Curriculum Guide

for

Geometry Honors Course Number 220

Department of Mathematics Belvidere High School

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, the development of formal proof, 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 provide an in depth understanding of Euclidean Geometry.

Course Description & Course Proficiency COURSE TITLE: Geometry Honors COURSE LENGTH: 36 weeks

COURSE # 220 PERIODS PER WK:

5

TOTAL CREDIT: 5 GRADE LEVEL: 9 10

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:

www.geometryonline.com 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.

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 email 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 2nd marking period. Plane or Solid Figure Project 4th 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 20-50 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

Parent/Guardian Signature

Teacher Signature

Parent/Guardian Email _______________________________________

Updated 05/07/2010

Course Content Weekly Curriculum Map: Geometry Honors Duration Content New Jersey of Unit Common Core

Assessment

Resources

Standards G-CO: 1, 2, 4, 5, 12 A-SSE: 1 A-CED: 4 G-GPE: 7

class discussion, homework, quiz, test

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

3 wks

Point, Lines, Planes, and Angles

3 wks

Reasoning and Proof

A-REI: 1 G-CO: 9, 10 A-CED: 1

class discussion, homework, quiz, test

3 wks

Parallel and Perpendicular Lines

G-CO: 1, 9, 12 G-GPE: 5

class discussion, homework, quiz, test

4 wks

Triangle Congruence

G-CO: 6,7,8,10 G-SRT: 5 G-MG: 3 G-GPE: 4,5,7

class discussion, homework, quiz, test

2 wks

Properties and Attributes of Triangles

G-CO: 9, 10, 12 G-SRT: 4, 6, 8 G-C: 3 G-MG: 2,3

class discussion, homework, quiz, test

3 wks

Quadrilaterals

G-CO: 11 G-GPE: 5 G-MG: 3 G-SRT: 5

class discussion, homework, quiz, test

1 wk

Review / Testing

3wks

Similarity

Mid-term Exam

G-SRT:2 G-MG: 3 G-C: 1 G: SRT: 1, 3, 4, 5 G-CO: 2

class discussion, homework, quiz, test

Text: Geometry: Holt McDougal Supplemental: Geometry: Glencoe Discovering Geometry An Inductive Approach: Key

Curriculum Press

G-GPE: 6 3 wks

Right Triangles and Trigonometry

G-SRT: 4, 6, 7, 8, 10, 11

class discussion, homework, quiz, test

1 wk

Transformations

G-CO: 3, 4 G-SRT 1

class discussion, homework, quiz, test

3wks

Areas of Polygons and Circles

class discussion, homework, quiz, test

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

1 wk

Review/Testing

class discussion, homework, quiz, test

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

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:

Common Core:

G-CO 1

1-1, 1-2, 1-3, Know precise definitions of angle, circle, perpendicular line, parallel line, and line 1-4, segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G-CO 12

1-3

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 1

1-5

Interpret expressions that represent a quantity in terms of its context.

A-CED 4

1-5

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

G-GPE 7

1-6

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

G-CO 4

1-7

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO 2

1-7

Represent transformations in the plane using e.g., transparencies and geometry software

G-CO 5

1-7

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? What is the necessity for undefined terms? What applications led to the development of coordinate

Enduring Understanding: Definitions will improve mathematical reasoning by being able to illustrate and examine geometric concepts. Euclidean geometry begins with the three undefined terms of point, line, and plane. The coordinate system helps locate objects on a plane.

systems?

Identifying locations on a map or globe using latitude and longitude in order to travel compares to coordinates in a plane.

Instructional Focus:. Identify, name, and draw points, lines, segments, rays, Review and apply the following Algebra 1 topics: and planes.  Simplify algebraic expressions Apply basic facts about points, lines, and planes.  Solve linear equations Identify collinear and coplanar points and intersecting  Simplify absolute value expressions lines and planes in space.  Operations with signed numbers Use length and midpoint of a segment.  Translate word problems into algebraic Construct midpoints and congruent segments. equations 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”

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)

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 1

2-5

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 10

2-7

Prove theorems about triangles.

A-CED 1

2-8

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?

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.

Instructional Focus: 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  Apply algebraic properties conditional statements. Write the inverse, converse, and contrapositive of a conditional statement. Apply the Law of Detachment and the Law of Syllogism in logical reasoning. Write and analyze biconditional statements. Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence.

