South Plainfield Public Schools Curriculum Guide Mathematics Geometry Authors: Anthony Emmons Kathy Zoda Curriculum Coordinator: Paul C. Rafalowski Board Approved on: June 13, 2012 1 Table of Contents South Plainfield Public Schools Board of Education Members and Administration Page: 3 Recognitions Page: 4 District Mission Statement Page: 5 Index of Courses Page: 6 Curriculum Guides Page: 7-42 Mathematics Practice Standards Page: 43-45 Common Core Standards Page: 46-48 Resources for State Assessments Page: 49 2 Members of the Board of Education Jim Giannakis, President Debbie Boyle, Vice President Carol Byrne John T. Farinella, Jr Christopher Hubner Sharon Miller William Seesselberg Joseph Sorrentino Gary Stevenson Central Office Administration Dr. Stephen Genco, Superintendent of Schools Dr. Frank Cocchiola, Interim Assistant Superintendent of Schools Mr. James Olobardi, Board Secretary/ BA Mrs. Laurie Hall, Supervisor of Student Personnel Services Mr. Vincent Parisi, Supervisor of Math and Science Mrs. Marlene Steele, Supervisor of Transportation Mrs. Annemarie Stoeckel, Supervisor of Technology Ms. Elaine Gallo, Director of Guidance Mr. Al Czech, Director of Athletics Mr. Paul Rafalowski, Curriculum Coordinator 3 Recognitions The following individuals are recognized for their support in developing this Curriculum Guide: Grade/Course Writer(s) Kindergarten: Ms. Joy Czaplinski and Ms. Pat Public Grade 1: Ms. Patti Schenck-Ratti, Ms. Kim Wolfskeil and Ms. Nicole Wrublevski Grade 2: Ms. Cate Bonanno, Ms. Shannon Colucci and Ms. Maureen Wilson Grade 3: Ms. Cate Bonanno and Ms. Theresa Luck Grade 4: Ms. Linda Downey and Ms. Kathy Simpson Grade 5: Mr. John Orfan and Ms. Carolyn White Grade 6: Ms. Joanne Haus and Ms. Cathy Pompilio Grade 7: Ms. Marianne Decker and Ms. Kathy Zoda Grade 8: Ms. Marianne Decker and Ms. Donna Tierney Algebra 1: Ms. Donna Tierney and Ms. Kathy Zoda Geometry: Mr. Anthony Emmons and Ms. Kathy Zoda Algebra 2: Mr. Anthony Emmons and Mr. John Greco Algebra 3/Trigonometry: Ms. Anu Garrison and Mr. David Knarr Senior Math Applications: Mr. John Greco Pre-Calculus: Ms. Anu Garrison and Mr. David Knarr Calculus: Mr. David Knarr Supervisors: Supervisor of Mathematics and Science: Mr. Vince Parisi Curriculum Coordinator: Mr. Paul C. Rafalowski Supervisor of Technology: Ms. Annemarie Stoeckel 4 South Plainfield Public Schools District Mission Statement To ensure that all pupils are equipped with essential skills necessary to acquire a common body of knowledge and understanding; To instill the desire to question and look for truth in order that pupils may become critical thinkers, life-long learners, and contributing members of society in an environment of mutual respect and consideration. It is the expectation of this school district that all pupils achieve the New Jersey Core Curriculum Content Standards at all grade levels. Adopted September, 2008 NOTE: The following pacing guide was developed during the creation of these curriculum units. The actual implementation of each unit may take more or less time. Time should also be dedicated to preparation for benchmark and State assessments, and analysis of student results on the same. A separate document is included at the end of this curriculum guide with suggestions and resources related to State Assessments (if applicable). The material in this document should be integrated throughout the school year, and with an awareness of the State Testing Schedule. It is highly recommended that teachers meet throughout the school year to coordinate their efforts in implementing the curriculum and preparing students for benchmark and State Assessments in consideration of both the School and District calendars. 5 Index of Mathematics Courses Elementary Schools (Franklin, Kennedy, Riley, Roosevelt) Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grant School Grade 5 Grade 6 Honors Grade 6 Middle School Grade 7 Honors Grade 7 Grade 8 Honors Grade 8 Grade 8 Algebra 1 High School Algebra 1 Academic Algebra 1 Honors Algebra 1 Geometry Academic Geometry Honors Geometry Algebra 2 Academic Algebra 2 Honors Algebra 2 Algebra 3/Trigonometry Senior Math Applications Pre-Calculus Honors Pre-Calculus Calculus Calculus AB Calculus BC 6 South Plainfield Public Schools Curriculum Guide Content Area: Mathematics Course Title: Geometry Grade Level: 9 and 10 Unit 0: Algebra Pre-Requisites for Geometry 2 Weeks Unit 1: Constructions and Congruence 4 Weeks Unit 2: Proving Geometric Theorems 4 Weeks Unit 3: Reasoning and Congruent Triangle Proofs 4 Weeks Unit 4: Similarity and Proofs 3 Weeks Unit 5: Special Right Triangles and Trigonometry 3 Weeks Unit 6: Coordinate Geometry and Proofs 4 Weeks Board Approved on: June 13, 2012 7 South Plainfield Public Schools Curriculum Guide Content Area: Mathematics Course Title: Geometry Grade Level: 9 and 10 Unit 7: Circles, lines, angles, arc relationships 3 Weeks Unit 8: Circles and Conic Sections on the Coordinate Plane 3 Weeks Unit 9: Geometry in Three Dimensions 3 Weeks Unit 10: Applications of Geometric Probability 3 Weeks Board Approved on: June 13, 2012 8 Unit 0 Overview Content Area – Mathematics Unit Title – Algebra Pre-Requisites for Geometry Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – In order to make sense of geometry, students must be proficient with certain skills that involve equations. Specifically, they must be able to evaluate expressions, write and solve linear and quadratic equations, solve a system of equations, be adept at identifying coordinates on the Cartesian Plane, be able to graph linear and quadratic equations using a variety of forms and methods, and be able to apply ratios and proportions accurately and precisely. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration –– Calculator and iPad, projector and/or SMART Board as available and/or appropriate 21st Century Themes – Civic Literacy Global Awareness 21st Century Skills – Critical Thinking & Problem Solving Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 9 Domain Standards: A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.REI.4: Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. A.REI.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A.REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A.REI.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. Unit Essential Questions Why is it necessary to use algebraic representation and terminology? Unit Enduring Understandings Algebra is a universal symbolic language necessary for mathematical communication Terminology: expression, equation, linear equation, quadratic equation, system of equations, solution set, graphic solution, function, inequality, Cartesian Plane, ordered pair, coordinate, domain, range, irrational number, ratio, proportion Goals/Objectives Students will be able to evaluate expressions write and solve linear and quadratic equations solve a system of equations Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Instructional Strategies Whole Group Lessons Small Group Explorations In-class group work Independent Practice Homework Class Discussion Formative: White Board Response Black Board Race Ticket Out Think/Pair/Share Open Discussion Homework Quizzes* 10 Notebook Quizzes* be adept at identifying coordinates on the Cartesian Plane be able to graph linear equations using a variety of forms and methods Activities Textbook Activities Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding Think Pair Share Summative: Multi-Lesson Quizzes Unit Test *based on unit or need Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument be able to write the equation of a line be able to graph quadratic equations using a variety of methods be able to apply ratios and proportions accurately and precisely. Diverse Learners (ELL, Special Ed, Gifted & Talented) - Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Teacher discretion due to prerequisite skills assessment. 11 Unit 1 Overview Content Area – Mathematics Unit 1: Constructions and Congruence Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – Students will not only learn the definitions of particular geometric figures but also develop a deeper understanding of them through constructions and measurements. In this unit, students will develop definitions of the transformations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Transformations will then be used to define congruent figures. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – Calculators will be used when deemed necessary. Geometer’s Sketchpad if available Ruler, Protractor, Compass 21st Century Themes – Civic Literacy Global Awareness 21st Century Skills – Creativity & Innovation Critical Thinking & Problem Solving Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 12 Domain Standards: G.CO.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. G.CO.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). G.CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.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. G.CO.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. G.CO.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. G.CO.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. G.CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.C.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Unit Essential Questions How do we recognize and apply transformations of shapes to solve problems? Unit Enduring Understandings All constructions are based on the properties of geometric figures. What are the basic geometric figures and how are they related to one another? Measurement quantifies attributes of objects according to various units, systems, and processes. How do geometric relationships and the application of measurement help us to solve real-world problems? Analyzing geometric relationships develops reasoning and justification skills. What are the characteristics and applications of symmetry? How does what we measure affect how we measure? Terminology: Ruler, Compass, Protractor, Points, Lines, Segments, Planes, Rays, Angles, Angle Bisector, Segment Bisector, Midpoint, Segment Addition Postulate, Angle Addition Postulate, Complementary, Supplementary 13 Goals/Objectives Students will be able to apply properties of transformation using coordinate geometry use transformations to move figures, create designs, and/or demonstrate geometric properties construct and/or validate properties of geometric figures using appropriate tools and technology solve problems using constructions Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Instructional Strategies Whole Group Lessons Small Group Explorations In-class group work Independent Practice Homework Class Discussion Formative: White Board Response Black Board Race Ticket Out Think/Pair/Share Open Discussion Homework