Geometry - South Plainfield Public Schools

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South Plainfield Public Schools
Curriculum Guide
Mathematics
Geometry
Authors:
Anthony Emmons
Kathy Zoda
Curriculum Coordinator:
Paul C. Rafalowski
Board Approved on: June 13, 2012
1
Table of Contents
South Plainfield Public Schools Board of Education Members
and Administration
Page: 3
Recognitions
Page: 4
District Mission Statement
Page: 5
Index of Courses
Page: 6
Curriculum Guides
Page: 7-42
Mathematics Practice Standards
Page: 43-45
Common Core Standards
Page: 46-48
Resources for State Assessments
Page: 49
2
Members of the Board of Education
Jim Giannakis, President
Debbie Boyle, Vice President
Carol Byrne
John T. Farinella, Jr
Christopher Hubner
Sharon Miller
William Seesselberg
Joseph Sorrentino
Gary Stevenson
Central Office Administration
Dr. Stephen Genco, Superintendent of Schools
Dr. Frank Cocchiola, Interim Assistant Superintendent of Schools
Mr. James Olobardi, Board Secretary/ BA
Mrs. Laurie Hall, Supervisor of Student Personnel Services
Mr. Vincent Parisi, Supervisor of Math and Science
Mrs. Marlene Steele, Supervisor of Transportation
Mrs. Annemarie Stoeckel, Supervisor of Technology
Ms. Elaine Gallo, Director of Guidance
Mr. Al Czech, Director of Athletics
Mr. Paul Rafalowski, Curriculum Coordinator
3
Recognitions
The following individuals are recognized for their support in developing this Curriculum Guide:
Grade/Course
Writer(s)
Kindergarten:
Ms. Joy Czaplinski and Ms. Pat Public
Grade 1:
Ms. Patti Schenck-Ratti, Ms. Kim Wolfskeil and
Ms. Nicole Wrublevski
Grade 2:
Ms. Cate Bonanno, Ms. Shannon Colucci and
Ms. Maureen Wilson
Grade 3:
Ms. Cate Bonanno and Ms. Theresa Luck
Grade 4:
Ms. Linda Downey and Ms. Kathy Simpson
Grade 5:
Mr. John Orfan and Ms. Carolyn White
Grade 6:
Ms. Joanne Haus and Ms. Cathy Pompilio
Grade 7:
Ms. Marianne Decker and Ms. Kathy Zoda
Grade 8:
Ms. Marianne Decker and Ms. Donna Tierney
Algebra 1:
Ms. Donna Tierney and Ms. Kathy Zoda
Geometry:
Mr. Anthony Emmons and Ms. Kathy Zoda
Algebra 2:
Mr. Anthony Emmons and Mr. John Greco
Algebra 3/Trigonometry:
Ms. Anu Garrison and Mr. David Knarr
Senior Math Applications:
Mr. John Greco
Pre-Calculus:
Ms. Anu Garrison and Mr. David Knarr
Calculus:
Mr. David Knarr
Supervisors:
Supervisor of Mathematics and Science:
Mr. Vince Parisi
Curriculum Coordinator:
Mr. Paul C. Rafalowski
Supervisor of Technology:
Ms. Annemarie Stoeckel
4
South Plainfield Public Schools
District Mission Statement
To ensure that all pupils are equipped with essential skills necessary to acquire a common body
of knowledge and understanding;
To instill the desire to question and look for truth in order that pupils may become critical
thinkers, life-long learners, and contributing members of society in an environment of mutual
respect and consideration.
It is the expectation of this school district that all pupils achieve the New Jersey Core
Curriculum Content Standards at all grade levels.
Adopted September, 2008
NOTE: The following pacing guide was developed during the creation of these curriculum units. The actual
implementation of each unit may take more or less time. Time should also be dedicated to preparation for
benchmark and State assessments, and analysis of student results on the same. A separate document is included
at the end of this curriculum guide with suggestions and resources related to State Assessments (if
applicable). The material in this document should be integrated throughout the school year, and with an
awareness of the State Testing Schedule. It is highly recommended that teachers meet throughout the school
year to coordinate their efforts in implementing the curriculum and preparing students for benchmark and State
Assessments in consideration of both the School and District calendars.
5
Index of Mathematics Courses
Elementary Schools
(Franklin, Kennedy, Riley, Roosevelt)
Kindergarten
Grade 1
Grade 2
Grade 3
Grade 4
Grant School
Grade 5
Grade 6
Honors Grade 6
Middle School
Grade 7
Honors Grade 7
Grade 8
Honors Grade 8
Grade 8 Algebra 1
High School
Algebra 1
Academic Algebra 1
Honors Algebra 1
Geometry
Academic Geometry
Honors Geometry
Algebra 2
Academic Algebra 2
Honors Algebra 2
Algebra 3/Trigonometry
Senior Math Applications
Pre-Calculus
Honors Pre-Calculus
Calculus
Calculus AB
Calculus BC
6
South Plainfield Public Schools Curriculum Guide
Content Area: Mathematics
Course Title: Geometry
Grade Level: 9 and 10
Unit 0: Algebra Pre-Requisites for Geometry
2 Weeks
Unit 1: Constructions and Congruence
4 Weeks
Unit 2: Proving Geometric Theorems
4 Weeks
Unit 3: Reasoning and Congruent Triangle
Proofs
4 Weeks
Unit 4: Similarity and Proofs
3 Weeks
Unit 5: Special Right Triangles and
Trigonometry
3 Weeks
Unit 6: Coordinate Geometry and Proofs
4 Weeks
Board Approved on: June 13, 2012
7
South Plainfield Public Schools Curriculum Guide
Content Area: Mathematics
Course Title: Geometry
Grade Level: 9 and 10
Unit 7: Circles, lines, angles, arc
relationships
3 Weeks
Unit 8: Circles and Conic Sections on the
Coordinate Plane
3 Weeks
Unit 9: Geometry in Three Dimensions
3 Weeks
Unit 10: Applications of Geometric
Probability
3 Weeks
Board Approved on: June 13, 2012
8
Unit 0 Overview
Content Area – Mathematics
Unit Title – Algebra Pre-Requisites for Geometry
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale – In order to make sense of geometry, students must be proficient with certain
