final geometry honors - monroe township school district · 2013. 3. 11. · 4"|page"...

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CURRICULUM MANAGEMENT SYSTEM

MONROE TOWNSHIP SCHOOLS

Course Name: Geometry (Honors) Grade: 8-‐9-‐10

For adoption by all regular education programs Board Approved: September 2012 as specified and for adoption or adaptation by all Special Education Programs in accordance with Board of Education Policy # 2220.

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Table of Contents

Monroe Township Schools Administration and Board of Education Members Page ….3

Mission, Vision, Beliefs, and Goals Page ….4

Philosophy Page ….5

Core Curriculum Content Standards Page ….6

Scope and Sequence Page …7

Core Content Overview Page …11

Goals/Essential Questions/Objectives/Instructional Tools/Activities Page …14

Course Benchmarks Page ….82

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Monroe Township Schools Administration and Board of Education Members

ADMINISTRATION Dr. Kenneth R. Hamilton, Superintendent

Dr. Jeff C. Gorman, Assistant Superintendent

BOARD OF EDUCATION Ms. Kathy Kolupanowich, Board President Mr. Ken Chiarella, Board Vice President

Ms. Amy Antelis Mr. Marvin I. Braverman

Mr. Lew Kaufman Mr. Mark Klein Mr. John Leary

Mr. Louis C. Masters Mr. Ira Tessler

Jamesburg Representative Ms. Patrice Faraone

WRITERS NAME

Ms. Samantha R. Grimaldi

MATHEMATICS CURRICULUM INCHARGE (9-‐12) Dr. Manjit K. Sran

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Mission, Vision, Beliefs, and Goals

Mission Statement

The Monroe Public Schools in collaboration with the members of the community shall ensure that all children receive an exemplary education by well-‐trained committed staff in a safe and orderly environment.

Vision Statement

The Monroe Township Board of Education commits itself to all children by preparing them to reach their full potential and to function in a global society through a preeminent education.

Beliefs

1. All decisions are made on the premise that children must come first. 2. All district decisions are made to ensure that practices and policies are developed to be inclusive, sensitive and meaningful to our diverse population.

3. We believe there is a sense of urgency about improving rigor and student achievement. 4. All members of our community are responsible for building capacity to reach excellence. 5. We are committed to a process for continuous improvement based on collecting, analyzing, and reflecting on data to guide our decisions. 6. We believe that collaboration maximizes the potential for improved outcomes. 7. We act with integrity, respect, and honesty with recognition that the schools serve as the social core of the community. 8. We believe that resources must be committed to address the population expansion in the community. 9. We believe that there are no disposable students in our community and every child means every child.

Board of Education Goals

1. Raise achievement for all students paying particular attention to disparities between subgroups. 2. Systematically collect, analyze, and evaluate available data to inform all decisions. 3. Improve business efficiencies where possible to reduce overall operating costs. 4. Provide support programs for students across the continuum of academic achievement with an emphasis on those who are in the middle. 5. Provide early interventions for all students who are at risk of not reaching their full potential. 6. To Create a 21st Century Environment of Learning that Promotes Inspiration, Motivation, Exploration, and Innovation.

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SOPHY

Philosophy

Monroe Township Schools are committed to providing all students with a quality education resulting in life -‐long learners who can succeed in a global society. The mathematics program, grades K -‐ 12, is predicated on that belief and is guided by the following six principles as stated by the National Council of Teachers of Mathematics (NCTM) in the Principles and Standards for School Mathematics, 2000. First, a mathematics education requires equity. All students will be given worthwhile opportunities and strong support to meet high mathematical expectations. Second, a coherent mathematics curriculum will effectively organize, integrate, and articulate important mathematical ideas across the grades. Third, effective mathematics teaching requires the following: a) knowing and understanding mathematics, students as learners, and pedagogical strategies b) having a challenging and supportive classroom environment and c) continually reflecting on and refining instructional practice. Fourth, students must learn mathematics with understanding. A student's prior experiences and knowledge will actively build new knowledge. Fifth, assessment should support the learning of important mathematics and provide useful information to both teachers and students. Lastly, technology enhances mathematics learning, supports effective mathematics teaching, and influences what mathematics is taught.

As students begin their mathematics education in Monroe Township, classroom instruction will reflect the best thinking of the day. Children will engage in a wide variety of learning activities designed to develop their ability to reason and solve complex problems. Calculators, computers, manipulatives, technology, and the Internet will be used as tools to enhance learning and assist in problem solving. Group work, projects, literature, and interdisciplinary activities will make mathematics more meaningful and aid understanding. Classroom instruction will be designed to meet the learning needs of all children and will reflect a variety of learning styles.

In this changing world those who have a good understanding of mathematics will have many opportunities and doors open to them throughout their lives. Mathematics is not for the select few but rather is for everyone. Monroe Township Schools are committed to providing all students with the opportunity and the support necessary to learn significant mathematics with depth and understanding.

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Common Core State Standards (CSSS)

The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

Links: 1. CCSS Home Page: http://www.corestandards.org 2. CCSS FAQ: http://www.corestandards.org/frequently-‐asked-‐questions 3. CCSS The Standards: http://www.corestandards.org/the-‐standards 4. NJDOE Link to CCSS: http://www.state.nj.us/education/sca 5. Partnership for Assessment of Readiness for College and Careers (PARCC): http://parcconline.org OE: Core Curriculum Content Standards

SCOPE AND SEQUENCE: HONORS GEOMETRY

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Unit I: Congruence, Proof, and Construction Big Idea I: Visualization and Measurement Big Idea II: Reasoning and Proof I. Points, Lines, Planes

a. Collinear and Coplanar b. Lines, Segments, and Rays c. Space

II. Distance a. Distance Formula b. Midpoint Formula c. Segment Addition

III. Angles and Bisectors a. Types of Angles b. Angle and Segment Bisectors c. Angle Pair Relationships

IV. Polygons a. Identification b. Convex vs. Concave c. Regular vs. Irregular d. Nets

I. Conditional Statements a. Conjectures based on patterns b. Hypothesis and Conclusion c. If-‐then form, Converse, Inverse, Contrapositive

II. Proofs a. Postulates vs. Theorems b. Algebraic Proofs c. Flow Proofs d. Two Column Proofs


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Unit I: Congruence, Proof, and Construction Big Idea III: Reasoning and Measurement Big Idea IV: Visualization, Congruency, and

Relationships I. Intersecting Lines

a. Transversals b. Identifying Angles

II. Parallel Lines and Transversals a. Angle Relationships b. Algebraic Reasoning c. Proof of Parallel Lines

III. Parallel vs. Perpendicular a. Slope of a Line b. Writing Equations in Slope-‐Intercept and Point Slope Form c. Distance between Two Parallel Lines

I. Triangle Classification a. Side Lengths b. Angle Measures c. Isosceles vs. Equilateral

II. Exterior Angles vs. Interior Angles a. Angle Sum Theorem b. Exterior Angle Theorem

III. Congruent Triangles a. Congruency Theorems-‐ SSS, SAS, ASA, AAS b. Right Triangle Congruency-‐LL, HA,LA, HL c. Proofs using above Theorems

IV. Bisectors, Medians, and Altitudes in Triangles a. Perpendicular Bisectors b. Angle Bisectors c. Medians d. Altitudes (inside and outside the triangle)

V. Inequalities a. Angle Measure vs. Side Measure b. Greatest Side or Angle of a Triangle c. Acute, Right or Obtuse by Side Measures


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Unit II: Similarity, Proof, and Trigonometry Big Idea V: Similarity and Visualization Big Idea VI: Measurement Relationships

I. Proportions a. Application and Reasoning b. Scale Factor

II. Similar Polygons and Similar Figures a. Writing and Solving Proportions b. Similarity Postulates for Triangles-‐ AA, SSS, SAS

Fractals

I. Radicals a. Perfect Squares b. Radical Simplification c. Geometric Mean

II. Pythagorean Theorem a. Proof b. Converse c. Special Right Triangles

III. Ratios a. Trigonometric Ratios-‐ Sine, Cosine, Tangent b. Angle of Elevations and Depression c. Law of Sines d. Law of Cosines

Unit III: Extending to Three Dimensions Big Idea VII: Two Dimensional Measurement Big Idea VIII: Three Dimensional Measurement

I. Perimeter a. Triangles b. Quadrilaterals c. Irregular Figures d. Regular Polygons

II. Area a. Triangles b. Quadrilaterals c. Irregular Figures d. Regular Polygons

III. Coordinate Geometry a. Classifying polygon b. Perimeter vs. Area

I. Nets a. Prisms b. Cylinders c. Pyramids d. Cones

II. Surface Area and Volume a. Prisms b. Cylinders c. Pyramids d. Cones e. Spheres


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Unit IV: Circles and Expressing Geometric Properties through Equations

Big Idea IX: Quadrilaterals Big Idea X: Circles I. Polygons

a. Regular vs. Irregular b. Sum of Interior and Exterior Angles

II. Parallelograms a. Properties b. Proofs c. Rectangles, Rhombi, and Squares

III. Trapezoids a. Properties b. Coordinate Geometry c. Medians (Mid-‐segments) d. Other Quadrilaterals (Kites):

I. Circles a. Parts and Properties b. Area and Circumference

II. Arcs a. Properties b. Arc Measure vs. Arc Length c. Arcs and Chords

III. Intersection a. Inscribed Angles b. Concentric Circles c. Tangents and Secants

IV. Equations

Unit V: Applications of Probability

Big Idea XI: Probability I. Permutations and Combinations (Brief Review)

a. Fundamental Counting Principal b. Finding Permutations and Combinations

II. Probability (Brief Review) a. Experimental Probability b. Theoretical Probability c. Probability of Multiple Events

III. Geometric Probability a. Length b. Area c. Using Segments d. Using Areas

CORE CONTENT OVERVIEW: HONORS GEOMETRY

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Core Content Overview

Big Idea I: CC9-‐12.G.CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. CC9-‐12.G.CO.4 Develop definitions of rotations, reflect6ions, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. CC9-‐12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). CC9-‐12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Big Idea II: CC9-‐12.G.CO.9 Prove theorems about lines and angles. CC9-‐12.G.CO.10 Prove theorems about triangles.

Big Idea III:

CC9-‐12.G.CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. CC9-‐12.G.CO.9 Prove theorems about lines and angles. CC9-‐12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). CC9-‐12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point)

Big Idea IV:

CC9-‐12.G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. CC9-‐12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motion. CC9-‐12.G.CO.9 Prove theorems about lines and angles. CC9-‐12.G.CO.10 Prove theorems about triangles CC9-‐12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. CC9-‐12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. CC9-‐12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorems to solve right triangles in applied problems. CC9-‐12.G.SRT.9 Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle

Big Idea V:

CC9-‐12.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not. CC9-‐12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. CC9-‐12.G.SRT.4 Prove theorems about triangles. CC9-‐12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.


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Big Idea VI:

CC9-‐12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

• CC9-‐12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. CC9-‐12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorems to solve right triangles in applied problems.

• CC9-‐12.G.SRT.10 Prove the Law of Sines and the Law of Cosines and use them to solve problems. CC9-‐12.G.SRT.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-‐right triangles (e.g., surveying problems, resultant forces)

Big Idea VII: CC9-‐12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. CC9-‐12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. CC9-‐12.G.MG.3 Apply geometric methods to solve design problems. CC9-‐12.S.CP.1 Describe events as subsets of a sample space using characteristics of the outcomes as unions, intersections, or complements of other events.

Big Idea VIII: CC9-‐12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. CC9-‐12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. CC9-‐12.G.GMD.4 Identify the shapes of two-‐dimensional cross sections of three-‐dimensional objects, and identify three-‐dimensional objects generated by rotations of two-‐dimensional objects.

