hcps geometry curriculum guide
TRANSCRIPT
Geometry Pacing Guide and Curriculum Reference Based on the 2009 Virginia Standards of Learning
2016-2017
Henrico County Public Schools
Henrico Curriculum Framework Geometry
Henrico County Public Schools Page 1 of 72
Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessments. It defines the content knowledge, skills, and
understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program
appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and
describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should
learn.
The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for
each objective. The Curriculum Guide is divided by unit and ordered to match the established HCPS pacing. Each unit is divided into two parts: a one page unit
overview and a Teacher Notes and Resource section. The unit overview contains the suggested lessons for the unit and all the DOE curriculum framework
information including the related SOL(s), strands, Essential Knowledge and Skills, and Essential Understandings. The Teacher Notes and Resource section is
divided by Resources, Key Vocabulary, Essential Questions, Teacher Notes and Elaborations, Honors/AP Extensions, and Sample Instructional Strategies and
Activities. The purpose of each section is explained below.
Vertical Articulation: This section includes the foundational objectives and the future objectives correlated to each SOL.
Unit Overview:
Curriculum Information: This section includes the SOL and SOL Reporting Category, focus or topic, and pacing guidelines.
Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined.
This is not meant to be an exhaustive list nor is a list that limits what taught in the classroom. This section is helpful to teachers when planning classroom
assessments as it is a guide to the knowledge and skills that define the objective. (Taken from the Curriculum Framework)
Essential Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an
understanding of the objectives. (Taken from the Curriculum Framework)
Teacher Notes and Resources:
Resources: This section gives textbook resources, links to related Investigating Geometry Online (IGO) modules, and links to VDOE’s Enhanced Scope
and Sequence lessons.
Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and
skills.
Essential Questions: This section explains what is meant to be the key knowledge and skills that define the standard.
Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this
objective and may extend the teachers’ knowledge of the objective beyond the current grade level.
Extensions: This section provides content and suggestions to differentiate for honors/Pre-AP level classes.
Sample Instructional Strategies and Activities: This section provides suggestions for varying instructional techniques within the classroom.
Special thanks to Prince William County Public Schools for allowing information from their curriculum documents to be included in this document.
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Geometry Pacing and Curriculum Guide
Course Outline First Marking Period at a Glance
Unit 1: Transformations (G.3cd)
Unit 2: Fundamentals (G.4abef)
Unit 3: Logic (G.1abcd)
Unit 4: Parallel and Perpendicular
Lines (G.2abc, G3ab)
Second Marking Period at a Glance
Unit 5: Triangle Fundamentals
(G.4, G.5, G.10)
Unit 6: Proofs with Congruent
Triangles (G.3a, G.6)
Unit 7: Similarity (G.7, G.14a)
Unit 8: Right Triangles (G.3a, G.8)
Third Marking Period at a Glance
Unit 9: Polygons (G.4, G.10)
Unit 10: Quadrilaterals (G.3a, G.9)
Unit 11: Circles (G.11ac)
Unit 12: Equation of Circles (G.12)
Fourth Marking Period at a Glance
Unit 13: Surface Area and
Volume (G.13, G.14bc)
SOL Review
View the online HCPS Pacing Guide for more details
Big Ideas
1. Logic & Reasoning 2. Parallel Lines 3. Properties of Triangles 4. Polygons 5. Proportional Reasoning
6. Quadrilaterals 7. Right Triangles 8. Circles 9. Surface Area & Volume 10. Coordinate Geometry
GEOMETRY SOL TEST BLUEPRINT (50 QUESTIONS TOTAL)
Resources
Text: Glencoe Geometry: Integration, Applications,
Connections, 1998, Glencoe McGraw-Hill HCPS Mathematics Website
http://blogs.henrico.k12.va.us/math/ HCPS Geometry Online
http://teachers.henrico.k12.va.us/math/HCPSgeo/
Reasoning, Lines, and Transformations 18 questions 36% of the Test
Triangles 14 questions 28% of the Test
Polygons, Circles, and Three-Dimensional Figures 18 questions 36% of the Test
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Virginia Department of Education Mathematics SOL Resources http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml
DOE Enhances Scope and Sequence Lesson Plans http://www.doe.virginia.gov/testing/sol/scope_sequence/mathematics_2009/
SOL Vertical Articulation Previous Standards Geometry Standard Future Standards
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.
Under Construction
A.6 graph linear equations/linear inequal (2 vars) ‐ a) determine slope of
line given equation of line/graph of line or two points on line ‐ slope as
rate of change; b) write equation of line given graph of line/two points
on line or slope‐point on line
8.6 a) verify/describe relationships among vertical/adjacent/
supplementary/complementary angles;
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.
A.6 graph linear equations/linear inequal (2 vars) ‐ a) determine slope of
line given equation of line/graph of line or two points on line ‐ slope as
rate of change; b) write equation of line given graph of line/two points
on line or slope‐point on line
8.8 a) apply transformations to plane figures; b) ID applications of
transformations
7.8 represent transformations (reflections, dilations, rotations, and
translations) of polygons in the coordinate plane by graphing
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; & d)
determining whether a figure has been translated, reflected, rotated, or
dilated, using coordinate methods.
6.12 determine congruence of segments/angles/polygons 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.5 The student, given information concerning the lengths of sides
and/or measures of angles in triangles, will a) order the sides by length,
given the angle measures; b) order the angles by degree measure, given
the side lengths; c) determine whether a triangle exists; & 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.
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7.6 determine similarity of plane figures and write proportions to express
relationships between similar quads and triangles
6.12 determine congruence of segments/angles/polygons
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.
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.
A.3 express sq roots/cube roots of whole numbers/the square root of
monomial alg exp (simplest radical form)
8.10 a) verify the Pythagorean Theorem; b) apply the Pythagorean
Theorem
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.
A.4 solve multistep linear/ quad equation (2 vars) ‐ a) solve literal
equation; b) justify steps used in simplifying expresessions and solving
equations; c) solve quad equations (alg/graph); d) solve multistep linear
equations (alg/graph)
7.7 compare/contrast quadrilaterals based on properties
6.13 ID/describe properties of quadrilaterals
G.9 The student will verify characteristics of quadrilaterals and use
properties of quadrilaterals to solve real‐world problems.
A.4 solve multistep linear/ quad equation (2 vars) ‐ a) solve literal
equation; b) justify steps used in simplifying expresessions and solving
equations; c) solve quad equations (alg/graph); d) solve multistep linear
equations (alg/graph)
6.12 determine congruence of segments/angles/polygons
G.10 The student will solve real‐world problems involving angles of
polygons.
6.10 a) define π; b) solve practical problems w/circumference/area of
circle; c) solve practical problems involving area and perimeter given
radius/diameter; d) describe/determine volume/surface area of
rectangular prism
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.
6.10 a) define π; b) solve practical problems w/ circumference/area of
circle; c) solve practical problems involving area and perimeter given
radius/diameter; d)describe/determine volume/surface area of
rectangular prism
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.
8.7 a) investigate/solve practical problems involving volume/surface area
of prisms, cylinders, cones, pyramids; b) describe how changes in
measured attribute affects volume/surface area
8.9 construct a 3‐D model given top or bottom/side/front views
7.5 a) describe volume/surface area of cylinders; b) solve practical
problems involving volume/surface area of rect. prims and cylinders; c)
describe how changes in measured attribute affects volume/surface area
G.13 The student will use formulas for surface area and volume of three
dimensional objects to solve real‐world problems.
8.7 a) investigate/solve practical problems involving volume/surface area
of prisms, cylinders, cones, pyramids; b) describe how changes in
measured attribute affects volume/ surface area
7.5 a) describe volume/surface area of cylinders; b) solve practical
problems involving volume/surface area of rect. prims and cylinders; c)
describe how changes in measured attribute affects volume/surface area
G.14 The student will use similar geometric objects in two‐ or three
dimensions to a) compare ratios between side lengths, perimeters, areas,
and volumes; b) determine how changes in one or more dimensions of an
object affect area and/or volume of the object; c) determine how changes
in area and/or volume of an object affect one or more dimensions of the
object; and d) solve real‐world problems about similar geometric objects.
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Fundamentals
Fir
st M
ark
ing
Peri
od
Lessons
Point, Line, Plane
Segments and Rays
Angles
Pairs of Angles
Strand:
Grade 4, 5, 6: Geometry
SOL 4.10 The student will
a) identify and describe representations of
points, lines, line segments, rays, and
angles, including endpoints and vertices
5.12 The student will classify
a) angles as right, acute, obtuse, or
straight
6.12 The student will determine
congruence of segments, angles, and
polygons.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Grade 4
Identify and describe representations of points,
lines, line segments, rays, and angles,
including endpoints and vertices.
Grade 5
Classify angles as right, acute, straight, or
obtuse.
Grade 6
Determine the congruence of segments,
angles, and polygons given their attributes.
Essential Understanding All students should
Grade 4
Understand that points, lines, line segments,
rays, and angles, including endpoints and
vertices are fundamental components of
noncircular geometric figures.
Understand that the shortest distance between
two points on a flat surface is a line segment.
Grade 5
Understand that angles can be classified as
right, acute, obtuse, or straight according
to their measures.
Grade 6
Given two congruent figures, what
inferences can be drawn about how the
figures are related? The congruent figures
will have exactly the same size and shape.
(continued)
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Fundamentals (continued) Resources
HCPS Geometry Online:
Unit 1 - Fundamental Concepts
Textbook: 1-2 Points, Lines, and Planes
1-4 Measuring Segments
1-6 Exploring Angles
1-7 Angle Relationships
Key Vocabulary
acute
angle
bisect
collinear
complementary
coplanar
horizontal
intersect
line
obtuse
plane
point
postulate
ray
right [angle]
segment
supplementary
theorem
vertex
vertical
Essential Questions
How are the concepts of points, lines, line segments, rays, angles,
endpoints, and vertices important when describing and comparing
geometric figures?
Where can we find points, lines, line segments, rays, and angles in
the world around us?
How can a set of intersecting lines be used to demonstrate the
relationship between and among points, lines, line segments, rays,
angles, and geometric figures?
How can visualizing a circle folded into halves and quarters help
Return to Course Outline
us classify angles?
Given two congruent figures, what inferences can be drawn about
how the figures are related?
Teacher Notes and Elaborations
A point is a location in space. It has no length, width, or height. A
point is usually named with a capital letter.
A line is a collection of points going on and on infinitely in both
directions. It has no endpoints. When a line is drawn, at least two
points on it can be marked and given capital letter names. Arrows
must be drawn to show that the line goes on in both directions
infinitely (e.g., AB , read as “the line AB”).
A line segment is part of a line. It has two endpoints and includes all
the points between those endpoints. To name a line segment,
name the endpoints (e.g., AB , read as “the line segment AB”).
A ray is part of a line. It has one endpoint and continues infinitely in
one direction. To name a ray, say the name of its endpoint first
and then say the name of one other point on the ray (e.g., AB ,
read as “the ray AB”).
Two rays that have the same endpoint form an angle. This endpoint is
called the vertex. Angles are found wherever lines and line
segments intersect. An angle can be named in three different ways
by using
- three letters to name, in this order, a point on one ray, the
vertex, and a point on the other ray;
- one letter at the vertex; or
- a number written inside the rays of the angle.
Intersecting lines have one point in common. (continued)
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Fundamentals (continued) Teacher Notes and Elaborations (continued)
Congruent figures are figures having exactly the same size and shape.
Opportunities for exploring figures that are congruent and/or
noncongruent can best be accomplished by using physical
models.
A right angle measures exactly 90.
An acute angle measures greater than 0 but less than 90.
An obtuse angle measures greater than 90 but less than 180.
A straight angle forms an angle that measures exactly 180°.
Congruent figures have exactly the same size and the same shape.
Noncongruent figures may have the same shape but not the same size.
The symbol for congruency is .
The determination of the congruence or noncongruence of two
figures can be accomplished by placing one figure on top of the
other or by comparing the measurements of all sides and angles.
Construction of congruent line segments, angles, and polygons helps
students understand congruency.
Return to Course Outline
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Logic
Fir
st M
ark
ing
Peri
od
Lessons
Conditional Statements
Logic
Strand: Reasoning, Lines and
Transformations
SOL G.1 The student will construct and
judge the validity of a logical argument
consisting of a set of premises and a
conclusion. This will include
a) identifying the converse, inverse, and
contrapositive of a conditional
statement;
b) translating a short verbal argument
into symbolic form;
c) using Venn diagrams to represent set
relationships; and
d) using deductive reasoning.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Identify the converse, inverse, and
contrapositive of a conditional statement.
