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PreAP GeometryCurriculum Guide

Curriculum Guide (Revised 2016)

GEOMETRY AND PreAP GEOMETRY CURRICULUM GUIDE (Revised 2016) PRINCE WILLIAM COUNTY SCHOOLS

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Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should learn. The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Guide is divided into sections: Curriculum Information, Essential Knowledge and Skills, Key Vocabulary, Essential Questions and Understandings, Teacher Notes and Elaborations, Resources, and Sample Instructional Strategies and Activities. The purpose of each section is explained below. Curriculum Information: This section includes the objective and SOL Reporting Category, focus or topic, and in some, not all, foundational objectives that are being built upon. Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective. Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills. Essential Questions and Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives. Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the objective beyond the current grade level. It may also contain definitions of key vocabulary to help facilitate student learning. Resources: This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources. Sample Instructional Strategies and Activities: This section lists ideas and suggestions that teachers may use when planning instruction.


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The following chart is the pacing guide for the Prince William County Geometry Curriculum. The chart outlines the recommended order in which the objectives should be taught; provides the suggested number blocks to teach each unit and organizes the objectives into Units of Study. The Prince William County cross-content vocabulary terms that are in this course are: analyze, compare and contrast, conclude, evaluate, explain, generalize, question/inquire, sequence, solve, summarize, and synthesize.

Geometry Objectives Approximate Pacing

Unit 1 – Coordinate Geometry and Equations of Circles 3a, 3b, 4b, 4c, 4d, 12 10 Blocks

Unit 2 – Logic 1 5 Blocks

Unit 3 – Angle Relationships with Intersecting & Parallel Lines 2, 4f, 4g 6 Blocks

Unit 4 – Triangle Inequalities & Relationships with Triangles 4a, 5 5 Blocks

Unit 5 – Triangle Properties and Similar Triangles 4e, 7, 14a (2-D) 7 Blocks

Unit 6 – Congruent Triangles 6 7 Blocks

Unit 7 – Right Triangles and Special Right Triangles 8 7 Blocks

Unit 8 – Circles 11, 4 (specific to unit) 7 Blocks

Unit 9 – Quadrilaterals and Polygons 9, 10, 4(specific to polygons) 8 Blocks

Unit 10 – 3-D Figures 13, 14 5 Blocks

Unit 11 – Transformations 3c, 3d 5 Blocks

GEOMETRY SOL TEST QUESTION BREAKDOWN (50 QUESTIONS TOTAL) (Based on 2009 SOL Objectives and Reporting Categories)

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


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Objective Page G.1 G.2 Page 13 G.3 Page 19 G.4 Page 25 G.5 Page 29 G.6 Page 33 G.7 Page 37 G.8 Page 41 G.9 Page 47 G.10 Page 51 G.11 Page 55 G.12 Page 59 G.13 Page 63 G.14 Page 67


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Curriculum Information

Essential Knowledge and Skills Key Vocabulary

Essential Questions and Understandings Teacher Notes and Elaborations

SOL Reporting Category Reasoning, Lines and Transformations Topic Reasoning, Lines and Transformations Virginia SOL G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a. identifying the converse, inverse,

and contrapositive of a conditional statement;

b. translating a short verbal argument into symbolic form;

c. using Venn diagrams to represent set relationships; and

d. using deductive reasoning.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Identify the converse, inverse, and

contrapositive of a conditional statement.

Translate verbal arguments into symbolic form such as (p → q), and (~p → ~q).

Determine the validity of a logical argument.

Use valid forms of deductive reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples.

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

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

Interpret Venn diagrams. Recognize and use the symbols of

formal logic, which include , , ~, , and .

Identify logically equivalent statements.

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Essential Questions What is the relationship between reasoning, justification, and proof in geometry? What is a truth-value? How does a truth-value apply to conditional statements? How do deductive reasoning and Venn diagrams help judge the validity of logical

arguments? Essential Understandings Inductive reasoning, deductive reasoning, and proof are critical in establishing general

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

definitions, postulates, and theorems. Inductive reasoning is the method of drawing conclusions from a limited set of

observations. Logical arguments consist of a set of premises or hypotheses and a conclusion. Proof is a justification that is logically valid and based on initial assumptions,

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

plane), postulates, and theorems. When a conditional and its converse are true, the statements can be written as a

biconditional (i.e., iff or if and only if). Logical arguments that are valid may not be true. Truth and validity are not

synonymous. 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). A mathematical statement is one in which a fact or complete idea is expressed. Because a mathematical statement states a fact, many of them can be judged to be “true” or “false”. Questions and phrases are not mathematical statements since they can not be judged as true or false. Terms associated with logical arguments are reasoning, justification, and proof. Reasoning is the drawing of conclusions or inferences from facts, observations, or hypotheses. Justification is a rationale or argument for some mathematical proposition. A conjecture is a statement that has not been proved true nor shown to be false. A proof is a justification that is logically valid and based on initial assumptions, definitions, and proven results. Proofs are developed so that each step in the argument is in proper chronological order in relation to earlier steps. When building a proof the argument must be clearly developed and each step must be supported by a property, theorem, postulate, or definition.

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(continued) Extension for PreAP Geometry Identify, create, and determine the

truth-value of the converse, inverse, and contrapositive of a conditional statement.

Use chain reasoning to make a logical conclusion given a set of statements.

Construct truth tables given statements (conditional, conjunction, disjunction, biconditionals, etc.).

Investigate the concept of an indirect proof.

Key Vocabulary assumption biconditional statement conclusion conditional statement conjecture conjunction contrapositive converse counterexample deductive reasoning disjunction hypothesis (premise) inductive reasoning intersection inverse Law of Detachment Law of Syllogism Law of the Contrapositive logic postulate (axiom) proof theorem union

Teacher Notes and Elaborations (continued) 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. 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). In a two-column proof (T-form proof) two columns are presented where the first column contains a numbered chronological list of steps or statements that lead to the desired conclusion. The second column contains a list of reasons which support each step in the proof. These reasons are properties, theorems, postulates and definitions. This method clearly displays each step in the argument and keeps ideas organized. A paragraph proof consists of a detailed paragraph explaining the proof process. The paragraph is lengthy and contains the steps and reasons which lead to the final conclusion. It is essential that critical steps (or supporting reasons) are not left out. Coordinate geometry applies algebraic principles to geometric situations. Coordinate geometry proofs employ the use of formulas such as the distance formula, the slope formula and/or the midpoint formula as well as postulates, theorems and definitions. To develop a coordinate geometry proof: draw and label the graph; state the formula to be used; show all work; and write a concluding sentence stating what has been proven and why it is true. 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. If p (hypothesis), then q (conclusion). p q q p

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

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

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

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

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

Sentences, or statements, that have the same truth value are said to be logically equivalent. The contrapositive and original conditional statements are logically equivalent (Law of Contrapositive). Since the statement and its contrapositve are both true or else both false, they are called logically equivalent. The following statements are logically equivalent. True statement: If a figure is a triangle, then it is a polygon. True contrapositive: If a figure is not a polygon, then it is not a triangle. The converse has the same truth value as the inverse of the original statement. The converse and the inverse of the original statement are logically equivalent. Symbolic form includes truth tables (tabular representation of the truth or falsehood of hypotheses and conclusions) and Venn diagrams. Deductive reasoning uses rules to make conclusions. Applying the Law of Detachment, if you accept “If p then q” as true and you accept p as true, then you must logically accept q as true. It also follows if you accept “If p then q” as true and you accept not q as true, then you must logically accept not p as true. According to the Law of Syllogism, if you accept “If p then q” as true and if you accept “If q then r” as true, then you must logically accept “If p then r” as true. A counterexample is an example used to prove an if-then statement false. For that counterexample, the hypothesis is true and the conclusion is false. Inductive reasoning is a kind of reasoning in which the conclusion is based on several past observations. Symbolically means “therefore”. Ex: m ABC is 90° ABC is a right angle.

