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RICHMOND PUBLIC SCHOOLS
Pulling It All Together
MATHEMATICS
Geometry
Department of Instruction & Accountability
2012 – 2013
RICHMOND PUBLIC SCHOOLS
DEPARTMENT OF INSTRUCTION
Mathematics
CURRICULUM COMPASS
COMMITTEE
Ronald Bradford, Jr.
Tinkhani Hargrove
Bland Campbell
Joanne Seaton
Pulling It All Together
Irma Mayo
Diane Williams
Aaron Dixon
Cassandra Willis
Instructional Specialist
Instructional Specialist – Title I
Maria Crenshaw
Director of Instruction
Victoria S. Oakley
Dr. Yvonne W. Brandon
Chief Academic Officer
Superintendent
COPYRIGHT©2012
Richmond City Public Schools
Richmond, Virginia
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS ELECTRONIC
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WITHOUT PERMISSION FROM RICHMOND CITY PUBLIC SCHOOLS.
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Blueprint for Geometry Test
No. of
Items
Reporting Categories
18/36%
Reasoning, Lines, and
Transformations
14/28%
Triangles
18/36%
Polygons, Circles, and
Three-Dimensional
Figures
Excluded from test
50/100%
10
60
Total Operational Items
Field Test Items
Total Items on Test
MATHEMATICS – Geometry
Focus
Standards
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Venn Diagrams, Deductive Reasoning, and Proof
Angles of Parallel Lines
Distance, Midpoint, Slope, Symmetry and Transformation
Constructions
Triangle Inequality
Congruent Triangles
Similar Triangles
Right Triangle Relationships
G.1a-d
G.2a-c
G.3a-d
G.4a-g
G.5a-d
G.6
G.7
G.8

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
Quadrilaterals
Polygons and Tessellations
Circles
Equation of a Circle
Surface Area and Volume
Similar Solids
G.9
G.10
G.11a-c
G.12
G.13
G.14a-d
 NONE
No Standards are excluded from this test
* These field-test items will not be used to compute students’ scores on the test.
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Mathematics Standards of Learning for Virginia Schools
Geometry
This course is designed for students who have successfully completed the standards for Algebra I. All students are expected to achieve the Geometry
standards. The course includes, among other things, properties of geometric figures, trigonometric relationships, and reasoning to justify conclusions.
Methods of justification will include paragraph proofs, two-column proofs, indirect proofs, coordinate proofs, algebraic methods, and verbal
arguments. A gradual development of formal proof will be encouraged. Inductive and intuitive approaches to proof as well as deductive axiomatic
methods should be used.
This set of standards includes emphasis on two- and three-dimensional reasoning skills, coordinate and transformational geometry, and the use of
geometric models to solve problems. A variety of applications and some general problem-solving techniques, including algebraic skills, should be
used to implement these standards. Calculators, computers, graphing utilities (graphing calculators or computer graphing simulators), dynamic
geometry software, and other appropriate technology tools will be used to assist in teaching and learning. Any technology that will enhance student
learning should be used.
Reasoning, Lines, and Transformations
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;
d) using deductive reasoning.
MATHEMATICS – Geometry
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.
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Reasoning, Lines, and 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
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.
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.
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.
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.
Triangles
G.5
G.6
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.
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.
MATHEMATICS – Geometry
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Polygons and Circles
Three-Dimensional Figures
G.9
The student will verify characteristics of quadrilaterals and use
properties of quadrilaterals to solve real-world problems.
G.13
The student will use formulas for surface area and volume of threedimensional objects to solve real-world problems.
G.10
The student will solve real-world problems involving angles of
polygons.
G.14
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.
The student will use similar geometric objects in two- or threedimensions 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.
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.
MATHEMATICS – Geometry
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Standard: G.1a-d
Pulling It All Together
Reasoning, Lines, and Transformations
The student will construct and judge the validity of a logical argument consisting of a set of premises and a
conclusion.
Understanding the Standard
Teacher Notes
 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.
 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.
 When a conditional and its converse are true, the statements can be written as a biconditional, i.e., iff or if and only if.
 Logical arguments that are valid may not be true. Truth and validity are not synonymous.
 A conditional statement is an If- Then statement. The “If” part is the given, or hypothesis, and the “Then” part is the conclusion.
 Some statements are in the form of “if p, then q.” This form is called a conditional statement, where p is the statement you know is true
(hypothesis), and q is the statement that you conclude is true (conclusion). The symbol form of “if p, then q” is p=>q and is read “p
implies q”.
 The negation of a statement can be written using the symbol ~ (example ~p).
MATHEMATICS – Geometry
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
 The converse of a statement switches the hypothesis and the conclusion.
 The inverse of a statement is the negation of both the hypothesis and the conclusion.
 The contrapositive of a statement combines the converse and inverse.
 A Venn diagram is a special kind of diagram that shows logic statements in diagram form.
 Venn diagrams show three basic relationships: Disjoint (non-overlapping) -use the words “no or “none”, Overlapping (intersecting)- use the
word “some”, and Subset (enclosed)- use the words all or some.
 The law of syllogism is a type of reasoning used to reach a valid conclusion and is similar to the transitive property: two things equal to a
third are equal to each other.
Essential Understanding
Students
All students should:

Understand how to convert a non-mathematical sentence into a conditional statement.

Understand how to use truth tables to determine the validity of the converse, inverse, and contrapositive.

Understand how to diagram arguments with Venn diagrams.

Understand how to create Venn diagrams.

Understand how to reach a valid conclusion using the law of syllogism.
MATHEMATICS – Geometry
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Reasoning, Lines, and Transformations
Standard: G.1a-d
The student will construct and judge the validity of a logical argument consisting of a set of premises and a
conclusion.
Essential Knowledge,
Skills and Processes
Objectives
The student will:
a) identify the
converse, inverse,
and contrapositive
of a conditional
statement
b) translate a short
verbal argument
into symbolic form
MATHEMATICS – Geometry
To be successful with this
standard, students are 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
Common Core State
Standards
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, p.4
 RPS-Teaching by Design, Lesson G.1
Suggested Projects:
 Inductive and Deductive Reasoning
http://ttaconline.org/staff/sol/sol_sol_lessons.
asp
 Logic and Conditional Statements
http://ttaconline.org/staff/sol/sol_sol_lessons.
asp
Books/Materials:
Prentice Hall Study Guide & Practice
Workbook, pp. 15 – 20
 Prentice Hall Reading & Math Literacy
Masters, pp. 5 – 7
 Prentice Hall Daily Notetaking Guide, pp.
25 – 33, 91 – 94
 Prentice Hall Skills & Concepts Review, pp.
120 – 123, 142

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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
reasoning, including the law
of syllogism, the law of the
contrapositive, the law of
detachment, and
counterexamples.

c) use Venn diagrams to
represent set
relationships
d) use deductive reasoning.

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 ∨. Use inductive
reasoning to make
conjectures.
Pulling It All Together
Prentice Hall VA SOL Test Prep Workbook,
p. vii
 RPS-Geometry Reteaching Lesson G.1
 Geometry SOL Coach, pp. 44 – 58
 Luster, Helen, Preparing for the Geometry
SOL Test, pp. 40-57
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MATHEMATICS – Geometry
Technology Integration:
Prentice Hall Interactive Student Text
(online or CD-ROM)
PHSuccess Net (teachers)
Prentice Hall Presentation Pro
Prentice Hall Computer Test Generator
Prentice Hall Resource Pro with Planning
Express
www.PHSchool.com (students)
TI-83/84 Graphing Calculator
CPS Jeopardy Game
http://regentsprep.org (Math A)
www.pen.k12.va.us/VDOE/Instruction/mat
hresource.html
The Geometry Center http://www.umn.edu/
NASA
http://spacelink.nasa.gov/.index.html
The Math Forum
http://forum.swarthmore.edu/
4teachers http://www.4teachers.org
Appalachia Educational Laboratory (AEL)
http://www.ael.org/pnp/index.htm
Eisenhower National Clearinghouse
http://www.enc.org/
http://education.jlab.org/solquiz/index.html
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Reasoning, Lines, and Transformations
Standard: G.1a-d
The student will construct and judge the validity of a logical argument consisting of a set of premises and a
conclusion.
conclusion
conditional statement
conjecture
deductive reasoning
contrapositive
converse
inverse
negation
if-then statement
inductive reasoning proof
Law of Detachment
counterexample
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
Student reports
EduAide Access
Test Wizard
Teacher created assessment
Mathematics Projects
Writing Assignment
PH ExamView Test Generator
Key Terms/
Vocabulary
MATHEMATICS – Geometry
hypothesis
Venn diagram
Law of Syllogism
Lesson By Design Assessment
Textbook Resource Kit activities
SOL Released Test Items
eduTest Assessment
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Reasoning, Lines, and Transformations
Standard: G.2a-c
The student use the relationships between angles formed by two lines cut by a transversal.
Understanding the Standard
Teacher Notes

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 non-Euclidean geometries such as spherical geometry and hyperbolic geometry.

Angles are formed by 2 rays that share the same endpoint, called a vertex.

Complementary angles are 2 acute angles whose measures add up to 90 degrees.

Complements of the same angle or congruent angles are congruent.

Supplementary angles are 2 angles whose measures add up to 180 degrees.

Vertical angles are the opposite angles formed by intersecting lines.

Adjacent angles are angles that share 1 ray and the same vertex.

A transversal is a line that intersects 2 other lines.

When 2 parallel lines are cut by a transversal, they form: congruent alternate interior angles, congruent alternate exterior angles,
congruent corresponding angles, supplementary same-side interior and exterior angles.

