Geometry Course Number 222

Transcription

1 Curriculum Guide for Geometry Course Number 222 Department of Mathematics Belvidere High School

2 Department Philosophy The principle reason for studying mathematics is to learn how to apply mathematical thought to problem solving. Students are expected to develop facility in performing the fundamental operations that are associated with each course and to acquire a set of meaningful concepts that they can use effectively to solve problems. The ultimate goal of the department is to provide students with a facility to deal with mathematical functions in daily life and for post-secondary education. Course Overview Geometry is a logical system of thought and proof that deals with visualization and also links algebraic concepts to geometric figures. This course extends understanding of theory and practice through analysis, reasoning, deduction, and problem solving. The goal is to help students appreciate the geometrical value of the universe as we know it while examining the relationships in geometry and applying it to other areas of math. This course has been designed to cover the traditional Euclidean Geometry and provides a thorough preparation for the geometry portion of the SAT and HSPA.

3 Course Description & Course Proficiency COURSE TITLE: Geometry COURSE # 222 TOTAL CREDIT: 5 COURSE LENGTH: 36 weeks PERIODS PER WK: 5 GRADE LEVEL: PREREQUISITE(S): Algebra 1, teacher recommendation, and successful application for Honors Program. Pursuant to the High School Graduation Act (NJSA 18A: 7, et. Seq.), expectations for this course of study are outlined below OVERVIEW: Geometry is a logical system of thought and proof that deals with visualization and also links algebraic concepts to geometric figures. This course extends students' understanding of theory and practice through analysis, informal reasoning, deduction, the development of a formal proof, problem solving and enrichment assignments. TEXTBOOK: Geometry, Glencoe 2004 Online at: User Name: geo Password: v7tr2swagu SUPPLEMENTARY MATERIAL: Teacher prepared skills review and practice materials. PROFICIENCIES: Successful completion of this course will require the student: Learn the language of geometry. 2. Organize the ideas of geometry into a logical system of thought and reasoning. 3. Learn the meaning of geometric proof by deductive reasoning for the following: a. triangles b. parallel and perpendicular lines c. polygons other than triangles d. circles, angles and arcs, e. similar polygons. 4. Apply the concepts and properties of geometric figures and integrate with the coordinate plane. 5. Learn how to solve problems involving: a. areas of polygons b. measurement of angles and arcs c. proportion and proportional line segments d. similarity. 6. Understand the Pythagorean Theorem, special right triangles, trigonometric ratios. 7. Learn to construct various geometric figures. 8. Area of plane figures. 9. Apply coordinate geometry. 10. Surface area and volume of solids. 11. Complete daily homework assignments.

4 STUDENT ACHIEVEMENT: STUDY STRATEGIES: Students should review materials given in class on a daily basis to reinforce skills and concepts presented. It is expected for students to review Algebra skills throughout course. HOMEWORK EXPECTATIONS: Each homework assignment will be worth up to 5 points. Late work will not be given any credit except in the case of an extended absence. Students must make arrangements with teacher immediately upon return to school in order to receive credit. Parents may be notified by phone or when student does not have assignment. PROCEDURES FOR MAKING UP WORK: The student is responsible for any make-up work missed from being out of class. This includes any homework, class work, tests/quizzes, and updating of notebooks. Any work that is not made up following above procedures will result in a grade of zero. Board policy applies. MAJOR PROJECTS TO EXPECT: Triangle Project 2 nd marking period. Plane or Solid Figure Project 4 th marking period. MEASUREMENTS OF STUDENT ACHIEVEMENT: The measurement of student achievement will be done through various evaluative criteria, which will include: 1. Quizzes Each quiz will count as points. There will be at least one quiz given per chapter/unit. These quizzes will be announced. Except in the case of an extended absence, if a student is absent on the day a quiz is given, the quiz must be taken on the day he returns. 2. Mini Quizzes Each mini quiz will count as 1-20 points. Mini quizzes may or may not be announced. They will occur periodically to check immediate understanding of a skill or topic. 3. Tests Each test will count as 100 points. Tests will be given at the end of each chapter/unit and will be announced at least three (3) days in advance. There will be a review prior to each test. Except in the case of an extended absence, if a student is absent on the day a test is given, the test must be taken on the day he returns. 4. Mid-term & Final Examinations- A mid-term will be given midway through the school year and a final at the end of the year. These two grades will be worth 20% of the student s final grade. PURPOSE AND METHODS OF ASSESSMENT: Tests and quizzes are given to accurately assess each student s skill level and understanding of concepts. CAREER OBJECTIVE: To help students appreciate the geometrical value of the universe as we know it. To help students see interrelationships between geometry and other areas of mathematics, particularly: arithmetic, algebra, trigonometry and coordinate geometry. ADDITIONAL NOTES: 1. Regular attendance at school is required of all students by the laws of the State of New Jersey. Failure to attend on a regular basis may result in poor achievement and/or loss of credit as per Board of Education Policy and as stated in the Student Handbook. 2. This list must be returned, signed by parent or guardian, no later than the last day in September. Student Signature Teacher Signature Parent/Guardian Signature Parent/Guardian Updated 05/07/2010

