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ACPS Curriculum Framework 2014-15

Introduction The Mathematics Curriculum Framework serves as a guide for teachers when planning instruction and assessments. It defines the content knowledge, skills, and understandings that are measured by the Virginia 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 Framework delineates in greater specificity the content that all teachers should teach and all students should learn. The format of the Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Framework is divided by unit and ordered to match a sample pacing. Each unit is divided into three parts: ACPS Standards, Curriculum Overview and a Teacher Notes. The ACPS Standards section is divided into concepts, enduring understandings, essential standards, and lifelong learner standards. The Curriculum Overview contains the DOE curriculum framework information including the related SOL(s), strands, Essential Knowledge and Skills, and Essential Understandings. The Teacher Notes section is divided by Key Vocabulary, Essential Questions, Teacher Notes and Elaborations, Extensions, and Sample Instructional Strategies and Activities. The purpose of each section is explained below. Vertical Articulation (VDOE): This section includes the foundational objectives and the future objectives correlated to each SOL. ACPS Standards:

• Concepts: Interdisciplinary concepts and mathematics concepts specific to the associated SOL’s are listed in support of a concept centered approach to learning encouraged by ACPS. Mental construct or organizing idea that categorizes a variety of examples. Concepts are timeless, universal, abstract, and broad.

• Enduring Understandings: Broad generalizations and principles that connect two or more concepts in a statement of relationship. These understandings are build upon K-12.

• Essential Standards: Essential standards are derived through making connections between topics and enduring understandings. These standards are associated with particular gradebands.

• Lifelong Learner Standards: A standard designed to provide students with a foundation for lifelong inquiry and learning. Unit Overview:

• Curriculum Information: This section includes the SOL and SOL Reporting Category and focus or topic. • Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined.

This is not meant to be an exhaustive list nor is a list that limits what taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective. (Taken from the VDOE Curriculum Framework)

• Essential Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives. (Taken from the VDOE Curriculum Framework)

Teacher Notes:

• Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills.

• Essential Questions: This section explains what is meant to be the key knowledge and skills that define the standard.

ACPS Curriculum Framework Geometry

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• Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the objective beyond the current grade level.

• Extensions: This section provides content and suggestions to differentiate for honors level classes. • Sample Instructional Strategies and Activities: This section provides suggestions for varying instructional techniques within the classroom.

Special thanks to Henrico County Public Schools for allowing information from their curriculum documents to be included in this document.


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Geometry Sample Pacing Guide First Marking Period at a Glance

Prerequisite - Fundamentals

G.1 - Logic

G.2 - Lines

G.4 - Constructions

Second Marking Period at a Glance

G.5/G.10 - Triangles and Polygons

G.6 - Proofs with Triangles

G.7 - Similarity

Third Marking Period at a Glance

G.9 - Quadrilaterals

G.8 - Right Triangles

G.11 - Circles

Fourth Marking Period at a Glance

G.13/G.14 - Surface Area and Volume

G.3/G.9/G.12 - Coordinate Geometry

GEOMETRY SOL TEST BLUEPRINT (50 QUESTIONS TOTAL)

Resources

Virginia Department of Education Mathematics SOL Resources http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml

DOE Enhances Scope and Sequence Lesson Plans http://www.doe.virginia.gov/testing/sol/scope_sequence/mathematics_2009/

Reasoning, Lines, and Transformations 18 questions 36% of the Test Triangles 14 questions 28% of the Test Polygons, Circles, and Three-Dimensional Figures 18 questions 36% of the Test


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SOL Vertical Articulation Previous Standards Geometry Standard Future Standards

G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; & d) using deductive reasoning.

Under Construction

A.6 graph linear equations/linear inequal (2 vars) -‐ a) determine slope of line given equation of line/graph of line or two points on line -‐ slope as rate of change; b) write equation of line given graph of line/two points on line or slope-‐point on line 8.6 a) verify/describe relationships among vertical/adjacent/ supplementary/complementary angles;

G.2 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-‐world problems involving angles formed when parallel lines are cut by a transversal.

A.6 graph linear equations/linear inequal (2 vars) -‐ a) determine slope of line given equation of line/graph of line or two points on line -‐ slope as rate of change; b) write equation of line given graph of line/two points on line or slope-‐point on line 8.8 a) apply transformations to plane figures; b) ID applications of transformations 7.8 represent transformations (reflections, dilations, rotations, and translations) of polygons in the coordinate plane by graphing

G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include a) investigating and using formulas for finding distance, midpoint, and slope; b) applying slope to verify and determine whether lines are parallel or perpendicular; c) investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; & d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods.

6.12 determine congruence of segments/angles/polygons G.4 The student will construct and justify the constructions of a) a line segment congruent to a given line segment; b) the perpendicular bisector of a line segment; c) a perpendicular to a given line from a point not on the line; d) a perpendicular to a given line at a given point on the line; e) the bisector of a given angle, f) an angle congruent to a given angle; and g) a line parallel to a given line through a point not on the given line.

G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a) order the sides by length, given the angle measures; b) order the angles by degree measure, given the side lengths; c) determine whether a triangle exists; & d) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-‐world situations.

7.6 determine similarity of plane figures and write proportions to express relationships between similar quads and triangles 6.12 determine congruence of segments/angles/polygons

G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. G.7 The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs.


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A.3 express sq roots/cube roots of whole numbers/the square root of monomial alg exp (simplest radical form) 8.10 a) verify the Pythagorean Theorem; b) apply the Pythagorean Theorem

G.8 The student will solve real-‐world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry.

A.4 solve multistep linear/ quad equation (2 vars) -‐ a) solve literal equation; b) justify steps used in simplifying expresessions and solving equations; c) solve quad equations (alg/graph); d) solve multistep linear equations (alg/graph) 7.7 compare/contrast quadrilaterals based on properties 6.13 ID/describe properties of quadrilaterals

G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-‐world problems.

A.4 solve multistep linear/ quad equation (2 vars) -‐ a) solve literal equation; b) justify steps used in simplifying expresessions and solving equations; c) solve quad equations (alg/graph); d) solve multistep linear equations (alg/graph) 6.12 determine congruence of segments/angles/polygons

G.10 The student will solve real-‐world problems involving angles of polygons.

6.10 a) define π; b) solve practical problems w/circumference/area of circle; c) solve practical problems involving area and perimeter given radius/diameter; d) describe/determine volume/surface area of rectangular prism

G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles; b) solve real-‐world problems involving properties of circles; and c) find arc lengths and areas of sectors in circles.

6.10 a) define π; b) solve practical problems w/ circumference/area of circle; c) solve practical problems involving area and perimeter given radius/diameter; d)describe/determine volume/surface area of rectangular prism

G.12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle.

8.7 a) investigate/solve practical problems involving volume/surface area of prisms, cylinders, cones, pyramids; b) describe how changes in measured attribute affects volume/surface area 8.9 construct a 3-‐D model given top or bottom/side/front views 7.5 a) describe volume/surface area of cylinders; b) solve practical problems involving volume/surface area of rect. prims and cylinders; c) describe how changes in measured attribute affects volume/surface area

G.13 The student will use formulas for surface area and volume of three dimensional objects to solve real-‐world problems.

8.7 a) investigate/solve practical problems involving volume/surface area of prisms, cylinders, cones, pyramids; b) describe how changes in measured attribute affects volume/ surface area 7.5 a) describe volume/surface area of cylinders; b) solve practical problems involving volume/surface area of rect. prims and cylinders; c) describe how changes in measured attribute affects volume/surface area

G.14 The student will use similar geometric objects in two-‐ or three dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; b) determine how changes in one or more dimensions of an object affect area and/or volume of the object; c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and d) solve real-‐world problems about similar geometric objects.


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Interdisciplinary Concept: Systems; Communication Math Concept: Relationships; Quantifying Representation; Reasoning and Justification; Theory ACPS Mathematics Enduring Understandings: 2. Spatial relationships can be described using coordinate geometry and other representational systems. 3. Attributes of objects can be measured using processes and quantified units, using appropriate techniques, tools, and formulas. 11. Characteristics, properties, and mathematical arguments

about geometric relationships can be analyzed and developed using logical and spatial reasoning. 12. Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations. ACPS Essential Standard in grade band 9-12 2. Investigate and use Cartesian coordinates and other coordinate systems to analyze geometric situations. 3. Make reasonable estimates and accurate predictions about measurement by applying appropriate technology, and understanding the limitations. 11. Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings and

authentic situations. 12. Understand and represent translations and reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors,

function notation, and matrices. Life Long Learner Standards 7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;

Fundamentals Strand: Grade 4, 5, 6: Geometry SOL 4.10 The student will a) identify and describe representations of points, lines, line segments, rays, and angles, including endpoints and vertices 5.12 The student will classify a) angles as right, acute, obtuse, or

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Grade 4

• Identify and describe representations of points, lines, line segments, rays, and angles, including endpoints and vertices.

Grade 5

• Classify angles as right, acute, straight, or

Essential Understanding All students should Grade 4

• Understand that points, lines, line segments, rays, and angles, including endpoints and vertices are fundamental components of noncircular geometric figures.

• Understand that the shortest distance between two points on a flat surface is a line segment.

Grade 5

• Understand that angles can be classified as right, acute, obtuse, or straight according to their measures.


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straight 6.12 The student will determine congruence of segments, angles, and polygons. Return to Course Outline

obtuse. Grade 6

• Determine the congruence of segments, angles, and polygons given their attributes.

Grade 6

• Given two congruent figures, what inferences can be drawn about how the figures are related? The congruent figures will have exactly the same size and shape.


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Key Vocabulary acute angle bisect collinear complementary coplanar horizontal intersect line obtuse

plane point postulate ray right [angle] segment supplementary theorem vertex vertical

Essential Questions • How are the concepts of points, lines, line segments, rays, angles,

endpoints, and vertices important when describing and comparing geometric figures?

• Where can we find points, lines, line segments, rays, and angles in the world around us?