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Teacher Notes and Elaborations (continued) In logic, letters are used to represent simple statements that are either true or false. Simple statements can be joined to form compound statements. A conjunction is a compound statement composed of two simple statements joined by the word “and”. The symbol , is used to represent the word “and”. A disjunction is a compound statement of two simple statements joined by the word “or”. The symbol , is used to represent the word “or”. “Intersection” is the set of elements that are “Union” is the set of elements that elements of two or more given sets. belong to either or both of a given pair of sets. p q p q A biconditional statement is the conjunction of a conditional and its converse. Symbolically: ( ) ( )p q q p is written ( )p q and is read p if and only if q or p iff q. p q

(continued)

qpqp


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Teacher Notes and Elaborations (continued) 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 Conjunction Disjunction If p then q p and q p or q

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

T T T T T T T T T

T F F T F F T F T

F T T F T F F T T

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

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

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


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Teacher Notes and Elaborations (continued)

Extension for PreAP Geometry An indirect proof is a proof that begins by assuming temporarily that the conclusion is not true; then reason logically until a contradiction of the hypothesis or another known fact is reached. Generally, the word “not” or the presence of a “not symbol” (such as the not equal sign) in a problem indicates the need for an indirect proof. When formulating an indirect proof first assume that the opposite of what is to be proven is true. Next, from this assumption, determine what conclusions can be drawn. These conclusions must be based upon the assumption and the use of valid statements. Search for a conclusion that is known to be false because it contradicts given or known information. Since the assumption leads to a false conclusion, the assumption must be false. Therefore if the assumption (which is the opposite of what is to be proven) is false, then what is being proven must be true. Indirect Proof Example:

ABC is not isosceles. Prove that if altitude BD is drawn, it will not bisect AC . B Given: ABC is not isosceles altitude BD Prove: BD does not bisect AC A D C

STATEMENTS REASONS 1. ABC is not isosceles

altitude BD 1. Given

2. Assume BD bisects AC 2. Assumption leading to a contradiction. 3. D is the midpoint of AC 3. Bisector of a segment divides the segment at its midpoint. 4. AD DC 4. Midpoint divides a segment into two congruent segments. 5. BD AC 5. The altitude of a triangle is a line segment extending from any vertex of a triangle

perpendicular to the line containing the opposite side. 6. ADB, BDC are right angles 6. Perpendicular lines meet to form right angles 7. ADB BDC 7. All right angles are congruent. 8. BD BD 8. Reflexive Property 9. ADB CDB 9. SAS - If two sides and the included angle of one triangle are congruent to the

corresponding parts of a second triangle, the two triangles are congruent. 10. AB BC 10 CPCTC - Corresponding parts of congruent triangles are congruent. 11. ABC is isosceles 11 An isosceles triangle is a triangle with two congruent sides. 12. BD does not bisect AC 12 Contradiction steps 1 and 11


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Resources Sample Instructional Strategies and Activities

SOL Reporting Category Reasoning, Lines and Transformations Topic Reasoning, Lines and Transformations Virginia SOL G.1 Foundational Objectives 8.2 The student will describe orally and in writing the relationships between the subsets of the real number system.

Text: Geometry and PreAP Geometry Prentice Hall Geometry, Virginia Edition, ©2012, Charles et al., Pearson Education IGCSE Geometry Extended Mathematics for IGCSE, Third Edition, ©2011, Rayner, Oxford University Press

PWC Mathematics Website http://pwcs.math.schoolfusion.us Virginia Department of Education Website http://www.doe.virginia.gov/instruction/mathematics/index.shtml Geometry reference http://www.mathopenref.com/

Students, working in cooperative learning groups, will solve logic problems to introduce the concept of deductive reasoning. Each group of students will give their solutions and describe their thought processes.


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SOL Reporting Category Reasoning, Lines and Transformations Topic Reasoning, Lines and Transformations Virginia SOL G.2 The student will use the relationships between angles formed by two lines cut by a transversal a. determine whether two lines are

parallel; b. verify the parallelism, using

algebraic and coordinate methods as well as deductive proofs; and

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

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Use properties, postulates, and

theorems to determine whether two lines are parallel.

Use algebraic and coordinate methods as well as deductive proofs to verify whether two lines are parallel.

State the relationships between angles that are a linear pair.

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

Solve real-world problems involving intersecting and parallel lines in a plane.

Identify lines as parallel, intersecting, perpendicular, or skew.

Use definitions, postulates, and theorems to complete two-column or paragraph proofs with at least five steps.

Extension for PreAP Geometry Write equations of parallel and

perpendicular lines. Investigate skew lines using real world

models.

(continued)

Essential Questions What is the relationship between lines and angles? What is the difference between parallel lines and perpendicular lines? How are lines proven parallel? What is the difference between parallel lines and intersecting lines? What are the relationships between the angles formed when two parallel lines are cut by

a transversal? Essential Understandings Parallel lines intersected by a transversal form angles with specific relationships. Some angle relationships may be used when proving two lines intersected by a

transversal are parallel. The Parallel Postulate differentiates Euclidean from non-Euclidean geometries such as

spherical geometry and hyperbolic geometry. 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. Extension for PreAP Geometry Skew lines are non-coplanar lines that do not intersect. Experiences with skew lines should include 3-dimensional models. Intersection is a point or set of points common to two or more figures.

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Key Vocabulary adjacent angles algebraic method alternate exterior angles alternate interior angles complementary angles coordinate method corresponding angles deductive proof Euclidean Geometry exterior angle interior angle intersection linear pair parallel Parallel Postulate same-side (consecutive) interior angles same-side (consecutive) exterior angles supplementary angles transversal vertical angles

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

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

Proving lines parallel implies determining whether necessary and sufficient conditions (properties, definitions, postulates, and theorems) exist for parallelism. A proof is a chain of logical statements starting with given information and leading to a conclusion. Two column deductive proofs (formal proofs) are examples of deductive reasoning. They contain statements and reasons organized in two columns. Each step is called a statement, and the properties that justify each step are called reasons. In a paragraph proof (informal proof) a paragraph is written to explain why a conjecture for a given situation is true. Essential parts of a good proof include:

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

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Teacher Notes and Elaborations (continued) The following is an example of a paragraph proof. Given: E is the midpoint of BD B C AE ED E Prove: AEB CED A D In the figure above the facts that E is the midpoint of BD and AE ED is given. Since E is the midpoint of BD , then BE ED because the midpoint of a segment divides the segment into two congruent segments. Since vertical angles are congruent BEA DEC , there is now sufficient information to satisfy the SAS method of proving triangles congruent. Therefore, AEB CED because if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. The Parallel Postulate is the axiom of Euclidean Geometry stating that if two straight lines are cut by a third, the two will meet on the side of the third on which the sum of the interior angles is less than two right angles. Equivalently, Playfair’s Axiom states: “If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.” In Euclidean Geometry, parallel lines lie in the same plane and never intersect. In spherical geometry, the sphere is the plane, and a great circle represents a line. Two nonvertical coplanar lines are parallel if and only if their slopes are equal. Two nonvertical coplanar lines are perpendicular if and only if the product of their slopes is 1 .

