 6-2 Properties of Parallelograms http://www.keymath.com/x3331.xml “if Geometry” page 67 G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • Solve problems, including real-world problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids. • Prove that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula. • Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods. 6-3 Proving that a Quadrilateral is a Parallelogram G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • Solve problems, including real-world problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids. • Prove that quadrilaterals have specific properties, 6-4 Properties of http://www.mathsisfun.com/geometry/quadrilaterals Rhombuses, Rectangles, and -interactive.html Squares “if Geometry” page 70 “Milliken Geometry” page 24 1 block “Geometry Plane & Simple” pages 68, 69, 70 1 block http://www.mathsisfun.com/geometry/quadrilaterals -interactive.html “if Geometry” page 68 and 69 “Geometry Plane & Simple” page 66, 67 1 block 19 Comments Geometry Readiness Curriculum Guide 2011 Polygons and Quadrilaterals Chapter: 6 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Comments using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula. • Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods. G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • Solve problems, including real-world problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids. • Prove that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula. • Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods. 6-5 Conditions for Rhombuses, Rectangles, and Squares G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • Solve problems, including real-world problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids. • Prove that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula. • Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods. 6-6 Trapezoids and Kites 1 block 1 block G.2 The student will use the relationships between 6-7 Polygons in the angles formed by two lines cut by a transversal to Coordinate Plane a.) determine whether two lines are parallel; b.) verify the parallelism, using algebraic and 1 block coordinate methods as well as deductive proofs; and c.) solve real-world problems involving angles formed when parallel lines are cut by a transversal. 20 “Geometry Teacher’s Actvities Kit” pages 172 – 188 “if Geometry” page71 “Milliken Geometry” page 25 “Geometry Plane & Simple” page 71, 74 Suggestion for Chapter Review: DOE ESS Sample Lesson Plan G.9 Properties of Quadrilaterals Geometry Readiness Curriculum Guide 2011 Polygons and Quadrilaterals Chapter: 6 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Comments • Use algebraic and coordinate methods as well as deductive proofs to verify whether two lines are parallel. G.2 The student will use the relationships between 6-8 Applying Coordinate angles formed by two lines cut by a transversal to Geometry a.) determine whether two lines are parallel; b.) verify the parallelism, using algebraic and 1 block coordinate methods as well as deductive proofs; and c.) solve real-world problems involving angles formed when parallel lines are cut by a transversal. • Use algebraic and coordinate methods as well as deductive proofs to verify whether two lines are parallel. Area Chapter: 10 SOL with Essential Knowledge and Skill G.14 The student will use similar geometric objects in two- or three-dimensions to a)compare ratios between side lengths, perimeters, areas, and volumes. • Solve real-world problems involving measured attributes of similar objects. Textbook Chapters/Sections/Topics 10-1 Areas of Parallelograms and Triangles Supporting Materials Comments http://teachers.henrico.k12.va.us/math/igo/WU_A Make copies of the SOL lgebra/WU9_1Alg.htm formula sheet and give one to each student. http://teachers.henrico.k12.va.us/math/igo/09Are aVolume/9-1Area2D/HandsOn9-1FreshCoat.pdf 1 block http://www.youtheducationservices.ca/secure/sub jects/geometry/pdfs/ 9_Area_Peri_WS.pdf “Milliken Geometry” page 28 G.14 The student will use similar geometric objects 10-2 Areas of Trapezoids, in two- or three-dimensions to Rhombuses, and Kites a) compare ratios between side lengths, perimeters, areas, and volumes. 1 block • Solve real-world problems involving measured attributes of similar objects. http://teachers.henrico.k12.va.us/math/igo/09Are aVolume/9-1Area2D/WUcurrent9-1.pdf “Milliken Geometry” page 29 21 Encourage students to use their SOL formula sheet. Geometry Readiness Curriculum Guide 2011 Area Chapter: 10 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Comments G.14 The student will use similar geometric objects 10-4 Perimeters and Areas in two- or three-dimensions to of Similar Figures a) compare ratios between side lengths, perimeters, areas, and volumes; 1 block c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and d) solve real-world problems about similar geometric objects. • Compare ratios between side lengths, perimeters, areas, and volumes, given two similar