5 Duration of Unit Course Content Weekly Curriculum Map: Geometry Content 3 wks Point, Lines, Planes, and Angles 3 wks Reasoning and Proof 3 wks Parallel and Perpendicular Lines 4 wks Triangle Congruence 2 wks Properties and Attributes of Triangles New Jersey Common Core Standards G-CO: 1, 2, 4, 5, 12 A-SSE: 1 A-CED: 4 G-GPE: 7 A-REI: 1 G-CO: 9, 10 A-CED: 1 G-CO: 1, 9, 12 G-GPE: 5 G-CO: 6,7,8,10 G-SRT: 5 G-MG: 3 G-GPE: 4,5,7 G-CO: 9, 10, 12 G-SRT: 4, 6, 8 G-C: 3 G-MG: 2,3 3 wks Quadrilaterals G-CO: 11 G-GPE: 5 G-MG: 3 G-SRT: 5 Assessment class discussion, homework, quiz, test class discussion, homework, quiz, test class discussion, homework, quiz, test class discussion, homework, quiz, test class discussion, homework, quiz, test class discussion, homework, quiz, test 1 wk Review / Testing Mid-term Exam Resources New Jersey Geometry Model Course Content- Draft 11/6/08 Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press 3wks Similarity G-SRT:2 G-MG: 3 G-C: 1 G: SRT: 1, 3, 4, 5 G-CO: 2 G-GPE: 6 class discussion, homework, quiz, test Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press

6 3 wks Right Triangles and Trigonometry G-SRT: 4, 6, 7, 8, 10, 11 1 wk Transformations G-CO: 3, 4 G-SRT 1 3wks Areas of Polygons and Circles 2 wks Surface Area and Volume A-SSE: 1 A-CED: 4 G-GPE:7 G-GMD: 1 G-MG: 3 G-SRT:9 S-CP:1 G-GMD: 1,2,3,4 G-MG: 1,2 4 wks Circles G-C:2,3,4,5 G-CO: 13 class discussion, homework, quiz, test class discussion, homework, quiz, test class discussion, homework, quiz, test class discussion, homework, quiz, test class discussion, homework, quiz, test 1 wk Review/Testing Final Exams Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key Curriculum Press

7 Unit I. Points, Lines, Planes, and Angles State Standard Area of Concentration: Mathematics/High School-Geometry Unit Summary: Identify points, lines, planes, angles, and their relationships. Measure segments, angles, and polygons using a variety of methods. Introduce the terms and symbols of geometry. Make conjectures about vertical angles, linear pairs of angles, midpoints, and distance. Explore these properties in the coordinate plane. Define the term polygon and many specific types of polygons. Practice using two tools of geometry, the compass and the straightedge. Unit Rationale: The three building blocks of geometry are points, lines, and planes. These terms remain undefined. A general description is used to give a sense of what is meant by point, line, and plane. Gaining an intuitive understanding of the meaning of these terms is essential to begin a study of geometry. The language of geometry is used to describe many real-world objects. Lines and angles are found all around us in nature. Knowledge of the terms in this unit will help students begin to appreciate the geometrical value of the universe as we know it. Standards: Congruence G-CO Seeing Structure in Expressions A-SSE Creating Equations A-CED Expressing Geometric Properties with Equations G-GPE Common Core #: Text: G-CO 1 1-1, 1-2, 1-3, 1-4, Common Core: 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 Make formal geometric constructions with a variety of tools and methods.(compass and straightedge, string, reflective devices, paper folding, dynamic geometry software, etc.) Copying a segment; copying an angle, bisecting a segment; bisecting an angle, and constructing perpendicular lines. A-SSE Interpret expressions that represent a quantity in terms of its context. A-CED Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. G-GPE Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G-CO Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G-CO 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., translations versus horizontal stretch). G-CO Given a geometric figure and rotation, reflection, or translations, draw the transformed figure Specify a sequence of transformations that will carry a given figure onto another. Essential Questions: How can geometric/algebraic relationships best be represented? What is the importance of definitions? Enduring Understanding: Definitions will improve mathematical reasoning by being able to illustrate and examine geometric concepts. Euclidean geometry begins with the three undefined

