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POMPTON LAKES SCHOOL DISTRICT GEOMETRY FOUNDATIONS COURSE OF STUDY June 2012 Submitted By The Mathematic Department Dr. Paul Amoroso, Superintendent Mr. Vincent Przybylinski, Principal Mr. Anthony Mattera, Vice Principal Frances J. Macdonald, District Mathematics Supervisor BOARD MEMBERS Mr. Jose A. Arroyo, Mrs. Catherine Brolsma, Mr. Shawn Dougherty, Mrs. Nancy Lohse-Schwartz, Mr. Garry Luciani, Mr. Carl Padula, Mr. Tom Salus, Mrs. Stephanie Shaw, Mr. Timothy Troast, Jr. Unit Overview Content Area: MATH UNIT 1 Unit Title: LINES AND ANGLES Target Course/Grade Level: GEOMETRY FOUNDATIONS 9-10 Unit Summary: Students learn about points, lines and planes, the building blocks of Geometry. Line segments, rays, angles, polygons, parallel lines and perpendicular lines are also introduced in this unit. Students explore congruent segments and angles and learn to construct them with a compass and straightedge. Students expand on their knowledge of the Pythagorean theorem to master the distance formula and use the midpoint formula to find the midpoint of a segment. Students also compute the perimeter of a given polygon. Students identify the special angle relationships that result when a transversal intersects parallel lines. Students solve problems by writing linear equations and use slope to determine whether two lines are parallel, perpendicular or neither. Primary interdisciplinary connections: Science, Business, Economics, History, Art 21st century themes: Mathematical Literacy Unit Rationale: The content and skills acquired in the unit are the tools necessary to study polygons. The terms and notation will help students progress throughout the different geometric topics. Learning Targets Standards: 4.2 (Geometry and Measurement) All students will develop spatial sense and the ability to use geometric properties, relationships and measurement to model, describe, and analyze phenomena. Content Statements: Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric modeling and measurement can help us describe and interpret our physical environment and to solve problems. CPI # 4.1.12C.1 Cumulative Progress Indicator (CPI) Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits. 4.2.12A.5 Perform basic geometric constructions using a variety of methods (e.g. straightedge and compass, patty/tracing paper, or technology). 4.5B.1 Use communication to organize and clarify their mathematical thinking. 4.5B.2 Communicate their mathematical thinking coherently and clearly to peers, teachers, and others both orally and in writing. 4.5B.3 Analyze and evaluate the mathematical thinking and strategies of others. 4.5B.4 Use the language of mathematics to express mathematical ideas precisely. 4.5C.1 Recognize recurring themes across mathematical domains (e.g. patterns in number, algebra, and geometry.) 4.5C.2 Use connections among mathematical ideas to explain concepts (e.g. two linear equations have a unique solution because the lines they represent intersect at a single point). 4.5D.1 Recognize that mathematical facts, procedures, and claims must be justified. 4.5D.2 Use reasoning to support their mathematical conclusions and problem solutions. 4.5D.5 Make and investigate mathematical conjectures. Unit Essential Questions: What is the meaning and representation for geometric terms? What is the relationship between slope and rate of change? Unit Enduring Understandings: Students will learn all the definition and how to identify and label all geometric terms. Students will be able to calculate slope and express its meaning and relationship to a rate of change of a quantity. Unit Learning Targets: Students will ... Solve problems by making models of points, lines, planes, and angles. Find the slope of various lines and interpret its meaning in terms of a rate of change. Calculate the midpoint and distance of a segment. Explain the relationship between different angle pairs. Evidence of Learning Summative Assessment: Students will work collaboratively to complete the task of solving real life problems involving angles and lines. This task consists of group or individual questioning, class discussions, teacher developing standardized tests/quizzes. Also, students will complete a written assignment to explain their findings. Equipment needed: graphing calculator, Smart Board, computer access, protractors, rulers Teacher Resources: Math websites, textbooks, and resource books Formative Assessments: Discussions Journal Entries Smart board presentation Evaluation Questions Tests/Quizzes Create/draw geometric models Lesson Plans Lesson Timeframe Lesson 1 Points, Lines, and Planes Lesson 2 Linear Measure and Precision Lesson 3 Distance and Midpoints Lesson 4 Angle Measure Lesson 5 Angle Relationships Lesson 6 Polygons Lesson 7 Parallel Lines and Transversals Lesson 8 Angles and Parallel Lines Lesson 9 Slopes of Lines Lesson 10 Equations of Lines Lesson 11 Proving Lines Parallel Lesson 12 Perpendiculars and Distance 1 Day 2 Days 2 Days 2 Days 2 Days 2 Days 2 Days 2 Days 2 Days 2 Days 2 Days 2 Days Teacher Notes: Students can seek input from their peers and teachers throughout collaborative assignments and activities. Curriculum