Download Mapping for Instruction - First Nine Weeks

Roanoke County Public Schools
Geometry Readiness
Curriculum Guide
Revised 2012
Geometry Readiness Curriculum Guide
2011
Mathematics Curriculum Guide
Revised 2011. Available at www.rcs.k12.va.us.
Roanoke County Public Schools does not discriminate with regard to race, color, age, national origin, gender, or handicapping condition in an
educational and/or employment policy or practice. Questions and/or complaints should be addressed to the Deputy Superintendent/Title IX
Coordinator at (540) 562-3900 ext. 10121 or the Director of Pupil Personnel Services/504 Coordinator at (540) 562-3900 ext. 10181.
Acknowledgements
The following people have made tremendous contributions to the completion of this curriculum guide and all are appreciated.
Theresa Hartley
Cave Spring High
Kevin Minnix
William Byrd High
Sherri Mays
William Byrd High
Barbara Smith
Cave Spring High/Cave Spring Middle
Christina Hall
Northside Middle
Brooke Haun
Cave Spring High
Roanoke County Public Schools Administration
Dr. Lorraine Lange
Superintendent
Cecil Snead
Director of Secondary Instruction
Rebecca Eastwood
Director of Elementary Instruction
Linda Bowden
Mathematics Coordinator
Preface
This curriculum guide is written for the teachers to assist them in using the textbooks/resources in a most effective way. This guide will assist the mathematics
teacher in preparing students for the challenges of the twenty-first century. As established by the National Council of Teachers of Mathematics Principles and
Standards for School Mathematics, educational goals for students are changing. Students should have many and varied experiences in their mathematical
training to help them learn to value mathematics, become confident in their ability to do mathematics, become problem solvers , and learn to communicate and
reason mathematically. This guide, along with the available textbook resources, other professional literature, alternative assessment methods, and varied
instruction in-service activities will assist the mathematics teacher in continuing to integrate these student goals into the curriculum.
Geometry Readiness Curriculum Guide
2011
Table of Contents
Introduction/General Comments ............................................................................................................................................. i
Textbook/Resources Overview ............................................................................................................................................... ii
Sequence of Instruction and Pacing Suggestions .............................................................................Error! Bookmark not defined.
Mapping for Instruction - First Nine Weeks ............................................................................................................................ 3
Mapping for Instruction - Second Nine Weeks ...................................................................................................................... 11
Mapping for Instruction - Third Nine Weeks ......................................................................................................................... 17
Mapping for Instruction - Fourth Nine Weeks ....................................................................................................................... 23
SOL Blueprints ....................................................................................................................................................................... 31
SOL Enhanced Scope and Sequence ...................................................................................................................................... 32
Supplemental Resources ....................................................................................................................................................... 32
SOL 2009 Framework ............................................................................................................................................................ 33
Geometry Readiness Curriculum Guide
2011
Introduction/General Comments
This curriculum guide follows the 2009 Virginia Geometry SOLs and uses the 2012 edition of Prentice Hall – Geometry Virginia Edition textbook as the primary
resource. It is extremely important and required that the Sequence of Instruction and Pacing be followed as presented in the curriculum guide. This will allow
the formative assessment tests to be an effective instructional tool.
Students will take a formative assessment test during the second, third, and fourth nine weeks which will contain materials from the previous nine weeks. Each
teacher-designed test will assess current 9-weeks skill levels of the SOLs in the Sequence of Instruction and Pacing. This tool will help teachers to pinpoint their
students' strengths and weaknesses so they can make more informed decisions for instruction.
The Mapping for Instruction has been set up according to each SOL that is taught in the specific 9-weeks. Note that only part of some SOLs may be taught in a
certain 9-weeks.
Each student should have a graphing calculator for use throughout this course. The instructor should use the calculator for investigation and verification of
results as well as a powerful tool in solving the real-life application problems that are provided in each section of the textbook.
Cooperative learning is very beneficial to students in learning many of the topics in geometry, particularly with proofs and application problem solving.
The Geometer's Sketchpad software, available in every school and on high school students' laptops, should be used to enhance instruction throughout the
course.
Please note the following, keeping in mind this curriculum guide is meant to encompass all the different versions of Geometry Roanoke County offers in their
middle and high schools:
1. Within the "Mapping for Instruction" the suggested pacing outlined in the "Comments" section was designed for high school geometry classes that meet for
95 minutes every other day. It is suggested that the allotted time be doubled for those teaching middle school Algebra courses.
2. Within the "Mapping for Instruction", under "Supporting Materials" section, there are suggestions for remediation and/or enrichment to help teachers meet
the needs of their students.
i
Geometry Readiness Curriculum Guide
2011
Textbook/Resources Overview
Course Title: Geometry, Virginia Edition
Course Text: Prenctice Hall
Publisher: Pearson
The 2012 edition of Prentice Hall Geometry is divided up into 12 chapters.
Each chapter of the teacher’s guide begins with Get Ready, a review of previously learned skills that are relevant for the chapter; Chapter Overview that
introduces the Vocabulary, Big Ideas, My Math Video, and Chapter Preview; Math Background that summarizes main concepts and common errors; and Pacing
and Assignment Guide that suggests how to differentiate pacing for Basic, Average, and Advanced students. A Resources page lists the supplemental materials
available for each section of the chapter.
Each section within a chapter is divided into 1) Interactive Learning; 2) Guided Instruction; 3) Lesson Check; 4) Practice, with suggested assignments
differentiated for different levels; and 5) Assess & Remediate, where differentiated resources for Intervention, On-Level, and Extension activities are pictured.
The end of each chapter contains Pull It All Together, where the “Big Ideas” of the chapter are recapped and connected to associated real-world applications;
Chapter Review, which links the “Big Ideas” to answering essential questions; Chapter Vocabulary; Quick Review; and Chapter Test.
Teachers are provided with the following resources available online (at PowerGeometry.com), in digital format (most are editable) and/or print format.
Interactive Learning & Guided Instruction
Lesson Check & Practice
Assess & Remediate
My Math Video
Student Companion
ExamView CD-ROM
Solve It!
Practice and Problem-Solving Workbook
Lesson Quiz
Student Companion
Practice (Forms G and K)
Quizzes and Tests (Forms G and K)
Dynamic Activity
Extra Practice
Reteaching
Online problems
Find the Errors
Performance Tasks
Additional problems
Enrichment
Cumulative Review
ELL support
Answers and Solutions CD-ROM
Progress Monitoring Assessments
Activities, Games and Puzzles
Standardized Test Prep
Teaching with TI Technology
TI-Nspire Support CD-ROM
ii
Geometry Readiness Curriculum Guide
2011
Sequence of Instruction and Pacing Suggestions
First Nine Weeks
SOL
G.4a,b,e,f
G.3a
G.1a, b, d
G.1 d, G.2 a,b,c
G.3 a, b, G.4 c, d, g
Chapter/Sections/Topic
*Time Frame
Chapter 1 Tools of Geometry; Lessons 2-8
VA-3
6.00 blocks
Chapter 2 Reasoning and Proof; Lessons 1 - 6
VA-1
5.50 blocks
Chapter 3 Parallel and Perpendicular Lines; Lessons 1 - 8
5.50 blocks
Reviews, quizzes, and tests
5.50 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45
days in the middle schools.
First Nine Weeks Total
22.5 blocks
Second Nine Weeks
SOL
Chapter/Sections/Topic
*Time Frame
G.4 a, G.6
Chapter 4 Congruent Triangles; Lessons 1 - 7
7.00 blocks
G.1 d, G.5 a, b, c, d
Chapter 5 Relationships Within Triangles; Lessons1,2,(3 optional),4, 6
5.50 blocks
G.8
Chapter 8 Right Triangles and Trigonometry; Lessons 1 - 4
4.50 blocks
Reviews, quizzes, and tests
5.50 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45
days in the middle schools.
1
Second Nine Weeks Total
22.5 blocks
Geometry Readiness Curriculum Guide
2011
Sequence of Instruction and Pacing Suggestions
Third Nine Weeks
SOL
Chapter/Sections/Topic
G.7, G.14 d
Chapter 7 Similarity; Lessons 1 - 5
*Time Frame
4.50 blocks
G.2 b, G.9, G.10
Chapter 6 Polygons and Quadrilaterals; Lessons 1 - 8
G.11 b, c, G.14 a, c, d
Chapter 10 Area; Lessons 1, 2, 4 - 7
Pre-AP include Lesson 10-3
8.00 blocks
4.50 blocks
Reviews, quizzes, and tests
Include a review on Constructions (Lessons 1-6 and 3-6)
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for
45 days in the middle schools.
Third Nine Weeks Total
5.5 blocks
22.5 blocks
Fourth Nine Weeks
SOL
Chapter/Sections/Topic
*Time Frame
G.4, G.11 a, b, G.12
Chapter 12 Circles; Lessons 1 - 5
5.50 blocks
Chapter 1 Nets and Drawings for Visualizing Geometry; Lesson 1(Optional)
0.00 blocks
G.13, G.14 a, b, c, d
Chapter 11 Surface Area and Volume; Lessons 2 - 7(be sure to include the defintions: face,
edge, and vertex)
3.50 blocks
G.3 c, d, G.10
Chapter 9 Transformations; Lessons 1 - 7
2.00 blocks
1.50 blocks
Reviews, quizzes, and tests
4.50 blocks
Sol Review
5.50 blocks
Exam Review
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for
45 days in the middle schools.
2
Fourth Nine Weeks Total
22.5 blocks
Geometry Readiness Curriculum Guide
2011
Mapping for Instruction - First Nine Weeks
Tools of Geometry
Chapter: 1
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
1-2 Points, Lines and
Planes
Supporting Materials
Comments
“Geometry Plane & Simple” pages 1, 3, 5, and 6
“Milliken Geometry Reproducibles” page 3
0 .5 block
1-3 Measuring Segments
“Geometry Plane & Simple” pages 2 and 4
“if Geometry” pages 7 and 8
0.5 block
1-4 Measuring Angles
1 block