Big Idea IX:

CC9-‐12.G.CO.11 Prove theorems about parallelograms. CC9-‐12.G.CO.13 Construct and equilateral triangle, a square, and a regular hexagon inscribed in a circle. CC9-‐12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. CC9-‐12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point) CC9-‐12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios.)

Big Idea X:

CC9-‐12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. CC9-‐12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. CC9-‐12.G.C.4 Construct a tangent line from a point outside a given circle to the circle. CC9-‐12.G.GPE.1 Derive the equation of a circle of given center and radius by using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Big Idea XI:

S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.4. Construct and interpret two-‐way frequency tables of data when two categories are associated with each object being classified. Use the two-‐way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school


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Big Idea XI:

Cont.

will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.7. Apply the Addition Rule, P (A or B) = P (A) + P (B) – P (A and B), and interpret the answer in terms of the model. S.CP.8 Apply the general Multiplication Rule in a uniform probability model, P (A and B) = P (A) P (B|A) = P (B) P (A|B), and interpret the answer in terms of the model. S.CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems. S.MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). S.MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of the game).

BIG IDEA I: VISUALIZATION AND MEASUREMENT

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BIG IDEA I: VISUALIZATION AND MEASUREMENT Curriculum Management System

COURSE NAME: HONORS G EOMETRY OVERARCHING GOALS

1. Identify, classify, and use various points, lines, segments, rays, and planes to understand their common and different traits as well as understand the impact of their intersections.

2. Measurement techniques can be used in various ways to find distance, midpoints, and lengths of segments. 3. Segments and Angles can be combined to make larger systems also known as polygons.

ESSENTIAL QUESTIONS What are the building blocks of geometry? How can you describe the attributes of a segment or angle? Why are units of measure important?

SUGGESTED BLOCKS FOR INSTRUCTION: 6


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KNOW UNDERSTAND DO Students will know that: Students will understand that: Students will be able to:

Definitions of a point, line, ray, segment, plane, collinear, coplanar and space. A line is made up of an infinite amount of points. Two lines intersect at a point, two planes intersect to form a line, and three planes intersect at a point in space.

-‐Geometry is a mathematical system built on accepted facts, basic terms, and definitions. -‐Segments, rays, and lines are very similar but each have their own properties and can be combined to form larger figures in the geometric world. -‐ Formulas can be used to find the midpoint and length of any segment on a coordinate plane. -‐Special angle pairs can be used to identify geometric relationships and to find angle measures. Sample Conceptual Understandings Directions: Use the road map below to answer the questions that follow. Assume all roads are lines, segments, or rays.

1. In the map above, Prospect Plains Road represents what kind of geometric figure?

2. Determine which set of roads creates the following angle pairs: vertical angles, linear pair, and complementary angles.

3. Joe drives 5 miles from the Crossroads to Schoolhouse Road. Jenny drives 3 miles from 522 to Perrineville Road. How far did they travel in total to meet up at the corner of Schoolhouse and Perrineville?

4. If Clearbrook Park is at the point (3,3) and The New Monroe Township High School is located at the point (5,7). What is the distance between them? What point would be the best

-‐Draw examples of each vocabulary -‐Model each concept using coordinate geometry. -‐Create a compare and contrast chart listing similarities and differences between each undefined term. -‐Complete a philosophical conversation on the idea of space and how it applies to that around them.

Linear measure is the distance between two points. The formal definition of “between” is used often in geometry and applies to segments directly. Segment addition is the idea of adding two connected segments together to get the length of the larger segment formed. Measurements should be as precise as possible to ensure the most accurate dimensions.

-‐Use various lengths of string to precisely measure and determine the length of the string when two ends are connected with glue. Emphases on various string lengths as to NOT show congruency. -‐Write and solve algebraic expressions using segment addition. -‐Demonstrate the definition of between-‐ness by drawing their own segments with accurate points and measure between the two endpoints. -‐Use the concept of congruency to decipher a bar graph as seen on page 16 in the McGraw Hill text.

The distance and midpoint formula as is applies to points on a coordinate plane.

-‐Compare the method of the distance formula to that of the Pythagorean Theorem. -‐Use the midpoint formula to find the exact middle of a line segment with two endpoints given without drawing a diagram. -‐Find the endpoint of a segment when the midpoint and one other endpoint are given.

A ray is a part of a line. Two opposite rays form a line which is 180 degrees. Two rays with the same endpoint form and angle. Angles can be measured using a protractor. An angle can be named three different ways (vertex, three points, or a

-‐Draw rays and use the proper symbols for each. -‐Name angle three different ways and determine if there is a best way when given various examples. -‐ Use protractors to determine the measure


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number). An angle bisector cuts the angle into two perfectly congruent angles. Angles can be congruent or can be added together to find the measure of the larger angle formed.

meeting point for two friends one at each location? 5. Identify roads that form a hexagon on the map above.

• Answer the following: a) A transformation maps ZOWE onto LFMA. Does the

transformation appear to be an isometry? Explain. b) What is the image of segment ZE? What is the preimage of M?

• Graph the polygon with the given vertices. Then graph its image

for a dilation with center (0,0) and the given scale factor. a) M(-‐3,4), A(-‐6,-‐1), T(0,0), H(3,2) scale factor 5 b) F(-‐4,0), U(5,0), N(-‐2,-‐5) scale factor ½

• Construct a perpendicular bisector using the given directions.

1. Begin with line segment XY.

2. Place the compass at point X. Adjust the compass radius so that it is more than (1/2) XY. Draw two arcs as shown here.

3. Without changing the compass radius, place the compass on point Y. Draw two arcs intersecting the previously drawn arcs. Label

of each angle. -‐Write and solve algebraic equations using the properties of an angle bisector. -‐Demonstrate congruency marks on congruent angles efficiently.

Define and identify right angles, obtuse angles, acute angles, adjacent angles, vertical angles, linear pair, complementary angles, and supplementary angles. A perpendicular is a line segment or ray which intersects another to form a 90 degree angle. Tick marks and arc marcs are a vital part of geometry and must be identified in order to solve problems effectively.

-‐Draw and identify which angle pairs are adjacent, vertical, linear, complementary, supplementary, and perpendicular or a combination of them. -‐Model these relationships on a map of Monroe Township. -‐Write and solve algebraic equations using these properties. -‐Complete error analysis on angle pairs incorrectly identified.

The definition of a polygon is a closed figure whose sides are all segments whose endpoints only intersect two other segments at their endpoints. A polygon can be concave, convex, regular, and irregular. Certain polygons have special names where others are referred to as n-‐gons. You can find the perimeter of a polygon by adding together the lengths of its sides.

-‐Identify polygons to be regular or irregular, convex or concave. -‐Draw regular convex polygons and name each. -‐Find the perimeter of a regular convex figure when given the length of one side. -‐Find the perimeter of a irregular figure on a coordinate plane by finding the length of e ach segment its formed by.


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the intersection points A and B.

4. Using the straightedge, draw line AB. Label the intersection point M. Point M is the midpoint of line segment XY, and line AB is perpendicular to line segment XY.


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21st Century Skills

Creativity and Innovation Critical Thinking and Problem Solving Communication and Collaboration Information Literacy Media Literacy ICT Literacy Life and Career Skills Technology Based Activities http://www.p21.org/index.php?option=com_content&task=view&id=254&Itemid=119 http://www.iste.org/standards/nets-‐for-‐students.aspx

Differentiated Learning Activities

Concept Activity: Chapter Project: Performance Assessment Task Sample GOAL: The goal of this assignment is to properly display the definitions given to you throughout chapter one as

they are the building blocks of the rest of the course. ROLE: You are an author writing a children’s novel to help students with their geometrical retention of definitions. AUDIENCE: The publishing company has asked to you complete your novel for a 5th grade class just being introduced to Points, Lines, Planes, Segments, and Angles. SITUATION: As a young adult approaching middle school students are often overwhelmed by the amount of work piling on top of them. Geometry is a topic build on definitions and understandings. Mrs. Smith has asked her husband’s publishing company to come out with a user friendly guide to help her Geometry students connect the definitions they have been learning to the real world. Since you are such a pro at this, your boss has asked you to write the novel! PRODUCT PERFORMACE AND PURPOSE: Your publishing company is requiring your novel to have several pictures and diagrams. This must be at least 10 pages long with a storyline relating the concepts to real life situations. Make sure you identify the terms you are using and accompany them with diagrams or the BOSS is going to fire you from this important task! Your book should be completely finished and bound by the deadline! STANDARDS AND CRITERIA FOR SUCCESS: Your novel should include: -‐Colored pictures and diagrams -‐Definitions and concepts from chapter 1. -‐Child friendly storyline, easy to read and understand!


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Assessment Models

NOTE: The assessment models provided in this document are suggestions for the teacher. If the teacher chooses to develop his/her own model, it must be of equal or better quality and at the same or higher cognitive levels (as noted in parentheses).

Depending upon the needs of the class, the assessment questions may be answered in the form of essays, quizzes, mobiles, PowerPoint, oral reports, booklets, or other formats of measurement used by the teacher.

Open-‐Ended Assessment:

Instructor will conduct informal open-‐ended assessments through quick-‐writes, learning logs, and daily discussion including eyes closed surveys as to assess student comfort level.

Open-‐Ended (Formative) Assessment:

Group and individual work is assigned daily, from various sources (Synthesis, Analysis, and Evaluation). Introductory and Closing Activities will be done every day to pre-‐assess student knowledge and assess understanding of topics (Synthesis, Analysis,

and Evaluation). Summative Assessment:

Assessment questions should be open-‐ended and should follow the general format illustrated in the Essential Questions/Sample Conceptual Understanding section. (Synthesis, Analysis, Evaluation)

Students will be given quizzes that provide a brief review of the concepts and skills in the previous lessons.

Additional

Resources

Teacher made Performance Assessment Tasks (PATs) Released PATs Online State Resources

BIG IDEA II: REASONING AND PROOF

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BIG IDEA II: REASONING AND PROOF Curriculum Management System

COURSE NAME: HONORS GEOMETRY OVERARCHING GOALS

1. The ability to decipher patterns and determine the next logical term requires the analysis and synthesis of numbers, figures, and various objects. Finding the truths in these can lead to the proof of the hypothesis for the next term.

2. Conditional statements are the first form of proofs, when the hypothesis and conclusion are moved around the statement may or may not still be true. 3. Postulates are geometric truths we assume to be true, they can be used to prove or disprove theories of general approaches.

ESSENTIAL QUESTIONS

How can you make a conjecture and prove that it’s true? Is there a “best practice” to proving the answer is correct?



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The definition of conjecture, counterexample, hypothesis and conclusion.

-‐Observation is the key to making a conjecture about a pattern. -‐Solving equations and stating reasoning for each step can be considered a simple informal proof. -‐You can prove geometric relationships using given information, definitions, properties, postulates, and theorems. Sample Conceptual Understandings • Find a pattern for each sequence, describe the pattern and use it

to show the next two terms.

a) 1000, 100, 10, ___________, ________________ b) 5, -‐5, 5, -‐5, 5,____________, ______________ c) 34, 27, 20, 13, _____________, ___________

• Find a counterexample to show each conjecture is false.

a) The product of an integer and 2 is greater than 2. b) The city of Portland is in Oregon.

• Rewrite each conditional statement as the converse, inverse, and

contrapositive. Determine the true value for each. a) If I have four quarters, then I have a dollar. b) If an angle is obtuse, then its measure is greater than 90

degrees and less than 180 degrees. c) If a figure is a square, then it has four sides.

• What is the name of the property that justifies going from the first

line to the second line?

A B and B CA C

∠ ≅ ∠ ∠ ≅ ∠∠ ≅ ∠

• Fill in the reason that justifies each step. Given: QS=42 x+3 2x Prove: x=13 Q R S

-‐Make conjectures about the next term in a pattern. -‐Create your own patterns using numbers, letters, symbols, and pictures. -‐Find counterexamples to prove statements to be false.

If-‐then statements are often referred to as conditional statements and can be transformed into the inverse, converse, and contrapositive by moving the hypothesis and conclusion around and using negations. These various statements can be true or false.

-‐Write the converse, inverse, and contrapositive of a sentence when given the conditional statement. -‐Determine whether each of these are true or false. -‐Write a conditional statement that proves true for all statements and one that shows false.

Postulates are geometric relationships assumed to be true that are used to solve various proofs including informal proofs called paragraph proofs.

-‐Use Postulates 2.1-‐2.7 as well as the Midpoint Theorem (page 89-‐91 McGraw Hill) to complete several paragraph proofs. -‐Keep a running log of postulates to be memorized at a later date. -‐Use the five essentials of a good proof as seen on page 90 in the McGraw Hill book to complete various proofs.

Properties of Equality for Real Numbers are used to complete formal two-‐ column algebraic and geometric proofs.

-‐Use Equality Properties as seen on page 94 in McGraw Hill to set up and complete a two-‐column algebraic proof. -‐Show angle congruency on a clock by using equality properties and definitions of congruency in a two-‐column proof.

Two-‐column proofs involving segments can be solved using the Segment Addition Postulate as well as the Segment Congruence Properties.

-‐Complete proofs involving segment addition by using the Segment Addition Postulate and Segment Congruence Postulates as seen on page 101-‐102 in the McGraw Hill text.


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Two-‐column proofs involving angle relationships can be solved using the Angle Addition Postulate as well as the Angle Congruence Properties, Supplement Theorem and Complement Theorem.

Statements Reasons

1) QS=42 1)______________________

2) QR+RS=QS 2)______________________

3) (x+3)+2x=42 3)______________________

4) 3x+3=42 4)______________________

5) 3x=39 5)_______________________

6) x=13 6)______________________

-‐Complete proofs angle relationships by using the Angle Addition Postulate and Angle Congruence Postulates, Supplement Theorem and Complement Theorem, as seen on page 107-‐111 in the McGraw Hill text.


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21st Century Skills



Concept Activity: Develop the proof of the distance formula first in an informal proof and then in a formal two-‐column proof. What properties did you have to use? Which method was more beneficial and the easiest to understand? Explain your reasoning in a short paragraph. Chapter Project: Performance Assessment Task Sample Consider the numbers in the pattern below:

The sum of the numbers 1-‐10 is 55. The sum of the numbers 11-‐20 is 155. The sum of the numbers 21-‐30 is 255. a) What is the sum of the numbers 31-‐40? b) What is the sum of the numbers 101-‐110? c) What kind of reasoning did you use in parts (a) and (b)? d) The following is a development of a formula for the sum of n consecutive integers

The sum of n integers x to y ( 1) ( 2) ... ( 2) ( 1)S x x x y y y= + + + + + + − + − + The same sum in reverse order ( 1) ( 2) ... ( 2) ( 1)S y y y x x x+ = + − + − + + + + + + Add the equations. 2 ( ) ( ) ( ) ... ( ) ( ) ( )S x y x y x y x y x y x y= + + + + + + + + + + + + There are n terms of (x + y). 2 ( )S n x y= + Divide each side by 2. ( )

2n x yS +=

Use the formula to find the sum of the numbers from 101-‐110. e) What kind of reasoning did you use in part (d)?


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Assessment Models










Additional

Resources

Teacher made Performance Assessment Tasks (PATs) Released PATs Online State resources

BIG IDEA III: REASONING AND MEASUREMENT

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BIG IDEA III:REASONING AND MEASUREMENT Curriculum Management System


1. Some attributes of geometric figures, such as length, area, volume, and angle measure are measurable. Units are used to describe these attributes. 2. Definitions establish meanings and remove possible misunderstanding. Other truths are more complex and difficult to see. It is often possible to verify complex

truths by reasoning from simpler one by using deductive reasoning. 3. A coordinate system on a line is a number line on which points are labeled, corresponding to real numbers. A coordinate system in a plane is formed by two

perpendicular number lines, called the x-‐axis and y-‐axis, and the quadrants they form. The coordinate plane can be use to graph many functions. It is possible to verify sine complex truths using deductive reasoning in combination with Distance, Midpoint, and Slope formulas.

ESSENTIAL QUESTIONS How do you write the equation of a line in a coordinate plane? How do you prove that two lines are parallel or perpendicular?



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Parallel means two objects going on forever without intersecting on the same plane. If not on the same plane they are considered skew lines. Planes can also be parallel. When two lines are intersected at two different places by a line this line is called a transversal. A transversal creates four special angle pair relationships, alternate exterior, alternate interior, corresponding, and consecutive interior.

Not all lines and all planes intersect. When a line intersects two or more lines, the angles formed at the intersection points create special angle pairs. -‐The special angle pairs formed by parallel lines and a transversal are either congruent or supplementary. -‐Certain angle pairs can be used to decide whether two lines are parallel. -‐The relationships of two lines to a third line can be used to decide whether two lines are parallel or perpendicular to each other. -‐The sum of the angles of a triangle is always the same. -‐ A line can be graphed and its equation written when certain facts about the line, such as its slope and a point on the line are known. -‐Comparing the slopes of two lines can show whether the lines are parallel or perpendicular. Sample Conceptual Understandings • Identify ALL the numbered angles for each special angle pair given:

• If 1 110m∠ = ° find the measure of each missing angle in the diagram

seen below.

• Identify the pairs of parallel and perpendicular lines in the diagram

below.

-‐Name segments that are skew and segments that are parallel when given a prism. -‐ Identify a transversal when given various sets of lines and points on them. -‐ Complete error analysis when identifying special angle pair relationships. -‐Create application problems in which they must use and identify special angle pair relationships within a diagram using a transversal.

Parallel lines intersected by a transversal from special angle pairs that are either congruent or supplementary. Alternate interior, alternate exterior and corresponding angles are congruent. Consecutive interior angles are supplementary. If the transversal is also a perpendicular all angles formed are 90 degrees.

-‐Identify parallel lines by using indicators and symbols. -‐Determine special angle pairs and their measure. -‐Write and use algebraic expressions to find the value of angle measures. -‐Create an auxiliary line to solve more complicated problems.

Slope is the ratio of its vertical rise to horizontal run. Slope can also be considered the rate of change describing how a quantity changes over time. The slope formula can be used to find the slope between two coordinate points. A vertical line has a slope which is undefined and a horizontal line has a slope of zero. We can use the slope formula to determine if two or more lines are parallel, perpendicular, or neither. Slopes with the same value are parallel, slopes that are opposite reciprocals of one another are perpendicular and slopes with no relation to one another are neither.

-‐Use the slope formula to determine the slope between two given points. -‐Use the rise over run method to count slope when given the graph of a line. -‐Recognize that slope is also considered the rate of change when completing a word problem. -‐Implement the slope formula or rise over run method to determine if more than one line is parallel, perpendicular or have no relation to each other.


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There are two formulas that can be used to write the equation of a line on a coordinate plane. Slope-‐intercept form is best used when the problem gives you a slope and the y-‐intercept of the line. Point-‐slope form can be used when given one point and the slope. Either form can be used when given two points, or when asked to write the equation of a line parallel or perpendicular to a given line through a given point.

• What is the value of x for which, m nP ?

• Find the slope of line the passing through points (-‐7, 2) and (-‐7,5). • Write an equation of the line:

a) 1 , 122

m b= − =

b) 3, (1, 9)m = − c) (4,2) (3, 2)and −

• What is the equation of the line perpendicular to y= 2x -‐ 5 containing (1,-‐3)?

• Given a point and a line, construct a line parallel to the line given through the point given.

1. Begin with point P and line k.

2. Draw an arbitrary line through point P, intersecting line k. Call the intersection point Q. Now the task is to construct an angle with vertex P, congruent to the angle of intersection.

-‐Recognize the best formula to use to write the equation when given specific information. -‐Use the slope formula along with the slope-‐intercept formula and point-‐slope formula to write equations of lines. -‐Write equation of a line when given the point and slope. -‐Write the equation of a line when given two points. -‐Write the equation of a line when given one point and an equation. -‐Decipher information and use linear equations to solve application problems in the everyday world.

When special angle pairs are shown to be congruent or supplementary, the lines that were intersected by a transversal are parallel. If they are not congruent and supplementary the lines intersected by the transversal are not parallel. Formal two column proofs can be written to prove lines parallel when given a diagram and specific information.

-‐Identify parallel lines by using the properties of alternate interior, alternate exterior, consecutive interior, and corresponding angles. -‐ Write and use algebraic equations to find the value of the variable in which makes the lines parallel. -‐Use the slope formula to determine if two lines are parallel within a coordinate plane. -‐Form a two column proof using definitions, postulates, and theorems to prove lines to be parallel in a given diagram.

The shortest distance from a point to a line is a perpendicular drawn from the point to that line. Two lines are parallel if every point on one line is equidistant from the corresponding point on the second line. We can find the distance between two parallel lines by writing the equation of the line perpendicular

-‐Draw altitudes to represent the shortest distance from a point to a line. -‐Find the distance between two parallel lines. -‐Use the slope formula to find the slope perpendicular.


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to them and using the distance formula.

3. Center the compass at point Q and draw an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc.

4. Set the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R.

5. Line PR is parallel to line k.


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21st Century Skills



Concept Activity: Create an art design in which parallel lines and a transversal are present. Label angles in your design with letters or color codes. Write a key describing the types of angle relationships shown. Chapter Project: Graphing Calculator Activity: Parallel Lines l and m are cut by a transversal t. The equations of l, m, and t are 1 14, 6,

2 2y x y x= − = − and 2 1y x= − + , respectively.

Use a graphing calculator to determine the points of intersection of t with l and m. Performance Assessment Task Sample Task One: You want to put tape on the ground to mark the lines of a volleyball court. What is the most efficient way

to make sure that the opposite sides of the court are parallel? Support your response with diagrams and a short essay explaining your process. Task Two: AB

contains points A(-‐6,-‐1) and B(1,4). CD

Contains point D (7, 2). If ABC BCD∠ ≅ ∠ and

90,m ABC∠ = what is the equation for CD

? (HINT: Sketch a graph)


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Assessment Models










Additional

Resources


BIG IDEA IV: VISUALIZATION, CONGRUENCY, AND RELATIONSHIPS

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BIG IDEA IV: VISUALIZATION, CONGRUENCY, AND RELATIONSHIPS Curriculum Management System


1. Visualization can help you connect properties of real objects with two-‐dimensional drawings of these object and classification as well. 2. Definitions establish meanings and remove possible misunderstanding. Other truths are more complex and difficult to understand. It is often possible to verify

complex truths by reasoning from simpler ones using deductive reasoning. 3. A coordinate system in a plane is formed by two perpendicular number lines, called the x-‐ and y-‐axes and the quadrants they form. It is possible to verify some

complex truths using deductive reasoning in combination with Distance, Midpoint, and Slope Formulas. ESSENTIAL QUESTIONS

Where do we see classification used in concepts involving mathematics? How do you identify corresponding parts of congruent triangles? How do you show that two triangles are congruent? How can you tell if a triangle is isosceles or equilateral? How do you use coordinate geometry to find relationships within triangles? How do you solve problems that involve measurements of triangles?



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Triangles are classified by their side lengths and angle measures. Equilateral equiangular triangles are often called an equilateral triangle, which means they have all congruent sides and 60-‐degree angles.

-‐Classification is used in all facets of mathematics and can be used as a first step to dissecting a problem. -‐The use of visualization, tick marks, and arc marks show corresponding sides and angles in congruent triangles. -‐You can prove two triangles congruent using the Congruence Postulates (SSS, SAS, ASA, AAS, HL). -‐Isosceles and equilateral triangles are extremely similar but have unique differences, which enable us to tell them apart. -‐Triangles play a key role in relationships involving perpendicular bisectors and angle bisectors. -‐There are special parts of a triangle that are always concurrent. A triangles three perpendicular bisectors are always concurrent, as are a triangles three angle bisectors, its three medians, and its three altitudes. -‐The measures of angles of a triangle are related to the lengths of the opposite sides. Sample Conceptual Understandings • Fill in the blanks given the congruency statement

.RSTUV KLMNO≅ 1) _____TS ≅ 2) _____N∠ ≅ 3) _____LM ≅ 4) _____VUTSR ≅ • Which postulate, if any, could you use to prove the two triangles

congruent? If there is not enough information to prove the triangles congruent write, not enough information.

-‐Classify triangles by using the side classification and angle classifications. -‐Contemplate the idea of a triangle that is impossible to make (ex. Right equilateral) -‐Use the properties of an equilateral or isosceles triangle to set up algebraic equations and solve for missing side measures. -‐ Recall the distance formula is used to find the lengths of the sides of a triangle on a coordinate plane and can assist in classifying the triangle by its side lengths.

The Angle Sum Theorem states that any triangle has three angles that will add up to 180 degrees. The Exterior Angle Theorem states the exterior angle of a triangle is congruent to the sum of the remote interior angles inside the triangle. Deductive reasoning can be used to determine the types of angles in a triangle when given one angle.

-‐Write and solve algebraic equations using the Angle Sum Theorem to find the measures of the angles in a triangle. -‐Write and solve algebraic equations using the Exterior Angle Theorem to find the measure of the exterior angle and remote interior angles. -‐Use deductive reasoning in informal and formal proofs to determine the classification of missing side and angle measures when given certain information.

4.3

Congruent triangles have corresponding congruent sides and angles. Congruency statements can be written to show which sides and angles are congruent. Triangle congruence have reflexive, symmetric, and transitive properties that can be used in formal proofs. Congruence is preserved through transformations such as slides, flips, and turns.

-‐Write congruency statements when given two congruent figures. -‐List congruent sides and angles in two congruent figures when given only the congruency statement. -‐Writing and solving algebraic equations after showing triangles congruent. -‐ Complete two column proofs to show three triangles congruent. -‐Verify triangle transformations on a coordinate plane by using the distance formula. -‐Use a protractor to measure the angles of a triangle transformation on a coordinate


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• Given : ,LN KM KL ML⊥ ≅ Prove: Triangle KLN is congruent to Triangle MLN

• P is the incenter of , 20 .XYZ m XYP∠ = °V Find the measure of the

indicated angles.

)))

a PXYb XYZc PZX

∠∠∠

• Triangle PQR has medians QM and PN that intersect at Z. If ZM = 4, find QZ and QM.

• In triangle ABC below, P is the centroid.

a) If PR=6, find AP and AR. b) If PB=6, find QP and QB. c) If SC=6, find CP and PS.

plane.

The Side-‐Side-‐Side (SSS) postulate can be used to prove two triangles congruent when no angles are given and only side measures are shown. The Side-‐Angle-‐Side (SAS) postulate can be used to prove two triangles congruent when two corresponding sides are congruent and their included angle is the same measure. The included angle is the angle formed by the two congruent sides.

-‐Recognize triangles, which can be proven congruent by using the SSS postulate. -‐ Use the SSS postulate and the distance formula to prove triangles congruent on a coordinate plane. -‐Recognize triangles, which can be proven congruent by using the SAS postulate. -‐Use the congruence postulates along with the congruence properties to develop two column proofs regarding congruent triangles.

The Angle-‐Side-‐Angle (ASA) postulate can be used to prove two triangles congruent when two corresponding congruent angles are given and the corresponding sides between them are congruent. The Angle-‐Angle-‐Side (AAS) postulate can be used to prove two triangles congruent when two corresponding angles are congruent and a corresponding side that follows is congruent. The Hypotenuse-‐Leg (HL) postulate is only used in right triangles. Two right triangles can be proven congruent if the corresponding hypotenuse and leg are congruent in both triangles.

-‐Recognize triangles, which can be proven congruent by using the ASA postulate. -‐ Use the ASA postulate and congruence properties to complete two column proofs. -‐Recognize triangles, which can be proven congruent by using the ASA postulate. -‐ Use the ASA postulate and congruence properties to complete two column proofs. -‐Utilize vertical angles and shared sides to demonstrate them as possible congruent parts. -‐Prove why HL only works in right triangles and cannot be referred to as Angle-‐Side-‐Side.

Isosceles triangles have two congruent sides called legs. The side not congruent is called the base. The angles across from the congruent angles are called base angles and are also congruent. The remaining angle is called the vertex angle. These properties can be used to prove a triangle to be isosceles.

-‐Complete a two-‐column proof, which proves the two angles across from the congruent sides are also congruent. -‐Use properties of isosceles base angles to find missing angle measures in triangles. -‐Write and solve algebraic equations using the properties of an isosceles triangle to find missing side and angle measures.


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A perpendicular bisector of a side of a triangle is a line, segment, or ray that passes through the midpoint of a side and is perpendicular to that side. When three or more lines intersect at common point these lines are called concurrent line and they meet at a point of concurrency. Three perpendicular bisectors meet at the circumcenter . The circumcenter is equidistance from each of the vertices of the triangle. Three angle bisectors intersect to meet at the incenter. The incenter is equidistant from each of the sides of the triangle. A median is a segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side. Three medians meet at a centroid. The centroid is located 2/3 the distance from the vertex to the midpoint of the opposite side. An altitude of a triangle is a segment from a vertex to the opposite side forming a right angle. Three altitudes meet at the orthocenter.

• Error Analysis: Point O is the incenter of a scalene triangle XYZ.

Your friend says that .m YXO m YZO∠ = ∠ Is your friend correct? Explain.

• In triangle RST 70m R∠ = and the 80.m S∠ = List the sides and angles in ascending order.

• Is it possible to have a triangle with the given side lengths? a) 5 in, 8 in, 15 in b) 10cm, 12cm, 20cm c) The lengths of two sides of a triangle are 12 and 13. Find the

range of possible side lengths for the third side. Use the figure below.

• Copy the diagram below. Then draw the image of triangle ZXY for a 90 degree rotation about P. Label the vertices of the image using prime notation.

• Given a line segment as one side, construct an equilateral triangle.

1. Begin with line segment TU.

2. Center the compass at point T, and set the compass radius to

TU. Draw an arc as shown

3. Keeping the same radius, center the compass at point U and draw another arc intersecting the first one. Let point V be the

-‐Identify different concurrent lines and their points of concurrency. -‐Use the properties of the points of concurrency to solve algebraic problems. -‐Use systems of equations, distance formula, slope formula, and midpoint formula to find points of concurrency on a coordinate plane. -‐Form a table in which they special segments in a triangle are listed, the type of segment is identified, and their point of concurrency is determined.

Inequalities can be written when comparing two quantities of different size. The Exterior Angle Inequality Theorem states that the measure of the exterior angle must be greater than each of the two remote interior angles individually. The length of the side corresponds to the size of the angle. The longest side is across from the largest angle, the shortest side is across from the smallest. An inequality can be written to compare side lengths and angle measures. -‐Use the relationship between side length and angle measure to complete real life

Write and use basic inequalities. -‐Perform algebraic operations on inequalities. -‐Use deductive reasoning to write inequalities involving angle measures and the exterior angle. -‐Order side lengths and angle measures in ascending or descending order. This should be shown in an inequality -‐Use the relationship between side length and angle measure to complete real life application problems.


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application problems.

point of intersection.

4. Draw line segments TV and UV. Triangle TUV is an equilateral triangle, and each of its interior angles has a measure of 60°.

The Triangle Inequality Theorem states the sum of any two sides of a triangle is greater than the length of the third side. An inequality can be written to determine the length of the third side of a triangle when given the other two sides.

-‐Identify whether measures given can be the side lengths of a triangle. -‐Use standardized test questions to determine a possible side length of a triangle when only given two measurements. -‐Write and solve application problems involving the triangle inequality.

Not all side lengths and angle measures need to be given to write an inequality comparing two triangles. The Hinge Theorem and SSS Inequality Theorem allow us to write inequalities with minimal but specific information given.

-‐Write an informal proof using the Hinge Theorem (SAS) to prove a comparison in two triangles. -‐Use the Triangle Inequality Theorem to write and solve an algebraic expression. -‐Complete an application problem involving writing and solving triangle inequalities within two triangles.


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21st Century Skills



Concept Activity:

Use the following instructions to construct the figure needed. Use Geometers SketchPad to construct AB

and AC

.Construct BC to form .ABCV Construct a line parallel

to BC that intersects AB

and AC

at points D and E to form .ADEV 1) Are the three angles in ABCV congruent to the three angles in ?ADEV

Manipulate the figure to change the positions of DE and BC 2) Do the corresponding angles of the triangles remain congruent? 3) Are the two new triangles congruent? 4) Can the two triangles be congruent?

Concept Activity: Complete the following activity using Geometer’s SketchPad. -‐Construct a triangle and the three perpendicular bisectors of its sides. -‐Construct a triangle and its three angle bisectors. -‐Construct a triangle. Through the vertex of the triangle construct a segment that is perpendicular to the line containing the side opposite that vertex. This is called the altitude. Construct altitudes for the other two vertices. -‐ Construct a triangle and the three medians of the triangle.

a) What property do the lines containing altitudes and the medians seem to have? b) State your conjectures about the lines containing the altitudes and about the medians. c) Think about acute, right, and obtuse triangles. Fill in the table below with inside, on, or outside to describe the location of each point of concurrency.

Perpendicular Bisectors

Angle Bisectors Lines Containing the Altitudes

Medians

Acute Triangle Right Triangle Obtuse Triangle d) What observations, if any, can you make about the special segments for isosceles triangles? For equilateral triangles?

Chapter Project: It’s your job to create a study guide for a friend who doesn’t quite understand triangles and congruence as much as you do! You are going to make a foldable for your friend to cover each topic in classifying and determining congruent triangles! To make a foldable:

1. Take a stack of paper and hold it long-‐ways.


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2. Fold right hand corner over to opposite side of the paper to create a right triangle. 3. Cut excess paper off bottom. 4. Staple across the hypotenuse.

You want to label each page of your foldable. Make sure you include notes, definitions, examples, and any helpful hints in your foldable journal so your friend has the ultimate study guide! You can also add your personal thoughts on each topic whether you thought it was easy and grasped it right away or had trouble and it’s something your friend should study very hard to understand. Make sure your foldable is neat, precise, and easily understood.


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Performance Assessment Task Sample Task One: Triangle GAB is isosceles with vertex angle A and triangle BCD is isosceles with vertex angle C. Is triangle BGH congruent to triangle BDH? Justify your reasoning in a short paragraph.

Task Two: You and some neighbors are landscaping a community park. The organizer of the project selects an area for two congruent triangular rock gardens. You agree to be in charge of placing the pieces of wood to outline the gardens. The only tools you have are a saw, a protractor, and two very long pieces of wood. Describe one way to guarantee that the triangular outlines will be congruent. Justify your answer in a informal proof. Task Three: Your math teacher manages a campground during summer vacation. He loves math so much that he has mapped the campground on the coordinate grid. The campsites have the following coordinates: Brighton Bluff at B(2,2), Pomona Peak at P(4,10) and Harmony Hill at H(12,2). He wants to build showers that are equidistant from all three campsites. Find the coordinates of the point where the showers should be placed. Task Four:

a) Draw triangle ABC with obtuse C∠ and construct its orthocenter O. Then find the orthocenter of triangles ABO, ACO, and BCO. What did you discover? Explain how you came to this result.

b) Will your conjecture be true for any acute or right triangle ABC? Explain your reasoning. Task Five: In triangle ABC, .AB BC≠ Show that there does not exist a point P on altitude BD that is equidistant from A and C.


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Assessment Models










Additional

Resources


BIG IDEA V: SIMILARITY AND VISUALIZATION

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BIG IDEA V: SIMILARITY AND VISUALIZATION Curriculum Management System


1. Two geometric figures are similar when corresponding lengths are proportional and corresponding angles congruent. 2. Definitions establish meanings and remove possible misunderstanding. Other truths are more complex and difficult to se. It is often possible to verify complex

truths by reasoning from simpler ones by using deductive reasoning. 3. Visualization can help you connect properties of real objects with two dimensional drawings of these.

ESSENTIAL QUESTIONS How do you use proportions to find side lengths in similar polygons? How do you show two triangles are similar? How do you identify corresponding parts of similar triangles?



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A ratio is a comparison of two quantities. A proportion is an equation stating that two ratios are equal. Proportions can set up and ratios cross-‐multiplied to determine if there is equivalence. Proportions can also be set up with a variable that can be solved for by cross-‐multiplying. This variable can represent a missing quantity we want to find.

-‐An equation can be written stating that two ratios are equal, and if the equation contains a variable, it can be solved to find the value of the variable. -‐ Ratios and proportions can be used to decide whether two polygons are similar and to find unknown side lengths of similar figures. -‐When two or more parallel lines intersect other lines, proportional segments are formed. -‐Triangles can be similar based on the relationship of two or three corresponding parts. Sample Conceptual Understandings • A high school has 16 math teacher for 1856 math students. What

is the ratio of math teachers to math students? • The polygons are similar. Write a similarity statement and give the

scale factor.

• A 3-‐ft vertical post casts a 24-‐in shadow at the same time a pine

tree cast a 30 foot shadow. How tall is the pine tree? • Are these triangles similar? How do you know?

• Find the value of x in the figure below

-‐Practice writing ratios as fractions and with a colon. -‐Use ratios to find angle measures in a triangle. -‐Use ratios to find the missing side lengths when given the perimeter of a triangle. -‐Cross multiply the means and the extremes to determine if the cross product of a proportion is equal. -‐Set up and solve proportions in which a variable needs to be found. -‐Solve application problems by setting up and solving algebraic proportions.

Polygons are similar if their corresponding angles are congruent and corresponding sides proportional. Similarity statements can be written to show congruent angles and corresponding proportional sides. Scale factor is the ratio by which two similar figures are compared.

-‐Write proportions of corresponding parts to determine if two figures are similar. -‐Use scale factor in application problems to determine the size of scaled models. -‐Solve for missing side lengths of similar figures by using scale factor and writing proportional statements.

Triangles can be similar when their angles are congruent and sides are proportional. We can determine similarity of triangles by writing proportions, using the AA, SSS, and SAS Similarity. Similarity of triangles is reflexive, symmetric, and transitive.

-‐Complete paragraph proofs comparing three triangles to see if they are similar. -‐Use the AA, SSS, and SAS Similarity Theorems and write proportions to find whether two or more triangles are similar. -‐Write and solve algebraic proportions in conjunction with the similarity theorems. -‐Complete application problems using indirect measurement.

Triangles can be formed using parallel lines and transversals. The parallel lines split the triangle into proportional parts. The converse of this is also true. A midsegment of a triangle is a segment whose endpoints are the midpoints of the

-‐Complete a two-‐column proof in which the Triangle Proportionality Theorem is used. -‐Find the length of a missing side by writing proportions. -‐Determine if two lines in a triangle are parallel by completing a proportion.


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sides of a triangle. The midsegment is one half the length of the side it is parallel to. When three or more parallel lines are intersected by two different transversals they cut the transversals proportionally.

•

Given: CD bisects ACB∠ , AE is parallel to CD

Prove: AD BDAC BC

=

Grph the polygon with the given vertices. Then graph its image for a dilation with center (0,0) and the given scale factor.

c) M(-‐3,4), A(-‐6,-‐1), T(0,0), H(3,2) scale factor 5 d) F(-‐4,0), U(5,0), N(-‐2,-‐5) scale factor ½

-‐Use coordinate geometry to find coordinates of the midsegment. -‐Use coordinate geometry to determine whether two lines are parallel in a triangle. -‐Verify using coordinate geometry that the midsegment is one half the length of its parallel side. -‐Complete application problems in which segments are divided proportionally.

6.5

When two triangles are similar their perimeters are proportional and the special segments inside of them are proportional. Angle bisectors can also create proportional parts.

-‐Use application problems to find the perimeter of two similar figures. -‐Use previous knowledge of special segments in triangles to write proportions and solve for corresponding sides of the triangles. -‐Write paragraph proofs to prove proportional parts. -‐Write and solve algebraic proportions involving medians to determine the length of corresponding sides. -‐Use the Angle Bisector Theorem in proofs of proportional parts.

C

A B

E

D


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21st Century Skills



Concept Activity: In the Fibonacci sequence, each term after the first two terms is the sum of the preceding two terms. The first six terms are 1, 1, 2, 3, 5, and 8.

a) What are the next nine terms in the Fibonacci sequence?

b) Starting with the second term, the ratios of each term to the previous term for the first six terms are 1 2 3 51, 2, 1.5, 1.666...,1 1 2 3= = = = and 8 1.6.

5= What are

the next nine ratios rounded to the nearest thousandth? c) Compare the ratios you found in question (b). What do you notice?

Use the coordinates given to graph the image of MNOP for dilation with center (0,0) and a scale factor of 2. Use coordinate geometry and the definition of similar polygons to prove that MNOP is similar to its image. Answer the questions that follow. M(3,2), N(3,0), O(0,0) ,P(1,3)

1. How can you find the new coordinates? 2. How can you verify the new coordinates are correct? 3. How can you prove the polygons are similar?

Performance Assessment Task Sample Task One: In the diagram below segment AC is parallel to DF is parallel to BH and segment CB is parallel to


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segment FE.

a) Find four similar triangles. Explain how you know that they are similar. b) Using the similar triangles you found in part (a), complete the following extended proportion.

AB DEAC DG

= = =

Task Two: You are making the kite shown below from five pairs of congruent panels. Use the given information to find the side lengths of the kite’s panels in (a)-‐(d) below.

ABCD is a kite. EB= 15 in, BC= 25 in Segment EX is perpendicular to segment BC and segments EX, YF, and GZ are parallel The extended ratio XY:YZ:YC is 3:1:4 Find the side lengths of each panel of the kite. a) Triangle BEX b) XEFY c) YFGZ d) Triangle ZGC


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Assessment Models





Open-‐Ended (Formative) Assessment: Group and individual work is assigned daily, from various sources (Synthesis, Analysis, and Evaluation). Introductory and Closing Activities will be done every day to pre-‐assess student knowledge and assess understanding of topics (Synthesis, Analysis,




Additional

Resources


BIG IDEA VI: MEASUREMENT RELATIONSHIPS

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BIG IDEA VI: MEASUREMENT RELATIONSHIPS Curriculum Management System


1. Some attributes of geometric figures, such as length, area, volume, and angle measure are measurable. Units are used to describe these attributes. 2. Two geometric figures are similar when corresponding lengths are proportional and corresponding angles are congruent.

ESSENTIAL QUESTIONS

How do you find a side length or angle measure in a right triangle? How do trigonometric ratios relate to similar right triangles?



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The geometric mean between two numbers is the positive square root of their product. When an altitude is drawn to the hypotenuse of one triangle three similar right triangles are formed.

-‐Geometric mean describes the relationship between the altitude and hypotenuse of a right triangle. -‐If lengths of any two sides of a right triangle are known, the length of the third side can be found by using the Pythagorean Theorem. -‐Certain right triangles have properties that allow their side lengths to be determined without using the Pythagorean Theorem. -‐If certain combinations of side lengths and angle measures of a right triangle are known, ratios can be used to find the other side lengths and angle measures. -‐Ratios can be used to find side lengths and angle measures of a right triangle when certain combinations of side lengths and angle measures are known. -‐The angles of elevation and depression are the acute angles of right triangles formed by a horizontal distance and a vertical height. -‐The side and angle measures of any triangle can be found using the Law of Sines or Law of Cosines when given a specific combination of sides and angles. Sample Conceptual Understandings • Find the geometric mean between the given numbers

a) 10 and 30 b) 4 and 16 c) 20 and 35

• Determine whether the given three sides form a right triangle. If yes, are they a Pythagorean Triple? a) 6,8,10 b) 15, 25, 40

c) 3 4, ,15 5

• Use special right triangles to find the missing side lengths of each.

-‐Find the geometric mean when given two numbers. -‐Identify and find the geometric mean when given a diagram. -‐Write a similarity statement for the three right triangles formed. -‐Find missing side lengths of a right triangle when an altitude is drawn.

The Pythagorean Theorem can be used on right triangles to determine the missing side lengths. If three lengths are given the converse of the Pythagorean Theorem can be used to determine if the triangle is right.

-‐Use the Pythagorean Theorem to determine the missing side lengths. -‐Use the Pythagorean Theorem in conjunction with geometric mean to find missing side lengths in a right triangle. -‐Complete a tiered activity in which the Pythagorean Theorem and its converse is used. -‐Verify that a triangle is right triangle on the coordinate plane. -‐Identify and use Pythagorean Triples to determine right triangles.

There are certain right triangles in which a pattern can be formed to find their side lengths. These triangles have angle measures of 45-‐45-‐90 or 30-‐60-‐90.

-‐Use properties of 45-‐45-‐90 triangles to find missing side lengths and the missing hypotenuse. -‐Use properties of 30-‐60-‐90 triangles to find missing side lengths and the missing hypotenuse. -‐Determine if a triangle on a coordinate plane is a special right triangle by using its side measures.

7.4

Trigonometry is used to find missing side lengths and angles measures when given a certain combination of sides and angles. A calculator is used to evaluate each expression.

-‐Identify and use trigonometric ratios to find side lengths when given an angle measure and one other side in a right triangle. -‐Find sine, cosine, and tangent ratios when given a triangle. -‐Use a calculator to evaluate expressions. -‐Use trigonometric ratios to find angle

x

3 3

3


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• Find the value of x. Round to the nearest tenth.

• While flying a kite Linda lets out 45 ft. of string and anchors it to

the ground. She determines that the angle of elevation of the kite is 58° . What is the height of the kite from the ground?

• A woman stands 15 ft from a statue. She looks up at an angle of 60° to see the top of her statue. Her eye level is 5 ft. above ground. How tall is the statue to the nearest foot?

• A whale watching boat leaves port and travels 12 miles due north. Then the boat travels 5 miles due east. In what direction should the boat head to return to port?

measures. -‐Find missing side and angle measures of a triangle on a coordinate plane.

-‐The angle of elevation is the angle between the horizontal and the line of sight when an observer is looking up. The angle of depression is the angle formed with the horizontal and the line of sight when the observer is looking down. The angle of elevation and angle of depression are congruent and can be used when solving application problems.

-‐ Solve application questions in which the angle of depression is needed or must be found. -‐ Solve application questions in which the angle of elevation is needed or must be found. -‐Effectively draw diagrams depicting the application problem. -‐Create their own problems to be shared with a group or the class as a whole.

-‐The Law of Sines and Law of Cosines can be used to find parts of ANY triangle that is not a right triangle. The Law of Sines and Law of Cosines require different types of information and can be used to find missing side lengths and angles measures. Solving a triangle means finding all the missing side and angle measures.

-‐Write and solve ratios with the proper amount of information using the Law of Sines (AAS. ASA, SSA). -‐Use the Law of Sines to solve problems involving indirect measurement. -‐Use the Law of Cosines to solve application problems when given 3 sides or one angle and two sides. -‐Solve triangles for all missing information using the Law of Sines and/or the Law of Cosines.

8

6431

x

x

22

32


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21st Century Skills



Concept Activity: Comparing Magnitude and Components of Vectors -‐Draw vector a in standard position. -‐Draw vector b in standard position with the same direction as a, but with a magnitude twice the magnitude of vector a.

1. Write vector a and vector b in component form. 2. What do you notice about the components of vector a and vector b? 3. Draw vector b so that its magnitude is three times that of vector a. how do the components of vector a and vector b compare? 4. Describe the vector magnitude and direction of a vector (x,y) after the components are multiplied by n.

Exploring Trigonometric Ratios

Use geometry software to construct AB

and AC

so that A∠ is acute. Through a point D on AB

, construct a line perpendicular to AB

that intersects AC

in point E. Moving point D changes the size of triangle ADE. Moving point C changes the size of A∠ . Exercises

1. –Measure A∠ to find the lengths of the sides of triangle ADE. -‐Calculate the ratio leg opposite A

hypotenuse∠ which is .ED

AE

-‐Move point D to change the side of triangle ADE without changing m A∠

2. –Move point C to change m A∠ a. What do you observe about the ratio as m A∠ changes? b. What does the ratio approach as m A∠ approaches 0? As m A∠ approaches 90?

3. -‐Make a table that shows the value s for m A∠ and the ratio of leg opposite Ahypotenuse

∠ . In your table, include 10, 20, 30,….80 for the m A∠ .

-‐Compare your table with a table of trigonometric ratios.

Do your values for leg opposite Ahypotenuse

∠ match the values in one of the columns of the table? What is the name of this ratio in the table?


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Performance Assessment Task Sample Task One: The diagram below shoes equilateral triangle ABC sharing a side with square ACDE. The square has side lengths of four. What is BE? Justify your answer.

Task Two: A construction crew wants to hoist a heavy beam so it is standing up straight. They tie a rope to the beam, secure the base, and pull the rope through a pulley to raise one end of the beam from the ground. When the beam makes an angle of 40 degrees with the ground, the top of the beam is 8ft above the ground. The construction site has some telephone wires crossing it. The workers are concerned that the beam may hit the wires. When the beam makes an angle of 60 degrees with the ground the wires are 2ft above the top of the beam. Will the beam clear the wires on its way to standing up straight? Explain.


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Assessment Models










Additional

Resources


BIG IDEA VII: TWO DIMENSIONAL MEASUREMENT

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BIG IDEA VII: TWO DIMENSIONAL MEASUREMENT Curriculum Management System


1. Some attributes of geometric figures, such as length, area, volume, and angle measure, are measureable. Units are used to describe these attributes. 2. Two geometric figures are similar when corresponding lengths are proportional and corresponding angles are congruent. Areas of similar figures are

proportional to the squares of their corresponding lengths. ESSENTIAL QUESTIONS

How do you find the area of a polygon or find the circumference or area of a circle? How do perimeters and areas of similar polygons compare?



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A parallelogram is a quadrilateral with both pairs of opposite sides parallel and congruent. Any side can be called a base. For each base, there is a corresponding altitude that is perpendicular to the base. The area of a parallelogram is found by multiplying its base and its height.

-‐The area of a parallelogram or a triangle can be found when the length of its base and height are known. -‐The area of a trapezoid can be found when the height and the lengths of its bases are known. -‐The area of a rhombus or kite can be found when lengths of the diagonals are known. -‐The area of a regular polygon is a function of the distance from the center to a side. -‐Trigonometry can be used to find the area of a regular polygon when the length of a side, radius, or apothem, is known or to find the area of a triangle when the length of two sides and the included angle is known. -‐The length of part of a circle’s circumference can be found by relating it to an angle in a circle. -‐The area of parts of a circle formed by radii and arcs can be found when the circle’s radius is known. -‐Ratios can be used to compare the perimeters and area of similar figures. Sample Conceptual Understandings • Find the area of each figure.

• Find the area of each regular polygon.

-‐Find the area of a parallelogram. -‐Find the perimeter of a parallelogram. -‐Use the area of a parallelogram to solve real world problems. -‐Find the area of a parallelogram on a coordinate plane.

Multiplying the base and the height then taking the product and dividing by two allows you to find the area of a triangle. The formulas for the areas of a trapezoid and a rhombus are related to that of a triangle. By taking the sum of the bases multiplying it by the height and dividing by tow the area of a trapezoid can be found. The rhombi area formula simply requires the product of the diagonals to be divided by two. Two figures are congruent if they have congruent angles.

-‐Find the areas of different triangles including those that are equilateral, right, scalene, and isosceles. -‐Find the area of a trapezoid and a rhombus. -‐Use coordinate geometry to find the area of a trapezoid and rhombus on a coordinate plane. -‐Use algebra to find the missing measures of a triangle, trapezoid, and rhombus when the area is given. -‐Use area formulas to determine if two figures in real life are congruent.

The area of any regular polygon can be found when it is inscribed in a circle. The apothem is required to find the area of any regular polygon. The area is found when you take the product of the perimeter of the polygon and the apothem then divide by two. The area of a circle uses a value of pi and the radius. The area can be found by finding the product of pi and the square of the radius.

-‐Find the area of a regular polygon when given the apothem. -‐Find the area of a regular polygon when given only the radius. -‐Find the area of a circle. -‐Use Pi-‐Form when finding the area of a circle. -‐Determine the radius of a circle when given the area. -‐Use the area of a circle to solve real world problems. -‐Use the area of regular polygon formula and area of a circle formula to find the area of the shaded region.

An irregular figure is a figure that cannot be classified into a specific shape. You must separate these figures into known

-‐Find the area of a irregular figure. -‐Use the area of a irregular figure to solve a real life problem.

5m

4m

10in

9in

60°

11mm

15mm6mm


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figures and find the area of each part. Once each part is found add them together to find the area of the entire irregular figure.

• Find the area of each circle.

• Find the area of the irregular figure below.

-‐Determine the area of a irregular figure on a coordinate plane.

6in 7m

7in

3cm

10

10

5


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21st Century Skills



Concept Activity: Area of a Circle Suppose each regular polygon is inscribed in a circle of radius r.

1. Copy and complete the following table. Round to the nearest hundredth. Number of Sides 3 5 8 10 20 50 Measure of a Side 1.73r 1.18r .77r .62r .31r .126r Measure of Apothem .5r .81r .92r .95r .99r .998r Area 2. What happens to the appearance of the polygons as the number of sides increases? 3. What happened to the areas as the number of sides increases? 4. Make a conjecture about the area of a circle.

Performance Assessment Task Sample Task One: A real estate company sells plots of land. The plot shown below costs $84,120. What is the price per square foot on the land? Explain how you found your answer in a brief paragraph.

Task Two: Regular hexagon ABCDEF has vertices at (4,4 3), (8,4 3), (10,2 3)A B C , (8,0), (4,0), (2,2 3)D E F . Suppose the sides of the hexagon are reduced by 40% to produce a similar regular hexagon. What he perimeter and area of the smaller regular hexagon? Round your answer to the nearest tenth. Explain how you came to your conclusion in a brief paragraph.

265 ft.

260 ft.150 ft.

125 ft.


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Assessment Models









Additional

Resources


BIG IDEA VIII: THREE DIMENSIONAL MEASUREMENT

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BIG IDEA VIII: THREE DIMENSIONAL MEASUREMENT Curriculum Management System


1. Visualization can help you connect properties of real objects with two-‐dimensional drawings of these objects. 2. Some attributes of geometric figures such as length, area, volume, and angle measure, are measurable. Units are used to describe these attributes

ESSENTIAL QUESTIONS

How can you determine the intersection of a solid and a plane? How do you find the surface area and volume of a solid?

Suggested Blocks for Instruction: 9


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A solid with all flat surfaces that enclose a single region of space is called a polyhedron. Each flat surface, or face, is a polygon. The line segments where the faces intersect are called edges. Edges intersect at a point called a vertex.. The shape of their faces name Polyhedra. Some solids are not Polyhedra. A net is a pattern for a three-‐dimensional figure if it was laid flat. Surface area is the sum of the area of each part of the net. Surface area is measured in squared units.

-‐A three dimensional figure can be analyzed by describing the relationship among its vertices, edges, and faces. -‐The area of three-‐dimensional figure is equal to the sum of the areas of each surface of the figure. -‐The volume of a prism and cylinder can be found when its height and area of its base are known. -‐The volume of a pyramid is related to the volume of a prism with the same base and height. -‐The surface area and the volume of a sphere can be found when the radius is known. Sample Conceptual Understandings Find the surface area and volume of the solids below:

-‐Identify and name three-‐dimensional figures. -‐Draw nets for any solid. -‐Use net to determine surface area.

A prism is a polyhedron with two parallel congruent bases. The rectangular faces that are not bases are called lateral faces. The lateral faces intersect at lateral edges. The height of the prism is the altitude that connects both bases. Lateral area is the sum of the area of the lateral faces. It can be found by multiplying the perimeter of the base by the height of the prism. The lateral area is used to find the surface area of the prism. The surface area can be found by adding the lateral area to two times the area of the base of the prism. A cylinder is a solid with congruent circular bases in a pair of parallel planes. The axis of a cylinder is the segment with endpoint that are centers of the circular bases. The lateral area of a cylinder can be found by taking the product of 2-‐pi the radius of the base and the height of the cylinder. Using the lateral area and adding two times the area of each base to it can find the surface area.

-‐Find the lateral area of all different types of prisms (ex. Triangular, pentagonal, etc.) -‐Find the surface area of various types of prisms. -‐Use the apothem to find the surface area of prisms with bases that are regular polygons. -‐Use surface area of a prism to solve real world problems. -‐Find the lateral area of a cylinder. -‐Use the lateral area of a cylinder to find the surface area. -‐Find missing dimensions of a cylinder when given the surface area.

4m3m

12m 20cm

12cm9cm


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A pyramid is a polyhedron in which all the lateral faces intersect at one vertex. Their bases can be any polygon. The altitude is from the vertex to the center of the base. The slant height is an altitude from the vertex to the edge of the base of the pyramid. The lateral area is found by taking half of the perimeter of the base multiplied by the slant height. The surface area is found by adding the lateral area to the area of the base. A cone has a circular base and a vertex. The axis (height) goes from the vertex of the center of the circular base. The slant height is an altitude from the vertex to the circumference of the circular base. The product of pi, the radius of the circular base, and the slant height find the lateral area. The surface area can be found by adding the area of the base to the lateral area.

-‐Find the lateral area of a regular pyramid. -‐Find the surface area of a regular pyramid of any base. (ex. Square, pentagonal, rectangular) -‐Find the lateral area of a right cone. -‐Use the lateral area to find the surface area of a right cone. -‐Use the surface area of a pyramid and a cone to solve real world problems.

A sphere is a set of points in space that are a given distance from a given point. The cross section of the circle through the center is called the “great circle.” Each great circle separates the sphere into two congruent halves or hemispheres. The surface area of a sphere can be found by taking four times the area of the great circle. Volume of a figure is the measure of space the figure encloses. It is measured in cubic units. Taking 4/3 of pi and cubing the radius finds the volume of a sphere.

-‐Find the area of the great circle. -‐Find the surface area of a sphere. -‐Use the surface area formula to solve real world problems involving spheres. -‐Find the volume of a sphere. -‐Use the volume of a sphere to solve real world problems.


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The volume of any prism is found by multiplying the area of the base of the figure times the height of the lateral faces. The volume of a cylinder is found by multiplying the area of the base ( 2rπ ) by the height of the figure.

-‐Find the volume of a triangular prism. -‐Find the volume of a rectangular prism. -‐Find the volume of any prism with a regular polygon as its base. -‐Find the volume of a right cylinder. -‐Find the volume of an oblique cylinder. -‐Use volume to solve real world problems.

The volume of a pyramid is the area of its base multiplied by its height. and divided by 3. The height can be found by using the lateral height. The volume of a cone is found by multiplying the area of the base ( 2rπ ) by the height of the cone.

-‐Find the volume of a pyramid with a square base. -‐Find the volume of a pyramid with a rectangular base. -‐Find the volume of any pyramid with a regular polygon as its base. -‐Find the volume of a right cone. -‐Find the volume of an oblique cylinder. -‐Use volume to solve real world problems.


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21st Century Skills



Concept Activity: Collect some empty cardboard containers shaped like prisms and cylinder.

1. Measure each container ad calculate its surface area. 2. Flatten each container by carefully separating the places where it has been glued together. Find the total area of the packaging material used. 3. For each container, find the percent by which the area of the packing material exceeds the surface area of the container.

-‐How does the unfolded prism-‐shaped package differ for the net of the prism? -‐What did you find out about the amount of extra material needed for the prism shaped containers? For the cylindrical? -‐Why would a manufacturer be concerned about the surface area of a package? About the amount of material used for the package?

Performance Assessment Task Sample Task One: Sketch the solids described below. The complete steps c through e. a) Sketch a triangular prism. b) Sketch a cylinder of the same height. c) Label both figures, giving only the dimensions necessary to calculate the surface area and the volume. d) Find the surface area and volume of the prism to the nearest tenth. e) Find the surface area and volume of the cylinder. Leave answer in Pi Form. Task Two Explore similar solids by working through the steps below. a) Are all cubes similar? If so, explain why. If not, give an example of two cubes that are not similar. b) Are all cones similar? If so, explain why. If not, give an example of two cones that are not similar. c) Are all spheres similar? If so, explain why. If not, give an example of two spheres that are not similar. d) Describe a family of pyramids that are similar.


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Assessment Models









Additional

Resources


BIG IDEA IX: QUADRILATERALS

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BIG IDEA IX: QUADRILATERALS Curriculum Management System


1. Some attributes of geometric figures, such as length, area, volume, and angle measure, are measurable. Units are used to describe these attributes. 2. Definitions establish meanings and remove possible misunderstandings. Other truths are more complex and difficult to see. It is often possible to verify complex

truths by reasoning from simpler ones using deductive reasoning. ESSENTIAL QUESTIONS

How can you find the sum of the measures of polygon angles? How can you classify quadrilaterals?



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A diagonal of a polygon is a segment that connects any two nonconsecutive vertices. You can find the sum of the interior angles by using the Interior Angle Sum Theorem. This theorem can be used to find an individual angle, the number of sides of a polygon, and the sum of all the interior angles. The Exterior Angle Sum Theorem states that all the exterior angles of a polygon will always add up to 360 degrees.

-‐The sum of the angle measures of a polygon depends on the number of sides the polygons has. -‐Parallelograms have special properties regarding sides, angles, and diagonals. -‐If a quadrilaterals sides, angles, and diagonals have certain properties; it can be shown that the quadrilateral is a parallelogram. -‐The special parallelograms, rhombus, rectangle, and square have basic properties of their sides, angles, and diagonals that help identify them. -‐The angles, sides, and diagonals of a trapezoid have certain properties. Sample Conceptual Understandings • Find the measure of the missing angle.

• Find the measure of the interior and exterior angle of the missing

polygon. a) Hexagon b) 16-‐gon c) Pentagon • Find the values of x and y in parallelogram ABCD.

a) AB=2y, BC=y+3, CD=5x-‐1, and DA=2x+4. b) AB=2y+1, BC=y+1, CD=7x-‐3, and DA=3x.

• Determine whether the statement is sometimes, always, or never true. a) A rhombus is a square. b) A square is a rectangle. c) A rhombus is a rectangle. d) The diagonals of a parallelogram are perpendicular. e) The diagonals of a parallelogram are congruent. f) Opposite angles of a parallelogram are congruent.

-‐Derive the Interior Angle Sum Theorem through an interactive activity. -‐Use the Interior Angle Sum Theorem to find the sum of the angles of regular polygons. Complete application problems in which the Interior Angle Sum Theorem is used. -‐Find the number of sides a polygons has when given the sum of the interior angles. -‐Derive the Exterior Angle Theorem by using the Geometer’s Sketchpad activity to investigate its properties. -‐Use the Exterior Angle Theorem to determine the amount of sides a polygon has when the angle measure is known.

Parallelograms are quadrilaterals with opposite sides parallel. All parallelograms have distinctive properties such as: opposite sides congruent and parallel, opposite angles congruent, consecutive interior angles are supplementary, and if the parallelogram contains one right angle then all four angles are also right. The diagonals in a parallelogram bisect one another, and cut the parallelogram into two congruent triangles. These properties can be used to prove a polygon to be parallelograms by showing them to be valid.

-‐Write a two-‐column proof in which they prove that the opposite angles of a parallelogram are congruent. -‐Recognize and use properties of parallelograms to write algebraic equations and solve for values throughout the parallelogram. -‐Complete standardized test questions involving the use of properties of parallelograms. -‐Determine if the given quadrilateral is a parallelogram by showing properties to be true. -‐Find the value of x and y to make the quadrilateral a parallelogram. -‐Use the slope and distance formula on a coordinate plane to determine whether the given points form a parallelogram.

A rectangle is a quadrilateral with four right angles. It is considered a special type of parallelogram since the opposite

-‐Recognize and use properties of rectangles to write algebraic equations and find the values of variables.


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sides are congruent. It has all the properties of a parallelogram except all angles are 90 degrees. The diagonals of a rectangle are congruent. These properties can be used to classify a parallelogram as a rectangle.

• Determine whether the following is a parallelogram.

• Find the measures of the numbered angles in the following isosceles trapezoid.

• A trapezoid has base lengths of 6x-‐1 units and 3 units. Its median

has a length of 5x-‐3 units. What is the value of x? • Determine if the given figure tessellates. If so, Draw a sketch. If

not, explain. a) kite b) regular 14-‐gon c)decagon

-‐ Prove parallelograms to be rectangles by using their properties in an application problem. -‐Use the slope formula and distance formula on a coordinate plane to prove a parallelogram to be a rectangle.

A square is a special type of parallelogram called a rhombus. Rhombi have four congruent sides and all the properties of a parallelogram. Rhombi have diagonals that are perpendicular, and diagonals that bisect the angles they intersect. If a parallelogram is both a rhombus and a rectangle then it is considered a square. A square has all the properties of a parallelogram, rectangle, and rhombus.

-‐Write a two-‐column proof in which they must prove the diagonals of a rhombus to be perpendicular. -‐Write and solve algebraic equations to find the measure of sides and angles in a rhombus. -‐Use coordinate geometry to determine if the parallelogram is a rectangle, rhombus, or a square. -‐Apply the properties of a square to an application problem.

A trapezoid is a quadrilateral with exactly two parallel sides. The parallel sides are called the bases and the other called the legs. When the legs are congruent the trapezoid is considered an isosceles trapezoid with base angles as well. The median of a trapezoid is a parallel segment connecting the midpoints of the legs. The median is half the sum of the bases.

-‐Prove diagonals of an isosceles trapezoid to be congruent through a flow proof. -‐Identify isosceles trapezoids using a protractor. -‐Determine a trapezoid to be isosceles or not on a coordinate plane. -‐Find the median of a trapezoid when both bases are given. -‐Find a base of a trapezoid when one base and the median is given.


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21st Century Skills



Concept Activity: Exterior Angles of Polygons Use geometry software. Construct a polygon using extended segments. Mark a point on each ray so you can measure the angles. -‐Measure each exterior angle. -‐Calculate the sum of the measures of the exterior angles. -‐Manipulate the polygon. Observe the sum of the measures of the exterior angles of the new polygon. Performance Assessment Task Sample Task One: ABCDEF is a regular hexagon. What is the most precise classification of quadrilateral GBHE? How do

you know? What are the angle measures of GBHE?

Task Two: JKLM is a parallelogram. If you extend each side by a distance x, what kind of quadrilateral is PQRS? How do you know?


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Assessment Models





Open-‐Ended (Formative) Assessment: Group and individual work is assigned daily, from various sources (Synthesis, Analysis, and Evaluation). Introductory and Closing Activities will be done every day to pre-‐assess student knowledge and assess understanding of topics (Synthesis,

Analysis, and Evaluation). Summative Assessment:



Additional

Resources


BIG IDEA X: CIRCLES

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BIG IDEA X: CIRCLES Curriculum Management System


1. Definitions establish meaning and remove possible misunderstanding. Other truths are more complex and difficult to see. It is often possible to verify complex truths by reasoning from simpler ones by using deductive reasoning.

2. Some attributes of geometric figures such as length, area, volume, and angle measure, are measurable. Units are used to describe these attributes. 3. It is possible to verify some complex truths on the coordinate plane using deductive reasoning in combination with Distance, Midpoint, and Slope Formulas.

ESSENTIAL QUESTIONS

How can you prove relationships between angles and arcs in a circle? When lines intersect a circle or within a circle, how do you find the measures of the resulting angles, arcs, and segments? How do you find the equation of a circle in the coordinate plane?


BIG IDEA X: CIRCLES

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A circle has many parts that can be measured and used in various ways. The radius is needed to find the circumference and area of a circle.

-‐A radius of a circle and the tangent that intersects the endpoint of the radius on the circle has a special relationship. -‐Information about congruent parts of a circle (or congruent circles) can be used to find information about other parts of the circle (or circles) -‐Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. This includes arcs formed by chords that inscribe angles, angles and arcs formed by lines intersecting either within a circle or outside a circle and intersecting chord, intersecting secants, or a secant that intersects a tangent. -‐The information in the equation of a circle allows the circle to be graphed. The equation of a circle can be written if its center and radius are known. Sample Conceptual Understandings • Find the missing parts of the circle.

• Use the properties of tangents to find the value of x in each figure.

-‐Identify parts of a circle. -‐Find the radius and diameter of a circle when given specific information. -‐Find measures of the radius and diameters in intersecting circles. -‐Determine the value of Pi by using the ratio between the circumference and the diameter of a circle. -‐Problem solve for the circumference, radius, and diameter of a figure when given a specific equation. Use other figures to find the circumference of a circle including those with special right triangles.

An angle formed by two radii that meet at the center is called a central angle. The central angles of a circle can be added to form 360 degrees. Connecting arcs form the outer rim of a circle. The central angle and the arc it intercepts have the same measure. A arc smaller than 180 degrees is a minor arc, one greater than 180 degrees is a major arc, and one equal to 180 degrees is a semi-‐circle. Arcs can be added together by using the Arc Addition Postulate. Ratios can be used to find the length of an arc.

-‐Write and use algebraic equations to find the measures of central angles and their intercepted arcs. -‐Identify various types of arcs in a given diagram and find the value of their measure. -‐Use central angles and arcs in circle graphs to determine the percentages of a given figure. -‐Find the length of the arcs of circle when given the central angle measure and circumference.

In circles two arcs are congruent if their intercepting chords are congruent. Polygons can be inscribed if all of their vertices lie on the circle. A circle is circumscribed if the polygons vertices lie on its perimeter. When a diameter and a chord are perpendicular to each other the chord is bisected as well as the arc it intercepts. Radii along with the

-‐Compare chords in a circle to a circular waffle maker. -‐Use proofs to show the Congruent Arcs and Chords Theorem are valid. -‐Determine the measure of interior angles of inscribed polygons. -‐Use circumscribed circles to determine the values of interior angles of a polygon. -‐Construct a circle and use folding techniques to prove a diameter bisects a

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Pythagorean Theorem and Trigonometric Ratios are used to solve these types of problems. Chords are congruent if they are equidistant from the center of the circle.

• Find the perimeter of triangle ABC if circle O is inscribed.

• Find x.

• Find the value of each variable.

chord and its intercepted arc if perpendicular to it. -‐Construct radii to use the Pythagorean Theorem and Trigonometric Ratios in solving problems.

Angles are inscribed in circles when their vertex lies on the circumference of the circle. Its intercepted arc is two times the value of the angle. If two inscribed angles intercept the same arc then they are congruent to one another. Every inscribed angle that intercepts a semicircle is a right angle. When a quadrilateral is inscribed in a circle then its opposite angles are supplementary.

-‐Complete a constructive activity in which students will determine the measure of inscribed angles and their intercepted arcs. -‐Write and solve algebraic equations using the properties of inscribed angles. -‐Prove two triangles to be congruent when their angles intercept the same arcs. -‐Complete application problems in which inscribed angles are used with probability. -‐ Write and solve equations in which the goal is to find the angles of an inscribed quadrilateral.

A tangent is a line which intersects the circle in exactly one place, called the point of tangency. If a tangent and a radius intersect at the point of tangency a right angle is formed. The converse is also true. If two tangents to one circle meet at the same exterior point then they are congruent. Polygons are circumscribed about the circle if each one of their sides intersects the circle just once at a point of tangency.

-‐Create a document using Geometer’s SketchPad in which they will determine the relationship between tangents and radii. -‐Find lengths of radii or tangents by using the Pythagorean Theorem. -‐Identify tangents using the Converse of the Pythagorean Theorem. -‐Write and solve algebraic equation using tangents. -‐Use various theorems and postulates to determine the perimeter of a triangle circumscribed about a circle.

A line that intersects a circle in exactly two points is called a secant. If two secants intersect at the interior of a circle then the measure of the angle formed is one half the sum of the intercepted arcs formed by the angle and its vertical angle. When a secant and a tangent intersect at

-‐Use a two column proof to show that when two secants intersect the value of the angle is one half the sum of the intercepted arcs. -‐Write and solve algebraic equations to find the missing angle measure or arc measure when two secants intersect inside the circle. -‐Write and solve algebraic equations to find

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the point of tangency then the angle is half the measure of the intercepted arc. When two secants, two tangents, or a secant and a tangent intersect in the exterior of the circle the angle formed is one half the difference of the values of the bigger arc and the smaller arc.

• Find the value of each variable.

missing angle and arc measures when tangents and secants intersect on the circumference of the circle. -‐Complete application problems in which two tangents intersect outside of the circle.

When two chords intersect the products of their parts are equal to one another. When two secants intersect outside of the circle the product of the whole secant and the exterior parts are congruent to one another. When a tangent and secant intersect outside of the circle, the square of the exterior tangent is equal to the exterior part of the secant multiplied by the whole secant.

-‐Use basic algebra to find the lengths of chords inside a circle. -‐Solve application problems involving two chords intersecting. -‐ Write equations to find the lengths of secants intersecting outside the circle. -‐ Write equations to find the lengths of secants and tangents intersecting outside the circle. -‐Use the quadratic formula to solve for variables when finding the lengths of secants and tangents.

You can find the equation of a circle on a coordinate plane when given the center point and either the radius or diameter of the circle.

-‐Write the equation of a circle when given the center and the radius. -‐Write the equation of a circle when given the center and a diameter. -‐Use tangents and the center of a circle to write equations. -‐Graph a circle when given the equation of it. -‐Write the equation of a circle when given three points in which it intersects.

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• What is the standard equation of the circle with center (5, -‐2) and a radius of 7?

• What is the standard equation of the circle with center (4,3) that passes through (-‐1,1)?

• Given three non-‐collinear points, construct the circle that includes all three points.

1. Begin with points A, B, and C.

2. Draw line segments AB and BC.

3. Construct the perpendicular bisectors of line segments AB and BC. Let point P be the intersection of the perpendicular bisectors.

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4. Center the compass on point P, and draw the circle through points A, B, and C.

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21st Century Skills



Concept Activity: Paper Folding with Circles: Part One:

1. Use a compass to draw a circle on tracing paper. 2. Use a straightedge to draw two radii. 3. Set your compass to distance shorter than the radii. Place its point at the center of the circle. Mark two congruent segments one on each radius. 4. Fold a line perpendicular to each radius at the point marked on the radius.

a) How do you measure the distance between a point and a line? b) Each perpendicular contains a chord. Compare the lengths of the chords. c) What is the relationship among the lengths of the chords that’s are equidistant from the center of a circle?

Part Two: 1. Use the compass and draw a circle on tracing paper. 2. Use a straightedge to draw two chords that are not diameters. 3. Fold the perpendicular bisector for each chord. d) Where do the perpendicular bisectors appear to intersect the other now? e) Draw a third chord and fold its perpendicular bisector. Where does it appear to intersect the other two? f) What is true about the perpendicular bisector of a chord?

Performance Assessment Task Sample A gardener wants three rosebushes in her garden to be watered by a rotating water sprinkler. The gardener draws

a diagram of the garden using a grid in which each unit represents 1ft The rosebushes are at (1,3), (5,11), and (11,4). She wants to position the sprinkler at a point equidistant from each rosebush. Where should the gardener place the sprinkler? What equation describes the boundary of the circular region that the circular region will cover?

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BIG IDEA XI: PROBABILITY

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BIG IDEA XI: PROBABILITY Curriculum Management System


1. Statistics and probability provide tools for describing variability in data and for making informed decisions that take it into account. 2. Definitions establish meanings and remove possible misunderstandings. Other truths are more complex and difficult to see. It is often possible to verify complex

truths by reasoning from simpler ones using deductive reasoning. 3. The ability to decipher patterns and determine the next logical term requires the analysis and synthesis of numbers, figures, and various objects. Finding the

truths in these can lead to the proof of the hypothesis for the next term.

ESSENTIAL QUESTIONS What is the difference between experimental and theoretical probability? How are the laws of probability used to predict outcomes in the real world? How is statistics used to analyze data in real world situations?



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The Fundamental Counting Principle describes the method of using multiplication to count. A permutation is an arrangement of items in a particular order. Using factorial notation allows you to write 3x2x1 as 3!. A selection in which order does not matter is called a combination.

-‐Various counting methods can help you analyze situations and develop theoretical probabilities. -‐You can use multiplication to quickly count the number of ways certain things can happen. -‐The probability of an impossible event is 0. The probability of a certain event is 1. Otherwise the probability of an event is a number between 0 and 1. -‐To find the probability of two events occurring you have to decide whether one event occurring affects the other event. -‐Conditional probability exists when two events are dependent. -‐ In geometric probability, numbers of favorable and possible outcomes are geometric measures such as lengths of segments or areas of regions. Sample Conceptual Understandings • Determine the number of possible license plates possible in 1912

in comparison to 2004 when given the following: • -‐In 2004 license plates had a three places for letters and three

places for digits. • -‐In 1912, license plates had places for only four digits. • In how many ways can you file 12 folders, one after another, in a

drawer? • Ten students are in a race. First, second, and third places will win

medals. In how many ways can 10 runners finish first, second, and third with no ties allowed?

• What is the number of combinations of 13 items taken 4 at a time?

• Of the 60 vehicles in the parking lot, 15 of them are pick up trucks. What is the experimental probability that a vehicle is a pick up?

• What is the probability of getting a 5 on a roll of a standard number cube?

• At a picnic there are 10 diet drinks and 5 regular drinks. There are also 8 bags of fat-‐free chips and 12 bags of regular chips. If you grab a drink and a bag of chips without looking, what is the probability that you get a diet soda and fat free chips?

-‐Use the Fundamental Counting Principle. -‐Finding the number of permutations of n items. -‐Use the permutation formula. -‐Use the combination formula. -‐Identify whether the order matters in a event.

When you gather data from observations, you calculate an experimental probability. Each observation is an experiment or a trial. A simulation is a model of the event. The set of all possible outcomes to an experiment or activity is a sample space. If the outcomes in a sample space have the same chance of occurring, the outcomes are called equally likely outcomes. Theoretical probability is when a sample space has equally likely outcomes to an event occurring.

-‐Find experimental probability. -‐Use a simulation. -‐Find theoretical probability. -‐Find probability using combinations.

When the occurrence of one event affects how a second event can occur, the events are dependents events. Otherwise the events are independent events. Two events that cannot happen at the same time are mutually exclusive events.

-‐Classifying events as independent or dependent. -‐Find the probability of independent events. -‐Find the probability of dependent events. Finding the probability of mutually exclusive events.

The probability of an event occurring given that another event occurs is called conditional probability. A contingency table is a frequency table that contains data from two different categories. Using the formula for conditional probability you can calculate the conditional probability from other probabilities.

-‐Find conditional probability using contingency tables. -‐Use conditional probability in statistics. -‐Use the conditional probability formula. -‐Use a tree diagram to find the conditional probability.


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In geometric probability, numbers of favorable and possible outcomes are geometric measures such as lengths of segments or areas of regions. The probability of a point being located on a specific part of a segment is the ratio of the specific length of the segment to the length of the whole segment. Probability can also be used when finding the chances of a point being in an inscribed figure.

• A utility company asked 50 customers whether they pay their bills online or by mail. Using the diagrams below determine what the probability that a customer pays the bill online is a male?

ONLINE BY MAIL Male 12 8 Female 24 6 • A point on AM is chosen at random. Find the probability the

point lies on the given segment.

a) DJ b) JL c) BE d) CK e) AJ f) BL • A Sunday night sports show is on from 10:00pm to 10:30pm. You

want to find out if your favorite team won this weekend, but forgot that the show was on. You turned it on at 10:14pm. The score will be announced at one random time during the show. What is the probability that you haven’t missed the report about your favorite team?

• A point in the figure is chosen at random. In the following figures find the probability that the point lies in the shaded region.

-‐Use segments to find probability. -‐Use area to find probability.

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21st Century Skills



Concept Activity: To win a prize at a carnival game, you must toss a quarter so that it lands within a 1-‐in circle as shown. Assume that the center of a tossed quarter is equally likely to land at any point within the 8-‐in square.

a. What is the probability that the quarter lands entirely in the circle in one toss? b. On average how many coins do you have to toss to win a prize? Explain.

1. In this problem, what represents the favorable outcome? 2. In this problem, what represents all the possible outcomes? 3. If a section of the quarter is in the circle, does this count as a favorable outcome? 4. How can you determine a smaller circle within which the center of the quarter must land for the quarter to be entirely within the 1-‐in circle? What is the radius

of the circle?

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5. Use words to write a probability ratio. Then rewrite the ratio using the appropriate formulas. Substitute the appropriate measures and find the probability. 6. Based on this, what is the average number of coins you must toss before you can expect to win a prize? Explain.

Performance Assessment Task Sample Task One:

Suppose you have n terms from which you choose r at a time. Explain why you must divide the number of

permutations !

( )!nn r−

by r! to find the number of combinations !

!( )!n

r n r−.

Task Two: Suppose you stack three identical number cubes. It is possible to have no sides, two sides, or all four sides of the stack showing all the same number. (Note that if one side of a stack shows all the same number, then the opposite side must as well.) How many ways are there to stack three standard number cubes so that at least two sides of the stack show all the same number? If you can rotate a stack so that it is the same as another, count them as the same arrangement. Explain your solution.


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COURSE BENCHMARKS

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HONORS GEOMETRY

1. The student will be able to identify and use parts and types of lines, angles, and planes in problems solving.

2. The student will be able to use logical reasoning and conditional statements to solve problems.

3. The student will be able to use angle relationships with parallel and perpendicular lines to solve problems.

4. The student will be able to use triangle classifications and congruent triangles to solve problems.

5. The student will be able to use the relationships of sides and angles in triangles to solve problems.

6. The student will be able to use proportions to determine similarity of triangles.

7. The student will be able to use right triangle trigonometry to solve problems.

8. The student will be able to use properties of quadrilaterals to solve problems.

9. The student will be able to use and apply properties of lines and angles in circles.

10. The student will be able to use properties of polygons to solve problems.

11. The student will be able to find the lateral area, surface area, and volume of three-‐dimensional figures.

12. The student will be able to recognize and apply properties of transformations.

13. The students will use geometric probability and statistics to analyze real life situations.

final geometry honors - monroe township school district · 2013. 3. 11. · 4"|page"...

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