GRADE 6 CURRICULUM GUIDE (Revised 2010) · Geometry Curriculum Guide ... (do one shape per week,...

Geometry

Math Curriculum Guide

Geometry Curriculum Guide

The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. 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 into sections: Curriculum Information, Essential Knowledge and Skills, Key

Vocabulary, Essential Questions and Understandings, Teacher Notes and Elaborations, Resources, and Sample Instructional Strategies and Activities.

The purpose of each section is explained below.

Curriculum Information:

This section includes the objective and SOL Reporting Category, focus or topic, and in some, not all, foundational objectives that are being built upon.

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 a list that limits what is 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.

Cognitive Level:

Blooms Taxonomy: What students must be able to do with what they know.

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 and Understandings:

This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the

objectives.

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. It may also contain definitions of key vocabulary to help facilitate student learning.

Resources:

This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources.

Sample Instructional Strategies and Activities:

The following chart is the pacing guide for the Dinwiddie Co Public Schools Geometry Curriculum. The chart provides the suggested number blocks to

teach each objective. The Dinwiddie Co Public Schools cross-content vocabulary terms that are in this course are: analyze, compare and contrast,

conclude, evaluate, explain, generalize, question/inquire, sequence, solve, summarize, and synthesize.

Resources:

This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources.

Sample Instructional Strategies and Activities:

This section lists ideas and suggestions that teachers may use when planning instruction.

The following chart is the pacing guide for the Dinwiddie Co Public Schools Geometry Curriculum. The chart outlines the order in which the objectives

should be taught; provides the suggested number blocks to teach each unit; and organizes the objectives into Units of Study. The Dinwiddie Co Public

Schools cross-content vocabulary terms that are in this course are: analyze, compare and contrast, conclude, evaluate, explain, generalize,

question/inquire, sequence, solve, summarize, and synthesize.

Geometry (Year Long) Pacing Guide

1st Nine Weeks (44D)

Sept 6 - Nov 4

# 2nd Nine Weeks (43D)

Nov 9 - Jan 27

# 3rd Nine Weeks (44D)

Jan 31 - Apr 4

# 4th Nine Weeks (34D)

Apr 5 – Jun 16

(SOLs May 25 & May 31)

#

G.1 Logic (Conditional,

Converse, Inverse,

Contrapositive, Venn Diagrams,

LOD, LOS, LOC, Symbolic

Form)

TEST REVIEW/TEST

Fundamentals of Geometry

(Basic Vocabulary)

TEST REVIEW/TEST

G.3a Midpoint/Distance

G.11 Circles

(All lines/segments within

circles as vocabulary)

TEST REVIEW/TEST

G.4 Constructions

TEST REVIEW/TEST

12

10

12

10

G.3b Slopes of Parallel and

Perpendicular

G.2 Parallel Lines cut by a

transversal

TEST REVIEW/TEST

G.3d Transformations

G.3c Symmetry

TEST REVIEW/TEST

G.5 Triangle Inequalities

(Triangle Angle-Sum, Isosceles

and Equilateral)

G.6 Proving Triangles

Congruent

(Congruency Statements,

Shortcuts)

TEST REVIEW/TEST

16

10

17

G.10 Polygons/G.9 Quads

(Angle-Sum, Interior/Exterior

Angles of Regular/Non-Regular,

Finding the number of sides given an

angle)

TEST REVIEW/TEST

G.13 Surface Area and Volume

TEST REVIEW/TEST

G.7 Similarity

G.14 Similarity of 3-D

TEST REVIEW/TEST

14

12

18

G.8 Right Triangles

TEST REVIEW/TEST

G.12 Equation of a Circle

G.11 Circles

TEST REVIEW TEST

12

15

Geometry (Semester) Pacing Guide

1st Nine Weeks

Sep 6—Nov 4 (first Semester)

Jan 31—Apr 4 (2nd Semester)

# of

days

44

2nd Nine Weeks

Nov 9—Jan 27 (first Semester)(SOLs Jan 11 & 12)

Apr 5—Jun 16 (second Semester) (SOLs May 25 & 31)

# of

days

43/45

G.1 Logic

(Conditional, Converse, Inverse, Contrapositive, Venn Diagrams,

LOD, LOS, LOC, Symbolic Form)

TEST REVIEW/TEST

Fundamentals of Geometry

(Basic Vocabulary)

G.3a Midpoint/Distance

G.11 Circles

(All lines/segments within circles as vocabulary)

TEST REVIEW/TEST

G.4 Constructions

(no test on this unit but include on every test for the rest of the

semester)

G.3b Slopes of Parallel and Perpendicular

G.2 Parallel Lines cut by a transversal

TEST REVIEW/TEST

G.3d Transformations

G.3c Symmetry

G.5 Triangle Inequalities

(Triangle Angle-Sum, Isosceles and Equilateral)

TEST REVIEW/TEST

9

12

4

9

10

**G.13 Surface Area and Volume

(do one shape per week, 5-10 minutes per day)**

G.6 Proving Triangles Congruent

(Congruency Statements, Shortcuts)

G.10 Polygons/G.9 Quads

(Angle-Sum, Interior/Exterior Angles of Regular/Non-Regular,

Finding the number of sides given an angle)

TEST

G.7 Similarity

G.14 Similarity of 3-D

G.8 Right Triangles

POST TEST

G.12 Equation of a Circle

G.11 Circles

9

12

6

GEOMETRY CURRICULUM GUIDE (Revised August 2016) Dinwiddie County Public Schools

. 3

GEOMETRY SOL TEST QUESTION BREAKDOWN (50 QUESTIONS TOTAL)

(Based on 2009 SOL Objectives and Reporting Categories)

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

Objective Page

G.1 Page 3

G.2 Page 11

G.3 Page 17

G.4 Page 23

G.5 Page 27

G.6 Page 31

G.7 Page 35

G.8 Page 39

G.9 Page 45

G.10 Page 49

G.11 3

G.12 Page 57

G.13 Page 61

G.14 Page 65

Curriculum Information

Essential Knowledge and Skills

Key Vocabulary

Essential Questions and Understandings

Teacher Notes and Elaborations

SOL Reporting Category

Reasoning, Lines and Transformations

Topic


Virginia 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.

