Greene County Public Schools Geometry Pacing and Curriculum Guide

Course Outline Quarter 1 Quarter 2 Quarter 3 Quarter 4 Prerequisite­Fundamentals G.6 Proofs with Congruent Triangles G.9 Quadrilaterals (Cont.) G.10/G.11 Polygons and Circles G.1 Logic G.4 More Constructions G.7 Similarity G.13/G.14 Surface Area and Volume G.2/G.3 Parallel and Perpendicular Lines G.5 Triangles G.8 Right Triangles G.3 Symmetry and Transformations G.4 Constructions G.9 Quadrilaterals G.11 Circles

G.12 Equation of a Circle

Advanced Geometry 8th Grade Advanced Geometry is required to include 10 or more topics from the Cross­Curricular

and Advanced Geometry Extensions Column in this document and utilize advanced algebraic equations with all applicable geometric concepts.

High School Advanced Geometry is required to include 5 or more topics from the Cross­Curricular and Advanced Geometry Extensions Column in this document and utilize advanced algebraic equations with all applicable geometric concepts.

Resources

Text: Geometry: Reasoning, Applying, Measuring , 2004, McDougal Littell

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

DOE Enhanced Scope and Sequence Lesson Plans

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

Common Assessment Pre/Post Assessments IA Benchmarks Tests

Common Unit Plans Meet weekly to create common lessons Collaborate on teaching strategies Base the lesson on the Curriculum Guide

Greene County Public Schools

Geometry Curriculum & Pacing Guide 2015­2016 Quarter 1 August 18, 2015 – October 16, 2015

Time/Date

s

SOL/Strand Objective/Content/Essential Knowledge and Skills/Cognitive Level

Vertical Alignment

Vocabulary Cross­Curricular and Advanced Geometry Extensions

8/18­9/4

Textbook

1­2 1­3 1­4 1­5 1­6

Prerequisite Fundamental

s

SOL G.3a Distance Midpoint Slope

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.

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

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

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

acute angles angle bisect collinear complementary congruent coordinate coplanar distance horizontal intersect line linear pair midpoint obtuse angle plane point postulate Pythagorean thm. ray right angle

Taxi­Cab Routes Distance Formula (Textbook pp 66 & 67)

Blooms = Apply

ruler segment slope straight angle supplementary theorem vertex vertical angles

8/18­9/4

SOL G.4abe† Construction

s

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;

e) The bisector of a given angle. †) A regular hexagon inscribed in a circle; †) An equilateral triangle inscribed in a

circle; †) A square inscribed in a circle.

Blooms = Create

SOL 6.12 Determine congruence of segments, angles and polygons

bisector compass construction equilateral hexagon perpendicular regular square straight edge triangle

History of Constructions Euclid’s Constructions with only a compass and a straightedge in his 13 volume work called The Elements. (Page 25) Science /Honeycombs Recognize that bees always use hexagons because they are perfect in saving on labor (effort and energy) and wax.

9/8­9/30

Textbook

2.1 2.2 2.3 2.4 2.5 2.6

SOL G.1 Logic

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;

biconditional conditional

statement contrapositive converse counterexamples deductive reasoning definition

Writing Conditional Statements in Advertising (Textbook page 77)

c) Using Venn diagrams to represent set relationships;

d) Using Deductive Reasoning The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

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

Determine the validity of a logical

argument. Use valid forms of deductive

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

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

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

Interpret Venn diagrams. Recognize and use the symbols of

formal logic, which include → ,↔, , and . ,,

Blooms = Evaluate

intersection invalid inverse law of contrapositive law of detachment law of syllogism postulate proof set symbolic form theorem translate truth value union valid Venn diagrams verbal arguments

10/1­10/17

Textbook

3­1 3­2 3­3 3­4 3­5

SOL G.2 Parallel & Perpendicula

r Lines

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 real­world problems involving angles formed when parallel lines are cut by a transversal.

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

Use 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 same­side (consecutive) interior angles.

Solve real­world problems involving intersecting and parallel lines in a plane.