Quizzes* Notebook Quizzes* Activities Textbook Activities Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding Think Pair Share Summative: Multi-Lesson Quizzes Unit Test *based on unit or need Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument construct angle bisectors construct segment bisectors use transformations to move figures, create designs, and demonstrate geometric properties Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 14 Unit 2 Overview Content Area – Mathematics Unit 2: Proving Geometric Theorems Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – Here students study deductive reasoning and conditional statements in preparation for the standards contained herein. Students will use deductive reasoning to construct arguments and write formal proofs for Geometric Theorems about lines, angles, triangles, and parallelograms. These arguments will contain Properties of Equality and Congruence (largely from Algebra 1), as well as applications of the geometric figures studied, measured, and constructed in unit one. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – Calculators will be used when deemed necessary. Geometer’s Sketchpad if available Ruler, Protractor, Compass 21st Century Themes – Global Awareness Environmental Literacy 21st Century Skills – Critical Thinking & Problem Solving Information Literacy Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 15 Domain Standards: G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.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. G.CO.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. G.CO.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. Unit Essential Questions What is the relationship between lines and angles? Unit Enduring Understandings Logical arguments consist of a set of premises or hypotheses and a conclusion. Two intersecting lines form angles with specific relationships. Parallel lines cut by a transversal form angles with specific relationships. Analyzing geometric relationships develops reasoning and justification skills. How do geometric relationships and the application of measurement help us to solve real-world problems? Terminology: Hypothesis, Conclusion, Law of Detachment, Law of Syllogism, Addition Subtraction Multiplication Division Properties of Equality, Reflexive Symmetric Transitive Substitution Property of Equality/Congruence, Conditional, Converse, Biconditional, Inverse, Contrapositive, Given, Conclusion, Statement, Justification, Proof, Counterexample, Vertical Angles, Linear Pair, Complementary, Supplementary 16 Goals/Objectives Students will be able to - Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) recognize conditional statements, the hypothesis and conclusion Instructional Strategies Whole Group Lessons Small Group Explorations In-class group work Independent Practice Homework Class Discussion Formative: White Board Response Black Board Race Ticket Out Think/Pair/Share Open Discussion Homework Quizzes* Notebook Quizzes* write and interpret conditional statements including the converse, inverse, and contrapositive use deductive reasoning, the laws of detachment and syllogism, to draw conclusions Activities Textbook Activities Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding Think Pair Share using deductive reasoning, draw logical conclusions from given information and known facts Summative: Multi-Lesson Quizzes Unit Test *based on unit or need Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument use counterexamples to disprove conditional statements Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 17 Unit 3 Overview Content Area – Mathematics Unit 3: Reasoning and Congruent Triangle Proofs Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – The concept of congruent triangles is explored by examining which parts of a triangle are necessary and sufficient to construct a unique triangle. Students prove congruence of triangles through two-column and/or paragraph proofs. Corresponding parts of congruent triangles is used to prove congruence of sides and/or angles. Compass and straightedge constructions are reviewed and congruent triangles are applied to justify the transformation. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – Calculators will be used when deemed necessary. Geometer’s Sketchpad if available Ruler, Protractor, Compass 21st Century Themes – Global Awareness Civic Literacy 21st Century Skills – Creativity & Innovation Critical Thinking & Problem Solving Communication Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 18 Domain Standards: G.CO.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. G.CO.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. G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.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. Unit Essential Questions How do geometric relationships and the application of measurements help us to solve real-world problems? Unit Enduring Understandings Proof is a justification that is logically valid based on definitions, postulates, and theorems. What is the difference between similarity and congruence? Analyzing geometric relationships develops reasoning and justification skills. Terminology: Congruent, Corresponding Parts, Congruence Statement, SSS, SAS, AAS, ASA, HL, Vertical Angles, Reflexive Property, Alternate Interior Angles, Corresponding Angles Goals/Objectives Students will be able to Identify Corresponding Parts of Triangles Determine Whether or Not Two Triangles are Congruent Write a Congruence Statement for Two Given Triangles Show that two triangles are congruent using SSS, SAS, AAS, ASA, HL. Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Instructional Strategies Whole Group Lessons Small Group Explorations In-class group work Independent Practice Homework Class Discussion Formative: White Board Response Black Board Race Ticket Out Think/Pair/Share Open Discussion Homework Quizzes* Notebook Quizzes* Activities Textbook Activities Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding Think Pair Share Summative: Multi-Lesson Quizzes Unit Test *based on unit or need Prove triangles congruent given information in the form 19 of a figure or statement using deductive proofs. Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument Recognize Vertical Angles and Reflexive Property, apply concepts to proofs. Recognize Angles Formed By Two Parallel Lines and a Transversal, apply concepts to proofs. use transformations to move figures, create designs, and demonstrate geometric properties Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 20 Unit 4 Overview Content Area – Mathematics Unit 4: Similarity and Proofs Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – Students investigate and apply dilations of polygons in the coordinate plane. Similarity of polygons and triangles are explored. Triangle similarity postulates and theorems are formally proven. The proportionality of corresponding sides of similar figures is applied. Similarity extends to side-splitting, proportional medians, altitudes, angle bisectors, and segment theorems. The geometric mean is defined and related to arithmetic mean. Ratios of areas of similar solids are also studied. Academic and Honors – Students will study the relationships that exist when an altitude is drawn to the hypotenuse of a right triangles. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – Calculators will be used when deemed necessary. Geometer’s Sketchpad if available Ruler, Protractor, Compass 21st Century Themes – Environmental Literacy Economic, Business Literacy 21st Century Skills – Creativity & Innovation Critical Thinking & Problem Solving Information Literacy Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 21 Domain Standards: G.SRT.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. G.SRT.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. G.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.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. G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.SRT.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. Unit Essential Questions How is similarity of geometric figures applied and verified? Unit Enduring Understandings Similar geometric figures have proportional attributes. What is the difference between similarity and congruence? What is the relationship between corresponding side lengths of similar figures? The measures of geometric figures can be calculated and analyzed using a variety of strategies, tools, and technologies. Analyzing geometric relationships develops reasoning and justification skills. Terminology: Side-Splitter, Angle Bisector, Median, Altitude, Centroid, Dilation, Scale Factor, Similarity Ratio, Center of Dilation, Geometric Mean, Postulate, Theorem, Proportion Goals/Objectives Students will be able to identify and/or verify congruent and similar figures apply proportionality of corresponding parts of similar figures apply the properties of similar figures to area problems Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Instructional Strategies Whole Group Lessons Small Group Explorations In-class group work Independent Practice Homework Class Discussion Formative: White Board Response Black Board Race Ticket Out Think/Pair/Share Open Discussion Homework Quizzes* Notebook Quizzes* Activities Textbook Activities 22 apply properties of similarity and scale factors to unit conversions (inches to feet vs. sq. inches to sq. feet) Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding Think Pair Share use algebraic and/or geometric properties to measure indirectly Summative: Multi-Lesson Quizzes Unit Test *based on unit or need Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument determine the positive geometric mean between two numbers use transformations to move figures, create designs, and demonstrate geometric properties Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 23 Unit 5 Overview Content Area – Mathematics Unit 5: Special Right Triangles and Trigonometry Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – Applications of the Pythagorean Theorem and its converse, Pythagorean triples and Pythagorean inequalities are explored. The special right triangles of 30-60-90 and 45-45-90 are also studied, as well as Sine, Cosine, and Tangent Ratios. Connections are made to similar triangles and proportionality of side lengths. Operations with Square Roots and Simplifying Square Roots, as well as Solving Proportions, (concepts from Algebra 1) are revisited in this unit. Academic and Honors - Vectors Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – Calculators will be used when deemed necessary. Geometer’s Sketchpad if available Ruler, Protractor, Compass 21st Century Themes – Global Awareness Environmental Literacy 21st Century Skills – Creativity & Innovation Critical Thinking & Problem Solving Flexibility & Adaptability Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 24 Domain Standards: G.SRT.