• How can a set of intersecting lines be used to demonstrate the relationship between and among points, lines, line segments, rays, angles, and geometric figures? How can visualizing a circle folded into halves and quarters help us classify angles?

• Given two congruent figures, what inferences can be drawn about how the figures are related?

Return to Course Outline

Teacher Notes and Elaborations A point is a location in space. It has no length, width, or height. A point is

usually named with a capital letter. A line is a collection of points going on and on infinitely in both directions.

It has no endpoints. When a line is drawn, at least two points on it can be marked and given capital letter names. Arrows must be drawn to show that the line goes on in both directions infinitely (e.g., AB

suur, read as “the

line AB”).

A line segment is part of a line. It has two endpoints and includes all the points between those endpoints. To name a line segment, name the endpoints (e.g., AB , read as “the line segment AB”).

A ray is part of a line. It has one endpoint and continues infinitely in one direction. To name a ray, say the name of its endpoint first and then say the name of one other point on the ray (e.g., AB

uuur, read as “the ray AB”).

Two rays that have the same endpoint form an angle. This endpoint is called the vertex. Angles are found wherever lines and line segments intersect. An angle can be named in three different ways by using

- three letters to name, in this order, a point on one ray, the vertex, and a point on the other ray;

- one letter at the vertex; or - a number written inside the rays of the angle.

Intersecting lines have one point in common. (continued)


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Teacher Notes and Elaborations (continued) Congruent figures are figures having exactly the same size and shape.

Opportunities for exploring figures that are congruent and/or noncongruent can best be accomplished by using physical models.

A right angle measures exactly 90°.

An acute angle measures greater than 0° but less than 90°.

An obtuse angle measures greater than 90° but less than 180°. A straight angle forms an angle that measures exactly 180°. Congruent figures have exactly the same size and the same shape.

Noncongruent figures may have the same shape but not the same size. The symbol for congruency is ≅ .

The determination of the congruence or noncongruence of two figures can be accomplished by placing one figure on top of the other or by comparing the measurements of all sides and angles.

Construction of congruent line segments, angles, and polygons helps students understand congruency.



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Interdisciplinary Concept: Communication Math Concept: Reasoning and Justification; Theory ACPS Mathematics Enduring Understandings: 11. Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial

reasoning. ACPS Essential Standard in grade band 9-12 5. Use various samples and testing methods to make logical inferences about data. Life Long Learner Standards 3. Think analytically, critically, and creatively to pursue new ideas, acquire new knowledge, and make decisions; 4. Understand and apply principles of logic and reasoning; develop,

evaluate, and defend arguments; 7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;

Logic Strand: Reasoning, Lines and Transformations SOL G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying the converse, inverse, and

contrapositive of a conditional statement;

b) translating a short verbal argument into symbolic form;

c) using Venn diagrams to represent set relationships; and

d) using deductive reasoning. Return to Course Outline

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Identify the converse, inverse, and

contrapositive of a conditional statement. • Translate verbal arguments into symbolic form

such as (p → q) and (~p → ~q). • Determine the validity of a logical argument. • Use valid forms of deductive reasoning,

including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples.

• Select and use various types of reasoning and methods of proof, as appropriate.

• Use Venn diagrams to represent set relationships, such as intersection, and union.

• Interpret Venn diagrams. • Recognize and use the symbols of formal logic,

Essential Understanding • Inductive reasoning, deductive reasoning, and proof

are critical in establishing general claims. • Deductive reasoning is the method that uses logic to

draw conclusions based on definitions, postulates, and theorems.

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

• Logical arguments consist of a set of premises or hypotheses and a conclusion.

• Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, and theorems.

• Euclidean geometry is an axiomatic system based on undefined terms (point, line, and plane), postulates, and theorems.

• When a conditional and its converse are true, the statements can be written as a biconditional (i.e., iff or if and only if).


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which include → , ↔ , ~, ∴, ᴧ and ᴠ.

• Logical arguments that are valid may not be true. Truth and validity are not synonymous.

Key Vocabulary biconditional statement conclusion conditional statement conjecture contrapositive converse

counterexample deductive reasoning hypothesis (premise) inductive reasoning inverse Law of Detachment Law of Syllogism Law of the Contrapositive

postulate (axiom) proof symbolic form Venn diagram


Essential Questions • When is a statement a lie? • What is the importance or need for symbolic representation of words? • What does it mean to be logical? • How can logic be represented visually? • What is the relationship between reasoning, justification, and proof in

geometry? • What is a truth-value? • How does a truth-value apply to conditional statements? • How do deductive reasoning and Venn diagrams help judge the validity

of logical arguments? Teacher Notes and Elaborations Logic is the study of the principles of reasoning. Logical arguments consist of a set of premises (hypotheses) and a conclusion (the last step in a reasoning process). Terms associated with logical arguments are reasoning, justification, and proof. Reasoning is the drawing of conclusions or inferences from facts, observations, or hypotheses. Justification is a rationale or argument for some mathematical proposition. A conjecture is a statement that has not been proved true nor shown to be false. A proof is a justification that is logically valid and based on initial assumptions, definitions, and proven results. A theorem is a statement that can be proved and a postulate or axiom is an assumption (a statement taken for granted) that is accepted without proof. A justification may be less formal than a proof. It may consist of a set of examples that seem to support the proposition or it may be an intuitive argument. The three concepts are related in that reasoning is used to seek a justification of a proposition, which, if possible, is turned into a proof.


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Teacher Notes and Elaborations (continued) Communication of reasoning and/or justification to complete a proof can be shown through symbolic form (truth tables or Venn diagrams) or written form (paragraph, indirect, two-column or coordinate method).

An if-then statement is called a conditional statement or simply a conditional. A conditional statement includes an initial condition or hypothesis (premise) and its corresponding outcome (conclusion). The conditional statement is written in the if (hypothesis) – then (conclusion) form. q

If p (hypothesis), then q (conclusion). p The converse (a proposition produced by reversing position or order) of the conditional statement is formed by interchanging the hypothesis and its conclusion. p

If q (conclusion), then p (hypothesis). q The inverse of the conditional statement is formed by negating both the hypothesis and the conclusion.

If not p (hypothesis), then not q (conclusion).

The contrapositive of the conditional statement is formed by interchanging and negating both the hypothesis and the conclusion.

If not q (conclusion), then not p (hypothesis).

The contrapositive and original conditional statements are logically equivalent (Law of Contrapositive). Return to Course Outline

Symbolic form includes truth tables (tabular representation of the truth or falsehood of hypotheses and conclusions) and Venn diagrams. Deductive reasoning uses rules to make conclusions. Applying the Law of Detachment, if you accept “If p then q” as true and you accept p as true, then you must logically accept q as true. It also follows if you accept “If p then q” as true and you accept not q as true, then you must logically accept not p as true. According to the Law of Syllogism, if you accept “If p then q” as true and if you accept “If q then r” as true, then you must logically accept “If p then r” as true. A counterexample is an example used to prove an if-then statement false. For that counterexample, the hypothesis is true and the conclusion is false. Inductive reasoning is a kind of reasoning in which the conclusion is based on several past observations. Symbolically ∴ means “therefore”. Ex: m ∠ABC is 90° ∴ m ∠ABC is a right angle. In logic, letters are used to represent simple statements that are either true or false. Simple statements can be joined to form compound statements. A conjunction is a compound statement composed of two simple statements joined by the word “and”. The symbol ∧ , is used to represent the word “and”. A disjunction is a compound statement x


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Extension • Identify, create, and determine the truth-value of the converse,

inverse, and contrapositive of a conditional statement. • Use chain reasoning to make a logical conclusion given a set of

statements. • Identify logically equivalent statements. • Construct truth tables given statements (conditional, conjunction,

disjunction, biconditionals, etc.). • Investigate the concept of an indirect proof. Extension for PreAP Geometry The truth value of a statement is either true or false. A truth table can be used to determine the conditions under which a statement is true. Truth Tables: Conditional If p then q

p q p q→

T T T T F F F T T F F T

Conjunction p and q

p q p q∧

T T T T F F F T F F F F


Disjunction p or q

p q p q∨

T T T T F T F T T F F F

An indirect proof is a proof that begins by assuming temporarily that the conclusion is not true; then reason logically until a contradiction of the hypothesis or another known fact is reached. Sample Instructional Strategies and Activities • Students, working in cooperative learning groups, will solve logic

problems to introduce the concept of deductive reasoning. Each group of students will give their solutions and describe their thought processes.


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Interdisciplinary Concept: Systems; Properties and Models; Communication; Math Concept: Relationships; Quantifying Representation; Models; Analysis and Evaluation; Reasoning and Justification ACPS Mathematics Enduring Understandings: 2. Spatial relationships can be described using coordinate geometry and other representational systems. 4. Situations and structures can be represented, modeled, and analyzed using algebraic symbols. 11. Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial

reasoning. ACPS Essential Standard in grade band 9-12 2. Investigate and use Cartesian coordinates and other coordinate systems to analyze geometric situations. 10. Approximate and interpret rates of change from graphical and numerical data. 11. Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings and

authentic situations. Life Long Learner Standards 2. Gather, organize, and analyze data, evaluate processes and products, and draw conclusions; 4. Understand and apply principles of logic and reasoning; develop, evaluate, and defend arguments; 6. Apply and adapt a variety of appropriate strategies to solve new and increasingly complex problems;

Lines Strand: Reasoning, Lines and Transformations SOL G.2 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are

parallel; b) verify the parallelism, using algebraic

and coordinate methods as well as deductive proofs; and

c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Use algebraic and coordinate methods as well

as deductive proofs to verify whether two lines are parallel.

• Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal including corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles.

• Solve real-world problems involving

Essential Understanding • Euclidean geometry is an axiomatic system based on

undefined terms (point, line, and plane), postulates, and theorems.

• When a conditional and its converse are true, the statements can be written as a biconditional (i.e., iff or if and only if).