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Teacher Notes and Elaborations (continued) Algebraic and coordinate methods should also be used to determine parallelism. Coordinate geometry establishes a correspondence between algebraic concepts and geometric concepts. For example, the distance formula is derived as an application of the Pythagorean Theorem. The Pythagorean Theorem in turn is used to develop the equation of a circle. The coordinate proof is often more convenient than a two-column proof. The following is an example of a coordinate proof involving parallelism. Prove: The segment that joins the midpoint of two sides of a triangle is parallel to the third side. Given: OAB and M and N the midpoints of OB and OA respectively. Prove: MN || BA Proof: Choose axes and coordinates as shown. y B (2 ,2 )b c M O N A (2 ,0)a x

1. Midpoints are 2 0 2 0 2 2M( , ) ( , ) ( , )2 2 2 2

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


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SOL Reporting Category Reasoning, Lines and Transformations Topic Reasoning, Lines and Transformations Virginia SOL G.2 Foundational Objectives A.4 The student will solve multi-step linear and quadratic equations in two variables, including a. solving literal equations (formulas)

for a given variable; and d. solving multi-step linear equations

algebraically and graphically. A.6 The student will graph linear equations and linear inequalities in two variables, including a. determining the slope of a line when

given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined; and

b. writing the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.

8.6 The student will a. verify by measuring and describe

the relationships among vertical angles, adjacent angles, supplementary angles, and complementary angles; and

b. measure angles of less than 360°.


PWC Mathematics Website http://pwcs.math.schoolfusion.us Virginia Department of Education Website http://www.doe.virginia.gov/instruction/mathematics/index.shtml Geometry reference http://www.mathopenref.com/ Foundational Objectives (continued) 8.10 The student will a. verify the Pythagorean Theorem; and b. apply the Pythagorean Theorem. 8.15 The student will a. solve multi-step linear equations in one

variable on one and two sides of the equation.

8.16 The student will graph a linear equation in two variables.

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

Use patty paper to trace and compare lines and angles. Have class look for parallel, intersecting, perpendicular, and skew lines in the

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

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

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

Have students use patty paper to discover congruent angles formed when parallel lines 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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SOL Reporting Category Reasoning, Lines and Transformations Topic Reasoning, Lines and Transformations Virginia SOL G.3 The student will use pictorial representations, including computer software, constructions, and coordinate methods to solve problems involving symmetry and transformation. This will include a. investigating and using formulas for

finding distance, midpoint, and slope;

b. applying slope to verify and determine whether lines are parallel or perpendicular;

c. investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and

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

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Given an image and preimage, identify

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

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

Find the coordinates of the midpoint of a segment, using the midpoint formula.

Use a formula to find the slope of a line.

Determine whether a figure has point symmetry, line symmetry, both, or neither.

Compare the slopes to determine whether two lines are parallel, perpendicular, or neither.

Use algebraic, coordinate, and deductive methods to determine if lines are perpendicular.

Draw on a coordinate plane the image that results from a geometric figure that has been reflected, rotated, or dilated.

Identify the coordinates of the image that results from a geometric figure that has been reflected, rotated, or dilated.

Find the coordinates of an endpoint of a segment given the coordinates of the midpoint and one endpoint.

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Essential Questions What is the relationship between the distance formula and the Pythagorean Theorem? How does the concept of midpoint and slope relate to symmetry and transformation? What is line symmetry? When is a figure symmetric about a point? What types of symmetrical problems are found in real-life? How is a figure translated, reflected, rotated, or dilated? Essential Understandings Transformations and combinations of transformations can be used to describe movement

of objects in a plane. The distance formula is an application of the Pythagorean Theorem. Geometric figures can be represented in the coordinate plane. Techniques for investigating symmetry may include paper folding, coordinate methods,

and dynamic geometry software. Parallel lines have the same slope. The product of the slopes of perpendicular lines is 1 . The image of an object or function graph after an isomorphic transformation is

congruent to the preimage of the object. Teacher Notes and Elaborations Transformations and combinations of transformations can be used to describe movement. The Pythagorean Theorem states that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the legs. Pythagorean Triples are three positive integers that satisfy the Pythagorean theorem. The converse of the Pythagorean Theorem guarantees that a, b, and c are lengths of the sides of a right triangle. Because of this, any such triple of integers is called a Pythagorean triple. For example, 3, 4, 5 is a Pythagorean triple since

2 2 23 4 5 . Another triple is 6, 8, 10, since 2 2 26 8 10 . The triple 3, 4, 5 is called a primitive Pythagorean triple because no factor (other than 1) is common to all three integers. 6, 8, 10 is not a primitive triple. Other primitive triples are 5, 12, 13; 8, 15, 17; and 7, 24, 25. Students should recognize these primitive triples in order to use them to create other triples such as 9, 12, 15, which is found by multiplying each measure in 3, 4, 5 by a factor of 3. Two situations must be considered when finding the distance between two points: the distance on a number line ( 2 1x x ) and the distance in the coordinate plane (distance formula or Pythagorean Theorem).

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(continued)

Extensions for PreAP Geometry Investigate the relationship between a

rotation and the composition of reflections.

Investigate point-slope form as it relates to the equation of a line (slope-intercept form) and the formula for slope.

Use slopes of parallel and perpendicular lines to write equations in standard, point-slope, and slope-intercept forms.

Represent translations, reflections, and rotations using algebraic and/or coordinate notation.

Apply the Pythagorean Theorem to a right triangle in the coordinate plane to derive the distance formula.

Key Vocabulary dilation distance formula image isometry isomorphism line symmetry midpoint midpoint formula parallel lines parallel planes perpendicular lines point symmetry preimage Pythagorean Theorem reflection rotation slope slope formula symmetry transformation translation

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

( 1) 5 22

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

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

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

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) .

(continued)


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Teacher Notes and Elaborations (continued) 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 2P P is M 1 2 1 2, 2 2

x x y y

.

Some students may have difficulty in extending the concept of finding the midpoint of a line segment on one number line to a line segment in the coordinate plane. Using models such as the one above will aid in developing this concept. The slope (effect of steepness) of a line containing two points in the coordinate plane can be found using the slope formula. The slope of a vertical line is undefined since x1 = x2. Parallel lines are lines that do not intersect and are coplanar. Parallel planes are planes that do not intersect. Nonvertical lines are parallel if they have the same slope and different y-intercepts. Any two vertical lines are parallel. Perpendicular lines are lines that intersect at right angles. Two non-vertical lines are perpendicular if and only if the product of their slopes is 1 . Students should have multiple experiences applying the following formulas. Given two points (x1, y1) and (x2, y2):

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

x x y y

;

- the distance formula is 2 22 1 2 1x x y y ; and

- the slope formula is

2 1

2 1

y yx x

.

(continued)


22








Teacher Notes and Elaborations (continued)

Extension for PreAP Geometry Point-slope form is an equation of the form 1 1( )y y m x x for the line passing through a point whose coordinates are 1 1( , )x y and having slope m . Regular polygons are frequently used to introduce the concepts of symmetry, transformations, and tessellation. A geometric configuration (curve, surface, etc.) is said to be symmetric (have symmetry) with respect to a point, a line, or a plane, when for every point on the configuration there is another point of the configuration such that the pair is symmetric with respect to the point, line, or plane. The point is the center of symmetry; the line is the axis of symmetry, and the plane is the plane of symmetry. A line of symmetry is a line that can be drawn so that the figure on one side is the reflection image of the figure on the opposite side. A figure has point symmetry if there is a symmetry point O such that the half-turn HO maps the figure onto itself. A figure has line symmetry if there is a symmetry line k such that the reflection Rk maps the figure onto itself. Extension for PreAP Geometry The composite of reflections with respect to two intersecting lines is a transformation called a rotation. The point of intersection, point P, is the center of rotation. The figure rotates or turns around the point P. Point symmetry is a rotational symmetry of 180°. A dilation is a similarity transformation that alters the size of a geometric figure, but does not change the shape. For each dilation, a scale factor enlarges the dilation image, reduces the dilation image, or maintains a congruence transformation. An isomorphism is a one-to-one mapping that preserves the relationship between two sets. The original figure is the preimage. The resulting figure is an image. An isometry is a transformation in which the preimage and image are congruent. Reflections, rotations, and translations are isometries. Dilations are not isometries. Reflection is a transformation in which a line acts like a mirror, reflecting points to their images. For many figures, a point can be found that is a point of reflection for all points on the figure. This point of reflection is called a point of symmetry. When a point is reflected across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite. When a point is reflected across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. When a point is reflected across the line y x , then the x-coordinate and the y-coordinate change places. When a point is reflected across the line y x , the x-coordinate and

the y-coordinate change places and are negated (the signs are changed). A rotation is a transformation suggested by a rotating paddle wheel. When the wheel moves, each paddle rotates to a new position. When the wheel stops, the position of a paddle ( P ) can be referred to mathematically as the image of the initial position of the paddle (P). A figure with rotational symmetry of 180° has point symmetry.