SOL 8.6a a) verify and describe relationships among vertical, adjacent, supplementary, and complementary angles

alternate exterior

angles alternate interior

angles corresponding

angles parallel same­side interior

angles transversal

Advanced Geometry The Parallel Postulate differentiates Euclidean from non­Euclidean geometries such as spherical geometry and hyperbolic geometry. Science Parallel paths of light passing through glass. (Textbook Page 163) Earth Science/History Eratosthenes, a Greek Scholar over 2000 years ago, he estimated Earth’s circumference by using the fact that the Sun’s rays are parallel and alternate interior angles of parallel lines are congruent. (Textbook Page 145)

Blooms = Evaluate

9/29­10/17 SOL G.4cfg Construction

s

The student will construct and justify the constructions of:

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

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.

Blooms = Create

Greene County Public Schools Geometry Curriculum & Pacing Guide 2015­2016 Quarter 2

October 19, 2015 – December 22, 2015

Time/Dates

SOL/Strand Objective/Content/Essential Knowledge and Skills/Cognitive Level

Vertical Alignment

Vocabulary Cross­Curricular

and Advanced Geometry Extensions

10/19­10/23 Textbook

3­6 3­7

SOL G.3 Parallel & Perpendicula

r Lines

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:

b) Investigating and use the Slope Formula;

c) Applying slope to verify and determine whether lines are parallel or perpendicular.

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

Use a formula to find the slope of a line. Compare the slopes to determine

whether two lines are parallel, perpendicular, or neither.

Blooms = Analyze

SOL A.6ab Graph linear equations and linear inequalities in two variables – a) determine slope of line given equation of the line, graph of line or two points on the line; discuss slope as rate of change. b) write equation of a line given the graph of a line, two points on a line or the slope and a point on the line.

algebraic method coordinate

method

10/26­11/1

7

SOL G.6 Congruent

The student, given information in the form of a figure or statement, will prove two

SOL 6.12 Determine the congruence of

AAS ASA

Logic Creating a

Textbook

4­1 4­2 4­3 4­4 4­5 4­6 4­7

Triangles triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

Use definitions, postulates, and theorems to prove triangles congruent.

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

Use algebraic methods to prove two triangles are congruent.

Blooms = Evaluate

segments, angles, and polygons.

congruent figures corollary corresponding

angles corresponding

sides exterior angle HL hypotenuse included angle included side interior angle isosceles triangle legs right triangle SAS SSS triangle

two­column proof without a bank of definitions, properties, postulates & theorems. Advanced Geometry Coordinate Proof with variables as the coordinates. Advanced Geometry Construction of a triangle congruent to a given triangle. (Page 218)

11/18­12/1

Textbook

5­1 5­2 5­5

SOL G.5 Triangles

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.

opposite angle opposite side

Advanced Geometry centroid orthocenter Advanced Geometry midsegment of triangle Advanced Geometry hinge theorem and it’s converse

These concepts will be considered in the context of real­world situations. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

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 real­world problems given information about the lengths of sides and/or measures of angles in triangles.

Blooms = Analyze

11/18­12/1 SOL G.4bd†

The student will construct and justify the constructions of:

Ь) The perpendicular bisector of a line segment;

d) A perpendicular to a given line at a given point on the line;

†) The inscribed circle of a triangle; and †) The circumscribed circle of a triangle.

Blooms = Create

circumcenter concurrent lines incenter angle bisector of

a triangle

12/2­12/22

Textbook

6­1 6­2

SOL G.9

Quadrilaterals

The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real­world problems.