• Logical arguments that are valid may not be true. Truth and validity are not synonymous.

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

• Some angle relationships may be used when proving two lines intersected by a transversal are parallel.

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


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intersecting and parallel lines in a plane. geometry and hyperbolic geometry.

Lines (continued)

Key Vocabulary adjacent angles algebraic method alternate exterior angles alternate interior angles complementary angles consecutive (same- side) interior angles

coordinate method corresponding angles deductive reasoning inductive reasoning exterior angle interior angle

linear pair parallel perpendicular skew supplementary angles transversal union vertical angles

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? Return to Course Outline

Teacher Notes and Elaborations Euclidean Geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose elements is the earliest known systematic discussion of geometry. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other theorems (propositions) from these. Angles with the same measure are congruent angles. Adjacent angles are two angles that share a common side and have the same vertex, but have no interior points in common. Vertical angles are two angles whose sides form two pairs of opposite rays. When two lines intersect, they form two pairs of vertical angles.

When two lines intersect, two types of angle pairs are formed: vertical angles and adjacent supplementary angles. Vertical angles are congruent and two adjacent angles are supplementary.

Parallel lines are lines that are in the same plane (coplanar) and never intersect because they are always the same distance apart. They have no points in common. The symbol || indicates parallel lines. Skew lines do not intersect and are not coplanar.

(continued)


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Lines (continued) Teacher Notes and Elaborations G.2 Intersection is a point or set of points common to two or more figures. A transversal is a line that intersects two or more coplanar lines in different points forming eight angles. Interior angles lie between the two lines. Alternate interior angles are on opposite sides of the transversal. Consecutive interior angles are on the same side of the transversal. Exterior angles lie outside the two lines. Alternate exterior angles are on opposite sides of the transversal. Corresponding angles are nonadjacent angles located on the same side of the transversal where one angle is an interior angle and the other is an exterior angle. If the sum of the measures of two angles is 180°, then the two angles are supplementary. If the two angles are adjacent and supplementary then they are a linear pair. If the sum of the measures of two angles is 90°, then the two angles are complementary. If the two angles are adjacent and complementary then they form a right angle. If two lines in a plane are cut by a transversal, the lines are parallel if:

- alternate interior angles are congruent, - alternate exterior angles are congruent, - corresponding angles are congruent, - same side (consecutive) interior angles are supplementary.

Proving lines parallel implies determining whether necessary and sufficient conditions (properties, definitions, postulates, and theorems) exist for parallelism. A proof is a chain of logical statements starting with given information and leading to a conclusion. Return to Course Outline

Two column deductive proofs (formal proofs) are examples of deductive reasoning. They contain statements and reasons organized in two columns. Each step is called a statement, and the properties that justify each step are called reasons. Essential parts of a good proof include:

1. state the theorem or conjecture to be proven; 2. list the given information; 3. if possible, draw a diagram to illustrate the given information; 4. state what is to be proved; and 5. develop a system of deductive reasoning.

The Parallel Postulate is the axiom of Euclidean Geometry stating that if two straight lines are cut by a third, the two will meet on the side of the third on which the sum of the interior angles is less than two right angles. Equivalently, Playfair’s Axiom states: “If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.” In Euclidean Geometry, parallel lines lie in the same plane and never intersect. In spherical geometry, the sphere is the plane, and a great circle represents a line. Two nonvertical coplanar lines are parallel if and only if their slopes are equal. Two nonvertical coplanar lines are perpendicular if and only if the product of their slopes is 1− . Algebraic and coordinate methods should also be used to determine parallelism. Coordinate geometry establishes a correspondence between algebraic concepts and geometric concepts. For example, the distance formula is derived as an application of the Pythagorean Theorem. The Pythagorean Theorem in turn is used to develop the equation of a circle. The coordinate proof is often more convenient

(continued)


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Lines (continued) Teacher Notes and Elaborations G.2 (continued) than a two-column proof. The following is an example of a coordinate proof involving parallelism.

Prove: The segment that joins the midpoint of two sides of a triangle is parallel to the third side.

Given: OABΔ 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 2M( , ) ( , ) ( , )2 2 2 2b c b c b c+ +

= = and

2 0 0 0 2 0N( , ) ( , ) ( ,0)2 2 2 2a a a+ +

= = ; by Midpoint Formula.

2. Slope of 0MN c ca b a b−

= = −− −

and the slope of

0 2 2BA2 2 2( )

c c ca b a b a b− −

= = = −− − −

; by definition of slope.

3. Slope of MN = slope of BA ; by Substitution Property.∴ MN || BA; two nonvertical lines are parallel if and only if their slopes are equal.


Extension Skew lines are non-coplanar lines that do not intersect. Experiences with skew lines should include 3-dimensional models. Extension • Use algebraic, coordinate, and deductive methods to determine if lines are

perpendicular. • Write equations of parallel and perpendicular lines. • Investigate skew lines using real world models. • Use definitions, postulates, and theorems to complete two-column or

paragraph proofs with at least five steps. Extension In a paragraph proof (informal proof) a paragraph is written to explain why a conjecture for a given situation is true. Sample Instructional Strategies and Activities • Have students pick two lines on notebook paper. Use straight edge and

pencil to darken lines chosen. Using a straight edge, draw a transversal. Label angles. Have students accurately measure pairs of special angles using a protractor. Perform the same procedures with two non-parallel lines cut by a transversal. Write conjectures for each special angle pair (corresponding, consecutive interior, alternate interior, and alternate exterior).

• Use patty paper to trace and compare lines and angles.

(continued)


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Lines (continued) Sample Instructional Strategies and Activities • Have class look for parallel, intersecting, perpendicular, and skew

lines in the classroom. In groups, students list as many pairs of them as they can find in ten minutes. Each group gives some examples from their list. This can be used as a competition.

o Have students pick two lines on notebook paper. Use straight edge and pencil to darken lines chosen. Using a straight edge, sketch a transversal. Label angles. Have students accurately measure pairs of special angles. Use the same procedure with two non-parallel lines cut by a transversal. Write conjectures for each special angle pair (corresponding, consecutive interior, alternate interior, and alternate exterior).

o Take class outside to look for parallel, intersecting, perpendicular, and skew lines and for identified angles. In groups, students list as many pairs of them as they can find in ten minutes. After returning to the classroom, each group gives some examples from their list. This can be used as a competition.

• Have students use patty paper to discover congruent angles formed when parallel lines are cut by a transversal.

• Have students build an angle log book. Students will draw pictures of various angles and label the angle. Students will relate the angle to an object in the room.



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Interdisciplinary Concept: Systems; Communication Math Concept: Relationships; Quantifying Representation; Reasoning and Justification ACPS Mathematics Enduring Understandings: 2. Spatial relationships can be described using coordinate geometry and other representational systems. 11. Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial

reasoning. 12. Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations. ACPS Essential Standard in grade band 9-12 2. Investigate and use Cartesian coordinates and other coordinate systems to analyze geometric situations. 11. Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings and

authentic situations. Life Long Learner Standards 5. Seek, recognize and understand systems, patterns, themes, and interactions; 6. Apply and adapt a variety of appropriate strategies to solve new and increasingly complex problems; 7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;

Constructions Strand: Reasoning, Lines, and Transformations SOL G.4 The student will construct and justify the constructions of a) a line segment congruent to a

given line segment; b) the perpendicular bisector of a line

segment; c) a perpendicular to a given line

from a point not on the line; d) a perpendicular to a given line at a

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Construct and justify the constructions of

- a line segment congruent to a given line segment;

- the perpendicular bisector of a line segment;

- a perpendicular to a given line from a point not on the line;

- a perpendicular to a given line at a point on the line;

- the bisector of a given angle; - an angle congruent to a given angle; and - a line parallel to a given line through a

Essential Understanding • Construction techniques are used to solve real-world

problems in engineering, architectural design, and building construction.

• Construction techniques include using a straightedge and compass, paper folding, and dynamic geometry software.


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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.

point not on the given line. • Construct an equilateral triangle, a square, and

a regular hexagon inscribed in a circle. • Construct the inscribed and circumscribed

circles of a triangle. • Construct a tangent line from a point outside a

given circle to the circle.

Key Vocabulary

bisector centroid circumcenter circumscribed compass congruence construction incenter

inscribed intersection parallel lines perpendicular bisector perpendicular lines bisector straightedge transversal

Essential Questions • What is the relationship between points, rays, and angles? • Why are constructions important? • How are constructions justified? Teacher Notes and Elaborations "Construction" in geometry means to draw shapes, angles or lines accurately. Constructions are done using tools including software programs such as Sketchpad, Geogebra, patty paper, a straightedge, and a compass. If students are using a ruler as a straightedge, they should be instructed to ignore its markings. Constructions help build an understanding of the relationships between lines and angles. Return to Course Outline

The seven basic constructions can be used to do more complicated constructions such as points of concurrency: centroid, incenter, circumcenter and orthocenter. The intersection of two figures is the set of points that is in both figures. A transversal is a line that intersects two or more coplanar lines in different points. Two angles are congruent if and only if they have equal measures. A ray is an angle bisector if and only if it divides the angle into two congruent adjacent angles. Parallel lines are lines that do not intersect and are coplanar. Perpendicular lines are lines that intersect at right angles. A segment bisector is a line, segment, ray, or plane that intersects the segment at its midpoint. A perpendicular bisector of a segment is a line, ray, or segment that is perpendicular to the segment at its midpoint. A circle is circumscribed about a triangle if the circle contains all the vertices of the triangle. A triangle is inscribed in a circle if each of its vertices lies on the circle. In a triangle, a median is a segment that joins a vertex of the triangle


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Teacher Notes and Elaborations (continued) and the midpoint of the side opposite that vertex. The medians of a triangle intersect at the balance point called the centroid.