(continued)


23








Teacher Notes and Elaborations (continued) 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. 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. Extension for PreAP Geometry Translations, reflections, and rotations can be represented using algebraic and/or coordinate notation. Line Reflections: Reflection in the x-axis: When a point is reflected across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite. ( , ) '( , ) or ( , ) ( , )x axisP x y P x y r x y x y Reflection in the y-axis: When a point is reflected across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. ( , ) '( , ) or r ( , ) ( , )y axisP x y P x y x y x y Reflection in y x : When a point is reflected across the line y x , then the x-coordinate and the y-coordinate change places. ( , ) '( , ) or ( , ) ( , )y xP x y P y x r x y y x Reflection in y x : When a point is reflected across the line y x , the x-coordinate and the y-coordinate change places and are negated (the signs are changed). ( , ) '( , ) or ( , ) ( , )y xP x y P y x r x y y x Rotations: (assuming center of rotation to be the origin) Rotation of 90 :

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

Rotation of 180 : 180

( , ) ( , )R x y x y

Rotation of 270 : 270

( , ) ( , )R x y y x


24



SOL Reporting Category Reasoning, Lines and Transformations Topic Reasoning, Lines and Transformations Virginia SOL G.3 Foundational Objectives A.4a, d, f The student will solve multi-step linear and quadratic equations in two variables, including a. solving literal equations (formulas)

for a given variable; d. solving multi-step linear equations

algebraically and graphically; and f. solving real-world problems

involving equations and systems of equations.

A.6 The student will graph linear equations and linear inequalities in two variables, including a. determining the slope of a line when



8.8 The student will a. apply transformations to plane

figures; and b. identify applications of

transformations.

(continued)


PWC Mathematics Website http://pwcs.math.schoolfusion.us Virginia Department of Education Website http://www.doe.virginia.gov/instruction/mathematics/index.shtml Geometry reference http://www.regentsprep.org/regents/math/geometry/math-GEOMETRY.htm Geometry reference http://www.mathopenref.com/ Foundational Objectives (continued) 8.10 The student will a. verify the Pythagorean Theorem; and b. apply the Pythagorean Theorem. 8.15 The student will a. solve multi-step linear equations in one

variable on one and two sides of the equation.

8.16 The student will graph a linear equation in two variables. 7.8 The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane.

Do activities from the Geometer’s Sketchpad by Key Curriculum Press. Use coordinate geometry as a tool for making conjectures about midpoints, slopes, and

distance. Each student is given a sheet of construction paper. Next, the teacher puts a few drops of

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

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

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

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

Use patty paper to demonstrate reflections, rotations, dilations, or translations. Use examples of advertisements to identify examples of transformations.


25




SOL Reporting Category Reasoning, Lines and Transformations Topic Reasoning, Lines, and Transformations Virginia SOL G.4 The student will construct and justify the constructions of a. a line segment congruent to a given

line segment; b. the perpendicular bisector of a line

segment; c. a perpendicular to a given line from

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

given point on the line; e. the bisector of a given angle; f. an angle congruent to a given angle;

and g. a line parallel to a given line

through a point not on the given line.

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


Construct and justify an equilateral triangle, a square, and a regular hexagon including those inscribed in a circle.

Construct the inscribed and circumscribed circles of a triangle.

Construct and justify a tangent line from a point outside a given circle to the circle.

(continued)

Essential Questions What is the relationship between points, rays, and angles? Why are constructions important? How are constructions justified? Essential Understandings Construction techniques are used to solve real-world problems in engineering,

architectural design, and building construction. Construction techniques include using a straightedge and compass, paper folding, and

dynamic geometry software. 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 Sketch Pad, patty paper, a straightedge, and a compass. If students are using a ruler as a straightedge, they should be instructed to ignore its markings. Constructions help build an understanding of the relationships between lines and angles. The seven basic constructions can be used to do more complicated constructions such as tangents, geometric mean, and proportional segments. The intersection of two figures is the set of points that is in both figures. A transversal is a line that intersects two or more coplanar lines in different points. Two angles are congruent if and only if they have equal measures. A ray is an angle bisector if and only if it divides the angle into two congruent adjacent angles. Parallel lines are lines that do not intersect and are coplanar. Perpendicular lines are lines that intersect at right angles. A segment bisector is a line, segment, ray, or plane that intersects the segment at its midpoint. A perpendicular bisector of a segment is a line, ray, or segment that is perpendicular to the segment at its midpoint. A circle is circumscribed about a triangle if the circle contains all the vertices of the triangle. A triangle is inscribed in a circle if each of its vertices lies on the circle. In a triangle, a median is a segment that joins a vertex of the triangle and the midpoint of the side opposite that vertex. The medians of a triangle intersect at the common point called the centroid.

(continued)


26











(continued)

Extension for PreAP Geometry Construct angles with measures of 15,

30, 45, 60, 75, and 135 degrees. Construct a tangent to a circle through a

point on the circle. Justify the constructions of:

- angles with measures of 15, 30, 45, 60, 75, and 135 degrees; and

- a tangent to a circle through a point on the circle.

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

Construct and justify a triangle similar to a given triangle on a given line segment as the base.

Key Vocabulary angle bisector centroid circumcenter circumscribed compass construction incenter inscribed intersection parallel lines perpendicular bisector perpendicular lines segment bisector straightedge transversal

Teacher Notes and Elaborations (continued) 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. Instruction should include experiences with completing constructions within the context of a more complex figure. For example: Given ABC inscribed in a circle with diameter AC , complete a construction to identify the center of the circle. Justification of constructions may involve application of postulates, theorems, definitions, and properties. Justification of constructions may differ depending upon the plan proposed, and the order in which concepts are taught. Construction Justification 1. Construct a line segment congruent Radii of equal circles are equal to a given a line segment 2. Construct an angle congruent to a given Radii of equal circles are equal angle SSS Postulate Corresponding parts of congruent triangles are congruent 3. Construct the bisector of a given angle Radii of equal circles are equal SSS Postulate Corresponding parts of congruent triangles are congruent Definition of an angle bisector

(continued)


27










Teacher Notes and Elaborations (continued) 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 4. Construct the perpendicular bisector Radii of equal circles are equal of a given segment Through any two points there is exactly one line If a point is equidistant from the endpoints of a line segment, then the point lies on the perpendicular bisector of the line segment 5. Construct the perpendicular to a line Radii of equal circles are equal at the given point on the line. Definition of a straight angle Definition of an angle bisector Definition of right angles and definition of perpendicular lines 6. Construct the perpendicular to the line from a point Radii of equal circles are equal not on the line. If a point is equidistant from the endpoints of a line segment, then the point lies on the perpendicular bisector of the line 7. Construct the parallel to a given line though a given point Radii of equal circles are equal not on the line. If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel


28



SOL Reporting Category Reasoning, Lines and Transformations Topic Reasoning, Lines, and Transformations Virginia SOL G.4 Foundational Objectives


PWC Mathematics Website http://pwcs.math.schoolfusion.us Virginia Department of Education Website http://www.doe.virginia.gov/instruction/mathematics/index.shtml Geometry reference http://www.mathopenref.com/ Animated geometric constructions http://www.mathsisfun.com/geometry/constructions.html


29




SOL Reporting Category Triangles Topic Triangles Virginia SOL G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a. order the sides by length, given the

angle measures; b. order the angles by degree measure,

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

and d. determine the range in which the

length of the third side must lie. These concepts will be considered in the context of real-world situations.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Order the sides of a triangle by their

lengths when given the measures of the angles.

