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RICHMOND PUBLIC SCHOOLS Pulling It All Together MATHEMATICS Geometry Department of Instruction & Accountability 2012 – 2013 RICHMOND PUBLIC SCHOOLS DEPARTMENT OF INSTRUCTION Mathematics CURRICULUM COMPASS COMMITTEE Ronald Bradford, Jr. Tinkhani Hargrove Bland Campbell Joanne Seaton Pulling It All Together Irma Mayo Diane Williams Aaron Dixon Cassandra Willis Instructional Specialist Instructional Specialist – Title I Maria Crenshaw Director of Instruction Victoria S. Oakley Dr. Yvonne W. Brandon Chief Academic Officer Superintendent COPYRIGHT©2012 Richmond City Public Schools Richmond, Virginia ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION FROM RICHMOND CITY PUBLIC SCHOOLS. RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Blueprint for Geometry Test No. of Items Reporting Categories 18/36% Reasoning, Lines, and Transformations 14/28% Triangles 18/36% Polygons, Circles, and Three-Dimensional Figures Excluded from test 50/100% 10 60 Total Operational Items Field Test Items Total Items on Test MATHEMATICS – Geometry Focus Standards Venn Diagrams, Deductive Reasoning, and Proof Angles of Parallel Lines Distance, Midpoint, Slope, Symmetry and Transformation Constructions Triangle Inequality Congruent Triangles Similar Triangles Right Triangle Relationships G.1a-d G.2a-c G.3a-d G.4a-g G.5a-d G.6 G.7 G.8 Quadrilaterals Polygons and Tessellations Circles Equation of a Circle Surface Area and Volume Similar Solids G.9 G.10 G.11a-c G.12 G.13 G.14a-d NONE No Standards are excluded from this test * These field-test items will not be used to compute students’ scores on the test. i RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Mathematics Standards of Learning for Virginia Schools Geometry This course is designed for students who have successfully completed the standards for Algebra I. All students are expected to achieve the Geometry standards. The course includes, among other things, properties of geometric figures, trigonometric relationships, and reasoning to justify conclusions. Methods of justification will include paragraph proofs, two-column proofs, indirect proofs, coordinate proofs, algebraic methods, and verbal arguments. A gradual development of formal proof will be encouraged. Inductive and intuitive approaches to proof as well as deductive axiomatic methods should be used. This set of standards includes emphasis on two- and three-dimensional reasoning skills, coordinate and transformational geometry, and the use of geometric models to solve problems. A variety of applications and some general problem-solving techniques, including algebraic skills, should be used to implement these standards. Calculators, computers, graphing utilities (graphing calculators or computer graphing simulators), dynamic geometry software, and other appropriate technology tools will be used to assist in teaching and learning. Any technology that will enhance student learning should be used. Reasoning, Lines, and Transformations G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; d) using deductive reasoning. MATHEMATICS – Geometry G.2 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. ii RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Reasoning, Lines, and Transformations 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. G.4 The student will construct and justify the constructions of a) a line segment congruent to a given line segment; b) the perpendicular bisector of a line segment; c) a perpendicular to a given line from a point not on the line; d) a perpendicular to a given line at a given point on the line; e) the bisector of a given angle, f) an angle congruent to a given angle; and g) a line parallel to a given line through a point not on the given line. G.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. 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. Triangles G.5 G.6 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, 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. MATHEMATICS – Geometry iii RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Polygons and Circles Three-Dimensional Figures G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. G.13 The student will use formulas for surface area and volume of threedimensional objects to solve real-world problems. G.10 The student will solve real-world problems involving angles of polygons. G.14 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 similar geometric objects in two- or threedimensions 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. 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. MATHEMATICS – Geometry iv RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Standard: G.1a-d Pulling It All Together Reasoning, Lines, and Transformations The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. Understanding the Standard Teacher Notes Inductive reasoning, deductive reasoning, and proof are critical in establishing general claims. Deductive reasoning is the method that uses logic to draw conclusions based on definitions, postulates, and theorems. Inductive reasoning is the method of drawing conclusions from a limited set of observations. Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, and theorems. Logical arguments consist of a set of premises or hypotheses and a conclusion. Euclidean geometry is an axiomatic system based on undefined terms (point, line and plane), postulates, and theorems. When a conditional and its converse are true, the statements can be written as a biconditional, i.e., iff or if and only if. Logical arguments that are valid may not be true. Truth and validity are not synonymous. A conditional statement is an If- Then statement. The “If” part is the given, or hypothesis, and the “Then” part is the conclusion. Some statements are in the form of “if p, then q.” This form is called a conditional statement, where p is the statement you know is true (hypothesis), and q is the statement that you conclude is true (conclusion). The symbol form of “if p, then q” is p=>q and is read “p implies q”. The negation of a statement can be written using the symbol ~ (example ~p). MATHEMATICS – Geometry 1 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together The converse of a statement switches the hypothesis and the conclusion. The inverse of a statement is the negation of both the hypothesis and the conclusion. The contrapositive of a statement combines the converse and inverse. A Venn diagram is a special kind of diagram that shows logic statements in diagram form. Venn diagrams show three basic relationships: Disjoint (non-overlapping) -use the words “no or “none”, Overlapping (intersecting)- use the word “some”, and Subset (enclosed)- use the words all or some. The law of syllogism is a type of reasoning used to reach a valid conclusion and is similar to the transitive property: two things equal to a third are equal to each other. Essential Understanding Students All students should: Understand how to convert a non-mathematical sentence into a conditional statement. Understand how to use truth tables to determine the validity of the converse, inverse, and contrapositive. Understand how to diagram arguments with Venn diagrams. Understand how to create Venn diagrams. Understand how to reach a valid conclusion using the law of syllogism. MATHEMATICS – Geometry 2 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Reasoning, Lines, and Transformations Standard: G.1a-d The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. Essential Knowledge, Skills and Processes Objectives The student will: a) identify the converse, inverse, and contrapositive of a conditional statement b) translate a short verbal argument into symbolic form MATHEMATICS – Geometry To be successful with this standard, students are 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 Common Core State Standards Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, p.4 RPS-Teaching by Design, Lesson G.1 Suggested Projects: Inductive and Deductive Reasoning http://ttaconline.org/staff/sol/sol_sol_lessons. asp Logic and Conditional Statements http://ttaconline.org/staff/sol/sol_sol_lessons. asp Books/Materials: Prentice Hall Study Guide & Practice Workbook, pp. 15 – 20 Prentice Hall Reading & Math Literacy Masters, pp. 5 – 7 Prentice Hall Daily Notetaking Guide, pp. 25 – 33, 91 – 94 Prentice Hall Skills & Concepts Review, pp. 120 – 123, 142 3 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS reasoning, including the law of syllogism, the law of the contrapositive, the law of detachment, and counterexamples. c) use Venn diagrams to represent set relationships d) use deductive reasoning. 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 ∨. Use inductive reasoning to make conjectures. Pulling It All Together Prentice Hall VA SOL Test Prep Workbook, p. vii RPS-Geometry Reteaching Lesson G.1 Geometry SOL Coach, pp. 44 – 58 Luster, Helen, Preparing for the Geometry SOL Test, pp. 40-57 MATHEMATICS – Geometry Technology Integration: Prentice Hall Interactive Student Text (online or CD-ROM) PHSuccess Net (teachers) Prentice Hall Presentation Pro Prentice Hall Computer Test Generator Prentice Hall Resource Pro with Planning Express www.PHSchool.com (students) TI-83/84 Graphing Calculator CPS Jeopardy Game http://regentsprep.org (Math A) www.pen.k12.va.us/VDOE/Instruction/mat hresource.html The Geometry Center http://www.umn.edu/ NASA http://spacelink.nasa.gov/.index.html The Math Forum http://forum.swarthmore.edu/ 4teachers http://www.4teachers.org Appalachia Educational Laboratory (AEL) http://www.ael.org/pnp/index.htm Eisenhower National Clearinghouse http://www.enc.org/ http://education.jlab.org/solquiz/index.html 4 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Reasoning, Lines, and Transformations Standard: G.1a-d The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. conclusion conditional statement conjecture deductive reasoning contrapositive converse inverse negation if-then statement inductive reasoning proof Law of Detachment counterexample Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment Student reports EduAide Access Test Wizard Teacher created assessment Mathematics Projects Writing Assignment PH ExamView Test Generator Key Terms/ Vocabulary MATHEMATICS – Geometry hypothesis Venn diagram Law of Syllogism Lesson By Design Assessment Textbook Resource Kit activities SOL Released Test Items eduTest Assessment 5 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Reasoning, Lines, and Transformations Standard: G.2a-c The student use the relationships between angles formed by two lines cut by a transversal. Understanding the Standard Teacher Notes 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. Angles are formed by 2 rays that share the same endpoint, called a