www.mathisfun.com/geometry/protractor-using.html
“Geometry Plane & Simple” pages 9 and 10
“Geometry Teacher’s Activities Kit” pages 29 – 31, 36 - 46
G.2. The student will use the relationships
1-5 Exploring Angle Pairs
between angles formed by two lines cut by a
transversal to
1 block
c) solve real-world problems involving
angles formed when parallel lines are cut by
a transversal.
• Solve real-world problems involving
intersecting and parallel lines in a plane
www.worksheetworks.com/math/geometry/measuringangles.html
http://education.ti.com/calculators/timath/US/Activities/?sa=
5024#
“Geometry Plane & Simple” pages 12, 13, 14,18, 19, and
20
“Milliken Geometry Reproducibles” pages 4 – 6
“if Geometry” pages 9, 11 – 13
“Geometry Teacher’s Activities Kit” pages 32 – 35, and 61
-64
G.4 The student will construct and justify the 1-6 Basic Constructions
constructions of
a) a line segment congruent to a given line 1 block
segment;
b) the perpendicular bisector of a line
segment
e) the bisector of a given angle
f) an angle congruent to a given angle
Justifications for constructions VA-3
G.3 The student will use pictorial
representations, including computer
software, constructions, and coordinate
1.7 Midpoint and Distance
in the Coordinate Plane
http://mathisfun.com/geometry/constructions.html
http://whistleralley.com/construction/reference.html
http://www.mathopenref.com/constructions.com
http://www.youtube.com/watch?v=s-atfsonr8w
“if Geometry” pages 45, 46, 49, 50
DOE ESS Sample Lesson Plan G.4 Constructions
“Milliken Geometry Reproducibles” pages 12 and 19
“if Geometry” page 37
DOE ESS Sample Lesson Plan G.3 Distance and Midpoint
3
Review how to use a
protractor
VA-3 is located in the
textbook between
chapters 10 and 11
Concept will appear on
FA#3
Geometry Readiness Curriculum Guide
2011
Tools of Geometry
Chapter: 1
SOL with Essential Knowledge and Skill
methods, to solve problems involving
symmetry and transformation. This will
include
a) investigating and using formulas for
finding distance, midpoint, and slope;
Textbook
Chapters/Sections/Topics
Supporting Materials
1 block
Formulas
1.8 Perimeter,
Circumference
and Area
“Milliken Geometry Reproducibles” pages 27, 28, 29, 32,
and 33
“if Geometry” page 72
1 block
4
Comments
Geometry Readiness Curriculum Guide
2011
Reasoning and Proof
Chapter: 2
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
G.1. The student will construct and judge the validity of a logical 2-1 Patterns and Inductive
argument consisting of a set of premises and a conclusion. This Reasoning
will include
• Select and use various types of reasoning and methods of
0.5 block
proof, as appropriate.
DOE ESS Sample Lesson Plan
G.1 Inductive and Deductive
Reasoning
G.1 The student will construct and judge the validity of a logical 2-2 Conditional Statements
argument consisting of a set of premises and a conclusion. This VA-1: Review Venn Diagrams
will include
a) identifying the converse, inverse, and contrapositive of a
1 block
conditional statement;
b) translating a short verbal argument into symbolic form;
c) using Venn diagrams to represent set relationships; and
• Identify the converse, inverse, and contrapositive of a
conditional statement.
“Geometry Plane & Simple” pages
37 and 38
“if Geometry” pages 91 and 92
“Geometry Teacher’s Activities Kit”
pages 65 - 67
Translate verbal arguments into symbolic form,
such as (p  q) and (~p  ~q).
• Determine the validity of a logical argument.
• Use valid forms of deductive reasoning, including the law of
syllogism, the law of the contrapositive, the law of detachment,
and counterexamples.
• Select and use various types of reasoning and methods of
proof, as appropriate.
• Use Venn diagrams to represent set relationships, such as
intersection and union.
• Interpret Venn diagrams.
• Recognize and use the symbols of formal logic, which include
→, ↔, ~,  ,  , and  . .
G.1 The student will construct and judge the validity of a logical 2-3 Biconditionals and Definitions
argument consisting of a set of premises and a conclusion. This
will include
1 block
a) identifying the converse, inverse, and contrapositive of a
conditional statement;
5
DOE ESS Sample Lesson Plan
G.1 Logic and Conditional
Statements
Comments
Need to
supplement logic
notation including
therefore 
Geometry Readiness Curriculum Guide
2011
Reasoning and Proof
Chapter: 2
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
b) translating a short verbal argument into symbolic form;
c) using Venn diagrams to represent set relationships; and
• Identify the converse, inverse, and contrapositive of a
conditional statement.
Translate verbal arguments into symbolic form, such
as (p  q) and (~p  ~q).
• Determine the validity of a logical argument.
• Use valid forms of deductive reasoning, including the law of
syllogism, the law of the contrapositive, the law of detachment,
and counterexamples.
• Select and use various types of reasoning and methods of
proof, as appropriate.
• Use Venn diagrams to represent set relationships, such as
intersection and union.
• Interpret Venn diagrams.
• Recognize and use the symbols of formal logic, which include
→, ↔, ~,  ,  , and  . .
G.1 The student will construct and judge the validity of a logical 2-4 Deductive Reasoning
argument consisting of a set of premises and a conclusion. This
will include
0 .5 block
a) identifying the converse, inverse, and contrapositive of a
conditional statement;
b) translating a short verbal argument into symbolic form;
c) using Venn diagrams to represent set relationships; and
• Identify the converse, inverse, and contrapositive of a
conditional statement.
Translate verbal arguments into symbolic form, such
as (p  q) and (~p  ~q).
• Determine the validity of a logical argument.
• Use valid forms of deductive reasoning, including the law of
syllogism, the law of the contrapositive, the law of detachment,
and counterexamples.
• Select and use various types of reasoning and methods of
proof, as appropriate.
• Use Venn diagrams to represent set relationships, such as
6
“Geometry Plane & Simple” pages
39 and 40
DOE ESS Sample Lesson Plan
G.1 Inductive and Deductive
Reasoning
Comments
Geometry Readiness Curriculum Guide
2011
Reasoning and Proof
Chapter: 2
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
intersection and union.
• Interpret Venn diagrams.
• Recognize and use the symbols of formal logic, which include
→, ↔, ~,  ,  , and  . .
G.1 The student will construct and judge the validity of a logical 2-5 Reasoning in Algebra and
argument consisting of a set of premises and a conclusion. This Geometry
will include
d) using deductive reasoning
1.5 block
• Use valid forms of deductive reasoning, including the law of
syllogism, the law of the contrapositive, the law of detachment,
and counterexamples.
• Select and use various types of reasoning and methods of
proof, as appropriate
G.1 The student will construct and judge the validity of a logical 2-6 Proving Angles Congruent
argument consisting of a set of premises and a conclusion. This
will include
1 block
d) using deductive reasoning
• Use valid forms of deductive reasoning, including the law of
syllogism, the law of the contrapositive, the law of detachment,
and counterexamples.
.
7
“Geometry Plane & Simple” pages
15 - 17
Comments
Geometry Readiness Curriculum Guide
2011
Parallel and Perpendicular Lines
Chapter: 3
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
3-1 Lines and Angles
G.2 The student will use the relationship
between angles formed by two lines cut by a
1 block
transversal to
a) determine whether two lines are parallel;
b) verify the parallelism, using algebraic and
coordinate methods as well as deductive proofs;
and
c) solve real-world problems involving angles
formed when parallel lines are cut by a
transversal.
http://www.curriki.org/xwiki/bin/view/MyCurriki/Collecti
ons?user=IsaacNewton
“Milliken Geometry Reproducibles” page 6
3-2 Properties of Parallel Lines
G.2 The student will use the relationship
between angles formed by two lines cut by a
1 block
transversal to
a) determine whether two
lines are parallel;
b) verify the parallelism, using algebraic and
coordinate methods as well as deductive proofs;
and
c) solve real-world problems involving angles
formed when parallel lines are cut by a
transversal.
 Solve problems by using the
relationships between pairs of angles
formed by the intersection of two parallel
lines and a transversal including
corresponding angles, alternate interior
angles, alternate exterior angles, and
same-side (consecutive) interior angles.
“Geometry Plane & Simple” page 60
“Geometry Teacher’s Activities Kit” pages 47 - 52 and
61 – 64
G.1 The student will construct and judge the
validity of a logical argument consisting of a set
of premises and a conclusion. This will include
d) using deductive reasoning.
G.2 The student will use the relationships
3-3 Proving Lines Parallel
“Geometry Plane & Simple” page 59
1 block
DOE ESS Sample Lesson Plan
G.2 Lines and Angles
8
DOE ESS Sample Lesson Plan
G.2 Lines and Angles
Comments
Geometry Readiness Curriculum Guide
2011
Parallel and Perpendicular Lines
Chapter: 3
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
between angles formed by two lines cut by a
transversal to
a) determine whether two lines are parallel;
b) verify the parallelism, using algebraic and
coordinate methods as well as deductive proofs;
and
c) solve real-world problems involving angles
formed when parallel lines are cut by a
transversa
l• Use algebraic and coordinate methods as well
as deductive proofs to verify whether two lines
are parallel.
“Geometry Teacher’s Activities Kit” pages 53 -57
G.2 The student will use the relationships
between angles formed by two lines cut by a
transversal to
c) solve real-world problems involving angles
formed when parallel lines are cut by a
transversal.
3-4 Parallel and Perpendicular
Lines
G.2 The student will use the relationships
between angles formed by two lines cut by a
transversal to
c) solve real-world problems involving angles
formed when parallel lines are cut by a
transversalThe student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
• Solve problems by using the relationhips
between pairs of angles formed bye the
intersection of two parallel lines and a
transversal including correspondings anles,
alternate intreior angles, alternate exterior
angles, and same-side interior angles
• Solve real-world problems involving
intersecting and parallel lines in a plane.
3-5 Parallel Lines and Triangles “Milliken Geometry Reproducibles” page 8
“Geometry Teacher’s Activities Kit” pages 91 - 95
0.5 block
G.4 The student will construct and justify the
constructions of
c) a perpendicular to a given line from a point
not on the line;
3-6 Constructing Parallel and
Perpendicular Lines
0.5 block
http://www.mathopenref.com/constructions.html
“if Geometry” pages 46, 47, 48,51
VA-3: Justifying a
9
VA-3 is located in the
textbook between
chapters 10 and 11.
Geometry Readiness Curriculum Guide
2011
Parallel and Perpendicular Lines
Chapter: 3
SOL with Essential Knowledge and Skill
d) a perpendicular to a given line at a given
point on the line;
f) an angle congruent to a given angle; and