Development Resources The completed Curriculum Design Template shows how this unit is situated within this district’s Math Course. Unit Overview Content Area: MATH UNIT 2 Unit Title: TRIANGLES Target Course/Grade Level: GEOMETRY FOUNDATIONS 9-10 Unit Summary: In this unit, students prove triangles congruent and similar using various methods. Students classify triangles angles according to their angles or sides and apply the angle sum theorem and the exterior angle theorem. Special segments of triangles including bisectors, medians, and altitudes are identified and explored. Students apply properties of inequalities relating to the measures of angles and sides of a triangle and then extend those properties to two triangles. Students learn to solve right triangles via various methods including Pythagorean Theorem, trigonometric ratios, and geometric mean. Students also use the Law of Sines and Cosines to solve non-right triangles. Primary interdisciplinary connections: Wood shop, Science, Art, 21st century themes: Mathematical Literacy Unit Rationale: Students’ knowledge of properties of triangles in essential in a discovery of properties of other polygons including quadrilaterals. This knowledge will help them in higher level mathematics courses including pre-calculus and calculus. Architects, surveyors, and civil engineers use trigonometric ratios in their work. Learning Targets Standards: 4.2 (Geometry and Measurement) All students will develop spatial sense and the ability to use geometric properties, relationships and measurement to model, describe, and analyze phenomena. Content Statements: Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric modeling and measurement can help us describe and interpret our physical environment and to solve problems. CPI # Cumulative Progress Indicator (CPI) Use geometric models to represent real-world situations and objects and to solve 4.2.12A.1 problems using those models (e.g. use Pythagorean theorem to determine if an object can fit through a door). 4.2.12A.3 Apply the properties of geometric shapes. 4.2.12A.5 Perform basic geometric constructions using a variety of methods. 4.2.12E.1 Use techniques of indirect measurement to represent and solve problems. 4.5B.1 Use communication to organize and clarify their mathematical thinking. 4.5B.2 Communicate their mathematical thinking coherently and clearly to peers, teachers, and others both orally and in writing. 4.5B.3 Analyze and evaluate the mathematical thinking and strategies of others. 4.5B.4 Use the language of mathematics to express mathematical ideas precisely. 4.5C.1 Recognize recurring themes across mathematical domains (e.g. patterns in number, algebra, and geometry.) 4.5C.2 Use connections among mathematical ideas to explain concepts (e.g. two linear equations have a unique solution because the lines they represent intersect at a single point). 4.5D.1 Recognize that mathematical facts, procedures, and claims must be justified. 4.5D.2 Use reasoning to support their mathematical conclusions and problem solutions. 4.5D.5 Make and investigate mathematical conjectures. Unit Essential Questions: What is the difference between congruent triangles and similar triangle? What method(s) is most appropriate for solving a right triangle or non-right triangle? Unit Enduring Understandings: Students will compare the sides and angles of triangles to determine whether they are congruent or similar. Student will use their knowledge of various methods learning included Pythagorean Theorem, geometric mean, trigonometry, Law of Sines and Cosines to solve a triangle. Unit Learning Targets: Students will ... Classify triangles according to their angles and sides. Determine whether triangles are congruent or similar. Prove triangles are congruent using various postulates and theorems for congruence. Identify special segments in triangles including median, angle bisector, perpendicular bisector, and altitude. Use triangle inequality theorem to determine if numbers can be lengths of a side of triangle. Determine relationships between angles and sides of a triangle. Use proportions to solve problems. Use relationships between proportional parts of triangles. Use appropriate methods to solve right triangles including geometric mean, Pythagorean Theorem, special right triangles, and trigonometry. Solve non-right triangles using the Law of Sines and Law of Cosines. Use Converse of Pythagorean Theorem to determine if a triangle is a right triangle. Evidence of Learning Summative Assessment: Students will work collaboratively to complete the task of solving real life problems involving triangles. This task consists of group or individual questioning, class discussions, teacher developing standardized tests/quizzes. Also, students will complete a written assignment to explain their findings. Equipment needed: Graphing Calculator, Smart Board, Computer Access, Protractors, Rulers Teacher Resources: Math Websites, textbooks, and resource books Formative Assessments: Discussions Journal Entries Smart board presentation Evaluation Questions Tests/Quizzes Create/draw congruent and similar triangles Lesson Plans Lesson Timeframe Lesson 1 1 Day Classify Triangles Lesson 2 2 Days Angles