To circumscribe a circle about a triangle, construct the perpendicular bisectors of each side. The point where these perpendicular bisectors meet is the circumcenter. Using the circumcenter and any vertex of the triangle as the radius, construct the circle about the triangle. To construct a circle inscribed inside a triangle, construct the angle bisectors. The incenter is the point where the angle bisectors meet. Construct a perpendicular from the incenter to one of the sides of the triangle. This perpendicular segment is the radius of the inscribed circle.

Justification of constructions may involve application of postulates, theorems, definitions, and properties. Justification of constructions may differ depending upon the plan proposed, and the order in which concepts are taught.

Construction Justification 1. Construct a line segment Radii of equal circles are equal congruent to a given a line segment 2. Construct an angle congruent Radii of equal circles are equal to a given angle SSS Postulate Corresponding parts of

congruent triangles are congruent

3. Construct the bisector of a Radii of equal circles are equal given angle SSS Postulate Corresponding parts of

congruent triangles are congruent

Definition of an angle bisector

4. Construct the perpendicular Radii of equal circles are equal bisector of a given segment Through any two points there is

exactly one line If a point is equidistant from the endpoints of a line segment, then the point lies on the perpendic-

ular bisector of the line segment

5. Construct the perpendicular Radii of equal circles are equal to a line at the given point on Definition of a straight angle the line. Definition of an angle bisector Definition of right angles and

definition of perpendicular lines

6. Construct the perpendicular to Radii of equal circles are equal the line from a point not on the If a point is equidistant from the

line. endpoints of a line segment, then the point lies on the perpendicular bisector of the line

7. Construct the parallel to a Radii of equal circles are equal given line though a given If two lines are cut by a

point not on the line. transversal and corresponding angles are congruent, then the lines are parallel

Constructions (continued)


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Extension (continued) • Justify the constructions of:

- equilateral triangles; - squares; - angles with measures of 15, 30, 45, 60, 75, and 135 degrees; - regular hexagons; - a tangent to a circle through a point on the circle; and - a tangent line from a point outside a given circle to the circle.

• Given a segment, by construction divide the segment into a given number of congruent parts.


Extension • 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.


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Interdisciplinary Concept: Systems; Communication Math Concept: Relationships; Quantifying Representation; Reasoning and Justification ACPS Mathematics Enduring Understandings: 2. Spatial relationships can be described using coordinate geometry and other representational systems. 3. Attributes of objects can be measured using processes and quantified units, using appropriate techniques, tools, and formulas. 11. Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial

reasoning. ACPS Essential Standard in grade band 9-12 2. Investigate and use Cartesian coordinates and other coordinate systems to analyze geometric situations. 3. Make reasonable estimates and accurate predictions about measurement by applying appropriate technology, and understanding the limitations. 11. Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings and

authentic situations. Life Long Learner Standards 3. Think analytically, critically, and creatively to pursue new ideas, acquire new knowledge, and make decisions; 4. Understand and apply principles of logic and reasoning; develop, evaluate, and defend arguments; 6. Apply and adapt a variety of appropriate strategies to solve new and increasingly complex problems; 7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;

Triangles and Polygons Strand: Triangles; Polygons and Circles SOL G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a) order the sides by length, given the

angle measures; Return to Course Outline b) order the angles by degree measure,

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Order the sides of a triangle by their lengths

when given the measures of the angles. • Order the angles of a triangle by their

measures when given the lengths of the sides. • Given the lengths of three segments, determine

whether a triangle could be formed. • Given the lengths of two sides of a triangle,

determine the range in which the length of the third side must lie.

Essential Understanding • The longest side of a triangle is opposite the largest

angle of the triangle and the shortest side is opposite the smallest angle.

• In a triangle, the length of two sides and the included angle determine the length of the side opposite the angle.

• In order for a triangle to exist, the length of each side must be within a range that is determined by the lengths of the other two sides.

• A regular polygon will tessellate the plane if the measure of an interior angle is a factor of 360.


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given the side lengths; c) determine whether a triangle exists;

and d) determine the range in which the

length of the third side must lie. These concepts will be considered in the context of real-world situations. SOL G.10 The student will solve real-world problems involving angles of polygons.

• Solve real-world problems given information about the lengths of sides and/or measures of angles in triangles.

• Solve real-world problems involving the measures of interior and exterior angles of polygons.

• Identify tessellations in art, construction, and nature.

• Find the sum of the measures of the interior and exterior angles of a convex polygon.

• Find the measure of each interior and exterior angle of a regular polygon.

• Find the number of sides of a regular polygon, given the measures of interior or exterior angles of the polygon.

• Both regular and nonregular polygons can tessellate the plane.

• Two intersecting lines form angles with specific relationships.

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

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

• The sum of the measures of the interior angles of a convex polygon may be found by dividing the interior of the polygon into nonoverlapping triangles.

Key Vocabulary

altitude apothem concave convex decagon diagonal dodecagon exterior angle heptagon hexagon interior angle isosceles linear pair median n-gon Return to Course Outline

nonagon octagon opposite pentagon polygon quadrilateral regular/irregular polygon scalene tessellation tiling triangle Triangle Inequality Theorem

Essential Questions • What are the angle relationships of a triangle? • What conditions must exist for a triangle to be formed? • What is the relationship between the measure of the angles and the lengths

of the opposite sides of a triangle? • What are the distinguishing characteristics of a polygon? • How do we verify that polygons can tile a plane? • What are the relationships between the sides of a polygon and the angles of

a polygon? Teacher Notes and Elaborations 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.


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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. 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

Teacher Notes and Elaborations 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. 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

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 Extension • 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.


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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.



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Triangles and Polygons (continued) Extension Using properties of triangles, inequalities can be written relating pairs of angles or segment measures.

Note: Figures are not drawn to scale

BCD CAB∠ > ∠

CD BC>

Hinge Theorem: (SAS Inequality) If two sides of a triangle are congruent to two sides of another triangle, and the included angle in one triangle is greater than the included angles in the other, then the third side of the first triangle is longer than the third side in the second triangle. Extension • Investigate and identify the regular polygons that tessellate. • Distinguish between pure and semi-pure tessellations. Return to Course Outline

Extension 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. Sample Instructional Strategies and Activities • Coordinate geometry can be used to investigate relationships among

triangles. • Use pieces of yarn, straws, sticks, or magnetic tape to see which

combinations of lengths can be used to make triangles. • Use Geo-Legs or Anglegs to illustrate combinations of lengths that can be

used to form triangles. • Cut out a triangle. Place a different color dot in each angle. Place the

triangle on the paper and trace around it in pencil. Slide triangle over and mark the color in each angle so that the colors correspond with the cardboard triangle. Place triangle back on top and rotate it so that it no longer overlaps. Repeat until the plane is filled. Have students identify parallel lines, vertical angles, etc. Students make conjectures about lines and angles in the tessellation. Students are given various polygons and asked if they tessellate a plane. Explain why or why not.

• 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


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reasoning. ACPS Essential Standard in grade band 9-12 11. Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings and

authentic situations. Life Long Learner Standards 3. Think analytically, critically, and creatively to pursue new ideas, acquire new knowledge, and make decisions; 4. Understand and apply principles of logic and reasoning; develop, evaluate, and defend arguments; 7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;

Proofs with Triangles Strand: Triangles SOL G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Return to Course Outline

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Use definitions, postulates, and theorems to

prove triangles congruent. • Use coordinate methods, such as the distance

formula and the slope formula, to prove two triangles are congruent.

• Use algebraic methods to prove two triangles are congruent.

Essential Understanding • Congruence has real-world applications in a variety

of areas, including art, architecture, and the sciences.

• Congruence does not depend on the position of the triangle.

• Concepts of logic can demonstrate congruence or similarity.

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

(continued)


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Proofs with Triangles (continued) Key Vocabulary

AAS Theorem algebraic methods altitude ASA Postulate coordinate methods corresponding parts deductive proof

definition distance formula HL Postulate hypotenuse included angle included side leg

postulate properties SAS Postulate SSS Postulate theorem

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

congruent? • How can congruent triangles assist in the proof of other geometric

ideas? Teacher Notes and Elaborations When two figures have exactly the same shape and size, they are said to be congruent. Using algebraic methods, if all corresponding parts Return to Course Outline

can be shown to be equal, then the figures are congruent. This can include coordinate methods such as distance formula and the slope formula. Congruent figures have corresponding parts (matching parts) that have equal measures. Corresponding parts of congruent triangles are congruent (CPCTC). Congruence does not depend on the position of the triangle. A theorem is a statement that can be proved and a postulate is an assumption that is accepted without proof. Definitions, postulates, and theorems are used in proofs. A proof is a chain of logical statements starting with given information and leading to a conclusion. Two column deductive proofs are examples of deductive reasoning. Properties (facts about real numbers and equality from algebra) can also be used to justify steps in proofs. A side of a triangle is said to be included (included side) between two angles if the vertices of the two angles are the endpoints of the side. An angle of a triangle is said to be included (included angle) between two sides if the angle is formed by the two sides. Triangles can be proven congruent with the following correspondences:

SSS Postulate: Three sides of one triangle are congruent to the corresponding sides of another triangle.

SAS Postulate: Two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle.

ASA Postulate: Two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle. (continued)


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Proofs with Triangles (continued) Teacher Notes and Elaborations (continued)

AAS Theorem: Two angles and a non-included side of one triangle are congruent to the corresponding two angles and a non-included side of a second triangle.

In a right triangle the side opposite the right angle is the hypotenuse and the other two sides are called legs. Right triangles can be proven congruent with the following correspondence:

HL Postulate: The hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle.

Medians, altitudes, and perpendicular bisectors are also used in proving triangles congruent. A median of a triangle is a segment that joins a vertex to the midpoint of the opposite side. An altitude of a triangle is a segment from a vertex and perpendicular segment from a vertex to the line containing the opposite side. Extension Use angle bisectors, medians, altitudes, perpendicular bisectors to prove triangles congruent. • Correlate LL, HA, LA to SAS, AAS, and ASA respectively. • Investigate the points of concurrency of the lines associated with

triangles (angle bisectors (incenter), perpendicular bisectors (circumcenter), altitudes (orthocenter), and medians (centroid)).