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 .

Identify logically equivalent

statements.

Cognitive Level (Bloom’s Taxonomy, Revised)

Knowledge – identify

Apply – Construct, use

Understand – Translate

Analyze – Conclusion

Essential Questions

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?

Essential Understandings

Inductive reasoning, deductive reasoning, and proof are critical in establishing general

claims.

Deductive reasoning is the method that uses logic to draw conclusions based on

definitions, postulates, and theorems.

Inductive reasoning is the method of drawing conclusions from a limited set of

observations.

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.


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). A

mathematical statement is one in which a fact or complete idea is expressed. Because a

mathematical statement states a fact, many of them can be judged to be ―true‖ or ―false‖.

Questions and phrases are not mathematical statements since they can not be judged as true

or false.

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.

Proofs are developed so that each step in the argument is in proper chronological order in

relation to earlier steps. When building a proof the argument must be clearly developed and

each step must be supported by a property, theorem, postulate, or definition.

(continued)


. 5






Topic


Virginia SOL G.1

The student will construct and judge the

validity of a logical argument





statement;


into symbolic form;




Teacher Notes and Elaborations (continued)

The converse (a proposition produced by reversing position or order) of the conditional statement is formed by interchanging the

hypothesis and its conclusion.

If q (conclusion), then p (hypothesis).

q p p

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). ~ ~p q

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). ~ ~q p

Sentences, or statements, that have the same truth value are said to be logically equivalent. The contrapositive and original conditional

statements are logically equivalent (Law of Contrapositive). Since the statement and its contrapositve are both true or else both false, they

are called logically equivalent. The following statements are logically equivalent.

True statement: If a figure is a triangle, then it is a polygon.

True contrapositive: If a figure is not a polygon, then it is not a triangle.

The converse has the same truth value as the inverse of the original statement. The converse and the inverse of the original statement are

logically equivalent.

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° ABC is a right angle.(continued)






Topic


Virginia SOL G.1







statement;


into symbolic form;





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 of two simple statements joined by the word ―or‖. The symbol , is

used to represent the word ―or‖.

―Intersection‖ is the set of elements that are ―Union‖ is the set of elements that

elements of two or more given sets. belong to either or both of a given pair of sets.

p q p q

A biconditional statement is the conjunction of a conditional and its converse.

Symbolically: ( ) ( )p q q p is written ( )p q and is read p if and only if q or p iff q.

p

q

(continued)

q p q p


. 7






Topic


Virginia SOL G.1







statement;


into symbolic form;





Extension for 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 Conjunction Disjunction

If p then q p and q p or q

p q p q p q p q p q p q

T T T T T T T T T

T F F T F F T F T

F T T F T F F T T

F F T F F F F F F

Instruction should include completion of truth tables for compound statements such as ~ ( )u v w .

u v w ~ u v w ~ ( )u v w

T T T F T F

T T F F T F

T F T F T F

T F F F F F

F T T T T T

F T F T T T

F F T T T T

F F F T F F






Topic


Virginia SOL G.1







statement;


into symbolic form;






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. Generally, the word ―not‖ or the presence of a ―not symbol‖ (such as the not equal

sign) in a problem indicates the need for an indirect proof. When formulating an indirect proof first assume that the opposite of what is to

be proven is true. Next, from this assumption, determine what conclusions can be drawn. These conclusions must be based upon the

assumption and the use of valid statements. Search for a conclusion that is known to be false because it contradicts given or known

information. Since the assumption leads to a false conclusion, the assumption must be false. Therefore if the assumption (which is the

opposite of what is to be proven) is false, then what is being proven must be true.

Indirect Proof Example:

ABC is not isosceles. Prove that if altitude BD is drawn, it will not bisect AC .

B

Given: ABC is not isosceles

altitude BD

Prove: BD does not bisect AC

A D C

STATEMENTS REASONS

1. ABC is not isosceles

altitude BD

1. Given

2. Assume BD bisects AC 2. Assumption leading to a contradiction.

3. D is the midpoint of AC 3. Bisector of a segment divides the segment at its midpoint.

4. AD DC 4. Midpoint divides a segment into two congruent segments.

5. BD AC 5. The altitude of a triangle is a line segment extending from any vertex of a triangle

perpendicular to the line containing the opposite side.

6. ADB, BDC are right angles 6. Perpendicular lines meet to form right angles

7. ADB BDC 7. All right angles are congruent.

8. BD BD 8. Reflexive Property

9. ADB CDB 9. SAS - If two sides and the included angle of one triangle are congruent to the

corresponding parts of a second triangle, the two triangles are congruent.

10. AB BC 10 CPCTC - Corresponding parts of congruent triangles are congruent.

11. ABC is isosceles 11 An isosceles triangle is a triangle with two congruent sides.

12. BD does not bisect AC 12 Contradiction steps 1 and 11


. 9


Resources Sample Instructional Strategies and Activities



Topic


Virginia SOL G.1

Foundational Objectives

8.2

The student will describe orally and in

writing the relationships between the

subsets of the real number system.