of Triangles Lesson 3 2 Days Congruent Triangles Lesson 4 2 Days Proving Congruence-SSS, SAS Lesson 5 2 Days Proving Congruence-ASA, AAS Lesson 6 2 Days Isosceles Triangles Lesson 7 3 Days Bisectors, Medians, and Altitudes Lesson 8 2 Days Inequalities and Triangles Lesson 9 2 Days The Triangle Inequality Lesson 10 1 Day Proportions Lesson 11 1 Day Similar Polygons Lesson 12 2 Days Similar Triangles Lesson 13 2 Days Parallel Lines and Proportional Parts Lesson 14 2 Days Parts of Similar Triangles Lesson 15 1 Day Geometric Mean Lesson 16 2 Days The Pythagorean Theorem and Its Converse Lesson 17 2 Days Special Right Triangles Lesson 18 2 Days Trigonometry Lesson 19 1 Day Angle of Elevation and Depression Lesson 20 2 Days The Law of Sines Lesson 21 2 Days The Law of Cosines Teacher Notes: Students can seek input from their peers and teachers throughout collaborative assignments and activities. Curriculum Development Resources: The completed Curriculum Design Template shows how this unit is situated within this district’s Math Course. Unit Overview Content Area: MATH UNIT 3 Unit Title: QUADRILATERALS AND CIRCLES Target Course/Grade Level: GEOMETRY FOUNDATIONS 9-10 Unit Summary: In this unit, students explore polygons by investigating the exterior and interior angles of polygons. Students learn to recognize and apply the properties of parallelograms, rectangles, rhombi, squares, and trapezoids. Students explore the different types of transformations: reflections, translations, rotations, and dilations. They learn to identify, draw, and recognize figures that have been transformed. Students identify the parts of a circle and solve problems involving circumference and area. Arc and angle measures and the measures of segments within a circle are explored. Equations of circles are derived and applied. Primary interdisciplinary connections: Art, Science 21st century themes: Mathematical Literacy Unit Rationale: The area of quadrilaterals is necessary for calculating the surface area of prisms. Understanding the properties is essential to success in engineering, architecture, and design. Circles are needed to understanding spheres. Learning Targets Standards: 4.2 (Geometry and Measurement) All students will develop spatial sense and the ability to use geometric properties, relationships and measurement to model, describe, and analyze phenomena. Content Statements: Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric modeling and measurement can help us describe and interpret our physical environment and to solve problems. CPI # 4.2.12B.1 Cumulative Progress Indicator (CPI) Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic representation, and conversely, determine whether and how one representation can be transformed to another by a transformation or sequence of transformations. 4.2.12B.4 Generate and analyze iterative geometric patterns. 4.2.12C.3 Find an equation of circle given its center and radius and, given an equation of a circle in standard form, find its center and radius. 4.2.5.D.1 Select and use appropriate units to measure angles and areas. 4.2.5.D.4 Use measurements and estimates to describe and compare phenomena. 4.5B.1 Use communication to organize and clarify their mathematical thinking. 4.5B.2 Communicate their mathematical thinking coherently and clearly to peers, teachers, and others both orally and in writing. 4.5B.3 Analyze and evaluate the mathematical thinking and strategies of others. 4.5B.4 Use the language of mathematics to express mathematical ideas precisely. 4.5C.1 Recognize recurring themes across mathematical domains (e.g. patterns in number, algebra, and geometry.) 4.5C.2 Use connections among mathematical ideas to explain concepts (e.g. two linear equations have a unique solution because the lines they represent intersect at a single point). 4.5D.1 Recognize that mathematical facts, procedures, and claims must be justified. 4.5D.2 Use reasoning to support their mathematical conclusions and problem solutions. 4.5D.5 Make and investigate mathematical conjectures. Unit Essential Questions: What are the similarities/differences of special quadrilaterals? What are the parts of a circle? What are the relationships betweens lines and circles? Unit Enduring Understandings: Trapezoid has only pair of parallel sides; squares and rectangles have congruent diagonals and four right angles; squares and rhombi have four congruent sides. Students will be able to identify and label diameter, radius, chord, secant, tangent, and center. Students will use various theorems involving chords, secants, and tangents to solve problems. Unit Learning Targets: Students will ... Find the interior and exterior angle sum for different convex polygons. Classify and compare different quadrilaterals based on their properties. Use the different properties of quadrilaterals to solve problems. Use slope, distance, and midpoint formulas to justify what type of quadrilateral it is. Create different transformations of figures. Identify parts of a circle. Determine relationships between segments and lines and circles. Identify the relationship of different angles to a circle. Write equations of circles and use to graph circles in coordinate plane. Evidence