To determine if 2 lines are parallel, you need to show that:
 Pairs of corresponding angles are equal
 Pairs of alternate interior angles are equal
 Pairs of alternate exterior angles are equal
 Pairs of same-side interior angles are supplementary
 Pairs of same-side exterior angles are supplementary
 The 2 lines have the same slope.
MATHEMATICS – Geometry
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Essential Understanding
Students
All students should understand:

How to identify angles formed by intersecting lines (linear pairs and vertical angles.

How to state the relationships between pairs of angles including: congruent, complementary and supplementary.

How to solve problems involving lines intersecting in a plane.

How to identify angles formed by lines cut by a transversal (such as: same-side interior, same-side exterior, alternate interior,
alternate exterior, and corresponding.

How to state the relationships between angles formed by parallel lines cut by a transversal.

How to use the relationships between pairs of angles (such as: same-side interior, same-side exterior, alternate interior, alternate
exterior, and corresponding angles) to solve practical problems.

How to perform constructions to verify congruent angles.

How to measure angles using protractors.

Parallel Lines have the same slope.

How to use angle relationship to determine if lines are parallel.

How to find the slope of a line.

How to use algebraic methods to determine if lines are parallel.
MATHEMATICS – Geometry
7
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Reasoning, Lines, and Transformations
Standard: G.2a-c
The student use the relationships between angles formed by two lines cut by a transversal.
Objectives
The student will:
a) determine whether
two lines are parallel;
b) verify the parallelism,
using algebraic and
coordinate methods as
well as deductive
proofs;
MATHEMATICS – Geometry
Essential Knowledge,
Skills and Processes
To be successful with this
standard, students are
expected 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.
Common Core State
Standards
CCSS for MathematicsGeometry
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.
4. Develop definitions of
rotations, reflections, and
translations in terms of
angles, circles,
perpendicular lines,
parallel lines, and line
segments.
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, p. 7
 RPS-Teaching By Design, Lesson G.4
Suggested Projects:
 Investigating Lines and Angles
http://ttaconline.org/staff/sol/sol_sol_lessons.asp
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 122-129
 Prentice Hall Study Guide & Practice Workbook,
pp. 27-28
 Prentice Hall Daily Notetaking Guide, pp. 43-45
 Prentice Hall Skills and Concepts Review, p. 126
 Prentice Hall VA SOL Test Prep Workbook, p. vii
 SOL Geometry Coach pp. 17-26
 Luster, Helen, Preparing for the Geometry SOL
Test, pp. 15-21
Technology Integration:
 Interactive Student Text (online and on CD-ROM)
 PHSuccessNet (for teacher)
 Prentice Hall Presentation Pro
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
 Prove geometric
theorems
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.
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.
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Pulling It All Together
Prentice Hall Computer Test Generator
Prentice Hall Resource Pro with Planning Express
www.PHSchool.com
TI-83/84 Graphing Calculators
Geometer’s Sketchpad
http://www.smv.org/pubs/index.html
http://tqd.advanced.org/2647/geometry/
angle/parallel.html
The Geometry Center http://www.umn.edu/
The Math Forum http://forum.swarthmore.edu/
4teachers http://www.4teachers.org
Eisenhower National Clearinghouse
http://www.enc.org/
http://www.pen.k12.va.us/VDOE/Instruction/mathresource.html
Expressing Geometric
Properties with Equations
G-GPE
 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
MATHEMATICS – Geometry
9
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
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).
MATHEMATICS – Geometry
10
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Reasoning, Lines, and Transformations
Standard: G.2a-c
The student use the relationships between angles formed by two lines cut by a transversal.
alternate interior
consecutive angles
linear pairs of angles
consecutive interior angles
alternate
angles
equilateral
parallel lines
right angles
interior angles
alternate exterior angles
equiangular
skew lines
transversal
same-side interior angles
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
eduTest Assessment
Key Terms/
Vocabulary
MATHEMATICS – Geometry
complementary angles
supplementary angles
slope
vertical angles
Student reports
EduAide Access
Test Wizard test
Mathematics Projects
SOL Released Test Items
11
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Reasoning, Lines, and Transformations
Standard: G.3a-d
The student will use pictorial representations, including computer software, constructions, and coordinate
methods, to solve problems involving symmetry and transformation.
Understanding the Standards
Teacher Notes
 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.
 You can find the distance between any two points on the coordinate plane by using the distance formula: (y-y) +(x-x).
 You can find the midpoint of a segment by using the formula: (x + x)/2, (y + y)/2.
 You can find the slope of a line by using the formula: (y-y)/(x-x).
 A figure has symmetry when it can be mapped, folded or rotated onto itself.
 A figure can have one or both of two basic symmetries: reflectional –a figure that folds onto itself, and rotational – if there is a rotation
of 180 degrees or less that maps the figure onto itself.
 Any time you transform a figure – that is move, shrink or enlarge the figure- you make a transformation.
 There are 4 transformations: reflect (flip), translate (slide), rotate (turn), and dilate (to enlarge or reduce).
MATHEMATICS – Geometry
12
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Essential Understanding
Students
All students should understand that:

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.
MATHEMATICS – Geometry
13
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Reasoning, Lines, and Transformations
Standard: G.3a-d
The student will use pictorial representations, including computer software, constructions, and coordinate methods,
to solve problems involving symmetry and transformation.
Objectives
The student will:
Essential
Knowledge, Skills
and Processes
To be successful with
this standard, students
are expected to:
a) investigate and use  Find the coordinates of
the midpoint of a
formulas for
segment, using the
finding distance,
midpoint formula.
midpoint, and
slope;
 Apply the distance
formula to find the
length of a line segment
when given the
coordinates of the
endpoints.
 Use a formula to find the
slope of a line.
MATHEMATICS – Geometry
Common Core State
Standards
CCSS for MathematicsGeometry
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.
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 RPS-Teaching By Design, Lesson G.2
 Prentice Hall Technology Activities 52, 94
 Prentice Hall Hands-On Activities 38, 39
 Serra, Michael, Patty Paper Geometry, pp. 145-154
Suggested Projects:
 Cabri Jr. The Equation of a Line
http://education.ti.com/educationportal/activityexchange/
Activity.do?cid=US&aId=6714
 Parallel/Perpendicular lines
http://www.algebralab.org/lessons/
lesson.aspx?file=geometry_coordparallelperpendicular.xml
 Graphic Line Designs
http://education.ti.com/educationportal/
activityexchange/Activity.do?cid=US&aId=3928
2. Represent
transformations in the
plane using, e.g.,
14
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
b) apply slope to
verify and
determine whether
lines are parallel or
perpendicular;
 Compare the slopes to
determine whether two
lines are parallel,
perpendicular, or
neither.
c) investigate
symmetry and
determine whether
lines are parallel or
perpendicular;
 Determine whether a
figure has point
symmetry, line
symmetry, both, or
neither.
d) determine 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.
 Solve real-world
problems involving
MATHEMATICS – Geometry
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).
Pulling It All Together
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 42-50, 158-164,
634-652, 662-666
 Prentice Hall Study Guide & Practice Workbook, pp. 11-12,
35-36, 143-152
 Prentice Hall Reading & Math Literacy Masters, pp. 3, 11
 Prentice Hall Daily Notetaking Guide, pp. 19-21, 55-57,
217-225, 229-231
 Prentice Hall Skills and Concepts Review, p.118
 Prentice Hall VA SOL Test Prep Workbook, p. ix
 VA SOL Coach-Geometry, pp. 192-233
 Luster, Helen, Preparing for the SOL Geometry Test, pp.
158-178
 RPS- Reteaching Lesson G.2
3. Given a rectangle,
parallelogram,
trapezoid, or regular
polygon, describe the
rotations and reflections
that carry it onto itself.
Technology Integration:
 Interactive Student Text (online or CD-ROM)
 PHSuccessNet (teachers)
 Prentice Hall Pro Presentation
 Prentice Hall Test Generator
 Prentice Hall Resource Pro with Planning Express
4. Develop definitions of
 www.PHSchool.com
rotations, reflections,
and translations in terms  TI-83/84 Graphing Calculator
 CPS Jeopardy Game
of angles, circles,
 http://www.smv.org/pubs/index.html
perpendicular lines,
 Geometer’s Sketchpad
parallel lines, and line
 NASA http://spacelink.nasa.gov/.index.html
segments.
 4teachers http://www.4teachers.org
 http://regentsprep.org
5. Given a geometric figure
and a rotation, reflection,
15
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
intersecting and parallel
lines in a plane.
Pulling It All Together
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.
Expressing Geometric
Properties with Equations
G-GPE

MATHEMATICS – Geometry
Use coordinates to
16
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
prove simple
geometric theorems
algebraically
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.
MATHEMATICS – Geometry
17
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Reasoning, Lines, and Transformations
Standard: G.3a-d
The student will use pictorial representations, including computer software, constructions, and coordinate
methods, to solve problems involving symmetry and transformation.
Key Terms/
Vocabulary
clockwise
plane
postulate
image
slide
distance
midpoint
counterclockwise
symmetry
point symmetry
intersection point
line postulate
line symmetry
theorem
segment
transformation
preimage
Pythagorean
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
eduTest Assessment
MATHEMATICS – Geometry
slope
plane ruler
dilation
translation
reflection
rotation
Student reports
EduAide Access
Test Wizard test
Mathematics Projects
SOL Released Test Items
18
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Standard: G.4a-g
Pulling It All Together
Reasoning, Lines, and Transformations
The student will construct and justify constructions.
Understanding the Standards
Teacher Notes

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.

A perpendicular bisector of a line segment is a line, line segment, or ray that forms a right angle with the line segment and divides the line
segment into two equal parts.

The bisector of an angle separates the interior of the angle into two congruent angles.

To bisect means to divide into two equal parts.

A line of symmetry for an angle or line segment bisects the angle or line segment.

In a construction you can only use a straightedge and a compass.