figures. • Describe how changes in one or more dimensions affect other derived measures (perimeter, area, total surface area, and volume) of an object. • Describe how changes in one or more measures (perimeter, area, total surface area, and volume) affect other measures of an object. • Solve real-world problems involving measured attributes of similar objects. http://teachers.henrico.k12.va.us/math/igo/WU_A Encourage students to lgebra/WU9_5Alg.htm use their SOL formula sheet. G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. • Another formula for the area of a triangle is A=1/2ab(sinC). http://teachers.henrico.k12.va.us/math/igo/07Rig htTriangles/7-4Trigonometry/7-4WU.htm 10-5 Trigonometry and Area 0.5 block Encourage students to use their SOL formula sheet. Theorem 10-8 Area of a Triangle Given SAS (page 645) must be covered as it is part of the essential understandings in the curriculum framework. G.11 The student will use angles, arcs, chords, 10-6 Circles and Arcs tangents, and secants to b) solve real-world problems involving properties of 0.5 block circles; and c) find arc lengths and areas of sectors in circles. • Find lengths, angle measures, and arc measures associated with central and inscribed angles. • Calculate the area of a sector and the length of an arc of a circle, using proportions. http://teachers.henrico.k12.va.us/math/igo/08 Circles/8-4ArcsChords/8-4WU.htm DOE ESS Sample Lesson Plan G.11 Arc Length and Area of a Sector 22 Geometry Readiness Curriculum Guide 2011 Area Chapter: 10 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Comments • Solve real-world problems associated with circles, using properties of angles, lines, and arcs. • Verify properties of circles, using deductive reasoning, algebraic, and coordinate methods. G.11 The student will use angles, arcs, chords, 10-7 Areas of Circles and tangents, and secants to Sectors c) find arc lengths and areas of sectors in circles. • Calculate the area of a sector and the length of an 0.5 block arc of a circle, using proportions. • Solve real-world problems associated with circles, using properties of angles, lines, and arcs. • Verify properties of circles, using deductive reasoning, algebraic, and coordinate methods. http://teachers.henrico.k12.va.us/math/igo/09Are aVolume/9-1Area2D/CW9-1.pdf http://teachers.henrico.k12.va.us/math/igo/09Are aVolume/9-1Area2D/HW9-1.pdf Section 10-8, which was omitted, may be used after SOL testing for enrichment. “Milliken Geometry” page 32, 33 DOE ESS Sample Lesson Plan G.11 Arc Length and Area of a Sector Mapping for Instruction - Fourth Nine Weeks Circles Chapter: 12 SOL with Essential Knowledge and Skill G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles; b)solve real-world problems involving properties of circles; • Find lengths, angle measures, and arc measures associated with two intersecting tangents; • Solve real-world problems associated with circles, using properties of angles, lines, and arcs. • Verify properties of circles, using deductive Textbook Chapters/Sections/Topics Supporting Materials “Geometry Teacher’s Activities Kit” pages 214 – 216 “Geometry Plane & Simple” page 104 12-1 Tangent Lines 1 block 23 Comments Geometry Readiness Curriculum Guide 2011 Circles Chapter: 12 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Comments reasoning, algebraic, and coordinate methods G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles; b)solve real-world problems involving properties of circles; • Find lengths, angle measures, and arc measures associated with two intersecting chords; • Solve real-world problems associated with circles, using properties of angles, lines, and arcs. • Verify properties of circles, using deductive reasoning, algebraic, and coordinate methods “Geometry Teacher’s Activities Kit” pages 217 – 219 “Geometry Plane & Simple” page 103 12-2 Chords and Arcs 1 block G.4 The student will construct and justify the 12-3 Inscribed Angles constructions of a) a line segment congruent to a given line segment; 1 block b) the perpendicular bisector of a line segment; c) a perpendicular to a given line from a point not on the line; d) a perpendicular to a given line at a given point on the line; e) the bisector of a given angle, f) an angle congruent to a given angle; • Construct an equilateral triangle , a square, and a regular hexagon inscribed in a circle. • Construct the inscribed and circumscribed circles of a triangle. • Construct a tangent line from a point outside a given circle to the circle http://www.mathopenref.com/constincircle.html http://www.mathsisfun.com/geometry/constructtrianglecircum.html http://www.mathopenref.com/constequilateral.ht ml “Milliken Geometry” page 35 “Geometry Plane & Simple” pages 106, 107, 108, 109 G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles • Find lengths, angle measures, and arc measures associated with – two intersecting chords; – central and inscribed angles. • Solve real-world problems associated with circles, using properties of angles, lines, and arcs. 24 Constructions of