skills that involve equations. Specifically, they must be able to evaluate expressions, write and solve linear
and quadratic equations, solve a system of equations, be adept at identifying coordinates on the Cartesian
Plane, be able to graph linear and quadratic equations using a variety of forms and methods, and be able to
apply ratios and proportions accurately and precisely.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration –– Calculator and iPad, projector and/or SMART Board as available and/or
appropriate
21st Century Themes –
Civic Literacy
Global Awareness
21st Century Skills –
Critical Thinking & Problem Solving
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
9
Domain Standards:
A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include
equations arising from linear and quadratic functions, and simple rational and exponential functions.
A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales.
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example,
represent inequalities describing nutritional and cost constraints on combinations of different foods.
A.CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
A.REI.4: Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation
of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the
square, the quadratic formula and factoring, as appropriate to the initial form of the equation.
Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
A.REI.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum
of that equation and a multiple of the other produces a system with the same solutions.
A.REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs
of linear equations in two variables.
A.REI.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables
algebraically and graphically. For example, find the points of intersection between the line y = –3x and the
circle x2 + y2 = 3.
Unit Essential Questions
 Why is it necessary to use algebraic
representation and terminology?
Unit Enduring Understandings
 Algebra is a universal symbolic language
necessary for mathematical communication
Terminology: expression, equation, linear equation, quadratic equation, system of equations, solution set,
graphic solution, function, inequality, Cartesian Plane, ordered pair, coordinate, domain, range, irrational
number, ratio, proportion
Goals/Objectives
Students will be able to evaluate expressions
write and solve linear and
quadratic equations
solve a system of
equations
Learning Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Whole Group Lessons
Small Group Explorations
In-class group work
Independent Practice
Homework
Class Discussion
Formative:
White Board Response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
Homework Quizzes*
10
Notebook Quizzes*
be adept at identifying
coordinates on the
Cartesian Plane
be able to graph linear
equations using a variety
of forms and methods
Activities
Textbook Activities
Group Math Tutor – Strong Student Teaching
Cooperatively With Students of Varied Levels
of Understanding
Think Pair Share
Summative:
Multi-Lesson Quizzes
Unit Test
*based on unit or need
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
be able to write the
equation of a line
be able to graph quadratic
equations using a variety
of methods
be able to apply ratios and
proportions accurately
and precisely.
Diverse Learners (ELL, Special Ed, Gifted & Talented) - Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources - Teacher discretion due to prerequisite skills assessment.
11
Unit 1 Overview
Content Area – Mathematics
Unit 1: Constructions and Congruence
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale – Students will not only learn the definitions of particular geometric figures but
also develop a deeper understanding of them through constructions and measurements.
In this unit, students will develop definitions of the transformations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments. Transformations will then be used to define congruent
figures.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration –
Calculators will be used when deemed necessary.
Geometer’s Sketchpad if available
Ruler, Protractor, Compass
21st Century Themes –
Civic Literacy
Global Awareness
21st Century Skills –
Creativity & Innovation
Critical Thinking & Problem Solving
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
12
Domain Standards:
G.CO.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.
G.CO.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe
transformations as functions that take points in the plane as inputs and give other points as outputs.
Compare transformations that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
G.CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
G.CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
G.CO.5: 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.
G.CO.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a
given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.
G.CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are
congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.12: 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.
G.CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
G.C.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
Unit Essential Questions
 How do we recognize and apply
transformations of shapes to solve problems?
Unit Enduring Understandings
 All constructions are based on the properties of
geometric figures.