Extension LL Theorem: The legs of one right triangle are congruent to the legs of another right triangle. HA Theorem: The hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and acute angle of the other Return to Course Outline

right triangle. LA Theorem: One leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle. The medians of a triangle intersect at the common point called the centroid. In a triangle, the point where the perpendicular bisectors of each side intersect is the circumcenter. In a triangle, the incenter is the point where the angle bisectors intersect. In a triangle, the orthocenter is the point of intersection of the three altitudes. Sample Instructional Strategies and Activities • Use coordinate geometry to investigate relationships among triangles. • Given specifications such as side lengths or angle measures, students

draw a triangle. Next, the students compare their drawings to see if they are congruent. This is done to test AAS, SSS, etc. before they are introduced.

• Students are given a printed deductive proof of theorem. Cut it up into a statement of theorem, given, prove, diagram, individual statements, and individual reasons. Each group of students is given a set of pieces and must put the proof together in correct order.

• Use pieces of yarn, straws, or sticks to see which combinations of lengths can be used to make triangles.

• Use patty paper to demonstrate congruent triangles.


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Similarity Strand: Triangles SOL G.7 The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Return to Course Outline

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Use definitions, postulates, and theorems to

prove triangles similar. • Use algebraic methods to prove that triangles

are similar. • Use coordinate methods, such as the distance

formula, to prove two triangles are similar. • Compare ratios between side lengths,

perimeters, areas, and volumes, given two similar figures.

• Solve real-world problems involving measured attributes of similar objects.

Essential Understanding • Similarity has real-world applications in a variety of

areas, including art, architecture, and the sciences. • Similarity does not depend on the position of the

triangle. • Congruent figures are also similar, but similar

figures are not necessarily congruent. • A constant ratio exists between corresponding

lengths of sides of similar figures. • Proportional reasoning is integral to comparing

attribute measures in similar objects (continued)

Similarity (continued)


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Key Vocabulary

area deductive proof perimeter proportion ratio scale factor

similar figures similar triangles AA Similarity SSS Similarity SAS Similarity volume

Essential Questions • When is a proportion necessary to solve a problem? • Are common units of measure necessary when solving

proportions? • How are similar triangles utilized in art, architecture and the

sciences? • What is the difference between congruence and similarity? • What is the relationship between similar triangles and proportions? • What are the one-to-one correspondences that prove triangles

similar? Return to Course Outline

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

Teacher Notes and Elaborations Congruent figures have corresponding parts that have equal measures while similar figures have corresponding angles congruent but corresponding sides with proportional measures. Coordinate methods such as distance formula and the slope formula can be used to prove triangles are similar. A theorem is a statement that can be proved and a postulate is an assumption that is accepted without proof. Definitions, postulates, and theorems are used in proofs. A proof is a chain of logical statements starting with given information and leading to a conclusion. Two column deductive proofs are examples of deductive reasoning. Properties (facts about real numbers and equality from algebra) can also be used to justify steps in proofs. A ratio is a comparison of two quantities. The ratio of a to b can be

expressed as ab

, where b ≠ 0. If two ratios are equal, then a proportion exists.

Therefore a cb d= is a proportion and the cross products are equal (ad = bc).

Similar figures are figures that have the same shape but not necessarily the same size. Two triangles are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. The ratio of the lengths of two corresponding sides of two similar polygons is called a scale factor. An angle of a triangle is said to be included (included angle) between two sides if the angle is formed by the two sides.

(continued)


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Similarity (continued) Teacher Notes and Elaborations (continued) There are three ways to determine whether two triangles are similar when all measurements of both triangles are not known:

AA Similarity: Show that two angles of one triangle are congruent to two angles of the other. SSS Similarity: Show that the measures of the corresponding sides of the triangles are proportional. SAS Similarity: Show that the measures of two sides of a triangle are proportional to the measures of the corresponding sides of the other triangle and that the included angles are congruent.

If a line is drawn parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of proportional lengths. a c a c

b d=

b d If two triangles are similar, then the measures of the lengths of the corresponding angle bisectors of the triangles are proportional to the measures of the lengths of the corresponding sides. a ~ c x y

x ay c=

A median of a triangle is a segment that joins a vertex to the midpoint of the opposite side. Return to Course Outline

If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. ~ c a x y

x ay c=

An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.

a b

e af b=

e f Extension Use definitions, postulates, and theorems to complete two-column or paragraph proofs with at least five steps. • Investigate proportionality in a triangle intersected by three or more

parallel lines. • Investigate the Golden Ratio. Extension If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

(continued)

Similarity (continued)


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Extension (continued) A B C D E F

AB DE=BC EF

, AC BC=DF EF

, AC DF=BC EF

If a line segment is divided into two lengths such that the ratio of the segments’ entire length to the longer length is equal to the ratio of the longer length to the shorter length, then the segment has been divided into the Golden Ratio. a b

a b aa b+

= (This golden ratio is approximately 1.618.)

In a rectangle, if the ratio of the longer side to the shorter approximates 1.618, the rectangle is called a Golden Rectangle Sample Instructional Strategies and Activities • Use coordinate geometry to investigate relationships among

triangles. • Students are given a printed deductive proof of theorem. Cut it up

into a statement of theorem, given, prove, diagram, individual Return to Course Outline

statements, and individual reasons. Each group of students is given a set of pieces and must put the proof together in correct order.

• Each group of students will measure the height of one of their members, the shadow of that member, and the shadow of a light pole or flagpole. Using similar triangles and proportions, each group calculates the height of the pole. Next, the groups compare their calculations.

• Given the pitch of a roof, the students will calculate the roof truss and using toothpicks will construct a model of the roof.

• Use patty paper to demonstrate similar triangles. • Using cylinders made from PVC pipe or empty cans determine the change

in volume with respect to changes in height or radius. Fill cylinders with water to compare the volumes.


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Quadrilaterals Strand: Polygons and Circles SOL G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. Return to Course Outline

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Solve problems, including real-world

problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids.

• Prove that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula.

• Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods.

• Prove properties of angles for a quadrilateral inscribed in a circle.

Essential Understanding • The terms characteristics and properties can be used

interchangeably to describe quadrilaterals. The term characteristics is used in elementary and middle school mathematics.

• Quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals.

• Characteristics of quadrilaterals can be used to identify the quadrilateral and to find the measures of sides and angles.

(continued)


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Quadrilaterals (continued) Key Vocabulary

base base angles characteristics diagonal isosceles trapezoid kite legs

median of a trapezoid parallelogram quadrilateral rectangle rhombus square trapezoid

Essential Questions • What are the distinguishing features of the different types of

quadrilaterals? • How are the properties of quadrilaterals used to solve real-life

problems? • What is the hierarchical nature among quadrilaterals? Return to Course Outline

Teacher Notes and Elaborations Algebraic methods and coordinate methods such as distance formula, midpoint formula, and the slope formula can be used to prove quadrilateral properties. A quadrilateral is a polygon with four sides. Quadrilaterals have a hierarchical nature based on relationships among their sides, their angles, and their diagonals. The diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. A parallelogram is a quadrilateral with opposite sides parallel and congruent. Consecutive angles of a parallelogram are supplementary; opposite angles are congruent; and the diagonals of a parallelogram bisect each other. A rectangle is a parallelogram with four right angles. The diagonals of a rectangle are congruent. A rhombus is a parallelogram with congruent sides. The diagonals of a rhombus are perpendicular and bisect each other and the opposite angles. A square is a parallelogram, a rectangle, and a rhombus. A trapezoid is a quadrilateral with exactly one pair of opposite sides parallel. An isosceles trapezoid has congruent legs (the non-parallel sides). Both pairs of base angles in an isosceles trapezoid are congruent and diagonals are congruent. The median of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases and has a length equal to half the sum of the lengths of the bases. A kite is a quadrilateral with two pairs of congruent adjacent sides.

(continued)


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Quadrilaterals (continued) Teacher Notes and Elaborations (continued) Characteristics of quadrilaterals are used to identify figures, and to find values for missing parts and areas. Areas of work that use quadrilaterals include art, construction, fabric design, and architecture. The hierarchical nature of quadrilaterals can be described as ranking based on characteristics.

If a quadrilateral is inscribed in a circle, its opposite angles are supplementary. This can be verified by considering that the arcs intercepted by opposite angles of an inscribed quadrilateral form a circle. Return to Course Outline

Example: Quadrilateral ABCD is inscribed in a circle. ª ª ª ªAB BC CD DA 360m m m m+ + + = °

The measure of º1DAB = BCD2

m m∠ and the measure of

º1BCD = DAB2

m m∠ . A B

º ºBCD = 2 A and DAB = 2 Cm m m m⋅ ∠ ⋅ ∠

º ºBCD DAB 360m m+ = ° 2 A+2 C = 360m m⋅ ∠ ⋅ ∠ ° D C A C 180m m∠ + ∠ = ° Sample Instructional Strategies and Activities • Use flowcharts or Venn diagrams to show relationships and properties of

quadrilaterals. • Use patty paper to show properties of the different quadrilaterals. • Use notecards to create models of different quadrilaterals. Discuss the

characteristics and have students record their findings on the back of the models.


Page 37 of 57




Right Triangles Strand: Triangles SOL G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. Return to Course Outline

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Determine whether a triangle formed with

three given lengths is a right triangle. • Solve for missing lengths in geometric figures,

using properties of 45°-45°-90° triangles. • Solve for missing lengths in geometric figures,

using properties of 30°-60°-90° triangles. • Solve problems involving right triangles, using

sine, cosine, and tangent ratios. • Solve real-world problems, using right triangle

trigonometry and properties of right triangles. • Explain and use the relationship between the

sine and cosine of complementary angles. • Use definitions, postulates, and theorems to

prove triangles similar. • Use algebraic methods to prove that triangles

Essential Understanding • The Pythagorean Theorem is essential for solving

problems involving right triangles. • Many historical and algebraic proofs of the

Pythagorean Theorem exist. • The relationships between the sides and angles of

right triangles are useful in many applied fields. • Some practical problems can be solved by choosing

an efficient representation of the problem. • Another formula for the area of a triangle is

1 sin2

A ab C= .