of Learning Summative Assessment: Students will work collaboratively to complete the task of solving real life problems involving quadrilaterals, circles, and transformations. This task consists of group or individual questioning, class discussions, teacher developing standardized tests/quizzes. Also, students will complete a written assignment to explain their findings. Equipment needed: Graphing calculator, Smart Board, Computer Access, Protractors, Rulers Teacher Resources: Math Web Sites, textbooks, and resource books Formative Assessments: Discussions Journal Entries Smart board presentation Evaluation Questions Tests/Quizzes Create/draw transformations Lesson Plans Lesson Timeframe Lesson 1 1 Day Angles of Polygons Lesson 2 2 Days Parallelograms Lesson 3 2 Days Tests for Parallelograms Lesson 4 1 Day Rectangles Lesson 5 2 Days Rhombi and Squares Lesson 6 2 Days Trapezoids Lesson 7 2 Days Reflections Lesson 8 1 Day Translations Lesson 9 2 Days Rotations Lesson 10 2 Days Dilations Lesson 11 1 Day Circles and Circumference Lesson 12 2 Days Angles and Arcs Lesson 13 2 Days Arcs and Chords Lesson 14 2 Days Inscribed Angles Lesson 15 2 Days Tangents Lesson 16 2 Days Secants, Tangents, and Angle Measures Lesson 17 2 Days Special Segments in a Circle Lesson 18 1 Day Equations of Circles Teacher Notes: Students can seek input from their peers and teachers throughout collaborative assignments and activities. Curriculum Development Resources: The completed Curriculum Design Template shows how this unit is situated within this district’s Math Course. Unit Overview Content Area: MATH UNIT 4 Unit Title: AREA AND VOLUME Target Course/Grade Level: GEOMETRY FOUNDATIONS 9-10 Unit Summary: Area and volume can be used to analyze real-world situations. In this unit, you will learn about formulas used to find the areas of two-dimensional figures and the surface and the surface areas and volumes of three-dimensional figures. Primary interdisciplinary connections: Science, Business, Art 21st century themes: Mathematical Literacy Unit Rationale: The Knowledge about Area and Volume that students gain while studying this unit will be important to them in the future mathematics courses, in physics, and in many careers that they might choose. Learning Targets Standards: 4.2 (Geometry and Measurement) All students will develop spatial sense and the ability to use geometric properties, relationships and measurement to model, describe, and analyze phenomena. Content Statements: Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric modeling and measurement can help us describe and interpret our physical environment and to solve problems. CPI # Cumulative Progress Indicator (CPI) Use geometric models to represent real-world situations and objects and to solve 4.2.12A.1 problems using those models. 4.2.12A.2 Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views.) 4.2.12E.2 Develop and apply strategies and formulas for finding perimeter and area of squares rectangles. Unit Essential Questions: What is the difference between Area and Volume? What are the units of measure of Area and Volume? Is the Area used to find the Volume of a solid figure? Unit Enduring Understandings: Area measures the surface of a two dimensional figure and the volume shows how much a sold figure can hold. The unit of measure for Area is square units and Volume is cubic units. Yes because the volume of any solid figure is based on the area of its base and height. Unit Learning Targets: Students will ... Solve problems by making a model. Find the area of various 2-dimensional figures such as parallelograms, rectangles, trapezoid, rhombus, squares, triangles, circles, and regular polygons. Find the lateral areas and surface areas and volumes of various solid figures such as rectangular prisms, cubes, pyramids, cones, and spheres Evidence of Learning Summative Assessment: Students will work collaboratively to complete the task of solving real life problems involving area and volume. This task consists of group or individual questioning, class discussions, teacher developing standardized tests/quizzes. Also, students will complete a written assignment to explain their findings. Equipment needed: Graphing calculator, Smart board, and computer access. Teacher Resources: Math web sites, textbooks and resource books. Formative Assessments: Discussions Journal Entries Smart board presentation Evaluation Questions Tests/Quizzes Create/draw/fold 3-dimensional figures. Lesson Plans Lesson Timeframe Lesson 1 Areas of Parallelograms 1 Day Lesson 2 Areas of Triangles, Trapezoids and Rhombi. 2 Days Lesson 3 Areas of regular polygons and circles. 2 Days Lesson 4 Areas of irregular figures. 2 Days Lesson 5 Three dimensional figures 1 Day Lesson 6 Surface Areas of prisms, cylinders, pyramids 2 Days Lesson 7 Volumes of Cones and Spheres 2 Days Lesson 8 Volumes of prisms and cylinders. 2 Days Lesson 9 Volumes of pyramids and cones. 2 Days Lesson 10 Volumes of Spheres 1 Day Teacher Notes: Students can seek input from their peers and/or teachers before developing their projects. Curriculum Development Resources The completed Curriculum Design Template shows how this unit is situated within this district’s Math Course.