In a construction, a point must either be given or be the intersection of a previously constructed figure.

If a figure is given you can assume as many points as necessary to make that figure

A straightedge can draw a line, through two points A and B.

A compass can draw a circle with the center at A and containing a second point B on its circumference.
MATHEMATICS – Geometry
19
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Essential Understanding
Students
All students should understand the following:

Construction techniques are used to solve real-life problems in engineering, architectural design, and building construction.

Construction techniques may include using a straightedge and compass, paper folding, and dynamic geometry software.
MATHEMATICS – Geometry
20
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Reasoning, Lines, and Transformations
Standard: G.4a-g
The student will construct and justify constructions.
Objectives
The student
will:
a) construct a
line segment
congruent to a
given line
segment
b) construct the
perpendicular
bisector of a
line segment;
c) construct a
perpendicular
to a given line
from a point
not on the
line;
d) construct a
perpendicular
Essential
Knowledge, Skills
and Processes
To be successful with
this standard, students
are expected to:
Common Core State
Standards
CCSS for MathematicsGeometry
Congruence G-CO
 Make geometric
 Construct and justify the
constructions
constructions of
– a line segment
12. Make formal geometric
congruent to a given
constructions with a variety
line segment;
of tools and methods
– the perpendicular
(compass and straightedge,
bisector of a line
string, reflective devices,
segment;
paper folding, dynamic
– a perpendicular to a
geometric software, etc.).
given line from a point
Copying a segment; copying
not on the line;
an angle; bisecting a
– a perpendicular to a
segment; bisecting an angle;
given line at a point on
constructing perpendicular
the line;
lines, including the
– the bisector of a given
perpendicular bisector of a
angle;
line segment; and
– an angle congruent to a
constructing a line parallel
given angle; and
to a given line through a
– a line parallel to a given
MATHEMATICS – Geometry
Pulling It All Together
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 RPS-Teaching By Design, Lesson G.11
 Prentice Hall Mathematics-Geometry, p. 41
 Enhanced Scope and Sequence, pp. 19 - 25
 Serra, Michael, Patty Paper Geometry,
Investigation Set 2, pp. 17 – 22
Suggested Projects:
 Perpendicular Bisector Theorem
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=4035
 Perpendicular Bisector of a Line Segment
http://education.ti.com/educationportal/
activityexchange/Activity.do?cid=US&aId=6856
 Points on the Perpendicular Bisector of a
Segment
http://education.ti.com/educationportal/
activityexchange/Activity.do?cid=US&aId=6860
21
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
to a given line
at a given
point on the
line;
line through a point not
on the given line.
point not on the line.
e) construct the
bisector of a
given angle;
f) construct an
angle
congruent to a
given angle;
and
g) construct a
line parallel to
a given line
through a point
not on the
given line.
Pulling It All Together
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 34 - 39
 Prentice Hall Study Guide and Practice Workbook,
pp. 9, 10
 Prentice Hall Daily Notetaking Guide, pp. 15 – 18
 Prentice Hall Skills and Concepts Review, p. 117
 Prentice Hall VA SOL Test Prep Workbook, p. vii
 SOL Geometry Coach, pp. 27 – 41
 Luster, Helen, Preparing for the SOL Geometry
Test, pp. 22 – 34
 RPS – Reteach Lessons, G.11
Technology Integration:
Prentice Hall Resource PRO CD-ROM
Prentice Hall Presentation PRO CD
Prentice Hall Interactive Text,
www.PHSchool.com
 Prentice Hall SuccessNet
 Prentice Hall Computer Test Generator
 Prentice Hall Resource Pro with Planning Express
 TI-83/84 Graphing Calculators
 Geometer’s Sketchpad
 http://www.smv.org/pubs/index.html
 http://tqd.advanced.org/2647/geometry/
sample2.html
 http://www.pen.k12.va.us/VDOE/Instruction/mathresource.html
 4teachers http://www.4teachers.org
 www.regentsprep.org
 http://forum.swarthmore.edu/
 http://spacelink.nasa.gov/.index.html



 Construct an equilateral
triangle, a square, and a
regular hexagon inscribed
in a circle.
13. Construct an equilateral
triangle, a square, and a
regular hexagon inscribed in
a circle.
Circles G-C
 Understand and apply
theorems about circles
 Construct the inscribed
MATHEMATICS – Geometry
3. Construct the inscribed and
22
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
and circumscribed circles
of a triangle.
 Construct a tangent line
from a point outside a
given circle to the circle.
MATHEMATICS – Geometry
Pulling It All Together
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.
23
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Reasoning, Lines, and Transformations
Standard: G.4a-g
The student will construct and justify constructions.
Pulling It All Together
Key Terms/
Vocabulary
construction
quadrilateral
inscribed
angle bisector
straightedge
regular hexagon
compass
congruent
equilateral triangle
circumscribed
vertex
perpendicular
midpoint
point
tangent
triangle
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
MATHEMATICS – Geometry
segment bisector
line segment
line
bisect
square
Student reports
EduAide Access
Test Wizard test
Mathematics Projects
SOL Released Test Items
24
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
Standard: G.5a-d
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.
Understanding the Standard
Teacher Notes

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.

A triangle can be formed if the sum of the lengths of any two sides of a triangle is greater than the length of the third side. (Triangle
Inequality Theorem)

When you know the lengths of two sides of a triangle, you can determine the range of possible lengths of the third side by applying the
Triangle Inequality Theorem.

If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side and the smaller angle lies opposite the
shorter side.

If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle and the shorter side lies opposite the
smaller angle.
MATHEMATICS – Geometry
25
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Essential Understanding
Students
All students should:

Understand that the longest side of a triangle is opposite the largest angle of the triangle and the shortest side is opposite the smallest
angle.

Understand that in a triangle, the measure of an angle and the lengths of the adjacent sides determine the length of the side opposite the
angle.

Understand that a triangle can be formed if the sum of two sides is greater than the third side.
MATHEMATICS – Geometry
26
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
Standard: G.5a-d
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
Knowledge, Skills
and Processes
Objectives
The student will:
To be successful with
this standard,
students are expected
to:
a) order the sides by
length, given the
angles measures;
 Order the sides of a
triangle by their lengths
when given the
measures of the angles.
b) order the angles by
degree measure,
given the side
lengths;
 Order the angles of a
triangle by their
measures when given
the lengths of the sides.
MATHEMATICS – Geometry
Common Core State
Standards
CCSS for MathematicsGeometry
Congruence G-CO
 Prove geometric theorems
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
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, pp. 15-16
 Mathematics Enhanced Scope and SequenceGeometry, pp. 37-40
 RPS-Teaching by Design, Lesson G.6
Suggested Projects:
 Triangles from Midpoints
http://ttaconline.org/staff/sol/sol_sol_lessons.asp
 How Many Triangles
http://ttaconline.org/staff/sol/sol_sol_lessons.asp
 Investigating the Triangle Inequality
http://education.ti.com/educationportal/
activityexchange/Activity.do?cid=US&aId=7777
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 273-278
27
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
c) determine whether a
triangle exists; and
d) determine the range
in which the length
of the third side must
lie.
 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 – Geometry
medians of a triangle meet
at a point.








Pulling It All Together
Prentice Hall Study Guide & Practice Workbook, pp.
61-62
Prentice Hall Reading and Math Literacy, p. 19
Prentice Hall Daily Notetaking Guide Workbook, pp.
95-97
Prentice Hall Skills and Concepts Review, p. 143
Prentice Hall VA SOL Test Prep Workbook, p. vii
The VA SOL Mathematics Coach, Geometry pp. 60-66
Luster, Helen, Preparing for the SOL Geometry Test,
pp. 75-78
RPS Reteaching Lesson G.6
Technology Integration:
 Prentice Hall Interactive Student Text
 Prentice Hall Presentation Pro CD-ROM
 Prentice Hall Resource Pro
 Prentice Hall SuccessNet (teacher)
 Prentice Hall Computer Test Generator
 Geometer’s Sketchpad
 TI-83/84 Graphing Calculator
 CPS Jeopardy Game
 http://www.pen.k12.va.us/VDOE/Instruction/
 math_resource.html
 Eisenhower National Clearinghouse
http://www.enc.org/
 4teachers http://www,4teachers.org
 The Geometry Center
 http://www.umn.edu/
 The Math Forum
 http://forum.swarthmore.edu/
 http://education.jlab.org/solquiz/index.html
28
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
MATHEMATICS – Geometry
Pulling It All Together
 http://regentsprep.org
 http://forum.swarthmore.edu/
 http://spacelink.nasa.gov/.index.html
29
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
Standard: G.5a-d
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.
Key Terms/
Vocabulary
Addition Property of Inequality
shortest side
less than
largest angle
opposite angles
inequality
longest side
smallest angle
Methods of
Instruction
Guided practice
Peer Tutoring
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
MATHEMATICS – Geometry
Direct Instruction
Small Group
opposite side
greater than
Student reports
EduAide Access
Test Wizard test
Mathematics Projects
SOL Released Test Items
30
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Standard: G.6
Pulling It All Together
Triangles
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.
Understanding the Standard
Teacher Notes






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.
Two triangles are congruent if and only if their corresponding parts are congruent.
Triangles can be proven congruent by the following postulates or theorems:
 If three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent by the SideSide-Side (SSS) Postulate.
 If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the
two triangles are congruent by the Side-Angle-Side (SAS) Postulate.
 If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the
two triangles are congruent by the Angle-Side-Angle Postulate.
 If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of
another triangle, then the two triangles are congruent by the Angle-Angle-Side Theorem.
 If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles
are congruent by the Hypotenuse-Leg Theorem.

Once you know that triangles are congruent, you can make conclusions about corresponding segments and angles because
corresponding parts of congruent triangles are congruent (CPCTC).
MATHEMATICS – Geometry
31
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Essential Understanding
Students
All students should understand the following concepts:

Congruence has real-life applications in art, architecture, and construction.