Write two-column proofs. Prove geometric theorems by using deductive reasoning. Write flowchart and paragraph proofs. Analyze the truth value of conjunctions and disjunctions. Construct truth tables to determine the truth value of logical statements. Evidence of Learning Summative Assessment Formative Assessment: Multiple-choice test Portfolio or Journal Free-Response test Definition and Conjecture List Performance Assessment Quizzes Cumulative Test Geometry Activities: “ExamView” a. If-then statements b. Matrix Logic c. Right Angles Mini-project: Traveling Networks Project: Euler’s Formula for Networks The Daffynition Game Cooperative Problem Solving: Lunar Survival; Patterns at the Lunar Colony Three-Peg Puzzle Group Activity: Number of Handshakes Envelope Proofs Equipment needed: Text Chapter 2 Patty Paper Protractor Scissors Rulers

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 1

3-1

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 12

3-3,3-4

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: Enduring Understanding: How can the relationship of angles formed by parallel There are many applications of the properties of or perpendicular lines be used in real-world situations? parallel and perpendicular lines in the real-world. For example, using a plumb bob or carpenter’s square, What are some applications of linear equations? 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. Instructional Focus: Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal.

Review and apply the following Algebra 1 topics:  Graph linear equations  Calculate slope

Prove and 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. Prove and 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. 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

  

Write equations of lines Solve systems of equations Translate word problems into equations

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 c. Non-Euclidean Geometry Constructions: Parallel & Perpendicular Lines Mini-Project: Avoiding Lattice Points Open-Ended Investigation: Parallel and Perpendicular Slope Conjecture Project: Making a Clinometer Cooperative Problem Solving: Geometrivia Geometer’s Sketchpad: Parallel Lines Conjecture and its converse Graphing Calculator Investigation: Line Designs; Intersections of Lines

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 6

4-1

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

G-CO 10

4-2, 4-3. 4-9 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180, base angles of isosceles triangles are congruent.

G-SRT 5

4-4, 4-5, 4-6, Use congruence and similarity criteria for triangles to solve problems and prove 4-7 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 5

4-7

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 4

4-8

Use coordinates to prove simple geometric theorems algebraically.

G-GPE 7

4-8

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motion.

Essential Questions: What makes shapes alike and different? How can the study of triangles help solve problems in architecture?

Enduring Understanding: Congruent shapes are also similar, but similar shapes may not be congruent. 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. Instructional Focus: Draw, identify, and describe transformations in the Review and apply the following Algebra 1 topics: coordinate plane.  Solve linear equations Use properties of rigid motions to determine whether  Translate word problems into equations figures are congruent and to prove figures congruent.  Solve systems of linear equations in two Classify triangles by their angle measures and side variables by substitution 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. Prove triangles congruent by using the definition of congruence. 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. Prove triangles congruent by using ASA, AAS, and HL. Use CPCTC to prove parts of triangles are congruent. Position figures in the coordinate plane for use in coordinate proofs.(Optional) Prove geometric concepts by using coordinate proof.(Optional) Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test “ExamView”

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. Congruence in Right Triangles e. Isosceles Triangles Mini-Project: Perimeters and Unknown Values; Folding Triangles Graphing Calculator Investigation: Lines and Isosceles Triangles

Open-Ended Investigation: Triangle Exterior Angle Conjecture; Congruent Triangle Conjectures; Vertex Angle Bisector Conjecture The Big Triangle Puzzles Equipment needed: Text Chapter 4 Graph paper Patty paper

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 following theorems about triangles: Perpendicular Bisector Theorem, Angle Bisector Theorem, Circumcenter Theorem, Centroid Theorem, and Triangle Midsegment 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 45-45-90 and a 30-60-90 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 9

5-1

Prove geometric theorems about lines and angles.

G-SRT 4

5-1, 5-7

Prove theorems about triangles.

G-C 3

5-2

Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.

G-CO 12

5-2, 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 2

5-2

Apply concepts of density based on area and volume in modeling situations…

G-CO 10

5-3, 5-4,5-5, Prove theorems about triangles.

5-6 G-MG 3

5-3

Apply geometric methods to solve design problems…

G-SRT 8

5-7

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

G-SRT 6

5-8

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 the lengths of the sides of a triangle? Why are triangles so useful in structures?

Enduring Understanding: The triangle is used in mechanisms that move by 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.