• The ratios of side lengths in similar right triangles (adjacent/hypotenuse or opposite/hypotenuse) are independent of the scale factor and depend only on the angle the hypotenuse makes with the adjacent side, thus justifying the definition and calculation of trigonometric functions using the ratios of side lengths for similar right triangles.


Page 38 of 57

are similar. • Use coordinate methods, such as the distance

formula, to prove two triangles are similar.

• Similarity has real-world applications in a variety of areas, including art, architecture, and the sciences.

• Similarity does not depend on the position of the triangle.

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

(continued) Key Vocabulary

angle of depression angle of elevation area of a triangle cosine hypotenuse Pythagorean Theorem

right triangle sine tangent trigonometry 45°-45°-90° triangle 30°-60°-90° triangle

Essential Questions • How can one determine a missing measurement of a right triangle? • How can one verify that a triangle is a right triangle? • What is a trigonometric ratio? • What is the relationship between sine and cosine in terms of

complementary angles? Teacher Notes and Elaborations Right triangles (any triangle with one 90° angle) are triangles with 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. Return to Course Outline

Special right triangles are the 30° - 60° - 90° and the 45° - 45° - 90°.

- In a 45° - 45° - 90° triangle, the hypotenuse is 2 times as long as one of the legs.

- In the 30° - 60° - 90° triangles, the hypotenuse is twice as long as the shorter leg and the longer leg is 3 times as long as the shorter leg.

The Pythagorean Theorem states that in a right triangle, the square of the measure of the hypotenuse equals the sum of the squares of the measures of the legs. The converse of the Pythagorean Theorem states that if the square of the measure of the longest side equals the sum of the squares of the measures of the other two sides of a triangle, then the triangle is a right triangle. If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle. If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle. Pythagorean Triples are three positive integers that satisfy the Pythagorean theorem. In a right triangle with the altitude drawn to the hypotenuse, the geometric mean can be used to find missing measures of that triangle.

(continued)


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Right Triangles (continued) Teacher Notes and Elaborations (continued)

If r, s, and t are positive numbers with r ss 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 13sin3018

= 13cos6018

=

13 18 sin 30 = cos60

30º The angle of elevation is the angle formed by a horizontal line and the line of sight to an object above that horizontal line. The angle of depression is the angle formed by a horizontal line and the line of sight to an object below that horizontal line. The angle of elevation and the angle of depression in the same diagram are always congruent. Return to Course Outline

Extension • Use the Law of Sines and the Law of Cosines to find missing measures in

triangles. • Find the geometric mean in right triangles. Extension The Law of Sines states that for any triangle with angles of measures A, B, and C, and sides of lengths a, b, and c (a opposite A∠ , opposite b B∠ , and

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

sin sin sinA B Ca b c

= =

The Law of Cosines states that for any triangle with sides of lengths a, b, and c then 2 2 2 2 cosc a b ab C= + − . This law is often used when at least two sides are known (SAS or SSS). The measures of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse, is the geometric mean between the measures of the two segments of the hypotenuse. h

x hh y=

x y If the altitude is drawn to the hypotenuse of a right triangle, then the measure of a leg of the triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

(continued)


Page 40 of 57

Right Triangles (continued) Extension (continued)

a h b x aa c= and y b

b c=

x ⎢ c ⎢

Sample Instructional Strategies and Activities • Use pieces of yarn, straws, or sticks to see which combinations of

lengths can be used to make acute, obtuse, and right triangles. • Have students make a hypsometer, then go outside and measure the

heights of buildings, trees, poles, etc., with the hypsometer. • The teacher prepares a set of clue cards containing trigonometry

word problems. Students work in groups of 4 or 5 draw a diagram of the problem, set up a trig equation, then solve the problem.



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Interdisciplinary Concept: Systems; Communication Math Concept: Relationships; Quantifying Representation; Reasoning and Justification ACPS Mathematics Enduring Understandings: 2. Spatial relationships can be described using coordinate geometry and other representational systems. 11. Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial

reasoning. 12. Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations. ACPS Essential Standard in grade band 9-12 2. Investigate and use Cartesian coordinates and other coordinate systems to analyze geometric situations. 11. Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings and

authentic situations. Life Long Learner Standards 5. Seek, recognize and understand systems, patterns, themes, and interactions; 6. Apply and adapt a variety of appropriate strategies to solve new and increasingly complex problems; 7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;

Circles Strand: Polygons and Circles SOL G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and

apply properties of circles; b) solve real-world problems

involving properties of circles; and

c) find arc lengths and areas of sectors in circles.


Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Find lengths, angle measures, and arc measures associated with – two intersecting chords; – two intersecting secants; – an intersecting secant and tangent; – two intersecting tangents; and – central and inscribed angles.

• Calculate the area of a sector and the length of an arc of a circle, using proportions. • Solve real-world problems associated with circles, using properties of angles, lines,

and arcs. • Verify properties of circles, using deductive reasoning, algebraic, and coordinate

methods.

Essential Understanding • Many relationships

exist between and among angles, arcs, secants, chords, and tangents of a circle.

• All circles are similar. • A chord is part of a

secant. • Real-world

applications may be drawn from architecture, art, and construction.

(continued)


Page 42 of 57

Circles (continued)

Key Vocabulary

arc arc length arc measure central angle chord circle circumscribed circumference common tangent concentric circles diameter

inscribed intercepted arc major arc minor arc point of tangency radius secant sector semicircles tangent

Essential Questions • How might geometric objects (points, segments, lines, etc.)

interact/intersect with circles? • What is area/circumference and how is it measured? What does the

value pi represent? • How does a tangent line relate to the circle? • How are the angle formulas of circles related to similar triangles? • What are the relationships between chords and arcs? • What is the difference between arc length and arc measure? Return to Course Outline

Teacher Notes and Elaborations A circle is the set of all points equidistant from a given point in a plane. The distance from the center of the circle to a point on the circle is the radius. The arc measure is the degree measure of its central angle. A central angle is an angle with its vertex at the circle’s center. A central angle separates a circle into two arcs called a major arc (measures greater than 180º but less than 360º), and a minor arc (measures greater than 0º but less than 180º). Semicircles are the two arcs of a circle that are cut off by a diameter. A semicircle measures 180º. An arc is an unbroken part of a curve of a circle. The central angle measures the same as its intercepted arc. The intercepted arc is the part of the circle that lies between the two lines that intersect the circle. A chord is a segment joining two points on the circle. A diameter is a chord that passes through the circle’s center. A secant is a line that contains a chord. A tangent is a line that intersects a circle in only one point. Measures of chords, secant segments, and tangent segments can be determined. An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle. The measure of an inscribed angle is equal to one-half the measure of its intercepted arc.

(continued)


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Circles (continued) Teacher Notes and Elaborations (continued) The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs. Use the properties of chords, secants, and tangents to determine missing lengths. When two chords intersect inside a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. When two secant segments are drawn to a circle from an exterior point, the product of the lengths of one secant segment and its exterior segment is equal to the product of the lengths of the other secant segment and its exterior segment. When a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its exterior segment. The length of an arc (arc length) is a linear measure and is part of the circumference (perimeter of a circle). A sector of a circle is that part of the circle bounded by two radii and an arc. Length of an arc and area of a sector can be calculated using the following formulas: In circle O, the measure of ªAB x= (This is a degree measure.) Return to Course Outline

Length of ªAB 2360x rπ= ⋅ (This is a linear measure.)

Area of sector 2AOB360x rπ= ⋅

Verifying the properties of circles may include definitions, postulates, theorems, algebraic methods, and coordinate methods. Wheels and gears are two important applications of circles. In the same circle or congruent circles:

- Congruent chords have congruent arcs and vice versa. - Congruent chords are equidistant from the center and vice versa. - A diameter that is perpendicular to a chord bisects the chord and its

arc. An angle inscribed in a semi-circle is a right angle. Opposite angles of an inscribed quadrilateral are supplementary. Extension • Find the area of a segment of a circle. • Find the area of an annulus. Extension A segment of a circle is the region between an arc and a chord of a circle. To find the area of a segment, find the area of the sector and subtract the area of the triangle.

(continued)


Page 44 of 57

Circles (continued) Extension (continued) An annulus is the region between two concentric circles. To find the area of an annulus, find the area of the larger circle and subtract the area of the smaller circle. Sample Instructional Strategies and Activities • Use the graphing calculator to show that a triangle inscribed in a

semicircle is a right triangle; to show that the product of the parts of one chord equal the product of the parts of the other chord; to graph and identify circles as tangent, intersecting, or concentric; and to graph and recognize tangents as internal or external.

• Use patty paper to demonstrate the properties of circles. • Students use post-it notes to identify intercepted arcs. • Students use post-it notes to find multiple angles and arc measures

in circle drawings. Return to Course Outline


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Interdisciplinary Concept: Systems; Properties and Models; Communication Math Concept: Relationships; Quantifying Representation; Models; Analysis and Evaluation; Reasoning and Justification ACPS Mathematics Enduring Understandings: 2. Spatial relationships can be described using coordinate geometry and other representational systems. 3. Attributes of objects can be measured using processes and quantified units, using appropriate techniques, tools, and formulas. 4. Situations and structures can be represented, modeled, and analyzed using algebraic symbols. 11. Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial


authentic situations. Life Long Learner Standards 2. Gather, organize, and analyze data, evaluate processes and products, and draw conclusions; 6. Apply and adapt a variety of appropriate strategies to solve new and increasingly complex problems; 7. Acquire and use precise language to clearly communicate ideas, knowledge, and processes;

Surface Area and Volume Strand: Three-Dimensional Figures SOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. SOL G.14 The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side

lengths, perimeters, areas, and volumes;

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Find the total surface area of cylinders, prisms,

pyramids, cones, and spheres, using the appropriate formulas.