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.
MATHEMATICS – Geometry
32
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
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
Knowledge, Skills
and Processes
Objectives
The student will:

name and label
corresponding
parts of congruent
triangles.
To be successful with
this standard, students
are expected to:

Use definitions,
postulates, and theorems
to prove triangles
congruent.
Common Core State
Standards
CCSS for MathematicsGeometry
Congruence G-CO
 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
MATHEMATICS – Geometry
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, pp. 8, 10, 11,
12-13, 22,
 Prentice Hall Technology Activities, pp. 81-83, 91
 Mathematics Enhanced Scope and SequenceGeometry, pp. 34-36
 RPS-Teaching by Design, Lesson G.5
 Serra, Michael, Patty Paper Geometry,
Investigation Set 8, pp. 125-131
Suggested Projects:
 ASA Triangle Congruence
http://education.ti.com/educationportal/
activityexchange/Activity.do?cid=US&aId=7839
 SAS Triangle Congruence
http://education.ti.com/educationportal/
activityexchange/Activity.do?cid=US&aId=7823
 SSS Triangle Congruence
 http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aI
33
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
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.

use SSS, SAS,
ASA, AAS, and
HL to test for
triangle
congruence.

Use coordinate methods,
such as the distance
formula and the slope
formula, to prove two
triangles 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 triangles
congruent.

Use algebraic methods to
prove two triangles are
congruent.
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.
MATHEMATICS – Geometry
Pulling It All Together
d=7818
 Congruent Triangles
http://education.ti.com/educationportal/
activityexchange/Activity.do?cid=US&aId=5967
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 179229, 416-452
 Prentice Hall Study Guide and Practice
Workbook, pp. 39-52, 93-102
 Prentice Hall Daily Notetaking Guide, pp. 61-80,
142-155
 Prentice Hall Reading and Math Literacy, pp.
13-15
 Prentice Hall Skills & Concepts Review, pp. 132138, 159-163
 Prentice Hall VA SOL Test Prep, p. vii
 RPS Reteaching Lesson G.5
 VA SOL Coach, Geometry, pp. 67 – 88
 Preparing for the SOL Geometry Test, pp. 57-74
Technology Integration:
 Prentice Hall Interactive Student Text
 Prentice Hall SuccessNet (teacher)
 Prentice Hall Presentation Pro CD-ROM
 Prentice Hall Resource Pro
 Prentice Hall Test Generator
 www.PHSchool (students)
34
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS

Similarity, Right Triangles, 

and Trigonometry G-SRT
 Prove theorems involving
similarity
5. Use congruence and
similarity criteria for
triangles to solve
problems and to prove
relationships in geometric
figures.










Pulling It All Together
TI- 83/84 Graphing Calculator
Geometer’s Sketchpad
www.pen.k12.va.us/VDOE/Instructions/wmstds/
geometry.shtml
Eisenhower National Clearinghouse
http://www.enc.org/
www.edhelper.com
http://mathforum.org/library
4teachers
http://www.4teachers.org
http://education.jlab.org/solquiz/index.html
http://regentsprep.org
The Geometry Center http://www.umn.edu/
Field Trips:
MATHEMATICS – Geometry
35
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
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.
Key Terms/
Vocabulary
acute triangle
auxiliary lines
base
base angles
congruent triangles
flow proof
isosceles triangle
legs
obtuse triangle
reflexive property
right triangle
scalene triangle
sides
transitive property
vertex angle
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
Student reports
EduAide Access
Test Wizard test
Textbook Resource Kit activities Mathematics Projects
SOL Released Test Items
eduTest Assessment
MATHEMATICS – Geometry
equiangular triangle
equilateral triangle
paragraph proof
vertices
Teacher created assessment
Lesson By Design Assessment
Writing Assignment
36
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
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.
Understanding the Standard
Teacher Notes

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.

A proportion is a statement that two ratios are equal.

Two polygons are similar () if and only if their corresponding angles are congruent and the measures of their corresponding sides are
proportional.

The ratio of the lengths of two corresponding sides of two similar polygons is called the scale factor.

Triangles can be proven similar by the following postulates or theorems:
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar by the Angle-Angle
Similarity Postulate (AA)
 If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the
included angles are congruent, then the triangles are similar by the Side-Angle-Side Similarity Theorem (SAS)
 If the corresponding sides of two triangles are proportional, then the triangles are similar by the Side-Side-Side Similarity
Theorem (SSS).
The geometric mean between two positive numbers a and b is the positive number x such that = .



When the altitude is drawn to the hypotenuse of a right triangle:
 the two triangles formed are similar to the original triangle and to each other;
MATHEMATICS – Geometry
37
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS


Pulling It All Together
the length of the altitude is the geometric mean of the lengths of the segments of the hypotenuse; and
the length of each leg is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is
adjacent to that leg.

If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into
segments of proportional lengths.

If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is
parallel to the third side.

A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is onehalf the length of the third side.
Essential Understanding
Students
All students should understand the following concepts:

Similarity has real-life applications in art, architecture, and construction.

Similarity does not depend on the position of the triangle.

Congruent figures are also similar, but similar figures are not necessarily congruent.
MATHEMATICS – Geometry
38
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
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
Knowledge, Skills
and Processes
Objectives
The student will:
To be successful with
this standard, students
are expected to:
Common Core State
Standards
CCSS for MathematicsGeometry
Similarity, Right Triangles,
and Trigonometry G-SRT

identify similar
figures and
solve problems
involving
similar figures,
 Use definitions,
postulates, and theorems
to prove triangles
similar.
 Use algebraic methods
to prove that triangles
are similar.

use proportional
parts of
triangles to
solve problems.
 Use coordinate methods,
such as the distance
formula, to prove two
triangles are similar.
MATHEMATICS – Geometry

Understand similarity in
terms of similarity
transformations
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
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, pp. 8, 10, 11,
12-13, 22,
 Prentice Hall Technology Activities, pp. 81-83,
91
 Mathematics Enhanced Scope and SequenceGeometry, pp. 34-36
 RPS-Teaching by Design, Lesson G.5
 Serra, Michael, Patty Paper Geometry,
Investigation Set 8, pp. 125-131
Suggested Projects:
 Constructing Similar Triangles
http://education.ti.com/educationportal/
activityexchange/Activity.do?cid=US&aId=8179
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 179229, 416-452
39
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS

Use algebraic methods to
prove two triangles are
congruent.

prove triangles
similar.

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.
MATHEMATICS – Geometry
Pulling It All Together






Prentice Hall Study Guide and Practice
Workbook, pp. 39-52, 93-102
Prentice Hall Daily Notetaking Guide, pp. 6180, 142-155
Prentice Hall Reading and Math Literacy, pp.
13-15
Prentice Hall Skills & Concepts Review, pp.
132-138, 159-163
Prentice Hall VA SOL Test Prep, p. vii
RPS Reteaching Lesson G.5
VA SOL Coach, Geometry, pp. 67 – 88
Preparing for the SOL Geometry Test, pp. 57-74
Technology Integration:
 Prentice Hall Interactive Student Text
 Prentice Hall SuccessNet (teacher)
 Prentice Hall Presentation Pro CD-ROM
 Prentice Hall Resource Pro
 Prentice Hall Test Generator
 www.PHSchool (students)
 TI- 83/84 Graphing Calculator
 Geometer’s Sketchpad
 www.pen.k12.va.us/VDOE/Instructions/wmstds/
 geometry.shtml
 Eisenhower National Clearinghouse
 http://www.enc.org/
 www.edhelper.com
 http://mathforum.org/library
 4teachers
 http://www.4teachers.org
40
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together



MATHEMATICS – Geometry
http://education.jlab.org/solquiz/index.html
http://regentsprep.org
The Geometry Center http://www.umn.edu/
41
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
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.
Key Terms/
Vocabulary
similar
angle bisector
geometric mean
cross-product property
scale factor
ratio
proportion
coordinate
ratio of similarity
midsegment
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
EduAide Access
Test Wizard test
Mathematics Projects
Student reports
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
SOL Released Test Items
MATHEMATICS – Geometry
proof
median
altitude
42
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
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.
Understanding the Standard
Teacher Notes

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 = 1/ 2 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.

In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the legs (altitude and base). This
relationship is known as the Pythagorean Theorem: a2 + b2 = c2.

The Pythagorean Theorem is used to find the measure of any one of the three sides of a right triangle if the measures of the other two
sides are known.

The converse of the Pythagorean Theorem states that if the square of the length of one side of a triangle is equal to the sum of the
squares of the lengths lf the other two sides, then the triangle is a right triangle.

In a triangle with the longest side c, if c a2 + b2, the triangle is obtuse, and if c2  a2 + b2, the triangle is acute.

Positive numbers a, b, and c form a Pythagorean triple of a2 + b2 = c2.
MATHEMATICS – Geometry
43
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS

Pulling It All Together
In a 45- 45- 90 triangle, both legs are congruent and the length of the hypotenuse is times the length of a leg.

In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is times the
length of the shorter leg.

A ratio of the lengths of sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine (sin), cosine
(cos), and tangent (tan).

An angle below a horizontal line is an angle of depression. An angle above a horizontal line is an angle of elevation.
Essential Understanding
Students
All students should:

Understand that the Pythagorean Theorem is essential for problem solving involving right triangles.

Understand the relationships between the sides and angles of right triangles are useful in many applied fields.