Text:

Geometry

Prentice Hall Geometry, Virginia

Edition, ©2012, Charles et al., Pearson

Education

VDOE Enhanced Scope and Sequence

Sample Lesson Plans

http://www.doe.virginia.gov/testing/sol/sco

pe_sequence/mathematics_2009/index.php

Virginia Department of Education Website

http://www.doe.virginia.gov/instruction/ma

thematics/index.shtml

Geometry reference

http://www.mathopenref.com/

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.

http://www.doe.virginia.gov/testing/sol/scope_sequence/mathematics_2009/index.php


http://www.doe.virginia.gov/instruction/mathematics/index.shtml



This page is intentionally left blank.


. 11



Key Vocabulary





Topic


Virginia SOL G.2

The student will use the relationships

between angles formed by two lines cut

by a transversal

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.



mathematical reasoning, connections and

representations to:

Use properties, postulates, and theorems

to determine whether two lines are

parallel.

Use algebraic and coordinate methods as

well as deductive proofs to verify

whether two lines are parallel.

State the relationships between angles

that are a linear pair.

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.

Identify lines as parallel, intersecting,

perpendicular, or skew.

Use definitions, postulates, and theorems

to complete two-column or paragraph

proofs with at least five steps.

Cognitive Level (Bloom’s Taxonomy,

Revised)

Remember – Add, Subtract, Multiply,

Divide

Understand – Use, Identify

Apply – Solve

Evaluate – Verify


Write equations of parallel and

perpendicular lines.

Investigate skew lines using real world

models.

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? Essential Understandings

Parallel lines intersected by a transversal form angles with specific relationships.

Some angle relationships may be used when proving two lines intersected by a

transversal are parallel.

The Parallel Postulate differentiates Euclidean from non-Euclidean geometries such as

spherical geometry and hyperbolic geometry.


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.


Skew lines are non-coplanar lines that do not intersect. Experiences with skew lines should

include 3-dimensional models.

Intersection is a point or set of points common to two or more figures.



Key Vocabulary





Topic


Virginia SOL G.2



by a transversal


parallel;







Key Vocabulary

adjacent angles

algebraic method

alternate exterior angles

alternate interior angles

complementary angles

coordinate method

corresponding angles

deductive proof

Euclidean Geometry

exterior angle

interior angle

intersection

linear pair

parallel lines

Parallel Postulate

consecutive interior angles

same-side (consecutive) exterior angles

supplementary angles

transversal

vertical angles


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. Consecutive exterior angles are on the same side 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,

- same side (consecutive) exterior 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.

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. In a paragraph proof (informal

proof) a paragraph is written to explain why a conjecture for a given situation is true.

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.

(continued)


. 13






Topic


Virginia SOL G.2



by a transversal


parallel;








The following is an example of a paragraph proof.

Given: E is the midpoint of BD B C

AE ED E

Prove: AEB CED

A D

In the figure above the facts that E is the midpoint of BD and AE ED is given. Since E is the midpoint of BD , then BE ED because

the midpoint of a segment divides the segment into two congruent segments. Since vertical angles are congruent BEA DEC , there is

now sufficient information to satisfy the SAS method of proving triangles congruent. Therefore, AEB CED because if two sides and

the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

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 .

(continued)






Topic


Virginia SOL G.2



by a transversal


parallel;








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 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.

4. MN || BA ; two nonvertical lines are parallel if and only if their slopes are equal.


. 15





Topic


Virginia SOL G.2


A.4

The student will solve multi-step linear

and quadratic equations in two

variables, including

a. solving literal equations (formulas)

for a given variable; and

d. solving multi-step linear equations

algebraically and graphically.

A.6

The student will graph linear equations

and linear inequalities in two variables,

including

a. determining the slope of a line when

given an equation of the line, the

graph of the line, or two points on

the line. Slope will be described as

rate of change and will be positive,

negative, zero, or undefined; and

b. writing the equation of a line when

given the graph of the line, two

points on the line, or the slope and a

point on the line.

8.6

The student will

a. verify by measuring and describe

the relationships among vertical

angles, adjacent angles,

supplementary angles, and

complementary angles; and

b. measure angles of less than 360°.

Text:

Geometry Prentice Hall Geometry,

Virginia Edition, ©2012, Charles et al.,

Pearson Education


Sample Lesson Plans






Geometry reference


Foundational Objectives (continued)

8.10

The student will

a. verify the Pythagorean Theorem; and

b. apply the Pythagorean Theorem.

8.15

The student will

a. solve multi-step linear equations in one

variable on one and two sides of the

equation.

8.16

The student will graph a linear equation in

two variables.

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.

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.

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).

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


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.







. 17



Key Vocabulary





Topic


Virginia 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.





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.

Find the coordinates of the midpoint of

a segment, using the midpoint formula.

Use a formula to find the slope of a

line.

Determine whether a figure has point

symmetry, line symmetry, both, or

neither.

Compare the slopes to determine

whether two lines are parallel,

perpendicular, or neither.

Use algebraic, coordinate, and

deductive methods to determine if

lines are perpendicular.

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.

Find the coordinates of an endpoint of

a segment given the coordinates of the

midpoint and one endpoint.


Understand – Identify, Use

Apply – Reflect

Analyze - Compare

Evaluate – Determine

Create - Draw

Essential Questions

What is the relationship between the distance formula and the Pythagorean Theorem?

How does the concept of midpoint and slope relate to symmetry and transformation?

What is line symmetry?

When is a figure symmetric about a point?

What types of symmetrical problems are found in real-life?

How is a figure translated, reflected, rotated, or dilated?


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.


Transformations and combinations of transformations can be used to describe movement.