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

Given the lengths of three segments, determine whether a triangle could be formed.

Given the lengths of two sides of a triangle, determine the range in which the length of the third side must lie.

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

Extension for PreAP Geometry Use the Hinge Theorem and its

converse to compare side lengths and angle measures in two triangles.

Given a quadrilateral with one diagonal, write inequalities relating pairs of angles or segment measures.

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

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

opposite sides? Essential Understandings The longest side of a triangle is opposite the largest angle of the triangle and the shortest

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

side opposite the angle. In order for a triangle to exist, the length of each side must be within a range that is

determined by the lengths of the other two sides. Teacher Notes and Elaborations Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. If one side of a triangle is longer than another side, then the angle opposite (across from) the longer side is larger than the angle opposite the shorter side. If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Sides of a triangle can be put in order when given the measures of the angles. If the sides of a triangle are ordered longest to shortest then the angles opposite must also be ordered largest to smallest.

(continued)


30



SOL Reporting Category Triangles Topic Triangles Virginia SOL G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a. order the sides by length, given the

angle measures; b. order the angles by degree measure,

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

and d. determine the range in which the

length of the third side must lie. These concepts will be considered in the context of real-world situations.

Teacher Notes and Elaborations (continued) Extension for PreAP Geometry Using properties of triangles, inequalities can be written relating pairs of angles or segment measures. A A 10 95° B 51° B 5 13 65° 34° C 18 C 55° 15 D 60° D Note: Figures are not drawn to scale BCD CAB CD BC Hinge Theorem: (SAS Inequality) If two sides of a triangle are congruent to two sides of another triangle, and the included angle in one triangle is greater than the included angles in the other, then the third side of the first triangle is longer than the third side in the second triangle.


31



SOL Reporting Category Triangles Topic Triangles Virginia SOL G.5 Foundational Objectives



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.


32



33




SOL Reporting Category Triangles Topic Triangles Virginia SOL G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Use definitions, postulates, and

theorems to prove triangles are congruent (including Hypotenuse-Leg Postulate).

Use algebraic methods to prove two triangles are congruent.

Use coordinate methods, such as the distance formula and the slope formula, to prove two triangles are congruent.

Use angle bisectors, medians, altitudes, perpendicular bisectors to prove triangles congruent.

Extension for PreAP Geometry Correlate LL, HA, LA to SAS, AAS,

and ASA respectively. Investigate the points of concurrency of

the lines associated with triangles (angle bisectors (incenter), perpendicular bisectors (circumcenter), altitudes (orthocenter), and medians (centroid)).

Key Vocabulary AAS Theorem algebraic methods altitude ASA Postulate coordinate methods corresponding parts deductive proof definition distance formula

(continued)

Essential Questions What are congruent triangles? What are the one-to-one correspondences that prove triangles congruent? What can be deduced from congruent triangles? Essential Understandings Congruence has real-world applications in a variety of areas, including art, architecture,

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

(continued)


34




SOL Reporting Category Triangles Topic Triangles Virginia SOL G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.

Key Vocabulary (continued) HL Postulate hypotenuse included angle included side leg median of a triangle postulate properties SAS Postulate slope formula SSS Postulate theorem

Teacher Notes and Elaborations (continued) Triangles can be proven congruent with the following correspondences: SSS Postulate: Three sides of one triangle are congruent to the corresponding sides of another triangle. SAS Postulate: Two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle. ASA Postulate: Two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle. AAS Theorem: Two angles and a non-included side of one triangle are congruent to the corresponding two angles and a non-included side of a second triangle. In a right triangle the side opposite the right angle is the hypotenuse and the other two sides are called legs. Right triangles can be proven congruent with the following correspondence: HL Postulate: The hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle. Extension for PreAP Geometry LL Theorem: The legs of one right triangle are congruent to the legs of another right triangle. HA Theorem: The hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and acute angle of the other right triangle. LA Theorem: One leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle. Medians, altitudes, and perpendicular bisectors are also used in proving triangles congruent. A median of a triangle is a segment that joins a vertex to the midpoint of the opposite side. An altitude of a triangle is a segment from a vertex and perpendicular segment from a vertex to the line containing the opposite side. Extension for PreAP Geometry The medians of a triangle intersect at the common point called the centroid. In a triangle, the point where the perpendicular bisectors of each side intersect is the circumcenter. In a triangle, the incenter is the point where the angle bisectors intersect. In a triangle, the orthocenter is the point of intersection of the three altitudes.


35



SOL Reporting Category Triangles Topic Triangles Virginia SOL G.6 Foundational Objectives A.4d The student will solve multi-step linear and quadratic equations in two variables, including d. solving multi-step linear equations

algebraically and graphically. 8.10 The student will a. verify the Pythagorean Theorem;

and b. apply the Pythagorean Theorem.



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.


36



37




SOL Reporting Category Triangles Topic Triangles Virginia 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.

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.

Use similar relationships between triangles to solve real-world problems.

Use definitions, postulates, and theorems to complete two-column proofs with at least five steps.

Extension for PreAP Geometry Use definitions, postulates, and

theorems to complete paragraph proofs with at least five steps.

Investigate proportionality in a triangle intersected by three or more parallel lines.

Investigate the Golden Ratio.

Key Vocabulary angle bisector congruent deductive proof definition distance formula included angle median of a triangle postulate properties proportion ratio scale factor similar triangles (AA Similarity, SSS Similarity, SAS Similarity) theorem

Essential Questions What is the difference between congruence and similarity? What is the relationship between similar triangles and proportions? What are the one-to-one correspondences that prove triangles similar? What is the relationship between segments when a line intersects two sides of a triangle

and is parallel to the third side? What is the relationship between segments when an angle is bisected? Essential Understandings Similarity has real-world applications in a variety of areas, including art, architecture,

and the sciences. Similarity does not depend on the position of the triangle. Congruent figures are also similar, but similar figures are not necessarily congruent. 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). 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)


38


Essential Questions and Understandings

Teacher Notes and Elaborations SOL Reporting Category Triangles Topic Triangles Virginia 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.

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 cb 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. 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

(continued)


39



Teacher Notes and Elaborations SOL Reporting Category Triangles Topic Triangles Virginia 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

Teacher Notes and Elaborations (continued) An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides. a b e f

e af b

Extension for PreAP Geometry If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. A B C D E F

AB DE=BC EF

, AC BC=DF EF

, AC DF=BC EF

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

a b a

a b

(This golden ratio is approximately 1.618.)

In a rectangle, if the ratio of the longer side to the shorter approximates 1.618, the rectangle is called a Golden Rectangle.


40



SOL Reporting Category Triangles Topic Triangles Virginia SOL G.7 Foundational Objectives 8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions 7.4 The student will solve single-step and multi-step practical problems, using proportional reasoning. 7.6 The student will determine whether plane figures (quadrilaterals and triangles) are similar and write proportions to express the relationships between corresponding sides of similar figures. 6.1 The student will describe and compare data, using ratios, and will use

appropriate notations such as ab

, a to b,

and a:b.



Use coordinate geometry to investigate relationships among triangles. Students are given a printed deductive proof of theorem. Cut it up into a statement of

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

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

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

Use patty paper to demonstrate similar triangles.