circumscribe and inscribed polygons may be presented here or later. Use the links to access online construction examples. Geometry Readiness Curriculum Guide 2011 Circles Chapter: 12 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Comments • Verify properties of circles, using deductive reasoning, algebraic, and coordinate methods G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; • Find lengths, angle measures, and arc measures associated with – two intersecting chords; – central and inscribed angles. • Solve real-world problems associated with circles, using properties of angles, lines, and arcs. • Verify properties of circles, using deductive reasoning, algebraic, and coordinate methods – two intersecting secants; – an intersecting secant and tangent; – two intersecting tangents; and 12-4 Angle Measures and Segment Lengths 1 block “Geometry Plane & Simple” page 110 “Milliken Geometry” page 37, 38 “Geometry Teacher’s Activities Kit” pages 220 – 230 DOE ESS Sample Lesson Plan G.11 Angles, Arcs, and Segments in Circles G.12 The student, given the coordinates of the 12-5 Circles in the Coordinate center of a circle and a point on the circle, will write the Plane equation of the circle. • Identify the center, radius, and diameter of a circle 1.5 block from a given standard equation. • Use the distance formula to find the radius of a circle. • Given the coordinates of the center and radius of the circle, identify a point on the circle. • Given the equation of a circle in standard form, identify the coordinates of the center and find the radius of the circle. • Given the coordinates of the endpoints of a diameter, find the equation of the circle. • Given the coordinates of the center and a point on the circle, find the equation of the circle. • Recognize that the equation of a circle of given center and radius is derived using the Pythagorean Theorem. 25 http://education.ti.com/calculators/timath/US/Acti vities/Detail?sa=502481d=12554 “Geometry Teacher’s Activities Kit” pages 231 – 233 DOE ESS Sample Lesson Plan G.12 Circles in the Coordinate Plane *Include 12-6 definition of a locus. Geometry Readiness Curriculum Guide 2011 Surface Area and Volume Chapter: 11 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics 1-1 Nets and Drawings for Visualizing Geometry Optional Supporting Materials NCTM Isometric Drawing Tool http://illuminations.nctm.org/activit ydetail.aspx?id=125 http://www.learner.org/interactives /geometry/index.html “Geometry Teacher’s Activities Kit” pages 295 - 297 G.13 The student will use formulas for surface area 11-2 Surface Areas of and volume of three-dimensional objects to solve real- Prisms and Cylinders world problems. • Find the total surface area of cylinders, prisms, 0.5 block pyramids, cones, and spheres, using the appropriate formulas. Comments The SAT exam and the state SOL test both provide formula sheets for students to use. Therefore, it is not necessary to have students memorize formulas. However, students need to be aware of the derivation of these formulas and their use in problem solving. The formulas for all chapters is on page 838 and 839. Teachers may want to make copies of the SOL formula sheet and distribute to students at the beginning of lesson 11-2. G.13 The student will use formulas for surface area 11-3 Surface Areas of and volume of three-dimensional objects to solve real- Pyramids and Cones world problems. • Find the total surface area of cylinders, prisms, 0.5 block pyramids, cones, and spheres, using the appropriate formulas • Solve problems, including real-world problems, involving total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-dimensional figures. • Calculators may be used to find decimal approximations for results. G.14 The student will use similar geometric objects in two- or three-dimensions to b) determine how changes in one or more dimensions of an object affect area and/or volume of the object. G.13 The student will use formulas for surface area 11-4 Volumes of Prims and and volume of three-dimensional objects to solve real- Cylinders 26 “if Geometry” pages 86, 87 “Geometry Teacher’s Activities Geometry Readiness Curriculum Guide 2011 Surface Area and Volume Chapter: 11 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials Kit” pages 298 - 301 world problems. • Find the total surface area of cylinders, prisms, 0.5 block pyramids, cones, and spheres, using the appropriate formulas • Solve problems, including real-world problems, involving total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-dimensional figures. • Calculators may be used to find decimal approximations for results. • Solve problems, including real-world problems, involving total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-dimensional figures. • Calculators may be used to find decimal approximations for results. • Calculate the volume of cylinders, prisms, pyramids, cones, and spheres, using the appropriate formulas. G.14 The student will use similar geometric objects in two- or three-dimensions to b) determine how changes in one or more dimensions of an object affect area and/or volume of the object. G.13 The student will use formulas for surface area 11-5 Volumes of Pyramids and volume of three-dimensional objects to solve real- and Cones world problems. • Solve problems, including real-world problems, 0.5 block involving total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-dimensional figures. • Calculators may be used to find decimal approximations for results. • Calculate the volume of cylinders, prisms, pyramids, cones, and spheres, using the appropriate formulas. G.14 The student will use similar geometric objects in two- or three-dimensions to b) determine how changes in one or more dimensions of an object affect area and/or volume of the object. 