8 What is the necessity for undefined terms? What applications led to the development of coordinate systems? Instructional Focus: Identify, name, and draw points, lines, segments, rays, and planes. Apply basic facts about points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in space. Use length and midpoint of a segment. Construct midpoints and congruent segments. Name and classify angles. Measure and construct angles and angle bisectors. Identify adjacent, vertical, complementary, and supplementary angles. Find measures of pairs of angles. Apply formulas for perimeter, area, and circumference. Develop and apply the formula for midpoint. Use the Distance Formula and the Pythagorean Theorem to find the distance between two points. Identify reflections, rotations, and translations. Graph transformations in the coordinate plane. Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView terms of point, line, and plane. The coordinate system helps locate objects on a plane. Identifying locations on a map or globe using latitude and longitude in order to travel compares to coordinates in a plane. Review and apply the following Algebra 1 topics: Simplify algebraic expressions Solve linear equations Simplify absolute value expressions Operations with signed numbers Translate word problems into algebraic equations Evidence of Learning Formative Assessment: Study Guide and Intervention Worksheets Skills Practice and Practice Worksheets Portfolio or Journal Definition and Conjecture List Quizzes Geometry Activities: a. Modeling Intersecting Planes b. Midpoint of a Segment c. Modeling the Pythagorean Theorem d. Angle Relationships Mini-Project Intersecting Planes Pythagorean Puzzle Project: Finding Treasure with Coordinates Open-Ended Investigation: Vertical Angles Conjecture and the Linear Pair Conjecture Equipment needed: Text Chapter 1 Compass Straightedge Protractor Landscape of Geometry Video (Program 1 The Shape of Things & Program 3 Lines that Cross)

9 Unit 2. Reasoning and Proof State Standard Area of Concentration: Mathematics/High School-Geometry Mathematics/High School-Algebra Unit Summary: Use reasoning to make conjectures and prove conjectures. If a conjecture is false find counterexamples. If a conjecture is true, verify using informal and formal proofs. Determine the truth values of compound statements and construct truth tables. Analyze conditional statements and write related conditionals. Introduce terms postulates and theorems. Algebraic properties of equality are applied to geometry. Write formal and informal proofs proving segment and angle relationships. Unit Rationale: Thinking logically is an important skill for daily living. Exploring different methods of reasoning helps us to think through situations logically. The ideas of geometry are organized into a logical system of thought and reasoning. Geometry uses and applies these methods to solve problems. Many different professions rely on these reasoning skills. For example, doctors use reasoning to diagnose and treat patients and meteorologists use patterns to make predictions. Topics in this unit will help develop clean thinking and the ability to weigh an argument critically and impartially. Standards: Reasoning with Equations and Inequalities A-REI Congruence G-CO Seeing Structure in Expressions A-SSE Creating Equations A-CED Common Core: Text: Cumulative Progress Indicators: A-REI Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. G-CO 9 2-6, 2-7 Prove theorems about lines and angles. G-CO Prove theorems about triangles. A-CED Create equations in one variable and use them to solve problems. Essential Questions: How can mathematical reasoning help you make generalizations? How do you know when you have proved something? Instructional Focus: Enduring Understanding: Reasoning allows us to make conjectures and prove conjectures. Many professions rely on reasoning and logic to solve problems and reach valid conclusions. Use inductive reasoning to identify patterns and make Review the following Algebra 1 topics: conjectures. Identify algebraic properties of equality Find counterexamples to disprove conjectures. Solve linear equations; with focus on fractions Identify, write, and analyze the truth value of conditional statements. (Optional) Write the inverse, converse, and contrapositive of a conditional statement. (Optional) Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Apply algebraic properties Evidence of Learning

10 Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView Equipment needed: Text Chapter 2 Patty Paper Protractor Scissors Rulers Formative Assessment: Portfolio or Journal Definition and Conjecture List Quizzes Geometry Activities: a. If-then statements b. Matrix Logic c. Right Angles Mini-project: Traveling Networks The Daffynition Game Cooperative Problem Solving: Patterns at the Lunar Colony Three-Peg Puzzle Group Activity: Number of Handshakes Envelope Proofs: Algebraic

