Mathematics. Geometry PreAP Geometry Curriculum Guide. Curriculum Guide (Revised 2016)

Transcription

1 Mathematics Geometry PreAP Geometry Curriculum Guide Curriculum Guide (Revised 2016)

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3 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:, Essential Knowledge and Skills, Key Vocabulary,,, Resources, and Sample Instructional Strategies and Activities. The purpose of each section is explained below. : This section includes the objective and, 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. : This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives. : 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. 1

4 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, 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 2

5 Objective G.1 Page 5 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 Page 3

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7 Essential Knowledge and Skills Key Vocabulary 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. Essential Questions What is the relationship between reasoning, justification, and proof in geometry? What is a truth-value? How does a truth-value apply to conditional statements? How do deductive reasoning and Venn diagrams help judge the validity of logical arguments? Essential Understandings Inductive reasoning, deductive reasoning, and proof are critical in establishing general claims. Deductive reasoning is the method that uses logic to draw conclusions based on definitions, postulates, and theorems. Inductive reasoning is the method of drawing conclusions from a limited set of observations. Logical arguments consist of a set of premises or hypotheses and a conclusion. Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, and theorems. Euclidean geometry is an axiomatic system based on undefined terms (point, line, and plane), postulates, and theorems. When a conditional and its converse are true, the statements can be written as a biconditional (i.e., iff or if and only if). Logical arguments that are valid may not be true. Truth and validity are not synonymous. Logic is the study of the principles of reasoning. Logical arguments consist of a set of premises (hypotheses) and a conclusion (the last step in a reasoning process). A mathematical statement is one in which a fact or complete idea is expressed. Because a mathematical statement states a fact, many of them can be judged to be true or false. Questions and phrases are not mathematical statements since they can not be judged as true or false. Terms associated with logical arguments are reasoning, justification, and proof. Reasoning is the drawing of conclusions or inferences from facts, observations, or hypotheses. Justification is a rationale or argument for some mathematical proposition. A conjecture is a statement that has not been proved true nor shown to be false. A proof is a justification that is logically valid and based on initial assumptions, definitions, and proven results. Proofs are developed so that each step in the argument is in proper chronological order in relation to earlier steps. When building a proof the argument must be clearly developed and each step must be supported by a property, theorem, postulate, or definition. 5

8 Essential Knowledge and Skills Key Vocabulary 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. 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 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, twocolumn 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 6

9 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 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 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. q 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. 7

10 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. 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 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 p q 8

11 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. 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 9

12 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. 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 1. Given altitude BD 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 10

13 Resources Sample Instructional Strategies and Activities 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 Virginia Department of Education Website thematics/index.shtml Geometry reference 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. 11

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15 Essential Knowledge and Skills Key Vocabulary 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. Essential Questions What is the relationship between lines and angles? What is the difference between parallel lines and perpendicular lines? How are lines proven parallel? What is the difference between parallel lines and intersecting lines? What are the relationships between the angles formed when two parallel lines are cut by a transversal? Essential Understandings Parallel lines intersected by a transversal form angles with specific relationships. Some angle relationships may be used when proving two lines intersected by a transversal are parallel. The Parallel Postulate differentiates Euclidean from non-euclidean geometries such as spherical geometry and hyperbolic geometry. Euclidean Geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose elements is the earliest known systematic discussion of geometry. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other theorems (propositions) from these. Angles with the same measure are congruent angles. Adjacent angles are two angles that share a common side and have the same vertex, but have no interior points in common. Vertical angles are two angles whose sides form two pairs of opposite rays. When two lines intersect, they form two pairs of vertical angles. When two lines intersect, two types of angle pairs are formed: vertical angles and adjacent supplementary angles. Vertical angles are congruent and two adjacent angles are supplementary. Extension for PreAP Geometry Write equations of parallel and perpendicular lines. Investigate skew lines using real world models. 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. 13

16 Essential Knowledge and Skills Key Vocabulary 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. 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 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. 14

17 The following is an example of a paragraph proof. 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. 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. 15

18 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. 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. Given: OAB and M and N the midpoints of OB and OA respectively. Prove: MN BA Proof: Choose axes and coordinates as shown. y M B(2 b,2 c ) O N A(2 a,0) x 2b 0 2c 0 2b 2c 2a a 0 1. Midpoints are M(, ) (, ) ( bc, ) and N(, ) (, ) ( a,0) ; by Midpoint Formula c c 0 2c 2c c 2. Slope of MN and the slope of BA ; by definition of slope. a b a b 2 a 2 b 2( a b ) a b 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. 16

19 Resources Sample Instructional Strategies and Activities 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. 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 Virginia Department of Education Website thematics/index.shtml Geometry reference Foundational Objectives 8.10 The student will a. verify the Pythagorean Theorem; and b. apply the Pythagorean Theorem The student will a. solve multi-step linear equations in one variable on one and two sides of the equation 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 nonparallel 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. 17

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21 Essential Knowledge and Skills Key Vocabulary 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. 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. 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 Another triple is 6, 8, 10, since 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 ( x2 x1 ) and the distance in the coordinate plane (distance formula or Pythagorean Theorem). 19

22 Essential Knowledge and Skills Key Vocabulary 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. 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 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 ( 1) 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 C 3 D E y x G H -8-9 The midpoint of CE is D (2,2). The midpoint of FH is G ( 3, 2). F y x 20

23 This idea is used twice to find the coordinates of the midpoint of a slanting segment with endpoints P( 1 x1, y 1) and P( 2 x2, y 2). 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 P1(x1, y1) -7 R T -8-9 y M P2(x2, y2) S x The midpoint of PP 1 2is M x x 1 2, 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 x 1 = x 2. 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 (x 1, y 1 ) and (x 2, y 2 ): the midpoint formula is x x, y y 2 2 ; the distance formula is x x y y - the slope formula is y 2 y1 x x 2 1 ; and. 21

24 Extension for PreAP Geometry Point-slope form is an equation of the form y y1 m( x x1) for the line passing through a point whose coordinates are ( x1, y 1) and having slope m. 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. 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 H O maps the figure onto itself. A figure has line symmetry if there is a symmetry line k such that the reflection R k 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. 22

25 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 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. as the image to pre-image. For example: 4 to 3 or 4 3 represents an enlargement. A composite of reflections is the transformation that results from performing one reflection after another. A translation (slide) is the composite of two reflections over parallel lines. 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. P(, xy) P'(, x y) or r (, xy) (, x y) x axis 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. Pxy (, ) P'( xy, ) or r ( xy, ) ( xy, ) y axis Reflection in y x: When a point is reflected across the line y x, then the x-coordinate and the y-coordinate change places. P( x, y) P'( y, x) or r ( x, y) ( y, x) 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). Pxy (, ) P'( y, x) or r ( xy, ) ( y, x) y x Rotations: (assuming center of rotation to be the origin) Rotation of 90 : R ( x, y) ( y, x) 90 Rotation of 180 : R ( x, y) ( x, y) 180 Rotation of 270 : R ( x, y) ( y, x)

26 Resources Sample Instructional Strategies and Activities 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 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. 8.8 The student will a. apply transformations to plane figures; and b. identify applications of transformations. 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 Virginia Department of Education Website thematics/index.shtml Geometry reference eometry/math-geometry.htm Geometry reference Foundational Objectives 8.10 The student will a. verify the Pythagorean Theorem; and b. apply the Pythagorean Theorem The student will a. solve multi-step linear equations in one variable on one and two sides of the equation 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. 24