What are the basic geometric figures and how
are they related to one another?

Measurement quantifies attributes of objects
according to various units, systems, and processes.

How do geometric relationships and the
application of measurement help us to solve
real-world problems?

Analyzing geometric relationships develops
reasoning and justification skills.

What are the characteristics and applications of
symmetry?

How does what we measure affect how we
measure?
Terminology: Ruler, Compass, Protractor, Points, Lines, Segments, Planes, Rays, Angles, Angle Bisector,
Segment Bisector, Midpoint, Segment Addition Postulate, Angle Addition Postulate, Complementary,
Supplementary
13
Goals/Objectives
Students will be able to apply properties of
transformation using
coordinate geometry
use transformations to move
figures, create designs, and/or
demonstrate geometric
properties
construct and/or validate
properties of geometric
figures using appropriate tools
and technology
solve problems using
constructions
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Whole Group Lessons
Small Group Explorations
In-class group work
Independent Practice
Homework
Class Discussion
Formative:
White Board Response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
Homework Quizzes*
Notebook Quizzes*
Activities
Textbook Activities
Group Math Tutor – Strong Student
Teaching Cooperatively With Students of
Varied Levels of Understanding
Think Pair Share
Summative:
Multi-Lesson Quizzes
Unit Test
*based on unit or need
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
construct angle bisectors
construct segment bisectors
use transformations to move
figures, create designs, and
demonstrate geometric
properties
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
14
Unit 2 Overview
Content Area – Mathematics
Unit 2: Proving Geometric Theorems
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale – Here students study deductive reasoning and conditional statements in
preparation for the standards contained herein. Students will use deductive reasoning to construct arguments
and write formal proofs for Geometric Theorems about lines, angles, triangles, and parallelograms. These
arguments will contain Properties of Equality and Congruence (largely from Algebra 1), as well as
applications of the geometric figures studied, measured, and constructed in unit one.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration –
Calculators will be used when deemed necessary.
Geometer’s Sketchpad if available
Ruler, Protractor, Compass
21st Century Themes –
Global Awareness
Environmental Literacy
21st Century Skills –
Critical Thinking & Problem Solving
Information Literacy
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
15
Domain Standards:
G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition
of congruence in terms of rigid motions.
G.CO.9: 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.
G.CO.10: 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.
G.CO.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite
angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are
parallelograms with congruent diagonals.
Unit Essential Questions
 What is the relationship between lines and
angles?
Unit Enduring Understandings
 Logical arguments consist of a set of premises or
hypotheses and a conclusion.


Two intersecting lines form angles with specific
relationships.

Parallel lines cut by a transversal form angles with
specific relationships.

Analyzing geometric relationships develops
reasoning and justification skills.
How do geometric relationships and the
application of measurement help us to solve
real-world problems?
Terminology: Hypothesis, Conclusion, Law of Detachment, Law of Syllogism, Addition Subtraction
Multiplication Division Properties of Equality, Reflexive Symmetric Transitive Substitution Property of
Equality/Congruence, Conditional, Converse, Biconditional, Inverse, Contrapositive, Given, Conclusion,
Statement, Justification, Proof, Counterexample, Vertical Angles, Linear Pair, Complementary,
Supplementary
16
Goals/Objectives
Students will be able to -
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
recognize conditional
statements, the hypothesis and
conclusion
Instructional Strategies
Whole Group Lessons
Small Group Explorations
In-class group work
Independent Practice
Homework
Class Discussion
Formative:
White Board Response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
Homework Quizzes*
Notebook Quizzes*
write and interpret conditional
statements including the
converse, inverse, and
contrapositive
use deductive reasoning, the
laws of detachment and
syllogism, to draw
conclusions
Activities
Textbook Activities
Group Math Tutor – Strong Student
Teaching Cooperatively With Students of
Varied Levels of Understanding
Think Pair Share
using deductive reasoning,
draw logical conclusions from
given information and known
facts
Summative:
Multi-Lesson Quizzes
Unit Test
*based on unit or need
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
use counterexamples to
disprove conditional
statements
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
17
Unit 3 Overview
Content Area – Mathematics
Unit 3: Reasoning and Congruent Triangle Proofs
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale – The concept of congruent triangles is explored by examining which parts of a
triangle are necessary and sufficient to construct a unique triangle. Students prove congruence of triangles
through two-column and/or paragraph proofs. Corresponding parts of congruent triangles is used to prove
congruence of sides and/or angles.
Compass and straightedge constructions are reviewed and congruent triangles are applied to justify the
transformation.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration –
Calculators will be used when deemed necessary.
Geometer’s Sketchpad if available
Ruler, Protractor, Compass
21st Century Themes –
Global Awareness
Civic Literacy
21st Century Skills –
Creativity & Innovation
Critical Thinking & Problem Solving
Communication
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
18
Domain Standards:
G.CO.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a
given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.
G.CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are
congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition
of congruence in terms of rigid motions.
G.CO.10: 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.
Unit Essential Questions
 How do geometric relationships and the
application of measurements help us to solve
real-world problems?

Unit Enduring Understandings
 Proof is a justification that is logically valid based
on definitions, postulates, and theorems.