g) a line parallel to a given line through a point
not on the given line.
G.3 The student will use pictorial
representations, including computer software,
constructions, and coordinate methods, to solve
problems involving symmetry and
transformation. This will include
a) investigating and using formulas for finding
distance, midpoint, and slope;
G.2 The student will use the relationships
between angles formed by two lines cut by a
transversal to
a) determine whether two lines are parallel;
b) verify the parallelism, using algebraic and
coordinate methods as well as deductive proofs;
and
• Use algebraic and coordinate methods as well
as deductive proofs to verify whether two lines
are parallel..
• Solve problems by using the relationships
between pairs of angles formed by the
intersection of two parallel lines and a
transversal including corresponding angles,
alternate interior angles, alternate exterior
angles, and same-side (consecutive) interior
angles.
• Solve real-world problems involving
intersecting and parallel lines in a plane.
Textbook
Chapters/Sections/Topics
Supporting Materials
Construction)
0.5 block
“if Geometry” pages 32 - 36
3.7 Equations of Lines in the
Coordinate Plane
0.5 block
3-8 Slopes of Parallel and
Perpendicular Lines
DOE ESS Sample Lesson Plan
G.3 Slope
0.5 block
10
Comments
Geometry Readiness Curriculum Guide
2011
Mapping for Instruction - Second Nine Weeks
Congruent Triangles
Chapter: 4
SOL with Essential Knowledge and Skill
G.6 The student, given information in the form of a
figure or statement, will prove two triangles are
congruent, using algebraic and coordinate methods
as well as deductive proofs.
Textbook
Chapters/Sections/Topics
4-1 Congruent Figures
1 block
Supporting Materials
Comments
http://www.mathsisfun.com/geometry/trianglescongruent.html
http://www.cliffsnotes.com/study_guide/CongruentTriangles.topicArticleId-18851,articleId-18788.html
http://www.mathopenref.com/congruenttriangles.html
All of these websites
review the triangle
congruences and
offer some activities.
“Geometry Plane & Simple” page 21
G.6 The student, given information in the form of a
figure or statement, will prove two triangles are
congruent, using algebraic and coordinate methods
as well as deductive proofs.
• Use definitions, postulates, and theorems to prove
triangles congruent.
• Use coordinate methods, such as the distance
formula and the slope formula, to prove two triangles
are congruent.
G.4a The student will construct and justify the
construction of a line segment congruent to a given
line segment.
4-2 Triangle Congruence
by SSS and SAS
VA - 4 Congruent and
Similar Triangles
G.6 The student, given information in the form of a
figure or statement, will prove two triangles are
congruent, using algebraic and coordinate methods
as well as deductive proofs.
• Use definitions, postulates, and theorems to prove
triangles congruent.
• Use coordinate methods, such as the distance
formula and the slope formula, to prove two triangles
are congruent.
4-3 Triangle Congruence
by ASA and AAS
G.6 The student, given information in the form of a
figure or statement, will prove two triangles are
congruent, using algebraic and coordinate methods
as well as deductive proofs.
• Use definitions, postulates, and theorems to prove
triangles congruent.
4-4 Using Corresponding
Parts of Congruent
Triangles
1 block
http://www.youtube.com/watch?v=TPL12Tk7L6U
This youtube video
www.uen.org/.../downloadFile.cgi?...Congruent_Trian describes an activity
gles_Activity.
you can use with
your students to
Section VA - 4 is found in the Virginia Section of the prove triangle
textbook between Chapters 10 and 11.
congruences.
“Geometry Plane & Simple” page 41, 42,44
“if Geometry” page 25
1 block
1 block
11
http://illuminations.nctm.org/ActivityDetail.aspx?id=4
“Geometry Plane & Simple” page 43
“Milliken Geometry” page 14, 15, 16
“if Geometry” pages 26, 27, 29 and 30
“Geometry Teacher’s Activities Kit” pages 160 - 163
“Geometry Plane & Simple” page 45, 46, 47,48
Geometry Readiness Curriculum Guide
2011
Congruent Triangles
Chapter: 4
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
• Use coordinate methods, such as the distance
formula and the slope formula, to prove two triangles
are congruent.
• Use algebraic methods to prove two triangles are
congruent.
G.6 The student, given information in the form of a
figure or statement, will prove two triangles are
congruent, using algebraic and coordinate methods
as well as deductive proofs.
• Use definitions, postulates, and theorems to prove
triangles congruent.
• Use algebraic methods to prove two triangles are
congruent.
4-5 Isosceles and
Equilateral Triangles
G.6 The student, given information in the form of a
figure or statement, will prove two triangles are
congruent, using algebraic and coordinate methods
as well as deductive proofs.
• Use definitions, postulates, and theorems to prove
triangles congruent.
• Use algebraic methods to prove two triangles are
congruent.
4-6 Congruence in Right
Triangles
G.6 The student, given information in the form of a
figure or statement, will prove two triangles are
congruent, using algebraic and coordinate methods
as well as deductive proofs.
• Use definitions, postulates, and theorems to prove
triangles congruent.
4-7 Congruence in
Overlapping Triangles
1 block
1 block
“Geometry Plane & Simple” pages 41, 42, 43, 44,46,
47
1 block
12
Comments
Geometry Readiness Curriculum Guide
2011
Relationships Within Triangles
Chapter: 5
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
G.5 The student, given information concerning the
lengths of sides and/or measures of angles in
triangles, will
a.) order the sides by length, given the angle
measures;
b.) order the angles by degree measure, given the
side lengths;
c.) determine whether a triangle exists; and
d.) determine the range in which the length of the
third side must lie.
These concepts will be considered in the context of
real-world situations.
5-1 Midsegments of
Triangles
G.5 The student, given information concerning the
lengths of sides and/or measures of angles in
triangles, will
a.) order the sides by length, given the angle
measures;
b.) order the angles by degree measure, given the
side lengths;
c.) determine whether a triangle exists; and
d.) determine the range in which the length of the
third side must lie.
These concepts will be considered in the context of
real-world situations.
5-2 Perpendicular and
Angle Bisectors
G.5 The student, given information concerning the
lengths of sides and/or measures of angles in
triangles, will
a.) order the sides by length, given the angle
measures;
b.) order the angles by degree measure, given the
side lengths;
c.) determine whether a triangle exists; and
d.) determine the range in which the length of the
third side must lie.
These concepts will be considered in the context of
real-world situations.
5-4 Medians and Altitudes
Supporting Materials
http://www.youtube.com/watch?v=cSZYvU_dsAw
1 block
2 blocks
0.5 block
13
“Geometry Teacher’s Activities Kit” pages 144 -,145
“Geometry Plane & Simple” page 73
Comments
Geometry Readiness Curriculum Guide
2011
Relationships Within Triangles
Chapter: 5
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
G.1d The student will construct and judge the
5-6 Inequalities in One
validity of a logical argument consisting of a set of
Triangle
premises and a conclusion. This will include
a.) identifying the converse, inverse, and
2 blocks
contrapositive of a conditional statement;
b.) translating a short verbal argument into
symbolic form;
c.) using Venn diagrams to represent set
relationships; and
d.) using deductive reasoning.
• Use valid forms of deductive reasoning, including
the law of syllogism, the law of the contrapositive,
the law of detachment, and counterexamples.
G.5 The student, given information concerning the
lengths of sides and/or measures of angles in
triangles, will
a.) order the sides by length, given the angle
measures;
b.) order the angles by degree measure, given the
side lengths;
c.) determine whether a triangle exists; and
d.) determine the range in which the length of the
third side must lie.
These concepts will be considered in the context of
real-world situations.
• Order the sides of a triangle by their lengths when
given the measures of the angles.
• Order the angles of a triangle by their measures
when given the lengths of the sides.
• Given the lengths of three segments, determine
whether a triangle could be formed.
• Given the lengths of two sides of a triangle,
determine the range in which the length of the third
side must lie.
• Solve real-world problems given information
about the lengths of sides and/or measures of
angles in triangles.
Supporting Materials
http://illuminations.nctm.org/lessondetail.aspx?ID=L68
1
http://www.mathwarehouse.com/geometry/triangles/tri
angle-inequality-theorem-rule-explained.php
http://www.mathwarehouse.com/geometry/triangles/
14
Comments
Geometry Readiness Curriculum Guide
G.8 The student will solve real-world
problems involving right triangles by using the
Pythagorean Theorem and its converse,
properties of special right triangles, and right
triangle trigonometry.
• Determine whether a triangle formed with
three given lengths is a right triangle.
• Solve real-world problems, using right
triangle trigonometry and properties of right
triangles.
2011
8-1 The Pythagorean
Theorem and its Converse
1 block
http://regentsprep.org/Regents/math/geometry/GP13/TRe Chapter VA in the
sourcePyth.htm
textbook provides
additional lessons and
http://teachers.henrico.k12.va.us/math/igo/WU_Algebra/ standards review to
WU7_4Alg.htm
get students ready for
the end of the year
http://teachers.henrico.k12.va.us/math/igo/07RightTriangl SOL test. VA
es/7-2PythagoreanThm/CW7-2.pdf
Standards Review
Pythagorean
http://teachers.henrico.k12.va.us/math/igo/07RightTriangl Theorem (page VA
es/7-2PythagoreanThm/HW7-2.pdf
14) could be used
after Chapter 8 is
http://www.math-videos-online.com/easy-pythagoreancompleted for
theorem-proofs.html
additional review for
the SOL test.
http://www.math-play.com/Geometry-Math-Games.html
http://regentsprep.org/Regents/math/geometry/GP13/Pra
cPyth.htm
DOE ESS Sample Lesson Plan G.8 The Pythagorean
Relationship
SOL with Essential Knowledge and Skill
G.8 The student will solve real-world
problems involving right triangles by using the
Pythagorean Theorem and its converse,
properties of special right triangles, and right
triangle trigonometry.
• Solve for missing lengths in geometric
figures, using properties of 45-45-90 triangles.
• Solve for missing lengths in geometric
figures, using properties of 30-60-90 triangles.
• Solve real-world problems, using right
triangle trigonometry and properties of right
triangles.
Textbook
Chapters/Sections/Topics
8-2 Special Right Triangles
Supporting Materials
http://teachers.henrico.k12.va.us/math/igo/07RightTriangl
es/7-3SpecialRtTriangles/CW7-3.pdf
1 block
http://teachers.henrico.k12.va.us/math/igo/07RightTriangl
es/7-3SpecialRtTriangles/HW7-3.pdf
“Geometry Plane & Simple” pages 80, 81, and 82
“Milliken Geoemtry” pages 17, 18
“if Geometry” page 62
“Geometry Teacher’s Activities Kit” pages 154 – 157