• Calculate the volume of cylinders, prisms, pyramids, cones, and spheres, using the appropriate formulas.

• Solve problems, including real-world problems, involving total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-

Essential Understanding • The surface area of a three-dimensional object is the

sum of the areas of all its faces. • The volume of a three-dimensional object is the

number of unit cubes that would fill the object. • A change in one dimension of an object results in

predictable changes in area and/or volume. • A constant ratio exists between corresponding

lengths of sides of similar figures. • Proportional reasoning is integral to comparing

attribute measures in similar objects.


Page 46 of 57

Return to Course Outline 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.


dimensional figures. • Calculators may be used to find decimal

approximations for results. • Compare ratios between side lengths,

perimeters, areas, and volumes, given two similar figures.

• Describe how changes in one or more dimensions affect other derived measures (perimeter, area, total surface area, and volume) of an object.

• Describe how changes in one or more measures (perimeter, area, total surface area, and volume) affect other measures of an object.

• Solve real-world problems involving measured attributes of similar objects.

(continued)


Page 47 of 57

Surface Area and Volume (continued) Key Vocabulary

altitude area base base area (B) base edge cone cube face

height lateral edge lateral area prism polygon pyramid similar figures

slant height sphere surface area three-dimensional two dimensional vertex volume

Essential Questions • What is area? • What is volume? • How are the lateral area, surface area, and volume of the following

figures determined: prisms, cylinders, pyramids, cones, and spheres?


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

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

Teacher Notes and Elaborations A dimension is the number of coordinates required to locate a point in a space. A flat surface is two-dimensional because two coordinates are needed to specify a point on it. Three-dimensional space is a geometric model of the physical universe in which we live. The three dimensions are commonly called length, width, and depth (or height), although any three directions can be chosen, provided that they do not lie in the same plane. A polygon is a geometric figure formed by three or more coplanar segments called sides. Each side intersects exactly two other sides, but only at their endpoints, and the intersecting sides must be noncollinear. A vertex of an angle is a point common to the two sides of the angle. In a polygon, a vertex is a point common to two sides of the polygon. The vertex of a polyhedron is a point common to the edges of a polyhedron. In a polyhedron the flat surfaces formed by the polygons and their interiors are called faces. Area is the number of square units in a region. Surface area is a measurement of coverage such as wallpaper. Lateral area is the area of the exterior surface (lateral surface) of a three-dimensional figure not including the area of the base(s). A prism is a three-dimensional figure whose lateral faces are

(continued)


Page 48 of 57

Surface Area and Volume (continued) Teacher Notes and Elaborations (continued) parallelograms. If the faces are rectangles, the prism is a right prism. A prism is classified by the shape of its base. A pyramid is a three-dimensional figure whose lateral faces are triangles. In regular pyramids, the base is a regular polygon, lateral edges are congruent, and all lateral faces are congruent isosceles triangles. Slant height in a pyramid is the distance from the vertex perpendicular to the base on a lateral face of the pyramid. Slant height on a cone is the distance from the vertex to the circle. Height is the perpendicular distance between bases or between a vertex and a base. A cone is a three-dimensional figure that has a circular base, a vertex not in the plane of the circle, and a curved lateral surface. In a right cone, the altitude is a perpendicular segment from the vertex to the center of the base. The height (h) is the length of the altitude. The slant height ( l ) is the distance from the vertex to a point on the edge of the base. Surface area is the lateral area plus the area of the base(s). Bases of prisms are congruent polygons lying in parallel planes. An altitude (height) of a prism is a segment joining the two base planes and perpendicular to both. The faces of a prism that are not its bases are called lateral faces. Adjacent lateral faces intersect in parallel segments called lateral edges. In right prisms the lateral edges are also altitudes. Volume is the capacity of a three-dimensional figure such as the amount of water in an aquarium. The volume of an irregularly shaped object can be found by measuring its displacement. When an object is placed in a liquid, it causes the liquid to rise. This volume is called the objects’ displacement. Return to Course Outline

The base of a three-dimensional figure could be a circle, a triangle, a square, a rectangle, a regular hexagon or another type of polygon. Many formulas use B to represent the area of the base of the solid figure. To find the area of a base (B) in three dimensional figures, use the area formula that applies. Formulas for those figures may need to be reviewed. A sphere is the set of all points in space equidistant from a given point. The center is the given point and the radius is the given distance. Surface area and volume of spheres will also be found. When determining surface area of combinations of solids, attention needs to be given to the possibility of shared faces. Similar figures are figures that have the same shape but not necessarily the same size. Scale factors (proportional reasoning) are used to compare perimeters, areas, and volumes of similar two-dimensional and three-dimensional geometric figures. A change in one dimension of an object results in changes in area and volume in specific patterns. Volumes, areas, and perimeters of similar polygons are examined to draw conclusions about how changes in one dimension affect both area and volume. If the given perimeter of a polygon is increased or decreased, the area will increase or decrease by the square of the change and the volume increases or decreases by the cube of the change. Similar solids are solids that have the same shape but not necessarily the same size. All spheres are similar.

(continued)


Page 49 of 57

Surface Area and Volume (continued) Teacher Notes and Elaborations (continued) If the scale factor of two similar solids is a:b, then:

– The ratio of corresponding perimeters is a:b. – The ratios of the base areas, of the lateral areas, and of the total

areas are a2:b2. – The ratio of the volumes is a3:b3.

Sample Instructional Strategies and Activities • Use strings, straws, toothpicks, etc. to make three-dimensional

objects. • Students make a three-dimensional object from any material they

choose. They calculate lateral area, total area, and volume and incorporate this into a written report, which includes their calculations, a sketch of their model, and a description of their procedure. Students give a brief oral report of their project.

• Using a geometric model kit, students will investigate relationships among volume formulas.

• Demonstrate a way that the formula for the surface area of a sphere might have been evolved.

• To demonstrate the formula for surface area of a sphere, cut an orange in half and trace the circumference of the orange on paper several times. Peel the orange and completely fill as many circles as possible. The result should be four filled circles, thus four times the area of the circle.

• Using items from a pantry have students measure and compute surface area and volume.

• When an object is placed in a liquid, it causes the liquid to rise. This volume is called the objects’ displacement. The volume of an irregularly shaped object can be found by measuring its displacement.


Example: A rock is placed into a rectangular prism containing water. The base of the container is 10 centimeters by 15 centimeters and when the rock is put in the prism, the water level rises 2 centimeters due to the displacement. This new “slice” of water has a volume of 300 cubic centimeters (10 15 2⋅ ⋅ ). Therefore, the volume of the rock is 300 cubic centimeters. • Each student is given a sheet of construction paper. Next, they are

instructed to cut a square from each corner and form an open top box with the maximum volume.

• Have students use string and a ruler to determine whether two solids are similar. If the figures are similar then use the measurements to compare areas and volumes.


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Interdisciplinary Concept: Systems; Communication Math Concept: Relationships; Quantifying Representation; Reasoning and Justification ACPS Mathematics Enduring Understandings: 2. Spatial relationships can be described using coordinate geometry and other representational systems. 3. Attributes of objects can be measured using processes and quantified units, using appropriate techniques, tools, and formulas. 11. Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial

reasoning. 12. Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations. ACPS Essential Standard in grade band 9-12 2. Investigate and use Cartesian coordinates and other coordinate systems to analyze geometric situations. 3. Make reasonable estimates and accurate predictions about measurement by applying appropriate technology, and understanding the limitations. 11. Analyze characteristics and properties of geometric shapes and develop mathematical arguments about these shapes in applied settings and

authentic situations. 12. Understand and represent translations and reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors,

function notation, and matrices. Life Long Learner Standards 2. Gather, organize, and analyze data, evaluate processes and products, and draw conclusions; 3. Think analytically, critically, and creatively to pursue new ideas, acquire new knowledge, and make decisions;

Coordinate Geometry Strand: Reasoning, Lines, and Transformations; Polygons and Circles SOL G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include a) investigating and using formulas for

finding distance, midpoint, and slope; Return to Course Outline

Essential Knowledge and Skills The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Find the coordinates of the midpoint of a segment, using

the midpoint formula. • Use a formula to find the slope of a line. • Compare the slopes to determine whether two lines are

parallel, perpendicular, or neither. • Determine whether a figure has point symmetry, line

symmetry, both, or neither. • Given an image and pre-image, identify the

transformation that has taken place as a reflection, rotation, dilation, or translation.

Essential Understanding • Transformations and combinations of transformations can be

used to describe movement of objects in a plane. • The distance formula is an application of the Pythagorean

Theorem. • Geometric figures can be represented in the coordinate plane. • Techniques for investigating symmetry may include paper

folding, coordinate methods, and dynamic geometry software. • Parallel lines have the same slope. • The product of the slopes of perpendicular lines is -1. • The image of an object or function graph after an isomorphic

transformation is congruent to the preimage of the object. • The terms characteristics and properties can be used

interchangeably to describe quadrilaterals. The term


Page 51 of 57

b) applying slope to verify and determine

whether lines are parallel or perpendicular; c) investigating symmetry and determining

whether a figure is symmetric with respect to a line or a point; and

d) determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods.

SOL G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. SOL G.12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. Return to Course Outline

• Apply the distance formula to find the length of a line segment when given the coordinates of the endpoints.

• Solve problems, including real-world problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids.

• Prove that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula.

• Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods.

• Prove properties of angles for a quadrilateral inscribed in a circle.

• Identify the center, radius, and diameter of a circle from a given standard equation.

• Use the distance formula to find the radius of a circle. • Given the coordinates of the center and radius of the

circle, identify a point on the circle. • Given the equation of a circle in standard form, identify

the coordinates of the center and find the radius of the circle.

• Given the coordinates of the endpoints of a diameter, find the equation of the circle.