Understand that some practical problems can be solved by choosing an efficient representation of the problem.
MATHEMATICS – Geometry
44
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Triangles
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
Knowledge, Skills
and Processes
Objectives
The student will:

use the
Pythagorean
Theorem and its
converse.
To be successful with
this standard,
students are expected
to:
Common Core State
Standards
CCSS for Mathematics –
Grade 8
Geometry 8.G
 Understand and apply
 Verify the Pythagorean
the Pythagorean Theorem
Theorem and its
converse using deductive
6. Explain a proof of the
arguments as well as
Pythagorean Theorem and
algebraic and coordinate
its converse.
methods.
 Determine whether a
triangle formed with
three given lengths is a
right triangle.
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, pp. 25, 26, 2829
 Prentice Hall Technology Activities, p. 90
 RPS-Teaching by Design Lesson Plans- SOL G.7
 Mathematics Scope and Sequence-Geometry, pp.
41-54
Suggested Projects:
 30-60-90 right triangles in Cabri Jr
http://education.ti.com/educationportal
7. Apply the Pythagorean
/activityexchange/Activity.do?cid=US&aId=6199
Theorem to determine
 Applying the Pythagorean Theorem
unknown side lengths in
http://education.ti.com/educationportal
right triangles in real world /activityexchange/Activity.do?cid=US&aId=1836
and mathematical
 Investigating Special Triangles
problems in two and three
http://education.ti.com/educationportal
dimensions.
/activityexchange/Activity.do?cid=US&aId=7896
Similarity, Right Triangles,
and Trigonometry G-SRT
MATHEMATICS – Geometry
45
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS

use the properties
of 45-45-90 and
30-60-90
triangles.

calculate the
sines, cosines,
and tangents of
acute angles in
right triangles.

use the sine,
cosine, and
tangent to
determine
unknown
measures in right
triangles.
Pulling It All Together
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 357 Define trigonometric
 Solve for missing
371, 470-486
ratios and solve problems
lengths in geometric
 Prentice Hall Study Guide & Practice Workbook,
involving right triangles
figures, using properties
pp. 79-82, 105-110
of 45-45-90°triangles.
6. Understand that by
 Prentice Hall Reading and Math Literacy
similarity, side ratios in
Masters, pp. 26, 33, 34
right
triangles
are

Prentice Hall Daily Notetaking Guide Workbook,
 Solve for missing lengths
properties of the angles in
pp. 121-126, 159-167
in geometric figures,
the
triangle,
leading
to

Prentice Hall Skills and Concepts Review, pp.
using properties of 30definitions of
152, 153, 165-167
60-90°triangles.
trigonometric ratios for
 Prentice Hall VA SOL Test Prep Workbook, p.vii
acute angles.
 The VA SOL Mathematics Coach, Geometry, pp.
89-119
 Solve problems
 Luster, Helen, Preparing for the SOL Geometry
involving right triangles,
Test, pp. 79-87
using sine, cosine, and
7. Explain and use the
 RPS- Reteaching Lesson G.7
tangent ratios.
relationship between the
sine and cosine of
Technology Integration:
complementary angles.
 Solve real-world

Prentice Hall Interactive Student Text
problems, using right

PHSuccessNet (teacher)
8. Use trigonometric ratios
triangle trigonometry and

Prentice Hall Presentation Pro
and the Pythagorean
properties of right

PH Resource Pro with Planning Express
Theorem to solve right
triangles.

www.PHSchool.com
triangles in applied

CPS Jeopardy Game
problems.

TI-83/84 Graphing Calculator
 Explain and use the

Geometer’s Sketchpad
 Apply trigonometry to
relationship between the

Understanding Math software
general triangles
sine and cosine of

http://forum.swarthmore.edu/
complementary angles.

http://education.jlab.org/solquiz/index/html
9. Derive the formula A= ½
ab sin(C) for the area of a  http://www.umn.edu/
MATHEMATICS – Geometry
46
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
triangle by drawing an
auxiliary line from a vertex
perpendicular to the
opposite side.





Pulling It All Together
www.edhelper.com
http://math.rice.edu/~lanius/Lessons/
http://illuminations.nctm.org/imath/
www.knowledge.state.va.us/main/lesson/les.cfm
http://regentsprep.org
CCSS for Mathematics.–
Functions
Trigonometric Functions FTF
 Extend the domain of
trigonometric functions
using the unit circle
3. Use special triangles to
determine geometrically
the values of sine,
cosine, tangent for pi/3,
pi/4 and pi/6, and use the
unit circle to express the
values of sine, cosines,
and tangent for x, pi + x,
and 2pi – x in terms of
their values for x, where
x is any real number.
MATHEMATICS – Geometry
47
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Standard: G.8
Pulling It All Together
Triangles
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.
Key Terms/
Vocabulary
angle of depression
angle of elevation
cosine
30-60-90 triangle
isosceles right triangle
hypotenuse
45-45-90 triangle
leg
Pythagorean Theorem
Pythagorean Triples
sine
tangent
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
EduAide Access
Test Wizard test
Mathematics Projects
Student reports
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
SOL Released Test Items
MATHEMATICS – Geometry
trigonometric ratios
trigonometry
48
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.9
The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world
problems.
Understanding the Standard
Teacher Notes

The terms characteristics and properties can be used interchangeably to describe quadrilaterals. The term characteristic is used in
elementary and middle school mathematics.

Quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals.

Characteristics of quadrilaterals can be used to identify the quadrilateral and to find the measures of sides and angles.

Properties of Parallelograms
a)
b)
c)
d)

Opposite sides and opposite angles are congruent.
The diagonals bisect each other. The point of intersection is the midpoint of both diagonals.
Opposite sides and angles are congruent.
Consecutive angles are supplementary.
A quadrilateral is a parallelogram if any of the following is true.
a)
b)
c)
d)
The diagonals of the quadrilateral bisect each other.
One pair of opposite sides of the quadrilateral is both congruent and parallel.
Both pairs of opposite sides of the quadrilateral are congruent.
Both pairs of opposite angles of the quadrilateral are congruent.

The diagonals of a rhombus are perpendicular. Each diagonal of a rhombus bisects a pair of opposite angles of the rhombus.

The diagonals of a rectangle are congruent.
MATHEMATICS – Geometry
49
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together

If one diagonal of a parallelogram bisects two angles of the parallelogram, it is a rhombus. If the diagonals of a parallelogram are
perpendicular, it is a rhombus. If the diagonals of a parallelogram are congruent, them it is a rectangle.

The nonparallel sides of a trapezoid are the legs. Each pair of angles adjacent to the base of a trapezoid are base angles of the
trapezoid.

Base angles of an isosceles trapezoid are congruent. The diagonals of an isosceles trapezoid are congruent.

The diagonals of kite are perpendicular.

The segment that joins the midpoints of the nonparallel sides of a trapezoid is the midsegment of the trapezoid. It is parallel to the
bases and half as long as the sum of the lengths of the bases.
Essential Understanding
Students
All students should:

Understand that quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals.

Understand that the properties of quadrilaterals can be used to identify the quadrilateral and find the measures of sides and angles.
MATHEMATICS – Geometry
50
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.9
The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve realworld problems.
Objectives
The student will:



discover the
relationships
among angles,
sides, and
diagonals of
parallelograms.
determine the
characteristics of
quadrilaterals that
indicate they are
parallelogram.
discover the
properties of
rectangles, rhombi,
MATHEMATICS – Geometry
Essential
Knowledge, Skills
and Processes
To be successful with
this standard, students
are expected 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,
Common Core State
Standards
CCSS for MathematicsGeometry
Congruence G-CO
 Prove geometric
theorems
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.
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, pp. 18 – 20
 Prentice Hall Technology Activities, pp. 85- 87
 RPS -Teaching by Design Lesson Plan, Lesson
G.8
 Mathematics Enhanced Scope and Sequence:
Properties of Quadrilaterals, pp. 63 – 69
 Patty Paper Geometry: Investigation Set 6.1 –
6.4. pp. 87 - 93
Suggested Projects:
 Classifying Quadrilaterals
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=5747
 Constructing Quadrilaterals
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=4058
 Constructing and Investigating Properties of a
Rhombus
http://education.ti.com/educationportal
51
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
squares,
trapezoids, and
kites.
slope, and midpoint
formula.
Circles G-C
 Understand and apply
theorems about circles
 Prove the characteristics
of quadrilaterals, using
deductive reasoning,
algebraic, and
coordinate methods.
 Prove properties of
angles for a quadrilateral
inscribed in a circle.
MATHEMATICS – Geometry
3. Construct the
inscribed and
circumscribed circles
of a triangle, and
prove properties of
angles for a
quadrilateral inscribed
in a circle.
Pulling It All Together
/activityexchange/Activity.do?cid=US&aId=8178
 Exploring Quadrilaterals with Cabri Jr.
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=6215
 Pearson Prentice Hall Geometry Chapter 6:
Quadrilaterals Lesson 6-1 to 6-4
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=4153
 Pearson Prentice Hall Geometry Chapter 6:
Quadrilaterals -- Lessons 6-5 to 6-7
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=4154
Books/Materials:
 Prentice Hall Geometry, pp. 288 – 325
 Prentice Hall Study Guide and Practice
Workbook, pp. 63 - 72
 Prentice Hall Reading and Math Literacy, pp. 21 –
24
 Prentice Hall Daily Notetaking Guide, pp. 98 –
112
 Prentice Hall Geometry VA SOL Test Prep
Workbook, p.viii
 Prentice Hall Skills and Concepts Review, pp. 144
- 148
 The VA SOL Mathematics Coach, Geometry , pp.
122 - 130
 Luster, Helen, Preparing for the SOL Geometry
SOL Test, pp. 91-109
 RPS-Reteaching Lesson G.8
52
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Technology Integration:
 Prentice Hall Resource PRO CD-ROM
 Prentice Hall Presentation PRO CD
 Prentice Hall Interactive Text,
www.PHSchool.com
 Prentice Hall SuccessNet
 Prentice Hall Resource Pro with Planning Express
 Prentice Hall ExamView test generator
 TI-83/84 Graphing Calclulator
 Understanding Math software
 CPS Jeopardy Game
 Geometer’s Sketchpad
 http://www.pen.k12.va.us/VDOE/Instruction/mathrersource.html
 Eisenhower National Clearinghouse www.enc/org
 http://www.4teachers.org
 The Geometry Center http://www.umn.edu/
 The Math Forum http://forum.swarthmore.edu/
 NCTM http://illuminations.nctm.org/imath/
 http://regents.prep.org
 http://education.jlab.org/solquiz/index.html
 www.edhelper.com
 www.math.rice.edu/~lanius/Lessons/
 www.iq-poquoson.org
MATHEMATICS – Geometry
53
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.9
The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve realworld problems.
Key Terms/
Vocabulary
base
kite
square
trapezoid
kite
base angle
isosceles trapezoid
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Methods of
Assessment
PH ExamView Test Generator
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
MATHEMATICS – Geometry
parallelogram
median
midsegment
rhombus
quadrilateral
rectangle
Cooperative Grouping
Group Discussions
Student reports
EduAide Access
Test Wizard test
Mathematics Projects
SOL Released Test Items
54
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.10
The student will solve real-world problems involving angles of polygons.
Understanding the Standard
Teacher Notes

A regular polygon will tessellate the plane if the measure of an interior angle is a factor of 360.

Both regular and non-regular polygons can tessellate the plane.

Two intersecting lines form angles with specific relationships.

An exterior angle is formed by extending a side of a polygon.

The exterior angle and the corresponding interior angle form a linear pair.

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.

A polygon is a closed plane figure with at least three sides.

A polygon is convex if no diagonal contains points outside the polygon. Otherwise it is concave.

A regular polygon is equilateral and equiangular.

The sum of the measures of the interior angles of an n-gon is (n  2)180.

The sum of the measures of the exterior angles (one at each vertex) of an n-gon is 360.

The measure of an interior angle of a regular n-gon =

The measure of an exterior angle of a regular n-gon =.

Patterns that cover a plane with repeating figures so that there are no overlapping or empty spaces are called tessellations. A regular
MATHEMATICS – Geometry
55
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
tessellation uses only one type of regular polygon.

If a regular polygon has an interior angle with a measure that is a factor of 360, then the polygon will tessellate the plane.

Uniform tessellations containing two or more regular polygons are called semi-regular.
Essential Understanding
Students
All students should:

Understand that a regular polygon will tessellate the plane if the measure of an interior angle is a factor of 360.

Understand that both regular and non-regular polygons will tessellate the plane.

Two intersecting lines form angles with specific relationships.

An exterior angle is formed by extending a side of a polygon.

The exterior angle and the corresponding interior angle form a linear pair.

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 – Geometry
56
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.10
The student will solve real-world problems involving angles of polygons.
Essential
Knowledge, Skills
and Processes
Objectives
The student
will:
To be successful with
this standard, students
are expected to:




name and
identify
polygons;
find the sum
of the
measures of
interior and
exterior
angles of
convex and
regular
polygons;
solve
problems
involving
Solve real-world
problems involving the
measures of interior and
exterior angles of
polygons.

Identify tessellations in
art, construction, and
nature.

Find the sum of the
measures of the interior
and exterior angles of a
convex polygon.
MATHEMATICS – Geometry
Common Core State
Standards
CCSS for MathematicsGeometry
Modeling with Geometry GMG

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).
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, pp. 8, 9, 36
 Prentice Hall Technology Activities, p. 81
 Enhanced Scope and Sequence, Polygons, pp. 57 62
 RPS -Teaching By Design, Lesson G.8
 Serrra, Michael, Patty Paper Geometry,
Investigation Set 4.1. pp. 56 - 57
Suggested Projects:
 Angles in a Polygon
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=7428
 Geomaster: Sum of Interior Angles of Polygon
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=4043
 Got exterior angles equal to 360? with Cabri
Jr.
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=7322
 Investigating the Angle-Sum Theorem of
57
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
angle
measures of
polygons.


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.
Pulling It All Together
Polygons
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=7303
Books/Materials:
 Prentice Hall Mathematics, Geometry, pp. 131 –
138, 142 – 150, 667- 673
 Prentice Hall Study Guide and Practice Workbook,
pp. 29 – 32, 153 – 154
 Prentice Hall Reading and Math Literacy Masters,
pp. 10, 12, 47
 Prentice Hall Daily Notetaking Guide, pp. 46 – 51,
232 – 234
 Prentice Hall Skills and Concepts Review, p. 128
 Prentice Hall VA SOL Test Prep Workbook, p. viii
 VA SOL Coach, Geometry, pp. 131 – 137
 Luster, Helen, Preparing for the SOL Geometry
Test, pp. 110 - 118
 RPS Reteaching Lesson G.9
Technology Integration
 Prentice Hall Resource PRO CD-ROM
 Prentice Hall Presentation PRO CD
 Prentice Hall Interactive Text, www.PHSchool.com
 Prentice Hall SuccessNet
 CPS Jeopardy Game
 Geometer’s Sketchpad
 Understanding Math software
 http://www.pen.k12.va.us/VDOE/
Instruction/mathresource.html
MATHEMATICS – Geometry
58
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together










MATHEMATICS – Geometry
Eisenhower National Clearinghouse,
http://www.enc.org/
4teachers, http://4teachers.org
The Geometry Center, http://umn.edu/
The Math Forum, http://forum.swarthmore.edu/
http://education.jlab.org/solquiz/index.htm;
http://regentsprep.org
www.edhelper
http://math.rice.edu/~lanius/Lessons/
http://illuminations.nctm.org.imath
www.knowledge.state.va.us/main/lesson/les.cfm
59
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.10
The student will solve real-world problems involving angles of polygons.
Key Terms/
Vocabulary
concave polygon
convex polygon
tessellation
vertices
polygon
hexagon
interior angle
equiangular
quilateral
regular polygon
regular tessellation
octagon
pentagon
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
EduAide Access
Test Wizard test
Mathematics Projects
Student reports
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
SOL Released Test Items
MATHEMATICS – Geometry
exterior angle
uniform
tiling a plane
tessellation
60
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.11a-c
The student will use angles, arcs, chords, tangents, and secants.
Pulling It All Together
Understanding the Standard
Teacher Notes

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.

A tangent to a circle is a line, ray, or segment in the plane of the circle that intersects the circle in exactly one point, the point of tangency.

If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

If a line is perpendicular to the radius at its endpoint on the circle, then the line is tangent to the circle.

Two segments tangent to a circle from a point outside the circle are congruent.

In the same circle or in congruent circles:
a) congruent central angles intercept congruent arcs;
b) congruent arcs have congruent central angles;
c) congruent chords have congruent arcs;
d) congruent arcs have congruent chords;
e) chords equidistant from the center are congruent; and
MATHEMATICS – Geometry
61
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
f) congruent chords are equidistant from the center of the circle.
 A diameter that is perpendicular to a chord bisects the chord and its arc.
 The perpendicular bisector of a chord contains the center of the circle.
 The vertex of an inscribed angle lies on a circle. Its sides intercept an arc of the circle. All the vertices of an inscribed polygon lie on a
circle.
 The measure of an inscribed angle is half the measure of its intercepted arc.
 The measure of an angle formed by a chord and a tangent that intersects on a circle is half the measure of the intercepted arc.
 Two inscribed angles that intercept the same arc are congruent.
 The opposite angles of a quadrilateral inscribed in a circle are supplementary.
 The measure of an angle formed by two intersecting chords in a circle is half the sun of the measures of the intercepted arcs.
 A secant is a line, ray, or segment that intersects a circle at two points.
 The measure of an angle formed by two secants, two tangents, or a secant and a tangent from a point outside a circle is half the difference
of the measures of the intercepted arcs
 Arc length of a circle = measure of the arc divided by 360 times the product of two, pi, and the radius.
 Area of a sector of a circle = measure of the intercepted arc divided by 360 times the product of the radius squared and pi.
MATHEMATICS – Geometry
62
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Essential Understanding
Students
All students should:

Understand and apply the many relationships that exist between and among the angles, arcs, secants, chords, and tangents of a circle.

Understand that all circles are similar.

Understand that a chord is a part of a secant.

Understand that many relationships exist between and among angles, arcs, secants, chords, and tangents of a circle.

Understand that all circles are similar.

Understand that a chord is part of a secant.

Understand that real-world applications may be drawn from architecture, art, and construction.
MATHEMATICS – Geometry
63
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.11a-c
a) investigate,
verify, and
apply
properties
of circles;
The student will use angles, arcs, chords, tangents, and secants.
Essential
Knowledge, Skills
and Processes
Objectives
The student
will:
Pulling It All Together
To be successful with
this standard, students
are expected to:

MATHEMATICS – Geometry
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.
Common Core State
Standards
CCSS for MathematicsGeometry
Congruence G-CO
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, pp. 35 - 37
 Lesson by Design, Lesson G.10
 Enhanced Scope and Sequence, pp. 73 - 86
 Experiment with
Suggested Projects:
transformations in the
 Angles Formed by Intersecting Chords, Secants,
plane.
and Tangents
1. Know precise
http://education.ti.com/educationportal
definitions of angle,
/activityexchange/Activity.do?cid=US&aId=4065
circle, perpendicular

Arc length and Area of Sectors
line, parallel line, and
http://education.ti.com/educationportal
line segment, based on
/activityexchange/Activity.do?cid=US&aId=5421
the undefined
notions

Evaluating the Products of Cords of a Circle
of point, line, distance
http://education.ti.com/educationportal
along a line, and
/activityexchange/Activity.do?cid=US&aId=7377
distance around a
circular arc.
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 386 – 393,
Circles G-C
395 – 400, 582 – 589, 590 – 595, 598 – 605, 606 613
 Understand and
64
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
b) solve realworld
problems
involving
properties of
circles; and
c) find arc
lengths and
areas of
sectors in
circles.
 Solve real-world
problems associated with
circles, using properties
of angles, lines, and arcs.
apply theorems about
circles
1. Prove that all circles
are similar.
Pulling It All Together



 Calculate the area of a
sector and the length of
an arc of a circle, using
proportions.
 Verify properties of
circles, using deductive
reasoning, algebraic, and
coordinate methods.
MATHEMATICS – Geometry
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.