vertex. Complementary angles are 2 acute angles whose measures add up to 90 degrees. Complements of the same angle or congruent angles are congruent. Supplementary angles are 2 angles whose measures add up to 180 degrees. Vertical angles are the opposite angles formed by intersecting lines. Adjacent angles are angles that share 1 ray and the same vertex. A transversal is a line that intersects 2 other lines. When 2 parallel lines are cut by a transversal, they form: congruent alternate interior angles, congruent alternate exterior angles, congruent corresponding angles, supplementary same-side interior and exterior angles. To determine if 2 lines are parallel, you need to show that: Pairs of corresponding angles are equal Pairs of alternate interior angles are equal Pairs of alternate exterior angles are equal Pairs of same-side interior angles are supplementary Pairs of same-side exterior angles are supplementary The 2 lines have the same slope. MATHEMATICS – Geometry 6 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Essential Understanding Students All students should understand: How to identify angles formed by intersecting lines (linear pairs and vertical angles. How to state the relationships between pairs of angles including: congruent, complementary and supplementary. How to solve problems involving lines intersecting in a plane. How to identify angles formed by lines cut by a transversal (such as: same-side interior, same-side exterior, alternate interior, alternate exterior, and corresponding. How to state the relationships between angles formed by parallel lines cut by a transversal. How to use the relationships between pairs of angles (such as: same-side interior, same-side exterior, alternate interior, alternate exterior, and corresponding angles) to solve practical problems. How to perform constructions to verify congruent angles. How to measure angles using protractors. Parallel Lines have the same slope. How to use angle relationship to determine if lines are parallel. How to find the slope of a line. How to use algebraic methods to determine if lines are parallel. MATHEMATICS – Geometry 7 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Reasoning, Lines, and Transformations Standard: G.2a-c The student use the relationships between angles formed by two lines cut by a transversal. Objectives The student will: a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; MATHEMATICS – Geometry Essential Knowledge, Skills and Processes To be successful with this standard, students are expected to: Use algebraic and coordinate methods as well as deductive proofs to verify whether two lines are parallel. Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal including corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles. Common Core State Standards CCSS for MathematicsGeometry Congruence G-CO Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, p. 7 RPS-Teaching By Design, Lesson G.4 Suggested Projects: Investigating Lines and Angles http://ttaconline.org/staff/sol/sol_sol_lessons.asp Books/Materials: Prentice Hall Mathematics-Geometry, pp. 122-129 Prentice Hall Study Guide & Practice Workbook, pp. 27-28 Prentice Hall Daily Notetaking Guide, pp. 43-45 Prentice Hall Skills and Concepts Review, p. 126 Prentice Hall VA SOL Test Prep Workbook, p. vii SOL Geometry Coach pp. 17-26 Luster, Helen, Preparing for the Geometry SOL Test, pp. 15-21 Technology Integration: Interactive Student Text (online and on CD-ROM) PHSuccessNet (for teacher) Prentice Hall Presentation Pro 8 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Prove geometric theorems c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. Solve real-world problems involving intersecting and parallel lines in a plane. 9. Prove theorems about lines and angles. Theorems include: Vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Pulling It All Together Prentice Hall Computer Test Generator Prentice Hall Resource Pro with Planning Express www.PHSchool.com TI-83/84 Graphing Calculators Geometer’s Sketchpad http://www.smv.org/pubs/index.html http://tqd.advanced.org/2647/geometry/ angle/parallel.html The Geometry Center http://www.umn.edu/ The Math Forum http://forum.swarthmore.edu/ 4teachers http://www.4teachers.org Eisenhower National Clearinghouse http://www.enc.org/ http://www.pen.k12.va.us/VDOE/Instruction/mathresource.html Expressing Geometric Properties with Equations G-GPE Use coordinates to prove simple geometric theorems algebraically 4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given MATHEMATICS – Geometry 9 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together points in the coordinate plane is a rectangle; prove or disprove that the point (1,3) lies on the circle centered at the origin and containing the point (0,2). 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). MATHEMATICS – Geometry 10 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Reasoning, Lines, and Transformations Standard: G.2a-c The student use the relationships between angles formed by two lines cut by a transversal. alternate interior consecutive angles linear pairs of angles consecutive interior angles alternate angles equilateral parallel lines right angles interior angles alternate exterior angles equiangular skew lines transversal same-side interior angles Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment eduTest Assessment Key Terms/ Vocabulary MATHEMATICS – Geometry complementary angles supplementary angles slope vertical angles Student reports EduAide Access Test Wizard test Mathematics Projects SOL Released Test Items 11 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Reasoning, Lines, and Transformations Standard: G.3a-d The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. Understanding the Standards Teacher Notes 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. You can find the distance between any two points on the coordinate plane by using the distance formula: (y-y) +(x-x). You can find the midpoint of a segment by using the formula: (x + x)/2, (y + y)/2. You can find the slope of a line by using the formula: (y-y)/(x-x). A figure has symmetry when it can be mapped, folded or rotated onto itself. A figure can have one or both of two basic symmetries: reflectional –a figure that folds onto itself, and rotational – if there is a rotation of 180 degrees or less that maps the figure onto itself. Any time you transform a figure – that is move, shrink or enlarge the figure- you make a transformation. There are 4 transformations: reflect (flip), translate (slide), rotate (turn), and dilate (to enlarge or reduce). MATHEMATICS – Geometry 12 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Essential Understanding Students All students should understand that: 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. MATHEMATICS – Geometry 13 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Reasoning, Lines, and Transformations Standard: G.3a-d The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. Objectives The student will: Essential Knowledge, Skills and Processes To be successful with this standard, students are expected to: a) investigate and use Find the coordinates of the midpoint of a formulas for segment, using the finding distance, midpoint formula. midpoint, and slope; Apply the distance formula to find the length of a line segment when given the coordinates of the endpoints. Use a formula to find the slope of a line. MATHEMATICS – Geometry Common Core State Standards CCSS for MathematicsGeometry Congruence G-CO Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: RPS-Teaching By Design, Lesson G.2 Prentice Hall Technology Activities 52, 94 Prentice Hall Hands-On Activities 38, 39 Serra, Michael, Patty Paper Geometry, pp. 145-154 Suggested Projects: Cabri Jr. The Equation of a Line http://education.ti.com/educationportal/activityexchange/ Activity.do?cid=US&aId=6714 Parallel/Perpendicular lines http://www.algebralab.org/lessons/ lesson.aspx?file=geometry_coordparallelperpendicular.xml Graphic Line Designs http://education.ti.com/educationportal/ activityexchange/Activity.do?cid=US&aId=3928 2. Represent transformations in the plane using, e.g., 14 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS b) apply slope to verify and determine whether lines are parallel or perpendicular; Compare the slopes to determine whether two lines are parallel, perpendicular, or neither. c) investigate symmetry and determine whether lines are parallel or perpendicular; Determine whether a figure has point symmetry, line symmetry, both, or neither. d) determine whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, dilation, or translation. Solve real-world problems involving MATHEMATICS – Geometry transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Pulling It All Together Books/Materials: Prentice Hall Mathematics-Geometry, pp. 42-50, 158-164, 634-652, 662-666 Prentice Hall Study Guide & Practice Workbook, pp. 11-12, 35-36, 143-152 Prentice Hall Reading & Math Literacy Masters, pp. 3, 11 Prentice Hall Daily Notetaking Guide, pp. 19-21, 55-57, 217-225, 229-231 Prentice Hall Skills and Concepts Review, p.118 Prentice Hall VA SOL Test Prep Workbook, p. ix VA SOL Coach-Geometry, pp. 192-233 Luster, Helen, Preparing for the SOL Geometry Test, pp. 158-178 RPS- Reteaching Lesson G.2 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Technology Integration: Interactive Student Text (online or CD-ROM) PHSuccessNet (teachers) Prentice Hall Pro Presentation Prentice Hall Test Generator Prentice Hall Resource Pro with Planning Express 4. Develop definitions of www.PHSchool.com rotations, reflections, and translations in terms TI-83/84 Graphing Calculator CPS Jeopardy Game of angles, circles, http://www.smv.org/pubs/index.html perpendicular lines, Geometer’s Sketchpad parallel lines, and line NASA http://spacelink.nasa.gov/.index.html segments. 