DOE ESS Sample Lesson Plan G.8 Special Right
Triangles and Right Triangles
G.8 The student will solve real-world
8-3 Trigonometry
http://teachers.henrico.k12.va.us/math/igo/07RightTriangl
15
Comments
Geometry Readiness Curriculum Guide
2011
problems involving right triangles by using the
Pythagorean Theorem and its converse,
1.5 blocks
properties of special right triangles, and right
triangle trigonometry.
• Solve problems involving right triangles,
using sine, cosine, and tangent ratios.
• Solve real-world problems, using right
triangle trigonometry and properties of right
triangles.
• Explain and use the relationship between the
sine and cosine of complementary angles.
es/7-4Trigonometry/CW7-4.pdf
G.8 The student will solve real-world
problems involving right triangles by using the
Pythagorean Theorem and its converse,
properties of special right triangles, and right
triangle trigonometry.
• Solve real-world problems, using right
triangle trigonometry and properties of right
triangles.
http://e-zgeometry.com/
8-4 Angles of Elevation and
Depression
http://teachers.henrico.k12.va.us/math/igo/07RightTriangl
es/7-4Trigonometry/HW7-4.pdf
http://teachers.henrico.k12.va.us/math/igo/07RightTriangl
es/7-4Trigonometry/7-4WU.htm
http://www.youtheducationservices.ca/trigonometry.html#
1 block
16
Geometry Readiness Curriculum Guide
2011
Mapping for Instruction - Third Nine Weeks
Similarity
Chapter: 7
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
7-1 Ratios and Proportions
http://e-zgeometry.com/
1 block
http://www.kutasoftware.com/freeige.html
http://teachers.henrico.k12.va.us/math/igo/05Similarity/51UsingProportions/CW5-1.pdf
http://teachers.henrico.k12.va.us/math/igo/05Similarity/51UsingProportions/HW5-1.pdf
G.14.d The student will use similar geometric
objects in two- or three-dimentions to solve
real-world problems about similar geometric
objects.
• Solve real-world problems involving
measured attributes of similar objects.
7-2 Similar Polygons
http://teachers.henrico.k12.va.us/math/igo/05Similarity/52SimilarPolygons/CW5-2.pdf
1 block
G.7 The student, given information in the form 7-3 Proving Triangles Similar
of a figure or statement, will prove two
triangles are similar, using algebraic and
1 block
coordinate methods as well as deductive
proofs.
• Use definitions, postulates, and theorems to
prove triangles similar.
• Use algebraic methods to prove that
triangles are similar.
• Use coordinate methods, such as the
distance formula, to prove two triangles are
similar.
http://teachers.henrico.k12.va.us/math/igo/05Similarity/52SimilarPolygons/HW5-2.pdf
http://teachers.henrico.k12.va.us/math/igo/05Similarity/53SimilarTriangles/CW5-3.pdf
http://teachers.henrico.k12.va.us/math/igo/05Similarity/53SimilarTriangles/HW5-3.pdf
G.7 The student, given information in the form 7-4 Similarity in Right
of a figure or statement, will prove two
Triangles
triangles are similar, using algebraic and
coordinate methods as well as deductive
0.5 block
proofs.
17
Comments
Geometry Readiness Curriculum Guide
2011
Similarity
Chapter: 7
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
• Use definitions, postulates, and theorems to
prove triangles similar.
• Use algebraic methods to prove that
triangles are similar.
• Use coordinate methods, such as the
distance formula, to prove two triangles are
similar.
7-5 Proportions in Triangles
http://teachers.henrico.k12.va.us/math/igo/05Similarity/54ProportionalParts/CW5-4.pdf
0.5 block
http://teachers.henrico.k12.va.us/math/igo/05Similarity/54ProportionalParts/HW5-4.pdf
G.7 The student, given information in the form VA-4 Congruent and Similar
of a figure or statement, will prove two
Triangles
triangles are similar, using algebraic and
coordinate methods as well as deductive
0.5 block
proofs.
• Use coordinate methods, such as the
distance formula, to prove two triangles are
similar.
Click on the Jeopardy Review on the left side of this
webpage
http://teachers.henrico.k12.va.us/math/igo/05Similarity/5_4
.html
18
Chapter VA in the
textbook provides
additional lessons
and standards
review to get
students ready for
the end of the year
SOL test. VA-4
Congruent and
Similar Triangles
(page VA 10) can be
used at the end of
Chapter 7 to teach
using coordinate
methods, such as
the distance formula,
to prove two
triangles are similar.
Geometry Readiness Curriculum Guide
2011
Polygons and Quadrilaterals
Chapter: 6
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
G.10 The student will solve real-world problems
6-1 The Polygon Angle-Sum
involving angles of polygons.
Theorems
• Find the sum of the measures of the interior and
exterior angles of a convex polygon.
1 block
• Find the measure of each interior and exterior angle
of a regular polygon.
• Find the number of sides of a regular polygon, given
the measures of interior or exterior angles of the
polygon.
http://homepage.mac.com/efithian/Geometry/Activit
y-04.html
“Milliken Geometry” page 23
“Geometry Teacher’s Activities Kit” pages 139 141
G.9 The student will verify characteristics of
quadrilaterals and use properties of quadrilaterals to
solve real-world problems.
• Solve problems, including real-world problems,
using the properties specific to parallelograms,
rectangles, rhombi, squares, isosceles trapezoids,
and trapezoids.
6-2 Properties of
Parallelograms
http://www.keymath.com/x3331.xml
“if Geometry” page 67
G.9 The student will verify characteristics of
quadrilaterals and use properties of quadrilaterals to
solve real-world problems.
• Solve problems, including real-world problems,
using the properties specific to parallelograms,
rectangles, rhombi, squares, isosceles trapezoids,
and trapezoids.
• Prove that quadrilaterals have specific properties,
using coordinate and algebraic methods, such as the
distance formula, slope, and midpoint formula.
• Prove the characteristics of quadrilaterals, using
deductive reasoning, algebraic, and coordinate
methods.
6-3 Proving that a
Quadrilateral is a
Parallelogram
G.9 The student will verify characteristics of
quadrilaterals and use properties of quadrilaterals to
solve real-world problems.
• Solve problems, including real-world problems,
using the properties specific to parallelograms,
rectangles, rhombi, squares, isosceles trapezoids,
and trapezoids.
• Prove that quadrilaterals have specific properties,
6-4 Properties of
http://www.mathsisfun.com/geometry/quadrilaterals
Rhombuses, Rectangles, and -interactive.html
Squares
“if Geometry” page 70
“Milliken Geometry” page 24
1 block
“Geometry Plane & Simple” pages 68, 69, 70
1 block
http://www.mathsisfun.com/geometry/quadrilaterals
-interactive.html
“if Geometry” page 68 and 69
“Geometry Plane & Simple” page 66, 67
1 block
19
Comments
Geometry Readiness Curriculum Guide
2011
Polygons and Quadrilaterals
Chapter: 6
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
using coordinate and algebraic methods, such as the
distance formula, slope, and midpoint formula.
• Prove the characteristics of quadrilaterals, using
deductive reasoning, algebraic, and coordinate
methods.
G.9 The student will verify characteristics of
quadrilaterals and use properties of quadrilaterals to
solve real-world problems.
• Solve problems, including real-world problems,
using the properties specific to parallelograms,
rectangles, rhombi, squares, isosceles trapezoids,
and trapezoids.
• Prove that quadrilaterals have specific properties,
using coordinate and algebraic methods, such as the
distance formula, slope, and midpoint formula.
• Prove the characteristics of quadrilaterals, using
deductive reasoning, algebraic, and coordinate
methods.
6-5 Conditions for
Rhombuses, Rectangles, and
Squares
G.9 The student will verify characteristics of
quadrilaterals and use properties of quadrilaterals to
solve real-world problems.
• Solve problems, including real-world problems,
using the properties specific to parallelograms,
rectangles, rhombi, squares, isosceles trapezoids,
and trapezoids.
• Prove that quadrilaterals have specific properties,
using coordinate and algebraic methods, such as the
distance formula, slope, and midpoint formula.
• Prove the characteristics of quadrilaterals, using
deductive reasoning, algebraic, and coordinate
methods.
6-6 Trapezoids and Kites
1 block
1 block
G.2 The student will use the relationships between
6-7 Polygons in the
angles formed by two lines cut by a transversal to
Coordinate Plane
a.) determine whether two lines are parallel;
b.) verify the parallelism, using algebraic and
1 block
coordinate methods as well as deductive proofs; and
c.) solve real-world problems involving angles formed
when parallel lines are cut by a transversal.
20
“Geometry Teacher’s Actvities Kit” pages 172 –
188
“if Geometry” page71
“Milliken Geometry” page 25
“Geometry Plane & Simple” page 71, 74
Suggestion for
Chapter Review:
DOE ESS Sample
Lesson Plan
G.9 Properties of
Quadrilaterals
Geometry Readiness Curriculum Guide
2011
Polygons and Quadrilaterals
Chapter: 6
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
• Use algebraic and coordinate methods as well as
deductive proofs to verify whether two lines are
parallel.
G.2 The student will use the relationships between
6-8 Applying Coordinate
angles formed by two lines cut by a transversal to
Geometry
a.) determine whether two lines are parallel;
b.) verify the parallelism, using algebraic and
1 block
coordinate methods as well as deductive proofs; and
c.) solve real-world problems involving angles formed
when parallel lines are cut by a transversal.
• Use algebraic and coordinate methods as well as
deductive proofs to verify whether two lines are
parallel.
Area
Chapter: 10
SOL with Essential Knowledge and Skill
G.14 The student will use similar geometric objects
in two- or three-dimensions to
a)compare ratios between side lengths, perimeters,
areas, and volumes.
• Solve real-world problems involving measured
attributes of similar objects.
Textbook
Chapters/Sections/Topics
10-1 Areas of
Parallelograms and
Triangles
Supporting Materials
Comments
http://teachers.henrico.k12.va.us/math/igo/WU_A Make copies of the SOL
lgebra/WU9_1Alg.htm
formula sheet and give
one to each student.
http://teachers.henrico.k12.va.us/math/igo/09Are
aVolume/9-1Area2D/HandsOn9-1FreshCoat.pdf
1 block
http://www.youtheducationservices.ca/secure/sub
jects/geometry/pdfs/
9_Area_Peri_WS.pdf
“Milliken Geometry” page 28
G.14 The student will use similar geometric objects 10-2 Areas of Trapezoids,
in two- or three-dimensions to
Rhombuses, and Kites
a) compare ratios between side lengths, perimeters,
areas, and volumes.
1 block
• Solve real-world problems involving measured
attributes of similar objects.
http://teachers.henrico.k12.va.us/math/igo/09Are
aVolume/9-1Area2D/WUcurrent9-1.pdf
“Milliken Geometry” page 29
21
Encourage students to
use their SOL formula
sheet.
Geometry Readiness Curriculum Guide
2011
Area
Chapter: 10
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
G.14 The student will use similar geometric objects 10-4 Perimeters and Areas
in two- or three-dimensions to
of Similar Figures
a) compare ratios between side lengths, perimeters,
areas, and volumes;
1 block
c) determine how changes in area and/or volume of