11 Unit 3. Parallel and Perpendicular Lines State Standard Area of Concentration: Mathematics/High School-Geometry Mathematics/High School-Algebra Unit Summary: Examine the special angle relationships that occur when lines are cut by transversals. Examine special properties that occur when those lines are parallel. Connect algebra to properties of lines. Review solving systems of equations, slope, and writing the equation of a line from Algebra 1. Use slope to determine whether two lines are parallel, perpendicular, or intersecting. Solve problems by writing linear equations. Unit Rationale: Many buildings are designed using basic shapes, lines, and planes. Examples of parallel, perpendicular, and skew lines can be seen in real world structures. Carpenters, designers, and construction managers must know the relationships of angles created by parallel lines and their transversals. When examining the relationship between algebra and the geometric topics in this unit, students will see the great impact geometry has had on some of the world s greatest architectural achievements. Standards: Congruence G-CO Expressing Geometric Properties with Equations G-GPE Common Core: Text: Cumulative Progress Indicators: G-CO Know precise definitions of angles, 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 9 3-2, 3-3, 3-4 Prove theorems about lines and angles. (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 ,3-4 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, paper folding, dynamic geometric software). 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-GPE 5 3-5, 3-6 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (eg find the equation of a line parallel or perpendicular to a given line that passes through a given point). Essential Questions: How can the relationship of angles formed by parallel or perpendicular lines be used in real-world situations? What are some applications of linear equations? Instructional Focus: Identify parallel, perpendicular, and skew lines. Enduring Understanding: There are many applications of the properties of parallel and perpendicular lines in the real-world. For example, using a plumb bob or carpenter s square, making perspective drawings, and strings on a piano or guitar. The graph of a line can be used to describe the speed (rate of change) when traveling. There are many real-world uses for slope including grade of a road, pitch of a roof, and incline of a ramp. Review and apply the following Algebra 1 topics:

12 Identify the angles formed by two lines and a transversal. Use theorems about the angles formed by parallel lines and a transversal. Use the angles formed by a transversal to prove two lines are parallel. Apply theorems about perpendicular lines. Find the slope of a line. Use slopes to identify parallel and perpendicular lines. Graph lines and write their equations in slope-intercept form and point-slope form. Classify lines as parallel, intersecting, or coinciding. Graph linear equations Calculate slope Write equations of lines Solve systems of equations Translate word problems into equations Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView Equipment needed: Text Chapter 3 Geometer s Sketchpad Software Graphing Calculator Compass Straightedge Protractor Pick up sticks Graphing Lines Game Board Number cubes Formative Assessment: Study Guide and Intervention Worksheets Skills Practice and Practice Worksheets Portfolio or Journal Definition and Conjecture List Quizzes Geometry Activities: a. Draw a Rectangular Prism b. Graphing Lines in the Coordinate Plane Constructions: Parallel & Perpendicular Lines Open-Ended Investigation: Parallel and Perpendicular Slope Conjecture Cooperative Problem Solving: Geometrivia Geometer s Sketchpad: Parallel Lines Conjecture and its converse Graphing Calculator Investigation: Line Designs; Intersections of Lines

13 Unit 4. Triangle Congruence State Standard Area of Concentration: Mathematics/High School-Geometry Unit Summary: Measure the lengths of the sides and angles of a triangle so that it can be classified and compared to other triangles. Apply the Triangle Sum Theorem and Exterior Angle Theorem. Test for triangle congruence and prove triangles are congruent by writing informal and formal proofs. Identify special properties of isosceles and equilateral triangles. Unit Rationale: Triangles can be found in nature, art, construction, and other areas of life. Triangles have properties that make them useful in structures such as buildings and bridges. Congruent triangles are used by surveyors by determining if one triangle can be mapped into another by means of a sequence of rigid transformations. This unit will allow students to have a better understanding of the purpose and importance of triangles in real-life. Standards: Congruence G-CO Similarity, Right Triangles, and Trigonometry G-SRT Modeling with Geometry G-MG Expressing Geometric Properties with Equations G-GPE Common Core: Text: Cumulative Progress Indicators: G-CO 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 4-1, 4-5, 4-6 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 4-5, 4-6 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motion. G-CO 10 G-SRT 5 4-4, 4-5, 4-6, , Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180, base angles of isosceles triangles are congruent. Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. G-MG 3 4-7, 4-8 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). G-GPE Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems(e.g. find the equation of line parallel or perpendicular to a given line that passes through a given point). G-GPE Use coordinates to prove simple geometric theorems algebraically. G-GPE Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Essential Questions: What makes shapes alike and different? Enduring Understanding: Congruent shapes are also similar, but similar shapes may not be congruent.