41


Essential Knowledge and Skills

Key Vocabulary Essential Questions and Understandings

Teacher Notes and Elaborations SOL Reporting Category Triangles Topic Triangles Virginia SOL G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry.

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, obtuse, or acute triangle.

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

Solve for missing lengths in geometric figures, using properties of 30° - 60° - 90° triangles.

Solve problems involving right triangles using sine, cosine, and tangent ratios.

Explain and use the relationship between the sine and cosine of complementary angles.

Solve real-world problems using right triangle trigonometry and properties of right triangles.

Express linear measurements as simplified radicals and decimal approximations.

Extension for PreAP Geometry Use the Law of Sines and the Law of

Cosines to find missing measures in triangles.

Find the geometric mean in right triangles.

Key Vocabulary acute triangle angle of depression angle of elevation area of a triangle

(continued)

Essential Questions What are the different ways of finding missing sides and angles of triangles? How do special right triangle theorems apply? What is geometric mean? What is a trigonometric ratio? Essential Understandings The Pythagorean Theorem is essential for solving problems involving right triangles. Many historical and algebraic proofs of the Pythagorean Theorem exist. The relationships between the sides and angles of right triangles are useful in many

applied fields. Some practical problems can be solved by choosing an efficient representation of the

problem.

Another formula for the area of a triangle is 1 sin2

A ab C .

The ratios of side lengths in similar right triangles adjacent opposite or

hypotenuse hypotenuse

are

independent of the scale factor and depend only on the angle the hypotenuse makes with the adjacent side, thus justifying the definition and calculation of trigonometric functions using the ratios of side lengths for similar right triangles.

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

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

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

When solving for lengths of sides in special right triangles rationalizing the denominator may be needed. (For most students this is the introduction to this topic.)

(continued)


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Teacher Notes and Elaborations SOL Reporting Category Triangles Topic Triangles Virginia SOL G.8 The student will solve real world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry.

Key Vocabulary (continued) cosine geometric mean hypotenuse obtuse triangle Pythagorean Theorem ratio right triangle similar right triangle sine tangent trigonometry 45°-45°-90° triangle 30°-60°-90° triangle

Teacher Notes and Elaborations (continued) Rationalizing a denominator is a procedure for transforming a quotient with a radical in the denominator into an expression with no radical in the denominator. The following are examples of rationalizing the denominator of radical expressions.

Example 1: 3

3 3 3x x

(Multiply by 1.)

3

3 3x

33

x

Example 2: 3 5 3 5 2

2 2 2x x

(Multiply by 1.)

3 102

x

Example 3: 2 2 1 3

1 3 1 3 1 3

(Use the conjugate of 1 3 to multiply

by 1.)

2 2 3

1 3 3 3

2 2 32

1 3 The Pythagorean Theorem states that in a right triangle, the square of the measure of the hypotenuse equals the sum of the squares of the measures of the legs. The converse of the Pythagorean Theorem states that if the square of the measure of the longest side equals the sum of the squares of the measures of the other two sides of a triangle, then the triangle is a right triangle.

(continued)


43




Teacher Notes and Elaborations (continued) If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle. If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle. Pythagorean Triples are three positive integers that satisfy the Pythagorean theorem. In a right triangle with the altitude drawn to the hypotenuse, the geometric mean can be used to find missing measures of that triangle. If r,

s, and t are positive numbers with r 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 13sin 3018

13cos6018

13 18 sin30 = cos 60 30º The angle of elevation is the angle formed by a horizontal line and the line of sight to an object above that horizontal line. The angle of depression is the angle formed by a horizontal line and the line of sight to an object below that horizontal line. The angle of elevation and the angle of depression in the same diagram are always congruent.

(continued)


44




Teacher Notes and Elaborations (continued) Extension for PreAP Geometry The Law of Sines states that for any triangle with angles of measures A, B, and C, and sides of lengths a, b, and c (a opposite A ,

opposite b B , and opposite c C ) sin sin sinA B C

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

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

h x 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.

a h b x aa c and

y bb c

x y c


45



SOL Reporting Category Triangles Topic Triangles Virginia SOL G.8 Foundational Objectives A.3 The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form. 8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions 8.5 The student will a. determine whether a given number

is a perfect square; and b. find the two consecutive whole

numbers between which a square root lies.

8.10 The student will a. verify the Pythagorean Theorem;

and b. apply the Pythagorean Theorem. In middle school, area of a triangle is found and applied using the formula

12

A bh .



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.


46



47




Teacher Notes and Elaborations SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems.

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 properties of angles for a quadrilateral inscribed in a circle.

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

Extension for PreAP Geometry Investigate and identify the

quadrilaterals formed by connecting the midpoints of the sides of a given quadrilateral.

Key Vocabulary base angles characteristics diagonal isosceles trapezoid kite legs median of a trapezoid parallelogram 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? Essential Understandings The terms characteristics and properties can be used interchangeably to describe

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

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

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

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.

(continued)


48




Teacher Notes and Elaborations (continued) A kite is a quadrilateral with two pairs of congruent adjacent sides. One pair of opposite angles is congruent. A rhombus may be considered a special case of a kite. If all 4 sides of a kite have the same length, then it must also be a rhombus with two pairs of opposite angles congruent; and if all 4 angles of the kite are equal, then it must also be a square. However not all kites are rhombi. Characteristics of quadrilaterals are used to identify figures, and to find values for missing parts and areas. The hierarchical nature of quadrilaterals can be described as ranking based on characteristics. Areas of work that use quadrilaterals include art, construction, fabric design, and architecture.

Quadrilateral Family Tree Quadrilateral 1 pair of opposite 2 pairs of opposite sides is parallel sides are parallel 2 pairs of adjacent 1 pair of opposite, congruent sides congruent angles Trapezoid Parallelogram Kite equiangular equilateral equilateral Right Trapezoid Isosceles Rectangle Rhombus Trapezoid 2 right angles Non-parallel sides (legs) are congruent equilateral equiangular Square

(continued)

The sum of the measures of the angles is 360°.

4-sided polygon


49




Teacher Notes and Elaborations (continued) If a quadrilateral is inscribed in a circle, its opposite angles are supplementary. This can be verified by considering that the arcs intercepted by opposite angles of an inscribed quadrilateral form a circle. Example: Quadrilateral ABCD is inscribed in a circle. AB BC CD DA 360m m m m .

The measure of 1DAB = BCD2

m m and the measure of 1BCD = DAB2

m m .

A B BCD = 2 A and DAB = 2 Cm m m m BCD DAB 360m m 2 A+2 C = 360m m A C 180m m D C


50



SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.9 Foundational Objectives 7.7 The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid. 6.13 The student will describe and identify properties of quadrilaterals.



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

Use flowcharts or Venn diagrams to show relationships and properties of quadrilaterals. Use patty paper to show properties of the different quadrilaterals. Use notecards to create models of different quadrilaterals. Discuss the characteristics

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


51




Teacher Notes and Elaborations SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.10 The student will solve real-world problems involving angles of polygons.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Solve real-world problems involving

the measures of interior and exterior angles of polygons.

Identify tessellations in art, construction, and nature.

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

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

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

Investigate and identify the regular polygons that tessellate.