Prentice Hall Study Guide and Practice Workbook,
pp. 87 – 90
Prentice Hall Reading and Math Literacy, pp. 132 138
Prentice Hall Skills and Concepts Review, pp. 156,
157
Prentice Hall Notetaking Guide, pp. 198 - 210
Prentice Hall VA SOL Test Prep Workbook, p. viii
The VA SOL Mathematics Coach, Geometry pp. 138 151
Luster, Helen, Preparing for the SOL Geometry Test,
pp. 119 - 131
Technology Integration:
 Prentice Hall Resource PRO CD-ROM
 Prentice Hall Presentation PRO CD
 Prentice Hall Interactive Text, www.PHSchool.com
 Prentice Hall SuccessNet
 Prentice Hall Resource Pro with Planning Express
 CPS Jeopardy Game
 Find arc lengths and
 Graphing Calculator, TI-83/84
areas of sectors of
 Geometer’s Sketchpad
circles
 http:www.enc.org/
5. Derive using similarity  http://www.4teachers.org
the fact that the length  http:www.umn.edu/
 http://forum.swarthmore.edu
of the arc intercepted
 http://education.jlab.org/solquiz/index.html
by an angle is
 http://regentsprep.org
proportional to the
 www.edhelper
radius, and define the
 http://math.rice.edu/~lanius/Lessons/
radian measure of the
 http://illuminations.nctm.org/imath
angle as the constant
65
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
of proportionality;
derive the formula for
the area of a sector.
Pulling It All Together


www.knowledge.state.va.us/main/lesson/les.cfm
http://www.pen.k12.va.us/VDOE/
Instruction/mathresource.html
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.
MATHEMATICS – Geometry
66
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Topic:
Pulling It All Together
Polygons, Circles, and Three-Dimensional Figures
Standard: G.11a-c
The student will use angles, arcs, chords, tangents, and secants.
Key Terms/
Vocabulary
annulus
segment of circle
radius
internal tangent
external tangent
point of tangency
circumscribed
arc length
semi-circle
secant
length of an arc
inscribed angle
inscribed polygon
concentric
center
tangent
sector
major arc
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
EduAide Access
Test Wizard test
Mathematics Projects
Student reports
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
SOL Released Test Items
MATHEMATICS – Geometry
minor arc
intercepted arc
diameter
central angle
chord
67
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.12
The student, given the coordinates of the center of the circle and a point on the circle, will write the equation
of the circle.
Understanding the Standard
Teacher Notes

A circle is a locus of points equidistant from a given point, the center.

Standard form for the equation of a circle is (x – h)2 + ( y − k )2 = r2 , where the coordinates of the center of the circle are (h, k) and r
is the length of the radius.

The circle is a conic section.

The standard form equation of a circle is a way to express the definition of a circle on the coordinate plane.

h is the x-coordinate of the center of the circle.

k is the y-coordinate of the center of the circle.

r is the radius of the circle.

The center of the circle is also called the vertex.

The vertex is equal to (x, y) = (h, k).
MATHEMATICS – Geometry
68
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Essential Understanding
Students
All students should:

Understand that a circle is a locus of points equidistant from a given point, the center.

Understand that the standard form for the equation of a circle is (x − h)2 + (y − k )2 = r2 , where the coordinates of the center of the
circle are (h, k) and r is the length of the radius.

Understand that the circle is a conic section.

Understand that the standard form equation of a circle is a way to express the definition of a circle on the coordinate plane.
MATHEMATICS – Geometry
69
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.12
The student, given the coordinates of the center of the circle and a point on the circle, will write the equation of
the circle.
Essential
Knowledge, Skills
and Processes
Objectives
The student
will:

Find the
coordinates of
the center and
the length of
the radius of a
circle whose
equation is
given in the
form (x – h)2
+ (y – k)2 = r2.
To be successful with
this standard,
students are expected
to:

Identify the center,
radius, and diameter
of a circle from a
given standard
equation.

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,
MATHEMATICS – Geometry
Common Core State
Standards
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
CCSS for MathematicsGeometry
Activities
 Notes on equation of a circle.
http://www.regentsprep.org/Regents/math/algtrig/ATC1/
Expressing Geometric
circlelesson.htm
Properties with
 Practice with equation of circle.
Equations G-GPE
http://www.regentsprep.org/Regents/math/algtrig/ATC1/
circlepractice.htm
 Translate between the  Lesson on equation of circle
geometric description
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra
and the equation for a _conics_circle.xml
conic section
http://www.mathsisfun.com/algebra/circle-equations.html
http://www2.randia.net:8080/instructional/math/matha3/0351. Derive the equation of a 036%20Equation%20of%20a%20Circle%20.pdf
circle of given center
 Video on equation of circle
and radius using the
http://www.onlinemathlearning.com/equation-circle-3.html
Pythagorean Theorem;
 An applet to explore the equation of a circle and the
complete the square to
properties of the circle
find the center and
http://www.analyzemath.com/CircleEq/CircleEq.html
radius of a circle given
 Geometer’s sketchpad activity
by an equation.
http://mathforum.org/sketchpad/ckcircle.html
70
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
identify the
coordinates of the
center and find the
radius of the circle.
 Write the
equation of a
circle, given
the coordinates
of its center
and the length
of its radius.
 Use the
distance
formula to
derive the
general
equation of a
 Use coordinates to
prove simple geometric
theorems algebraically
4. Use coordinates to
 Given the
coordinates of the
prove simple geometric
endpoints of a
theorems algebraically.
diameter, find the
For example, prove or
equation of the circle.
disprove that a figure
defined by four given
points in the coordinate
 Given the
plane is a rectangle;
coordinates of the
prove or disprove that
center and a point on
the point (1,3) lies on
the circle, find the
the circle centered at
equation of the circle.
the origin and
containing the point
 Recognize that the
(0,2).
equation of a circle
of given center and
radius is derived
using the
Pythagorean
Theorem.
 Use the distance
formula to find the
radius of a circle.
MATHEMATICS – Geometry
Pulling It All Together
Suggested Projects:
 Circle Equations
http://education.ti.com/educationportal/activityexchange/
Activity.do?cid=US&aId=5736
 Match the Graph (circles)
http://education.ti.com/educationportal/activityexchange/
Activity.do?cid=US&aId=5538
Books/Materials:
 Prentice Hall Mathematics Algebra 2 pages 532-582
 Prentice Hall Mathematics Algebra 2 Daily Note Taking
guide workbook, Pages 193-207
 Prentice Hall Mathematics Algebra 2 Study Guide and
PracticeWorkbook Pages 129-140
 AMSCO Preparing for the Virginia SOL Algebra II Test
pages 146-155
Technology Integration:
 Prentice Hall Resource PRO CD-ROM
 Prentice Hall Presentation PRO CD
 Prentice Hall Interactive Text, www.PHSchool.com
 Prentice Hall SuccessNet
 Prentice Hall Resource Pro with Planning Express
 CPS Jeopardy Game
 Graphing Calculator, TI-83/84
 Geometer’s Sketchpad
 http:www.enc.org/
 http://www.4teachers.org
 http:www.umn.edu/
 http://forum.swarthmore.edu
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RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
circle whose
center is any
given point in
the plane and
whose radius
has a length r.
MATHEMATICS – Geometry







Pulling It All Together
http://education.jlab.org/solquiz/index.html
http://regentsprep.org
www.edhelper
http://math.rice.edu/~lanius/Lessons/
http://illuminations.nctm.org/imath
www.knowledge.state.va.us/main/lesson/les.cfm
http://www.pen.k12.va.us/VDOE/Instruction/
mathresource.html
72
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.12
The student, given the coordinates of the center of the circle and a point on the circle, will write the
equation of the circle.
Key Terms/
Vocabulary
conic section
circle
center
radius
diameter
vertex
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator
EduAide Access
Test Wizard test
Mathematics Projects
Student reports
MATHEMATICS – Geometry
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
SOL Released Test Items
73
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.13
The student will use formulas for surface area and volume of three-dimensional objects to solve real-world
problems.
Understanding the Standard
Teacher Notes

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.

The surface area is the sum of the area of the area of each face (including the bases).

A pyramid is a polyhedron with one base. All other faces are triangles.

A cylinder has 2 congruent circles as bases, and its lateral surface is a rectangle.

A cone is a 3-D figure with one circular base and a curved surface that connects at a point.

A prism is a solid that has 2 congruent polygonal bases that are parallel.

Volume is the amount of space inside a 3-dimensional object.