Instructional Focus: Prove and apply theorems about perpendicular Review and apply the following Algebra 1 topics: bisectors.  Solve linear equations Prove and apply theorems about angle bisectors.  Solve inequalities Prove and apply properties of perpendicular bisectors  Simplify radical expressions of a triangle.  Rationalize the denominator Prove and apply properties of angle bisectors of a  Multiply and divide radicals triangle.  Add and subtract radicals Apply properties of medians of a triangle. Apply properties of altitudes of a triangle. Prove and use properties of triangle midsegments. Write indirect proofs. 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 45-45-90 triangles. Justify and apply properties of 30-60-90 triangles. Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test “ExamView” Equipment needed: Text Chapter 5 Graph paper Linguine

Formative Assessment: Study Guide and Intervention Worksheets Portfolio or Journal Definition and Conjecture List Geometry activity: Inequalities for Sides and Angles of Triangles Project: Triangles at Work; Napoleon’s Theorem

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 11

6-1, 6-2, 6-3, Prove theorems about parallelograms. Theorems include: opposite sides are 6-4, 6-5 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 3

6-3

Apply geometric methods to solve design problems…

G- SRT 5

6-5

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?

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.

Instructional Focus: Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons. Prove and apply properties of parallelograms.

Review and apply the following Algebra 1 topics:  Use Laws of Exponents to simplify  Factor polynomials  Multiply polynomials

Use properties of parallelograms to solve problems.  Multiply and divide rational expressions Prove that a given quadrilateral is a parallelogram.  Solve quadratic equations by factoring Prove and apply properties of rectangles, rhombuses, and squares. Use properties of rectangles, rhombuses, and squares to solve problems. Prove that a given quadrilateral is a rectangle, rhombus, or square. Use properties of kites to solve problems. Use properties of trapezoids to solve problems. Evidence of Learning Summative Assessment Formative Assessment: Multiple-choice test Study Guide and Intervention Worksheets Free-Response test Skills Practice and Practice Worksheets Performance Assessment Portfolio or Journal Cumulative Test Definition and Conjecture List “ExamView” 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 Equipment needed: Text Chapter 6 Compass Straightedge Protractor Pipe cleaners Straws Patty Paper Geoboard/Geobands

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 real-

world 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:

Text:

Cumulative Progress Indicators:

G-SRT 2

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 3

7-1

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 1

7-2

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 3

7-3

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 2

7-6

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 6

7-6

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Essential Questions: How are similarity and congruence related? How is similarity important in industry?

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.

Instructional Focus: 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  Translate word problems into equations 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. Use coordinate proof to prove figures similar. Divide a directed line segment into partitions. Evidence of Learning Summative Assessment Formative Assessment: Multiple-choice test Study Guide and Intervention Worksheets Free-Response test Skills Practice and Practice Worksheets Performance Assessment Portfolio or Journal Cumulative Test Definition and Conjecture List “ExamView” 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 Equipment needed: Text Chapter 7 Video: M! Project Math: Similarity Protractor, Ruler Compass & Straightedge Isometric dot paper

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:

G-SRT 4

Cumulative Progress Indicators: 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 7

8-2

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.

G-SRT 10 (optional)

8-5

Prove the Laws of Sines and Cosines and use them to solve problems.

G-SRT 11 (optional)

8-5

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

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?

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 3-4-5 triangle to build the pyramids. Measures of distances can be found indirectly by writing and solving proportions and using trigonometry.

Instructional Focus: 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 2nd degree equations triangles and to solve real-world problems. Use the relationship between the sine and cosine of complementary angles.

Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Solve problems involving angles of elevation and depression. Use Law of Sines & Law of Cosines to solve triangles.(optional) Summative Assessment Multiple-choice test Free-Response test Performance Assessment Cumulative Test “ExamView”

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. The Pythagorean Theorem b. Trigonometric Ratios c. Trigonometry Mini-Project: The Pythagorean Theorem Project: Creating a Geometry Flip Book; Indirect Measurement Geometry Software Investigation: Right Triangles Formed by the Altitude, A Pythagorean Fractal, The Ambiguous Case of the Law of Sines Graphing Calculator Investigation: The Height Reached by a Ladder Cooperative Problem Solving: Pythagoras in Space Open-Ended Investigation: Pythagorean Proposition, Converse of the Pythagorean Theorem, 30-60-90 Right Triangle Conjecture, Pythagorean Triples

Equipment needed: Text Chapter 8 Video: M! Project Math The Theorem of Pythagoras Patty paper Rulers Scissors

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 3

9-5

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, Given a geometric figure and a rotation, refection, or translation, draw the 9-4 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 1

9-7

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.

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 Formative Assessment: Multiple-choice test Study Guide and Intervention Worksheets Free-Response test Skills Practice and Practice Worksheets Performance Assessment Portfolio or Journal Cumulative Test Definition and Conjecture List “ExamView” 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 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

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 1

10-1

Interpret expressions that represent a quantity in terms of its context.

A-CED 4

10-1

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

G-GPE 7

10-4, 10-5

Use coordinates to compute areas of triangles and rectangles.