Extension for PreAP Geometry Distinguish between pure and semi-

pure tessellations. Key Vocabulary concave convex decagon diagonal dodecagon exterior angle heptagon hexagon interior angle linear pair n-gon nonagon

(continued)

Essential Questions What are the distinguishing characteristics of a polygon? What is a regular polygon? What is the relationship between the interior and exterior angles of polygons? What is the relationship between the number of sides of a polygon and its angles? What are tessellations? Essential Understandings A regular polygon will tessellate the plane if the measure of an interior angle is a factor

of 360. Both regular and non-regular polygons can tessellate the plane. Two intersecting lines form angles with specific relationships. An exterior angle is formed by extending a side of a polygon. The exterior angle and the corresponding interior angle form a linear pair. The sum of the measures of the interior angles of a convex polygon may be found by

dividing the interior of the polygon into non-overlapping triangles. Teacher Notes and Elaborations A polygon is a plane figure formed by coplanar segments (sides) such that (1) each segment intersects exactly two other segments, one at each endpoint; and (2) no two points with a common endpoint are collinear. Polygons are named by their number of sides and classified as convex (a line containing a side of a polygon contains no interior points of that polygon) or concave (a line containing a side of a polygon also contains interior points of the polygon). Common polygons: 3 sides: triangle 7 sides: heptagon 10 sides: decagon 4 sides: quadrilateral 8 sides: octagon 12 sides: dodecagon 5 sides: pentagon 9 sides: nonagon n sides: n-gon 6 sides: hexagon A segment joining two nonconsecutive vertices is a diagonal of the polygon. Two angles that are adjacent (share a leg) and supplementary (add up to 180°) form a linear pair.

(continued)


52



Teacher Notes and Elaborations SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.10 The student will solve real-world problems involving angles of polygons.

Key Vocabulary (continued) octagon pentagon polygon quadrilateral regular polygon tessellation triangle

Teacher Notes and Elaborations (continued) Polygons have interior angles (angles formed by the sides of the polygon and enclosed by the polygon) and exterior angles (angles formed by extending an existing side). The exterior angle and the corresponding interior angle form a linear pair. The sum of the measures of the interior angles of a polygon is found by multiplying two less than the number of sides by 180°, [ ( 2)180n ]. The sum of the measures of the exterior angles, one at each vertex, is 360°. A regular polygon is a convex polygon with all sides congruent and all angles congruent. The center of a regular polygon is the center of the circumscribed circle. Given the measure of an exterior angle of a regular polygon, the number of sides can be determined by dividing 360° by the measure of that angle. The central angle of a regular polygon is an angle formed by two radii drawn to consecutive vertices. Its measure can be determined by dividing 360° by the number of sides. A polygon will tessellate the plane if the interior angles at a vertex add to 360°. Tessellations are repeated copies of a figure that completely fill a plane without overlapping. The hexagon pattern in a honeycomb is a tessellation of regular hexagons. Both regular and non-regular polygons can tessellate the plane. When a tessellation uses only one shape it is called a pure tessellation. The three regular polygons that create pure tessellations are triangle, square, and hexagon. Regular polygon tessellation Non-regular polygon tessellation Extension for PreAP Geometry 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.


53



SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.10 Foundational Objectives A.4 The student will solve multi-step linear and quadratic equations in two variables, including d. solving multi-step linear equations





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


54



55




Teacher Notes and Elaborations SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.11 The student will use angles, arcs, chords, tangents, and secants to a. investigate, verify, and apply

properties of circles; b. solve real-world problems involving

properties of circles; and c. find arc lengths and areas of sectors

in circles.

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.

Find the area of the region between concentric circles.

Extension for PreAP Geometry Find the area of a segment of a circle. Find the probability that a point chosen

at random in a figure is in a shaded region.

Key Vocabulary arc arc length arc measure central angle chord circle circumference

(continued)

Essential Questions What are the relationships between angle and arc measures? What are the relationships among the lengths of secant segments, tangent segments, and

chords? What are the relationships between chords and arcs? What is the relationship between a central angle and the area of a sector? What is the relationship between a central angle and the length of an arc? What is the difference between arc length and arc measure? Essential Understandings Many relationships exist between and among angles, arcs, secants, chords, and tangents

of a circle. All circles are similar. A chord is part of a secant. Real-world applications may be drawn from architecture, art, and construction. 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. The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

(continued)


56




Teacher Notes and Elaborations SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.11 The student will use angles, arcs, chords, tangents, and secants to a. investigate, verify, and apply



in circles.

Key Vocabulary (continued) diameter inscribed angle intercepted arc major arc minor arc secant sector semicircles tangent

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

Length of AB 2360

x r (This is a linear measure.)

Area of sector 2AOB360

x r

Experiences using a measure of one part of the circle to find measures of other parts of the circle should be included. Verifying the properties of circles may include definitions, postulates, theorems, algebraic methods, and coordinate methods. In the same circle or congruent circles:

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


57

(continued)



Teacher Notes and Elaborations


58

SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.11 The student will use angles, arcs, chords, tangents, and secants to a. investigate, verify, and apply



in circles.

Teacher Notes and Elaborations (continued) An angle inscribed in a semi-circle is a right angle. Opposite angles of an inscribed quadrilateral are supplementary. An annulus is the region between two concentric circles. To find the area of an annulus, find the area of the larger circle and subtract the area of the smaller circle. Real world problems do not always include figures. Experiences drawing a figure to represent the problem should be provided. Extension for PreAP Geometry A segment of a circle is the region between an arc and a chord of a circle. To find the area of a segment, find the area of the sector and subtract the area of the triangle Given a figure with a shaded region, students will find the probability that a point chosen at random will be in the shaded region. Find the area of the figures then write a ratio of the area of the shaded region to the area of the entire figure. Figures should include circles as well as polygons such as the following. Probabilities can be written as ratios, decimals, or percents.




59

SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.11 Foundational Objectives A.4 The student will solve multi-step linear and quadratic equations in two variables, including d. solving multi-step linear equations



8.11 The student will solve practical area and perimeter problems involving composite plane figures. 6.10a, b, c The student will a. define pi (π) as the ratio of the

circumference of a circle to its diameter;

b. solve practical problems involving circumference and area of a circle, given the diameter or radius; and

c. solve practical problems involving area and perimeter.



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.


60




SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G.12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Identify the center, radius, and diameter

of a circle from a given standard equation.

Use the distance formula to find the radius of a circle.

Given the coordinates of the center and radius of the circle, identify a point on the circle.

Given the equation of a circle in standard form, identify the coordinates of the center and find the radius of the circle.

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

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

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

Extension for PreAP Geometry Investigate and identify points that lie

inside or outside a circle. Write inequality statements for regions

either inside or outside a circle and sketch these graphs.

Investigate and write the equation of a circle given three points on the circle.

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

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

circle? What is the relationship between distance formula and the equation of a circle? What is a conic section? Essential Understandings A circle is a locus of points equidistant from a given point, the center. Standard form for the equation of a circle is, 2 2 2( ) ( )x h y k r where the

coordinates of the center of the circle are ( , )h k and r is the length of the radius. The circle is a conic section. Teacher Notes and Elaborations Locus means a figure that is the set of all points, and only those points, that satisfy one or more conditions. The Pythagorean Theorem (distance formula) can be used to develop an equation of a circle. Let P(x, y) represent any point on the circle. The distance between C(h, k) and P(x, y) is r. y

2 2( ) ( )x h y k r 2 2 2( ) ( )x h y k r P(x, y) r C(h, k) x Given the coordinates of the center of the circle (h, k) and a radius r, four easily identified points on the circle are: ( , )h r k , ( , )h r k , ( , )h k r , ( , )h k r

(continued)


61




Teacher Notes and Elaborations (continued) Example: Given the coordinates of the center of a circle, ( 2,6) , with radius 3 four points on the circle are ( 2 3,6) , ( 2 3,6) , ( 2,6 3) , ( 2,6 3) or (1,6) ( 5,6) ( 2,9) ( 2,3) Given an equation for a circle, substitute coordinates of a point to determine the location of the point in reference to the circle. The locations of points are inside a circle, on the circle, or outside the circle. Example: Given the center of the circle ( 7,5) and the radius of the circle is 6, determine whether the following points are inside, on, or outside the circle. Find the equation of a circle: 2 2( 7) ( 5) 36x y Given point ( 3,2) 25 < 36 therefore ( 3,2) is inside the circle. Given point (6, 4) 250 > 36 therefore (6, 4) is outside the circle. Given point ( 1,5) 36 = 36 therefore ( 1,5) is on the circle. Given the coordinates of the endpoints of a diameter, midpoint formula can be used to find the center of the circle and distance formula can be used to find the radius. A conic section is one of a group of curves formed by the intersection of a plane and a right circular cone. The curve is a circle if the plane is parallel to the base of the cone.