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. G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.SRT.7: Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Unit Essential Questions What are the trigonometric ratios and how are they used to solve real world problems? How are the other facts and properties of triangles in general, related to trigonometry and the Pythagorean Theorem? Unit Enduring Understandings Right Triangles have special properties, making it such that all possible ratios among the side lengths are known quantities. We know so many facts about triangles, all of which must hold true as we solve problems and consider the appropriateness of our answers. Analyzing geometric relationships develops reasoning and justification skills. Terminology: SOHCAHTOA, Sine, Cosine, Tangent, Opposite Leg, Adjacent Leg, Hypotenuse, Trigonometric Ratio, Inverse, Pythagorean Theorem, Pythagorean Triple, Right Triangle, Right Angle, Acute Angle, Square Roots, Perfect Square, Rational, Irrational, Rationalize Goals/Objectives Students will be able to identify the types of right triangles state and apply the Pythagorean theorem and its converse to solve problems identify and apply common Pythagorean triples apply the properties of special right triangles Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Instructional Strategies Whole Group Lessons Small Group Explorations In-class group work Independent Practice Homework Class Discussion Activities Textbook Activities Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding Think Pair Share Formative: White Board Response Black Board Race Ticket Out Think/Pair/Share Open Discussion Homework Quizzes* Notebook Quizzes* define and apply trigonometric ratios to real Summative: Multi-Lesson Quizzes Unit Test *based on unit or need 25 world situations Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument use trigonometric functions to find the measures of angles of a right triangle Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 26 Unit 6 Overview Content Area – Mathematics Unit 6: Coordinate Geometry and Proofs Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – Students will build on their work with the Pythagorean Theorem in 8th grade to find distances and use a rectangular coordinate system to verify geometric relationships. Such relationships include properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. Students will continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. (Academic Geometry and Honors Geometry will also study conic sections in this unit) Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – calculators, as necessary. Geometer sketch pad, if available. Rulers, graph paper 21st Century Themes – Civic Literacy Global Awareness 21st Century Skills – Creativity & Innovation Critical Thinking & Problem Solving Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 27 Domain Standards: G-GPE.2: Derive the equation of a parabola given a focus and directrix. G.GPE.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). G.GPE.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). G.GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ Unit Essential Questions Unit Enduring Understandings How can the coordinate plane help us better understand the properties and theorems for geometric shapes? How can algebraic equations be used to solve real problems in engineering and design? The coordinate plane is a useful tool for visualizing and interpreting shapes in our physical environment. Algebraic representations are used to communicate and generalize patterns in geometry. Perpendicular and parallel lines are the basis for all building. Terminology: base angles of a trapezoid, consecutive angles, isosceles trapezoid, kite, mid-segment of a trapezoid, parallelogram, rectangles, rhombus, square, trapezoid Goals/Objectives Students will be able to Define, identify, and classify special types of quadrilaterals Use relationships among sides and among angles of parallelograms to prove the specific type of quadrilateral or to find missing measures in a quadrilateral Use relationships involving diagonals of parallelograms or Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Instructional Strategies Whole group instruction Small group exploration Independent Practice Class Discussion Formative : White Board response Black Board Race Ticket Out Think/Pair/Share Open Discussion (Homework Quizzes Notebook Quizzes – based on unit or need) Activities Text book activities Supplemental worksheets, Individual and group activities for: Skill development 28 Summative: transversals to prove the type of quadrilateral or to find missing measures. Multi-Lesson Quizzes Unit Test Word Problems Geometric representation of data Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument Identify figures in the coordinate plane by determining slope for parallel and perpendicular lines. Identify coordinates of special figures by using their properties. Find a point that divides a given segment on the coordinate plane into a specified ratio. Prove theorems using figures in the coordinate plane Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 29 Unit 7 Overview Content Area – Mathematics Unit 7: Circles: lines, angles, arc relationships Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – Students will prove basic theorems about circles, with particular attention to perpendicularity and inscribed angles, in order to see symmetry in circles and as an application of triangle congruence criteria. Other angle relationships will also be studied. Students will study relationships among segments on chords, secants, and tangents as an application of similarity. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – calculators, as necessary. Geometer sketch pad, if available. Rulers, graph paper, compass 21st Century Themes – Financial, Economic, Business and Entrepreneurial Literacy 21st Century Skills – Learning and Innovation Life and Career Skills Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 30 Domain Standards: G.C.1: Prove that all circles are similar. G.C.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. G.C.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. Unit Essential Questions Unit Enduring Understandings Why is it important to understand the unique relationships among the lines, segments, angles, and arcs in a circle? Analyzing geometric relationships develops reasoning and justification skills. Circles allow us to model a wide variety of natural phenomena. Terminology: arc, adjacent arc, center, central angle, chord, chord segment, circle circumference, circumscribed, common tangent, compound locus, concentric circles, congruent arcs, congruent circles, diameter, external segment, inscribed, inscribed angle, intercepted arc, major arc, minor arc, pi, point of tangency, radius, secant, secant segment, semicircle, tangent Goals/Objectives Students will be able to Identify central angles, major arcs, minor arcs, and semicircles, and find their measures Find arc lengths Recognize and use the relationships between arcs, chords, and diameters. Find measures of inscribed angles and inscribed polygons Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Instructional Strategies Whole group instruction Small group exploration Independent Practice Class Discussion Formative : White Board response Black Board Race Ticket Out Think/Pair/Share Open Discussion (Homework Quizzes Notebook Quizzes – based on unit or need) Activities Text book activities Supplemental worksheets, Individual and group activities for: Skill development Word Problems Geometric representation of data Use properties of tangents to 31 Summative: Multi-Lesson Quizzes Unit Test Research Project solve problems involving circumscribed polygons. Possible Short Research Project: Find measures of angles formed by lines intersecting on or inside a circle. Research the use of radar by an air traffic controller. Specifically relate parts of a circle to the responsibilities of the job. Find measures of angles formed by lines intersecting outside the circle. Or Find measures of segments that intersect in the interior or exterior of a circle. Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument Research the use of sonar in submarines. Specifically relate parts of a circle to the responsibilities of the job. Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 32 Unit 8 Overview Content Area – Mathematics Basic Geometry Unit 8: Circles on the Coordinate Plane Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw a graph in the coordinate plane and apply techniques for solving quadratic equations to determine intersections between lines and circles or parabolas and circles, as well as the points of intersection between two circles. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – calculators, as necessary. Geometer sketch pad, if available. Rulers, graph paper 21st Century Themes – Financial, Economic, Business and Entrepreneurial Literacy 21st Century Skills – Life and Career Skills Technology Skills Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 33 Domain Standards: G.GPE.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. G.GPE.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). G.MG.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). A-REI.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. Unit Essential Questions Unit Enduring Understandings What does the graph of an equation in two variables look like? Graphing equations provides a tangible way to understand abstract concepts. How can we determine the shape of a graph from its equation? It is possible to know the shape of a graph by analyzing its equation. Terminology: center, chord, equation of a circle, equation of a parabola in vertex form, external segment, common tangent, compound locus, point(s) of intersection, point of tangency, radius, secant, tangent. Goals/Objectives Students will be able to use the Pythagorean Theorem to derive the equation of a circle of a given radius and center write the equation of a circle in the coordinate plane from a given center and radius -identify the center and radius of a circle given its equation use the equation of a circle to prove geometric theorems Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Instructional Strategies Whole group instruction Small group exploration Independent Practice Class Discussion Formative : White Board response Black Board Race Ticket Out Think/Pair/Share Open Discussion (Homework Quizzes Notebook Quizzes – based on unit or need) Activities Text book