14 How can the study of triangles help solve problems in architecture? Instructional Focus: Draw, identify, and describe transformations in the coordinate plane. Use properties of rigid motions to determine whether figures are congruent and to prove figures congruent. Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. Use properties of congruent triangles. Apply SSS and SAS to construct triangles and solve problems. Prove triangles congruent by using SSS and SAS. Apply ASA, AAS, and HL to construct triangles and solve problems. Apply properties of isosceles and equilateral triangles. Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView Equipment needed: Text Chapter 4 Properties of triangles are used to design and build bridges, towers, and other structures. Builders of ancient Egypt used a tool called a plumb level in building architectural wonders such as the Egyptian pyramids using basic properties of isosceles triangles to determine if a surface is level. Review and apply the following Algebra 1 topics: Solve linear equations Translate word problems into equations Solve systems of linear equations in two variables by substitution Evidence of Learning Formative Assessment: Study Guide and Intervention Worksheets Skills Practice and Practice Worksheets Portfolio or Journal Definition and Conjecture List Quizzes Geometry Activities: a. Equilateral Triangles b. Angles of Triangles c. Congruent Triangles d. Isosceles Triangles Mini-Project: Folding Triangles Graphing Calculator Investigation: Lines and Isosceles Triangles Open-Ended Investigation: Triangle Exterior Angle Conjecture; Congruent Triangle Conjectures The Big Triangle Puzzles

15 Unit 5. Properties and Attributes of Triangles State Standard Area of Concentration: Mathematics/High School-Geometry Mathematics/High School-Algebra Unit Summary: Investigate and apply the Triangle Midsegmet Theorem. Define equidistant, concurrency, and locus. Review from algebra 1 solving inequalities and compound inequalities. Explore triangle inequalities. Review the Pythagorean Theorem and how to use it to find an unknown side length in a right triangle. Review how to simplify square roots and radicals. Discuss Pythagorean triples and how to identify them by using the converse of the Pythagorean theorem. Then explain how to use the Pythagorean Inequalities Theorem to classify triangles by their angle measures. Examine the relationships between the side lengths of a and a triangle by using the Pythagorean Theorem. Use these relationships to find unknown side lengths of special right triangles. Unit Rationale: The triangle inequality theorem can be used to find the shortest distance when traveling. Because the triangle is rigid it is used to add strength to structures. But the triangle is also used in mechanisms that move by changing the length of one side. Some examples of mechanisms that take advantage of the rigidity of the triangle while allowing one side to vary in length are dump trucks, reclining deck chairs, and car jacks. Standards: Congruence G-CO Similarity, Right Triangles, and Trigonometry G-SRT Circles G-C Modeling with Geometry G-MG Common Core: Text: Cumulative Progress Indicators: G-CO Prove geometric theorems about lines and angles. G-SRT 4 5-1, 5-7 Prove theorems about triangles. G-C Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle. G-CO , 5-3 Make formal geometric constructions with a variety of tools and methods(compass and straight edge, string, reflective devices, paper folding, dynamic geometry software, etc). G-MG Apply concepts of density based on area and volume in modeling situations G-CO , 5-4,5-5, 5-6 Prove theorems about triangles. G-MG Apply geometric methods to solve design problems G-SRT Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G-SRT 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. Essential Questions: Is there a relationship between the size of angles and Enduring Understanding: The triangle is used in mechanisms that move by

16 the lengths of the sides of a triangle? Why are triangles so useful in structures? Instructional Focus: Apply properties of medians of a triangle. (Optional) Apply properties of altitudes of a triangle.(optional) Prove and use properties of triangle midsegments.(optional) Apply inequalities in one triangle. Apply inequalities in two triangles. Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles. Justify and apply properties of triangles. Justify and apply properties of triangles. Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView Equipment needed: Text Chapter 5 Graph paper Linguine changing the length of one side. The lengths of the sides of a triangle are in relationship to the size of angles. Because the triangle is rigid it is used to add strength to structures. Review and apply the following Algebra 1 topics: Solve linear equations Solve inequalities Simplify radical expressions Rationalize the denominator Multiply and divide radicals Add and subtract radicals Formative Assessment: Study Guide and Intervention Worksheets Portfolio or Journal Definition and Conjecture List Geometry activity: a) Inequalities for Sides and Angles of Triangles b) The Pythagorean Theorem Mini-Project: The Pythagorean Theorem Project: Triangles at Work Cooperative Problem Solving: Pythagoras in Space Open-Ended Investigation: Pythagorean Proposition, Converse of the Pythagorean Theorem, Right Triangle Conjecture, Pythagorean Triples