27 “if Geometry” pages 88, 89 Comments Geometry Readiness Curriculum Guide 2011 Surface Area and Volume Chapter: 11 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials “Geometry Teacher’s Activities Kit” pages 302 – 308 G.13 The student will use formulas for surface area 11-6 Surface Areas and and volume of three-dimensional objects to solve real- Volumes of Spheres world problems. • Find the total surface area of cylinders, prisms, 0.5 block pyramids, cones, and spheres, using the appropriate formulas. • Calculate the volume of cylinders, prisms, pyramids, cones, and spheres, using the appropriate formulas. • Solve problems, including real-world problems, involving total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-dimensional figures. • Calculators may be used to find decimal approximations for results. DOE ESS Sample Lesson Plan G.13 Surface Area and Volume G.13 The student will use formulas for surface area 11-7 Areas and Volumes of DOE ESS Sample Lesson Plan and volume of three-dimensional objects to solve real- Similar Solids G.14 Similar Solids and world problems Proportional Reasoing • Compare ratios between side lengths, perimeters, 1 block areas, and volumes, given two similar figures. • Describe how changes in one or more dimensions affect other derived measures (perimeter, area, total surface area, and volume) of an object. • Describe how changes in one or more measures (perimeter, area, total surface area, and volume) affect other measures of an object. • Solve real-world problems involving measured attributes of similar objects.. 28 Comments Geometry Readiness Curriculum Guide 2011 Transformations Chapter: 9 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics Supporting Materials G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. • Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, dilation, or translation. 9-1 Translations http://www.mathisfun.com/geometry.transla tion.html G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. • Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, dilation, or translation. 9-2 Reflections G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. • Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, dilation, or translation. 9-3 Rotations G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve 9-4 Symmetry “if Geometry” page 16 0.25 block DOE ESS Sample Lesson Plan 0.25 block “if Geometry” page 20 http://www.mathisfun.com/geometry.reflecti on.html 0.25 block “if Geometry” page 18 http://www.mathisfun.com/geometry.rotatio n.html 0.25 block “Geometry Teacher’s Activities Kit” pages 189 – 195 “if Geometry” page 19, 21,22, 23 29 Comments Geometry Readiness Curriculum Guide 2011 Transformations Chapter: 9 SOL with Essential Knowledge and Skill Textbook Chapters/Sections/Topics problems involving symmetry and transformation. This will include c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and • Determine whether a figure has point symmetry, line symmetry, both, or neither. Supporting Materials Comments G.3 Symmetry G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. • Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, dilation, or translation. 9-5 Dilations http://www.mathisfun.com/geometry.resizin Suggested Review: g.html DOE ESS Sample Lesson Plan G.3 Transformations G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. • Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, dilation, or translation. 9-6 Compositions of Reflections G.10 The student will solve real-world problems involving angles of polygons. • Identify tessellations in art, construction, and nature. 9-7 Tesselations 0.25 block 0.25 block http://math.pppst.com/tessellations.html 0.5 block 30 Geometry Readiness Curriculum Guide 2011 SOL Blueprints http://www.doe.virginia.gov/testing/sol/blueprints/mathematics_blueprints/2009/blueprint_geometry.pdf This revised test blueprint will be 2011-2012 Mathematics Standards of effective with the administration of the Learning tests. 