an object affect one or more dimensions of the
object; and
d) solve real-world problems about similar
geometric objects.
• Compare ratios between side lengths, perimeters,
areas, and volumes, given two similar figures.
• Describe how changes in one or more dimensions
affect other derived measures (perimeter, area, total
surface area, and volume) of an object.
• Describe how changes in one or more measures
(perimeter, area, total surface area, and volume)
affect other measures of an object.
• Solve real-world problems involving measured
attributes of similar objects.
http://teachers.henrico.k12.va.us/math/igo/WU_A Encourage students to
lgebra/WU9_5Alg.htm
use their SOL formula
sheet.
G.8 The student will solve real-world problems
involving right triangles by using the Pythagorean
Theorem and its converse, properties of special
right triangles, and right triangle trigonometry.
• Another formula for the area of a triangle is
A=1/2ab(sinC).
http://teachers.henrico.k12.va.us/math/igo/07Rig
htTriangles/7-4Trigonometry/7-4WU.htm
10-5 Trigonometry and Area
0.5 block
Encourage students to
use their SOL formula
sheet.
Theorem 10-8 Area of a
Triangle Given SAS (page
645) must be covered as
it is part of the essential
understandings in the
curriculum framework.
G.11 The student will use angles, arcs, chords,
10-6 Circles and Arcs
tangents, and secants to
b) solve real-world problems involving properties of 0.5 block
circles; and
c) find arc lengths and areas of sectors in circles.
• Find lengths, angle measures, and arc measures
associated with central and inscribed angles.
• Calculate the area of a sector and the length of an
arc of a circle, using proportions.
http://teachers.henrico.k12.va.us/math/igo/08
Circles/8-4ArcsChords/8-4WU.htm
DOE ESS Sample Lesson Plan G.11 Arc Length
and Area of a Sector
22
Geometry Readiness Curriculum Guide
2011
Area
Chapter: 10
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
• Solve real-world problems associated with circles,
using properties of angles, lines, and arcs.
• Verify properties of circles, using deductive
reasoning, algebraic, and coordinate methods.
G.11 The student will use angles, arcs, chords,
10-7 Areas of Circles and
tangents, and secants to
Sectors
c) find arc lengths and areas of sectors in circles.
• Calculate the area of a sector and the length of an 0.5 block
arc of a circle, using proportions.
• Solve real-world problems associated with circles,
using properties of angles, lines, and arcs.
• Verify properties of circles, using deductive
reasoning, algebraic, and coordinate
methods.
http://teachers.henrico.k12.va.us/math/igo/09Are
aVolume/9-1Area2D/CW9-1.pdf
http://teachers.henrico.k12.va.us/math/igo/09Are
aVolume/9-1Area2D/HW9-1.pdf
Section 10-8, which was
omitted, may be used
after SOL testing for
enrichment.
“Milliken Geometry” page 32, 33
DOE ESS Sample Lesson Plan
G.11 Arc Length and Area of a Sector
Mapping for Instruction - Fourth Nine Weeks
Circles
Chapter: 12
SOL with Essential Knowledge and Skill
G.11 The student will use angles, arcs, chords,
tangents, and secants to
a) investigate, verify, and apply properties of circles;
b)solve real-world problems involving properties of
circles;
• Find lengths, angle measures, and arc measures
associated with two intersecting tangents;
• Solve real-world problems associated with circles,
using properties of angles, lines, and arcs.
• Verify properties of circles, using deductive
Textbook
Chapters/Sections/Topics
Supporting Materials
“Geometry Teacher’s Activities Kit” pages 214 –
216
“Geometry Plane & Simple” page 104
12-1 Tangent Lines
1 block
23
Comments
Geometry Readiness Curriculum Guide
2011
Circles
Chapter: 12
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
reasoning, algebraic, and coordinate methods
G.11 The student will use angles, arcs, chords,
tangents, and secants to
a) investigate, verify, and apply properties of circles;
b)solve real-world problems involving properties of
circles;
• Find lengths, angle measures, and arc measures
associated with two intersecting chords;
• Solve real-world problems associated with circles,
using properties of angles, lines, and arcs.
• Verify properties of circles, using deductive
reasoning, algebraic, and coordinate methods
“Geometry Teacher’s Activities Kit” pages 217 –
219
“Geometry Plane & Simple” page 103
12-2 Chords and Arcs
1 block
G.4
The student will construct and justify the
12-3 Inscribed Angles
constructions of
a) a line segment congruent to a given line segment;
1 block
b) the perpendicular bisector of a line segment;
c) a perpendicular to a given line from a point not on
the line;
d) a perpendicular to a given line at a given point on
the line;
e) the bisector of a given angle,
f) an angle congruent to a given angle;
• Construct an equilateral triangle , a square, and a
regular hexagon inscribed in a circle.
• Construct the inscribed and circumscribed circles of a
triangle.
• Construct a tangent line from a point outside a given
circle to the circle
http://www.mathopenref.com/constincircle.html
http://www.mathsisfun.com/geometry/constructtrianglecircum.html
http://www.mathopenref.com/constequilateral.ht
ml
“Milliken Geometry” page 35
“Geometry Plane & Simple” pages 106, 107, 108,
109
G.11 The student will use angles, arcs, chords,
tangents, and secants to
a) investigate, verify, and apply properties of circles
• Find lengths, angle measures, and arc measures
associated with
– two intersecting chords;
– central and inscribed angles.
• Solve real-world problems associated with circles,
using properties of angles, lines, and arcs.
24
Constructions of
circumscribe and
inscribed polygons
may be presented
here or later. Use the
links to access online
construction
examples.
Geometry Readiness Curriculum Guide
2011
Circles
Chapter: 12
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
• Verify properties of circles, using deductive
reasoning, algebraic, and coordinate methods
G.11 The student will use angles, arcs, chords,
tangents, and secants to
a) investigate, verify, and apply properties of circles;
b) solve real-world problems involving properties of
circles;
• Find lengths, angle measures, and arc measures
associated with
– two intersecting chords;
– central and inscribed angles.
• Solve real-world problems associated with circles,
using properties of angles, lines, and arcs.
• Verify properties of circles, using deductive
reasoning, algebraic, and coordinate methods
– two intersecting secants;
– an intersecting secant and tangent;
– two intersecting tangents; and
12-4 Angle Measures and
Segment Lengths
1 block
“Geometry Plane & Simple” page 110
“Milliken Geometry” page 37, 38
“Geometry Teacher’s Activities Kit” pages 220 –
230
DOE ESS Sample Lesson Plan
G.11 Angles, Arcs, and Segments in Circles
G.12 The student, given the coordinates of the
12-5 Circles in the Coordinate
center of a circle and a point on the circle, will write the Plane
equation of the circle.
• Identify the center, radius, and diameter of a circle
1.5 block
from a given standard equation.
• Use the distance formula to find the radius of a
circle.
• Given the coordinates of the center and radius of the
circle, identify a point on the circle.
• Given the equation of a circle in standard form,
identify the coordinates of the center and find the
radius of the circle.
• Given the coordinates of the endpoints of a diameter,
find the equation of the circle.
• Given the coordinates of the center and a point on
the circle, find the equation of the circle.
• Recognize that the equation of a circle of given
center and radius is derived using the Pythagorean
Theorem.
25
http://education.ti.com/calculators/timath/US/Acti
vities/Detail?sa=502481d=12554
“Geometry Teacher’s Activities Kit” pages 231 –
233
DOE ESS Sample Lesson Plan
G.12 Circles in the Coordinate Plane
*Include 12-6
definition of a locus.
Geometry Readiness Curriculum Guide
2011
Surface Area and Volume
Chapter: 11
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
1-1 Nets and Drawings for
Visualizing Geometry
Optional
Supporting Materials
NCTM Isometric Drawing Tool
http://illuminations.nctm.org/activit
ydetail.aspx?id=125
http://www.learner.org/interactives
/geometry/index.html
“Geometry Teacher’s Activities
Kit” pages 295 - 297
G.13 The student will use formulas for surface area 11-2 Surface Areas of
and volume of three-dimensional objects to solve real- Prisms and Cylinders
world problems.
• Find the total surface area of cylinders, prisms,
0.5 block
pyramids, cones, and spheres, using the appropriate
formulas.
Comments
The SAT exam and the state SOL test
both provide formula sheets for students
to use. Therefore, it is not necessary to
have students memorize formulas.
However, students need to be aware of
the derivation of these formulas and
their use in problem solving.
The formulas for all chapters is on page
838 and 839. Teachers may want to
make copies of the SOL formula sheet
and distribute to students at the
beginning of lesson 11-2.
G.13 The student will use formulas for surface area 11-3 Surface Areas of
and volume of three-dimensional objects to solve real- Pyramids and Cones
world problems.
• Find the total surface area of cylinders, prisms,
0.5 block
pyramids, cones, and spheres, using the appropriate
formulas
• Solve problems, including real-world problems,
involving total surface area and volume of cylinders,
prisms, pyramids, cones, and spheres as well as
combinations of three-dimensional figures.
• Calculators may be used to find decimal
approximations for results.
G.14 The student will use similar geometric objects
in two- or three-dimensions to
b) determine how changes in one or more dimensions
of an object affect area and/or volume of the object.
G.13 The student will use formulas for surface area 11-4 Volumes of Prims and
and volume of three-dimensional objects to solve real- Cylinders
26
“if Geometry” pages 86, 87
“Geometry Teacher’s Activities
Geometry Readiness Curriculum Guide
2011
Surface Area and Volume
Chapter: 11
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
Kit” pages 298 - 301
world problems.
• Find the total surface area of cylinders, prisms,
0.5 block
pyramids, cones, and spheres, using the appropriate
formulas
• Solve problems, including real-world problems,
involving total surface area and volume of cylinders,
prisms, pyramids, cones, and spheres as well as
combinations of three-dimensional figures.
• Calculators may be used to find decimal
approximations for results.
• Solve problems, including real-world problems,
involving total surface area and volume of cylinders,
prisms, pyramids, cones, and spheres as well as
combinations of three-dimensional figures.
• Calculators may be used to find decimal
approximations for results.
• Calculate the volume of cylinders, prisms, pyramids,
cones, and spheres, using the appropriate formulas.
G.14 The student will use similar geometric objects
in two- or three-dimensions to
b) determine how changes in one or more
dimensions of an object affect area and/or volume of
the object.
G.13 The student will use formulas for surface area 11-5 Volumes of Pyramids
and volume of three-dimensional objects to solve real- and Cones