What is the difference between similarity and
congruence?
Analyzing geometric relationships develops
reasoning and justification skills.
Terminology: Congruent, Corresponding Parts, Congruence Statement, SSS, SAS, AAS, ASA, HL,
Vertical Angles, Reflexive Property, Alternate Interior Angles, Corresponding Angles
Goals/Objectives
Students will be able to Identify Corresponding Parts
of Triangles
Determine Whether or Not
Two Triangles are Congruent
Write a Congruence
Statement for Two Given
Triangles
Show that two triangles are
congruent using SSS, SAS,
AAS, ASA, HL.
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Whole Group Lessons
Small Group Explorations
In-class group work
Independent Practice
Homework
Class Discussion
Formative:
White Board Response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
Homework Quizzes*
Notebook Quizzes*
Activities
Textbook Activities
Group Math Tutor – Strong Student
Teaching Cooperatively With Students of
Varied Levels of Understanding
Think Pair Share
Summative:
Multi-Lesson Quizzes
Unit Test
*based on unit or need
Prove triangles congruent
given information in the form
19
of a figure or statement using
deductive proofs.
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
Recognize Vertical Angles
and Reflexive Property, apply
concepts to proofs.
Recognize Angles Formed By
Two Parallel Lines and a
Transversal, apply concepts to
proofs.
use transformations to move
figures, create designs, and
demonstrate geometric
properties
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
20
Unit 4 Overview
Content Area – Mathematics
Unit 4: Similarity and Proofs
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale –
Students investigate and apply dilations of polygons in the coordinate plane. Similarity of polygons and
triangles are explored. Triangle similarity postulates and theorems are formally proven. The proportionality
of corresponding sides of similar figures is applied. Similarity extends to side-splitting, proportional
medians, altitudes, angle bisectors, and segment theorems. The geometric mean is defined and related to
arithmetic mean. Ratios of areas of similar solids are also studied.
Academic and Honors – Students will study the relationships that exist when an altitude is drawn to the
hypotenuse of a right triangles.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration –
Calculators will be used when deemed necessary.
Geometer’s Sketchpad if available
Ruler, Protractor, Compass
21st Century Themes –
Environmental Literacy
Economic, Business Literacy
21st Century Skills –
Creativity & Innovation
Critical Thinking & Problem Solving
Information Literacy
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
21
Domain Standards:
G.SRT.1: Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a
line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.SRT.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide
if they are similar; explain using similarity transformations the meaning of similarity for triangles as the
equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to
be similar.
G.SRT.4: 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.
G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships
in geometric figures.
G.SRT.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios for acute angles.
Unit Essential Questions
 How is similarity of geometric figures applied
and verified?
Unit Enduring Understandings
 Similar geometric figures have proportional
attributes.

What is the difference between similarity and
congruence?


What is the relationship between
corresponding side lengths of similar figures?
The measures of geometric figures can be
calculated and analyzed using a variety of
strategies, tools, and technologies.

Analyzing geometric relationships develops
reasoning and justification skills.
Terminology: Side-Splitter, Angle Bisector, Median, Altitude, Centroid, Dilation, Scale Factor, Similarity
Ratio, Center of Dilation, Geometric Mean, Postulate, Theorem, Proportion
Goals/Objectives
Students will be able to identify and/or verify
congruent and similar figures
apply proportionality of
corresponding parts of similar
figures
apply the properties of similar
figures to area problems
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Whole Group Lessons
Small Group Explorations
In-class group work
Independent Practice
Homework
Class Discussion
Formative:
White Board Response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
Homework Quizzes*
Notebook Quizzes*
Activities
Textbook Activities
22
apply properties of similarity
and scale factors to unit
conversions (inches to feet vs.
sq. inches to sq. feet)
Group Math Tutor – Strong Student
Teaching Cooperatively With Students of
Varied Levels of Understanding
Think Pair Share
use algebraic and/or geometric
properties to measure
indirectly
Summative:
Multi-Lesson Quizzes
Unit Test
*based on unit or need
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
determine the positive
geometric mean between two
numbers
use transformations to move
figures, create designs, and
demonstrate geometric
properties
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
23
Unit 5 Overview
Content Area – Mathematics
Unit 5: Special Right Triangles and Trigonometry
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale –
Applications of the Pythagorean Theorem and its converse, Pythagorean triples and Pythagorean
inequalities are explored. The special right triangles of 30-60-90 and 45-45-90 are also studied, as well as
Sine, Cosine, and Tangent Ratios. Connections are made to similar triangles and proportionality of side
lengths. Operations with Square Roots and Simplifying Square Roots, as well as Solving Proportions,
(concepts from Algebra 1) are revisited in this unit.
Academic and Honors - Vectors
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration –
Calculators will be used when deemed necessary.
Geometer’s Sketchpad if available
Ruler, Protractor, Compass
21st Century Themes –
Global Awareness
Environmental Literacy
21st Century Skills –
Creativity & Innovation
Critical Thinking & Problem Solving
Flexibility & Adaptability
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
24
Domain Standards:
G.SRT.4: 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.
G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships
in geometric figures.
G.SRT.7: Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
Unit Essential Questions
 What are the trigonometric ratios and how are
they used to solve real world problems?

How are the other facts and properties of
triangles in general, related to trigonometry
and the Pythagorean Theorem?
Unit Enduring Understandings
 Right Triangles have special properties, making it
such that all possible ratios among the side lengths
are known quantities.

We know so many facts about triangles, all of
which must hold true as we solve problems and
consider the appropriateness of our answers.