Translate verbal arguments into symbolic form
such as (p → q) and (~p → ~q).
Determine the validity of a logical argument.
Use valid forms of deductive reasoning,
including the law of syllogism, the law of the
contrapositive, the law of detachment, and
counterexamples.
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 , , ~, , ᴧ and ᴠ.
Essential Understanding 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.
Logical arguments consist of a set of premises
or hypotheses and a conclusion.
Proof is a justification that is logically valid
and based on initial assumptions, definitions,
postulates, and theorems.
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.
(continued)
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Logic (continued) Resources
HCPS Geometry Online:
Reasoning
Textbook: 2-2 If-Then Statements and Postulates
2-3 Deductive Reasoning
DOE ESS Lesson Plans:
Lines and Angles (PDF) (Word)
Key Vocabulary biconditional statement
conclusion
conditional statement
conjecture
contrapositive
converse
counterexample
deductive reasoning
hypothesis (premise)
inductive reasoning
inverse
Law of Detachment
Law of Syllogism
Law of the
Contrapositive
postulate (axiom)
proof
symbolic form
Venn diagram
Return to Course Outline
Essential Questions
When is a statement a lie?
What is the importance or need for symbolic representation of
words?
What does it mean to be logical?
How can logic be represented visually?
What is the relationship between reasoning, justification, and
proof in geometry?
What is a truth-value?
How does a truth-value apply to conditional statements?
How do deductive reasoning and Venn diagrams help judge the
validity of logical arguments?
Teacher Notes and Elaborations
Logic is the study of the principles of reasoning. Logical arguments
consist of a set of premises (hypotheses) and a conclusion (the last
step in a reasoning process).
Terms associated with logical arguments are reasoning, justification,
and proof. Reasoning is the drawing of conclusions or inferences from
facts, observations, or hypotheses. Justification is a rationale or
argument for some mathematical proposition. A conjecture is a
statement that has not been proved true nor shown to be false. A proof
is a justification that is logically valid and based on initial
assumptions, definitions, and proven results. A theorem is a statement
that can be proved and a postulate or axiom is an assumption (a
statement taken for granted) that is accepted without proof. A
justification may be less formal than a proof. It may consist of a set of
examples that seem to support the proposition or it may be an
intuitive argument. The three concepts are related in that reasoning is
used to seek a justification of a proposition, which, if possible, is
turned into a proof. (continued)
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Logic (continued) Teacher Notes and Elaborations (continued)
Communication of reasoning and/or justification to complete a proof
can be shown through symbolic form (truth tables or Venn diagrams)
or written form (paragraph, indirect, two-column or coordinate
method).
An if-then statement is called a conditional statement or simply a
conditional. A conditional statement includes an initial condition or
hypothesis (premise) and its corresponding outcome (conclusion). The
conditional statement is written in the if (hypothesis) – then
(conclusion) form.
q
If p (hypothesis), then q (conclusion). p
The converse (a proposition produced by reversing position or order)
of the conditional statement is formed by interchanging the hypothesis
and its conclusion.
p
If q (conclusion), then p (hypothesis).
q
The inverse of the conditional statement is formed by negating both
the hypothesis and the conclusion.
If not p (hypothesis), then not q (conclusion).
The contrapositive of the conditional statement is formed by
interchanging and negating both the hypothesis and the conclusion.
If not q (conclusion), then not p (hypothesis).
The contrapositive and original conditional statements are logically
equivalent (Law of Contrapositive).
Return to Course Outline
Symbolic form includes truth tables (tabular representation of the
truth or falsehood of hypotheses and conclusions) and Venn diagrams.
Deductive reasoning uses rules to make conclusions. Applying the
Law of Detachment, if you accept “If p then q” as true and you accept
p as true, then you must logically accept q as true. It also follows if
you accept “If p then q” as true and you accept not q as true, then you
must logically accept not p as true. According to the Law of
Syllogism, if you accept “If p then q” as true and if you accept “If q
then r” as true, then you must logically accept “If p then r” as true. A
counterexample is an example used to prove an if-then statement
false. For that counterexample, the hypothesis is true and the
conclusion is false.
Inductive reasoning is a kind of reasoning in which the conclusion is
based on several past observations.
Symbolically means “therefore”. Ex: m ABC is 90° m ABCis a right angle.
In logic, letters are used to represent simple statements that are either
true or false. Simple statements can be joined to form compound
statements. A conjunction is a compound statement composed of two
simple statements joined by the word “and”. The symbol , is used to
represent the word “and”. A disjunction is a compound statement x
(continued)
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Logic (continued)
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Extension for PreAP Geometry
Identify, create, and determine the truth-value of the converse,
inverse, and contrapositive of a conditional statement.
Use chain reasoning to make a logical conclusion given a set of
statements.
Identify logically equivalent statements.
Construct truth tables given statements (conditional, conjunction,
disjunction, biconditionals, etc.).
Investigate the concept of an indirect proof.
Extension for PreAP Geometry
The truth value of a statement is either true or false. A truth table can
be used to determine the conditions under which a statement is true.
Truth Tables:
Conditional
If p then q
p q p q
T T T
T F F
F T T
F F T
Conjunction
p and q
p q p q
T T T
T F F
F T F
F F F
Return to Course Outline
Disjunction
p or q p q p q
T T T
T F T
F T T
F F F
An indirect proof is a proof that begins by assuming temporarily that
the conclusion is not true; then reason logically until a contradiction
of the hypothesis or another known fact is reached.
Sample Instructional Strategies and Activities
Students, working in cooperative learning groups, will solve logic
problems to introduce the concept of deductive reasoning. Each
group of students will give their solutions and describe their
thought processes.
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Lines
Fir
st M
ark
ing
Peri
od
Lessons
Pairs of Lines
Angles and Parallel Lines
Proving Lines Parallel
Strand: Reasoning, Lines and
Transformations
SOL 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.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Use algebraic and coordinate methods as well
as deductive proofs to verify whether two lines
are parallel.
Solve problems by using the relationships
between pairs of angles formed by the
intersection of two parallel lines and a
transversal including corresponding angles,
alternate interior angles, alternate exterior
angles, and same-side (consecutive) interior
angles.
Solve real-world problems involving
intersecting and parallel lines in a plane.
Essential Understanding 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.
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.
(continued)
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Lines (continued) Resources
HCPS Geometry Online:
Lines
Textbook: 3-1 Parallel Lines and Transversals
3-2 Angles and Parallel Lines
3-3 Slopes of Lines
3-4 Proving Lines Parallel
DOE ESS Lesson Plans:
Lines and Angles (PDF) (Word)
Key Vocabulary adjacent angles
algebraic method
alternate exterior
angles
alternate interior angles
complementary angles
consecutive (same-
side) interior angles
coordinate method
corresponding angles
deductive reasoning
inductive reasoning
exterior angle
interior angle
linear pair
parallel
perpendicular
skew
supplementary angles
transversal
union
vertical angles
Return to Course Outline
Essential Questions
What is the relationship between lines and angles?
What is the difference between parallel lines and perpendicular
lines?
How are lines proven parallel?
What is the difference between parallel lines and intersecting
lines?
What are the relationships between the angles formed when two
parallel lines are cut by a transversal?
Teacher Notes and Elaborations Euclidean Geometry is a
mathematical system attributed to the Alexandrian Greek
mathematician Euclid, whose elements is the earliest known
systematic discussion of geometry. Euclid's method consists in
assuming a small set of intuitively appealing axioms, and deducing
many other theorems (propositions) from these.
Angles with the same measure are congruent angles. Adjacent angles
are two angles that share a common side and have the same vertex,
but have no interior points in common. Vertical angles are two angles
whose sides form two pairs of opposite rays. When two lines
intersect, they form two pairs of vertical angles.
When two lines intersect, two types of angle pairs are formed: vertical
angles and adjacent supplementary angles. Vertical angles are
congruent and two adjacent angles are supplementary.
Parallel lines are lines that are in the same plane (coplanar) and never
intersect because they are always the same distance apart. They have
no points in common. The symbol || indicates parallel lines. Skew
lines do not intersect and are not coplanar.
(continued)
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Lines (continued) Teacher Notes and Elaborations G.2
Intersection is a point or set of points common to two or more figures.
A transversal is a line that intersects two or more coplanar lines in
different points forming eight angles. Interior angles lie between the
two lines. Alternate interior angles are on opposite sides of the
transversal. Consecutive interior angles are on the same side of the
transversal. Exterior angles lie outside the two lines. Alternate
exterior angles are on opposite sides of the transversal.
Corresponding angles are nonadjacent angles located on the same
side of the transversal where one angle is an interior angle and the
other is an exterior angle.
If the sum of the measures of two angles is 180°, then the two angles
are supplementary. If the two angles are adjacent and supplementary
then they are a linear pair.
If the sum of the measures of two angles is 90°, then the two angles
are complementary. If the two angles are adjacent and complementary
then they form a right angle.
If two lines in a plane are cut by a transversal, the lines are parallel if:
- alternate interior angles are congruent,
- alternate exterior angles are congruent,
- corresponding angles are congruent,
- same side (consecutive) interior angles are supplementary.
Proving lines parallel implies determining whether necessary and
sufficient conditions (properties, definitions, postulates, and
theorems) exist for parallelism. A proof is a chain of logical
statements starting with given information and leading to a
conclusion.
Return to Course Outline
Two column deductive proofs (formal proofs) are examples of
deductive reasoning. They contain statements and reasons organized
in two columns. Each step is called a statement, and the properties
that justify each step are called reasons.
Essential parts of a good proof include:
1. state the theorem or conjecture to be proven;
2. list the given information;
3. if possible, draw a diagram to illustrate the given
information;
4. state what is to be proved; and
5. develop a system of deductive reasoning.
The Parallel Postulate is the axiom of Euclidean Geometry stating
that if two straight lines are cut by a third, the two will meet on the
side of the third on which the sum of the interior angles is less than
two right angles. Equivalently, Playfair’s Axiom states: “If given a
line and a point not on the line, then there exists exactly one line
through the point that is parallel to the given line.” In Euclidean
Geometry, parallel lines lie in the same plane and never intersect. In
spherical geometry, the sphere is the plane, and a great circle
represents a line.
Two nonvertical coplanar lines are parallel if and only if their slopes
are equal. Two nonvertical coplanar lines are perpendicular if and
only if the product of their slopes is 1 .
Algebraic and coordinate methods should also be used to determine
parallelism. Coordinate geometry establishes a correspondence
between algebraic concepts and geometric concepts. For example, the
distance formula is derived as an application of the Pythagorean
Theorem. The Pythagorean Theorem in turn is used to develop the
equation of a circle. The coordinate proof is often more convenient
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Lines (continued) Teacher Notes and Elaborations G.2 (continued)
than a two-column proof. The following is an example of a
coordinate proof involving parallelism.
Prove: The segment that joins the midpoint of two sides of a
triangle is parallel to the third side.
Given: OAB and M and N the midpoints of OB and OA
respectively.
Prove: MN || BA
Proof: Choose axes and coordinates as shown.
y
B (2 ,2 )b c
M
O N A (2 ,0)a x
1. Midpoints are 2 0 2 0 2 2
M( , ) ( , ) ( , )2 2 2 2
b c b cb c
and
2 0 0 0 2 0N( , ) ( , ) ( ,0)
2 2 2 2
a aa
; by Midpoint Formula.
2. Slope of 0
MNc c
a b a b
and the slope of
0 2 2BA
2 2 2( )
c c c
a b a b a b
; by definition of slope.
3. Slope of MN = slope of BA ; by Substitution Property. MN || BA ;
two nonvertical lines are parallel if and only if their slopes are
equal.
Return to Course Outline
Extension for PreAP Geometry
Skew lines are non-coplanar lines that do not intersect. Experiences
with skew lines should include 3-dimensional models.
Extension for PreAP Geometry Use algebraic, coordinate, and deductive methods to determine if
lines are perpendicular.
Write equations of parallel and perpendicular lines.
Investigate skew lines using real world models.
Use definitions, postulates, and theorems to complete two-column
or paragraph proofs with at least five steps.
Extension for PreAP Geometry In a paragraph proof (informal proof) a paragraph is written to explain
why a conjecture for a given situation is true.