4teachers http://www.4teachers.org http://regentsprep.org 5. Given a geometric figure and a rotation, reflection, 15 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS intersecting and parallel lines in a plane. Pulling It All Together or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Expressing Geometric Properties with Equations G-GPE MATHEMATICS – Geometry Use coordinates to 16 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together prove simple geometric theorems algebraically 5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. MATHEMATICS – Geometry 17 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Reasoning, Lines, and Transformations Standard: G.3a-d The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. Key Terms/ Vocabulary clockwise plane postulate image slide distance midpoint counterclockwise symmetry point symmetry intersection point line postulate line symmetry theorem segment transformation preimage Pythagorean Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment eduTest Assessment MATHEMATICS – Geometry slope plane ruler dilation translation reflection rotation Student reports EduAide Access Test Wizard test Mathematics Projects SOL Released Test Items 18 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Standard: G.4a-g Pulling It All Together Reasoning, Lines, and Transformations The student will construct and justify constructions. Understanding the Standards Teacher Notes 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. A perpendicular bisector of a line segment is a line, line segment, or ray that forms a right angle with the line segment and divides the line segment into two equal parts. The bisector of an angle separates the interior of the angle into two congruent angles. To bisect means to divide into two equal parts. A line of symmetry for an angle or line segment bisects the angle or line segment. In a construction you can only use a straightedge and a compass. In a construction, a point must either be given or be the intersection of a previously constructed figure. If a figure is given you can assume as many points as necessary to make that figure A straightedge can draw a line, through two points A and B. A compass can draw a circle with the center at A and containing a second point B on its circumference. MATHEMATICS – Geometry 19 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Essential Understanding Students All students should understand the following: Construction techniques are used to solve real-life problems in engineering, architectural design, and building construction. Construction techniques may include using a straightedge and compass, paper folding, and dynamic geometry software. MATHEMATICS – Geometry 20 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Reasoning, Lines, and Transformations Standard: G.4a-g The student will construct and justify constructions. Objectives The student will: a) construct a line segment congruent to a given line segment b) construct the perpendicular bisector of a line segment; c) construct a perpendicular to a given line from a point not on the line; d) construct a perpendicular Essential Knowledge, Skills and Processes To be successful with this standard, students are expected to: Common Core State Standards CCSS for MathematicsGeometry Congruence G-CO Make geometric Construct and justify the constructions constructions of – a line segment 12. Make formal geometric congruent to a given constructions with a variety line segment; of tools and methods – the perpendicular (compass and straightedge, bisector of a line string, reflective devices, segment; paper folding, dynamic – a perpendicular to a geometric software, etc.). given line from a point Copying a segment; copying not on the line; an angle; bisecting a – a perpendicular to a segment; bisecting an angle; given line at a point on constructing perpendicular the line; lines, including the – the bisector of a given perpendicular bisector of a angle; line segment; and – an angle congruent to a constructing a line parallel given angle; and to a given line through a – a line parallel to a given MATHEMATICS – Geometry Pulling It All Together Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: RPS-Teaching By Design, Lesson G.11 Prentice Hall Mathematics-Geometry, p. 41 Enhanced Scope and Sequence, pp. 19 - 25 Serra, Michael, Patty Paper Geometry, Investigation Set 2, pp. 17 – 22 Suggested Projects: Perpendicular Bisector Theorem http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=4035 Perpendicular Bisector of a Line Segment http://education.ti.com/educationportal/ activityexchange/Activity.do?cid=US&aId=6856 Points on the Perpendicular Bisector of a Segment http://education.ti.com/educationportal/ activityexchange/Activity.do?cid=US&aId=6860 21 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS to a given line at a given point on the line; line through a point not on the given line. point not on the line. e) construct the bisector of a given angle; f) construct an angle congruent to a given angle; and g) construct a line parallel to a given line through a point not on the given line. Pulling It All Together Books/Materials: Prentice Hall Mathematics-Geometry, pp. 34 - 39 Prentice Hall Study Guide and Practice Workbook, pp. 9, 10 Prentice Hall Daily Notetaking Guide, pp. 15 – 18 Prentice Hall Skills and Concepts Review, p. 117 Prentice Hall VA SOL Test Prep Workbook, p. vii SOL Geometry Coach, pp. 27 – 41 Luster, Helen, Preparing for the SOL Geometry Test, pp. 22 – 34 RPS – Reteach Lessons, G.11 Technology Integration: Prentice Hall Resource PRO CD-ROM Prentice Hall Presentation PRO CD Prentice Hall Interactive Text, www.PHSchool.com Prentice Hall SuccessNet Prentice Hall Computer Test Generator Prentice Hall Resource Pro with Planning Express TI-83/84 Graphing Calculators Geometer’s Sketchpad http://www.smv.org/pubs/index.html http://tqd.advanced.org/2647/geometry/ sample2.html http://www.pen.k12.va.us/VDOE/Instruction/mathresource.html 4teachers http://www.4teachers.org www.regentsprep.org http://forum.swarthmore.edu/ http://spacelink.nasa.gov/.index.html Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Circles G-C Understand and apply theorems about circles Construct the inscribed MATHEMATICS – Geometry 3. Construct the inscribed and 22 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS and circumscribed circles of a triangle. Construct a tangent line from a point outside a given circle to the circle. MATHEMATICS – Geometry Pulling It All Together circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 4. Construct a tangent line from a point outside a given circle to the circle. 23 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Reasoning, Lines, and Transformations Standard: G.4a-g The student will construct and justify constructions. Pulling It All Together Key Terms/ Vocabulary construction quadrilateral inscribed angle bisector straightedge regular hexagon compass congruent equilateral triangle circumscribed vertex perpendicular midpoint point tangent triangle Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment MATHEMATICS – Geometry segment bisector line segment line bisect square Student reports EduAide Access Test Wizard test Mathematics Projects SOL Released Test Items 24 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.5a-d 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. Understanding the Standard Teacher Notes The longest side of a triangle is opposite the largest angle of the triangle and the shortest side is opposite the smallest angle. In a triangle, the length of two sides and the included angle determine the length of the side opposite the angle. In order for a triangle to exist, the length of each side must be within a range that is determined by the lengths of the other two sides. A triangle can be formed if the sum of the lengths of any two sides of a triangle is greater than the length of the third side. (Triangle Inequality Theorem) When you know the lengths of two sides of a triangle, you can determine the range of possible lengths of the third side by applying the Triangle Inequality Theorem. If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side and the smaller angle lies opposite the shorter side. If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle and the shorter side lies opposite the smaller angle. MATHEMATICS – Geometry 25 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Essential Understanding Students All students should: Understand that the longest side of a triangle is opposite the largest angle of the triangle and the shortest side is opposite the smallest angle. Understand that in a triangle, the measure of an angle and the lengths of the adjacent sides determine the length of the side opposite the angle. Understand that a triangle can be formed if the sum of two sides is greater than the third side. MATHEMATICS – Geometry 26 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.5a-d The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a) order the sides by length, given the angle measures; b) order the angles by degree measure, given the side lengths; c) determine whether a triangle exists; and d) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. Essential Knowledge, Skills and Processes Objectives The student will: To be successful with this standard, students are expected to: a) order the sides by length, given the angles measures; Order the sides of a triangle by their lengths when given the measures of the angles. b) order the angles by degree measure, given the side lengths; Order the angles of a triangle by their measures when given the lengths of the sides. MATHEMATICS – Geometry Common Core State Standards CCSS for MathematicsGeometry Congruence G-CO Prove geometric theorems 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, pp. 15-16 Mathematics Enhanced Scope and SequenceGeometry, pp. 37-40 RPS-Teaching by Design, Lesson G.6 Suggested Projects: Triangles from Midpoints http://ttaconline.org/staff/sol/sol_sol_lessons.asp How Many Triangles http://ttaconline.org/staff/sol/sol_sol_lessons.asp Investigating the Triangle Inequality http://education.ti.com/educationportal/ activityexchange/Activity.do?cid=US&aId=7777 Books/Materials: Prentice Hall Mathematics-Geometry, pp. 273-278 27 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS c) determine whether a triangle exists; and d) determine the range in which the length of the third side must lie. Given the lengths of three segments, determine whether a triangle could be formed. Given the lengths of two sides of a triangle, determine the range in which the length of the third side must lie. Solve real-world problems given information about the lengths of sides and/or measures of angles in triangles. MATHEMATICS – Geometry medians of a triangle meet at a point. Pulling It All Together Prentice Hall Study Guide & Practice Workbook, pp. 61-62 Prentice Hall Reading and Math Literacy, p. 19 Prentice Hall Daily Notetaking Guide Workbook, pp. 95-97 Prentice Hall Skills and Concepts Review, p. 143 Prentice Hall VA SOL Test Prep Workbook, p. vii The VA SOL Mathematics Coach, Geometry pp. 60-66 Luster, Helen, Preparing for the SOL Geometry Test, pp. 75-78 RPS Reteaching Lesson G.6 Technology Integration: Prentice Hall Interactive Student Text Prentice Hall Presentation Pro CD-ROM Prentice Hall Resource Pro Prentice Hall SuccessNet (teacher) Prentice Hall Computer Test Generator Geometer’s Sketchpad TI-83/84 Graphing Calculator CPS Jeopardy Game http://www.pen.k12.va.us/VDOE/Instruction/ math_resource.html Eisenhower National Clearinghouse http://www.enc.org/ 4teachers http://www,4teachers.org The Geometry Center http://www.umn.edu/ The Math Forum http://forum.swarthmore.edu/ http://education.jlab.org/solquiz/index.html 28 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS MATHEMATICS – Geometry Pulling It All Together http://regentsprep.org http://forum.swarthmore.edu/ http://spacelink.nasa.gov/.index.html 29 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.5a-d 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. Key Terms/ Vocabulary Addition Property of Inequality shortest side less than largest angle opposite angles inequality longest side smallest angle Methods of Instruction Guided practice Peer Tutoring Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment MATHEMATICS – Geometry Direct Instruction Small Group opposite side greater than Student reports EduAide Access Test Wizard test Mathematics Projects SOL Released Test Items 30 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Standard: G.6 Pulling It All Together Triangles 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. Understanding the Standard Teacher Notes 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. Two triangles are congruent if and only if their corresponding parts are congruent. Triangles can be proven congruent by the following postulates or theorems: If three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent by the SideSide-Side (SSS) Postulate. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent by the Side-Angle-Side (SAS) Postulate. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent by the Angle-Side-Angle Postulate. If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent by the Angle-Angle-Side Theorem. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent by the Hypotenuse-Leg Theorem. Once you know that triangles are congruent, you can make conclusions about corresponding segments and angles because corresponding parts of congruent triangles are congruent (CPCTC). MATHEMATICS – Geometry 31 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Essential Understanding Students All students should understand the following concepts: Congruence has real-life applications in art, architecture, and construction. 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. MATHEMATICS – Geometry 32 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Essential Knowledge, Skills and Processes Objectives The student will: name and label corresponding parts of congruent triangles. To be successful with this standard, students are expected to: Use definitions, postulates, and theorems to prove triangles congruent. Common Core State Standards CCSS for MathematicsGeometry Congruence G-CO Understand congruence in terms of rigid motions 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of MATHEMATICS – Geometry Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, pp. 8, 10, 11, 12-13, 22, Prentice Hall Technology Activities, pp. 81-83, 91 Mathematics Enhanced Scope and SequenceGeometry, pp. 34-36 RPS-Teaching by Design, Lesson G.5 Serra, Michael, Patty Paper Geometry, Investigation Set 8, pp. 125-131 Suggested Projects: ASA Triangle Congruence http://education.ti.com/educationportal/ activityexchange/Activity.do?cid=US&aId=7839 SAS Triangle Congruence http://education.ti.com/educationportal/ activityexchange/Activity.do?cid=US&aId=7823 SSS Triangle Congruence http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aI 33 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. use SSS, SAS, ASA, AAS, and HL to test for triangle congruence. Use coordinate methods, such as the distance formula and the slope formula, to prove two triangles are congruent. 8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. prove triangles congruent. Use algebraic methods to prove two triangles are congruent. 10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MATHEMATICS – Geometry Pulling It All Together d=7818 Congruent Triangles http://education.ti.com/educationportal/ activityexchange/Activity.do?cid=US&aId=5967 Books/Materials: Prentice Hall Mathematics-Geometry, pp. 179229, 416-452 Prentice Hall Study Guide and Practice Workbook, pp. 39-52, 93-102 Prentice Hall Daily Notetaking Guide, pp. 61-80, 142-155 Prentice Hall Reading and Math Literacy, pp. 13-15 Prentice Hall Skills & Concepts Review, pp. 132138, 159-163 Prentice Hall VA SOL Test Prep, p. vii RPS Reteaching Lesson G.5 VA SOL Coach, Geometry, pp. 67 – 88 Preparing for the SOL Geometry Test, pp. 57-74 Technology Integration: Prentice Hall Interactive Student Text Prentice Hall SuccessNet (teacher) Prentice Hall Presentation Pro CD-ROM Prentice Hall Resource Pro Prentice Hall Test Generator www.PHSchool (students) 34 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Similarity, Right Triangles, and Trigonometry G-SRT Prove theorems involving similarity 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Pulling It All Together TI- 83/84 Graphing Calculator Geometer’s Sketchpad www.pen.k12.va.us/VDOE/Instructions/wmstds/ geometry.shtml Eisenhower National Clearinghouse http://www.enc.org/ www.edhelper.com http://mathforum.org/library 4teachers http://www.4teachers.org http://education.jlab.org/solquiz/index.html http://regentsprep.org The Geometry Center http://www.umn.edu/ Field Trips: MATHEMATICS – Geometry 35 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Key Terms/ Vocabulary acute triangle auxiliary lines base base angles congruent triangles flow proof isosceles triangle legs obtuse triangle reflexive property right triangle scalene triangle sides transitive property vertex angle Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator Student reports EduAide Access Test Wizard test Textbook Resource Kit activities Mathematics Projects SOL Released Test Items eduTest Assessment MATHEMATICS – Geometry equiangular triangle equilateral triangle paragraph proof vertices Teacher created assessment Lesson By Design Assessment Writing Assignment 36 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.7 The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Understanding the Standard Teacher Notes Similarity has real-world applications in a variety of areas, including art, architecture, and the sciences. Similarity does not depend on the position of the triangle. Congruent figures are also similar, but similar figures are not necessarily congruent. A proportion is a statement that two ratios are equal. Two polygons 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 the scale factor. Triangles can be proven similar by the following postulates or theorems: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar by the Angle-Angle Similarity Postulate (AA) If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar by the Side-Angle-Side Similarity Theorem (SAS) If the corresponding sides of two triangles are proportional, then the triangles are similar by the Side-Side-Side Similarity Theorem (SSS). The geometric mean between two positive numbers a and b is the positive number x such that = . When the altitude is drawn to the hypotenuse of a right triangle: the two triangles formed are similar to the original triangle and to each other; MATHEMATICS – Geometry 37 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together the length of the altitude is the geometric mean of the lengths of the segments of the hypotenuse; and the length of each leg is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to that leg. If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths. If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is onehalf the length of the third side. Essential Understanding Students All students should understand the following concepts: Similarity has real-life applications in art, architecture, and construction. Similarity does not depend on the position of the triangle. Congruent figures are also similar, but similar figures are not necessarily congruent. MATHEMATICS – Geometry 38 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.7 The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Essential Knowledge, Skills and Processes Objectives The student will: To be successful with this standard, students are expected to: Common Core State Standards CCSS for MathematicsGeometry Similarity, Right Triangles, and Trigonometry G-SRT identify similar figures and solve problems involving similar figures, Use definitions, postulates, and theorems to prove triangles similar. Use algebraic methods to prove that triangles are similar. use proportional parts of triangles to solve problems. Use coordinate methods, such as the distance formula, to prove two triangles are similar. MATHEMATICS – Geometry Understand similarity in terms of similarity transformations 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, pp. 8, 10, 11, 12-13, 22, Prentice Hall Technology Activities, pp. 81-83, 91 Mathematics Enhanced Scope and SequenceGeometry, pp. 34-36 RPS-Teaching by Design, Lesson G.5 Serra, Michael, Patty Paper Geometry, Investigation Set 8, pp. 125-131 Suggested Projects: Constructing Similar Triangles http://education.ti.com/educationportal/ activityexchange/Activity.do?cid=US&aId=8179 Books/Materials: Prentice Hall Mathematics-Geometry, pp. 179229, 416-452 39 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Use algebraic methods to prove two triangles are congruent. prove triangles similar. proportionality of all corresponding pairs of sides. 