17 Unit 6 Quadrilaterals State Standard Area of Concentration: Mathematics/High School-Geometry Mathematics/High School-Algebra Unit Summary: Investigate the interior and exterior angles of polygons. Apply the properties of regular polygons to solve realworld problems. Several different geometric shapes are examples of quadrilaterals. These shapes each have individual characteristics. Recognize and apply the properties of parallelograms. Recognize and apply the properties of parallelograms extending to rectangles, rhombi, and squares. Explore properties of quadrilaterals that do not have both pairs of opposite side parallel, such as trapezoids and kites. Unit Rationale: Polygons are seen in nature, in everyday objects, in art, and in architecture. Examining some of these objects while looking for patterns and developing conjectures about polygons can be an important opportunity for critical thinking. Quadrilaterals played an important part in the history of the parallel postulate. Properties of parallelograms can be examined when using a pantograph to copy a drawing or using a parallel rule to plot a course on a navigation chart. Because of many characteristics of special quadrilaterals, discoveries have been made that helped civilizations advance such as the role of trapezoids in arches, mechanisms that use quadrilateral linkages, and squaring the frame of a window or frame of a house. Standards: Congruence G-CO Expressing Geometric Properties with Equations G-GPE Modeling with Geometry G-MG Similarity, Right Triangles, and Trigonometry G-SRT Common Core: Text: Cumulative Progress Indicators: G-CO , 6-2, 6-3, 6-4, 6-5 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. G-GPE 5 6-2, 6-3 Use coordinates to prove simple geometric theorems algebraically. For example: prove or disprove that a figure define by four given points in the coordinate plane is a rectangle. G-MG 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). G- SRT Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. Essential Questions: How can knowledge of polygons be used in solving real-world situations? How can coordinates be used to describe and analyze geometric objects? Instructional Focus: Enduring Understanding: The properties of quadrilaterals are essential to success in engineering, architecture, or design. An object s location on a plane or in space can be described quantitatively. The position of any point on a surface can be specified by tow numbers. Computations with these numbers allow us to describe and measure geometric objects.

18 Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons. Apply properties of parallelograms. Use properties of parallelograms to solve problems. Apply properties of rectangles, rhombuses, and squares. Use properties of rectangles, rhombuses, and squares to solve problems. Identify the best name of a given quadrilateral i.e. a rectangle, rhombus, or square. Use properties of kites to solve problems. Use properties of trapezoids to solve problems. Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView Equipment needed: Text Chapter 6 Compass Straightedge Protractor Pipe cleaners Straws Patty Paper Geoboard/Geobands Review and apply the following Algebra 1 topics: Use Laws of Exponents to simplify Factor polynomials Multiply polynomials Multiply and divide rational expressions Solve quadratic equations by factoring Evidence of Learning Formative Assessment: Study Guide and Intervention Worksheets Skills Practice and Practice Worksheets Portfolio or Journal Definition and Conjecture List Quizzes Geometry Activities: a. Sum of the Exterior Angles of a Polygon b. Properties of Parallelograms c. Testing for a Parallelogram d. Kites e. Construct Median of a Trapezoid f. Linear Equations Mini-Project: Quadrilateral Sort; Square Search Constructions: Rectangle and Rhombus Graphing Calculator Investigation: Drawing Regular Polygons Geometer s Sketchpad Project: Star Polygons Project: Quadrilateral Linkages; Building an Arch Cooperative Problem Solving: The Geometry Scavenger Hunt The Big Quadrilateral Puzzles

19 Unit 7 Similarity State Standard Area of Concentration: Mathematics/High School-Geometry Mathematics/High School-Algebra Unit Summary: Extend knowledge of ratios and proportions to similar figures. Solve problems using cross products of proportions. Justify conditions for triangle similarity. Apply similarity to solve real-world problems. Unit Rationale: Similar polygons and their properties can be used to model and analyze many realworld situations. Similar figures are used to read maps accurately, work with blueprints, make movies, or even adapt recipes. Solving similarity proportions is a useful skill for working in the fields of chemistry, physics, and medicine. Standards: Similarity, Right Triangles, and Trigonometry G-SRT Modeling with Geometry G-MG Circles G-C Expressing Geometric Properties with Equations G-GPE Congruence G-CO Common Core: G-SRT 2 Text: Cumulative Progress Indicators: 7-1, 7-3, 7-4 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- MG 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). G- C Prove that all circles are similar. G SRT 1 7-2, 7-6 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 enter unchanged, b) The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G-SRT Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G-SRT 4 7-3, 7-4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely G-SRT 5 7-3, 7-4, 7-5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G- CO 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., translations versus horizontal stretch). G-GPE Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