• Given the coordinates of the center and a point on the circle, find the equation of the circle.

• Recognize that the equation of a circle of given center and radius is derived using the Pythagorean Theorem.

characteristics is used in elementary and middle school mathematics.

• Quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals.

• Characteristics of quadrilaterals can be used to identify the quadrilateral and to find the measures of sides and angles.

• A circle is a locus of points equidistant from a given point, the center.

• Standard form for the equation of a circle is

( ) ( )2 2 2x h y k r− + − = , where the coordinates of the center of the circle are ( , )h k and r is the length of the radius.

• The circle is a conic section.

(continued)


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Coordinate Geometry (continued) Key Vocabulary

dilation distance formula image isometry isomorphism line symmetry

midpoint formula point symmetry pre-image reflection rotation slope

slope formula standard form for the equation of a circle symmetry transformation translation

Essential Questions • What is the relationship between the distance formula, the

Pythagorean Theorem, and the equation of a circle? • How does the concept of midpoint and slope relate to symmetry

and transformation? • What is line symmetry? • What is point symmetry? • How can symmetry be used to describe naturally occurring

phenomena? • How is a figure translated, reflected, rotated, or dilated? • What is the relationship between the center, the radius, and the

standard equation of a circle? Return to Course Outline

Teacher Notes and Elaborations Like finding distance, two situations must be considered to find the midpoint of the line and the congruence of the two line segments. The two situations that must be considered are the midpoint on a number line and midpoint in the coordinate plane. The midpoint of a segment is the point that divides the segment into two congruent segments. The midpoint of AB is the average of the coordinates of A and B. A M B − 3 − 2 − 1 0 1 2 3 4 5 6 7

( 1) 5 22

− +=

The Midpoint Formula uses the idea that the midpoint of a horizontal or vertical line is the average of the coordinates of the endpoints. To find the midpoint of a horizontal line segment, find the average of the x endpoint coordinates; the y coordinate will be the same for all the points. To find the midpoint of a vertical line segment the x coordinate; will be the same for all points; the y coordinate will be the average of the y endpoint coordinates.

(continued)


Page 53 of 57

Coordinate Geometry (continued) Teacher Notes and Elaborations (continued)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

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

123456789

x

y

F

G

H

The midpoint of CE is D (2,2) . The midpoint of FH is G ( 3, 2)− −

. This idea is used twice to find the coordinates of the midpoint of a slanting segment with endpoints 1 1 1P ( , )x y and 2 2 2P ( , )x y .

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9-8-7-6-5-4-3-2-1

123456789

x

y

P1(x1, y1)

P2(x2, y2)

M S

R T

The midpoint of 1 2PP is M 1 2 1 2, 2 2

x x y y+ +⎛ ⎞⎜ ⎟⎝ ⎠

.

Some students may have difficulty in extending the concept of finding the midpoint of a line segment on one number line to a line segment in the coordinate plane. Using models such as the one above will aid in developing this concept. Return to Course Outline

The slope (effect of steepness) of a line containing two points in the coordinate plane can be found using the slope formula. The slope of a vertical line is undefined since x1 = x2. Parallel lines are lines that do not intersect and are coplanar. Parallel planes are planes that do not intersect. Nonvertical lines are parallel if they have the same slope and different y-intercepts. Any two vertical lines are parallel. Perpendicular lines are lines that intersect at right angles. Two non-vertical lines are perpendicular if and only if the product of their slopes is 1− . Students should have multiple experiences applying the following formulas. Given two points (x1, y1) and (x2, y2):

- the midpoint formula is 1 2 1 2,2 2

x x y y+ +⎛ ⎞⎜ ⎟⎝ ⎠

;

- the distance formula is ( ) ( )2 22 1 2 1x x y y− + − ; and

- the slope formula is ( )( )2 1

2 1

y yx x−

−.

Regular polygons are frequently used to introduce the concepts of symmetry, transformations, and tessellation. A geometric configuration (curve, surface, etc.) is said to be symmetric (have symmetry) with respect to a point, a line, or a plane, when for every point on the configuration there is another point of the configuration such that the pair is symmetric with respect to the point, line, or plane. The point is the center of symmetry; the line is the axis of symmetry, and the plane is the plane of symmetry. A line of symmetry is a line

(continued)


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Coordinate Geometry (continued) Teacher Notes and Elaborations (continued) that can be drawn so that the figure on one side is the reflection image of the figure on the opposite side. A figure has point symmetry if there is a symmetry point O such that the half-turn HO maps the figure onto itself. A figure has line symmetry if there is a symmetry line k such that the reflection Rk maps the figure onto itself. An isomorphism is a one-to-one mapping that preserves the relationship between two sets. The original figure is the preimage. The resulting figure is an image. An isometry is a transformation in which the preimage and image are congruent. Reflections, rotations, and translations are isometries. Dilations are not isometry. Reflection is a transformation in which a line acts like a mirror, reflecting points to their images. For many figures, a point can be found that is a point of reflection for all points on the figure. This point of reflection is called a point of symmetry. A rotation is a transformation suggested by a rotating paddle wheel. When the wheel moves, each paddle rotates to a new position. When the wheel stops, the position of a paddle ( Pʹ′ ) can be referred to mathematically as the image of the initial position of the paddle (P). A figure with rotational symmetry of 180° has point symmetry. A geometric transformation in a plane is a one-to-one correspondence between two sets of points. It is a change in its position, shape, or size. It maps a figure onto its image and may be described with arrow (→) notation. A reflection is a type of transformation that can be described by folding over a line of reflection or line of symmetry. For some figures, a point can be found that is a point of reflection for all points on the figure. Return to Course Outline

A dilation is a transformation that may change the size of a figure. It requires a center point and a scale factor. The scale factor is defined as the image to

pre-image. For example: 4 to 3 or 43

represents an enlargement.

A composite of reflections is the transformation that results from performing one reflection after another. A translation (slide) is the composite of two reflections over parallel lines. The Pythagorean Theorem (distance formula) can be used to develop an equation of a circle. y Let P(x, y) represent any point on the circle. The distance between C(h, k) and P(x, y) is r. 2 2( ) ( )x h y k r− + − = P(x, y)

2 2 2( ) ( )x h y k r− + − = C(h, k) x Given the coordinates of the center of the circle (h, k) and a radius r, four easily identified points on the circle are: ( , )h r k+ , ( , )h r k− , ( , )h k r+ , ( , )h k r− Given the coordinates of the endpoints of a diameter, midpoint formula can be used to find the center of the circle and distance formula can be used to find the radius

(continued)

r


Page 55 of 57

Coordinate Geometry (continued) Extension • Reflect triangles over horizontal and vertical lines in the coordinate

plane and the line y = x. • Draw on a coordinate plane the image that results from a geometric

figure that has been reflected, rotated, or dilated. • Investigate the relationship between a rotation and the composition

of reflections. • Investigate point-slope form as it relates to the equation of a line

(slope-intercept form) and the formula for slope. • Use slopes of parallel and perpendicular lines to write equations in

standard, point-slope, and slope-intercept forms. • Find the coordinates of an endpoint of a segment given the

coordinates of the midpoint and one endpoint. Extension Point-slope form is an equation of the form 1 1( )y y m x x− = − for the line passing through a point whose coordinates are 1 1( , )x y and having slope m. The composite of reflections with respect to two intersecting lines is a transformation called a rotation. The point of intersection, point P, is the center of rotation. The figure rotates or turns around the point P. Point symmetry is a rotational symmetry of 180°. A dilation is a similarity transformation that alters the size of a geometric figure, but does not change the shape. For each dilation, a scale factor enlarges the dilation image, reduces the dilation image, or maintains a congruence transformation. Extension • Investigate and identify points that lie inside, on, or outside a

circle. • Write inequality statements for regions either inside or outside a Return to Course Outline

circle and sketch these graphs. Extension An example of an inequality that describes the points (x, y) outside the circle that are more than three units from center (4, 2− ) is 2 2( 4) ( 2) 9x y− + + > . The graph would be a broken circle and shaded outside the circle. An example of an inequality that describes the points (x, y) inside the circle that are less than or equal to four units from center ( 3, 5− − ) is

2 2( 3) ( 5) 16x y+ + + ≤ . The graph would be a circle and shaded inside the circle. Sample Instructional Strategies and Activities • Do activities from the Geometer’s Sketchpad by Key Curriculum Press. • Use coordinate geometry as a tool for making conjectures about

midpoints, slopes, and distance. • Each student is given a sheet of construction paper. Next, the teacher puts

a few drops of finger paint, etc. on each paper. Each student folds his/her papers to illustrate symmetry with respect to a line.

• Demonstrate symmetry by using patty paper. • Cut out a triangle. Place a different color dot in each angle. Place the

triangle on the paper and trace around it in pencil. Slide triangle over and mark the color in each angle so that the colors correspond with the cardboard triangle. Place triangle back on top and rotate it so that it no longer overlaps. Repeat until the plane is filled. Have students identify parallel lines, vertical angles, etc. Students make conjectures about lines and angles in the tessellation. Students are given various polygons and asked if they tessellate a plane. Explain why or why not.

(continued)


Page 56 of 57

Coordinate Geometry (continued) Sample Instructional Strategies and Activities (continued) • Place a shape on the overhead projector. Have a student trace the

image on the blackboard. Move the projector away from the board and trace the new image. Take the original shape and compare the angles of the original with the angles of the images. Students can measure the lengths of the sides and compare ratios.

• Use patty paper to demonstrate reflections, rotations, dilations, or translations.

• Use examples of advertisements to identify examples of transformations.

• Give students coordinates of the vertices of a rectangle. Have students find the lengths of the diagonals, the midpoints of the diagonals, and the slopes of the diagonals. Have students make conjectures about the diagonals of the rectangle. Repeat with square, rhombus, parallelogram, isosceles trapezoid, trapezoid, and quadrilateral. Have students make conjectures about the diagonals of each.


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