Analyzing geometric relationships develops
reasoning and justification skills.
Terminology: SOHCAHTOA, Sine, Cosine, Tangent, Opposite Leg, Adjacent Leg, Hypotenuse,
Trigonometric Ratio, Inverse, Pythagorean Theorem, Pythagorean Triple, Right Triangle, Right Angle,
Acute Angle, Square Roots, Perfect Square, Rational, Irrational, Rationalize
Goals/Objectives
Students will be able to identify the types of right
triangles
state and apply the
Pythagorean theorem and its
converse to solve problems
identify and apply common
Pythagorean triples
apply the properties of special
right triangles
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Whole Group Lessons
Small Group Explorations
In-class group work
Independent Practice
Homework
Class Discussion
Activities
Textbook Activities
Group Math Tutor – Strong Student
Teaching Cooperatively With Students of
Varied Levels of Understanding
Think Pair Share
Formative:
White Board Response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
Homework Quizzes*
Notebook Quizzes*
define and apply
trigonometric ratios to real
Summative:
Multi-Lesson Quizzes
Unit Test
*based on unit or need
25
world situations
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
use trigonometric functions to
find the measures of angles of
a right triangle
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
26
Unit 6 Overview
Content Area – Mathematics
Unit 6: Coordinate Geometry and Proofs
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale –
Students will build on their work with the Pythagorean Theorem in 8th grade to find distances and use a
rectangular coordinate system to verify geometric relationships. Such relationships include properties of
special triangles and quadrilaterals and slopes of parallel and perpendicular lines. Students will continue
their study of quadratics by connecting the geometric and algebraic definitions of the parabola. (Academic
Geometry and Honors Geometry will also study conic sections in this unit)
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration – calculators, as necessary. Geometer sketch pad, if available.
Rulers, graph paper
21st Century Themes –
Civic Literacy
Global Awareness
21st Century Skills –
Creativity & Innovation
Critical Thinking & Problem Solving
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
27
Domain Standards:
G-GPE.2: Derive the equation of a parabola given a focus and directrix.
G.GPE.4: 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).
G.GPE.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a
given point).
G.GPE.6: Find the point on a directed line segment between two given points that partitions the segment in
a given ratio.
G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
using the distance formula.★
Unit Essential Questions
Unit Enduring Understandings



How can the coordinate plane help us better
understand the properties and theorems for
geometric shapes?
How can algebraic equations be used to solve
real problems in engineering and design?


The coordinate plane is a useful tool for
visualizing and interpreting shapes in our physical
environment.
Algebraic representations are used to
communicate and generalize patterns in geometry.
Perpendicular and parallel lines are the basis for
all building.
Terminology: base angles of a trapezoid, consecutive angles, isosceles trapezoid, kite, mid-segment of a
trapezoid, parallelogram, rectangles, rhombus, square, trapezoid
Goals/Objectives
Students will be able to Define, identify, and classify
special types of quadrilaterals
Use relationships among sides
and among angles of
parallelograms to prove the
specific type of quadrilateral
or to find missing measures in
a quadrilateral
Use relationships involving
diagonals of parallelograms or
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Whole group instruction
Small group exploration
Independent Practice
Class Discussion
Formative :
White Board response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
(Homework Quizzes
Notebook Quizzes – based on
unit or need)
Activities
Text book activities
Supplemental worksheets,
Individual and group activities for:
 Skill development
28
Summative:
transversals to prove the type
of quadrilateral or to find
missing measures.


Multi-Lesson Quizzes
Unit Test
Word Problems
Geometric representation of data
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
Identify figures in the
coordinate plane by
determining slope for parallel
and perpendicular lines.
Identify coordinates of special
figures by using their
properties.
Find a point that divides a
given segment on the
coordinate plane into a
specified ratio.
Prove theorems using figures
in the coordinate plane
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
29
Unit 7 Overview
Content Area – Mathematics
Unit 7: Circles: lines, angles, arc relationships
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale –
Students will prove basic theorems about circles, with particular attention to perpendicularity and inscribed
angles, in order to see symmetry in circles and as an application of triangle congruence criteria. Other angle
relationships will also be studied. Students will study relationships among segments on chords, secants, and
tangents as an application of similarity.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration – calculators, as necessary. Geometer sketch pad, if available.
Rulers, graph paper, compass
21st Century Themes –
Financial, Economic, Business and Entrepreneurial
Literacy
21st Century Skills –
Learning and Innovation
Life and Career Skills
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
30
Domain Standards:
G.C.1: Prove that all circles are similar.
G.C.2: 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.
G.C.5: 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.
Unit Essential Questions
Unit Enduring Understandings


Why is it important to understand the unique
relationships among the lines, segments,
angles, and arcs in a circle?