G-GMD 1

10-2

Give an informal argument for the formula for area of a circle.

G-MG 3

10-3

Apply geometric methods to solve design problems.

G-SRT 9

10-3

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 1

10-6

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?

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.

Instructional Focus: Develop and apply the formulas for the area of

Review and apply the following Algebra 1 topics:

triangles and special quadrilaterals.  Solve linear and 2nd degree equations Solve problems involving perimeters and areas of  Translate word problems into equations triangles and special quadrilateral.  Use formulas Use the Area Addition Postulate to find the area of  Solve and simplify radicals composite figures.  Solve ratio and proportion problems Use composite figures to estimate the areas of irregular shapes. Find the perimeters and areas of figures in a coordinate plane. Describe the effect on perimeter and area when one or more dimensions of a figure are changed. Apply the relationship between perimeter and area in problem solving. Calculate geometric probabilities. Evidence of Learning Summative Assessment Formative Assessment: Multiple-choice test Study Guide and Intervention Worksheets Free-Response test Skills Practice and Practice Worksheets Performance Assessment Portfolio or Journal Cumulative Test Definition and Conjecture List “ExamView” Quizzes Geometry Activities: a. Area of a Parallelogram b. Area of a Triangle c. Area of a Trapezoid d. Area of a Circle e. Area of a Regular Polygon Constructions: Regular Polygon Mini-Project: Areas of Circular Regions Open-Ended Investigation: Maximizing Area, Area vs Perimeter Project: Quilt Making Cooperative Problem Solving: Discovering New Area Formulas Geometer’s Sketchpad: Calculating Area in Ancient Egypt Graphing Calculator Investigation: Maximizing Area Equipment needed: Text Chapter 10 Grid Paper Straightedge Scissors Tape Compass

Unit 11. Surface Area and Volume State Standard Area of Concentration: Mathematics/High School-Geometry Unit Summary: Students represent 3D figures using nets. Create representations of three-dimensional figures. The basics types of geometric figures are described and their characteristics are discussed. Find the Lateral Area, Surface Area, and Volume of prisms, cylinders, pyramids, and cones. Identify the parts of a sphere and find its surface area and volume. Apply formulas for volume to real-world figures. Unit Rationale: Solids have three dimensions: length, width, and height. Solids occur in nature such as viruses, oranges, crystals, the earth itself, or man-made objects such as books, buildings, baseballs, soup cans, or even ice cream cones. At the molecular level, 3D geometry plays a very important role in a number of common substances such as carbon or even water. Many solid objects have shapes that can be easily described using common geometric terms. The volume of a figure is the measure of the amount of space that a figure encloses. Volume is measured in cubic units. Solids can be created from different views of the figure to investigate its volume. The formula for the volume of a rectangular prism(V=Bh) is the starting point for developing the volume formulas for other three-dimensional figures. Another important part of developing volume formulas is Cavalieri’s Principle, which says that two 3D figures with the same height and same cross-sectional area at every level have the same volume. Standards: Geometric Measurement and Dimension G-GMD Modeling with Geometry G-MG Common Core:

Text:

Cumulative Progress Indicators:

G-GMD 1

11-2, 11-3

Give an informal argument for the formulas for volume of a cylinder, pyramid, and cone.

G-GMD 2

11-2

Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

G-GMD 3

11-2, 11-3, 11-4

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

G-GMD 4

11-1

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

11-2

Use geometric shapes, their measures, and their properties to describe objects…

G-MG 2

11-2

Apply concepts of density based on area and volume in modeling situations.

G-MG 3

11-1

Apply geometric methods to solve design problems.

Essential Questions: How are 1, 2, and 3 dimensional shapes related? How can 3-dimensional objects be represented in 2dimensions? How is volume important in a real world context? What type of shapes should be used to increase or decrease the surface area for a given volume?

Enduring Understanding: Shape can be seen from different perspectives. Nets, perspective drawings, and projections are ways of representing 3D figures in two dimensions. It is important to know what the capacity of a given figure is.