Sample Instructional Strategies and Activities
Have students pick two lines on notebook paper. Use straight edge
and pencil to darken lines chosen. Using a straight edge, draw a
transversal. Label angles. Have students accurately measure pairs
of special angles using a protractor. Perform the same procedures
with two non-parallel lines cut by a transversal. Write conjectures
for each special angle pair (corresponding, consecutive interior,
alternate interior, and alternate exterior).
Use patty paper to trace and compare lines and angles.
(continued)
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Lines (continued) Sample Instructional Strategies and Activities
Have class look for parallel, intersecting, perpendicular, and skew
lines in the classroom. In groups, students list as many pairs of
them as they can find in ten minutes. Each group gives some
examples from their list. This can be used as a competition.
o Have students pick two lines on notebook paper. Use
straight edge and pencil to darken lines chosen. Using a
straight edge, sketch a transversal. Label angles. Have
students accurately measure pairs of special angles. Use
the same procedure with two non-parallel lines cut by a
transversal. Write conjectures for each special angle pair
(corresponding, consecutive interior, alternate interior, and
alternate exterior).
o Take class outside to look for parallel, intersecting,
perpendicular, and skew lines and for identified angles. In
groups, students list as many pairs of them as they can find
in ten minutes. After returning to the classroom, each
group gives some examples from their list. This can be
used as a competition.
Have students use patty paper to discover congruent angles
formed when parallel lines are cut by a transversal.
Have students build an angle log book. Students will draw
pictures of various angles and label the angle. Students will relate
the angle to an object in the room.
Return to Course Outline
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Constructions
Fir
st M
ark
ing
Peri
od
Lessons
Constructions
Strand: Reasoning, Lines, and
Transformations
SOL 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.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Construct and justify the constructions of
- a line segment congruent to a given line
segment;
- the perpendicular bisector of a line
segment;
- a perpendicular to a given line from a
point not on the line;
- a perpendicular to a given line at a point
on the line;
- the bisector of a given angle;
- an angle congruent to a given angle; and
- a line parallel to a given line through a
point not on the given line.
Construct an equilateral triangle, a square, and
a regular hexagon inscribed in a circle.
Construct the inscribed and circumscribed
circles of a triangle.
Construct a tangent line from a point outside a
given circle to the circle.
Essential Understanding
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.
(continued)
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Constructions (continued) Resources
HCPS Geometry Online:
Constructions
Textbook: pg. 31 A line segment congruent to a given line segment
pg. 39 The perpendicular bisector of a line segment
pg. 56 A perpendicular to a given line from a point not on the line
pg. 56 A perpendicular to a given line at a point on the line
pg. 48 The bisector of a given angle
pg. 47 An angle congruent to a given angle
pg. 146 A line parallel to a given line through a point not on the
given line
DOE ESS Lesson Plans:
Constructions (PDF) (Word)
Key Vocabulary
bisector
centroid
circumcenter
circumscribed
compass
congruence
construction
incenter
inscribed
intersection
parallel lines
perpendicular bisector
perpendicular lines
bisector
straightedge
transversal
Essential Questions
What is the relationship between points, rays, and angles?
Why are constructions important?
How are constructions justified?
Return to Course Outline
Teacher Notes and Elaborations
"Construction" in geometry means to draw shapes, angles or lines
accurately. Constructions are done using tools including software
programs such as Sketchpad, Geogebra, patty paper, a straightedge,
and a compass. If students are using a ruler as a straightedge, they
should be instructed to ignore its markings. Constructions help build
an understanding of the relationships between lines and angles. The
seven basic constructions can be used to do more complicated
constructions such as points of concurrency: centroid, incenter,
circumcenter and orthocenter.
The intersection of two figures is the set of points that is in both
figures.
A transversal is a line that intersects two or more coplanar lines in
different points.
Two angles are congruent if and only if they have equal measures. A
ray is an angle bisector if and only if it divides the angle into two
congruent adjacent angles.
Parallel lines are lines that do not intersect and are coplanar.
Perpendicular lines are lines that intersect at right angles. A segment
bisector is a line, segment, ray, or plane that intersects the segment at
its midpoint. A perpendicular bisector of a segment is a line, ray, or
segment that is perpendicular to the segment at its midpoint.
A circle is circumscribed about a triangle if the circle contains all the
vertices of the triangle. A triangle is inscribed in a circle if each of its
vertices lies on the circle.
In a triangle, a median is a segment that joins a vertex of the triangle (continued)
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Constructions (continued) Teacher Notes and Elaborations (continued) and the midpoint of the side opposite that vertex. The medians of a
triangle intersect at the balance point called the centroid.
To circumscribe a circle about a triangle, construct the perpendicular
bisectors of each side. The point where these perpendicular bisectors
meet is the circumcenter. Using the circumcenter and any vertex of
the triangle as the radius, construct the circle about the triangle.
To construct a circle inscribed inside a triangle, construct the angle
bisectors. The incenter is the point where the angle bisectors meet.
Construct a perpendicular from the incenter to one of the sides of the
triangle. This perpendicular segment is the radius of the inscribed
circle.
Justification of constructions may involve application of postulates,
theorems, definitions, and properties. Justification of constructions
may differ depending upon the plan proposed, and the order in which
concepts are taught.
Construction Justification 1. Construct a line segment Radii of equal circles are equal
congruent to a given a line
segment
2. Construct an angle congruent Radii of equal circles are equal
to a given angle SSS Postulate
Corresponding parts of
congruent triangles are
congruent
3. Construct the bisector of a Radii of equal circles are equal
given angle SSS Postulate
Corresponding parts of
congruent triangles are
congruent
Definition of an angle bisector Return to Course Outline
4. Construct the perpendicular Radii of equal circles are equal
bisector of a given segment Through any two points there is
exactly one line
If a point is equidistant from the
endpoints of a line segment, then
the point lies on the perpendic-
ular bisector of the line segment
5. Construct the perpendicular Radii of equal circles are equal
to a line at the given point on Definition of a straight angle
the line. Definition of an angle bisector
Definition of right angles and
definition of perpendicular
lines
6. Construct the perpendicular to Radii of equal circles are equal
the line from a point not on the If a point is equidistant from the
line. endpoints of a line segment,
then the point lies on the
perpendicular bisector of the
line
7. Construct the parallel to a Radii of equal circles are equal
given line though a given If two lines are cut by a
point not on the line. transversal and corresponding
angles are congruent, then the
lines are parallel
Extension for PreAP Geometry
Construct angles with measures of 15, 30, 45, 60, 75, and 135
degrees.
Construct a tangent to a circle through a point on the circle.
(continued)
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Constructions (continued) Extension for PreAP Geometry (continued)
Justify the constructions of:
- equilateral triangles;
- squares;
- angles with measures of 15, 30, 45, 60, 75, and 135 degrees;
- regular hexagons;
- a tangent to a circle through a point on the circle; and
- a tangent line from a point outside a given circle to the circle.
Given a segment, by construction divide the segment into a given
number of congruent parts.
Return to Course Outline
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Triangle Fundamentals
Sec
on
d M
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Per
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Lessons
Triangle Fundamentals
Isosceles Triangles
Triangle Inequalities
Strand: Triangles; Polygons and
Circles
SOL G.5
The student, given information
concerning the lengths of sides and/or
measures of angles in triangles, will
a) order the sides by length, given the
angle measures;
b) order the angles by degree
measure, given the side lengths;
c) determine whether a triangle
exists; and
d) determine the range in which the
length of the third side must lie.
These concepts will be considered in
the context of real-world situations.
SOL G.10 The student will solve real-world
problems involving angles of
polygons.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Order the sides of a triangle by their lengths
when given the measures of the angles.
Order the angles of a triangle by their
measures when given the lengths of the sides.
Given the lengths of three segments, determine
whether a triangle could be formed.
Given the lengths of two sides of a triangle,
determine the range in which the length of the
third side must lie.
Solve real-world problems given information
about the lengths of sides and/or measures of
angles in triangles.
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.
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.
Essential Understanding
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.
Two intersecting lines form angles with
specific relationships.
The exterior angle and the corresponding
interior angle form a linear pair.
(continued)
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Triangle Fundamentals (continued) Resources
HCPS Geometry Online:
Triangle Inequalities
Textbook: 4-1 Classifying Triangles
4-2 Measuring Angles in Triangles
4-6 Analyzing Isosceles Triangles
5-4 Inequalities for Sides and Angles of a Triangle
10-1 Polygons
10-2 Tessellations
DOE ESS Lesson Plans:
How Many Triangles? (PDF) (Word)
Angles in Polygons (PDF) (Word)
Key Vocabulary
altitude
apothem
concave
convex
decagon
diagonal
dodecagon
exterior angle
heptagon
hexagon
interior angle
isosceles
linear pair
median
n-gon
nonagon
octagon
opposite
pentagon
polygon
quadrilateral
regular/irregular polygon
scalene
tessellation
tiling
triangle
Triangle Inequality Theorem
Return to Course Outline
Essential Questions
What are the angle relationships of a triangle?
What conditions must exist for a triangle to be formed?
What is the relationship between the measure of the angles and the
lengths of the opposite sides of a triangle?
Teacher Notes and Elaborations
Triangle Inequality Theorem: The sum of the lengths of any two sides
of a triangle is greater than the length of the third side.
If one side of a triangle is longer than another side, then the angle
opposite (across from) the longer side is larger than the angle opposite
the shorter side.
If one angle of a triangle is larger than another angle, then the side
opposite the larger angle is longer than the side opposite the smaller
angle.
Sides of a triangle can be put in order when given the measures of the
angles. If the sides of a triangle are ordered longest to shortest then the
angles opposite must also be ordered largest to smallest.
Extension for PreAP Geometry
Use the Hinge Theorem and its converse to compare side lengths
and angle measures in two triangles.
Given a quadrilateral with one diagonal, write inequalities relating
pairs of angles or segment measures.
(continued)
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Triangle Fundamentals (continued) Extension for PreAP Geometry
Using properties of triangles, inequalities can be written relating pairs
of angles or segment measures. Note: Figures are not drawn to scale
BCD CAB
CD BC
Hinge Theorem: (SAS Inequality) If two sides of a triangle are
congruent to two sides of another triangle, and the included angle in
one triangle is greater than the included angles in the other, then the
third side of the first triangle is longer than the third side in the second
triangle.
Return to Course Outline
Sample Instructional Strategies and Activities
Coordinate geometry can be used to investigate relationships
among triangles.
Use pieces of yarn, straws, sticks, or magnetic tape to see which
combinations of lengths can be used to make triangles.
Use Geo-Legs or Anglegs to illustrate combinations of lengths that
can be used to form triangles.
Cut out a triangle. Place a different color dot in each angle. Place
the triangle on the paper and trace around it in pencil. Slide triangle
over and mark the color in each angle so that the colors correspond
with the cardboard triangle. Place triangle back on top and rotate it
so that it no longer overlaps. Repeat until the plane is filled. Have
students identify parallel lines, vertical angles, etc. Students make
conjectures about lines and angles in the tessellation. Students are
given various polygons and asked if they tessellate a plane. Explain
why or why not.
\
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Proofs with Congruent Triangles
Sec
on
d M
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ing
Per
iod
Lessons
Congruent Triangles
Proving Triangles Congruent –
SSS, SAS, ASA
Proving Triangles Congruent –
AAS, HL
Strand: Triangles
SOL 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.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Use definitions, postulates, and theorems to
prove triangles congruent.
Use coordinate methods, such as the distance
formula and the slope formula, to prove two
triangles are congruent.
Use algebraic methods to prove two triangles
are congruent.
Essential Understanding
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.
(continued)
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Proofs with Congruent Triangles (continued) Resources
HCPS Geometry Online:
Proofs
Textbook: 4-3 Exploring Congruent Triangles
4-4 Proving Triangles Congruent
4-5 More Congruent Triangles
5-2 Right Triangles
DOE ESS Lesson Plans:
Congruent Triangles (PDF) (Word)
Key Vocabulary
AAS Theorem
algebraic methods
altitude
ASA Postulate
coordinate methods
corresponding parts
deductive proof
definition
distance formula
HL Postulate
hypotenuse
included angle
included side
leg
postulate
properties
SAS Postulate
SSS Postulate
theorem
Essential Questions
What are congruent triangles?
What are the one-to-one correspondences that prove triangles
congruent?
How can congruent triangles assist in the proof of other geometric
ideas?