20 Essential Questions: How are similarity and congruence related? How is similarity important in industry? Instructional Focus: Enduring Understanding: Congruent shapes are also similar, but similar shapes may not be congruent. Artists use similarity and proportionality to give paintings an illusion of depth. Engineers us similar triangles when designing buildings. The mathematics of similarity and perspective are key to making realistic movie images. Identify similar polygons. Review and apply the following Algebra 1 topics: Apply properties of similar polygons to solve Solve ratio and proportion problems problems. Solve linear equations Draw and describe similarity transformations in the coordinate plane. Use properties of similarity transformations to determine whether polygons are similar and to prove circles are similar. Prove certain triangles are similar by using AA, SSS, and SAS. Use triangle similarity to solve problems. Use properties of similar triangles to find segment lengths. Apply proportionality and triangle angle bisector theorems. Use ratios to make indirect measurements. Use scale drawings to solve problems. Apply similarity properties in the coordinate plane. Divide a directed line segment into partitions. (Optional) Translate word problems into equations Evidence of Learning Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView Equipment needed: Text Chapter 7 Video: M! Project Math: Similarity Protractor, Ruler, Compass & Straightedge Isometric dot paper Formative Assessment: Study Guide and Intervention Worksheets Skills Practice and Practice Worksheets Portfolio or Journal Definition and Conjecture List Quizzes Geometry Activities: a. Similar Triangles b. Sierpinski s Triangle Mini-Project: Measuring Height Project: Making a Mural; The Shadow Knows; Why Elephants Have Big Ears Cooperative Problem Solving: Similarity in Space Open-Ended Investigation: Similarity in an Eclipse; Scale Factor; Fractals

21 Unit 8 Right Triangles and Trigonometry State Standard Area of Concentration: Mathematics/High School-Geometry Mathematics/High School-Algebra Unit Summary: Solve problems using the similarity relationships of right triangles. Apply trigonometric ratios to real-world situations. Unit Rationale: Trigonometry was developed by astronomers who wished to map the stars. Since measurements of stars and planets are inherently hard to make by direct methods, indirect measurement paved the way for study in this field. Properties and applications of right triangle trigonometry are used to calculate these distances that are difficult or impossible to measure directly as seen also in architecture, engineering, navigation, and surveying. Standards: Similarity, Right Triangles, and Trigonometry G-SRT Common Core: Text: Cumulative Progress Indicators: G-SRT 4 8-1,8-2 Prove Theorems about triangles. Theorems include: the Pythagorean Theorem proved using triangle similarity. G-SRT 6 8-1, 8-2 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. G-SRT Explain and use the relationship between the sine and cosine of complementary angles. G-SRT 8 8-3, 8-4 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Essential Questions: What is the historical impact of knowledge of right triangles and its properties? How can you find distance between two objects when you cannot use direct measurement? Instructional Focus: Enduring Understanding: The knowledge of right triangles and its properties allows us to lay the foundations of buildings accurately. The rope stretchers of Egypt used the properties of the triangle to build the pyramids. Measures of distances can be found indirectly by writing and solving proportions and using trigonometry. Use geometric mean to find segment lengths in right Review and apply the following Algebra 1 topics: triangles. Solve linear equations Apply similarity relationships in right triangles to solve Translate word problems into equations problems. Simplify radicals and apply Laws of Radicals Find the sine, cosine, and tangent of an acute angle. Solve equations with radicals Use trigonometric ratios to find side lengths in right Solve 2 nd degree equations triangles and to solve real-world problems. Evidence of Learning Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView Formative Assessment: Study Guide and Intervention Worksheets Skills Practice and Practice Worksheets Portfolio or Journal Definition and Conjecture List

22 Equipment needed: Text Chapter 8 Video: M! Project Math The Theorem of Pythagoras Patty paper Rulers Scissors Quizzes Geometry Activities: a. Trigonometric Ratios Project: Creating a Geometry Flip Book; Indirect Measurement Geometry Software Investigation: Right Triangles Formed by the Altitude, A Pythagorean Fractal, The Graphing Calculator Investigation: The Height Reached by a Ladder