A formula sheet will be provided by the teacher for students to use and find the surface area and volume of all 3-D objects and to
solve practical problems.
Essential Understanding
Students
All students should:

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.
MATHEMATICS – Geometry
74
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.13
The student will use formulas for surface area and volume of three-dimensional objects to solve real-world
problems.
Essential
Knowledge, Skills
and Processes
Objectives
The student
will:

use formulas
for surface
area and
volume of
threedimensional
objects to
solve practical
problems.
To be successful with
this standard, students
are expected to:



MATHEMATICS – Geometry
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
Common Core
State Standards
CCSS for
MathematicsGeometry
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,
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities 29, 30, pp. 32 – 33, 34
 Prentice Hall Technology Activities 51, p. 93
 RPS-Teaching By Design, Lesson G.13
 Mathematics Scope and Sequence, pp. 99 - 105
Suggested Projects:
 Exploring Surface Area and Volume
http://ttaconline.org/staff/sol/sol_sol_lessons.asp
 Designing a Soda Can
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=3182
 Pyramid Height Exploration
http://education.ti.com/educationportal
/activityexchange/Activity.do?cid=US&aId=6055
Books/Materials:
 Prentice Hall Mathematics-Geometry, pp. 528 – 564
 Prentice Hall Study Guide & Practice Workbook, pp. 119 –
128
75
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
total surface area and
volume of cylinders,
prisms, pyramids,
cones, and spheres as
well as combinations
of three-dimensional
figures.

Use calculators
will to find
decimal
approximations
for results.

Calculators may be
used to find decimal
approximations for
results.
and cone. Use
dissection
arguments,
Cavalieri.’s
principle, and
informal limit
arguments.
3. Use volume
formulas for
cylinders, pyramids,
cones, and spheres
to solve problems.
Modeling with
Geometry G-MG
 Apply geometric
concepts in
modeling
situations






Pulling It All Together
Prentice Hall Reading & Math Literacy, pp. 38 – 40
Prentice Hall Daily Notetaking Guide, pp. 179 – 194
Prentice Hall VA SOL Test Prep, p. viii
Prentice Hall Skills & Concepts Review, pp. 172 - 176
VA SOL Geometry Coach, pp. 163 – 178
Luster, Helen, Preparing for the Geometry SOL Test, pp. 141
- 148
Technology Integration:
 TI-83/84 Graphing Calculators
 Geometer’s Sketchpad
 www.regentsprep.org
 http://www.pen.k12.va/VDOE/Instruction/mathresource.html
 http:www.jeffersonlab/SOLquiz
 http://www.smv.org/pubs/index.html
 http://tqd.advanced.org/2647/geometry/ sample2.html
 4teachers http://www.4teachers.org
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)
MATHEMATICS – Geometry
76
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
2. Apply concepts of
density based on
area and volume in
modeling situations
(e.g., persons per
square mile, BTUs
per cubic foot).
MATHEMATICS – Geometry
77
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.13
The student will use formulas for surface area and volume of three-dimensional objects to solve real-world
problems.
Key Terms/
Vocabulary
apothem
axis
base
base area
cone
cube
cylinder
sphere
triangular prism
dodecahedron
edge
great circle
hexagon
net
rectangular prism
right cone
square pyramid
vertex
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Methods of
Assessment
PH ExamView Test Generator
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
Writing Assignment
MATHEMATICS – Geometry
face
hemisphere
oblique
octahedron
rectangular solid
regular polyhedron
right cylinder
surface area
volume
icosahedrons
lateral area
polyhedron
pyramid
regular pyramid
right prism
slant height
tetrahedron
Cooperative Grouping
Group Discussions
Student reports
EduAide Access
Test Wizard test
Mathematics Projects
SOL Released Test Items
78
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.14a-d
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;
d) solve real-world problems about similar geometric objects.
Understanding the Standard
Teacher Notes

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.

Ratios are a way of comparing two numbers, quantities or variables.

A proportion is a statement that two ratios are equal.

A scale is a ratio that compares the drawing measures to real world measures.

The scale factor of similar figures is the size of the change from the original figure.

Use proportions to compare perimeters, areas, and volumes of similar two-dimensional figures.

Use proportions to compare surface area and volumes of three-dimensional geometric figures.

Describe how a change in one measure affects other measures of an object.

When two figures are similar, the ratio of the areas is equal to the ratio of any two corresponding lengths squared and the ratio of
the volumes is equal to the ratio of any two corresponding lengths cubed.
MATHEMATICS – Geometry
79
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Essential Understanding
Students
All students should understand that:

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.
MATHEMATICS – Geometry
80
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.14a-d
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;
d) solve real-world problems about similar geometric objects.
Essential
Knowledge, Skills
and Processes
Objectives
The student
will:
a) compare ratios
between side
lengths,
perimeters,
areas, and
volumes;
b) determine
how changes
in one or more
dimensions of
an object
To be successful with
this standard, students
are expected to:


Compare ratios between
side lengths, perimeters,
areas, and volumes,
given two similar
figures.
Common Core State
Standards
CCSS for MathematicsGeometry
Similarity, Right
Triangles, and
Trigonometry G-SRT
 Understand similarity in
terms of similarity
transformations
Resources
Suggested Activities - Books & Materials
Technology Integration - Field Trips
Activities:
 Prentice Hall Hands-On Activities, pp. 24, 26, 27
 Prentice Hall Technology Activities 50, p. 91
 RPS-Teaching By Design, Lesson G.14
Suggested Projects:
 Polygons
 http://ttaconline.org/staff/sol/sol_sol_lessons.asp
 Gulliver’s Travels and Proportional Reasoning
 http://ttaconline.org/staff/sol/sol_sol_lessons.asp
2. Given two figures, use the
definition of similarity in
Books/Materials:
terms of similarity
 Prentice Hall Mathematics-Geometry, pp. 415 – 459,
Describe how changes in
transformations to decide
565 – 571
one or more dimensions
if they are similar; explain  Prentice Hall Study Guide & Practice Workbook, pp. 93
affect other derived
using similarity
– 103, 129 – 130
measures (perimeter,
transformations the
 Prentice Hall Reading & Math Literacy, pp. 29 – 32
area, total surface area,
meaning of similarity for
MATHEMATICS – Geometry
81
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
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 realworld
problems
about similar
geometric
objects.
and volume) of an
object.

triangles as the equality of
all corresponding pairs of
angles and the
proportionality of all
corresponding pairs of
sides.
Describe how changes in  Prove theorems
one or more measures
involving similarity
(perimeter, area, total
surface area, and
5. Use congruence and
volume) affect other
similarity criteria for
measures of an object.
triangles to solve
problems and to prove
relationships in geometric
figures.
Geometric Measurement
and Dimension G-GMD

Solve real-world
problems involving
measured attributes of
similar objects.
 Explain volume formulas
and use them to solve
problems





Pulling It All Together
Prentice Hall Daily Notetaking Guide, pp. 142 – 158
Prentice Hall VA SOL Test Prep Workbook, p.viii
Prentice Hall Skills & Concepts Review, pp. 159, 164
RPS- Reteaching Lessons G.14
SOL Geometry Coach, pp.180 - 189
Technology Integration:
 Prentice Hall Resource PRO CD-ROM
 Prentice Hall Presentation PRO CD
 Prentice Hall Interactive Text (student)
 www.PHSchool.com
 Prentice Hall SuccessNet
 Prentice Hall Computer Test Generator
 Prentice Hall Resource Pro with Planning Express
 TI-83/84 Graphing Calculator
 CPS Jeopardy Game
 Geometer’s Sketchpad
 www.regents.prep.org
 http://www.smv.org/pubs/index.html
 http://tqd.advanced.org/2647/geometry/ sample2.html
 http://www.pen.k12.va.us/VDOE/Instruction/mathresource.html
 4teachershttp://www.4teachers.org
2. Give an informal
argument using
Cavalieri.’s principle for
the formulas for the
volume of a sphere and
other� solid figures.
Modeling with Geometry
MATHEMATICS – Geometry
82
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
G-MG
 Apply geometric
concepts in modeling
situations
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).
MATHEMATICS – Geometry
83
RICHMOND PUBLIC SCHOOLS
CURRICULUM COMPASS
MATHEMATICS
Pulling It All Together
Topic:
Polygons, Circles, and Three-Dimensional Figures
Standard: G.14a-d
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;
d) solve real-world problems about similar geometric objects.
Key Terms/
Vocabulary
perimeter
cross product
proportion
total surface area
area
similar
ratio
scale factor
scale
extremes
means
Methods of
Instruction
Guided practice
Peer Tutoring
Direct Instruction
Small Group
Cooperative Grouping
Group Discussions
Methods of
Assessment
PH ExamView Test Generator Student reports
EduAide Access
Test Wizard test
Mathematics Projects
Writing Assignment
MATHEMATICS – Geometry
volume
exchange means
reciprocal
Teacher created assessment
Lesson By Design Assessment
Textbook Resource Kit activities
SOL Released Test Items
84
A Publication of Richmond Public Schools
Richmond, Virginia
In accordance with federal laws, the laws of the Commonwealth of Virginia and the policies of the School Board of the City of
Richmond, the Richmond Public Schools does not discriminate on the basis of sex, race, color, age, religion, disabilities or
national origin in the provision of employment and services. The Richmond Public Schools operates equal opportunity and
affirmative action programs for students and staff. The Richmond Public Schools is an equal opportunity/affirmative action
employer. The Title IX Officer is Ms. Angela C. Lewis, Clerk of the School Board, 301 North 9th St., Richmond, VA 232191927, (804) 780-7716. The Section 504 Coordinator is Mrs. Michelle Boyd, Director of Exceptional Education and Student
Services, 301 North 9th St., Richmond, VA, 23219-1927, (804) 780-7911. The ADA Coordinator is Ms. Valarie Abbott Jones,
2015 Seddon Way, Richmond, VA 23230-4117, (804) 780-6211. The United States Department of Education’s Office of
Civil Rights may also be contacted at 550 12th Street SW, PCP-6093 Washington, DC 20202, (202) 245-6700.
School Board
Dawn C. Page, Chair
Maurice A. Henderson, Vice Chair
Kimberly M. Bridges
Kimberly B. Gray
Norma H. Murdoch-Kitt
Adria A. Graham Scott
Chandra H. Smith
Donald L. Coleman
Evette L. Wilson
Dr. Yvonne W. Brandon, Superintendent