The Pythagorean Theorem states that in a right triangle the square of the hypotenuse is

equal to the sum of the squares of the legs. Pythagorean Triples are three positive integers

that satisfy the Pythagorean theorem. The converse of the Pythagorean Theorem guarantees

that a, b, and c are lengths of the sides of a right triangle. Because of this, any such triple of

integers is called a Pythagorean triple. For example, 3, 4, 5 is a Pythagorean triple since 2 2 23 4 5 . Another triple is 6, 8, 10, since 2 2 26 8 10 . The triple 3, 4, 5 is called a

primitive Pythagorean triple because no factor (other than 1) is common to all three

integers. 6, 8, 10 is not a primitive triple. Other primitive triples are 5, 12, 13; 8, 15, 17; and

7, 24, 25. Students should recognize these primitive triples in order to use them to create

other triples such as 9, 12, 15, which is found by multiplying each measure in 3, 4, 5 by a

factor of 3.

Two situations must be considered when finding the distance between two points: the

distance on a number line (2 1x x ) and the distance in the coordinate plane (distance

formula or Pythagorean Theorem). (continued)



Key Vocabulary





Topic


Virginia SOL G.3






include



slope;



or perpendicular;




point; and




(continued)

Extensions for Geometry

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.

Represent translations, reflections, and

rotations using algebraic and/or

coordinate notation.

Apply the Pythagorean Theorem to a

right triangle in the coordinate plane to

derive the distance formula.

Key Vocabulary dilation

distance formula

image

line symmetry

midpoint

midpoint formula

parallel lines

parallel planes

perpendicular lines

point symmetry

pre-image

Pythagorean Theorem

reflection

rotation

slope

slope formula

symmetry

transformation

translation


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.

-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) .

(continued)


. 19






Topic


Virginia SOL G.3






include



slope;



or perpendicular;




point; and





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.

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

.


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 . (continued)






Topic


Virginia SOL G.3






include



slope;



or perpendicular;




point; and





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

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.


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.

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. When a point is reflected

across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite. When a point is reflected across

the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. When a point is reflected across the line

y x , then the x-coordinate and the y-coordinate change places. When a point is reflected across the line y x , the x-coordinate and the

y-coordinate change places and are negated (the signs are changed).

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.

(continued)


. 21






Topic


Virginia SOL G.3






include



slope;



or perpendicular;




point; and





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.


Translations, reflections, and rotations can be represented using algebraic and/or coordinate notation.

Line Reflections:

Reflection in the x-axis: When a point is reflected across the x-axis, the x-coordinate remains the same, but the y-coordinate is

transformed into its opposite.

( , ) '( , ) or ( , ) ( , )x axisP x y P x y r x y x y

Reflection in the y-axis: When a point is reflected across the y-axis, the y-coordinate remains the same, but the x-coordinate is

transformed into its opposite.

( , ) '( , ) or r ( , ) ( , )y axisP x y P x y x y x y

Reflection in y x : When a point is reflected across the line y x , then the x-coordinate and the y-coordinate change places.

( , ) '( , ) or ( , ) ( , )y xP x y P y x r x y y x

Reflection in y x : When a point is reflected across the line y x , the x-coordinate and the y-coordinate change places and

are negated (the signs are changed).

( , ) '( , ) or ( , ) ( , )y xP x y P y x r x y y x

Rotations: (assuming center of rotation to be the origin)

Rotation of 90:

90( , ) ( , )R x y y x

Rotation of 180:

180( , ) ( , )R x y x y

Rotation of 270:

270( , ) ( , )R x y y x





Topic


Virginia SOL G.3


A.4a, d, f




a. solving literal equations (formulas)

for a given variable;


algebraically and graphically; and

f. solving real-world problems

involving equations and systems of

equations.

A.6



including










point on the line.

8.8

The student will

a. apply transformations to plane

figures; and

b. identify applications of

transformations.

(continued)

Text:



Pearson Education


Sample Lesson Plans

http://www.doe.virginia.gov/testing/sol/

scope_sequence/mathematics_2009/ind

ex.php




Geometry reference

http://www.regentsprep.org/regents/math/g

eometry/math-GEOMETRY.htm

Geometry reference


Foundational Objectives (continued)

8.10

The student will

a. verify the Pythagorean Theorem; and


8.15

The student will

a. solve multi-step linear equations in one

variable on one and two sides of the

equation.

8.16

The student will graph a linear equation in

two variables.

7.8

The student, given a polygon in the

coordinate plane, will represent

transformations (reflections, dilations,

rotations, and translations) by graphing in

the coordinate plane.

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.

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.






http://www.regentsprep.org/regents/math/geometry/math-GEOMETRY.htm

http://www.regentsprep.org/regents/math/geometry/math-GEOMETRY.htm



. 23



Key Vocabulary





Topic Reasoning, Lines, and Transformations

Virginia 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.





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 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


line.

Construct and justify an equilateral

triangle, a square, and a regular

hexagon including those inscribed in a

circle.

Construct the inscribed and

circumscribed circles of a triangle.

Construct and justify a tangent line

from a point outside a given circle to

the circle.


Create - Construct

Evaluate – Justify

(continued)

Essential Questions

What is the relationship between points, rays, and angles?

Why are constructions important?

How are constructions justified? Essential Understandings

Construction techniques are used to solve real-world problems in engineering,

architectural design, and building construction.

Construction techniques include using a straightedge and compass, paper folding, and

dynamic geometry software.


"Construction" in geometry means to draw shapes, angles or lines accurately. Constructions

are done using tools including software programs such as Sketch Pad, 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 tangents, geometric mean, and proportional

segments.

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 and the midpoint of the

side opposite that vertex. The medians of a triangle intersect at the common point called the

centroid. (continued)



Key Vocabulary






Virginia SOL G.4




line segment;


segment;







and



line.

(continued)


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.