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MATHEMATICS – Geometry Pulling It All Together Prentice Hall Study Guide and Practice Workbook, pp. 39-52, 93-102 Prentice Hall Daily Notetaking Guide, pp. 6180, 142-155 Prentice Hall Reading and Math Literacy, pp. 13-15 Prentice Hall Skills & Concepts Review, pp. 132-138, 159-163 Prentice Hall VA SOL Test Prep, p. vii RPS Reteaching Lesson G.5 VA SOL Coach, Geometry, pp. 67 – 88 Preparing for the SOL Geometry Test, pp. 57-74 Technology Integration: Prentice Hall Interactive Student Text Prentice Hall SuccessNet (teacher) Prentice Hall Presentation Pro CD-ROM Prentice Hall Resource Pro Prentice Hall Test Generator www.PHSchool (students) TI- 83/84 Graphing Calculator Geometer’s Sketchpad www.pen.k12.va.us/VDOE/Instructions/wmstds/ geometry.shtml Eisenhower National Clearinghouse http://www.enc.org/ www.edhelper.com http://mathforum.org/library 4teachers http://www.4teachers.org 40 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together MATHEMATICS – Geometry http://education.jlab.org/solquiz/index.html http://regentsprep.org The Geometry Center http://www.umn.edu/ 41 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.7 The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Key Terms/ Vocabulary similar angle bisector geometric mean cross-product property scale factor ratio proportion coordinate ratio of similarity midsegment Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator EduAide Access Test Wizard test Mathematics Projects Student reports Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment SOL Released Test Items MATHEMATICS – Geometry proof median altitude 42 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. Understanding the Standard Teacher Notes 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 A = 1/ 2 ab sin C . The ratios of side lengths in similar right triangles (adjacent/hypotenuse or opposite/hypotenuse) are independent of the scale factor and depend only on the angle the hypotenuse makes with the adjacent side, thus justifying the definition and calculation of trigonometric functions using the ratios of side lengths for similar right triangles. In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the legs (altitude and base). This relationship is known as the Pythagorean Theorem: a2 + b2 = c2. The Pythagorean Theorem is used to find the measure of any one of the three sides of a right triangle if the measures of the other two sides are known. The converse of the Pythagorean Theorem states that if the square of the length of one side of a triangle is equal to the sum of the squares of the lengths lf the other two sides, then the triangle is a right triangle. In a triangle with the longest side c, if c a2 + b2, the triangle is obtuse, and if c2 a2 + b2, the triangle is acute. Positive numbers a, b, and c form a Pythagorean triple of a2 + b2 = c2. MATHEMATICS – Geometry 43 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together In a 45- 45- 90 triangle, both legs are congruent and the length of the hypotenuse is times the length of a leg. In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is times the length of the shorter leg. A ratio of the lengths of sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine (sin), cosine (cos), and tangent (tan). An angle below a horizontal line is an angle of depression. An angle above a horizontal line is an angle of elevation. Essential Understanding Students All students should: Understand that the Pythagorean Theorem is essential for problem solving involving right triangles. Understand the relationships between the sides and angles of right triangles are useful in many applied fields. Understand that some practical problems can be solved by choosing an efficient representation of the problem. MATHEMATICS – Geometry 44 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Triangles Standard: G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. Essential Knowledge, Skills and Processes Objectives The student will: use the Pythagorean Theorem and its converse. To be successful with this standard, students are expected to: Common Core State Standards CCSS for Mathematics – Grade 8 Geometry 8.G Understand and apply Verify the Pythagorean the Pythagorean Theorem Theorem and its converse using deductive 6. Explain a proof of the arguments as well as Pythagorean Theorem and algebraic and coordinate its converse. methods. Determine whether a triangle formed with three given lengths is a right triangle. Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, pp. 25, 26, 2829 Prentice Hall Technology Activities, p. 90 RPS-Teaching by Design Lesson Plans- SOL G.7 Mathematics Scope and Sequence-Geometry, pp. 41-54 Suggested Projects: 30-60-90 right triangles in Cabri Jr http://education.ti.com/educationportal 7. Apply the Pythagorean /activityexchange/Activity.do?cid=US&aId=6199 Theorem to determine Applying the Pythagorean Theorem unknown side lengths in http://education.ti.com/educationportal right triangles in real world /activityexchange/Activity.do?cid=US&aId=1836 and mathematical Investigating Special Triangles problems in two and three http://education.ti.com/educationportal dimensions. /activityexchange/Activity.do?cid=US&aId=7896 Similarity, Right Triangles, and Trigonometry G-SRT MATHEMATICS – Geometry 45 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS use the properties of 45-45-90 and 30-60-90 triangles. calculate the sines, cosines, and tangents of acute angles in right triangles. use the sine, cosine, and tangent to determine unknown measures in right triangles. Pulling It All Together Books/Materials: Prentice Hall Mathematics-Geometry, pp. 357 Define trigonometric Solve for missing 371, 470-486 ratios and solve problems lengths in geometric Prentice Hall Study Guide & Practice Workbook, involving right triangles figures, using properties pp. 79-82, 105-110 of 45-45-90°triangles. 6. Understand that by Prentice Hall Reading and Math Literacy similarity, side ratios in Masters, pp. 26, 33, 34 right triangles are Prentice Hall Daily Notetaking Guide Workbook, Solve for missing lengths properties of the angles in pp. 121-126, 159-167 in geometric figures, the triangle, leading to Prentice Hall Skills and Concepts Review, pp. using properties of 30definitions of 152, 153, 165-167 60-90°triangles. trigonometric ratios for Prentice Hall VA SOL Test Prep Workbook, p.vii acute angles. The VA SOL Mathematics Coach, Geometry, pp. 89-119 Solve problems Luster, Helen, Preparing for the SOL Geometry involving right triangles, Test, pp. 79-87 using sine, cosine, and 7. Explain and use the RPS- Reteaching Lesson G.7 tangent ratios. relationship between the sine and cosine of Technology Integration: complementary angles. Solve real-world Prentice Hall Interactive Student Text problems, using right PHSuccessNet (teacher) 8. Use trigonometric ratios triangle trigonometry and Prentice Hall Presentation Pro and the Pythagorean properties of right PH Resource Pro with Planning Express Theorem to solve right triangles. www.PHSchool.com triangles in applied CPS Jeopardy Game problems. TI-83/84 Graphing Calculator Explain and use the Geometer’s Sketchpad Apply trigonometry to relationship between the Understanding Math software general triangles sine and cosine of http://forum.swarthmore.edu/ complementary angles. http://education.jlab.org/solquiz/index/html 9. Derive the formula A= ½ ab sin(C) for the area of a http://www.umn.edu/ MATHEMATICS – Geometry 46 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Pulling It All Together www.edhelper.com http://math.rice.edu/~lanius/Lessons/ http://illuminations.nctm.org/imath/ www.knowledge.state.va.us/main/lesson/les.cfm http://regentsprep.org CCSS for Mathematics.– Functions Trigonometric Functions FTF Extend the domain of trigonometric functions using the unit circle 3. Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosines, and tangent for x, pi + x, and 2pi – x in terms of their values for x, where x is any real number. MATHEMATICS – Geometry 47 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Standard: G.8 Pulling It All Together Triangles 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 Terms/ Vocabulary angle of depression angle of elevation cosine 30-60-90 triangle isosceles right triangle hypotenuse 45-45-90 triangle leg Pythagorean Theorem Pythagorean Triples sine tangent Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator EduAide Access Test Wizard test Mathematics Projects Student reports Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment SOL Released Test Items MATHEMATICS – Geometry trigonometric ratios trigonometry 48 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. Understanding the Standard Teacher Notes The terms characteristics and properties can be used interchangeably to describe quadrilaterals. The term characteristic 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. Properties of Parallelograms a) b) c) d) Opposite sides and opposite angles are congruent. The diagonals bisect each other. The point of intersection is the midpoint of both diagonals. Opposite sides and angles are congruent. Consecutive angles are supplementary. A quadrilateral is a parallelogram if any of the following is true. a) b) c) d) The diagonals of the quadrilateral bisect each other. One pair of opposite sides of the quadrilateral is both congruent and parallel. Both pairs of opposite sides of the quadrilateral are congruent. Both pairs of opposite angles of the quadrilateral are congruent. The diagonals of a rhombus are perpendicular. Each diagonal of a rhombus bisects a pair of opposite angles of the rhombus. The diagonals of a rectangle are congruent. MATHEMATICS – Geometry 49 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together If one diagonal of a parallelogram bisects two angles of the parallelogram, it is a rhombus. If the diagonals of a parallelogram are perpendicular, it is a rhombus. If the diagonals of a parallelogram are congruent, them it is a rectangle. The nonparallel sides of a trapezoid are the legs. Each pair of angles adjacent to the base of a trapezoid are base angles of the trapezoid. Base angles of an isosceles trapezoid are congruent. The diagonals of an isosceles trapezoid are congruent. The diagonals of kite are perpendicular. The segment that joins the midpoints of the nonparallel sides of a trapezoid is the midsegment of the trapezoid. It is parallel to the bases and half as long as the sum of the lengths of the bases. Essential Understanding Students All students should: Understand that quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals. Understand that the properties of quadrilaterals can be used to identify the quadrilateral and find the measures of sides and angles. MATHEMATICS – Geometry 50 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve realworld problems. Objectives The student will: discover the relationships among angles, sides, and diagonals of parallelograms. determine the characteristics of quadrilaterals that indicate they are parallelogram. discover the properties of rectangles, rhombi, MATHEMATICS – Geometry Essential Knowledge, Skills and Processes To be successful with this standard, students are expected 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, Common Core State Standards CCSS for MathematicsGeometry Congruence G-CO Prove geometric theorems 11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, pp. 18 – 20 Prentice Hall Technology Activities, pp. 85- 87 RPS -Teaching by Design Lesson Plan, Lesson G.8 Mathematics Enhanced Scope and Sequence: Properties of Quadrilaterals, pp. 63 – 69 Patty Paper Geometry: Investigation Set 6.1 – 6.4. pp. 87 - 93 Suggested Projects: Classifying Quadrilaterals http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=5747 Constructing Quadrilaterals http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=4058 Constructing and Investigating Properties of a Rhombus http://education.ti.com/educationportal 51 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS squares, trapezoids, and kites. slope, and midpoint formula. Circles G-C Understand and apply theorems about circles Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods. Prove properties of angles for a quadrilateral inscribed in a circle. MATHEMATICS – Geometry 