Teacher Notes and Elaborations
When two figures have exactly the same shape and size, they are said
to be congruent. Using algebraic methods, if all corresponding parts
Return to Course Outline
can be shown to be equal, then the figures are congruent. This can
include coordinate methods such as distance formula and the slope
formula.
Congruent figures have corresponding parts (matching parts) that
have equal measures. Corresponding parts of congruent triangles are
congruent (CPCTC).
Congruence does not depend on the position of the triangle.
A theorem is a statement that can be proved and a postulate is an
assumption that is accepted without proof. Definitions, postulates, and
theorems are used in proofs. A proof is a chain of logical statements
starting with given information and leading to a conclusion. Two
column deductive proofs are examples of deductive reasoning.
Properties (facts about real numbers and equality from algebra) can
also be used to justify steps in proofs.
A side of a triangle is said to be included (included side) between two
angles if the vertices of the two angles are the endpoints of the side.
An angle of a triangle is said to be included (included angle) between
two sides if the angle is formed by the two sides.
Triangles can be proven congruent with the following
correspondences:
SSS Postulate: Three sides of one triangle are congruent to the
corresponding sides of another triangle.
SAS Postulate: Two sides and the included angle of one triangle are
congruent to the corresponding two sides and included angle of
another triangle.
ASA Postulate: Two angles and the included side of one triangle
are congruent to the corresponding two angles and included side of
another triangle. (continued)
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Proofs with Congruent Triangles (continued) Teacher Notes and Elaborations (continued)
AAS Theorem: Two angles and a non-included side of one triangle
are congruent to the corresponding two angles and a non-included
side of a second triangle.
In a right triangle the side opposite the right angle is the hypotenuse
and the other two sides are called legs.
Right triangles can be proven congruent with the following
correspondence:
HL Postulate: The hypotenuse and a leg of one right triangle are
congruent to the hypotenuse and leg of another right triangle.
Medians, altitudes, and perpendicular bisectors are also used in
proving triangles congruent. A median of a triangle is a segment that
joins a vertex to the midpoint of the opposite side. An altitude of a
triangle is a segment from a vertex and perpendicular segment from a
vertex to the line containing the opposite side.
Extension for PreAP Geometry
Use angle bisectors, medians, altitudes, perpendicular bisectors to
prove triangles congruent.
Correlate LL, HA, LA to SAS, AAS, and ASA respectively.
Investigate the points of concurrency of the lines associated with
triangles (angle bisectors (incenter), perpendicular bisectors
(circumcenter), altitudes (orthocenter), and medians (centroid)).
Extension for PreAP Geometry
LL Theorem: The legs of one right triangle are congruent to the legs
of another right triangle.
HA Theorem: The hypotenuse and an acute angle of one right
triangle are congruent to the hypotenuse and acute angle of the other Return to Course Outline
right triangle.
LA Theorem: One leg and an acute angle of one right triangle are
congruent to the corresponding parts of another right triangle.
The medians of a triangle intersect at the common point called the
centroid.
In a triangle, the point where the perpendicular bisectors of each side
intersect is the circumcenter.
In a triangle, the incenter is the point where the angle bisectors
intersect.
In a triangle, the orthocenter is the point of intersection of the three
altitudes.
Sample Instructional Strategies and Activities Use coordinate geometry to investigate relationships among
triangles.
Given specifications such as side lengths or angle measures,
students draw a triangle. Next, the students compare their drawings
to see if they are congruent. This is done to test AAS, SSS, etc.
before they are introduced.
Students are given a printed deductive proof of theorem. Cut it up
into a statement of theorem, given, prove, diagram, individual
statements, and individual reasons. Each group of students is given
a set of pieces and must put the proof together in correct order.
Use pieces of yarn, straws, or sticks to see which combinations of
lengths can be used to make triangles.
Use patty paper to demonstrate congruent triangles.
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Similarity
Sec
on
d M
ark
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Per
iod
Lessons
Using Proportions
Exploring Similar Polygons
Identifying Similar Triangles
Strand: Triangles
SOL 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.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Use definitions, postulates, and theorems to
prove triangles similar.
Use algebraic methods to prove that triangles
are similar.
Use coordinate methods, such as the distance
formula, to prove two triangles are similar.
Compare ratios between side lengths,
perimeters, areas, and volumes, given two
similar figures.
Solve real-world problems involving measured
attributes of similar objects.
Essential Understanding
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 constant ratio exists between corresponding
lengths of sides of similar figures.
Proportional reasoning is integral to
comparing attribute measures in similar
objects
(continued)
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Similarity (continued) Resources
HCPS Geometry Online:
Similarity
Textbook: 7-1 Using Proportions
7-2 Exploring Similar Polygons
7-3 Identifying Similar Triangles
7-4 Parallel Lines and Proportional Parts
7-5 Parts of Similar Triangles
DOE ESS Lesson Plans:
Similar Triangles (PDF) (Word)
Key Vocabulary
area
deductive proof
perimeter
proportion
ratio
scale factor
similar figures
similar triangles
AA Similarity
SSS Similarity
SAS Similarity
volume
Essential Questions
When is a proportion necessary to solve a problem?
Are common units of measure necessary when solving
proportions?
How are similar triangles utilized in art, architecture and the
sciences?
What is the difference between congruence and similarity?
What is the relationship between similar triangles and proportions?
What are the one-to-one correspondences that prove triangles
similar? Return to Course Outline
In similar figures, how does a change of one measurement affect
perimeter, area, or volume?
Teacher Notes and Elaborations Congruent figures have corresponding parts that have equal measures
while similar figures have corresponding angles congruent but
corresponding sides with proportional measures. Coordinate methods
such as distance formula and the slope formula can be used to prove
triangles are similar.
A theorem is a statement that can be proved and a postulate is an
assumption that is accepted without proof. Definitions, postulates, and
theorems are used in proofs. A proof is a chain of logical statements
starting with given information and leading to a conclusion. Two
column deductive proofs are examples of deductive reasoning.
Properties (facts about real numbers and equality from algebra) can
also be used to justify steps in proofs.
A ratio is a comparison of two quantities. The ratio of a to b can be
expressed as a
b, where b 0. If two ratios are equal, then a proportion
exists. Therefore a c
b d is a proportion and the cross products are
equal (ad = bc).
Similar figures are figures that have the same shape but not
necessarily the same size. Two triangles 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 a scale factor.
An angle of a triangle is said to be included (included angle) between
two sides if the angle is formed by the two sides. (continued)
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Similarity (continued) Teacher Notes and Elaborations (continued) There are three ways to determine whether two triangles are similar
when all measurements of both triangles are not known:
AA Similarity: Show that two angles of one triangle are
congruent to two angles of the other.
SSS Similarity: Show that the measures of the corresponding
sides of the triangles are proportional.
SAS Similarity: Show that the measures of two sides of a triangle
are proportional to the measures of the corresponding sides of the
other triangle and that the included angles are congruent.
If a line is drawn parallel to one side of a triangle and intersects the
other two sides, then it separates the sides into segments of
proportional lengths.
a c a c
b d
b d
If two triangles are similar, then the measures of the lengths of the
corresponding angle bisectors of the triangles are proportional to the
measures of the lengths of the corresponding sides.
a ~ c
x y
x a
y c
A median of a triangle is a segment that joins a vertex to the midpoint
of the opposite side. Return to Course Outline
If two triangles are similar, then the measures of the corresponding
medians are proportional to the measures of the corresponding sides.
~ c
a x y
x a
y c
An angle bisector in a triangle separates the opposite side into
segments that have the same ratio as the other two sides.
a b
e a
f b
e f
Extension for PreAP Geometry
Use definitions, postulates, and theorems to complete two-column
or paragraph proofs with at least five steps.
Investigate proportionality in a triangle intersected by three or more
parallel lines.
Investigate the Golden Ratio.
Extension for PreAP Geometry
If three or more parallel lines intersect two transversals, then they cut
off the transversals proportionally.
(continued)
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Similarity (continued) Extension for PreAP Geometry (continued)
A B C
D
E
F
AB DE
=BC EF
, AC BC
=DF EF
, AC DF
=BC EF
If a line segment is divided into two lengths such that the ratio of the
segments’ entire length to the longer length is equal to the ratio of the
longer length to the shorter length, then the segment has been divided
into the Golden Ratio.
a b
a b a
a b
(This golden ratio is approximately 1.618.)
In a rectangle, if the ratio of the longer side to the shorter
approximates 1.618, the rectangle is called a Golden Rectangle
Sample Instructional Strategies and Activities
Use coordinate geometry to investigate relationships among
triangles.
Students are given a printed deductive proof of theorem. Cut it up
into a statement of theorem, given, prove, diagram, individual Return to Course Outline
statements, and individual reasons. Each group of students is given
a set of pieces and must put the proof together in correct order.
Each group of students will measure the height of one of their
members, the shadow of that member, and the shadow of a light
pole or flagpole. Using similar triangles and proportions, each
group calculates the height of the pole. Next, the groups compare
their calculations.
Given the pitch of a roof, the students will calculate the roof truss
and using toothpicks will construct a model of the roof.
Use patty paper to demonstrate similar triangles.
Using cylinders made from PVC pipe or empty cans determine the
change in volume with respect to changes in height or radius. Fill
cylinders with water to compare the volumes.
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Right Triangles
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Geometric Mean (optional)
The Pythagorean Theorem
Special Right Triangles
Trigonometry
Strand: Triangles
SOL 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.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Determine whether a triangle formed with
three given lengths is a right triangle.
Solve for missing lengths in geometric figures,
using properties of 45-45-90 triangles.
Solve for missing lengths in geometric figures,
using properties of 30-60-90 triangles.
Solve problems involving right triangles, using
sine, cosine, and tangent ratios.
Solve real-world problems, using right triangle
trigonometry and properties of right triangles.
Explain and use the relationship between the
sine and cosine of complementary angles.
Use definitions, postulates, and theorems to
prove triangles similar.
Use algebraic methods to prove that triangles
are similar.
Use coordinate methods, such as the distance
formula, to prove two triangles are similar.
Essential Understanding
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
1sin
2A ab 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.
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.
(continued)
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Right Triangles (continued) Resources
HCPS Geometry Online:
Right Triangles
Textbook: 8-1 Geometric Mean and the Pythagorean Theorem
8-2 Special Right Triangles
8-3 Ratios in Right Triangles
DOE ESS Lesson Plans:
The Pythagorean Relationship (PDF) (Word)
Special Right Triangles and Right Triangle Trigonometry (PDF)
(Word)
Key Vocabulary
angle of depression
angle of elevation
area of a triangle
cosine
hypotenuse
Pythagorean Theorem
right triangle
sine
tangent
trigonometry
45°-45°-90° triangle
30°-60°-90° triangle
Essential Questions
How can one determine a missing measurement of a right triangle?
How can one verify that a triangle is a right triangle?
What is a trigonometric ratio?
What is the relationship between sine and cosine in terms of
complementary angles?
Teacher Notes and Elaborations
Right triangles (any triangle with one 90° angle) are triangles with
Return to Course Outline
specific relationships.
The side opposite the right angle in a right triangle is the hypotenuse.
It is always the longest side of a right triangle.
Special right triangles are the 30° - 60° - 90° and the 45° - 45° - 90°.
- In a 45° - 45° - 90° triangle, the hypotenuse is 2 times as long
as one of the legs.
- In the 30° - 60° - 90° triangles, the hypotenuse is twice as long
as the shorter leg and the longer leg is 3 times as long as the
shorter leg.
The Pythagorean Theorem states that in a right triangle, the square of
the measure of the hypotenuse equals the sum of the squares of the
measures of the legs. The converse of the Pythagorean Theorem states
that if the square of the measure of the longest side equals the sum of
the squares of the measures of the other two sides of a triangle, then
the triangle is a right triangle.
If the square of the longest side of a triangle is greater than the sum of
the squares of the other two sides, then the triangle is an obtuse
triangle.
If the square of the longest side of a triangle is less than the sum of the
squares of the other two sides, then the triangle is an acute triangle.
Pythagorean Triples are three positive integers that satisfy the
Pythagorean theorem.
In a right triangle with the altitude drawn to the hypotenuse, the
geometric mean can be used to find missing measures of that triangle.
(continued)
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Right Triangles (continued) Teacher Notes and Elaborations (continued)
If r, s, and t are positive numbers with r s
s t , then s is the geometric
mean between r and t. Similar right triangles have the same shape but
not necessarily the same size. They can be used to find missing
triangle segments.