Analyzing geometric relationships develops
reasoning and justification skills.
Circles allow us to model a wide variety of natural
phenomena.
Terminology: arc, adjacent arc, center, central angle, chord, chord segment, circle circumference,
circumscribed, common tangent, compound locus, concentric circles, congruent arcs, congruent circles,
diameter, external segment, inscribed, inscribed angle, intercepted arc, major arc, minor arc, pi, point of
tangency, radius, secant, secant segment, semicircle, tangent
Goals/Objectives
Students will be able to Identify central angles, major
arcs, minor arcs, and
semicircles, and find their
measures
Find arc lengths
Recognize and use the
relationships between arcs,
chords, and diameters.
Find measures of inscribed
angles and inscribed polygons
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Whole group instruction
Small group exploration
Independent Practice
Class Discussion
Formative :
White Board response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
(Homework Quizzes
Notebook Quizzes – based on
unit or need)
Activities
Text book activities
Supplemental worksheets,
Individual and group activities for:
 Skill development
 Word Problems
 Geometric representation of data
Use properties of tangents to
31
Summative:
Multi-Lesson Quizzes
Unit Test
Research Project
solve problems involving
circumscribed polygons.
Possible Short Research Project:
Find measures of angles
formed by lines intersecting
on or inside a circle.
Research the use of radar by an air traffic
controller. Specifically relate parts of a
circle to the responsibilities of the job.
Find measures of angles
formed by lines intersecting
outside the circle.
Or
Find measures of segments
that intersect in the interior or
exterior of a circle.
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
Research the use of sonar in submarines.
Specifically relate parts of a circle to the
responsibilities of the job.
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
32
Unit 8 Overview
Content Area – Mathematics Basic Geometry
Unit 8: Circles on the Coordinate Plane
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale – In the Cartesian coordinate system, students use the distance formula to write
the equation of a circle when given the radius and the coordinates of its center. Given an equation of a
circle, they draw a graph in the coordinate plane and apply techniques for solving quadratic equations to
determine intersections between lines and circles or parabolas and circles, as well as the points of
intersection between two circles.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration – calculators, as necessary. Geometer sketch pad, if available.
Rulers, graph paper
21st Century Themes –
Financial, Economic, Business and Entrepreneurial
Literacy
21st Century Skills –
Life and Career Skills
Technology Skills
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
33
Domain Standards:
G.GPE.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem;
complete the square to find the center and radius of a circle given by an equation.
G.GPE.4: 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).
G.MG.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).
A-REI.7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables
algebraically and graphically. For example, find the points of intersection between the line y = –3x and the
circle x2 + y2 = 3.
Unit Essential Questions
Unit Enduring Understandings

What does the graph of an equation in two
variables look like?

Graphing equations provides a tangible way to
understand abstract concepts.

How can we determine the shape of a graph
from its equation?

It is possible to know the shape of a graph by
analyzing its equation.
Terminology: center, chord, equation of a circle, equation of a parabola in vertex form, external segment,
common tangent, compound locus, point(s) of intersection, point of tangency, radius, secant, tangent.
Goals/Objectives
Students will be able to use the Pythagorean Theorem
to derive the equation of a
circle of a given radius and
center
write the equation of a circle
in the coordinate plane from a
given center and radius
-identify the center and radius
of a circle given its equation
use the equation of a circle to
prove geometric theorems
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Instructional Strategies
Whole group instruction
Small group exploration
Independent Practice
Class Discussion
Formative :
White Board response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
(Homework Quizzes
Notebook Quizzes – based on
unit or need)
Activities
Text book activities
Supplemental worksheets,
Individual and group activities for:
 Skill development
 Word Problems
34
Summative:
Multi-Lesson Quizzes
such as the location of a given
point (interior, exterior, or on
the circle) or the type of
polygon inscribed in a circle

Unit Test
Geometric representation of data
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
solve a system of equations
by drawing a graph in the
coordinate plane and applying
techniques for solving
quadratic equations to
determine intersections
between lines and circles
or parabolas and circles, as
well as the points of
intersection between two
circles.
use the equation of a circle to
solve real world problems
involving such topics as
meteorology, aerodynamics,
and search and rescue
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
35
Unit 9 Overview
Content Area – Mathematics
Unit 9: Geometry in Three Dimensions
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale –
Students’ experience with two-dimensional and three-dimensional objects is extended to include
explanations of circumference and perimeter of the base of three dimensional objects in order to develop
surface area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes
to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration –
21st Century Themes –
Financial, Economic, Business and Entrepreneurial
Literacy
21st Century Skills –
Life and Career Skills
Technology Skills
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
36
Domain Standards:
G.GMD.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal
limit arguments.
G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
G.GMD.4: Identify the shapes of two-dimensional cross-sections of three dimensional objects, and identify
three-dimensional objects generated by rotations of two-dimensional objects.
G.MG.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).*
G.MG.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per
square mile, BTUs per cubic foot).★
G.MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to
satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★
Unit Essential Questions
Unit Enduring Understandings


Conservation of surface area

Conservation of volume

Can two different three-dimensional objects
have the same surface area?
Can two different three-dimensional objects
have the same volume?
Terminology: altitude, axis, base Cavalieri’s Principal, center, chord, circular cone, cone, congruent solids,
corner view, cross section, cube, cylinder, diameter, edge, face, great circle, height, hemisphere, lateral area,
lateral edge, lateral face, net, oblique cone, oblique cylinder, oblique prism, perspective view, Platonic
Solids, polyhedron, prism, pyramid, radius, regular polyhedron, regular prism, regular pyramid, right cone,
right cylinder, right prism, scale factor, similar solids, slant height, slice, solid, sphere, surface area, tangent,
triangular prism, vertex, volume
Goals/Objectives
Students will be able to -
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Use top, front, side, and
corner views of threedimensional solids to make
models and describe and draw
cross sections and other slices
of three dimensional figures to
give a better understanding of
art and architecture.
Instructional Strategies
Whole group instruction
Small group exploration
Independent Practice
Class Discussion
Formative :
White Board response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
(Homework Quizzes
Notebook Quizzes – based on
unit or need)
Draw three dimensional
figures on isometric dot paper,
Activities
Text book activities
Supplemental worksheets,
Individual and group activities for:
37
make two dimensional nets
for three-dimensional solids,
and find surface areas for
these solids.