3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Pulling It All Together /activityexchange/Activity.do?cid=US&aId=8178 Exploring Quadrilaterals with Cabri Jr. http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=6215 Pearson Prentice Hall Geometry Chapter 6: Quadrilaterals Lesson 6-1 to 6-4 http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=4153 Pearson Prentice Hall Geometry Chapter 6: Quadrilaterals -- Lessons 6-5 to 6-7 http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=4154 Books/Materials: Prentice Hall Geometry, pp. 288 – 325 Prentice Hall Study Guide and Practice Workbook, pp. 63 - 72 Prentice Hall Reading and Math Literacy, pp. 21 – 24 Prentice Hall Daily Notetaking Guide, pp. 98 – 112 Prentice Hall Geometry VA SOL Test Prep Workbook, p.viii Prentice Hall Skills and Concepts Review, pp. 144 - 148 The VA SOL Mathematics Coach, Geometry , pp. 122 - 130 Luster, Helen, Preparing for the SOL Geometry SOL Test, pp. 91-109 RPS-Reteaching Lesson G.8 52 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Technology Integration: Prentice Hall Resource PRO CD-ROM Prentice Hall Presentation PRO CD Prentice Hall Interactive Text, www.PHSchool.com Prentice Hall SuccessNet Prentice Hall Resource Pro with Planning Express Prentice Hall ExamView test generator TI-83/84 Graphing Calclulator Understanding Math software CPS Jeopardy Game Geometer’s Sketchpad http://www.pen.k12.va.us/VDOE/Instruction/mathrersource.html Eisenhower National Clearinghouse www.enc/org http://www.4teachers.org The Geometry Center http://www.umn.edu/ The Math Forum http://forum.swarthmore.edu/ NCTM http://illuminations.nctm.org/imath/ http://regents.prep.org http://education.jlab.org/solquiz/index.html www.edhelper.com www.math.rice.edu/~lanius/Lessons/ www.iq-poquoson.org MATHEMATICS – Geometry 53 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve realworld problems. Key Terms/ Vocabulary base kite square trapezoid kite base angle isosceles trapezoid Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Methods of Assessment PH ExamView Test Generator Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment MATHEMATICS – Geometry parallelogram median midsegment rhombus quadrilateral rectangle Cooperative Grouping Group Discussions Student reports EduAide Access Test Wizard test Mathematics Projects SOL Released Test Items 54 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.10 The student will solve real-world problems involving angles of polygons. Understanding the Standard Teacher Notes 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 nonoverlapping triangles. A polygon is a closed plane figure with at least three sides. A polygon is convex if no diagonal contains points outside the polygon. Otherwise it is concave. A regular polygon is equilateral and equiangular. The sum of the measures of the interior angles of an n-gon is (n 2)180. The sum of the measures of the exterior angles (one at each vertex) of an n-gon is 360. The measure of an interior angle of a regular n-gon = The measure of an exterior angle of a regular n-gon =. Patterns that cover a plane with repeating figures so that there are no overlapping or empty spaces are called tessellations. A regular MATHEMATICS – Geometry 55 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together tessellation uses only one type of regular polygon. If a regular polygon has an interior angle with a measure that is a factor of 360, then the polygon will tessellate the plane. Uniform tessellations containing two or more regular polygons are called semi-regular. Essential Understanding Students All students should: Understand that a regular polygon will tessellate the plane if the measure of an interior angle is a factor of 360. Understand that both regular and non-regular polygons will tessellate the plane. Two intersecting lines form angles with specific relationships. An exterior angle is formed by extending a side of a polygon. The exterior angle and the corresponding interior angle form a linear pair. The sum of the measures of the interior angles of a convex polygon may be found by dividing the interior of the polygon into nonoverlapping triangles. MATHEMATICS – Geometry 56 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.10 The student will solve real-world problems involving angles of polygons. Essential Knowledge, Skills and Processes Objectives The student will: To be successful with this standard, students are expected to: name and identify polygons; find the sum of the measures of interior and exterior angles of convex and regular polygons; solve problems involving 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. MATHEMATICS – Geometry Common Core State Standards CCSS for MathematicsGeometry Modeling with Geometry GMG Apply geometric concepts in modeling situations 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, pp. 8, 9, 36 Prentice Hall Technology Activities, p. 81 Enhanced Scope and Sequence, Polygons, pp. 57 62 RPS -Teaching By Design, Lesson G.8 Serrra, Michael, Patty Paper Geometry, Investigation Set 4.1. pp. 56 - 57 Suggested Projects: Angles in a Polygon http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=7428 Geomaster: Sum of Interior Angles of Polygon http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=4043 Got exterior angles equal to 360? with Cabri Jr. http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=7322 Investigating the Angle-Sum Theorem of 57 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS angle measures of polygons. 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. Pulling It All Together Polygons http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=7303 Books/Materials: Prentice Hall Mathematics, Geometry, pp. 131 – 138, 142 – 150, 667- 673 Prentice Hall Study Guide and Practice Workbook, pp. 29 – 32, 153 – 154 Prentice Hall Reading and Math Literacy Masters, pp. 10, 12, 47 Prentice Hall Daily Notetaking Guide, pp. 46 – 51, 232 – 234 Prentice Hall Skills and Concepts Review, p. 128 Prentice Hall VA SOL Test Prep Workbook, p. viii VA SOL Coach, Geometry, pp. 131 – 137 Luster, Helen, Preparing for the SOL Geometry Test, pp. 110 - 118 RPS Reteaching Lesson G.9 Technology Integration Prentice Hall Resource PRO CD-ROM Prentice Hall Presentation PRO CD Prentice Hall Interactive Text, www.PHSchool.com Prentice Hall SuccessNet CPS Jeopardy Game Geometer’s Sketchpad Understanding Math software http://www.pen.k12.va.us/VDOE/ Instruction/mathresource.html MATHEMATICS – Geometry 58 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together MATHEMATICS – Geometry Eisenhower National Clearinghouse, http://www.enc.org/ 4teachers, http://4teachers.org The Geometry Center, http://umn.edu/ The Math Forum, http://forum.swarthmore.edu/ http://education.jlab.org/solquiz/index.htm; http://regentsprep.org www.edhelper http://math.rice.edu/~lanius/Lessons/ http://illuminations.nctm.org.imath www.knowledge.state.va.us/main/lesson/les.cfm 59 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.10 The student will solve real-world problems involving angles of polygons. Key Terms/ Vocabulary concave polygon convex polygon tessellation vertices polygon hexagon interior angle equiangular quilateral regular polygon regular tessellation octagon pentagon Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator EduAide Access Test Wizard test Mathematics Projects Student reports Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment SOL Released Test Items MATHEMATICS – Geometry exterior angle uniform tiling a plane tessellation 60 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.11a-c The student will use angles, arcs, chords, tangents, and secants. Pulling It All Together Understanding the Standard Teacher Notes Many relationships exist between and among angles, arcs, secants, chords, and tangents of a circle. All circles are similar. A chord is part of a secant. Real-world applications may be drawn from architecture, art, and construction. A tangent to a circle is a line, ray, or segment in the plane of the circle that intersects the circle in exactly one point, the point of tangency. If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. If a line is perpendicular to the radius at its endpoint on the circle, then the line is tangent to the circle. Two segments tangent to a circle from a point outside the circle are congruent. In the same circle or in congruent circles: a) congruent central angles intercept congruent arcs; b) congruent arcs have congruent central angles; c) congruent chords have congruent arcs; d) congruent arcs have congruent chords; e) chords equidistant from the center are congruent; and MATHEMATICS – Geometry 61 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together f) congruent chords are equidistant from the center of the circle. A diameter that is perpendicular to a chord bisects the chord and its arc. The perpendicular bisector of a chord contains the center of the circle. The vertex of an inscribed angle lies on a circle. Its sides intercept an arc of the circle. All the vertices of an inscribed polygon lie on a circle. The measure of an inscribed angle is half the measure of its intercepted arc. The measure of an angle formed by a chord and a tangent that intersects on a circle is half the measure of the intercepted arc. Two inscribed angles that intercept the same arc are congruent. The opposite angles of a quadrilateral inscribed in a circle are supplementary. The measure of an angle formed by two intersecting chords in a circle is half the sun of the measures of the intercepted arcs. A secant is a line, ray, or segment that intersects a circle at two points. The measure of an angle formed by two secants, two tangents, or a secant and a tangent from a point outside a circle is half the difference of the measures of the intercepted arcs Arc length of a circle = measure of the arc divided by 360 times the product of two, pi, and the radius. Area of a sector of a circle = measure of the intercepted arc divided by 360 times the product of the radius squared and pi. MATHEMATICS – Geometry 62 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Essential Understanding Students All students should: Understand and apply the many relationships that exist between and among the angles, arcs, secants, chords, and tangents of a circle. Understand that all circles are similar. Understand that a chord is a part of a secant. Understand that many relationships exist between and among angles, arcs, secants, chords, and tangents of a circle. Understand that all circles are similar. Understand that a chord is part of a secant. Understand that real-world applications may be drawn from architecture, art, and construction. MATHEMATICS – Geometry 63 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.11a-c a) investigate, verify, and apply