Trigonometry is a branch of mathematics that combines arithmetic,
algebra, and geometry. The right triangle is the basis of trigonometry.
In any right triangle, the ratio (quotient) of the lengths of two sides is
called a trigonometric ratio. Sine is the ratio of the side opposite an
acute angle to the hypotenuse. Cosine is the ratio of the side adjacent
an acute angle to the hypotenuse. Tangent is the ratio of the side
opposite an acute angle to the adjacent side. Sine and cosine relate an
angle measure to the ratio of the measures of a triangle’s leg to its
hypotenuse. The sine of one acute angle in a right triangle and cosine
of its complement is the same.
Example:
60º 13
sin 3018
13
cos6018
13 18 sin30 = cos60
30º
The angle of elevation is the angle formed by a horizontal line and the
line of sight to an object above that horizontal line. The angle of
depression is the angle formed by a horizontal line and the line of
sight to an object below that horizontal line. The angle of elevation
and the angle of depression in the same diagram are always congruent. Return to Course Outline
Extension for PreAP Geometry
Use the Law of Sines and the Law of Cosines to find missing
measures in triangles.
Find the geometric mean in right triangles.
Extension for PreAP Geometry The Law of Sines states that for any triangle with angles of measures
A, B, and C, and sides of lengths a, b, and c (a opposite A ,
opposite b B , and opposite c C ). This law is often used if two angles
and a side are known (AAS or ASA).
sin sin sinA B C
a b c
The Law of Cosines states that for any triangle with sides of lengths a,
b, and c then 2 2 2 2 cosc a b ab C . This law is often used when at least
two sides are known (SAS or SSS).
The measures of the altitude drawn from the vertex of the right angle
of a right triangle to its hypotenuse, is the geometric mean between
the measures of the two segments of the hypotenuse.
h
x h
h y
x y
If the altitude is drawn to the hypotenuse of a right triangle, then the
measure of a leg of the triangle is the geometric mean between the
measures of the hypotenuse and the segment of the hypotenuse
adjacent to that leg.
(continued)
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Right Triangles (continued) Extension for PreAP Geometry (continued)
a h b x a
a c and
y b
b c
x
c
Sample Instructional Strategies and Activities Use pieces of yarn, straws, or sticks to see which combinations of
lengths can be used to make acute, obtuse, and right triangles.
Have students make a hypsometer, then go outside and measure the
heights of buildings, trees, poles, etc., with the hypsometer.
The teacher prepares a set of clue cards containing trigonometry
word problems. Students work in groups of 4 or 5 draw a diagram
of the problem, set up a trig equation, then solve the problem.
Return to Course Outline
(continued)
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Polygons
Sec
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Per
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Lessons
Polygons
Strand: Triangles; Polygons and
Circles
SOL G.10 The student will solve real-world
problems involving angles of
polygons.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Solve real-world problems involving the
measures of interior and exterior angles of
polygons.
Identify tessellations in art, construction, and
nature.
Find the sum of the measures of the interior
and exterior angles of a convex polygon.
Find the measure of each interior and exterior
angle of a regular polygon.
Find the number of sides of a regular polygon,
given the measures of interior or exterior
angles of the polygon.
Essential Understanding
A regular polygon will tessellate the plane if
the measure of an interior angle is a factor of
360.
Both regular and nonregular polygons can
tessellate the plane.
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.
(continued)
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Polygons (continued) Resources
HCPS Geometry Online:
Tessellations
Textbook: 4-1 Classifying Triangles
4-2 Measuring Angles in Triangles
4-6 Analyzing Isosceles Triangles
5-4 Inequalities for Sides and Angles of a Triangle
10-1 Polygons
10-2 Tessellations
DOE ESS Lesson Plans:
How Many Triangles? (PDF) (Word)
Angles in Polygons (PDF) (Word)
Key Vocabulary
altitude
apothem
concave
convex
decagon
diagonal
dodecagon
exterior angle
heptagon
hexagon
interior angle
isosceles
linear pair
median
n-gon
nonagon
octagon
opposite
pentagon
polygon
quadrilateral
regular/irregular polygon
scalene
tessellation
tiling
triangle
Triangle Inequality Theorem
Return to Course Outline
Essential Questions
What are the distinguishing characteristics of a polygon?
How do we verify that polygons can tile a plane?
What are the relationships between the sides of a polygon and the
angles of a polygon?
Teacher Notes and Elaborations
A polygon is a plane figure formed by coplanar segments (sides) such
that (1) each segment intersects exactly two other segments, one at
each endpoint; and (2) no two points with a common endpoint are
collinear.
Polygons are named by their number of sides and classified as convex
(a line containing a side of a polygon contains no interior points of
that polygon) or concave (a line containing a side of a polygon also
contains interior points of the polygon).
Common polygons:
3 sides: triangle 7 sides: heptagon 10 sides: decagon
4 sides: quadrilateral 8 sides: octagon 12 sides: dodecagon
5 sides: pentagon 9 sides: nonagon n sides: n-gon
6 sides: hexagon
A segment joining two nonconsecutive vertices is a diagonal of the
polygon.
Two angles that are adjacent (share a leg) and supplementary (add up
to 180°) form a linear pair.
(continued)
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Polygons (continued) Teacher Notes and Elaborations (continued)
Polygons have interior angles (angles formed by the sides of the
polygon and enclosed by the polygon) and exterior angles (angles
formed by extending an existing side). The exterior angle and the
corresponding interior angle form a linear pair. The sum of the
measures of the interior angles of a polygon is found by multiplying
two less than the number of sides by 180°, [ ( 2)180n ]. The sum of
the measures of the exterior angles, one at each vertex, is 360°.
A regular polygon is a convex polygon with all sides congruent and
all angles congruent. The center of a regular polygon is the center of
the circumscribed circle. Given the measure of an exterior angle of a
regular polygon, the number of sides can be determined by dividing
360° by the measure of that angle. The central angle of a regular
polygon is an angle formed by two radii drawn to consecutive
vertices. Its measure can be determined by dividing 360° by the
number of sides.
A polygon will tessellate the plane if the interior angles at a vertex add
to 360°. Tessellations are repeated copies of a figure that completely
fill a plane without overlapping. The hexagon pattern in a honeycomb
is a tessellation of regular hexagons. Both regular and non-regular
polygons can tessellate the plane.
Return to Course Outline
When a tessellation uses only one shape it is called a pure tessellation.
The three regular polygons that create pure tessellations are triangle,
square, and hexagon.
Regular polygon tessellation Non-regular polygon tessellation
Extension for PreAP Geometry
Investigate and identify the regular polygons that tessellate.
Distinguish between pure and semi-pure tessellations.
Extension for PreAP Geometry
Tessellations that involve more than one type of shape are called
semi-pure tessellations. For example, in an octagon – square
tessellation, two regular octagons, and a square meet at each vertex
point.
Students, using materials of their choice, will make mobiles with
different polygons.
Students bring in photographs of regular polygons in art, nature, or
architecture.
Find tessellations in real world situations such as in art and
architecture.
Pattern blocks may be used to create tessellations.
Students can design a book cover using tessellations
(continued)
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Quadrilaterals
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Parallelograms
Rectangles
Rhombi and Squares
Trapezoids and Kites
Strand: Polygons and Circles
SOL G.9 The student will verify characteristics
of quadrilaterals and use properties of
quadrilaterals to solve real-world
problems.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Solve problems, including real-world
problems, using the properties specific to
parallelograms, rectangles, rhombi, squares,
isosceles trapezoids, and trapezoids.
Prove that quadrilaterals have specific
properties, using coordinate and algebraic
methods, such as the distance formula, slope,
and midpoint formula.
Prove the characteristics of quadrilaterals,
using deductive reasoning, algebraic, and
coordinate methods.
Prove properties of angles for a quadrilateral
inscribed in a circle.
Essential Understanding
The terms characteristics and properties can
be used interchangeably to describe
quadrilaterals. The term characteristics is
used in elementary and middle school
mathematics.
Quadrilaterals have a hierarchical nature
based on the relationships between their sides,
angles, and diagonals.
Characteristics of quadrilaterals can be used to
identify the quadrilateral and to find the
measures of sides and angles.
(continued)
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Quadrilaterals (continued) Resources
HCPS Geometry Online:
Quadrilaterals
Textbook: 6-1 Parallelograms
6-2 Tests for Parallelograms
6-3 Rectangles
6-4 Squares and Rhombi
6-4B Extension - Kites
6-5 Trapezoids
DOE ESS Lesson Plans:
Properties of Quadrilaterals (PDF) (Word)
Key Vocabulary
base
base angles
characteristics
diagonal
isosceles trapezoid
kite
legs
median of a trapezoid
parallelogram
quadrilateral
rectangle
rhombus
square
trapezoid
Essential Questions
What are the distinguishing features of the different types of
quadrilaterals?
How are the properties of quadrilaterals used to solve real-life
problems?
What is the hierarchical nature among quadrilaterals?
Return to Course Outline
Teacher Notes and Elaborations
Algebraic methods and coordinate methods such as distance formula,
midpoint formula, and the slope formula can be used to prove
quadrilateral properties.
A quadrilateral is a polygon with four sides. Quadrilaterals have a
hierarchical nature based on relationships among their sides, their
angles, and their diagonals. The diagonal of a polygon is a segment
joining two nonconsecutive vertices of the polygon.
A parallelogram is a quadrilateral with opposite sides parallel and
congruent. Consecutive angles of a parallelogram are supplementary;
opposite angles are congruent; and the diagonals of a parallelogram
bisect each other.
A rectangle is a parallelogram with four right angles. The diagonals of
a rectangle are congruent.
A rhombus is a parallelogram with congruent sides. The diagonals of a
rhombus are perpendicular and bisect each other and the opposite
angles.
A square is a parallelogram, a rectangle, and a rhombus.
A trapezoid is a quadrilateral with exactly one pair of opposite sides
parallel. An isosceles trapezoid has congruent legs (the non-parallel
sides). Both pairs of base angles in an isosceles trapezoid are
congruent and diagonals are congruent. The median of a trapezoid is
the segment that joins the midpoints of the legs. It is parallel to the
bases and has a length equal to half the sum of the lengths of the bases.
A kite is a quadrilateral with two pairs of congruent adjacent sides. (continued)
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Quadrilaterals (continued) Teacher Notes and Elaborations (continued) Characteristics of quadrilaterals are used to identify figures, and to find
values for missing parts and areas.
Areas of work that use quadrilaterals include art, construction, fabric
design, and architecture.
The hierarchical nature of quadrilaterals can be described as ranking
based on characteristics.
If a quadrilateral is inscribed in a circle, its opposite angles are
supplementary. This can be verified by considering that the arcs
intercepted by opposite angles of an inscribed quadrilateral form a
circle.
Return to Course Outline
Example:
Quadrilateral ABCD is inscribed in a circle.
AB BC CD DA 360m m m m
The measure of 1
DAB = BCD2
m m and the measure of
1
BCD = DAB2
m m . A B
BCD = 2 A and DAB = 2 Cm m m m
BCD DAB 360m m
2 A+2 C = 360m m D C
A C 180m m
Sample Instructional Strategies and Activities Use flowcharts or Venn diagrams to show relationships and
properties of quadrilaterals.
Use patty paper to show properties of the different quadrilaterals.
Use notecards to create models of different quadrilaterals. Discuss
the characteristics and have students record their findings on the
back of the models.
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Circles
Th
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Lessons
Terminology
Area and Circumference
Tangents
Arcs and Circles
Angle Formula
Segment Formula
Strand: Polygons and Circles
SOL 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.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Find lengths, angle measures, and arc
measures associated with
– two intersecting chords;
– two intersecting secants;
– an intersecting secant and tangent;
– two intersecting tangents; and
– central and inscribed angles.
Calculate the area of a sector and the
length of an arc of a circle, using
proportions.
Solve real-world problems associated with
circles, using properties of angles, lines,
and arcs.
Verify properties of circles, using
deductive reasoning, algebraic, and
coordinate methods.
Essential Understanding
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.