31 Geometry Readiness Curriculum Guide 2011 SOL Enhanced Scope and Sequence Supplemental Resources All supplemental material provided with the textbook should be used at the instructor's discretion to extend, enhance and/or review student knowledge of essential geometric skills. Examview software and Interactive Achievement may be used as a source of pre-made assessment materials, or to custom design assessments for this course. The following pages include supplemental worksheets referenced in this Geometry Curriculum Guide. Study Island (www.studyisland.com) can be used to enhance instruction. Interactive Achievement (www.interactiveachievement.com) can be used as a resource to create and administer quizzes and tests. If teachers find they need to supplement sections of the new textbook to adequately teach a SOL and/or possess materials that enhance that instruction, please contact Linda Bowden. Geometry Reproducibles by Milliken Publishing Company Geometry by Instructional Fair Geometry Teacher’s Activities Kit by Jossey-Bass Geometry Plane & Simple by Creative Publications (now Glencoe) http://www.doe.virginia.gov/testing/sol/practice_items/index.shtml 32 Geometry Readiness Curriculum Guide 2011 SOL 2009 Framework Geometry 33 Copyright © 2009 by the Virginia Department of Education P.O. Box 2120 Richmond, Virginia 23218-2120 http://www.doe.virginia.gov All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted. Superintendent of Public Instruction Patricia I. Wright, Ed.D. Assistant Superintendent for Instruction Linda M. Wallinger, Ph.D. Office of Elementary Instruction Mark R. Allan, Ph.D., Director Deborah P. Wickham, Ph.D., Mathematics Specialist Office of Middle and High School Instruction Michael F. Bolling, Mathematics Coordinator Acknowledgements The Virginia Department of Education wishes to express sincere thanks to Deborah Kiger Bliss, Lois A. Williams, Ed.D., and Felicia Dyke, Ph.D. who assisted in the development of the 2009 Mathematics Standards of Learning Curriculum Framework. NOTICE The Virginia Department of Education does not unlawfully discriminate on the basis of race, color, sex, national origin, age, or disability in employment or in its educational programs or services. The 2009 Mathematics Curriculum Framework can be found in PDF and Microsoft Word file formats on the Virginia Department of Education’s Web site at http://www.doe.virginia.gov. Virginia Mathematics Standards of Learning Curriculum Framework 2009 Introduction The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies the Mathematics Standards of Learning by defining the content knowledge, skills, and understandings that are measured by the Standards of Learning assessments. The Curriculum Framework provides additional guidance to school divisions and their teachers as they develop an instructional program appropriate for their students. It assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This supplemental framework delineates in greater specificity the content that all teachers should teach and all students should learn. Each topic in the Mathematics Standards of Learning Curriculum Framework is developed around the Standards of Learning. The format of the Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge and skills that should be the focus of instruction for each standard. The Curriculum Framework is divided into two columns: Essential Understandings and Essential Knowledge and Skills. The purpose of each column is explained below. Essential Understandings This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the Standards of Learning. Essential Knowledge and Skills Each standard is expanded in the Essential Knowledge and Skills column. What each student should know and be able to do in each standard is outlined. This is not meant to be an exhaustive list nor is a list that limits what taught in the classroom. It is meant to be the key knowledge and skills that define the standard. The Curriculum Framework serves as a guide for Standards of Learning assessment development. Assessment items may not and should not be a verbatim reflection of the information presented in the Curriculum Framework. Students are expected to continue to apply knowledge and skills from Standards of Learning presented in previous grades as they build mathematical expertise. TOPIC: REASONING, LINES, AND TRANSFORMATIONS GEOMETRY STANDARD G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; and d) using deductive reasoning. ESSENTIAL UNDERSTANDINGS Inductive reasoning, deductive reasoning, and proof are critical in establishing general claims. Deductive reasoning is the method that uses logic to draw conclusions based on definitions, postulates, and theorems. Inductive reasoning is the method of drawing conclusions from a limited set of observations. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Identify the converse, inverse, and contrapositive of a conditional statement. Translate verbal arguments into symbolic form, such as (p q) and (~p ~q). Determine the validity of a logical argument. Use valid forms of deductive reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples. Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, and theorems. Logical arguments consist of a set of premises or hypotheses and a conclusion. Euclidean geometry is an axiomatic system based on undefined terms (point, line and plane), postulates, and theorems. Select and use various types of reasoning and methods of proof, as appropriate. When a conditional and its converse are true, the statements can be written as a biconditional, i.e., iff or if and only if. Use Venn diagrams to represent set relationships, such as intersection and union. Interpret Venn diagrams. Recognize and use the symbols of formal logic, which include →, ↔, ~, , , and . Logical arguments that are valid may not be true. Truth and validity are not synonymous. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 1 TOPIC: REASONING, LINES, AND TRANSFORMATIONS GEOMETRY STANDARD G.2 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. ESSENTIAL UNDERSTANDINGS Parallel lines intersected by a transversal form angles with specific relationships. Some angle relationships may be used when proving two lines intersected by a transversal are parallel. The Parallel Postulate differentiates Euclidean from nonEuclidean geometries such as spherical geometry and hyperbolic geometry. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Use algebraic and coordinate methods as well as deductive proofs to verify whether two lines are parallel. Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal including corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles. Solve real-world problems involving intersecting and parallel lines in a plane. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 2 TOPIC: REASONING, LINES, AND TRANSFORMATIONS GEOMETRY STANDARD G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include a) investigating and using formulas for finding distance, midpoint, and slope; b) applying slope to verify and determine whether lines are parallel or perpendicular; c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. ESSENTIAL UNDERSTANDINGS Transformations and combinations of transformations can be used to describe movement of objects in a plane. The distance formula is an application of the Pythagorean Theorem. Geometric figures can be represented in the coordinate plane. Techniques for investigating symmetry may include paper folding, coordinate methods, and dynamic geometry software. Parallel lines have the same slope. The product of the slopes of perpendicular lines is -1. The image of an object or function graph after an isomorphic transformation is congruent to the preimage of the object. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Find the coordinates of the midpoint of a segment, using the midpoint formula. Use a formula to find the slope of a line. Compare the slopes to determine whether two lines are parallel, perpendicular, or neither. Determine whether a figure has point symmetry, line symmetry, both, or neither. Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, dilation, or translation. Apply the distance formula to find the length of a line segment when given the coordinates of the endpoints. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 3 TOPIC: REASONING, LINES, AND TRANSFORMATIONS GEOMETRY STANDARD G.4 The student will construct and justify the constructions of a) a line segment congruent to a given line segment; b) the perpendicular bisector of a line segment; c) a perpendicular to a given line from a point not on the line; d) a perpendicular to a given line at a given point on the line; e) the bisector of a given angle; f) an angle congruent to a given angle; and g) a line parallel to a given line through a point not on the given line. ESSENTIAL UNDERSTANDINGS Construction techniques are used to solve real-world problems in engineering, architectural design, and building construction. Construction techniques include using a straightedge and compass, paper folding, and dynamic geometry software. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Construct and justify the constructions of – a line segment congruent to a given line segment; – the perpendicular bisector of a line segment; – a perpendicular to a given line from a point not on the line; – a perpendicular to a given line at a point on the line; – the bisector of a given angle; – an angle congruent to a given angle; and – a line parallel to a given line through a point not on the given line. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.† Construct the inscribed and circumscribed circles of a triangle.† Construct a tangent line from a point outside a given circle to the circle.† † Revised March 2011 Mathematics Standards of Learning Curriculum Framework 2009: Geometry 4 TOPIC: TRIANGLES GEOMETRY STANDARD G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a) order the sides by length, given the angle measures; b) order the angles by degree measure, given the side lengths; c) determine whether a triangle exists; and d) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. ESSENTIAL UNDERSTANDINGS The longest side of a triangle is opposite the largest angle of the triangle and the shortest side is opposite the smallest angle. In a triangle, the length of two sides and the included angle determine the length of the side opposite the angle. In order for a triangle to exist, the length of each side must be within a range that is determined by the lengths of the other two sides. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Order the sides of a triangle by their lengths when given the measures of the angles. Order the angles of a triangle by their measures when given the lengths of the sides. Given the lengths of three segments, determine whether a triangle could be formed. Given the lengths of two sides of a triangle, determine the range in which the length of the third side must lie. Solve real-world problems given information about the lengths of sides and/or measures of angles in triangles. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 5 TOPIC: TRIANGLES GEOMETRY STANDARD G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. ESSENTIAL UNDERSTANDINGS Congruence has real-world applications in a variety of areas, including art, architecture, and the sciences. Congruence does not depend on the position of the triangle. Concepts of logic can demonstrate congruence or similarity. Congruent figures are also similar, but similar figures are not necessarily congruent. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Use definitions, postulates, and theorems to prove triangles congruent. Use coordinate methods, such as the distance formula and the slope formula, to prove two triangles are congruent. Use algebraic methods to prove two triangles are congruent. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 6 TOPIC: TRIANGLES GEOMETRY STANDARD G.7 The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. ESSENTIAL UNDERSTANDINGS Similarity has real-world applications in a variety of areas, including art, architecture, and the sciences. Similarity does not depend on the position of the triangle. Congruent figures are also similar, but similar figures are not necessarily congruent. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Use definitions, postulates, and theorems to prove triangles similar. Use algebraic methods to prove that triangles are similar. Use coordinate methods, such as the distance formula, to prove two triangles are similar. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 7 TOPIC: TRIANGLES GEOMETRY STANDARD G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. ESSENTIAL UNDERSTANDINGS The Pythagorean Theorem is essential for solving problems involving right triangles. Many historical and algebraic proofs of the Pythagorean Theorem exist. The relationships between the sides and angles of right triangles are useful in many applied fields. Some practical problems can be solved by choosing an efficient representation of the problem. Another formula for the area of a triangle is A ab sin C . The ratios of side lengths in similar right triangles (adjacent/hypotenuse or opposite/hypotenuse) are independent of the scale factor and depend only on the angle the hypotenuse makes with the adjacent side, thus justifying the definition and calculation of trigonometric functions using the ratios of side lengths for similar right triangles. 1 2 ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Determine whether a triangle formed with three given lengths is a right triangle. Solve for missing lengths in geometric figures, using properties of 45-45-90 triangles. Solve for missing lengths in geometric figures, using properties of 30-60-90 triangles. Solve problems involving right triangles, using sine, cosine, and tangent ratios. Solve real-world problems, using right triangle trigonometry and properties of right triangles. Explain and use the relationship between the sine and cosine of complementary angles.† † Mathematics Standards of Learning Curriculum Framework 2009: Geometry Revised March 2011 8 TOPIC: POLYGONS AND CIRCLES GEOMETRY STANDARD G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. ESSENTIAL UNDERSTANDINGS The terms characteristics and properties can be used interchangeably to describe quadrilaterals. The term characteristics is used in elementary and middle school mathematics. Quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Solve problems, including real-world problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids. Prove that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula. Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods. Prove properties of angles for a quadrilateral inscribed in a circle.† Characteristics of quadrilaterals can be used to identify the quadrilateral and to find the measures of sides and angles. † Mathematics Standards of Learning Curriculum Framework 2009: Geometry Revised March 2011 9 TOPIC: POLYGONS AND CIRCLES GEOMETRY STANDARD G.10 The student will solve real-world problems involving angles of polygons. ESSENTIAL UNDERSTANDINGS A regular polygon will tessellate the plane if the measure of an interior angle is a factor of 360. Both regular and nonregular polygons can tessellate the plane. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Solve real-world problems involving the measures of interior and exterior angles of polygons. Two intersecting lines form angles with specific relationships. Identify tessellations in art, construction, and nature. An exterior angle is formed by extending a side of a polygon. The exterior angle and the corresponding interior angle form a linear pair. Find the sum of the measures of the interior and exterior angles of a convex polygon. Find the measure of each interior and exterior angle of a regular polygon. Find the number of sides of a regular polygon, given the measures of interior or exterior angles of the polygon. The sum of the measures of the interior angles of a convex polygon may be found by dividing the interior of the polygon into nonoverlapping triangles. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 10 TOPIC: POLYGONS AND CIRCLES GEOMETRY STANDARD G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and c) find arc lengths and areas of sectors in circles. ESSENTIAL UNDERSTANDINGS Many relationships exist between and among angles, arcs, secants, chords, and tangents of a circle. All circles are similar. A chord is part of a secant. Real-world applications may be drawn from architecture, art, and construction. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Find lengths, angle measures, and arc measures associated with – two intersecting chords; – two intersecting secants; – an intersecting secant and tangent; – two intersecting tangents; and – central and inscribed angles. Calculate the area of a sector and the length of an arc of a circle, using proportions. Solve real-world problems associated with circles, using properties of angles, lines, and arcs. Verify properties of circles, using deductive reasoning, algebraic, and coordinate methods. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 11 TOPIC: POLYGONS AND CIRCLES GEOMETRY STANDARD G.12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. ESSENTIAL UNDERSTANDINGS A circle is a locus of points equidistant from a given point, the center. Standard form for the equation of a circle is 2 2 x h y k r 2 , where the coordinates of the center of the ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Identify the center, radius, and diameter of a circle from a given standard equation. circle are ( h, k ) and r is the length of the radius. Use the distance formula to find the radius of a circle. The circle is a conic section. Given the coordinates of the center and radius of the circle, identify a point on the circle. Given the equation of a circle in standard form, identify the coordinates of the center and find the radius of the circle. Given the coordinates of the endpoints of a diameter, find the equation of the circle. Given the coordinates of the center and a point on the circle, find the equation of the circle. Recognize that the equation of a circle of given center and radius is derived using the Pythagorean Theorem.† † Mathematics Standards of Learning Curriculum Framework 2009: Geometry Revised March 2011 12 TOPIC: THREE-DIMENSIONAL FIGURES GEOMETRY STANDARD G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. ESSENTIAL UNDERSTANDINGS The surface area of a three-dimensional object is the sum of the areas of all its faces. The volume of a three-dimensional object is the number of unit cubes that would fill the object. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Find the total surface area of cylinders, prisms, pyramids, cones, and spheres, using the appropriate formulas. Calculate the volume of cylinders, prisms, pyramids, cones, and spheres, using the appropriate formulas. Solve problems, including real-world problems, involving total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-dimensional figures. Calculators may be used to find decimal approximations for results. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 13 TOPIC: THREE-DIMENSIONAL FIGURES GEOMETRY STANDARD G.14 The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; b) determine how changes in one or more dimensions of an object affect area and/or volume of the object; c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and d) solve real-world problems about similar geometric objects. ESSENTIAL UNDERSTANDINGS A change in one dimension of an object results in predictable changes in area and/or volume. A constant ratio exists between corresponding lengths of sides of similar figures. Proportional reasoning is integral to comparing attribute measures in similar objects. ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Compare ratios between side lengths, perimeters, areas, and volumes, given two similar figures. Describe how changes in one or more dimensions affect other derived measures (perimeter, area, total surface area, and volume) of an object. Describe how changes in one or more measures (perimeter, area, total surface area, and volume) affect other measures of an object. Solve real-world problems involving measured attributes of similar objects. Mathematics Standards of Learning Curriculum Framework 2009: Geometry 14