world problems.
• Solve problems, including real-world problems,
0.5 block
involving total surface area and volume of cylinders,
prisms, pyramids, cones, and spheres as well as
combinations of three-dimensional figures.
• Calculators may be used to find decimal
approximations for results.
• Calculate the volume of cylinders, prisms, pyramids,
cones, and spheres, using the appropriate formulas.
G.14 The student will use similar geometric objects
in two- or three-dimensions to
b) determine how changes in one or more dimensions
of an object affect area and/or volume of the object.
27
“if Geometry” pages 88, 89
Comments
Geometry Readiness Curriculum Guide
2011
Surface Area and Volume
Chapter: 11
SOL with Essential Knowledge and Skill
Textbook
Chapters/Sections/Topics
Supporting Materials
“Geometry Teacher’s Activities
Kit” pages 302 – 308
G.13 The student will use formulas for surface area 11-6 Surface Areas and
and volume of three-dimensional objects to solve real- Volumes of Spheres
world problems.
• Find the total surface area of cylinders, prisms,
0.5 block
pyramids, cones, and spheres, using the appropriate
formulas.
• Calculate the volume of cylinders, prisms, pyramids,
cones, and spheres, using the appropriate formulas.
• Solve problems, including real-world problems,
involving total surface area and volume of cylinders,
prisms, pyramids, cones, and spheres as well as
combinations of three-dimensional figures.
• Calculators may be used to find decimal
approximations for results.
DOE ESS Sample Lesson Plan
G.13 Surface Area and Volume
G.13 The student will use formulas for surface area 11-7 Areas and Volumes of DOE ESS Sample Lesson Plan
and volume of three-dimensional objects to solve real- Similar Solids
G.14 Similar Solids and
world problems
Proportional Reasoing
• Compare ratios between side lengths, perimeters,
1 block
areas, and volumes, given two similar figures.
• Describe how changes in one or more dimensions
affect other derived measures (perimeter, area, total
surface area, and volume) of an object.
• Describe how changes in one or more measures
(perimeter, area, total surface area, and volume)
affect other measures of an object.
• Solve real-world problems involving measured
attributes of similar objects..
28
Comments
Geometry Readiness Curriculum Guide
2011
Transformations
Chapter: 9
SOL with Essential Knowledge and Skill
Textbook Chapters/Sections/Topics
Supporting Materials
G.3
The student will use pictorial
representations, including computer software,
constructions, and coordinate methods, to solve
problems involving symmetry and transformation.
This will include
d) determining whether a figure has been
translated, reflected, rotated, or dilated, using
coordinate methods.
• Given an image and preimage, identify the
transformation that has taken place as a
reflection, rotation, dilation, or translation.
9-1 Translations
http://www.mathisfun.com/geometry.transla
tion.html
G.3
The student will use pictorial
representations, including computer software,
constructions, and coordinate methods, to solve
problems involving symmetry and transformation.
This will include
d) determining whether a figure has been
translated, reflected, rotated, or dilated, using
coordinate methods.
• Given an image and preimage, identify the
transformation that has taken place as a
reflection, rotation, dilation, or translation.
9-2 Reflections
G.3
The student will use pictorial
representations, including computer software,
constructions, and coordinate methods, to solve
problems involving symmetry and transformation.
This will include
d) determining whether a figure has been
translated, reflected, rotated, or dilated, using
coordinate methods.
• Given an image and preimage, identify the
transformation that has taken place as a
reflection, rotation, dilation, or translation.
9-3 Rotations
G.3
The student will use pictorial
representations, including computer software,
constructions, and coordinate methods, to solve
9-4 Symmetry
“if Geometry” page 16
0.25 block
DOE ESS Sample Lesson Plan
0.25 block
“if Geometry” page 20
http://www.mathisfun.com/geometry.reflecti
on.html
0.25 block
“if Geometry” page 18
http://www.mathisfun.com/geometry.rotatio
n.html
0.25 block
“Geometry Teacher’s Activities Kit” pages
189 – 195
“if Geometry” page 19, 21,22, 23
29
Comments
Geometry Readiness Curriculum Guide
2011
Transformations
Chapter: 9
SOL with Essential Knowledge and Skill
Textbook Chapters/Sections/Topics
problems involving symmetry and transformation.
This will include
c) investigating symmetry and determining
whether a figure is symmetric with respect to a
line or a point; and
• Determine whether a figure has point symmetry,
line symmetry, both, or neither.
Supporting Materials
Comments
G.3 Symmetry
G.3
The student will use pictorial
representations, including computer software,
constructions, and coordinate methods, to solve
problems involving symmetry and transformation.
This will include
d) determining whether a figure has been
translated, reflected, rotated, or dilated, using
coordinate methods.
• Given an image and preimage, identify the
transformation that has taken place as a
reflection, rotation, dilation, or translation.
9-5 Dilations
http://www.mathisfun.com/geometry.resizin Suggested Review:
g.html
DOE ESS Sample
Lesson Plan
G.3
Transformations
G.3
The student will use pictorial
representations, including computer software,
constructions, and coordinate methods, to solve
problems involving symmetry and transformation.
This will include
d) determining whether a figure has been
translated, reflected, rotated, or dilated, using
coordinate methods.
• Given an image and preimage, identify the
transformation that has taken place as a
reflection, rotation, dilation, or translation.
9-6 Compositions of Reflections
G.10 The student will solve real-world
problems involving angles of polygons.
• Identify tessellations in art, construction, and
nature.
9-7 Tesselations
0.25 block
0.25 block
http://math.pppst.com/tessellations.html
0.5 block
30
Geometry Readiness Curriculum Guide
2011
SOL Blueprints
http://www.doe.virginia.gov/testing/sol/blueprints/mathematics_blueprints/2009/blueprint_geometry.pdf
This revised test blueprint will be
2011-2012 Mathematics Standards of
effective with the administration of the
Learning tests.
31
Geometry Readiness Curriculum Guide
2011
SOL Enhanced Scope and Sequence
Supplemental Resources
All supplemental material provided with the textbook should be used at the instructor's discretion to extend, enhance and/or review student knowledge
of essential geometric skills. Examview software and Interactive Achievement may be used as a source of pre-made assessment materials, or to
custom design assessments for this course.
The following pages include supplemental worksheets referenced in this Geometry Curriculum Guide.
Study Island (www.studyisland.com) can be used to enhance instruction.
Interactive Achievement (www.interactiveachievement.com) can be used as a resource to create and administer quizzes and tests.
If teachers find they need to supplement sections of the new textbook to adequately teach a SOL and/or possess materials that enhance that
instruction, please contact Linda Bowden.
Geometry Reproducibles by Milliken Publishing Company
Geometry by Instructional Fair
Geometry Teacher’s Activities Kit by Jossey-Bass
Geometry Plane & Simple by Creative Publications (now Glencoe)
http://www.doe.virginia.gov/testing/sol/practice_items/index.shtml
32
Geometry Readiness Curriculum Guide
2011
SOL 2009 Framework
Geometry
33
Copyright © 2009
by the
Virginia Department of Education
P.O. Box 2120
Richmond, Virginia 23218-2120
http://www.doe.virginia.gov
All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted.
Superintendent of Public Instruction
Patricia I. Wright, Ed.D.
Assistant Superintendent for Instruction
Linda M. Wallinger, Ph.D.
Office of Elementary Instruction
Mark R. Allan, Ph.D., Director
Deborah P. Wickham, Ph.D., Mathematics Specialist
Office of Middle and High School Instruction
Michael F. Bolling, Mathematics Coordinator
Acknowledgements
The Virginia Department of Education wishes to express sincere thanks to Deborah Kiger Bliss, Lois A. Williams, Ed.D., and Felicia Dyke, Ph.D.
who assisted in the development of the 2009 Mathematics Standards of Learning Curriculum Framework.
NOTICE
The Virginia Department of Education does not unlawfully discriminate on the basis of race, color, sex, national origin, age, or disability in
employment or in its educational programs or services.
The 2009 Mathematics Curriculum Framework can be found in PDF and Microsoft Word file formats on the Virginia Department of Education’s
Web site at http://www.doe.virginia.gov.
Virginia Mathematics Standards of Learning Curriculum Framework 2009
Introduction
The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and
amplifies the Mathematics Standards of Learning by defining the content knowledge, skills, and understandings that are measured by the Standards
of Learning assessments. The Curriculum Framework provides additional guidance to school divisions and their teachers as they develop an
instructional program appropriate for their students. It assists teachers in their lesson planning by identifying essential understandings, defining
essential content knowledge, and describing the intellectual skills students need to use. This supplemental framework delineates in greater specificity
the content that all teachers should teach and all students should learn.
Each topic in the Mathematics Standards of Learning Curriculum Framework is developed around the Standards of Learning. The format of the
Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge and skills that should be the focus of instruction for
each standard. The Curriculum Framework is divided into two columns: Essential Understandings and Essential Knowledge and Skills. The purpose
of each column is explained below.
Essential Understandings
This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the
Standards of Learning.
Essential Knowledge and Skills
Each standard is expanded in the Essential Knowledge and Skills column. What each student should know and be able to do in each standard is
outlined. This is not meant to be an exhaustive list nor is a list that limits what taught in the classroom. It is meant to be the key knowledge and skills
that define the standard.
The Curriculum Framework serves as a guide for Standards of Learning assessment development. Assessment items may not and should not be a
verbatim reflection of the information presented in the Curriculum Framework. Students are expected to continue to apply knowledge and skills
from Standards of Learning presented in previous grades as they build mathematical expertise.
TOPIC: REASONING, LINES, AND TRANSFORMATIONS
GEOMETRY
STANDARD G.1
The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include
a) identifying the converse, inverse, and contrapositive of a conditional statement;
b) translating a short verbal argument into symbolic form;
c) using Venn diagrams to represent set relationships; and
d) using deductive reasoning.
ESSENTIAL UNDERSTANDINGS