(continued)
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Circles (continued) Resources
HCPS Geometry Online:
Circles
Textbook: 9-1 Exploring Circles
9-2 Angles and Arcs
9-3 Arcs and Chords
9-4 Inscribed Angles
9-5 Tangents
9-6 Secants, Tangents, and Angle Measures
9-7 Special Segments in a Circle
DOE ESS Lesson Plans:
Angles, Arcs, and Segments in Circles (PDF) (Word)
Arc Length and Area of a Sector (PDF) (Word)
Key Vocabulary
arc
arc length
arc measure
central angle
chord
circle
circumscribed
circumference
common tangent
concentric circles
diameter
inscribed
intercepted arc
major arc
minor arc
point of tangency
radius
secant
sector
semicircles
tangent
Essential Questions
How might geometric objects (points, segments, lines, etc.)
interact/intersect with circles? Return to Course Outline
What is area/circumference and how is it measured? What does the
value pi represent?
How does a tangent line relate to the circle?
How are the angle formulas of circles related to similar triangles?
What are the relationships between chords and arcs?
What is the difference between arc length and arc measure?
Teacher Notes and Elaborations
A circle is the set of all points equidistant from a given point in a
plane. The distance from the center of the circle to a point on the circle
is the radius.
The arc measure is the degree measure of its central angle. A central
angle is an angle with its vertex at the circle’s center. A central angle
separates a circle into two arcs called a major arc (measures greater
than 180º but less than 360º), and a minor arc (measures greater than
0º but less than 180º). Semicircles are the two arcs of a circle that are
cut off by a diameter. A semicircle measures 180º. An arc is an
unbroken part of a curve of a circle. The central angle measures the
same as its intercepted arc. The intercepted arc is the part of the circle
that lies between the two lines that intersect the circle.
A chord is a segment joining two points on the circle. A diameter is a
chord that passes through the circle’s center. A secant is a line that
contains a chord. A tangent is a line that intersects a circle in only one
point. Measures of chords, secant segments, and tangent segments can
be determined.
An inscribed angle is an angle whose vertex is on the circle and
whose sides are chords of the circle. The measure of an inscribed
angle is equal to one-half the measure of its intercepted arc.
(continued)
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Circles (continued) Teacher Notes and Elaborations (continued) The measure of an angle formed by two chords that intersect inside a
circle is equal to half the sum of the measures of the intercepted arcs.
The measure of an angle formed by a chord and a tangent is equal to
half the measure of the intercepted arc.
The measure of an angle formed by two secants, two tangents, or a
secant and a tangent drawn from a point outside a circle is equal to
half the difference of the measures of the intercepted arcs.
Use the properties of chords, secants, and tangents to determine
missing lengths.
When two chords intersect inside a circle, the product of the lengths of
the segments of one chord equals the product of the lengths of the
segments of the other chord.
When two secant segments are drawn to a circle from an exterior
point, the product of the lengths of one secant segment and its exterior
segment is equal to the product of the lengths of the other secant
segment and its exterior segment.
When a tangent segment and a secant segment are drawn to a circle
from an exterior point, the square of the length of the tangent segment
is equal to the product of the lengths of the secant segment and its
exterior segment.
The length of an arc (arc length) is a linear measure and is part of the
circumference (perimeter of a circle). A sector of a circle is that part
of the circle bounded by two radii and an arc. Length of an arc and
area of a sector can be calculated using the following formulas:
In circle O, the measure of AB x (This is a degree measure.) Return to Course Outline
Length of AB 2360
xr (This is a linear measure.)
Area of sector 2AOB360
xr
Verifying the properties of circles may include definitions, postulates,
theorems, algebraic methods, and coordinate methods.
Wheels and gears are two important applications of circles.
In the same circle or congruent circles:
- Congruent chords have congruent arcs and vice versa.
- Congruent chords are equidistant from the center and vice versa.
- A diameter that is perpendicular to a chord bisects the chord and
its arc.
An angle inscribed in a semi-circle is a right angle. Opposite angles of
an inscribed quadrilateral are supplementary.
Extension for PreAP Geometry
Find the area of a segment of a circle.
Find the area of an annulus.
Extension for PreAP Geometry
A segment of a circle is the region between an arc and a chord of a
circle.
To find the area of a segment, find the area of
the sector and subtract the area of the triangle.
(continued)
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Circles (continued) Extension for PreAP Geometry (continued)
An annulus is the region between two concentric circles.
To find the area of an annulus, find the area of the larger circle and
subtract the area of the smaller circle.
Sample Instructional Strategies and Activities Use the graphing calculator to show that a triangle inscribed in a
semicircle is a right triangle; to show that the product of the parts
of one chord equal the product of the parts of the other chord; to
graph and identify circles as tangent, intersecting, or concentric;
and to graph and recognize tangents as internal or external.
Use patty paper to demonstrate the properties of circles.
Students use post-it notes to identify intercepted arcs.
Students use post-it notes to find multiple angles and arc measures
in circle drawings.
Return to Course Outline
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Surface Area and Volume
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Area of 2-D Shapes (optional)
Prisms and Pyramids
Cylinders and Cones
Spheres
Similar Objects
Strand: Three-Dimensional Figures
SOL G.13
The student will use formulas for
surface area and volume of three-
dimensional objects to solve real-
world problems.
SOL G.14
The student will use similar geometric
objects in two- or three-dimensions to
a) compare ratios between side
lengths, perimeters, areas, and
volumes;
b) determine how changes in one or
more dimensions of an object affect
area and/or volume of the object;
c) determine how changes in area
and/or volume of an object affect
one or more dimensions of the
object; and
d) solve real-world problems about
similar geometric objects.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving,
mathematical communication, mathematical
reasoning, connections, and representations to
Find the total surface area of cylinders, prisms,
pyramids, cones, and spheres, using the
appropriate formulas.
Calculate the volume of cylinders, prisms,
pyramids, cones, and spheres, using the
appropriate formulas.
Solve problems, including real-world
problems, involving total surface area and
volume of cylinders, prisms, pyramids, cones,
and spheres as well as combinations of three-
dimensional figures.
Calculators may be used to find decimal
approximations for results.
Compare ratios between side lengths,
perimeters, areas, and volumes, given two
similar figures.
Describe how changes in one or more
dimensions affect other derived measures
(perimeter, area, total surface area, and
volume) of an object.
Describe how changes in one or more
measures (perimeter, area, total surface area,
and volume) affect other measures of an
object.
Solve real-world problems involving measured
attributes of similar objects.
Essential Understanding
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.
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.
(continued)
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Surface Area and Volume (continued) Resources
HCPS Geometry Online:
Surface Area and Volume
Similar Geometric Objects
Textbook: 11-1 Exploring Three-Dimensional Figures
11-2 Nets and Surface Area
11-3 Surface Area of Prisms and Cylinders
11-4 Surface Area of Pyramids and Cones
11-5 Volume of Prisms and Cylinders
11-6 Volume of Pyramids and Cones
11-7 Surface Area and Volume of Spheres
11-8 Congruent and Similar Solids
DOE ESS Lesson Plans:
Surface Area and Volume (PDF) (Word)
Similar Solids and Proportional Reasoning (PDF) (Word)
Key Vocabulary
altitude
area
base
base area (B)
base edge
cone
cube
face
height
lateral edge
lateral area
prism
polygon
pyramid
similar figures
slant height
sphere
surface area
three-dimensional
two dimensional
vertex
volume
Essential Questions
What is area?
What is volume?
How are the lateral area, surface area, and volume of the following
figures determined: prisms, cylinders, pyramids, cones, and
spheres? Return to Course Outline
How does a change in dimensions affect the area and/or volume of
the object?
In similar figures, how does a change of one measurement affect
perimeter, area, or volume?
Teacher Notes and Elaborations
A dimension is the number of coordinates required to locate a point in
a space. A flat surface is two-dimensional because two coordinates are
needed to specify a point on it. Three-dimensional space is a
geometric model of the physical universe in which we live. The three
dimensions are commonly called length, width, and depth (or height),
although any three directions can be chosen, provided that they do not
lie in the same plane.
A polygon is a geometric figure formed by three or more coplanar
segments called sides. Each side intersects exactly two other sides, but
only at their endpoints, and the intersecting sides must be
noncollinear.
A vertex of an angle is a point common to the two sides of the angle.
In a polygon, a vertex is a point common to two sides of the polygon.
The vertex of a polyhedron is a point common to the edges of a
polyhedron. In a polyhedron the flat surfaces formed by the polygons
and their interiors are called faces.
Area is the number of square units in a region. Surface area is a
measurement of coverage such as wallpaper.
Lateral area is the area of the exterior surface (lateral surface) of a
three-dimensional figure not including the area of the base(s).
A prism is a three-dimensional figure whose lateral faces are (continued)
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Surface Area and Volume (continued) Teacher Notes and Elaborations (continued) parallelograms. If the faces are rectangles, the prism is a right prism.
A prism is classified by the shape of its base.
A pyramid is a three-dimensional figure whose lateral faces are
triangles. In regular pyramids, the base is a regular polygon, lateral
edges are congruent, and all lateral faces are congruent isosceles
triangles. Slant height in a pyramid is the distance from the vertex
perpendicular to the base on a lateral face of the pyramid. Slant height
on a cone is the distance from the vertex to the circle. Height is the
perpendicular distance between bases or between a vertex and a base.
A cone is a three-dimensional figure that has a circular base, a vertex
not in the plane of the circle, and a curved lateral surface. In a right
cone, the altitude is a perpendicular segment from the vertex to the
center of the base. The height (h) is the length of the altitude. The
slant height ( ) is the distance from the vertex to a point on the edge
of the base.
Surface area is the lateral area plus the area of the base(s). Bases of
prisms are congruent polygons lying in parallel planes. An altitude
(height) of a prism is a segment joining the two base planes and
perpendicular to both. The faces of a prism that are not its bases are
called lateral faces. Adjacent lateral faces intersect in parallel
segments called lateral edges. In right prisms the lateral edges are also
altitudes.
Volume is the capacity of a three-dimensional figure such as the
amount of water in an aquarium.
The volume of an irregularly shaped object can be found by
measuring its displacement. When an object is placed in a liquid, it
causes the liquid to rise. This volume is called the objects’
displacement. Return to Course Outline
The base of a three-dimensional figure could be a circle, a triangle, a
square, a rectangle, a regular hexagon or another type of polygon.
Many formulas use B to represent the area of the base of the solid
figure. To find the area of a base (B) in three dimensional figures, use
the area formula that applies. Formulas for those figures may need to
be reviewed.
A sphere is the set of all points in space equidistant from a given
point. The center is the given point and the radius is the given
distance. Surface area and volume of spheres will also be found.
When determining surface area of combinations of solids, attention
needs to be given to the possibility of shared faces.
Similar figures are figures that have the same shape but not
necessarily the same size.
Scale factors (proportional reasoning) are used to compare perimeters,
areas, and volumes of similar two-dimensional and three-dimensional
geometric figures. A change in one dimension of an object results in
changes in area and volume in specific patterns.
Volumes, areas, and perimeters of similar polygons are examined to
draw conclusions about how changes in one dimension affect both
area and volume.
If the given perimeter of a polygon is increased or decreased, the area
will increase or decrease by the square of the change and the volume
increases or decreases by the cube of the change.
Similar solids are solids that have the same shape but not necessarily
the same size. All spheres are similar.
(continued)
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Surface Area and Volume (continued) Teacher Notes and Elaborations (continued) If the scale factor of two similar solids is a:b, then:
– The ratio of corresponding perimeters is a:b.
– The ratios of the base areas, of the lateral areas, and of the total
areas are a2:b2.
– The ratio of the volumes is a3:b3.
Sample Instructional Strategies and Activities
Use strings, straws, toothpicks, etc. to make three-dimensional
objects.
Students make a three-dimensional object from any material they
choose. They calculate lateral area, total area, and volume and
incorporate this into a written report, which includes their
calculations, a sketch of their model, and a description of their
procedure. Students give a brief oral report of their project.
Using a geometric model kit, students will investigate relationships
among volume formulas.
Demonstrate a way that the formula for the surface area of a sphere
might have been evolved.
To demonstrate the formula for surface area of a sphere, cut an
orange in half and trace the circumference of the orange on paper
several times. Peel the orange and completely fill as many circles
as possible. The result should be four filled circles, thus four times
the area of the circle.
Using items from a pantry have students measure and compute
surface area and volume.
When an object is placed in a liquid, it causes the liquid to rise.
This volume is called the objects’ displacement. The volume of an
irregularly shaped object can be found by measuring its
displacement.
Return to Course Outline
Example: A rock is placed into a rectangular prism containing water.
The base of the container is 10 centimeters by 15 centimeters and
when the rock is put in the prism, the water level rises 2 centimeters
due to the displacement. This new “slice” of water has a volume of
300 cubic centimeters (10 15 2 ). Therefore, the volume of the rock is
300 cubic centimeters.