activities Supplemental worksheets, Individual and group activities for: Skill development Word Problems 34 Summative: Multi-Lesson Quizzes such as the location of a given point (interior, exterior, or on the circle) or the type of polygon inscribed in a circle Unit Test Geometric representation of data Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument solve a system of equations by drawing a graph in the coordinate plane and applying techniques for solving quadratic equations to determine intersections between lines and circles or parabolas and circles, as well as the points of intersection between two circles. use the equation of a circle to solve real world problems involving such topics as meteorology, aerodynamics, and search and rescue Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 35 Unit 9 Overview Content Area – Mathematics Unit 9: Geometry in Three Dimensions Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – Students’ experience with two-dimensional and three-dimensional objects is extended to include explanations of circumference and perimeter of the base of three dimensional objects in order to develop surface area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – 21st Century Themes – Financial, Economic, Business and Entrepreneurial Literacy 21st Century Skills – Life and Career Skills Technology Skills Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 36 Domain Standards: G.GMD.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. G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ G.GMD.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. G.MG.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).* G.MG.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★ G.MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★ Unit Essential Questions Unit Enduring Understandings Conservation of surface area Conservation of volume Can two different three-dimensional objects have the same surface area? Can two different three-dimensional objects have the same volume? Terminology: altitude, axis, base Cavalieri’s Principal, center, chord, circular cone, cone, congruent solids, corner view, cross section, cube, cylinder, diameter, edge, face, great circle, height, hemisphere, lateral area, lateral edge, lateral face, net, oblique cone, oblique cylinder, oblique prism, perspective view, Platonic Solids, polyhedron, prism, pyramid, radius, regular polyhedron, regular prism, regular pyramid, right cone, right cylinder, right prism, scale factor, similar solids, slant height, slice, solid, sphere, surface area, tangent, triangular prism, vertex, volume Goals/Objectives Students will be able to - Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Use top, front, side, and corner views of threedimensional solids to make models and describe and draw cross sections and other slices of three dimensional figures to give a better understanding of art and architecture. Instructional Strategies Whole group instruction Small group exploration Independent Practice Class Discussion Formative : White Board response Black Board Race Ticket Out Think/Pair/Share Open Discussion (Homework Quizzes Notebook Quizzes – based on unit or need) Draw three dimensional figures on isometric dot paper, Activities Text book activities Supplemental worksheets, Individual and group activities for: 37 make two dimensional nets for three-dimensional solids, and find surface areas for these solids. Skill development Word Problems Geometric representation of data Be able to distinguish between Possible Project: a right prism or pyramid and Students will create package for a fictitious an oblique prism or pyramid. product. In preparation, have students examine Understand Cavalieri’s household packaging for cereal, drink mix, Principle. powdered laundry detergent, or any other cardboard package. It is helpful to be able Find the volume for any three to show students such unique packaging as the Toblerone triangular prism, or a dimensional solid. trapezoidal prism (print cartridge package). Find the surface area and volume for a sphere. Identify congruent or similar solids by identifying their various properties. Assign or choose groups of no more than four. Assign or choose from a variety of household products. Depending on the level of the class, groups will either Draw a net, make the package, find the surface area and volume or Draw a net and make the package. - - The project can be expanded to include creative marketing of the package. It can be done as a competition. 38 Summative: Multi-Lesson Quizzes Unit Test Project Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 39 Unit 10 Overview Content Area – Mathematics Unit 10: Applications of Geometric Probability Target Course/Grade Level – Geometry for Grades 9 and 10 Unit Summary/Rationale – Probability concepts which were introduced in the middle grades are expanded to include the language of set theory, and the ability to compute and interpret theoretical and experimental probabilities for compound events. Students will also expand prior knowledge to incorporate mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology Integration – Calculator and iPad, projector and/or SMART Board as available and/or appropriate 21st Century Themes – Global Awareness Environmental Literacy Financial, Economic, Business and Entrepreneurial Literacy 21st Century Skills – Critical Thinking & Problem Solving Communication Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning 40 Domain Standards: S.CP.