properties of circles; The student will use angles, arcs, chords, tangents, and secants. Essential Knowledge, Skills and Processes Objectives The student will: Pulling It All Together To be successful with this standard, students are expected to: MATHEMATICS – Geometry 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. Common Core State Standards CCSS for MathematicsGeometry Congruence G-CO Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, pp. 35 - 37 Lesson by Design, Lesson G.10 Enhanced Scope and Sequence, pp. 73 - 86 Experiment with Suggested Projects: transformations in the Angles Formed by Intersecting Chords, Secants, plane. and Tangents 1. Know precise http://education.ti.com/educationportal definitions of angle, /activityexchange/Activity.do?cid=US&aId=4065 circle, perpendicular Arc length and Area of Sectors line, parallel line, and http://education.ti.com/educationportal line segment, based on /activityexchange/Activity.do?cid=US&aId=5421 the undefined notions Evaluating the Products of Cords of a Circle of point, line, distance http://education.ti.com/educationportal along a line, and /activityexchange/Activity.do?cid=US&aId=7377 distance around a circular arc. Books/Materials: Prentice Hall Mathematics-Geometry, pp. 386 – 393, Circles G-C 395 – 400, 582 – 589, 590 – 595, 598 – 605, 606 613 Understand and 64 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS b) solve realworld problems involving properties of circles; and c) find arc lengths and areas of sectors in circles. Solve real-world problems associated with circles, using properties of angles, lines, and arcs. apply theorems about circles 1. Prove that all circles are similar. Pulling It All Together Calculate the area of a sector and the length of an arc of a circle, using proportions. Verify properties of circles, using deductive reasoning, algebraic, and coordinate methods. MATHEMATICS – Geometry 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Prentice Hall Study Guide and Practice Workbook, pp. 87 – 90 Prentice Hall Reading and Math Literacy, pp. 132 138 Prentice Hall Skills and Concepts Review, pp. 156, 157 Prentice Hall Notetaking Guide, pp. 198 - 210 Prentice Hall VA SOL Test Prep Workbook, p. viii The VA SOL Mathematics Coach, Geometry pp. 138 151 Luster, Helen, Preparing for the SOL Geometry Test, pp. 119 - 131 Technology Integration: Prentice Hall Resource PRO CD-ROM Prentice Hall Presentation PRO CD Prentice Hall Interactive Text, www.PHSchool.com Prentice Hall SuccessNet Prentice Hall Resource Pro with Planning Express CPS Jeopardy Game Find arc lengths and Graphing Calculator, TI-83/84 areas of sectors of Geometer’s Sketchpad circles http:www.enc.org/ 5. Derive using similarity http://www.4teachers.org the fact that the length http:www.umn.edu/ http://forum.swarthmore.edu of the arc intercepted http://education.jlab.org/solquiz/index.html by an angle is http://regentsprep.org proportional to the www.edhelper radius, and define the http://math.rice.edu/~lanius/Lessons/ radian measure of the http://illuminations.nctm.org/imath angle as the constant 65 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS of proportionality; derive the formula for the area of a sector. Pulling It All Together www.knowledge.state.va.us/main/lesson/les.cfm http://www.pen.k12.va.us/VDOE/ Instruction/mathresource.html Geometric Measurement and Dimension G-GMD Explain volume formulas and use them to solve problems 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. MATHEMATICS – Geometry 66 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Topic: Pulling It All Together Polygons, Circles, and Three-Dimensional Figures Standard: G.11a-c The student will use angles, arcs, chords, tangents, and secants. Key Terms/ Vocabulary annulus segment of circle radius internal tangent external tangent point of tangency circumscribed arc length semi-circle secant length of an arc inscribed angle inscribed polygon concentric center tangent sector major arc Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator EduAide Access Test Wizard test Mathematics Projects Student reports Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment SOL Released Test Items MATHEMATICS – Geometry minor arc intercepted arc diameter central angle chord 67 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.12 The student, given the coordinates of the center of the circle and a point on the circle, will write the equation of the circle. Understanding the Standard Teacher Notes A circle is a locus of points equidistant from a given point, the center. Standard form for the equation of a circle is (x – h)2 + ( y − k )2 = r2 , where the coordinates of the center of the circle are (h, k) and r is the length of the radius. The circle is a conic section. The standard form equation of a circle is a way to express the definition of a circle on the coordinate plane. h is the x-coordinate of the center of the circle. k is the y-coordinate of the center of the circle. r is the radius of the circle. The center of the circle is also called the vertex. The vertex is equal to (x, y) = (h, k). MATHEMATICS – Geometry 68 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Essential Understanding Students All students should: Understand that a circle is a locus of points equidistant from a given point, the center. Understand that the standard form for the equation of a circle is (x − h)2 + (y − k )2 = r2 , where the coordinates of the center of the circle are (h, k) and r is the length of the radius. Understand that the circle is a conic section. Understand that the standard form equation of a circle is a way to express the definition of a circle on the coordinate plane. MATHEMATICS – Geometry 69 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.12 The student, given the coordinates of the center of the circle and a point on the circle, will write the equation of the circle. Essential Knowledge, Skills and Processes Objectives The student will: Find the coordinates of the center and the length of the radius of a circle whose equation is given in the form (x – h)2 + (y – k)2 = r2. To be successful with this standard, students are expected to: Identify the center, radius, and diameter of a circle from a given standard equation. 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, MATHEMATICS – Geometry Common Core State Standards Resources Suggested Activities - Books & Materials Technology Integration - Field Trips CCSS for MathematicsGeometry Activities Notes on equation of a circle. http://www.regentsprep.org/Regents/math/algtrig/ATC1/ Expressing Geometric circlelesson.htm Properties with Practice with equation of circle. Equations G-GPE http://www.regentsprep.org/Regents/math/algtrig/ATC1/ circlepractice.htm Translate between the Lesson on equation of circle geometric description http://www.algebralab.org/lessons/lesson.aspx?file=Algebra and the equation for a _conics_circle.xml conic section http://www.mathsisfun.com/algebra/circle-equations.html http://www2.randia.net:8080/instructional/math/matha3/0351. Derive the equation of a 036%20Equation%20of%20a%20Circle%20.pdf circle of given center Video on equation of circle and radius using the http://www.onlinemathlearning.com/equation-circle-3.html Pythagorean Theorem; An applet to explore the equation of a circle and the complete the square to properties of the circle find the center and http://www.analyzemath.com/CircleEq/CircleEq.html radius of a circle given Geometer’s sketchpad activity by an equation. http://mathforum.org/sketchpad/ckcircle.html 70 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS identify the coordinates of the center and find the radius of the circle. Write the equation of a circle, given the coordinates of its center and the length of its radius. Use the distance formula to derive the general equation of a Use coordinates to prove simple geometric theorems algebraically 4. Use coordinates to Given the coordinates of the prove simple geometric endpoints of a theorems algebraically. diameter, find the For example, prove or equation of the circle. disprove that a figure defined by four given points in the coordinate Given the plane is a rectangle; coordinates of the prove or disprove that center and a point on the point (1,3) lies on the circle, find the the circle centered at equation of the circle. the origin and containing the point Recognize that the (0,2). equation of a circle of given center and radius is derived using the Pythagorean Theorem. Use the distance formula to find the radius of a circle. MATHEMATICS – Geometry Pulling It All Together Suggested Projects: Circle Equations http://education.ti.com/educationportal/activityexchange/ Activity.do?cid=US&aId=5736 Match the Graph (circles) http://education.ti.com/educationportal/activityexchange/ Activity.do?cid=US&aId=5538 Books/Materials: Prentice Hall Mathematics Algebra 2 pages 532-582 Prentice Hall Mathematics Algebra 2 Daily Note Taking guide workbook, Pages 193-207 Prentice Hall Mathematics Algebra 2 Study Guide and PracticeWorkbook Pages 129-140 AMSCO Preparing for the Virginia SOL Algebra II Test pages 146-155 Technology Integration: Prentice Hall Resource PRO CD-ROM Prentice Hall Presentation PRO CD Prentice Hall Interactive Text, www.PHSchool.com Prentice Hall SuccessNet Prentice Hall Resource Pro with Planning Express CPS Jeopardy Game Graphing Calculator, TI-83/84 Geometer’s Sketchpad http:www.enc.org/ http://www.4teachers.org http:www.umn.edu/ http://forum.swarthmore.edu 71 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS circle whose center is any given point in the plane and whose radius has a length r. MATHEMATICS – Geometry Pulling It All Together http://education.jlab.org/solquiz/index.html http://regentsprep.org www.edhelper http://math.rice.edu/~lanius/Lessons/ http://illuminations.nctm.org/imath www.knowledge.state.va.us/main/lesson/les.cfm http://www.pen.k12.va.us/VDOE/Instruction/ mathresource.html 72 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.12 The student, given the coordinates of the center of the circle and a point on the circle, will write the equation of the circle. Key Terms/ Vocabulary conic section circle center radius diameter vertex Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator EduAide Access Test Wizard test Mathematics Projects Student reports MATHEMATICS – Geometry Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment SOL Released Test Items 73 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. Understanding the Standard Teacher Notes The surface area of a three-dimensional object is the sum of the areas of all its faces. The volume of a three-dimensional object is the number of unit cubes that would fill the object. The surface area is the sum of the area of the area of each face (including the bases). A pyramid