23 Unit 9. Transformations State Standard Area of Concentration: Mathematics/High School-Geometry Unit Summary: Explore different types of transformations: reflections, translations, rotations, and dilations. Identify, draw, and recognize figures that have been transformed. Unit Rationale: Geometry is not only the study of figures; it is also the study of the movement of figures. If you move all the points of a geometric figure according to set rules, you can create a new geometric figure. A transformation that preserves size and shape is called an isometry. Three types of isometries are translation, rotation, and reflection. Symmetry is an integral part of nature and of the arts and crafts of cultures worldwide. Symmetry can be found in art, architecture, crafts, poetry, music, dance, chemistry, biology, and mathematics. These geometric procedures and characteristics make objects more visually pleasing. Tessellations can be created when applying the principles of symmetry and isometries. Standards: Congruence G-CO Similarity, Right Triangles, and Trigonometry G-SRT Common Core: Text: Cumulative Progress Indicators: G-CO Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G-CO 4 9-1, 9-2, 9-3, 9-4 Given a geometric figure and a rotation, refection, or translation, draw the transformed figure using, eg. Graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G-SRT 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. Essential Questions: How can transformations be described mathematically? What are the types of transformations seen in realworld situations? Instructional Focus: Identify and draw reflections. Identify and draw translations. Identify and draw rotations. Apply theorems about isometries. Identify and draw compositions of transformations, such as glide reflections. Identify and describe symmetry in geometric figures. Understand how solids can be produced by rotating a two dimensional figure through space. Enduring Understanding: Transformations can be represented and verified using coordinate geometry. Reflections, translations, rotations, and dilations can all be seen in real-world situations.

24 Use transformations to draw tessellations.(optional) Identify regular and semiregular tessellations and figures that will tessellate.(optional) Identify and draw dilations. Evidence of Learning Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test ExamView Equipment needed: Text Chapter 9 Patty Paper Dot paper Straightedge Coordinate Grids Geomirror Pattern Blocks Video: Landscape of Geometry Program 8: The Range of Change Formative Assessment: Study Guide and Intervention Worksheets Skills Practice and Practice Worksheets Portfolio or Journal Definition and Conjecture List Quizzes Geometry Activities: a. Transformations b. Reflections and Translations c. Tessellations of Regular Polygons Mini-Project: Graphing and Translations; Geomirror Graphing Calculator Investigation: Transforming Triangles Cooperative Problem Solving: Poolroom Math; Miniature Golf Math Geometer s Sketchpad Project: Tessellating with the Conway Criterion Constructions: Reflections in a Line

25 Unit 10 Areas of Polygons and Circles State Standard Area of Concentration: Mathematics/High School-Geometry Unit Summary: Find the areas of parallelograms, rhombi, trapezoids, and triangles. Identify the apothem of a regular polygon and use that measure to find the areas of regular polygons. Find the areas of irregular figures, circles, sectors, and segments of circles. Develop and apply area formulas for circles, polygons, and composite figures. Unit Rationale: People in many occupations work with areas. Carpenters calculate areas to order materials for construction, painters calculate the area of surfaces to be painted, decorators need to know areas when installing materials in homes, and gardeners may use area to find the maximum area given perimeter. The area of a figure is measured by the number of squares of a unit length that can be arranged to completely fill that area. The fundamental idea in developing area formulas is the Area Addition Postulate: the area of a region is equal to the sum of the areas of the region s nonoverlapping parts. Historically, methods of measuring the area of people s property was needed in order for governments to tax land. The Babylonians and Egyptians developed some of the earliest mathematics, partly to keep track of land and finances. Standards: Expressing Geometric Properties with Equations G-GPE Geometric Measurement and Dimension G-GMD Modeling with Geometry G-MG Congruence G-GO Circles G-C Common Core: Text: Cumulative Progress Indicators: A-SSE Interpret expressions that represent a quantity in terms of its context. A-CED Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. G-GPE , 10-5 Use coordinates to compute areas of triangles and rectangles. G-GMD Give an informal argument for the formula for area of a circle. G-MG Apply geometric methods to solve design problems. G-SRT Derive the formula A=1/2 absin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. S-CP Describe events as subsets of 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 Essential Questions: What is the advantage of deriving formulas for area? How can area formulas be used in real-world situations? Instructional Focus: Develop and apply the formulas for the area of Enduring Understanding: Deriving formulas for area helps strengthen understanding of spatial relationships. Calculating area and perimeter can be seen in real life situations such as finding the amount of grass seed to needed to cover various shaped regions, finding the area of a banner, or finding area of figures in construction. Review and apply the following Algebra 1 topics: