CurriculumManagementSystem! - Monroe Township … · 2013-07-02 · Goals/Essential!Questions/Objectives/Instructional!Tools/Activities! ! Pages! ... V.Properties!of! ... Trapezoids!and!Kites!

Page 1

Curriculum Management System

MONROE TOWNSHIP SCHOOLS

Course Name: Geometry Grade: 9-‐10

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

Page 2

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 Pages …7

Goals/Essential Questions/Objectives/Instructional Tools/Activities Pages …11

Quarterly Benchmark Assessment Page ….87

Page 3

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. Louis C. Masters Mr. Doug Poye

Mr. Anthony Prezioso Mr. Ira Tessler

Jamesburg Representative Mr. Robert Czarneski

WRITER’S NAME Ms. Beth Goldstein

MATHEMATICS CURRICULUM INCHARGE (9-‐12)

Dr. Manjit K. Sran

Page 4

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.

Page 5

PHILOSOPHY

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.

Page 6

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

Page 7

Scope and Sequence

Quarter 1

Unit Topic(s)

I. Lines, Angles, and Planes a. Undefined and Basic Defined terms b. Segments and Their Measures c. Angles and Their Measures d. Segment and Angle Bisectors e. Special Angle Pairs

II. Reasoning and Proofs a. Logical Reasoning and Conditional Statements b. Number and Visual Patterns c. Conditional Statements d. Postulates about Points, Lines, and Segments e. Theorems about Special Pairs of Angles

III. Parallel and Perpendicular Lines a. Angles Formed by a Transversal b. Angle relationships Formed by Parallel Lines and a

Transversal c. Slopes of Parallel and Perpendicular Lines d. Proving Parallel Lines

Page 8

Scope and Sequence

Quarter 2

Unit Topic(s)

IV. Triangles

a. Classifying Triangles b. Angles and triangles c. Congruent Figures and Corresponding Parts d. Proving Triangles Congruent e. Isosceles and Equilateral Triangles

V. Properties of Triangles a. Medians, Angle Bisectors, Altitudes, and Perpendicular

Bisectors b. Mid-‐segment of a Triangle c. Inequalities in One Triangle

VI. Similarity a. Ratio and Proportions b. Similar Polygons c. Proving Triangles Similar d. Proportionality Theorems

Page 9

Scope and Sequence

Quarter 3

Unit Topic(s)

VII. Right Triangles and Trigonometry

a. Simplify Radicals and Geometric Mean b. The Pythagorean Theorem and Its Converse c. Special Right Triangles d. Trigonometric Ratios e. Law of Sines and Cosines

VIII. Quadrilaterals a. Interior Angles of a Quadrilateral b. Properties of Parallelograms c. Proving Parallelograms d. Rectangles, Rhombi, and Squares e. Trapezoids and Kites

IX. Polygons a. Classifying Polygons b. Interior and Exterior Angles of a Polygon

X. Circles a. Parts of a Circle b. Tangents c. Arcs and Chords d. Angle Relationships e. Segment Relationships

Page 10

Scope and Sequence

Quarter 4

Unit Topic(s)

XI. Area

a. Squares, Rectangles, and Parallelograms b. Triangles c. Rhombi and Kites d. Trapezoids e. Circles f. Regular and Irregular Figures g. Area of the Shaded Region h. Geometric Probability

XII. Three-‐Dimensional Figures a. Nets b. Surface area of Prisms, Cylinders, Pyramids, Cones, and

Spheres c. Volume of Prisms, Cylinders, Pyramids, Cones, and

Spheres XIII. Transformations

a. Reflection b. Translation c. Rotation d. Dilation

XIV. Probability a. Fundamental Counting Principal b. Permutations and Combinations c. Experimental Probability d. Theoretical Probability

Page 11

Unit I – Points, Lines, and Planes Stage 1 Desired Results

ESTABLISHED GOALS CCSS The Standards: http://www.corestandards.org/the-‐standards Common Core State Standards for Mathematics Mathematical Practices Make sense of problems and

persevere in solving them. Reason abstractly and

quantitatively. Construct viable arguments and

critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in

repeated reasoning.

Congruence G-‐CO Experiment with transformations in the plane 1. Know precise definitions of angle,

circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line distance along a line, and distance around a circular arc.

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

Modeling with Geometry G-‐MG

Transfer Students will be able to independently use their learning to … Understand and apply the basic undefined and defined terms of geometry.

Meaning UNDERSTANDINGS Students will understand that… • 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 in the coordinate plane.

• Number operations can be used to find and compare the lengths of segments and the measures of angles.

• Special angle pairs can be used to identify geometric relationships and to find angle measures.

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?

Acquisition Students will know… • A point represents a location in space. • A line is a set of points that extend indefinitely in two

directions. • A plane is represented by a flat surface that extends

indefinitely in all • Directions. • Collinear points are points that lie on the same line. • Coplanar points are points that lie on the same plane.

• Line segment AB consists of points A and B and the set of points between them.

• If point B is between points A and C , then AB BC AC+ = .

Students will be skilled at… Identifying and using points, lines,

and planes in space Finding the length of a segment

using the Segment Addition Postulate and the distance between two points using the Distance Formula.

Naming, measuring, and classifying angles.

Identifying adjacent angles and finding the measure of an angle using the Angle Addition Postulate.

Page 12

Apply geometric concepts in modeling situations 1. Use geometric shapes, their

measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (*)

• The distance between two points is defined as the length of a line segment.

• The distance between two points on a number line is given by the formula a b− , where a and b are the coordinates of the two endpoints.

• If the coordinates of point A are 1 1( , )x y and the

coordinates of point B are 2 2( , )x y , then the distance

between points A and B is 2 21 2 1 2( ) ( )x x y y− + − .

• An angle is the union of two rays that intersect at their endpoint. An angle is named using three points, one point from one side of the angle, the vertex of the angle, and the other point from the other side of the angle. Emphasize that the vertex must be named in the middle when naming an angle.

• An angle can be named three ways, using one letter (the vertex) when appropriate, three letters, and a number.

• An angle can be named using the vertex of the angle when there is only one angle at that vertex.

• Adjacent angles have a common vertex, a common side, and no common interior points.

• An acute angle is an angle whose measure is less than 090 (0 90m≤ ≤ ).

• A right angle has a measure of 090 . • An obtuse angle is an angle whose measure is more than

090 and less than 0180 (90 180m≤ ≤ ).

• A straight angle has a measure of 0180 .

• If point C is in the interior of AOBS , then m AOC m BOC m AOB+ =S S S .

• Point M is the midpoint of line segment AB if AM MB≅ .


Identifying congruent segments and angles.

Identifying and using midpoints and segment bisectors.

Applying the Midpoint Formula. Identifying and using angle

bisectors. Identifying and applying the

properties of vertical angles, a linear pair, complementary angles, and supplementary angles.

Performing basic constructions.

Page 13

coordinates of point B are 2 2( , )x y , then the coordinates of

the midpoint of 1 2 1 2,2 2

x x y yAB + +⎛ ⎞= ⎜ ⎟⎝ ⎠. The coordinates of

the midpoint are the average of the x -‐coordinates and the average of the y -‐coordinates.

• Ray OC

bisects AOB if AOC ≅BOC . • Review solving linear equations with the variable on both

sides of the equation. • Complementary angles are two angles whose sum is 90, and

supplementary angles are two angles whose sum is 180. • Help students remember the difference between

complementary angles and supplementary angles by explaining that alphabetically the letter c is before the letter s , and numerically 90 is before 180 .

• The Vertical Angles Theorem: Vertical angles are congruent. Unit I – Points, Lines, and Planes

Stage 2 -‐ Evidence Evaluative Criteria Assessment Evidence

RUBRIC/SCALE

3 -‐ POINT RESPONSE The response shows complete understanding of the problem's essential mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.

2 -‐ POINT RESPONSE

PERFORMANCE TASK (S): Directions: Use the road map below to answer the questions that follow. Assume all roads are lines, segments, or rays.

Page 14

The response shows nearly complete understanding of the problem's essential mathematical concepts. The student executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear, causing the reader to make some inferences.

1 -‐ POINT RESPONSE The response shows limited understanding of the problem's essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved may contribute to questions as to how and why decisions were made.

0 -‐ POINT RESPONSE The response shows insufficient understanding of the problem's essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.

Copyright © State of New Jersey, 2006 NJ Department of Education http://www.nj.gov/education/njpep/assessment/TestSpecs/MathNJASK/rubrics.html

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 meeting point for two friends one at each location?

5. Identify roads that form a hexagon on the map above.

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!

Page 15

Student Responses should be: Accurate Clear Effective Organized Thorough Thoughtful

OTHER EVIDENCE: Students will show they have achieved Stage 1 goals by . . .

• Providing written or oral response to one of the essential questions. • The students will keep an ongoing journal all year of accumulating insight about which rules and

properties. Include examples that show the rule or property correctly applied, as well as common mistakes.

• Passing all quizzes and tests relating to the unit. Diagnostic/Pre – Assessment:

Pre-‐test (5-‐10 open ended questions) will be given covering multiple concepts. Open-‐Ended (Formative) Assessment:

Homework is assigned daily, from the textbook, Chapter Resource Practice Workbook, or other sources. (Synthesis, Analysis, Evaluation)

Introductory and Closing Activities will be done every day to pre-‐assess student knowledge and assess understanding of topics.(Synthesis, Analysis, Evaluation)

Excerpts from previous HSPA exams including multiple choice and open-‐ended problems should be given every class to assess student understanding and measure their individual skills.

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.

Students will be given a unit test that provides a review of the concepts and skills in the unit.

Unit I – Points, Lines, and Planes Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction • Use index cards and toothpicks to make three-‐dimensional models of points, lines, and planes to use as a visual aid. (analysis) • Find the midpoint and segment bisector of a line segment by performing Activity 1.5, Folding Bisectors, McDougal Littell, p.33. On a piece of patty

paper, draw AB . Fold the paper so that B is on top of A . Label the point where the fold intersects AB as point M . Measure AM and MB in centimeters. Verify that AM MB≅ . (Analysis)

• Find the angle bisector of an angle by performing Activity 1.5, Folding Bisectors, McDougal Littell, p.33. On a piece of patty paper, draw ACB .

Page 16

Fold the paper so that CB

is on top of CA

. Draw any point on the fold and label the point D . Measure ACD and BCD with a protractor. Verify that ACD ≅ BCD . (analysis)

• To investigate the angles formed by intersecting lines, perform Activity 1.6, Angles and Intersecting Lines, McDougal Littell, p.43. Using geometry

software, construct intersecting lines AB

and CD

. Label the point of intersection point E . Measure the four angles formed by the intersecting lines, AEC , AED , BEC , and BED . Make a conjecture. Calculate the sum of any two adjacent angles. Make a conjecture. Move the lines into different positions by dragging the points to confirm conjectures. (analysis)

• 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 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.

Page 17

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

Page 18

Unit II – Reasoning and Proofs Stage 1 Desired Results





repeated reasoning.

Congruence G-‐CO Prove geometric theorems • Prove theorems about lines and

angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Make geometric constructions

Transfer Students will be able to independently use their learning to… Use inductive and deductive reasoning to draw conclusions.

Meaning UNDERSTANDINGS Students will understand that… • Patterns in some number sequences and some sequences of

geometric figures can be used to discover relationships. • Some mathematical relationships can be described using a

variety of if-‐then statements. • A definition is good if it can be written as a biconditional. • Given true statements, deductive reasoning can be used to

make a valid or true conclusion. • Algebraic properties of equality are used in geometry to

solve problems and justify reasoning. • Given information, definitions, properties, postulates, and

previously proven theorems can be used as reasons in a proof.

ESSENTIAL QUESTIONS • How can you make a conjecture and

prove that it is true? • What is the next number in a

number pattern or next figure in a visual pattern?

• What is the validity of a conditional? Of its converse? Inverse? Contrapositive?

• How do the basic postulates between points, lines, and planes relate to real-‐life applications?

• How are the properties between right, complementary, supplementary, and vertical angles applied when solving problems?

• Is there a “best practice” to proving the answer is correct?

Acquisition Students will know… • Inductive reasoning uses observed patterns to draw

conclusions. • Deductive reasoning uses accepted properties, postulates, or

theorems to create a string of logically connected statements.

• Proofs can be written in two-‐column, flow chart, or

Students will be skilled at… Describing and determining the next

number in a number pattern. Describing and sketching the next

figure in a visual pattern. Identifying the hypothesis and

conclusion in an if-‐then statement.

Page 19

• Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (*)

paragraph formats. • An arithmetic sequence is a sequence where the difference

between consecutive terms is constant. • A geometric sequence is a sequence where the ratio of any

term to the previous term is constant. • In an if-‐then statement, the part after the if is the hypothesis,

and the part after the then is the conclusion. • Through any two points there exists exactly one line. • If two lines intersect, then their intersection is exactly one

point. • Through any three noncollinear points there exists exactly

one plane. • If two planes intersect, then their intersection is a line. • The intersection of a plane and a line not contained in the

plane is a point. • The Right Angle Congruence Theorem: All right angles are

congruent. • The Congruent Supplements Theorem and the Congruent

Complements Theorem: If two angles are supplementary (complementary) to two congruent angles, then they are congruent, or supplements (complements) of congruent angles are congruent.

• If two angles are supplementary (complementary) to the same angle, then they are congruent.

• The Vertical Angles Theorem: Vertical angles are congruent.

Analyzing conditional statements. Stating a counterexample for a false

conditional statement. Writing the converse of a

conditional statement. Writing the inverse and

contrapositive of a conditional statement.

Writing a proof in two-‐column, flow chart, or paragraph format.

Identifying and applying basic postulates about points, lines, and planes.

Applying theorems involving right, complementary, supplementary, and vertical angles.

Unit II – Reasoning and Proofs Stage 2 -‐ Evidence

Evaluative Criteria Assessment Evidence RUBRIC/SCALE

3 -‐ POINT RESPONSE The response shows complete understanding of the problem's essential

PERFORMANCE TASK (S): Sample Assessment questions

• Find a pattern for each sequence, describe the pattern and use it to show the next two terms. a) 1000, 100, 10, ___________, ________________

Page 20

mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.

2 -‐ POINT RESPONSE The response shows nearly complete understanding of the problem's essential mathematical concepts. The student executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear, causing the reader to make some inferences.


0 -‐ POINT RESPONSE The response shows insufficient understanding of the problem's essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the

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

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)______________________

Page 21

reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.














Page 22

Unit II – Reasoning and Proofs Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction • What is the next number in the sequence 2,5,8,11,...? What type of sequence is this? (Arithmetic) • What is the next number in the sequence 2,6,18,54,...? What type of sequence is this? (Geometric) • Write in if-‐then format: A square is a rectangle.

• State the converse: If B is a midpoint of AC , then AB BC≅ . Determine if the converse is true or false. If false, provide a counterexample.

• If AB

and CD

intersect at point O and mAOD = 30 , find mDOB , mBOC , and mAOC .

• If AB

and CD

intersect at point O and mAOD = 2x +10 and mDOB = 3x − 20 , find x .

• If AB

and CD

intersect at point O and mAOD = 2x +10 and mBOC = 3x − 20 , find x . Teacher made Performance Assessment Tasks (PATs) Released PATs Online State Resources

Page 23

Unit III – Parallel and Perpendicular Lines Stage 1 Desired Results





repeated reasoning.

Congruence G-‐CO Experiment with transformations in the plane • Know precise definitions of angle,

circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

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

Transfer Students will be able to independently use their learning to… Apply angle relationships with parallel and perpendicular lines.

Meaning UNDERSTANDINGS Students will understand that… • Not all lines and not 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.

• A line can be graphed and its equation can be written when certain facts about the line, such as the slope and a point on the line, are known.

• Comparing the slopes of two lines can show whether the lines are parallel or perpendicular.

ESSENTIAL QUESTIONS • How are skew lines and parallel

lines alike? How are they different? • How are the relationships between

the angles formed by two parallel lines and by a transversal applied when finding angle measures?

• How do you prove that lines are parallel or perpendicular?

• How do you write an equation of a line in the coordinate plane?

Acquisition Students will know… • The sides of each pair of corresponding angles form the

letter F. • The sides of each pair of alternate interior angles form the

letter Z. • The sides of each pair of consecutive interior angles form the

letter C. • State the Corresponding Angles Postulate, and informally

prove and apply the Alternate Interior Angles Theorem, the

Students will be skilled at… Identifying parallel, perpendicular,

and skew lines and planes. Identifying angles formed by two

lines and a transversal. Identifying and using angle

relationships formed by two parallel lines and a transversal.

Proving lines parallel.

Page 24

Prove geometric theorems • Prove theorems about lines and


Expressing Geometric Properties with Equations G-‐PE Use coordinates to prove simple geometric theorems algebraically • 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)

Consecutive Interior Angles Theorem, the Alternate Exterior Angles Theorem, and the Perpendicular Transversal Theorem.

• State the Corresponding Angles Converse, and informally prove and apply the Alternate Interior Angles Converse, the Consecutive Interior Angles Converse, and the Alternate Exterior Angles Converse.


coordinates of point B are 2 2( , )x y , then the slope of line

AB

=y1 − y2

x1 − x2

or 2 1

2 1

y yx x

−−

.

• Parallel lines have the same slope. • Review rewriting a linear equation in standard form into

slope-‐intercept form. • Perpendicular lines have slopes that are opposite

reciprocals. The product of the slopes of perpendicular lines is -‐1.

Finding slopes of lines and using the slopes to identify parallel and perpendicular lines.

Unit III – Parallel and Perpendicular Lines Stage 2 -‐ Evidence


3 -‐ POINT RESPONSE The response shows complete understanding of the problem's essential mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response

PERFORMANCE TASK (S): Draw lines l , m , and t such that l mand t is the transversal. Number the eight angles 1-‐8. List the pairs of corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. List all the angles that are supplementary to 2 . List all the angles that are congruent to 3 . Sample Assessment Questions: • Find the slope of the line that passes through the points (0,3) and (3,1) . • Find the slope of the line that passes through the points (0,3) and ( 4, 3)− − .

Page 25

contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.




• Lines l and m are perpendicular. The slope of line l is 43

− . What is the slope of line m ?

• Are the lines with the equations 2 1y x= − and 2 1y x= − + perpendicular? Why or why not?

• Rewrite the equation 2 3 6x y− + = in slope-‐intercept form. • Are the lines with the equations 6 2y x− = and 6 12y x+ = perpendicular? Why or why not?

Page 26

Copyright © State of New Jersey, 2006 NJ Department of Education http://www.nj.gov/education/njpep/assessment/TestSpecs/MathNJASK/rubrics.html Student Responses should be: Accurate Clear Effective Organized Thorough Thoughtful












Unit III – Parallel and Perpendicular Lines Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction • Illustrate the relationships between angles formed by two parallel lines and a transversal, by performing Activity 3.3, Parallel Lines and angles,

Page 27

McDougal Littell, p.142. Construct parallel lines AB

and CD

. Construct transversal EF

. Label the point of intersection of AB

and EF

point Gand the point of intersection of CD

and EF

point H . Measure all eight of the angles formed by the three lines. Make a conjecture about the measures of corresponding angles, alternate interior angles, alternate interior angles, and the consecutive angles. Drag point E or F to change the angle the transversal makes with the parallel lines to verify conjectures.

• Illustrate the relationship between the slopes of perpendicular lines by performing Activity 3.6, Investigating Slopes of Perpendicular Lines, McDougal Littell, p.172. On a piece of coordinate graph paper, place an index card so that the corner of the index card is on a lattice point (a lattice point is a point on the coordinate graph paper where the grid lines intersect). Mark the lattice point on the coordinate graph paper. Rotate the index card so that each edge passes through another lattice point but neither edge is vertical. Mark each of the other lattice points. Find the slope of each line using the marked points. Multiply the slopes. Make a conjecture.

• 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.

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.

Page 28

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.


Page 29

Unit IV –Triangles Stage 1 Desired Results


persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and


repeated reasoning.

Congruence G-‐CO Understand congruence in terms of rigid motions • 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.

• Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are

Transfer Students will be able to independently use their learning to…

Use triangle classifications, properties of triangles, and congruent triangles.

Meaning UNDERSTANDINGS Students will understand that… • The sum of the angle measures of a triangle is always the

same. • Comparing the corresponding parts of two triangles can

show whether the figures are congruent. • Two triangles can be proven congruent without having to

show that all corresponding parts are congruent. • If two triangles are congruent, then every pair of their

corresponding parts is also congruent. • Than angles and sides of isosceles and equilateral triangles

have special relationships. • Congruent corresponding parts of one pair of congruent

triangles can sometimes be used to prove another pair of triangles congruent. This often involves overlapping triangles.

ESSENTIAL QUESTIONS • Where do we see classification

used in concepts involving mathematics?

• How do you explain the different possible types of triangles?

• How do you identify corresponding parts of congruent triangles?

• How do you show that two triangles are congruent?

• Why isn’t SSA a way to prove two triangles congruent?

• How can you tell whether a triangle is isosceles or equilateral?

Acquisition Students will know… • A scalene triangle is a triangle in which no sides have the

same length. • An isosceles triangle is a triangle with at least two equal

sides. • An equilateral triangle is a triangle with three equal sides. • An equilateral triangle is a special type of isosceles triangle

and that the two types of triangles are not separate and distinct.

• In an isosceles triangle, the legs are the two congruent sides,

Students will be skilled at… Classifing triangles by their sides

and angles. Applying the Triangle Sum

Theorem and the Exterior Angle Theorem.

Naming congruent figures and identifying corresponding parts.

Proving triangles congruent by SSS, SAS, ASA, AAS, and HL, without and

Page 30

congruent. • Explain how the criteria for triangle

congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove geometric theorems • Prove theorems about triangles.

Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Expressing Geometric Properties with Equations G-‐PE Use coordinates to prove simple geometric theorems algebraically • Use coordinates to prove simple

geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

the base is the third side, the base angles are opposite the legs, and the vertex angle is the angle that has the legs as its sides.

• An acute triangle is a triangle with three acute angles. • A right triangle is a triangle with one right angle. • An obtuse triangle is a triangle with one obtuse angle. • In a right triangle, the hypotenuse is the side opposite the

right angle, and the legs are the sides of the right angle. • There is only one right angle in a right triangle, and only one

obtuse angle in an obtuse triangle. • An equilateral triangle is equiangular, and an equiangular

triangle is equilateral. • The Triangle Sum Theorem: the measures of the interior

angles of a triangle have a sum of 0180 . • Prove, and apply the Triangle Sum Theorem with numeric

and algebraic examples. • The measure of each of the angles in an equilateral triangle

is 060 . • The Exterior Angle Theorem: the measure of an exterior

angle of a triangle equals the sum of the measures of the remote interior angles.

• Prove, and apply the Exterior Angle Theorem with numeric and algebraic examples.

• CPCTC – Corresponding Parts of Congruent Triangles are Congruent.

• The Third Angle Theorem: If two angles in one triangle are congruent to two angles in another triangle, then the third angle in the first triangle is congruent to the third angle in the other triangle.

• The Base Angles Theorem: If two sides in a triangle are congruent, then the angles opposite those sides are congruent. If two angles in a triangle are congruent, then the sides opposite those angles are congruent.

• Prove, and apply the Base Angles Theorem and its converse

with formal proofs. Using the properties of isosceles

and equilateral triangles.

Page 31

with numeric and algebraic examples.

Unit IV –Triangles Stage 2 -‐ Evidence




1 -‐ POINT RESPONSE The response shows limited understanding of the problem's essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete

PERFORMANCE TASK (S): • Draw and classify as many different types of triangles as possible (such as a scalene right triangle, a

scalene obtuse triangle, a scalene acute triangle, an isosceles right triangle, an isosceles obtuse triangle, an isosceles acute triangle, an equilateral, equiangular, acute triangle). List triangles that are not possible (such as an equilateral right triangle and an equilateral obtuse triangle).

• 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.

Page 32

explanation of how the problem was solved may contribute to questions as to how and why decisions were made.



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









Excerpts from previous HSPA exams including multiple choice and open-‐ended problems should be given every class to assess student understanding and measure their individual

Page 33

skills. 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)



Unit IV –Triangles

Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

• To apply the definition of congruent figures, perform the Visual Approach Lesson Opener, McDougal Littell Chapter 4 Resource Book, p.26. Look for

congruent geometric figures in a quilt design. Color the quilt design so that all congruent figures are the same color. • Illustrate the Triangle Sum Theorem by performing Activity 4.1, Investigating Angles of Triangles, McDougal Littell, p.193. Draw and cut out a paper

triangle. Tear off the three corners and place them adjacent to each other to form a line. • Illustrate the Exterior Angle Theorem by performing Activity 4.1, Investigating Angles of Triangles, McDougal Littell, p.193. Draw and cut out a

paper triangle. Place the triangle on a piece of paper and extend one side to form an exterior angle. Tear off the corners that are not adjacent to an exterior angle and place them adjacent to each other to form the exterior angle.

• Illustrate the SSS Congruence Postulate by performing Activity 4.3, Investigating Congruent Triangles, McDougal Littell, p.211. On a piece of paper, place three pencils of different lengths so they make a triangle. Mark each vertex of the triangle by pressing the pencil points to the paper. Remove the pencils and draw the sides of the triangle. Repeat the steps and try to make a triangle that is not congruent to the one that is drawn.

• Illustrate the SAS Congruent Postulate by performing Activity 4.3, Investigating Congruent Triangles, McDougal Littell, p.211. On a piece of paper, place two pencils so their erasers are at the center of a protractor. Arrange them to form a 045 angle. Mark two vertices of the triangle by pressing the pencil points to the paper. Mark the center of the protractor as the third vertex. Remove the pencils and protractor and draw the sides of the triangle. Repeat the steps and try to make a triangle that has a 045 angle but is not congruent to the one that is drawn.

• Illustrate that SSA is not a way to prove triangles congruent by performing Activity 4.4, Investigating Triangles and Congruence, McDougal Littell,

p.228. Draw a segment and label it AB . Draw another point not on AB . Label this point E and draw AE

. Draw a circle with center at point Bthat intersects AE

in two points. Label the intersection G and H . Draw BG and BH . Compare ABG and ABH .

• Illustrate the Base Angle Theorem by performing Activity 4.6, Investigating Isosceles Triangles, McDougal Littell, p.236. Construct and cut out a paper acute isosceles triangle. Then fold the triangle along a line that bisects the vertex angle. Repeat with an obtuse isosceles triangle.

Page 34

• 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 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°.


Page 35

Unit V – Properties of Triangles Stage 1 Desired results




repeated reasoning.

Congruence G-‐CO Prove geometric theorems • Prove theorems about lines and


• Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to


Identify and use the properties of sides and angles in triangles.

Meaning UNDERSTANDINGS Students will understand that… • Triangles play a key role in relationships involving

perpendicular bisectors and angle bisectors. • There are special parts of a triangle that are always

concurrent. • In direct reasoning, all possibilities are considered and then

all but one are proved false. The remaining possibility must be true.

• The measures of the angles of a triangle are related to the lengths of the opposite sides.

• In triangles that have two pairs of congruent sides, there is a relationship between the included angles and the third pairs of sides.

ESSENTIAL QUESTIONS • What properties do the points of

concurrency of the medians, angle bisectors, altitudes, and perpendicular bisectors in acute, right, and obtuse triangles possess?

• What relationships exist between the measures of the angles and the lengths of the sides of a triangle?

• What relationships exist between the lengths of sides of a triangle?

Acquisition Students will know… • A median is a segment that connects a vertex to the

midpoint of the opposite side. • The centroid of a triangle is two-‐thirds of the distance from

each vertex to the midpoint of the opposite side. • An angle bisector is a segment that bisects an angle. • Any point on an angle bisector is equidistant from the sides

of the angle. • The incenter of a triangle is equidistant from the sides of a

triangle. • A perpendicular bisector is a segment that bisects a side and

Students will be skilled at… • Drawing, identifying, and using the

properties of medians of a triangle. • Drawing, identifying, and using the

properties of angle bisectors of a triangle.

• Drawing, identifying, and using the properties of altitudes of a triangle.

• Drawing, identifying, and using the properties of perpendicular bisectors of a triangle.

• Drawing, identifying, and using the

Page 36

180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Make geometric constructions • Make formal geometric constructions

with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Similarity, Right Triangles, and Trigonometry G-‐SRT Prove theorems involving similarity • Prove theorems about triangles.

Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Circles G-‐C • Construct the inscribed and

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

is perpendicular to that side. • Any point on a perpendicular bisector is equidistant from

the endpoints of a segment. • The circumcenter of a triangle is equidistant from the

vertices of a triangle. • An altitude is a segment from a vertex, perpendicular to the

opposite side. • The centroid and incenter of a triangle always lie in the

interior of a triangle. The orthocenter of an obtuse triangle lies in the exterior of the triangle and on the vertex of the right angle in a right triangle. The circumcenter of an obtuse triangle lies in the exterior of the triangle and on the midpoint of the hypotenuse of a right triangle.

• The midsegment of a triangle is a segment that joins the midpoints of two sides of a triangle.

• The Midsegment Theorem: the midsegment of a triangle is parallel to and half the length of the third side of the triangle.

• The longest side in a triangle is opposite the largest angle and the shortest side is opposite the smallest angle.

• The largest angle in a triangle is opposite the longest side and the smallest angle is opposite the smallest side.

• The sum of two sides of a triangle must be greater than the third side.

• The length of the third side of a triangle is smaller than the sum and larger than the difference of the other two sides of the triangle.

properties of a midsegment of a triangle.

• Writing an indirect proof. • Using angle measurements of a

triangle to order the lengths of the sides of a triangle, and using the lengths of the sides to order the angles of a triangle.

• Determining if the lengths of sides determine a triangle.

• Determining the possible length of the third side of a triangle given the lengths of the other two sides.

Unit V – Properties of Triangles Stage 2 -‐ Evidence

Evaluative Criteria Assessment Evidence

Page 37

RUBRIC/SCALE




0 -‐ POINT RESPONSE The response shows insufficient

PERFORMANCE TASK (S): • Construct the medians of an acute, right, and obtuse triangle. Construct the angle bisectors of an acute,

obtuse, and right triangle. Construct the altitudes of an acute, obtuse, and right triangle. Construct the perpendicular bisectors of an acute, obtuse, and right triangle. State the similarities and differences in the points of concurrency.

• 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.

• 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

Page 38

understanding of the problem's essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.


b) 10cm, 12cm, 20cm 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












Page 39


Unit V – Properties of Triangles

Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction

• Illustrate the Concurrency of Medians of a Triangle Theorem by performing Activity 5.3, McDougal Littell, p.286. Draw ABC . Locate the midpoint of AB , BC , and AC , and label the midpointsD ,E , and F , respectively. Label the intersection of the three medians pointG . Measure the distance from each vertex to the centroid and from the centroid to each midpoint, and compare the ratios.

• Construct the angle bisector of an angle. Pick a point on the angle bisector and measure the distance from that point to each of the sides of the angle. • Illustrate the Concurrency of Angle Bisectors Theorem by drawing ABC and constructing the angle bisectors of each angle. Label the incenter

D and draw a line segment perpendicular to each side of the triangle from point D . Inscribe a circle in the triangle with center D .

• Illustrate the Perpendicular Bisector Theorem by performing Activity 5.1, Investigating Perpendicular Bisectors, p.263. Draw AB on a piece of paper. Fold the paper so that point B lies directly on point A . Draw a line along the crease in the paper. Label the point where the line intersects

AB as point M . Label another point on the line C . Draw CA and CB . Measure MA , MB , CMA , CA , and CB . • Illustrate the Concurrency of Perpendicular Bisectors of a Triangle Theorem by performing Activity 5.2, McDougal Littell, p.272. Cut a large paper

acute scalene triangle. Label vertices A , B , and C . Fold the triangle to form the perpendicular bisectors of the sides. Label the point of

intersection of the perpendicular bisectors as P . Measure AP , BP , and CP . • Illustrate the relationship between the lengths of the sides of a triangle and the measures of the angles by performing Activity 5.5, McDougal Littell,

p.294. Draw any scalene triangle ABC Find the measure of each angle of the triangle. Find the length of each side of the triangle. Determine the relationship between the largest angle and the longest side of the triangle, and the smallest angle and shortest side of the triangle.


Page 40

Unit VI -‐ Similarity Stage 1 Desired Results




repeated reasoning.

Similarity, Right Triangles, and Trig G-‐SRT Understand similarity in terms of similarity transformations • 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.

• Use the properties of similarity transformations to establish the AA

Transfer Students will be able to independently use their learning to… Identify and apply properties of similar figures.

Meaning UNDERSTANDINGS Students will understand that… • 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 prove whether two polygons are similar and to find unknown side lengths. Triangles can be shown to be similar based on the relationship of two or three pairs of corresponding parts.

• When two or more parallel lines intersect other lines, proportional segments are formed.

ESSENTIAL QUESTIONS • How do you show two triangles are

similar? • How do you identify corresponding

parts of similar triangles? • How do you use proportions to find

side lengths in similar polygons?

Acquisition Students will know… • The scale factor is the ratio of two corresponding sides of

similar figures. • The corresponding angles in similar figures are congruent,

and the lengths of the sides are proportional. • The ratio of the perimeters of two similar figures is the same

as the scale factor. • Angle-‐Angle for Similar Triangles: If two angles in one

triangle are congruent to two angles in another triangle, then the two triangles are similar.

• Side-‐Angle-‐Side for Similar Triangles: If two sides of one triangle are proportional to two sides in another triangle, and the included angle in the first triangle is congruent to the included angle in the second triangle, then the two triangles are similar.

Students will be skilled at… Using ratios and proportions to

solve problems. Defining and identify similar

polygons. Determining the measures of

corresponding angles of similar figures.

Using proportions of corresponding sides to determine the lengths of sides in similar figures.

Determining if two triangles are similar using the AA, SSS, or SAS Similarity Theorems.

Page 41

criterion for two triangles to be similar

Prove theorems involving similarity • Use congruence and similarity

criteria for triangles to solve problems and to prove relationships in geometric figures.

• Side-‐Side-‐Side for Similar Triangles: If each side in one triangle is proportional to its corresponding side in another triangle, then the two triangles are similar.

Using proportionality theorems to calculate segment lengths.

Unit VI -‐ Similarity Stage 2 -‐ Evidence





PERFORMANCE TASK (S): • Construct two similar polygons. List all the pairs of congruent angles. Write the ratios of the

corresponding sides in a statement of proportionality. Find the scale factor. Find the perimeter of each polygon. Find the ratios of the perimeters.

• Have students create a diorama of the classroom or a room in their homes. They should provide a scale factor that they used to create the diorama.

Sample Assessment Questions:

• Simplify 6 ft 18in • Solve: 2 24

3 x=

• Solve: 2 29 15

x x− +=

• If the ratio of the angles in a triangle is 2:3:4, what are the measures of the angles? • True or false: All regular polygons are similar. • 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.

Page 42

The response shows limited understanding of the problem's essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved may contribute to questions as to how and why decisions were made.



• 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







Homework is assigned daily, from the textbook, Chapter Resource Practice Workbook, or

Page 43

other sources. (Synthesis, Analysis, Evaluation) Introductory and Closing Activities will be done every day to pre-‐assess student

knowledge and assess understanding of topics.(Synthesis, Analysis, Evaluation) Excerpts from previous HSPA exams including multiple choice and open-‐ended problems

should be given every class to assess student understanding and measure their individual skills.




Unit VI -‐ Similarity Stage 3 – Learning Plan


• To “discover” the relationship between corresponding sides and angles in similar figures, perform Activity 8.3, Making Conjectures about Similarity,

McDougal Littell, p. 472. Using a ruler and a protractor, find the lengths of corresponding sides and measures of corresponding angles in two photographs, one an enlarged version of the other, and complete the chart. Make conjectures.

• To “discover” the Triangle Proportionality Theorem, perform Activity 8.6, Investigating Proportional Segments, McDougal Littell, p.497. Using

geometry software, construct ABC and DE such that DE AC and point D lies on AB and E lies on BC . Measure BD , DA , BE , and

EC , and calculate BDDA and BE

EC. Make a conjecture.

• To “discover” the relationship between the sides of a triangle when a ray bisects an angle of the triangle, perform Activity 8.6, Investigating Proportional Segments, McDougal Littell, p.497. Construct PQR and the angle bisector of P . Label the intersection of the angle bisector and

QR as point B . Measure BR , RP , BQ , and QP , and calculate BRBQ

and RPQP

. Make a conjecture.


Page 44

Unit VII – Right Triangles and Trigonometry Stage 1 Desired Results




repeated reasoning.

Similarity, Right Triangles, and Trig G-‐SRT Define trigonometric ratios and solve problems involving right triangles • 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.

• Explain and use the relationship between the sine and cosine of complementary angles.

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

Transfer Students will be able to independently use their learning to… Identify and apply the properties of right triangles.

Meaning UNDERSTANDINGS Students will understand that… • Drawing in the altitude to the hypotenuse of a right triangle

forms three pairs of similar right triangles. • If the 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 other side lengths and angle measures.

• If certain combinations of side lengths and angle measures of any triangle are known, the Law of Sines or the Law of Cosines can be used to find other side lengths and angle measures.

• The angles of elevation and depression are the acute angles of right triangles formed by a horizontal distance and a vertical height.

ESSENTIAL QUESTIONS • How do you determine the best

method to find a side length or angle measure in a right triangle?

• How do trigonometric ratios relate to similar right triangles?

• How do you determine the best method to find a side length or angle measure in any triangle?

Acquisition Students will know… • The Pythagorean Theorem: If a and b represent the lengths

of the legs of a right triangle, and c represents the length of the hypotenuse, then 2 2 2a b c+ = .

• The most common Pythagorean triplets are the 3,4,5 and the 5,12,13 right triangles.

• In a 45-‐45-‐90 triangle, if given the length of the leg, multiply

Students will be skilled at… Simplify ingnumeric expressions

with radicals. Applying the properties of the

similar triangles when the altitude is drawn to the hypotenuse of a right triangle.

Page 45

Apply trigonometry to general triangles • (+) 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).

by 2 to find the length of the hypotenuse, or, if given the length of the hypotenuse, divide by 2 to find the length of the leg.

• In a 30-‐60-‐90 triangle, if given the length of the shorter leg, multiply by 2 to find the length of the hypotenuse and multiply by 3 to find the length of the longer leg. If given the length of the hypotenuse, divide by 2 to find the length of the shorter leg. If given the length of the longer leg divide by

3 to find the length of the shorter leg. • The sine of an angle is equal to the ratio of the opposite leg

and the hypotenuse. • The cosine of an angle is equal to the ratio of the adjacent leg

and the hypotenuse. • The tangent of an angle is equal to the ratio of the opposite

leg and the adjacent leg. • Using SOH-‐CAH-‐TOA will help to remember the definitions

of sine, cosine, and tangent. • Special right triangles that are solved using trigonometric

ratios instead of using the properties between the sides will not have exact answers but approximations.

• The angle of elevation is the angle formed by the horizon and the line of sight when an observer is looking up. The angle of depression is the angle formed by the horizon and the line of sight when the observer is looking down. The angle of elevation and the angle of depression are congruent.

• The Law of Sines: In any ABC , sin A

a= sin B

b= sinC

c .

• The Law of Cosines: in any ABC ,

c2 = a2 + b2 − 2abcosC • For any triangle, use the Law of Cosines, given SAS, to find

the third side of a triangle. • For any triangle, use the Law of Cosines, given SSS, to find

the measure of any angle.

Using the Pythagorean Theorem to determine the missing length of a side of a right triangle.

Using the converse of the Pythagorean Theorem to verify right triangles.

Finding the missing side lengths in special right triangles (45-‐45-‐90 and 30-‐60-‐90 triangles).

Finding the sine, cosine, and tangent of an acute angle in a right triangle.

Identifying and performing the appropriate trigonometric ratio in varying right triangle situations to find missing values.

Solving problems involving the angle of elevation or depression.

Finding the measure of a missing angle or missing side using the Law of Sines.

Finding the measure of a missing angle or missing side using the Law of Sines or the Law of Cosines.

Applying right triangle relationships to real-‐life situations to find unknown measurements.

Page 46

• For any triangle, use the Law of Sines, given ASA or AAS, to find the length of either of the two remaining sides.

• For any triangle, use the Law of Sines, given ASS, to find the angle opposite one of the given sides. Note that 0, 1, or 2 triangles are possible.

Unit VII – Right Triangles and Trigonometry Stage 2 -‐ Evidence




1 -‐ POINT RESPONSE The response shows limited

PERFORMANCE TASK (S): Sample Assessment Questions:

• Simplify 75 ; 32

• Simplify 5 2⋅ ; 6 3⋅

• Simplify 13;

62

• If the legs of a right triangle have lengths 9 and 12, find the length of the hypotenuse. • If a leg of a right triangle has a length of 8 and the hypotenuse has a length of 17, find the length of the

hypotenuse. • The length of each of the legs of an isosceles right triangle is 6. Find the length of the hypotenuse. • The length of the hypotenuse of an isosceles right triangle is 6. Find the length of each of the legs. • The length of the shorter leg in a 30-‐60-‐90 triangle is 6. Find the length of the longer leg and the

hypotenuse. • The length of the hypotenuse in a 30-‐60-‐90 triangle is 6. Find the length of the shorter leg and the longer

leg. • The length of the longer leg in a 30-‐60-‐90 triangle is 6. Find the length of the shorter leg and the

hypotenuse. • In triangle ABC with right angle B , 4AB = and 5BC = . Find mA . • In triangle ABC with right angle B , mA = 40 and 50AC = . Find BC .

• In triangle ABC with right angle B , mA = 50 and 40AB = . Find AC . • While flying a kite Linda lets out 45 ft. of string and anchors it to the ground. She determines that the

Page 47

understanding of the problem's essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved may contribute to questions as to how and why decisions were made.



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?







Homework is assigned daily, from the textbook, Chapter Resource Practice Workbook, or

Page 48

other sources. (Synthesis, Analysis, Evaluation) Introductory and Closing Activities will be done every day to pre-‐assess student knowledge

and assess understanding of topics.(Synthesis, Analysis, Evaluation) Excerpts from previous HSPA exams including multiple choice and open-‐ended problems

should be given every class to assess student understanding and measure their individual skills.




Unit VII – Right Triangles and Trigonometry


Summary of Key Learning Events and Instruction • Visually illustrate the Pythagorean Theorem by performing the following activity. Construct a right scalene triangle. Construct squares on all three

sides of the right triangle. Construct the center of the square of the longer leg. (One way to do this is to construct the diagonals of that square and then erase the diagonals, but leave the point of intersection.) Construct a line through this center parallel to the hypotenuse. Construct another line through the center, perpendicular to the hypotenuse. Then cut out the two smaller squares, and divide the medium square into the four pieces using the drawn lines. Place the five pieces on the square drawn from the hypotenuse so that they cover this square and do not overlap.

• To “discover” the relationship between the legs and the hypotenuse in an isosceles right triangle, perform Activity 9.4, Investigating Special Right Triangles, McDougal Littell, p.550. The length of each leg in an isosceles right triangle is 3, 4, or 5. Each person in a group should choose a different length and, using the Pythagorean Theorem, find the length of the hypotenuse, in simplest radical form. Compare results with others in the group and make a conjecture.

• To “discover” the relationship between the shorter leg, longer leg, and hypotenuse in a 30-‐60-‐90 triangle, perform Activity 9.4, Investigating Special Right Triangles, p.550. Construct an equilateral right triangle with side lengths 4, 6, or 8. Construct the altitude from one of the vertices. Find the side lengths, in simplest radical form, of one of the 30-‐60-‐90 triangles, with each person in a group choosing a different length. Compare results with others in the group and make a conjecture.

• To illustrate that the sine, cosine, and tangent of an angle is independent of the size of the right triangle but dependent upon the measure of an angle and the ratios of the sides, perform the Geometer’s Sketchpad Activity, Exploring Geometry with the Geometer’s Sketchpad, 1999 Key Curriculum Press, p.196. Construct a right triangle ABC where B is the right angle. Measure A and label BC as opposite, AB as adjacent,

and AC as hypotenuse. Measure the ratios opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent. Drag point A and examine the ratios and mA .

Page 49

• Have students research the construction of ramps for handicap access, loading docks, or other uses. Have them determine building codes for the standard angle measures and lengths of the ramps.

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?


Page 50

Unit VIII -‐ Quadrilaterals Stage 1 Desired Results




repeated reasoning.

Congruence G-‐CO Experiment with transformations in the plane • Given a rectangle, parallelogram,

trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Prove geometric theorems • Prove theorems about

parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely,


Identify and apply the properties of quadrilaterals.

Meaning UNDERSTANDINGS Students will understand that… • The sum of the angle measures of a polygon depends on the

number of sides the polygon has. • Parallelograms have special properties regarding their sides,

angles, and diagonals. • If a quadrilateral’s 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.

ESSENTIAL QUESTIONS • How can you find the sum of the

measures of polygon angles? • How can you classify quadrilaterals?

Acquisition Students will know… • The five properties of a parallelogram are:

1. Both pairs of opposite sides are parallel. 2. Both pairs of opposite sides are congruent. 3. Both pairs of opposite angles are congruent. 4. Consecutive angles are supplementary. 5. Diagonals bisect each other.

• The five ways to prove that a quadrilateral is a parallelogram are: 1. Both pairs of opposite sides are parallel. 2. Both pairs of opposite sides are congruent.

Students will be skilled at… Using the Interior Angles of a

Quadrilateral Theorem to find the measures of angles in a quadrilateral.

Proving and applying the properties of a parallelogram.

Proving that a quadrilateral is a parallelogram.

Proving and applying the properties of rectangles, rhombi, and squares.

Proving and applying the properties

Page 51

rectangles are parallelograms with congruent diagonals

Make geometric constructions • Make formal geometric constructions

with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Expressing Geometric Properties with equations G-‐GPE Use coordinates to prove simple geometric theorems algebraically • Use coordinates to prove simple

geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

• 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).

3. Both pairs of opposite angles are congruent. 4. Diagonals bisect each other. 5. One pair of opposite sides are parallel and congruent.

• A rectangle has all the properties of a parallelogram and 1. four right angles 2. congruent diagonals

• When both diagonals are drawn in a rectangle, the four resulting non-‐overlapping triangles are isosceles.

• A rhombus has all the properties of a parallelogram and, in addition: 1. four congruent sides 2. the diagonals are perpendicular 3. each diagonal bisects two opposite angles

• When both diagonals are drawn in a rhombus, the resulting four non-‐overlapping triangles are congruent right triangles.

• A square is both a rectangle and a rhombus, and therefore has the properties of both figures. Illustrate the relationship between parallelograms, rectangles, rhombi, and squares with a Venn diagram.

• A trapezoid is a quadrilateral with only one pair of parallel sides (bases).

• A median of a trapezoid is: 1. Parallel to each base. 2. Half the sum of the lengths of the bases.

• An isosceles trapezoid is a trapezoid with congruent legs and possesses the following properties: 1. Each pair of base angles are congruent. 2. Diagonals are congruent.

of a trapezoid and an isosceles trapezoid.

Applying the properties of a kite.

Unit VIII -‐ Quadrilaterals Stage 2 -‐ Evidence


Page 52

RUBRIC/SCALE





PERFORMANCE TASK (S): • Complete a chart with a list of all the properties of quadrilaterals and check off the shape

(parallelogram, rectangle, rhombus, square, trapezoid, isosceles trapezoid, kite) that always has the given property. Compare and contrast properties of the different quadrilaterals.

• Have students research unique buildings. They should describe the shape of the building and the properties of that shape. Students can make a presentation to the class about the challenges faced by architects in creating the building.

Sample Assessment Questions: • The measures of three angles of a quadrilateral are 70, 80, and 90. What is the measure of the fourth

angle? • The measures of the angles in a quadrilateral are x , 2x , 3x , and 4x . Find the value of x . • In ABCD , the mA = 50 . Find mB , mC , and mD . • In ABCD , the mA = 5x +10 and mC = 10x − 20 . Find x. • In rectangle ABCD , mA = 10x − 20 . Find x.

• In rectangle ABCD , AC and BD intersect at point E . If 4 12AE x= − and 6BD x= , find x.

• In rhombus ABCD , 5 15AB x= + and 10BC x= . Find CD .

• In rhombus ABCD , 6AC = and 8BD = , find AB .

• In trapezoid ABCD , the mA = 4x + 6 and mB = 6x + 4 . Find x.

• In isosceles trapezoid ABCDwith bases BC and AD , the mA = 50 . Find mB , mC , and mD .

• The bases of a trapezoid are 8 and 12. What is the length of the median? • One of the bases of a trapezoid is 8 and the median is 12. What is the length of the other base?

Page 53













Page 54


Unit VIII -‐ Quadrilaterals Stage 3 – Learning Plan


• To “discover” the properties of a parallelogram, perform Activity 6.2, Investigating Parallelograms, McDougal Littell, p.329. Construct a

parallelogram using geometry software. Measure the lengths of the sides and the angles. Make conjectures. Construct the diagonals. Measure the distance from the intersection of the diagonals to each vertex of the parallelogram. Make conjectures.

• To “discover” the properties of a rectangle, construct a rectangle and its diagonals. Measure the diagonals. Make conjectures. Measure the angles formed by the intersection of the two diagonals. Make conjectures.

• To “discover” the properties of a rhombus, construct a rhombus and its diagonals. Measure the diagonals. Make conjectures. Measure the angles formed by the intersection of the two diagonals. Make conjectures. Measure the angles at each vertex formed by a diagonal and a side of the rhombus. Make conjectures.

• To “discover” the properties of a trapezoid, construct a trapezoid. Measure its angles and make conjectures. Construct the diagonals and measure the lengths of the diagonals and the distance from the intersection of the diagonals to each vertex of the trapezoid. Make conjectures.

• To “discover” the properties of an isosceles trapezoid, construct an isosceles trapezoid. Measure its angles and make conjectures. Construct the diagonals and measure the lengths of the diagonals and the distance from the intersection of the diagonals to each vertex of the isosceles trapezoid. Make conjectures.


Page 55

Unit IX-‐ Polygons

Stage 1 Desired Results ESTABLISHED GOALS CCSS The Standards: http://www.corestandards.org/the-‐standards Common Core State Standards for Mathematics Mathematical Practices Make sense of problems and




repeated reasoning.

Congruence G-‐CO Experiment with transformations in the plane • Given a rectangle, parallelogram,

trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Transfer Students will be able to independently use their learning to… Identify and determine angle measures of convex polygons.

Meaning UNDERSTANDINGS Students will understand that… • The sum of the interior angles of a polygon is related to the

number of sides of the polygon.

ESSENTIAL QUESTIONS • How do we identify and classify

polygons? • How is the sum of the measures of

the interior angles of a polygon related to the number of its sides?

Acquisition Students will know… • Polygons are classified by the number of sides: 3 sides, a

triangle; 4 sides, a quadrilateral; 5 sides, a pentagon; 6 sides, a hexagon; 7 sides, a heptagon; 8 sides, an octagon; 9 sides, a nonagon; 10 sides, a decagon.

• A polygon is concave if a segment joining any two points in the interior of the polygon is in the exterior of the polygon.

• A polygon is convex if it is not concave. • A regular polygons is equilateral and equiangular.. • The sum of the measures of the interior angles of a polygon,

with n sides is ( 2)180n− .

• The sum of the exterior angles of any polygon (regardless of the number of sides) is 360.

Students will be skilled at… Identifying, naming, and describing

polygons. Discovering and applying the

formula for the sum of the interior angles in a convex polygon.

Unit IX-‐ Polygons Stage 2 -‐ Evidence


Page 56

RUBRIC/SCALE





PERFORMANCE TASK(S):

• Determine the sum of the interior angles of a triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon. Explain what happens to the sum of the interior angles as the number of sides increase.

• Determine the sum of the exterior angles of a triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon. Explain what happens to the sum of the exterior angles as the number of sides increase.

• Determine the measure of each interior and exterior angle of a regular triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon. Explain what happens to each interior and exterior angle of a regular polygon as the number of sides increase.

• What is the relationship between each interior and exterior angle in any polygon?

Page 57

The response shows insufficient understanding of the problem's essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.











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

Page 58

previous lessons. Students will be given a unit test that provides a review of the concepts and skills in the

unit. Unit IX-‐ Polygons


Summary of Key Learning Events and Instruction To “discover” the sum of the angles in a convex polygon perform Activity 11.1, Investigating the Sum of Polygon Angle Measures, McDougal Littell, p.661. Complete a chart with the number of sides of a convex polygon, the number of triangles formed when all the diagonals are drawn from one of the vertices, and the sum of the interior angles. Generalize the formula: sum of the interior angles of a polygon = ( 2)180n− , where n is the number of sides in the polygon. Teacher made Performance Assessment Tasks (PATs) Released PATs Online State Resources

Page 59

Unit X-‐ Circles

Stage 1 Desired Results ESTABLISHED GOALS CCSS The Standards: http://www.corestandards.org/the-‐standards Common Core State Standards for Mathematics Mathematical Practices Make sense of problems and persevere

in solving them. Reason abstractly and quantitatively. Construct viable arguments and


repeated reasoning.

Congruence G-‐CO Experiment with transformations in the plane • Know precise definitions of angle,

circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

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

Transfer Students will be able to independently use their learning to… Identify and apply the properties of lines and angles in circles.

Meaning UNDERSTANDINGS Students will understand that… • A radius of a circle and the tangent that intersects the

endpoints of the radius on the circle have 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 (1) arcs formed by chords that inscribe angles, (2) angles and arcs formed by lines intersecting either within a circle or outside a circle, and (3) intersecting chords, 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.

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 resulting angles, arcs, and segments?

• How do you find the equation of a circle in the coordinate plane?

Acquisition Students will know…

• 2 radius diameter⋅ = and 12

radius diameter=

• All radii of the same circle or of congruent circles are congruent.

• A tangent is perpendicular to the radius of a circle drawn to the point of tangency.

Students will be skilled at… Identifying segments and lines

related to circles. Using properties of a tangent of a

circle. Using properties of chords of

circles. Identifying and using properties of

Page 60

Make geometric constructions • Construct an equilateral triangle, a

square, and a regular hexagon inscribed in a circle.

Circles G-‐C Understand and apply theorems about circles • Prove that all circles are similar. • Identify and describe relationships

among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

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

Find arc lengths and areas of sectors of circles • Derive using similarity the fact that the

length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

• Two segments from the same exterior point tangent to a circle are congruent.

• The measure of a central angle is equal to the measure of its intercepted arc.

• The measure of an inscribed angle is half the measure of its intercepted arc.

• Angles that intercept the same arc are congruent. • The opposite angles in an inscribed quadrilateral are

supplementary. • The measure of an angle formed by a tangent and a chord

that intersect at a point on the circle is one-‐half the measure of its intercepted arc.

• The measure of an angle formed by two chords that intersect in the interior of the circle is one-‐half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

• The measure of an angle formed by a tangent and a secant, two tangents, or two secants that intersect in the exterior of a circle is one-‐half the difference of the measures of the intercepted arcs.

• The standard form of a circle on a coordinate plane is 2 2 2( ) ( )x h y k r− + − = , where ( , )h k are the

coordinates of the center of the circle and r is the length of the radius.

arcs of circles. Identifying and using properties of

inscribed angles. Identifying and using properties of

angles formed by chords, secants, and tangents.

Applying the properties of intersecting chords, intersecting secants, and a secant that intersects a tangent.

Writing the equation of a circle in the coordinate plane.

Unit X-‐ Circles Stage 2 -‐ Evidence


Page 61

RUBRIC/SCALE




0 -‐ POINT RESPONSE The response shows insufficient understanding of the problem's essential mathematical concepts. The procedures, if

PERFORMANCE TASK(S):

Task 1: • Find the missing parts of the circle.

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

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

Page 62

any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.


• Find x.

• Find the value of each variable.

• Find the value of each variable.

• What is the standard equation of the circle with center (5, -‐2) and radius of 7? • What is the standard equation of the circle with center (4,3) that passes through (-‐1,1)? Task 2: • Make a chart. Explain the relationship between chords, secants, and tangents and the measures of the

Page 63

angles formed by theses segments or lines.












Students will be given a unit test that provides a review of the concepts and skills in the unit. Unit X-‐ Circles


Summary of Key Learning Events and Instruction To illustrate the relationship between the measure of an inscribed angle and its corresponding central angle, perform Activity 10.3, Investigating Inscribed Angles, McDougal Littell, p.612. Construct a circle with center P . Construct a central angle labeled RPSR . Locate three points on Pe in the exterior of RPSR and label them T ,U , and V . Draw the inscribed angles RTS , RUS , and RVS . Use a protractor to measure RPS , RTS , RUS , and RVS . Make conjectures.

Page 64

• Have students research the Hubble telescope to gather information about the history of the telescope and its purpose for the NASA program. (Two tangent lines that extend from Hubble telescope to the farthest point it can see on the Earth create the angle of sight for the Hubble telescope.)

• 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.

Page 65

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


Page 66

Unit XI-‐ Area





repeated reasoning.

Circles G-‐C Find arc lengths and areas of sectors of circles • Derive using similarity the fact that

the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Geometric Measurement and

Transfer Students will be able to independently use their learning to… Find the area of plane figures.

Meaning UNDERSTANDINGS Students will understand that… • The area of a parallelogram or a triangle can be found when

the length of its base and its 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 a kite can be found when the lengths of its diagonal are known.

• The area of a regular polygon is a function of the distance from the center to a side and the perimeter.

• 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 the 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.

ESSENTIAL QUESTIONS • How are the formulas for the area

of triangles and quadrilaterals applied when solving problems?

• How are the formulas for the circumference and arc length, and area of a circle and sector applied when solving problems?

• Do two-‐dimensional figures with the same area have the same perimeter? Why or why not?

• Do two-‐dimensional figures with the same perimeter have the same area? Why or why not?

• How do perimeters and areas of similar polygons compare?

Acquisition Students will know…

• Area of a square = 2s

• Area of a rectangle = length width⋅

• Area of a parallelogram = base height⋅

Students will be skilled at… • Applying the formulas for the area

of a square, rectangle, parallelogram, triangle, rhombus, trapezoid, and kite.

• Applying the formula for the

Page 67

Dimension G-‐GMD Explain volume formulas and use them to solve problems • Give an informal argument for the

formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments

Expressing Geometric Properties with Equations G-‐GPE Use coordinates to prove simple geometric theorems algebraically • Use coordinates to compute

perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

• Area of a triangle = 12

base height⋅ ⋅

• Area of a rhombus = 1 212

diagonal diagonal⋅ ⋅

• Area of a trapezoid = 1 21 ( )2

base base height⋅ + ⋅

• Circumference of a circle = 2 rπ or = dπ

• Arc length 2360m rπ= ⋅

• Area of a circle = 2rπ

• Area of a sector 2

360m rπ= ⋅

• It is not essential that students memorize the area formulas. It is given to students on most standardized tests (such as the HSPA and the SAT).

circumference and the arc length of a circle.

• Applying the formula for the area of a circle and the area of a sector of a circle.

• Applying the relationship between the perimeters and area of similar figures.

Unit XI-‐ Area Stage 2 -‐ Evidence


3 -‐ POINT RESPONSE The response shows complete understanding of the problem's essential mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer

PERFORMANCE TASK (S): • Have students construct a picture with a triangle, square, rectangle, parallelogram, rhombus, trapezoid,

circle, and sector of a circle, and find the area of each shape, with mathematical explanations. • You are asked to design a garden with the maximum area with a given amount of fencing. Draw,

determine, and explain the area of at least three regions. • Have students hypothetically “redecorate” a room in their home, with carpeting or floor tiles, paint or

wallpaper, and/or window treatments. All measurements need to be determined, and all material costs need to be researched. A detailed summary needs to be presented.

Sample Assessment Questions: • If a square has a side with a length of 8, what is the area of the square? • If the area of a square is 100, what is the length of each side of the square?

Page 68

how and why decisions were made.




Copyright © State of New Jersey, 2006 NJ Department of Education http://www.nj.gov/education/njpep/a

• If the length of a rectangle is 12, and the width is 5, what is the area of the rectangle? • If the area of a rectangle is 48 and the width is 8, what is the length of the rectangle? • If the base of a triangle is 12, and the height is 5, what is the area of the triangle? • If the area of a triangle is 48 and the base is 8, what is the height of the triangle? • If the radius of a circle is 8, what is the circumference of the circle, in terms of π ? • If the diameter of a circle is 8, what is the circumference of the circle, in terms of π ?

• If 36C π= , what is the radius of the circle? • If the radius of a circle is 10, what is the area of the circle, in terms of π ? • If the diameter of a circle is 10, what is the area of the circle, in terms of π ?

• If 36A π= , what is the radius of the circle? • Find the area of the shaded region of a circle inscribed in a square with side 6. Find the area of four

circles that are tangent to each other inscribed in a square with side 6. Find the area of nine circles that are tangent to each other inscribed in a square with side 6. (The area of the shaded region in all three situations is 36 9π− .)

Page 69

ssessment/TestSpecs/MathNJASK/rubrics.html Student Responses should be: Accurate Clear Effective Organized Thorough Thoughtful












Unit XI-‐ Area Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction • To “discover” the formulas for the area of parallelograms, triangles, and trapezoids, perform Activity 6.7, Areas of Quadrilaterals, McDougal Littell,

p.371. To illustrate the area of a parallelogram, draw a line through one of the vertices of an index card. Cut off the triangle and tape it to the opposite side to form a parallelogram. To illustrate the area of a triangle, fold a piece of paper, draw a scalene triangle, and cut through both thicknesses to create two congruent triangle. Align corresponding sides of the two triangles to form a parallelogram. To illustrate the area of a

Page 70

trapezoid, fold a piece of paper, draw a trapezoid, and cut through both thicknesses to create two congruent trapezoids. Align corresponding sides of the two trapezoids to form a parallelogram.

• To “discover” and define π , measure the circumference and diameter of a circle and divide the circumference by the diameter of the circle. • Find the area of each figure.

• Find the area of each regular polygon.

• Find the area of each circle.

10in

9in

60°

11mm

15mm6mm

Page 71

• Find the area of the irregular figure below.


Page 72

Unit XII-‐ Three-‐dimensional Figures





repeated reasoning.

Geometric Measurement and Dimension G-‐GMD Explain volume formulas and use them to solve problems • Give an informal argument for the

formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

Transfer Students will be able to independently use their learning to… Identify and find the surface area and volume of three-‐dimensional figures.

Meaning UNDERSTANDINGS Students will understand that… • Nets can be used to build solids. • A three-‐dimensional figure can be analyzed by describing the

relationships among its vertices, edges, and faces. • The surface area of a three-‐dimensional figure is equal to the

sum of the areas of each surface of the figure. • The volume of a prism and a cylinder can be found when its

height and the area of the 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 its radius is known. • Ratios can be used to compare the areas and volumes of

similar solids.

ESSENTIAL QUESTIONS • How do we use the net of a three-‐

dimensional figure in order to find the surface area of that figure?

• How can you determine the surface area and volume of a solid?

• What life situations might require us to calculate surface area or volume?

• Do three-‐dimensional shapes with the same volume have the same surface area? Why or why not?

• How do the surface areas and volumes of similar solids compare?

Acquisition Students will know… • Use the nets of a cube, rectangular and triangular prism, and

cylinder to find the surface area. Can also use the formula 2S B Ph= + for a prism and the formula 22 2S rh rπ π= +

for a cylinder. • Use the nets of a square and triangular pyramid to find the

surface area. Can also use the formula 12

S B Pl= + .

• Emphasize the difference between surface area and volume.

Students will be skilled at… • Identifying and classifying

polyhedra. • Identifying the net a three-‐

dimensional shape. • Finding the lateral and surface area

of a cube, rectangular and triangular prism and cylinder.

• Finding the lateral and surface area of a square and triangular pyramid.

Page 73

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

Visualize relationships between two-‐dimensional and three dimensional objects • Identify the shapes of two-‐

dimensional cross-‐sections of three dimensional objects, and identify three-‐dimensional objects generated by rotations of two-‐dimensional objects

• Volume of a cube: 3V e=

• Volume of a rectangular prism: V lwh=

• Volume of a cylinder: 2V r hπ=

• Volume of a triangular prism: 12 triangle triangle prismV b h h=

• Illustrate that the relationship between the volume of a prism and pyramid with the same base and height is

13pyramid prismV V= by pouring water from the pyramid into

the prism.

• Volume of a pyramid: 13

V Bh=

• Finding the volume of a cube, rectangular and triangular prism, and a cylinder.

• Finding the volume of a square and triangular pyramid.

• Finding the volume of a cone. • Finding the measures of the

missing height, slant height, edge, or radius given the surface area or the volume of a pyramid or a cone.

• Finding the volume of a sphere and hemisphere

Unit XII-‐ Three-‐dimensional Figures Stage 2 -‐ Evidence



2 -‐ POINT RESPONSE The response shows nearly complete understanding of the problem's essential mathematical concepts. The student

PERFORMANCE TASK (S): • Construct a prism, cylinder, and a pyramid out of construction paper. Find the surface area and volume

of each shape with a complete explanation. • You are asked to design a cereal box that is the most efficient with regards to the cardboard used with a

given volume. Draw diagrams or construct three prisms out of construction paper. Determine and explain the surface area of each prism.

Sample Assessment Questions: • The edge of a cube is 4. Find the surface area of the cube. • The surface area of a cube is 150. Find the length of an edge of the cube. • The edge of a cube is 5. Find the volume of the cube. • The volume of a cube is 64. Find the length of an edge of the cube. • A rectangular prism has length of 5, width of 4, and height of 3. Find the surface area of the rectangular

prism. • A rectangular prism has length of 5, width of 4, and height of 3. Find the volume of the rectangular

prism. • A cylinder has a base with a radius of 4 and a height of 5. Find the surface area (in terms of π ) of the

cylinder.

Page 74

executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear, causing the reader to make some inferences.




• A cylinder has a base with a radius of 4 and a height of 5. Find the volume (in terms of π ) of the cylinder.

• A square pyramid has a base with each side 5 and a height of 6. Find the volume of the square pyramid.

Student Responses should be: Accurate Clear

OTHER EVIDENCE:

Page 75

Effective Organized Thorough Thoughtful

Students will show they have achieved Stage 1 goals by . . . • Providing written or oral response to one of the essential questions. • The students will keep an ongoing journal all year of accumulating insight about which rules and










Unit XII-‐ Three-‐dimensional Figures Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction • Complete a chart listing different prisms and pyramids (for example, triangular prism, rectangular prism, pentagonal prism, …, triangular pyramid,

square pyramid, pentagonal pyramid, …), the number of faces (F), the number of vertices (V), and the number of edges (E). “Discover” Euler’s Theorem ( 2)F V E+ = +

Find the surface area and volume of the solids below:

Page 76


Page 77

Unit XIII-‐ Transformations




repeated reasoning.

Congruence G-‐CO Experiment with transformations in the plane

• Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Transfer Students will be able to independently use their learning to… Recognize and apply properties of transformations.

Meaning UNDERSTANDINGS Students will understand that… • The size and shape of a geometric figure stay the same when

(1) its location and orientation changes, (2) it is flipped across a line, or (3) it is turned about a point.

• A scale factor can be used to make a larger or smaller copy of a figure that is also similar to the original figure.

• If two figures in a plane are congruent, one can be mapped onto the other using a composition of reflections.

• Some shapes can fit together in a repeating pattern that fills a plane, or tessellates. The angle measures of polygons that fit together in this way have a special relationship.

ESSENTIAL QUESTIONS • How can you change a figure’s

position without changing its size and shape? How can you change a figure’s size without changing its shape?

• How can you represent a transformation in the coordinate plane?

• How do you recognize symmetry in a figure?

Acquisition Students will know… • A translation can be referred to as a slide, a reflection as a

flip, and a rotation as a turn. • Translations, reflections, and rotations preserve shape and

size.

Students will be skilled at… • Recognizing and applying

properties of translations. • Recognizing and applying

properties of reflections. • Recognizing and applying

properties of rotations. • Identifying symmetric figures and

drawing lines of symmetry.

Page 78

• Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another

Unit XIII-‐ Transformations Stage 2 -‐ Evidence




PERFORMANCE TASK (S): • On a coordinate paper, have students draw a triangle and state the coordinates of the vertices. Then

have students translate the triangle, reflect the triangle about the x -‐ and y -‐ axis, and rotate the triangle, and state the coordinates of the vertices for each triangle.

• Have students draw all the lines of symmetry in each of the following geometric figures: a scalene triangle, an isosceles triangle, an equilateral triangle, a parallelogram, a rectangle, a rhombus, a square, a trapezoid, an isosceles trapezoid, and a circle.

Page 79









Page 80







Unit XIII-‐ Transformations Stage 3 – Learning Plan Summary of Key Learning Events and Instruction

Perform “alphabet symmetry”. Identify the lines of symmetry in capital letters of the alphabet (mostly vertical and horizontal lines). Teacher made Performance Assessment Tasks (PATs) Released PATs Online State Resources

Page 81

Unit XIV -‐ Probability Stage 1 Desired Results




repeated reasoning.

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

Transfer Students will be able to independently use their learning to… Use statistics and determine the probability of various events.

Meaning UNDERSTANDINGS Students will understand that… • Various counting methods can help you analyze situations

and develop theoretical probabilities. • 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.

• In geometric probability, numbers of favorable and possible outcomes are geometric measures such as lengths of segments or areas of regions.

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?

Acquisition Students will know… • The Fundamental Counting Principle: If one event can occur

in n ways, and another event can occur in m ways, then the number of ways that both events can occur is n ⋅m

• Permutations of n objects taken r at a time: The number of permutations of r objects taken from a group of n distinct

objects is denoted by n pr and is given by n!

n − r( )! • Combinations of n objects taken r at a time: The number of

combinations of r objects taken from a group of n distinct

objects is denoted by nCr and is given by n!

r! n − r( )! . • A permutation is an arrangement of items in a particular

order. A selection in which order does not matter is called a

Students will be skilled at… • Using the Fundamental Counting

Principle. • Finding the number of

permutations of n items. • Finding the number of

combinations of n items. • Using the permutation formula. • Using the combination formula. • Identifying whether the order

matters in an event. • Finding experimental probability. • Using a simulation. • Finding theoretical probability.

Page 82

Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.4. Construct and interpret two-‐way frequency tables of data when two categories are associated with each object being classified. Use the two-‐way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S.CP.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.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

combination. • The Theoretical Probability of an Event: When all outcomes

are equally likely, the theoretical probability that an event A will occur is P(A) = (number of outcomes in A)/(total number of outcomes). The theoretical probability of an event is often simply called the probability of an event.

• Two events are independent if the occurrence of one has no effect on the occurrence of the other. If A and B are independent events, then the probability of both A and B occur is P(A and B) = P(A) ⋅ P(B).

• Two events A and B are dependent events if the occurrence of one affects the occurrence of the other. If A and B are dependent events, then the probability that A and B occur is P(A and B) = P(A) ⋅ P(B/A).

• The union or intersection of two events is called a compound event. If A and B are two events, then the probability of A or B is: P(A or B) = P(A) + P(B) – P(A and B). If A and B are mutually exclusive, then the probability of A or B is: P(A or B) = P(A) + P(B).

• Finding probability using combinations.

• Classifying events as independent or dependent.

• Finding the probability of independent events.

• Finding the probability of dependent events.

• Finding the probability of compound events.

• Using a tree diagram to find the conditional probability

• Using segments to find probability. • Using area to find probability.

Unit XIV -‐ Probability Stage 2 -‐ Evidence


Page 83

RUBRIC/SCALE





PERFORMANCE TASK (S): 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?

Page 84



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.












Students will be given a unit test that provides a review of the concepts and skills in the

Page 85

unit. Unit XIV -‐ Probability Stage 3 – Learning Plan

Summary of Key Learning Events and Instruction • 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? • 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.

1211109876543210

A B C D E F G H I J K L M

Page 86

Page 87

Benchmark Assessment Quarter 1 1. Students will be able to understand and apply the basic undefined and defined terms of geometry. 2. Students will be able to use inductive and deductive reasoning to draw conclusions. 3. Students will be able to apply angle relationships with parallel and perpendicular lines.

Benchmark Assessment Quarter 2 1. Students will be able to use triangle classifications, properties of triangles, and congruent triangles. 2. Students will be able to identify and use the properties of sides and angles in triangles. 3. Students will be able to identify and apply properties of similar figures.

Benchmark Assessment Quarter 3 1. Students will be able to identify and apply the properties of right triangles. 2. Students will be able to identify and apply the properties of quadrilaterals. 3. Students will be able to identify and determine angle measures of convex polygons. 4. Students will be able to identify and apply the properties of lines and angles in circles.

Benchmark Assessment Quarter 4 5. Students will be able to find the area of plane figures. 6. Students will be able to identify and find the surface area and volume of three-‐dimensional figures. 7. Students will be able to recognize and apply properties of transformations. 8. Students will be able to use statistics and determine the probability of various events.

Documents

CurriculumManagementSystem! - Monroe Township … · 2013-07-02 · Goals/Essential!Questions/Objectives/Instructional!Tools/Activities! ! Pages! ... V.Properties!of! ... Trapezoids!and!Kites!