A cube is the rectangular prism with the minimum surface area for a given volume, and with the maximum volume for a given surface area. The

cylinder with the minimum surface area for a given volume, and with the maximum volume for a given surface area will be the one in which the height is equal to its diameter. Instructional Focus: Classify three-dimensional figures according to their Review and apply the following Algebra 1 topics: properties.  Solve linear equations Use nets and cross sections to analyze three Use formulas dimensional figures.  Simplify ratios Learn and apply the formula for the volume of a prism. Learn and apply the formula for the volume of a cylinder. Learn and apply the formula for the volume of a pyramid. Learn and apply the formula for the volume of a cone. Learn and apply the formula for the volume of a sphere. Learn and apply the formula for the surface area of a sphere. Evidence of Learning Summative Assessment Formative Assessment: Multiple-choice test Study Guide and Intervention Worksheets Free-Response test Skills Practice and Practice Worksheets Performance Assessment Portfolio or Journal Cumulative Test Definition and Conjecture List “ExamView” Quizzes Geometry Activities: a. Surface Area of Cylinders and Cones b. Surface Area of a Sphere c. Locus and Spheres d. Volume of a Rectangular Prism e. Investigating the Volume of a Pyramid Constructions: Intersection of Loci, Constructing the Platonic Solids Mini-Project: Cone Patterns; Volume of Cylinders Open-Ended Investigation: Euler’s Formula for Solids Project: The Five Platonic Solids; The World’s Largest Pyramid Cooperative Problem Solving: Once Upon A Time Graphing Calculator Investigation: A Maximum Volume Box Equipment needed: Text Chapter 11 Isometric dot paper Straightedge Compass Polystyrene ball Scissors, Tape, Glue Centimeter cubes Card stock Ruler

Rice

Unit 12 Circle State Standard Area of Concentration: Mathematics/High School-Geometry Mathematics/High School-Algebra Unit Summary: Identify parts of a circle. Examine the special relationship of angles, arcs, and segments intersecting a circle. Find arc and angle measures and the measures of segments in a circle. Solve problems involving circumference. Explore special properties of circles. Unit Rationale: Circles are geometric shapes common to everyday life such as wheels on a vehicle. Circular wheels gave rise to circular gears, which helped bring on the industrial revolution. Potter’s wheels, clocks, and windmills which are all based on applications of wheels, represent great advances in civilization. Archaeologists use properties of circles to study our past when finding artifacts that are circular in design. The Tangent Conjecture can be seen in many applications related to circular motion such as satellites and circular orbits. Circle theorems are used in the study of the laws of motion, exploring cells in biology, creating images in art and marketing, calculating distances to the horizon, and many other areas of real life. Standards: Circles G-C Geometric Measurement and Dimension G-GMD Common Core:

Text:

Cumulative Progress Indicators:

G-C 2

12-1, 12-2, 12-4, 12-5, 12-6

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 angels, the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G-C 3

12-4

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G-C 4

12-1

Construct a tangent line from a point outside a given circle to the circle.

G-C 5

12-3

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius.

G-CO 13

12-4

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Essential Questions: What geometric relationships can be found in circles? How can these relationships help describe real-world situations?

Enduring Understanding: Geometric relationships in circles involve angles, arcs, and segments. Some real-world examples are determining the length of the line of sight on the earth’s surface, identifying the coverage of an overlapping circular pattern of irrigation, and finding the area or arc length of a slice of pie.

Instructional Focus:. Identify tangents, secants, and chords. Use properties of tangents to solve problems. Apply properties of arcs. Apply properties of chords. Find the area of sectors.

Review and apply the following Algebra 1 topics:  Solve linear equations  Translate word problems into equations  Solve 2nd degree equations

Find arc lengths.  Solve inequalities Find the measure of an inscribed angle.  Solve radical equations Use inscribed angles and their properties to solve  Simplify radical expressions problems.  Simplify ratios Find the measures of angles formed by lines that intersect circles. Use angle measures to solve problems. Find the lengths of segments formed by lines that intersect circles. Use the length of segments in circles to solve problems. Write equations and graph circles in the coordinate plane.(Optional) Evidence of Learning Summative Assessment Formative Assessment: Multiple-choice test Study Guide and Intervention Worksheets Free-Response test Skills Practice and Practice Worksheets Performance Assessment Portfolio or Journal Cumulative Test Definition and Conjecture List “ExamView” Quizzes Geometry Activities: a. Circumference Ratio b. Congruent Chords and Distance c. Measure of Inscribed Angles d. Inscribed Angles e. Inscribed and Circumscribed Triangles f. Pi Day Constructions: Construct a Circle to Inscribe a Triangle; Construct a Tangent; Inscribe a Circle in a Triangle Mini-Project: Locating the Center of a Circle Open-Ended Investigation: Elton Notle’s ceramic plate; Locate crater’s center; Tangent Conjectures Project: Racetrack Geometry Cooperative Problem Solving: Designing a Theater for Galileo Geometer’s Sketchpad Project: Turning Wheels Graphing Calculator Investigation: Graphing Circles and Tangents Equipment needed: Text Chapter 12 Video: M! Project Math: The Story of Pi