Skill development
Word Problems
Geometric representation of data
Be able to distinguish between Possible Project:
a right prism or pyramid and
Students will create package for a fictitious
an oblique prism or pyramid.
product.
In preparation, have students examine
Understand Cavalieri’s
household packaging for cereal, drink mix,
Principle.
powdered laundry detergent, or any other
cardboard package. It is helpful to be able
Find the volume for any three to show students such unique packaging as
the Toblerone triangular prism, or a
dimensional solid.
trapezoidal prism (print cartridge package).
Find the surface area and
volume for a sphere.
Identify congruent or similar
solids by identifying their
various properties.

Assign or choose groups of no more
than four.

Assign or choose from a variety of
household products.

Depending on the level of the class,
groups will either
Draw a net, make the package, find
the surface area and volume
or
Draw a net and make the package.
-
-
The project can be expanded to include
creative marketing of the package. It can
be done as a competition.
38
Summative:
Multi-Lesson Quizzes
Unit Test
Project
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
39
Unit 10 Overview
Content Area – Mathematics
Unit 10: Applications of Geometric Probability
Target Course/Grade Level – Geometry for Grades 9 and 10
Unit Summary/Rationale – Probability concepts which were introduced in the middle grades are expanded
to include the language of set theory, and the ability to compute and interpret theoretical and experimental
probabilities for compound events. Students will also expand prior knowledge to incorporate mutually
exclusive events, independent events, and conditional probability. Students should make use of geometric
probability models wherever possible. They use probability to make informed decisions.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology Integration – Calculator and iPad, projector and/or SMART Board as available and/or
appropriate
21st Century Themes –
Global Awareness
Environmental Literacy
Financial, Economic, Business and Entrepreneurial
Literacy
21st Century Skills –
Critical Thinking & Problem Solving
Communication
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning
40
Domain Standards:
S.CP.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,”
“not”).
S.CP.2: Understand that two events A and B are independent if the probability of A and B occurring
together is the product of their probabilities, and use this characterization to determine if they are
independent.
S.CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the conditional probability of A given B is the same as the
probability of A, and the conditional probability of B given A is the same as the probability of B.
S.CP.4: Construct and interpret two-way frequency tables of data when two categories are associated with
each object being classified. Use the two-way table as a sample space to decide if events are independent
and to approximate conditional probabilities. For example, collect data from a random sample of students in
your school on their favorite subject among math, science, and English. Estimate the probability that a
randomly selected student from your school will favor science given that the student is in tenth grade. Do
the same for other subjects and compare the results.
S.CP.5: Recognize and explain the concepts of conditional probability and independence in everyday
language and everyday situations. For example, compare the chance of having lung cancer if you are a
smoker with the chance of being a smoker if you have lung cancer.
S.CP.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A,
and interpret the answer in terms of the model.
S.CP.7: Apply the Addition Rule, P (A or B) = P (A) + P (B) – P (A and B), and interpret the answer in
terms of the model.
Unit Essential Questions
Unit Enduring Understandings

How can probability be modeled by plane
figures?


How can number lines be used to model time
or distance?
Some probability situations can be modeled using
one, two or three dimensional figures.
Terminology: compound probability, conditional probability, geometric probability, mutually exclusive
events, theoretical probability, experimental probability
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Goals/Objectives
Students will be able to -
Recommended Learning
Activities/Instructional Strategies
Evidence of Learning
(Formative & Summative)
Use segments and area models
to find probabilities of events
Instructional Strategies
Whole group instruction
Small group exploration
Independent Practice
Class Discussion
Formative :
White Board response
Black Board Race
Ticket Out
Think/Pair/Share
Open Discussion
(Homework Quizzes
Notebook Quizzes – based on
unit or need)
Understand independence and
conditional probability and
use them to interpret data.
Use rules of probability to
compute probabilities of
compound events in a uniform
probability model.
Activities
Text book activities
Supplemental worksheets,
Individual and group activities for:
 Skill development
 Word Problems
 Geometric representation of data