Justify the constructions of:

- angles with measures of 15, 30, 45,

60, 75, and 135 degrees; and

- a tangent to a circle through a point

on the circle.

Given a segment, by construction,

divide the segment into a given number

of congruent parts.

Construct and justify a triangle similar

to a given triangle on a given line

segment as the base.

Key Vocabulary

angle bisector

centroid (concurrency of medians of a

triangle)

circumcenter

circumscribed

compass

construction

incenter

inscribed

intersection

parallel lines

perpendicular bisector

perpendicular lines

segment bisector

straightedge

transversal


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 congruent Radii of equal circles are equal

to a given a line segment

2. Construct an angle congruent to a given Radii of equal circles are equal

angle SSS Postulate

Corresponding parts of congruent

triangles are congruent

3. Construct the bisector of a given angle Radii of equal circles are equal

SSS Postulate

Corresponding parts of congruent

triangles are congruent

Definition of an angle bisector

4. Construct the perpendicular bisector Radii of equal circles are equal

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 perpendicular

bisector of the line segment

(continued)


. 25







Virginia SOL G.4




line segment;


segment;







and



line.


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

5. Construct the perpendicular to a line Radii of equal circles are equal

at the given point on the line. Definition of a straight angle

Definition of an angle bisector

Definition of right angles and definition of perpendicular lines

6. Construct the perpendicular to the line from a point Radii of equal circles are equal

not on the line. If a point is equidistant from the endpoints of a line segment, then

the point lies on the perpendicular bisector of the line

7. Construct the parallel to a given line though a given point Radii of equal circles are equal

not on the line. If two lines are cut by a transversal and corresponding angles are

congruent, then the lines are parallel






Virginia SOL G.4


Text:



Pearson Education


Sample Lesson Plans






Geometry reference


Animated geometric constructions

http://www.mathsisfun.com/geometry/cons

tructions.html






http://www.mathsisfun.com/geometry/constructions.html

http://www.mathsisfun.com/geometry/constructions.html


. 27



Key Vocabulary




Triangles

Topic Triangles

Virginia 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.





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.


Apply – Simplify, Factor

Analyze - Order

Evaluate – Verify


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.

Key Vocabulary opposite ordering sides and angles of triangles Triangle Inequality Theorem

Scalene

Isosceles

Equilateral

Acute

Essential Questions

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?


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.


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.

(continued)

Obtuse

Right

Straight

Triangle Sum Theorem

Exterior Angle Theroem


. 29





Triangles

Topic Triangles

Virginia 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.



Using properties of triangles, inequalities can be written relating pairs of angles or segment measures.

A

A 10 95°

B 51° B

5 13 65°

34°

C 18

C 55°

15

D 60°

D

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.




Triangles

Topic Triangles

Virginia SOL G.5


Text:



Pearson Education


Sample Lesson Plans






Geometry reference


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.







. 31




Key Vocabulary




Triangles

Topic Triangles

Virginia SOL G.6

The student, given information in the

form of a figure or statement, will prove

two triangles are congruent, using


well as deductive proofs.





Use definitions, postulates, and

theorems to prove triangles are

congruent (including Hypotenuse-Leg

Postulate).

Use algebraic methods to prove two

triangles are congruent.

Use coordinate methods, such as the

distance formula and the slope formula,

to prove two triangles are congruent.

Use angle bisectors, medians, altitudes,

perpendicular bisectors to prove

triangles congruent.


Understand – Use

Evaluate – Prove


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)).

Key Vocabulary

AAS Theorem

algebraic methods

altitude

ASA Postulate

Congruent triangles

coordinate methods

corresponding parts

deductive proof

definition (continued)

Essential Questions

What are congruent triangles?

What are the one-to-one correspondences that prove triangles congruent?

What can be deduced from congruent triangles? Essential Understandings

Congruence has real-world applications in a variety of areas, including art, architecture,

and the sciences.

Congruence does not depend on the position of the triangle.

Concepts of logic can demonstrate congruence or similarity.

Congruent figures are also similar, but similar figures are not necessarily congruent.


When two figures have exactly the same shape and size, they are said to be congruent.

Using algebraic methods, if all corresponding parts 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.

(continued)


. 33



Key Vocabulary




Triangles

Topic Triangles

Virginia SOL G.6



two triangles are congruent, using



Key Vocabulary (continued)

distance formula

HL Postulate

hypotenuse

included angle

included side

leg

median of a triangle

postulate

properties

SAS Postulate

slope formula

SSS Postulate

theorem


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.

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.


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 right triangle.

LA Theorem: One leg and an acute angle of one right triangle are congruent to the

corresponding parts 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

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.




Triangles

Topic Triangles

Virginia SOL G.6


A.4d





algebraically and graphically.

8.10

The student will

a. verify the Pythagorean Theorem;

and


Text:



Pearson Education


Sample Lesson Plans






Geometry reference


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.







. 35




Key Vocabulary




Triangles

Topic Triangles

Virginia SOL G.7



two triangles are similar, using








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.

Use similar relationships between

triangles to solve real-world problems.


theorems to complete two-column

proofs with at least five steps.


Understand – Use

Apply – Solve

Evaluate – Prove



theorems to complete paragraph proofs

with at least five steps.

Investigate proportionality in a triangle

intersected by three or more parallel

lines.

Investigate the Golden Ratio.

Key Vocabulary angle bisector

congruent

deductive proof

definition

distance formula

included angle

median of a triangle

postulate

properties

proportion

ratio

scale factor

similar triangles (AA Similarity, SSS

Similarity, SAS Similarity) theorem

Essential Questions

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?

What is the relationship between segments when a line intersects two sides of a triangle

and is parallel to the third side?

What is the relationship between segments when an angle is bisected?


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.


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).