Each student is given a sheet of construction paper. Next, they are
instructed to cut a square from each corner and form an open top
box with the maximum volume.
Have students use string and a ruler to determine whether two
solids are similar. If the figures are similar then use the
measurements to compare areas and volumes.
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Coordinate Geometry
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Formulas
Quadrilaterals
Transformations
o Including dilations
Writing the Equation of a Circle
Strand: Reasoning, Lines, and
Transformations; Polygons and Circles
SOL 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.
SOL G.9 The student will verify characteristics of
quadrilaterals and use properties of
quadrilaterals to solve real-world problems.
SOL 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.
Return to Course Outline
Essential Knowledge and Skills
The student will use problem solving, mathematical
communication, mathematical reasoning, connections, and
representations to
Find the coordinates of the midpoint of a segment, using
the midpoint formula.
Use a formula to find the slope of a line.
Compare the slopes to determine whether two lines are
parallel, perpendicular, or neither.
Determine whether a figure has point symmetry, line
symmetry, both, or neither.
Given an image and pre-image, identify the
transformation that has taken place as a reflection,
rotation, dilation, or translation.
Apply the distance formula to find the length of a line
segment when given the coordinates of the endpoints.
Solve problems, including real-world problems, using
the properties specific to parallelograms, rectangles,
rhombi, squares, isosceles trapezoids, and trapezoids.
Prove that quadrilaterals have specific properties, using
coordinate and algebraic methods, such as the distance
formula, slope, and midpoint formula.
Prove the characteristics of quadrilaterals, using
deductive reasoning, algebraic, and coordinate methods.
Prove properties of angles for a quadrilateral inscribed
in a circle.
Identify the center, radius, and diameter of a circle from
a given standard equation.
Use the distance formula to find the radius of a circle.
Given the coordinates of the center and radius of the
circle, identify a point on the circle.
Given the equation of a circle in standard form, identify
the coordinates of the center and find the radius of the
circle.
Given the coordinates of the endpoints of a diameter,
find the equation of the circle.
Given the coordinates of the center and a point on the
circle, find the equation of the circle.
Recognize that the equation of a circle of given center
and radius is derived using the Pythagorean Theorem.
Essential Understanding 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.
The terms characteristics and properties can be used
interchangeably to describe quadrilaterals. The term
characteristics is used in elementary and middle school
mathematics.
Quadrilaterals have a hierarchical nature based on the
relationships between their sides, angles, and diagonals.
Characteristics of quadrilaterals can be used to identify
the quadrilateral and to find the measures of sides and
angles.
A circle is a locus of points equidistant from a given
point, the center.
Standard form for the equation of a circle is
2 2 2x h y k r , 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.
(continued)
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Coordinate Geometry (continued) Resources
HCPS Geometry Online:
G.3 - Symmetry & Transformations
G.9 - Quadrilaterals
G.12 – Equation of a Circle
Textbook: 1-4 Measuring Segments
1-5 Midpoints and Segment Congruence
3-3 Slopes of Lines
Ch. 6 Exploring Quadrilaterals
13-4 Mappings
13-5 Reflections
13-6 Translations
13-7 Rotations
13-8 Dilations
DOE ESS Lesson Plans:
Distance and Midpoint Formulas (PDF) (Word)
Slope (PDF) (Word)
Symmetry (PDF) (Word)
Transformations (PDF) (Word)
Circles in the Coordinate Plane (PDF) (Word)
Key Vocabulary
dilation
distance formula
image
isometry
isomorphism
line symmetry
midpoint formula
point symmetry
pre-image
reflection
rotation
slope
slope formula
standard form for the
equation of a circle
symmetry
transformation
translation
Essential Questions
What is the relationship between the distance formula, the
Pythagorean Theorem, and the equation of a circle? Return to Course Outline
How does the concept of midpoint and slope relate to symmetry
and transformation?
What is line symmetry?
What is point symmetry?
How can symmetry be used to describe naturally occurring
phenomena?
How is a figure translated, reflected, rotated, or dilated?
What is the relationship between the center, the radius, and the
standard equation of a circle?
Teacher Notes and Elaborations
Like finding distance, two situations must be considered to find the
midpoint of the line and the congruence of the two line segments. The
two situations that must be considered are the midpoint on a number
line and midpoint in the coordinate plane. The midpoint of a segment
is the point that divides the segment into two congruent segments. The
midpoint of AB is the average of the coordinates of A and B.
A M B
3 2 1 0 1 2 3 4 5 6 7
( 1) 5
22
The Midpoint Formula uses the idea that the midpoint of a horizontal
or vertical line is the average of the coordinates of the endpoints. To
find the midpoint of a horizontal line segment, find the average of the
x endpoint coordinates; the y coordinate will be the same for all the
points. To find the midpoint of a vertical line segment the x
coordinate; will be the same for all points; the y coordinate will be the
average of the y endpoint coordinates.
(continued)
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Coordinate Geometry (continued) Teacher Notes and Elaborations (continued)
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
y
C D E
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
y
F
G
H
The midpoint of CE is D (2,2) . The midpoint of FH is G ( 3, 2)
.
This idea is used twice to find the coordinates of the midpoint of a
slanting segment with endpoints 1 1 1P ( , )x y and
2 2 2P ( , )x y .
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
y
P1(x1, y1)
P2(x2, y2)
M S
R T
The midpoint of 1 2P P is M 1 2 1 2,
2 2
x x y y
.
Some students may have difficulty in extending the concept of finding
the midpoint of a line segment on one number line to a line segment in
the coordinate plane. Using models such as the one above will aid in
developing this concept. Return to Course Outline
The slope (effect of steepness) of a line containing two points in the
coordinate plane can be found using the slope formula. The slope of a
vertical line is undefined since x1 = x2. Parallel lines are lines that do
not intersect and are coplanar. Parallel planes are planes that do not
intersect. Nonvertical lines are parallel if they have the same slope and
different y-intercepts. Any two vertical lines are parallel.
Perpendicular lines are lines that intersect at right angles. Two non-
vertical lines are perpendicular if and only if the product of their
slopes is 1 .
Students should have multiple experiences applying the following
formulas.
Given two points (x1, y1) and (x2, y2):
- the midpoint formula is 1 2 1 2,2 2
x x y y
;
- the distance formula is 2 2
2 1 2 1x x y y ; and
- the slope formula is
2 1
2 1
y y
x x
.
Regular polygons are frequently used to introduce the concepts of
symmetry, transformations, and tessellation. A geometric
configuration (curve, surface, etc.) is said to be symmetric (have
symmetry) with respect to a point, a line, or a plane, when for every
point on the configuration there is another point of the configuration
such that the pair is symmetric with respect to the point, line, or plane.
The point is the center of symmetry; the line is the axis of symmetry,
and the plane is the plane of symmetry. A line of symmetry is a line
(continued)
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Coordinate Geometry (continued) Teacher Notes and Elaborations (continued) that can be drawn so that the figure on one side is the reflection image
of the figure on the opposite side.
A figure has point symmetry if there is a symmetry point O such that
the half-turn HO maps the figure onto itself. A figure has line
symmetry if there is a symmetry line k such that the reflection Rk maps
the figure onto itself.
An isomorphism is a one-to-one mapping that preserves the
relationship between two sets. The original figure is the preimage. The
resulting figure is an image. An isometry is a transformation in which
the preimage and image are congruent. Reflections, rotations, and
translations are isometries. Dilations are not isometry.
Reflection is a transformation in which a line acts like a mirror,
reflecting points to their images. For many figures, a point can be
found that is a point of reflection for all points on the figure. This
point of reflection is called a point of symmetry.
A rotation is a transformation suggested by a rotating paddle wheel.
When the wheel moves, each paddle rotates to a new position. When
the wheel stops, the position of a paddle ( P ) can be referred to
mathematically as the image of the initial position of the paddle (P). A
figure with rotational symmetry of 180° has point symmetry.
A geometric transformation in a plane is a one-to-one correspondence
between two sets of points. It is a change in its position, shape, or size.
It maps a figure onto its image and may be described with arrow (→)
notation. A reflection is a type of transformation that can be described
by folding over a line of reflection or line of symmetry. For some
figures, a point can be found that is a point of reflection for all points
on the figure.
Return to Course Outline
A dilation is a transformation that may change the size of a figure. It
requires a center point and a scale factor. The scale factor is defined as
the image to pre-image. For example: 4 to 3 or 4
3 represents an
enlargement.
A composite of reflections is the transformation that results from
performing one reflection after another. A translation (slide) is the
composite of two reflections over parallel lines. The Pythagorean Theorem (distance formula) can be used to develop
an equation of a circle.
y
Let P(x, y) represent any point on the circle.
The distance between C(h, k) and P(x, y) is r.
2 2( ) ( )x h y k r P(x, y)
2 2 2( ) ( )x h y k r C(h, k)
x
Given the coordinates of the center of the circle (h, k) and a radius r,
four easily identified points on the circle are:
( , )h r k , ( , )h r k , ( , )h k r , ( , )h k r
Given the coordinates of the endpoints of a diameter, midpoint
formula can be used to find the center of the circle and distance
formula can be used to find the radius
(continued)
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Coordinate Geometry (continued) Extension for PreAP Geometry
Reflect triangles over horizontal and vertical lines in the coordinate
plane and the line y = x.
Draw on a coordinate plane the image that results from a geometric
figure that has been reflected, rotated, or dilated.
Investigate the relationship between a rotation and the composition
of reflections.
Investigate point-slope form as it relates to the equation of a line
(slope-intercept form) and the formula for slope.
Use slopes of parallel and perpendicular lines to write equations in
standard, point-slope, and slope-intercept forms.
Find the coordinates of an endpoint of a segment given the
coordinates of the midpoint and one endpoint.
Extension for PreAP Geometry
Point-slope form is an equation of the form 1 1( )y y m x x for the
line passing through a point whose coordinates are 1 1( , )x y and having
slope m.
The composite of reflections with respect to two intersecting lines is a
transformation called a rotation. The point of intersection, point P, is
the center of rotation. The figure rotates or turns around the point P.
Point symmetry is a rotational symmetry of 180°.
A dilation is a similarity transformation that alters the size of a
geometric figure, but does not change the shape. For each dilation, a
scale factor enlarges the dilation image, reduces the dilation image, or
maintains a congruence transformation.
Extension for PreAP Geometry
Investigate and identify points that lie inside, on, or outside a
circle.
Write inequality statements for regions either inside or outside a Return to Course Outline
circle and sketch these graphs.
Extension for PreAP Geometry
An example of an inequality that describes the points (x, y) outside the
circle that are more than three units from center (4, 2 ) is 2 2( 4) ( 2) 9x y . The graph would be a broken circle and shaded
outside the circle.
An example of an inequality that describes the points (x, y) inside the
circle that are less than or equal to four units from center ( 3, 5 ) is 2 2( 3) ( 5) 16x y . The graph would be a circle and shaded inside the
circle.
Sample Instructional Strategies and Activities
Do activities from the Geometer’s Sketchpad by Key Curriculum
Press.
Use coordinate geometry as a tool for making conjectures about
midpoints, slopes, and distance.
Each student is given a sheet of construction paper. Next, the
teacher puts a few drops of finger paint, etc. on each paper. Each
student folds his/her papers to illustrate symmetry with respect to a
line.
Demonstrate symmetry by using patty paper.
Cut out a triangle. Place a different color dot in each angle. Place
the triangle on the paper and trace around it in pencil. Slide
triangle over and mark the color in each angle so that the colors
correspond with the cardboard triangle. Place triangle back on top
and rotate it so that it no longer overlaps. Repeat until the plane is
filled. Have students identify parallel lines, vertical angles, etc.
Students make conjectures about lines and angles in the
tessellation. Students are given various polygons and asked if they
tessellate a plane. Explain why or why not. (continued)
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Coordinate Geometry (continued) Sample Instructional Strategies and Activities (continued)
Place a shape on the overhead projector. Have a student trace the
image on the blackboard. Move the projector away from the board
and trace the new image. Take the original shape and compare the
angles of the original with the angles of the images. Students can
measure the lengths of the sides and compare ratios.
Use patty paper to demonstrate reflections, rotations, dilations, or
translations.
Use examples of advertisements to identify examples of
transformations.