Inductive reasoning, deductive reasoning, and proof are critical in
establishing general claims.

Deductive reasoning is the method that uses logic to draw
conclusions based on definitions, postulates, and theorems.

Inductive reasoning is the method of drawing conclusions from a
limited set of observations.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Identify the converse, inverse, and contrapositive of a conditional
statement.

Translate verbal arguments into symbolic form, such as
(p  q) and (~p  ~q).

Determine the validity of a logical argument.

Use valid forms of deductive reasoning, including the law of
syllogism, the law of the contrapositive, the law of detachment,
and counterexamples.

Proof is a justification that is logically valid and based on initial
assumptions, definitions, postulates, and theorems.

Logical arguments consist of a set of premises or hypotheses and
a conclusion.

Euclidean geometry is an axiomatic system based on undefined
terms (point, line and plane), postulates, and theorems.
Select and use various types of reasoning and methods of proof,
as appropriate.

When a conditional and its converse are true, the statements can
be written as a biconditional, i.e., iff or if and only if.
Use Venn diagrams to represent set relationships, such as
intersection and union.

Interpret Venn diagrams.

Recognize and use the symbols of formal logic, which include →,
↔, ~,  ,  , and  .



Logical arguments that are valid may not be true. Truth and
validity are not synonymous.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
1
TOPIC: REASONING, LINES, AND TRANSFORMATIONS
GEOMETRY
STANDARD G.2
The student will use the relationships between angles formed by two lines cut by a transversal to
a) determine whether two lines are parallel;
b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and
c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.
ESSENTIAL UNDERSTANDINGS

Parallel lines intersected by a transversal form angles with
specific relationships.

Some angle relationships may be used when proving two lines
intersected by a transversal are parallel.

The Parallel Postulate differentiates Euclidean from nonEuclidean geometries such as spherical geometry and hyperbolic
geometry.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Use algebraic and coordinate methods as well as deductive proofs
to verify whether two lines are parallel.

Solve problems by using the relationships between pairs of angles
formed by the intersection of two parallel lines and a transversal
including corresponding angles, alternate interior angles, alternate
exterior angles, and same-side (consecutive) interior angles.

Solve real-world problems involving intersecting and parallel
lines in a plane.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
2
TOPIC: REASONING, LINES, AND TRANSFORMATIONS
GEOMETRY
STANDARD G.3
The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems
involving symmetry and transformation. This will include
a) investigating and using formulas for finding distance, midpoint, and slope;
b) applying slope to verify and determine whether lines are parallel or perpendicular;
c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and
d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods.
ESSENTIAL UNDERSTANDINGS

Transformations and combinations of transformations can be used
to describe movement of objects in a plane.

The distance formula is an application of the Pythagorean
Theorem.

Geometric figures can be represented in the coordinate plane.

Techniques for investigating symmetry may include paper
folding, coordinate methods, and dynamic geometry software.

Parallel lines have the same slope.

The product of the slopes of perpendicular lines is -1.

The image of an object or function graph after an isomorphic
transformation is congruent to the preimage of the object.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Find the coordinates of the midpoint of a segment, using the
midpoint formula.

Use a formula to find the slope of a line.

Compare the slopes to determine whether two lines are parallel,
perpendicular, or neither.

Determine whether a figure has point symmetry, line symmetry,
both, or neither.

Given an image and preimage, identify the transformation that has
taken place as a reflection, rotation, dilation, or translation.

Apply the distance formula to find the length of a line segment
when given the coordinates of the endpoints.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
3
TOPIC: REASONING, LINES, AND TRANSFORMATIONS
GEOMETRY
STANDARD G.4
The student will construct and justify the constructions of
a) a line segment congruent to a given line segment;
b) the perpendicular bisector of a line segment;
c) a perpendicular to a given line from a point not on the line;
d) a perpendicular to a given line at a given point on the line;
e) the bisector of a given angle;
f) an angle congruent to a given angle; and
g) a line parallel to a given line through a point not on the given line.
ESSENTIAL UNDERSTANDINGS

Construction techniques are used to solve real-world problems in
engineering, architectural design, and building construction.

Construction techniques include using a straightedge and
compass, paper folding, and dynamic geometry software.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
 Construct and justify the constructions of
– a line segment congruent to a given line segment;
– the perpendicular bisector of a line segment;
– a perpendicular to a given line from a point not on the line;
– a perpendicular to a given line at a point on the line;
– the bisector of a given angle;
– an angle congruent to a given angle; and
– a line parallel to a given line through a point not on the
given line.