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)


. 37






Triangles

Topic Triangles

Virginia SOL G.7







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.

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 (continued)





. 39


Triangles

Topic Triangles

Virginia SOL G.7





well as deductive proofs


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 f

e a

f b


If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

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.




Triangles

Topic Triangles

Virginia SOL G.7


8.3

The student will

solve practical problems involving

rational numbers, percents, ratios, and

proportions

7.4

The student will solve single-step and

multi-step practical problems, using

proportional reasoning.

7.6

The student will determine whether

plane figures (quadrilaterals and

triangles) are similar and write

proportions to express the relationships

between corresponding sides of similar

figures.

6.1

The student will describe and compare

data, using ratios, and will use

appropriate notations such as a

b, a to b,

and a:b.

Text:



Pearson Education


Sample Lesson Plans






Geometry reference


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 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.







. 41



Key Vocabulary




Triangles

Topic Triangles

Virginia 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.





Determine whether a triangle formed

with three given lengths is a right,

obtuse, or acute 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.

Explain and use the relationship

between the sine and cosine of

complementary angles.

Solve real-world problems using right

triangle trigonometry and properties of

right triangles.

Express linear measurements as

simplified radicals and decimal

approximations.


Remember – Express

Apply – Solve

Evaluate – Verify

Create - Explain


Use the Law of Sines and the Law of

Cosines to find missing measures in

triangles.

Find the geometric mean in right

triangles.

Essential Questions

What are the different ways of finding missing sides and angles of triangles?

How do special right triangle theorems apply?

What is geometric mean?

What is a trigonometric ratio?


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 1

sin2

A ab C .

The ratios of side lengths in similar right triangles adjacent opposite

or hypotenuse 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.


Right triangles (any triangle with one 90° angle) are triangles with 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.

When solving for lengths of sides in special right triangles rationalizing the denominator

may be needed. (For most students this is the introduction to this topic.)

Key Vocabulary

acute triangle (continued)

angle of depression

angle of elevation

area of a triangle

Special Right Triangles (30-60-90, 45-90)

(continued)


. 43



Key Vocabulary




Triangles

Topic Triangles

Virginia SOL G.8

The student will solve real world





trigonometry.


cosine

geometric mean

hypotenuse

obtuse triangle

Pythagorean Theorem

ratio

right triangle

similar right triangle

sine

tangent

trigonometry

45°-45°-90° triangle

30°-60°-90° triangle


Rationalizing a denominator is a procedure for transforming a quotient with a radical in the

denominator into an expression with no radical in the denominator. The following are

examples of rationalizing the denominator of radical expressions.

Example 1: 3

3 3 3

x x

(Multiply by 1.)

3

3 3

x

3

3

x

Example 2: 3 5 3 5 2

2 2 2

x x (Multiply by 1.)

3 10

2

x

Example 3: 2 2 1 3

1 3 1 3 1 3

(Use the conjugate of 1 3 to multiply

by 1.)

2 2 3

1 3 3 3

2 2 3

2

1 3

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.

(continued)





. 45


Triangles

Topic Triangles

Virginia SOL G.8






trigonometry.


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. 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.

(continued)





Triangles

Topic Triangles

Virginia SOL G.8






trigonometry.


Extension for 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 ) sin sin sinA B C

a b c . This law is often used if two angles and a side are known (AAS or ASA).

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.

a h b x a

a c and

y b

b c

x y

c




. 47


Triangles

Topic Triangles

Virginia SOL G.8


A.3

The student will express the square

roots and cube roots of whole numbers

and the square root of a monomial

algebraic expression in simplest radical

form.

8.3

The student will



proportions

8.5 The student will

a. determine whether a given number

is a perfect square; and

b. find the two consecutive whole

numbers between which a square

root lies.

8.10

The student will


and


In middle school, area of a triangle is

found and applied using the formula

1

2A bh .

Text:



Pearson Education


Sample Lesson Plans






Geometry reference


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.







. 49



Key Vocabulary




Polygons, Circles, and Three-

Dimensional Figures

Topic

Polygons and Circles

Virginia SOL G.9

The student will verify characteristics

of quadrilaterals and use properties of

quadrilaterals to solve real-world

problems.





Solve problems, including real-world

problems using the properties specific

to parallelograms, rectangles, rhombi,

squares, isosceles trapezoids and

trapezoids.

Prove that quadrilaterals have specific

properties, using coordinate and

algebraic methods, such as the distance

formula, slope and midpoint formula.

Prove properties of angles for a

quadrilateral inscribed in a circle.

Prove the characteristics of

quadrilaterals, using deductive

reasoning, algebraic, and coordinate

methods.


Apply – Solve

Evaluate – Prove


Investigate and identify the

quadrilaterals formed by connecting the

midpoints of the sides of a given

quadrilateral.

Key Vocabulary

base angles

characteristics

diagonal

isosceles trapezoid

kite

legs

median of a trapezoid

parallelogram (properties)

quadrilateral

rectangle square

rhombus 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? Essential Understandings

The terms characteristics and properties can be used interchangeably to describe

quadrilaterals. The term characteristics is used in elementary and middle school

mathematics.

Quadrilaterals have a hierarchical nature based on the relationships between their sides,

angles, and diagonals.

Characteristics of quadrilaterals can be used to identify the quadrilateral and to find the

measures of sides and angles.


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.

(continued)


. 51






Dimensional Figures

Topic


Virginia SOL G.9

The student will verify characteristics

of quadrilaterals and use properties of

quadrilaterals to solve real-world

problems.


A kite is a quadrilateral with two pairs of congruent adjacent sides.

Characteristics of quadrilaterals are used to identify figures, and to find values for missing parts and areas.