(continued)

2 2

2 2

( 3 7) (2 5) 4 ( 3) 16 9 25

2 2 2 2(6 7) ( 4 5) 13 ( 9) 169 81 = 250

2 2 2( 1 7) (5 5) 6 0 36 0

=36


62




Teacher Notes and Elaborations (continued) Experiences should include determining any of the following from given information:

the coordinates of the center; the length of a radius; coordinate endpoints of a radius; the length of a diameter; coordinate endpoints of a diameter; the coordinates of a point on the circle; and the equation of a circle.

Extension for PreAP Geometry An example of an inequality that describes the points (x, y) outside the circle that are more than three units from center (4, 2 ) is

2 2( 4) ( 2) 9x y . The graph would be a broken circle and shaded outside the circle. An example of an inequality that describes the points (x, y) inside the circle that are less than or equal to four units from center ( 3, 5 ) is 2 2( 3) ( 5) 16x y . The graph would be a circle and shaded inside the circle. Given three points on a circle, students investigate how to use the slope formula, midpoint formula, equation of a line formula, and distance formula to find the equation of the circle. A source for this investigation can be found at http://www.regentsprep.org/regents/math/geometry/GCG6/RCir.htm


63



SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Polygons and Circles Virginia SOL G. 12 Foundational Objectives A.6 The student will graph linear equations and linear inequalities in two variables, including a. determining the slope of a line when



8.10 The student will a. verify the Pythagorean Theorem;

and b. apply the Pythagorean Theorem.



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


64




SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Three-Dimensional Figures Virginia SOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Find the total surface area of cylinders,

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

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

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

Calculators may be used to find decimal approximations for results.

Key Vocabulary altitude area area of the base (B) base cone face height lateral edge lateral area prism polygon pyramid slant height sphere surface area three-dimensional two dimensional vertex volume

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

cylinders, pyramids, cones, and spheres? Essential Understandings The surface area of a three-dimensional object is the sum of the areas of all its faces. The volume of a three-dimensional object is the number of unit cubes that would fill the

object. 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 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.

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SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Three-Dimensional Figures Virginia SOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.

Teacher Notes and Elaborations (continued) A cone is a three-dimensional figure that has a circular base, a vertex not in the plane of the circle, and a curved lateral surface. In a right cone, the altitude is a perpendicular segment from the vertex to the center of the base. The height (h) is the length of the altitude. The slant height ( ) is the distance from the vertex to a point on the edge of the base. Surface area is the lateral area plus the area of the base(s). Bases of prisms are congruent polygons lying in parallel planes. An altitude (height) of a prism is a segment joining the two base planes and perpendicular to both. The faces of a prism that are not its bases are called lateral faces. Adjacent lateral faces intersect in parallel segments called lateral edges. In right prisms the lateral edges are also altitudes. Volume is the capacity of a three-dimensional figure such as the amount of water in an aquarium. The volume of an irregularly shaped object can be found by measuring its displacement. When an object is placed in a liquid, it causes the liquid to rise. This volume is called the objects’ displacement. The base of a three-dimensional figure could be a circle, a triangle, a square, a rectangle, a regular hexagon or another type of polygon. Many formulas use B to represent the area of the base of the solid figure. To find the area of a base (B) in three dimensional figures, use the area formula that applies. Formulas for those figures may need to be reviewed. A sphere is the set of all points in space equidistant from a given point. The center is the given point and the radius is the given distance. Surface area and volume of spheres will also be found. When determining surface area of combinations of solids, attention needs to be given to the possibility of shared faces.


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SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Three-Dimensional Figures Virginia SOL G.13 Foundational Objectives 8.7 The student will a. investigate and solve practical

problems involving volume and surface area of prisms, cylinders, cones, and pyramids; and

b. describe how changing one measured attribute of the figure affects the volume and surface area.

8.9 The student will construct a three-dimensional model given the top or bottom, side and front views. 7.5 The student will a. describe volume and surface area of

cylinders; b. solve practical problems involving

the volume and surface area of rectangular prisms and cylinders;

c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area;

6.10d The student will d. describe and determine the volume

and surface area of a rectangular prism.



Students will draw and cut out regular polygons and tape them together to make three-dimensional objects. Colored paper may be used for effect.

Use strings, straws, toothpicks, etc. to make three-dimensional objects. Students make a three-dimensional object from any material they choose. They calculate

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

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

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

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

Using items from a pantry have students measure and compute surface area and volume. When an object is placed in a liquid, it causes the liquid to rise. This volume is called

the objects’ displacement. The volume of an irregularly shaped object can be found by measuring its displacement.

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


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SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Three-Dimensional Figures Virginia SOL G.14 The student will use similar geometric objects in two- or three-dimensions to a. compare ratios between side

lengths, perimeters, areas, and volumes;

b. determine how changes in one or more dimensions of an object affect area and/or volume of the object;

c. determine how changes in area and/or volume of an object affect one or more dimensions of the object; and

d. solve real-world problems about similar geometric objects.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: 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.

Compare ratios between side lengths, perimeters, areas, and volumes, given two similar figures.

Key Vocabulary constant ratio similar figures

Essential Questions How does a change in dimensions affect the area and/or volume of the object? How does a change in area and/or volume affect other measures? In similar figures, how does a change of one measurement affect perimeter, area, or

volume? Essential Understandings A change in one dimension of an object results in predictable changes in area and/or

volume. A constant ratio exists between corresponding lengths of sides of similar figures. Proportional reasoning is integral to comparing attribute measures in similar objects. Teacher Notes and Elaborations Similar figures are figures that have the same shape but not necessarily the same size. Scale factors (proportional reasoning) are used to compare perimeters, areas, and volumes of similar two-dimensional and three-dimensional geometric figures. A change in one dimension of an object results in changes in area and volume in specific patterns. Volumes, areas, and perimeters of similar polygons are examined to draw conclusions about how changes in one dimension affect both area and volume. If the given perimeter of a polygon is increased or decreased, the area will increase or decrease by the square of the change and the volume increases or decreases by the cube of the change. Similar solids are solids that have the same shape but not necessarily the same size. All spheres are similar. If the scale factor of two similar solids is a:b, then:

– The ratio of corresponding perimeters is a:b. – The ratios of the base areas, of the lateral areas, and of the total areas are a2:b2. – The ratio of the volumes is a3:b3.


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SOL Reporting Category Polygons, Circles, and Three-Dimensional Figures Topic Three-Dimensional Figures Virginia SOL G.14 Foundational Objectives 8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions. 8.7 The student will a. investigate and solve practical

problems involving volume and surface area of prisms, cylinders, cones, and pyramids; and

b. describe how changing one measured attribute of the figure affects the volume and surface area.

7.5 The student will a. describe volume and surface area of

cylinders; b. solve practical problems involving

the volume and surface area of rectangular prisms and cylinders; and

c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

7.6 The student will determine whether plane figures (quadrilaterals and triangles) are similar and write proportions to express the relationships between corresponding sides of similar figures.



Using cylinders made from PVC pipe or empty cans determine the change in volume with respect to changes in height or radius. Fill cylinders with water to compare the volumes.

Each student is given a sheet of construction paper. Next, they are instructed to cut a square from each corner and form an open top box with the maximum volume.

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


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