Construct an equilateral triangle, a square, and a regular hexagon
inscribed in a circle.†

Construct the inscribed and circumscribed circles of a triangle.†

Construct a tangent line from a point outside a given circle to the
circle.†
†
Revised March 2011
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
4
TOPIC: TRIANGLES
GEOMETRY
STANDARD G.5
The student, given information concerning the lengths of sides and/or measures of angles in triangles, will
a) order the sides by length, given the angle measures;
b) order the angles by degree measure, given the side lengths;
c) determine whether a triangle exists; and
d) determine the range in which the length of the third side must lie.
These concepts will be considered in the context of real-world situations.
ESSENTIAL UNDERSTANDINGS

The longest side of a triangle is opposite the largest angle of the
triangle and the shortest side is opposite the smallest angle.

In a triangle, the length of two sides and the included angle
determine the length of the side opposite the angle.

In order for a triangle to exist, the length of each side must be
within a range that is determined by the lengths of the other two
sides.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Order the sides of a triangle by their lengths when given the
measures of the angles.

Order the angles of a triangle by their measures when given the
lengths of the sides.

Given the lengths of three segments, determine whether a triangle
could be formed.

Given the lengths of two sides of a triangle, determine the range
in which the length of the third side must lie.

Solve real-world problems given information about the lengths of
sides and/or measures of angles in triangles.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
5
TOPIC: TRIANGLES
GEOMETRY
STANDARD G.6
The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate
methods as well as deductive proofs.
ESSENTIAL UNDERSTANDINGS

Congruence has real-world applications in a variety of areas,
including art, architecture, and the sciences.

Congruence does not depend on the position of the triangle.

Concepts of logic can demonstrate congruence or similarity.

Congruent figures are also similar, but similar figures are not
necessarily congruent.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Use definitions, postulates, and theorems to prove triangles
congruent.

Use coordinate methods, such as the distance formula and the
slope formula, to prove two triangles are congruent.

Use algebraic methods to prove two triangles are congruent.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
6
TOPIC: TRIANGLES
GEOMETRY
STANDARD G.7
The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate
methods as well as deductive proofs.
ESSENTIAL UNDERSTANDINGS

Similarity has real-world applications in a variety of areas,
including art, architecture, and the sciences.

Similarity does not depend on the position of the triangle.

Congruent figures are also similar, but similar figures are not
necessarily congruent.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Use definitions, postulates, and theorems to prove triangles
similar.

Use algebraic methods to prove that triangles are similar.

Use coordinate methods, such as the distance formula, to prove
two triangles are similar.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
7
TOPIC: TRIANGLES
GEOMETRY
STANDARD G.8
The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of
special right triangles, and right triangle trigonometry.
ESSENTIAL UNDERSTANDINGS

The Pythagorean Theorem is essential for solving problems
involving right triangles.

Many historical and algebraic proofs of the Pythagorean Theorem
exist.

The relationships between the sides and angles of right triangles
are useful in many applied fields.

Some practical problems can be solved by choosing an efficient
representation of the problem.

Another formula for the area of a triangle is A  ab sin C .

The ratios of side lengths in similar right triangles
(adjacent/hypotenuse or opposite/hypotenuse) are independent of
the scale factor and depend only on the angle the hypotenuse
makes with the adjacent side, thus justifying the definition and
calculation of trigonometric functions using the ratios of side
lengths for similar right triangles.
1
2
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Determine whether a triangle formed with three given lengths is a
right triangle.

Solve for missing lengths in geometric figures, using properties of
45-45-90 triangles.

Solve for missing lengths in geometric figures, using properties of
30-60-90 triangles.

Solve problems involving right triangles, using sine, cosine, and
tangent ratios.

Solve real-world problems, using right triangle trigonometry and
properties of right triangles.

Explain and use the relationship between the sine and cosine of
complementary angles.†
†
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
Revised March 2011
8
TOPIC: POLYGONS AND CIRCLES
GEOMETRY
STANDARD G.9
The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems.
ESSENTIAL UNDERSTANDINGS



The terms characteristics and properties can be used
interchangeably to describe quadrilaterals. The term
characteristics is used in elementary and middle school
mathematics.
Quadrilaterals have a hierarchical nature based on the
relationships between their sides, angles, and diagonals.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Solve problems, including real-world problems, using the
properties specific to parallelograms, rectangles, rhombi, squares,
isosceles trapezoids, and trapezoids.

Prove that quadrilaterals have specific properties, using
coordinate and algebraic methods, such as the distance formula,
slope, and midpoint formula.

Prove the characteristics of quadrilaterals, using deductive
reasoning, algebraic, and coordinate methods.

Prove properties of angles for a quadrilateral inscribed in a circle.†
Characteristics of quadrilaterals can be used to identify the
quadrilateral and to find the measures of sides and angles.
†
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
Revised March 2011
9
TOPIC: POLYGONS AND CIRCLES
GEOMETRY
STANDARD G.10
The student will solve real-world problems involving angles of polygons.
ESSENTIAL UNDERSTANDINGS

A regular polygon will tessellate the plane if the measure of an
interior angle is a factor of 360.

Both regular and nonregular polygons can tessellate the plane.

ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Solve real-world problems involving the measures of interior and
exterior angles of polygons.
Two intersecting lines form angles with specific relationships.

Identify tessellations in art, construction, and nature.

An exterior angle is formed by extending a side of a polygon.


The exterior angle and the corresponding interior angle form a
linear pair.
Find the sum of the measures of the interior and exterior angles of
a convex polygon.

Find the measure of each interior and exterior angle of a regular
polygon.

Find the number of sides of a regular polygon, given the measures
of interior or exterior angles of the polygon.

The sum of the measures of the interior angles of a convex
polygon may be found by dividing the interior of the polygon into
nonoverlapping triangles.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
10
TOPIC: POLYGONS AND CIRCLES
GEOMETRY
STANDARD G.11
The student will use angles, arcs, chords, tangents, and secants to
a) investigate, verify, and apply properties of circles;
b) solve real-world problems involving properties of circles; and
c) find arc lengths and areas of sectors in circles.
ESSENTIAL UNDERSTANDINGS

Many relationships exist between and among angles, arcs,
secants, chords, and tangents of a circle.

All circles are similar.

A chord is part of a secant.

Real-world applications may be drawn from architecture, art, and
construction.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Find lengths, angle measures, and arc measures associated with
– two intersecting chords;
– two intersecting secants;
– an intersecting secant and tangent;
– two intersecting tangents; and
– central and inscribed angles.

Calculate the area of a sector and the length of an arc of a circle,
using proportions.

Solve real-world problems associated with circles, using
properties of angles, lines, and arcs.

Verify properties of circles, using deductive reasoning, algebraic,
and coordinate methods.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
11
TOPIC: POLYGONS AND CIRCLES
GEOMETRY
STANDARD G.12
The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle.
ESSENTIAL UNDERSTANDINGS

A circle is a locus of points equidistant from a given point, the
center.

Standard form for the equation of a circle is
2
2
 x  h    y  k   r 2 , where the coordinates of the center of the

ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Identify the center, radius, and diameter of a circle from a given
standard equation.
circle are ( h, k ) and r is the length of the radius.

Use the distance formula to find the radius of a circle.
The circle is a conic section.

Given the coordinates of the center and radius of the circle,
identify a point on the circle.

Given the equation of a circle in standard form, identify the
coordinates of the center and find the radius of the circle.

Given the coordinates of the endpoints of a diameter, find the
equation of the circle.

Given the coordinates of the center and a point on the circle, find
the equation of the circle.

Recognize that the equation of a circle of given center and radius
is derived using the Pythagorean Theorem.†
†
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
Revised March 2011
12
TOPIC: THREE-DIMENSIONAL FIGURES
GEOMETRY
STANDARD G.13
The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.
ESSENTIAL UNDERSTANDINGS

The surface area of a three-dimensional object is the sum of the
areas of all its faces.

The volume of a three-dimensional object is the number of unit
cubes that would fill the object.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Find the total surface area of cylinders, prisms, pyramids, cones,
and spheres, using the appropriate formulas.

Calculate the volume of cylinders, prisms, pyramids, cones, and
spheres, using the appropriate formulas.

Solve problems, including real-world problems, involving total
surface area and volume of cylinders, prisms, pyramids, cones,
and spheres as well as combinations of three-dimensional figures.

Calculators may be used to find decimal approximations for
results.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
13
TOPIC: THREE-DIMENSIONAL FIGURES
GEOMETRY
STANDARD G.14
The student will use similar geometric objects in two- or three-dimensions to
a) compare ratios between side lengths, perimeters, areas, and volumes;
b) determine how changes in one or more dimensions of an object affect area and/or volume of the object;
c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and
d) solve real-world problems about similar geometric objects.
ESSENTIAL UNDERSTANDINGS

A change in one dimension of an object results in predictable
changes in area and/or volume.

A constant ratio exists between corresponding lengths of sides of
similar figures.

Proportional reasoning is integral to comparing attribute measures
in similar objects.
ESSENTIAL KNOWLEDGE AND SKILLS
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to

Compare ratios between side lengths, perimeters, areas, and
volumes, given two similar figures.

Describe how changes in one or more dimensions affect other
derived measures (perimeter, area, total surface area, and volume)
of an object.

Describe how changes in one or more measures (perimeter, area,
total surface area, and volume) affect other measures of an object.

Solve real-world problems involving measured attributes of
similar objects.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry
14