The hierarchical nature of quadrilaterals can be described as ranking based on characteristics.

Areas of work that use quadrilaterals include art, construction, fabric design, and architecture.

Quadrilaterals

Parallelograms

Trapezoids

Rectangles Squares Rhombi Kites

Isosceles

Trapezoids

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.

Example:

Quadrilateral ABCD is inscribed in a circle. AB BC CD DA 360m m m m .

The measure of 1DAB = BCD

2m 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

A C 180m m

D C





Dimensional Figures

Topic


Virginia SOL G.9


7.7

The student will compare and contrast

the following quadrilaterals based on

properties: parallelogram, rectangle,

square, rhombus, and trapezoid.

6.13

The student will describe and identify

properties of quadrilaterals.

Text:



Pearson Education


Sample Lesson Plans



PWC Mathematics Website

http://pwcs.math.schoolfusion.us




Geometry reference


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.

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.



http://pwcs.math.schoolfusion.us/





. 53




Key Vocabulary





Dimensional Figures

Topic


Virginia SOL G.10


problems involving angles of polygons.






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.

Investigate and identify the regular

polygons that tessellate.


Remember – Find

Understand – Identify

Apply – Solve

Evaluate – Investigate


Distinguish between pure and semi-

pure tessellations.

Key Vocabulary

concave heptagon

convex hexagon

decagon exterior angle

diagonal interior angle

dodecagon

linear pair

n-gon

nonagon

(continued)

Essential Questions

What are the distinguishing characteristics of a polygon?

What is a regular polygon?

What is the relationship between the interior and exterior angles of polygons?

What is the relationship between the number of sides of a polygon and its angles?

What are tessellations?


A regular polygon will tessellate the plane if the measure of an interior angle is a factor

of 360.

Both regular and non-regular polygons can tessellate the plane.

Two intersecting lines form angles with specific relationships.

An exterior angle is formed by extending a side of a polygon.

The exterior angle and the corresponding interior angle form a linear pair.

The sum of the measures of the interior angles of a convex polygon may be found by

dividing the interior of the polygon into non-overlapping triangles.


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)


. 55






Dimensional Figures

Topic


Virginia SOL G.10


problems involving angles of polygons.


Octagon

pentagon

polygon

quadrilateral

regular polygon

Similar polygons

tessellation

triangle

Polygon Exterior Angle Sum Theorem

Polygon Interior Angle Sum Theorem


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.

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


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.





Dimensional Figures

Topic


Virginia SOL G.10


A.4








equations.

Text:



Pearson Education


Sample Lesson Plans






Geometry reference


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.

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







. 57




Key Vocabulary





Dimensional Figures

Topic


Virginia 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.





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.

Find the area of the region between

concentric circles.


Remember – Find

Apply – Simplify, Factor

Analyze - Calculate

Evaluate – Verify


Find the area of a segment of a circle.

Find the probability that a point chosen

at random in a figure is in a shaded

region.

Key Vocabulary

arc chord

arc length circle

arc measure

central angle

circumference (continued)

Essential Questions

What are the relationships between angle and arc measures?

What are the relationships among the lengths of secant segments, tangent segments, and

chords?

What are the relationships between chords and arcs?

What is the relationship between a central angle and the area of a sector?

What is the relationship between a central angle and the length of an arc?

What is the difference between arc length and arc measure?


Many relationships exist between and among angles, arcs, secants, chords, and tangents

of a circle.

All circles are similar.

A chord is part of a secant.

Real-world applications may be drawn from architecture, art, and construction.


A 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.

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.

(continued)


. 59



Key Vocabulary





Dimensional Figures

Topic


Virginia SOL G.11








in circles.


diameter

inscribed angle

intercepted arc

major arc

minor arc

secant

Are of a sector

semicircles

tangent

central angels

segments in circles

Segments of Secants Theorem

Segment of Secants and Tangents Theorem


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.)

Length of AB 2360

xr (This is a linear measure.)

Area of sector 2AOB

360

xr

Experiences using a measure of one part of the circle to find measures of other parts of the

circle should be included.

Verifying the properties of circles may include definitions, postulates, theorems, algebraic

methods, and coordinate methods.

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. (continued)






Dimensional Figures

Topic


Virginia SOL G.11








in circles.


An angle inscribed in a semi-circle is a right angle. Opposite angles of an inscribed quadrilateral are supplementary.

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.

Real world problems do not always include figures. Experiences drawing a figure to represent the problem should be provided.


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

Given a figure with a shaded region, students will find the probability that a point chosen at random will be in the shaded region. Find the

area of the figures then write a ratio of the area of the shaded region to the area of the entire figure. Figures should include circles as well

as polygons such as the following.

Probabilities can be written as ratios, decimals, or percents.


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Dimensional Figures

Topic


Virginia SOL G.11


A.4








equations.

8.11 The student will solve practical area

and perimeter problems involving

composite plane figures.

6.10a, b, c

The student will

a. define pi (π) as the ratio of the

circumference of a circle to its

diameter;

b. solve practical problems involving

circumference and area of a circle,

given the diameter or radius; and

c. solve practical problems involving

area and perimeter.

Text:



Pearson Education


Sample Lesson Plans






Geometry reference


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.








Key Vocabulary





Dimensional Figures

Topic


Virginia 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.





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 a given center and radius is derived

using the Pythagorean Theorem.


Investigate and identify points that lie

inside or outside a circle.

Write inequality statements for regions

either inside or outside a circle and

sketch these graphs.

Investigate and write the equation of a

circle given three points on the circle.

Key Vocabulary conic section coordinates of the center locus standard form for the equation of a circle

Essential Questions

What is the relationship between the center, the radius, and the standard equation of a

circle?

What is the relationship between distance formula and the equation of a circle?