Possible Project:
Students will create a carnival game based
on geometric probability. Students will
determine various probabilities so that the
carnival has an advantage.
Summative:
Multi-Lesson Quizzes
Unit Test
Alternative or project-based
assessments will be
evaluated using a teacherselected or created rubric or
other instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented) – Differentiation strategies may include, but are
not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous
groups, depending on the learning objectives and the number of students that need further support and
scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as
they relate to the special needs of students in accordance with their Individualized Education Programs
(IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to,
extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or
suggestions from special education or ELL teachers.
Resources Textbooks: McDougal Littell Geometry: Concepts and Skills, 2003 - Geometry
Prentice Hall Mathematics Geometry, 2004 – Academic Geometry
Glencoe McGraw-Hill Geometry, 2010 – Honors Geometry
Websites:
42
Mathematics: Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all
levels should seek to develop in their students. These practices rest on important “processes and
proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM
process standards of problem solving, reasoning and proof, communication, representation, and connections.
The second are the strands of mathematical proficiency specified in the National Research Council’s report
Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of
mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics
as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking
for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make
conjectures about the form and meaning of the solution and plan a solution pathway rather than simply
jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of
the original problem in order to gain insight into its solution. They monitor and evaluate their progress and
change course if necessary. Older students might, depending on the context of the problem, transform
algebraic expressions or change the viewing window on their graphing calculator to get the information they
need. Mathematically proficient students can explain correspondences between equations, verbal
descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and
search for regularity or trends. Younger students might rely on using concrete objects or pictures to help
conceptualize and solve a problem. Mathematically proficient students check their answers to problems using
a different method, and they continually ask themselves, “Does this make sense?” They can understand the
approaches of others to solving complex problems and identify correspondences between different
approaches.
2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of the quantities and their relationships in problem situations.
Students bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—and
the ability to contextualize, to pause as needed during the manipulation process in order to probe into the
referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation
of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to
compute them; and knowing and flexibly using different properties of operations and objects.
3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
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statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into
cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to
others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient students
are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning
from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can
construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such
arguments can make sense and be correct, even though they are not generalized or made formal until later
grades. Later, students learn to determine domains to which an argument applies. Students at all grades can
listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or
improve the arguments.
4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to solve a
design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions and
approximations to simplify a complicated situation, realizing that these may need revision later. They are able
to identify important quantities in a practical situation and map their relationships using such tools as
diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the
situation and reflect on whether the results make sense, possibly improving the model if it has not served its
purpose.
5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These
tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry software. Proficient students are
sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each
of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example,
mathematically proficient high school students analyze graphs of functions and solutions generated using a
graphing calculator. They detect possible errors by strategically using estimation and other mathematical
knowledge. When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data. Mathematically
proficient students at various grade levels are able to identify relevant external mathematical resources, such
as digital content located on a website, and use them to pose or solve problems. They are able to use
technological tools to explore and deepen their understanding of concepts.
6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in
discussion with others and in their own reasoning. They state the meaning of the symbols they choose,
44
including using the equal sign consistently and appropriately. They are careful about specifying units of
measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate
accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem
context. In the elementary grades, students give carefully formulated explanations to each other. By the time
they reach high school they have learned to examine claims and make explicit use of definitions.
7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example,
might notice that three and seven more is the same amount as seven and three more, or they may sort a
collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the
well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2
+ 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an
existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.
They also can step back for an overview and shift perspective. They can see complicated things, such as some
algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 –
3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more
than 5 for any real numbers x and y.
8 Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and
for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the
same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the
calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle
school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel
when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general
formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient
students maintain oversight of the process, while attending to the details. They continually evaluate the
reasonableness of their intermediate results.
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Common Core Standards
(Bold Apply to Geometry)
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. 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).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto
itself.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel
lines, and line segments.
5. 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.
Understand congruence in terms of rigid motions.
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a
given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent.
8. 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.
9. 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.
10. 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.
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the
diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Make geometric constructions.
12. 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.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Similarity, Right Triangles, and Trigonometry G-SRT
Understand similarity in terms of similarity transformations.
1. Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing
through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar;
explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs
of angles and the proportionality of all corresponding pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity.
4. 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.
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles.
46
6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to
definitions of trigonometric ratios for acute angles.
7. Explain and use the relationship between the sine and cosine of complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★
Apply trigonometry to general triangles.
9. (+) Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces).
Modeling with Geometry G-MG
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).★
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per
cubic foot).★
3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints
or minimize cost; working with typographic grid systems based on ratios). ★
Geometric Measurement and Dimension G-GMD
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder,
pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★
Visualize relationships between two-dimensional and three-dimensional objects.
4.
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional
objects generated by rotations of two-dimensional objects.
Expressing Geometric Properties with Equations G-GPE
Translate between the geometric description and the equation for a conic section.
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find
the center and radius of a circle given by an equation.
2. Derive the equation of a parabola given a focus and directrix.
3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the
foci is constant.
Use coordinates to prove simple geometric theorems algebraically.
4. 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).
5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the
equation of a line parallel or perpendicular to a given line that passes through a given point).
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
★
Circles G-C
Understand and apply theorems about circles.
1. Prove that all circles are similar.
2. 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.
47
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
4. (+) Construct a tangent line from a point outside a given circle to the circle.
Find arc lengths and areas of sectors of circles.
5. 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.
Conditional Probability and the Rules of Probability S-CP
Understand independence and conditional probability and use them to interpret data.
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”).
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.
3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as
saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of
B given A is the same as the probability of B.
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.
5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday
situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if
you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the
answer in terms of the model.
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.
8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and
interpret the answer in terms of the model
9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Using Probability to Make Decisions S-MD
Calculate expected values and use them to solve problems.
1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the
corresponding probability distribution using the same graphical displays as for data distributions.
2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be
calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers
obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected
grade under various grading schemes.
4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned
empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the
United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100
randomly selected households?
Use probability to evaluate outcomes of decisions.
5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a
game at a fast-food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a lowdeductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey
goalie at the end of a game).
Resources for State Assessments
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