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S.CP.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S.CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S.CP.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S.CP.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.7: Apply the Addition Rule, P (A or B) = P (A) + P (B) – P (A and B), and interpret the answer in terms of the model. Unit Essential Questions Unit Enduring Understandings How can probability be modeled by plane figures? How can number lines be used to model time or distance? Some probability situations can be modeled using one, two or three dimensional figures. Terminology: compound probability, conditional probability, geometric probability, mutually exclusive events, theoretical probability, experimental probability 41 Goals/Objectives Students will be able to - Recommended Learning Activities/Instructional Strategies Evidence of Learning (Formative & Summative) Use segments and area models to find probabilities of events Instructional Strategies Whole group instruction Small group exploration Independent Practice Class Discussion Formative : White Board response Black Board Race Ticket Out Think/Pair/Share Open Discussion (Homework Quizzes Notebook Quizzes – based on unit or need) Understand independence and conditional probability and use them to interpret data. Use rules of probability to compute probabilities of compound events in a uniform probability model. Activities Text book activities Supplemental worksheets, Individual and group activities for: Skill development Word Problems Geometric representation of data Possible Project: Students will create a carnival game based on geometric probability. Students will determine various probabilities so that the carnival has an advantage. Summative: Multi-Lesson Quizzes Unit Test Alternative or project-based assessments will be evaluated using a teacherselected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry Prentice Hall Mathematics Geometry, 2004 – Academic Geometry Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry Websites: 42 Mathematics: Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of 43 statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, 44 including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 45 Common Core Standards (Bold Apply to Geometry) Congruence G-CO 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. 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. 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. 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. 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. 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. Similarity, Right Triangles, and Trigonometry G-SRT 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. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 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. Define trigonometric ratios and solve problems involving right triangles. 46 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. ★ 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 nonright triangles (e.g., surveying problems, resultant forces). Modeling with Geometry G-MG Apply geometric concepts in modeling situations. 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).★ 3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★ Geometric Measurement and Dimension G-GMD 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. ★ 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. Expressing Geometric Properties with Equations G-GPE 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. 3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. 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). 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. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★ Circles G-C 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. 47 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. 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. Conditional Probability and the Rules of Probability S-CP Understand independence and conditional probability and use them to interpret data. 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). 2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. 3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. 5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Use the rules of probability to compute probabilities of compound events in a uniform probability model. 6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. 7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. 8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model 9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Using Probability to Make Decisions S-MD Calculate expected values and use them to solve problems. 1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. 2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. 3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. 4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Use probability to evaluate outcomes of decisions. 5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a lowdeductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. 6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Resources for State Assessments 48