is a polyhedron with one base. All other faces are triangles. A cylinder has 2 congruent circles as bases, and its lateral surface is a rectangle. A cone is a 3-D figure with one circular base and a curved surface that connects at a point. A prism is a solid that has 2 congruent polygonal bases that are parallel. Volume is the amount of space inside a 3-dimensional object. A formula sheet will be provided by the teacher for students to use and find the surface area and volume of all 3-D objects and to solve practical problems. Essential Understanding Students All students should: 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. MATHEMATICS – Geometry 74 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. Essential Knowledge, Skills and Processes Objectives The student will: use formulas for surface area and volume of threedimensional objects to solve practical problems. To be successful with this standard, students are expected to: MATHEMATICS – Geometry 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 Common Core State Standards CCSS for MathematicsGeometry Geometric Measurement and Dimension G-GMD Explain volume formulas and use them to solve problems 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities 29, 30, pp. 32 – 33, 34 Prentice Hall Technology Activities 51, p. 93 RPS-Teaching By Design, Lesson G.13 Mathematics Scope and Sequence, pp. 99 - 105 Suggested Projects: Exploring Surface Area and Volume http://ttaconline.org/staff/sol/sol_sol_lessons.asp Designing a Soda Can http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=3182 Pyramid Height Exploration http://education.ti.com/educationportal /activityexchange/Activity.do?cid=US&aId=6055 Books/Materials: Prentice Hall Mathematics-Geometry, pp. 528 – 564 Prentice Hall Study Guide & Practice Workbook, pp. 119 – 128 75 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three-dimensional figures. Use calculators will to find decimal approximations for results. Calculators may be used to find decimal approximations for results. and cone. Use dissection arguments, Cavalieri.’s principle, and informal limit arguments. 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Modeling with Geometry G-MG Apply geometric concepts in modeling situations Pulling It All Together Prentice Hall Reading & Math Literacy, pp. 38 – 40 Prentice Hall Daily Notetaking Guide, pp. 179 – 194 Prentice Hall VA SOL Test Prep, p. viii Prentice Hall Skills & Concepts Review, pp. 172 - 176 VA SOL Geometry Coach, pp. 163 – 178 Luster, Helen, Preparing for the Geometry SOL Test, pp. 141 - 148 Technology Integration: TI-83/84 Graphing Calculators Geometer’s Sketchpad www.regentsprep.org http://www.pen.k12.va/VDOE/Instruction/mathresource.html http:www.jeffersonlab/SOLquiz http://www.smv.org/pubs/index.html http://tqd.advanced.org/2647/geometry/ sample2.html 4teachers http://www.4teachers.org 1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder) MATHEMATICS – Geometry 76 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together 2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). MATHEMATICS – Geometry 77 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. Key Terms/ Vocabulary apothem axis base base area cone cube cylinder sphere triangular prism dodecahedron edge great circle hexagon net rectangular prism right cone square pyramid vertex Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Methods of Assessment PH ExamView Test Generator Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities Writing Assignment MATHEMATICS – Geometry face hemisphere oblique octahedron rectangular solid regular polyhedron right cylinder surface area volume icosahedrons lateral area polyhedron pyramid regular pyramid right prism slant height tetrahedron Cooperative Grouping Group Discussions Student reports EduAide Access Test Wizard test Mathematics Projects SOL Released Test Items 78 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.14a-d 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; d) solve real-world problems about similar geometric objects. Understanding the Standard Teacher Notes 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. Ratios are a way of comparing two numbers, quantities or variables. A proportion is a statement that two ratios are equal. A scale is a ratio that compares the drawing measures to real world measures. The scale factor of similar figures is the size of the change from the original figure. Use proportions to compare perimeters, areas, and volumes of similar two-dimensional figures. Use proportions to compare surface area and volumes of three-dimensional geometric figures. Describe how a change in one measure affects other measures of an object. When two figures are similar, the ratio of the areas is equal to the ratio of any two corresponding lengths squared and the ratio of the volumes is equal to the ratio of any two corresponding lengths cubed. MATHEMATICS – Geometry 79 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Essential Understanding Students All students should understand that: 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. MATHEMATICS – Geometry 80 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.14a-d 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; d) solve real-world problems about similar geometric objects. Essential Knowledge, Skills and Processes Objectives The student will: a) compare ratios between side lengths, perimeters, areas, and volumes; b) determine how changes in one or more dimensions of an object To be successful with this standard, students are expected to: Compare ratios between side lengths, perimeters, areas, and volumes, given two similar figures. Common Core State Standards CCSS for MathematicsGeometry Similarity, Right Triangles, and Trigonometry G-SRT Understand similarity in terms of similarity transformations Resources Suggested Activities - Books & Materials Technology Integration - Field Trips Activities: Prentice Hall Hands-On Activities, pp. 24, 26, 27 Prentice Hall Technology Activities 50, p. 91 RPS-Teaching By Design, Lesson G.14 Suggested Projects: Polygons http://ttaconline.org/staff/sol/sol_sol_lessons.asp Gulliver’s Travels and Proportional Reasoning http://ttaconline.org/staff/sol/sol_sol_lessons.asp 2. Given two figures, use the definition of similarity in Books/Materials: terms of similarity Prentice Hall Mathematics-Geometry, pp. 415 – 459, Describe how changes in transformations to decide 565 – 571 one or more dimensions if they are similar; explain Prentice Hall Study Guide & Practice Workbook, pp. 93 affect other derived using similarity – 103, 129 – 130 measures (perimeter, transformations the Prentice Hall Reading & Math Literacy, pp. 29 – 32 area, total surface area, meaning of similarity for MATHEMATICS – Geometry 81 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS 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 realworld problems about similar geometric objects. and volume) of an object. triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Describe how changes in Prove theorems one or more measures involving similarity (perimeter, area, total surface area, and 5. Use congruence and volume) affect other similarity criteria for measures of an object. triangles to solve problems and to prove relationships in geometric figures. Geometric Measurement and Dimension G-GMD Solve real-world problems involving measured attributes of similar objects. Explain volume formulas and use them to solve problems Pulling It All Together Prentice Hall Daily Notetaking Guide, pp. 142 – 158 Prentice Hall VA SOL Test Prep Workbook, p.viii Prentice Hall Skills & Concepts Review, pp. 159, 164 RPS- Reteaching Lessons G.14 SOL Geometry Coach, pp.180 - 189 Technology Integration: Prentice Hall Resource PRO CD-ROM Prentice Hall Presentation PRO CD Prentice Hall Interactive Text (student) www.PHSchool.com Prentice Hall SuccessNet Prentice Hall Computer Test Generator Prentice Hall Resource Pro with Planning Express TI-83/84 Graphing Calculator CPS Jeopardy Game Geometer’s Sketchpad www.regents.prep.org http://www.smv.org/pubs/index.html http://tqd.advanced.org/2647/geometry/ sample2.html http://www.pen.k12.va.us/VDOE/Instruction/mathresource.html 4teachershttp://www.4teachers.org 2. Give an informal argument using Cavalieri.’s principle for the formulas for the volume of a sphere and other� solid figures. Modeling with Geometry MATHEMATICS – Geometry 82 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together G-MG Apply geometric concepts in modeling situations 3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). MATHEMATICS – Geometry 83 RICHMOND PUBLIC SCHOOLS CURRICULUM COMPASS MATHEMATICS Pulling It All Together Topic: Polygons, Circles, and Three-Dimensional Figures Standard: G.14a-d 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; d) solve real-world problems about similar geometric objects. Key Terms/ Vocabulary perimeter cross product proportion total surface area area similar ratio scale factor scale extremes means Methods of Instruction Guided practice Peer Tutoring Direct Instruction Small Group Cooperative Grouping Group Discussions Methods of Assessment PH ExamView Test Generator Student reports EduAide Access Test Wizard test Mathematics Projects Writing Assignment MATHEMATICS – Geometry volume exchange means reciprocal Teacher created assessment Lesson By Design Assessment Textbook Resource Kit activities SOL Released Test Items 84 A Publication of Richmond Public Schools Richmond, Virginia In accordance with federal laws, the laws of the Commonwealth of Virginia and the policies of the School Board of the City of Richmond, the Richmond Public Schools does not discriminate on the basis of sex, race, color, age, religion, disabilities or national origin in the provision of employment and services. The Richmond Public Schools operates equal opportunity and affirmative action programs for students and staff. The Richmond Public Schools is an equal opportunity/affirmative action employer. The Title IX Officer is Ms. Angela C. Lewis, Clerk of the School Board, 301 North 9th St., Richmond, VA 232191927, (804) 780-7716. The Section 504 Coordinator is Mrs. Michelle Boyd, Director of Exceptional Education and Student Services, 301 North 9th St., Richmond, VA, 23219-1927, (804) 780-7911. The ADA Coordinator is Ms. Valarie Abbott Jones, 2015 Seddon Way, Richmond, VA 23230-4117, (804) 780-6211. The United States Department of Education’s Office of Civil Rights may also be contacted at 550 12th Street SW, PCP-6093 Washington, DC 20202, (202) 245-6700. School Board Dawn C. Page, Chair Maurice A. Henderson, Vice Chair Kimberly M. Bridges Kimberly B. Gray Norma H. Murdoch-Kitt Adria A. Graham Scott Chandra H. Smith Donald L. Coleman Evette L. Wilson Dr. Yvonne W. Brandon, Superintendent