Give students coordinates of the vertices of a rectangle. Have
students find the lengths of the diagonals, the midpoints of the
diagonals, and the slopes of the diagonals. Have students make
conjectures about the diagonals of the rectangle. Repeat with
square, rhombus, parallelogram, isosceles trapezoid, trapezoid,
and quadrilateral. Have students make conjectures about the
diagonals of each.
Return to Course Outline
http://www.doe.virginia.gov/testing/test_administration/ancilliary_materials/mathematics/2009/2009_sol_formula_sheet_geometry.pdf
Henrico Curriculum Framework Geometry
Geometry
Copyright © 2009
by the
Virginia Department of Education
P.O. Box 2120
Richmond, Virginia 23218-2120
http://www.doe.virginia.gov
All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted.
Superintendent of Public Instruction
Patricia I. Wright, Ed.D.
Assistant Superintendent for Instruction
Linda M. Wallinger, Ph.D.
Office of Elementary Instruction
Mark R. Allan, Ph.D., Director
Deborah P. Wickham, Ph.D., Mathematics Specialist
Office of Middle and High School Instruction
Michael F. Bolling, Mathematics Coordinator
Acknowledgements
The Virginia Department of Education wishes to express sincere thanks to Deborah Kiger Bliss, Lois A. Williams, Ed.D., and Felicia Dyke, Ph.D.
who assisted in the development of the 2009 Mathematics Standards of Learning Curriculum Framework.
NOTICE
The Virginia Department of Education does not unlawfully discriminate on the basis of race, color, sex, national origin, age, or disability in
employment or in its educational programs or services.
The 2009 Mathematics Curriculum Framework can be found in PDF and Microsoft Word file formats on the Virginia Department of Education’s
Web site at http://www.doe.virginia.gov.
Virginia Mathematics Standards of Learning Curriculum Framework 2009
Introduction
The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and
amplifies the Mathematics Standards of Learning by defining the content knowledge, skills, and understandings that are measured by the Standards
of Learning assessments. The Curriculum Framework provides additional guidance to school divisions and their teachers as they develop an
instructional program appropriate for their students. It assists teachers in their lesson planning by identifying essential understandings, defining
essential content knowledge, and describing the intellectual skills students need to use. This supplemental framework delineates in greater specificity
the content that all teachers should teach and all students should learn.
Each topic in the Mathematics Standards of Learning Curriculum Framework is developed around the Standards of Learning. The format of the
Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge and skills that should be the focus of instruction for
each standard. The Curriculum Framework is divided into two columns: Essential Understandings and Essential Knowledge and Skills. The purpose
of each column is explained below.
Essential Understandings
This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the
Standards of Learning.
Essential Knowledge and Skills
Each standard is expanded in the Essential Knowledge and Skills column. What each student should know and be able to do in each standard is
outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. It is meant to be the key knowledge and skills
that define the standard.
The Curriculum Framework serves as a guide for Standards of Learning assessment development. Assessment items may not and should not be a
verbatim reflection of the information presented in the Curriculum Framework. Students are expected to continue to apply knowledge and skills
from Standards of Learning presented in previous grades as they build mathematical expertise.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 1
TOPIC: REASONING, LINES, AND TRANSFORMATIONS
GEOMETRY
STANDARD G.1
The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include
a) identifying the converse, inverse, and contrapositive of a conditional statement;
b) translating a short verbal argument into symbolic form;
c) using Venn diagrams to represent set relationships; and
d) using deductive reasoning.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Identify the converse, inverse, and contrapositive of a conditional
statement.
Translate verbal arguments into symbolic form, such as
(p q) and (~p ~q).
Determine the validity of a logical argument.
Use valid forms of deductive reasoning, including the law of
syllogism, the law of the contrapositive, the law of detachment,
and counterexamples.
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 .
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 2
TOPIC: REASONING, LINES, AND TRANSFORMATIONS
GEOMETRY
STANDARD G.2
The student will use the relationships between angles formed by two lines cut by a transversal to
a) determine whether two lines are parallel;
b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and
c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Use algebraic and coordinate methods as well as deductive proofs
to verify whether two lines are parallel.
Solve problems by using the relationships between pairs of angles
formed by the intersection of two parallel lines and a transversal
including corresponding angles, alternate interior angles, alternate
exterior angles, and same-side (consecutive) interior angles.
Solve real-world problems involving intersecting and parallel
lines in a plane.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 3
TOPIC: REASONING, LINES, AND TRANSFORMATIONS
GEOMETRY
STANDARD G.3
The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems
involving symmetry and transformation. This will include
a) investigating and using formulas for finding distance, midpoint, and slope;
b) applying slope to verify and determine whether lines are parallel or perpendicular;
c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and
d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Find the coordinates of the midpoint of a segment, using the
midpoint formula.
Use a formula to find the slope of a line.
Compare the slopes to determine whether two lines are parallel,
perpendicular, or neither.
Determine whether a figure has point symmetry, line symmetry,
both, or neither.
Given an image and preimage, identify the transformation that has
taken place as a reflection, rotation, dilation, or translation.
Apply the distance formula to find the length of a line segment
when given the coordinates of the endpoints.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 4
TOPIC: REASONING, LINES, AND TRANSFORMATIONS
GEOMETRY
STANDARD G.4
The student will construct and justify the constructions of
a) a line segment congruent to a given line segment;
b) the perpendicular bisector of a line segment;
c) a perpendicular to a given line from a point not on the line;
d) a perpendicular to a given line at a given point on the line;
e) the bisector of a given angle;
f) an angle congruent to a given angle; and
g) a line parallel to a given line through a point not on the given line.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Construct and justify the constructions of
– a line segment congruent to a given line segment;
– the perpendicular bisector of a line segment;
– a perpendicular to a given line from a point not on the line;
– a perpendicular to a given line at a point on the line;
– the bisector of a given angle;
– an angle congruent to a given angle; and
– a line parallel to a given line through a point not on the
given line.
Construct an equilateral triangle, a square, and a regular hexagon
inscribed in a circle.†
Construct the inscribed and circumscribed circles of a triangle.†
Construct a tangent line from a point outside a given circle to the
circle.†
†Revised March 2011
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 5
TOPIC: TRIANGLES
GEOMETRY
STANDARD G.5
The student, given information concerning the lengths of sides and/or measures of angles in triangles, will
a) order the sides by length, given the angle measures;
b) order the angles by degree measure, given the side lengths;
c) determine whether a triangle exists; and
d) determine the range in which the length of the third side must lie.
These concepts will be considered in the context of real-world situations.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Order the sides of a triangle by their lengths when given the
measures of the angles.
Order the angles of a triangle by their measures when given the
lengths of the sides.
Given the lengths of three segments, determine whether a triangle
could be formed.
Given the lengths of two sides of a triangle, determine the range
in which the length of the third side must lie.
Solve real-world problems given information about the lengths of
sides and/or measures of angles in triangles.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 6
TOPIC: TRIANGLES
GEOMETRY
STANDARD G.6
The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate
methods as well as deductive proofs.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Use definitions, postulates, and theorems to prove triangles
congruent.
Use coordinate methods, such as the distance formula and the
slope formula, to prove two triangles are congruent.
Use algebraic methods to prove two triangles are congruent.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 7
TOPIC: TRIANGLES
GEOMETRY
STANDARD G.7
The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate
methods as well as deductive proofs.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Use definitions, postulates, and theorems to prove triangles
similar.
Use algebraic methods to prove that triangles are similar.
Use coordinate methods, such as the distance formula, to prove
two triangles are similar.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 8
TOPIC: TRIANGLES
GEOMETRY
STANDARD G.8
The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of
special right triangles, and right triangle trigonometry.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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 is1
sin2
A ab 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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Determine whether a triangle formed with three given lengths is a
right triangle.
Solve for missing lengths in geometric figures, using properties of
45-45-90 triangles.
Solve for missing lengths in geometric figures, using properties of
30-60-90 triangles.
Solve problems involving right triangles, using sine, cosine, and
tangent ratios.
Solve real-world problems, using right triangle trigonometry and
properties of right triangles.
Explain and use the relationship between the sine and cosine of
complementary angles.†
†Revised March 2011
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 9
TOPIC: POLYGONS AND CIRCLES
GEOMETRY
STANDARD G.9
The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
The terms characteristics and properties can be used
interchangeably to describe quadrilaterals. The term
characteristics is used in elementary and middle school
mathematics.
Quadrilaterals have a hierarchical nature based on the
relationships between their sides, angles, and diagonals.
Characteristics of quadrilaterals can be used to identify the
quadrilateral and to find the measures of sides and angles.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Solve problems, including real-world problems, using the
properties specific to parallelograms, rectangles, rhombi, squares,
isosceles trapezoids, and trapezoids.
Prove that quadrilaterals have specific properties, using
coordinate and algebraic methods, such as the distance formula,
slope, and midpoint formula.
Prove the characteristics of quadrilaterals, using deductive
reasoning, algebraic, and coordinate methods.
Prove properties of angles for a quadrilateral inscribed in a circle.†
†Revised March 2011
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 10
TOPIC: POLYGONS AND CIRCLES
GEOMETRY
STANDARD G.10
The student will solve real-world problems involving angles of polygons.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A regular polygon will tessellate the plane if the measure of an
interior angle is a factor of 360.
Both regular and nonregular polygons can tessellate the plane.
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Solve real-world problems involving the measures of interior and
exterior angles of polygons.
Identify tessellations in art, construction, and nature.
Find the sum of the measures of the interior and exterior angles of
a convex polygon.
Find the measure of each interior and exterior angle of a regular
polygon.
Find the number of sides of a regular polygon, given the measures
of interior or exterior angles of the polygon.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 11
TOPIC: POLYGONS AND CIRCLES
GEOMETRY
STANDARD G.11 The student will use angles, arcs, chords, tangents, and secants to
a) investigate, verify, and apply properties of circles;
b) solve real-world problems involving properties of circles; and
c) find arc lengths and areas of sectors in circles.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Find lengths, angle measures, and arc measures associated with
– two intersecting chords;
– two intersecting secants;
– an intersecting secant and tangent;
– two intersecting tangents; and
– central and inscribed angles.
Calculate the area of a sector and the length of an arc of a circle,
using proportions.
Solve real-world problems associated with circles, using
properties of angles, lines, and arcs.
Verify properties of circles, using deductive reasoning, algebraic,
and coordinate methods.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 12
TOPIC: POLYGONS AND CIRCLES
GEOMETRY
STANDARD G.12
The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
A circle is a locus of points equidistant from a given point, the
center.
Standard form for the equation of a circle is
2 2 2x h y k r , 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 student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Identify the center, radius, and diameter of a circle from a given
standard equation.
Use the distance formula to find the radius of a circle.
Given the coordinates of the center and radius of the circle,
identify a point on the circle.
Given the equation of a circle in standard form, identify the
coordinates of the center and find the radius of the circle.
Given the coordinates of the endpoints of a diameter, find the
equation of the circle.
Given the coordinates of the center and a point on the circle, find
the equation of the circle.
Recognize that the equation of a circle of given center and radius
is derived using the Pythagorean Theorem.†
†Revised March 2011
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 13
TOPIC: THREE-DIMENSIONAL FIGURES
GEOMETRY
STANDARD G.13
The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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 student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Find the total surface area of cylinders, prisms, pyramids, cones,
and spheres, using the appropriate formulas.
Calculate the volume of cylinders, prisms, pyramids, cones, and
spheres, using the appropriate formulas.
Solve problems, including real-world problems, involving total
surface area and volume of cylinders, prisms, pyramids, cones,
and spheres as well as combinations of three-dimensional figures.
Calculators may be used to find decimal approximations for
results.
Mathematics Standards of Learning Curriculum Framework 2009: Geometry 14
TOPIC: THREE-DIMENSIONAL FIGURES
GEOMETRY
STANDARD G.14
The student will use similar geometric objects in two- or three-dimensions to
a) compare ratios between side lengths, perimeters, areas, and volumes;
b) determine how changes in one or more dimensions of an object affect area and/or volume of the object;
c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and
d) solve real-world problems about similar geometric objects.
ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
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.
The student will use problem solving, mathematical communication,
mathematical reasoning, connections, and representations to
Compare ratios between side lengths, perimeters, areas, and
volumes, given two similar figures.
Describe how changes in one or more dimensions affect other
derived measures (perimeter, area, total surface area, and volume)
of an object.
Describe how changes in one or more measures (perimeter, area,
total surface area, and volume) affect other measures of an object.
Solve real-world problems involving measured attributes of
similar objects.