What is a conic section?


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 2( ) ( )x 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.

Teacher Notes and Elaborations Locus means a figure that is the set of all points, and only those points, that satisfy one or more conditions. The Pythagorean Theorem (distance formula) can be used to develop an equation of a

circle.

Let P(x, y) represent any point on the circle.

The distance between C(h, k) and P(x, y) is r.

y

2 2( ) ( )x h y k r

2 2 2( ) ( )x h y k r

P(x, y)

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

(continued)


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Dimensional Figures

Topic


Virginia SOL G.12




circle.


Example: Given the coordinates of the center of a circle, ( 2,6) , with radius 3

four points on the circle are ( 2 3,6) , ( 2 3,6) , ( 2,6 3) , ( 2,6 3) or

(1,6) ( 5,6) ( 2,9) ( 2,3)

Given an equation for a circle, substitute coordinates of a point to determine the location of the point in reference to the circle. The

locations of points are inside a circle, on the circle, or outside the circle.

Example: Given the center of the circle ( 7,5) and the radius of the circle is 6, determine whether the following points are inside, on, or

outside the circle.

Find the equation of a circle: 2 2( 7) ( 5) 36x y

Given point ( 3,2) 25 < 36 therefore ( 3,2) is inside the circle.

Given point (6, 4) 250 > 36 therefore (6, 4) is outside the circle.

Given point ( 1,5) 36 = 36 therefore ( 1,5) is on the circle.

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.

A conic section is one of a group of curves formed by the intersection of a plane and a right circular cone. The curve is a circle if the plane

is parallel to the base of the cone.

2 2

2 2

( 3 7) (2 5)

4 ( 3)

16 9

25

2 2 2 2(6 7) ( 4 5) 13 ( 9)

169 81

= 250

2 2 2( 1 7) (5 5) 6 0

36 0

=36

(continued)





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Dimensional Figures

Topic


Virginia SOL G.12




circle.



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.

Given three points on a circle, students investigate how to use the slope formula, midpoint formula, equation of a line formula, and

distance formula to find the equation of the circle. A source for this investigation can be found at

http://www.regentsprep.org/regents/math/geometry/GCG6/RCir.htm

http://www.regentsprep.org/regents/math/geometry/GCG6/RCir.htm





Dimensional Figures

Topic


Virginia SOL G. 12


A.6



including










point on the line.

8.10

The student will


and


Text:



Pearson Education


Sample Lesson Plans






Geometry reference


Use conic models to demonstrate that a circle is the result of cutting a cone with a plane

parallel to the base.



Key Vocabulary









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Dimensional Figures

Topic

Three-Dimensional Figures

Virginia SOL G.13

The student will use formulas for

surface area and volume of three-

dimensional objects to solve real-world

problems.





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.


Remember – Find

Apply – Solve

Analyze - Calculate

Key Vocabulary

altitude surface area

area vertex

area of the base (B) volume

base three-dimensional

cone two-dimensional

cylinder

face

height

lateral edge

lateral area

prism

polygon

polyhedron

pyramid

Similar Solids Theorem

slant height

sphere

Essential Questions

What are the lateral area, surface area, and volume of the following figures: prisms,

cylinders, pyramids, cones, and spheres?


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 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 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.

(continued)






Dimensional Figures

Topic


Virginia SOL G.13

The student will use formulas for

surface area and volume of three-

dimensional objects to solve real-world

problems.


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.

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.




. 69



Dimensional Figures

Topic


Virginia SOL G.13



a. investigate and solve practical

problems involving volume and

surface area of prisms, cylinders,

cones, and pyramids; and

b. describe how changing one

measured attribute of the figure

affects the volume and surface area.

8.9

The student will construct a three-

dimensional model given the top or

bottom, side and front views.

7.5

The student will

a. describe volume and surface area of

cylinders;


the volume and surface area of

rectangular prisms and cylinders;

c. describe how changing one

measured attribute of a rectangular

prism affects its volume and surface

area;

6.10d

The student will

d. describe and determine the volume

and surface area of a rectangular

prism.

Text:



Pearson Education


Sample Lesson Plans






Geometry reference


Students will draw and cut out regular polygons and tape them together to make three-

dimensional objects. Colored paper may be used for effect.

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.

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.







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Key Vocabulary





Dimensional Figures

Topic


Virginia 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.





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.


measured attributes of similar objects.

Compare ratios between side lengths,

perimeters, areas, and volumes, given

two similar figures.


Remember – Describe

Apply – Solve

Analyze - Compare

Key Vocabulary constant ratio similar figures

Essential Questions

How does a change in dimensions affect the area and/or volume of the object?

How does a change in area and/or volume affect other measures?

In similar figures, how does a change of one measurement affect perimeter, area, or

volume?


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.


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.

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:b

2.

– The ratio of the volumes is a3:b

3.


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Dimensional Figures

Topic


Virginia SOL G.14


8.3

The student will



proportions.


a. investigate and solve practical

problems involving volume and

surface area of prisms, cylinders,

cones, and pyramids; and

b. describe how changing one

measured attribute of the figure

affects the volume and surface area.


a. describe volume and surface area of

cylinders;


the volume and surface area of

rectangular prisms and cylinders;

and

c. describe how changing one

measured attribute of a rectangular

prism affects its volume and surface

area.

7.6 The student will determine whether

plane figures (quadrilaterals and

triangles) are similar and write

proportions to express the relationships

between corresponding sides of similar

figures.

Text:



Pearson Education


Sample Lesson Plans






Geometry reference


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.

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.







. 75

NOTES

GRADE 6 CURRICULUM GUIDE (Revised 2010) · Geometry Curriculum Guide ... (do one shape per week,...

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