SOL 6.13 Identify and describe properties of quadrilaterals.

concave convex diagonal equiangular

Language The roots of Geometric prefixes. (Page 327)

6­3 6­4

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

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

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

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

Blooms = Evaluate

SOL 7.7 Compare and contrast quadrilaterals based on properties.

opposite angles parallelogram polygons quadrilateral rectangle rhombus

Greene County Public Schools Geometry Curriculum & Pacing Guide 2015­2016 Quarter 3

January 5, 2016 – March 11, 2016

Time/Dates

SOL/Strand Objective/Content/Essential Knowledge and Skills/Cognitive Level

Vertical Alignment

Vocabulary Cross­Curricular

and Advanced Geometry Extensions

1/5 ­1/8

Textbook

6­5 6­6

SOL G.9

Quadrilaterals

The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real­world problems. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

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

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

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

Blooms = Evaluate

SOL 6.13 Identify and describe properties of quadrilaterals. SOL 7.7 Compare and contrast quadrilaterals based on properties.

base angles bases diagonals isosceles

trapezoid trapezoid

Advanced Geometry midsegment of a trapezoid

Advanced Geometry kites

1/11­2/3

Textbook

8­1 8­2 8­3 8­4 8­5 8­6

SOL G.7 Similarity

The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representation to

Use definitions, postulates, and theorems to prove triangles similar.

Use algebraic methods to prove that triangles are similar.

Use coordinate methods, such as the distance formula, to prove two triangles are similar.

Blooms = Evaluate

SOL 7.6 Determine similarity of plane figures and write proportions to express relationships between similar quadrilaterals and triangles.

extremes means proportion ratio scale factor similar statement of

proportionality

Advanced Geometry Geometric mean History & Maps Part of the Lewis and Clark Trail on which Sacagawea acted as guide is now known as the Lolo Trail. The map, which shows a portion of the trail, has a scale of 1 inch = 6.7 miles. (Textbook Page 470)

2/4­2/26

Textbook

9­1 9­2 9­3 9­4 9­5 9­6

SOL G.8 Right

Triangles

The student will solve real­world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

Determine whether a triangle formed with three given lengths is a right triangle.

SOL 8.10ab a) verify the Pythagorean Theorem; b) apply the Pythagorean Theorem SOL A.3 Express square roots and cube roots of whole numbers and the square root of monomial algebra

angle of

depression angle of

elevation converse of

Pythagorean theorm

special right triangles

trigonometric ratio

Advanced Geometry Constructing 30°­60°­90° and 45°­45°­90° triangles Soccer Using Trig to calculate scoring angles in soccer. History A Babylonian clay tablet made around 350 B.C. contains Pythagorean Triples

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

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

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

Solve real­world problems, using right triangle trigonometry and properties of right triangles.

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

Blooms = Analyze

expressions in simplest radical form SOL A.4abcd Solve multistep linear and quadratic equations in two variables – a)solve literal equations; b) justify steps used in simplifying expressions and solving equations; c) solve quadratic equations algebraically and graphically; d) solve multistep linear equations algebraically and graphically.

trigonometry written in cuneiform characters. (Textbook page 547) History Pythagorean Theorem proofs by various mathematicians over time. (Textbook page 557) Science Using trigonometry and photos of shadows to determine the depth of crators on the moon. (Textbook page 564) Forestry Using a clinometer to find the height of a tree. (Textbook page 561)

2/29­3/11

Textbook 10­1 10­2 10­3 10­4 10­5 10­6

SOL

G.9/11/12 Circles

G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real­world problems.

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

Blooms = Evaluate G.11 The student will use angles, arcs, chords, tangents, and secants to:

a) Investigate, verify, and apply properties of circles;

6.10abc a) define π; b) solve practical problems with circumference and area of a circle; c) solve practical problems involving area and perimeter when given the radius or diameter.

arc center of a circle central angle chord circle circumscribed

polygon congruent arcs inscribed angle inscribed

quadrilateral inscribed triangle

Advanced Geometry Tangent Circles Satellites and Cell Phones Tangents Pi Days March 14 & 15

b) Solve real­world problems involving properties of circles.

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

Find lengths, angle measures, and arc measures associated with

a) two intersecting chords; b) two intersecting secants; c) an intersecting secant and tangent; d) two intersecting tangents; and e) central and inscribed angles. Solve real­world problems associated

with circles, using properties of angles, lines, and arcs.

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

Blooms = Evaluate G.12 The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

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

intercepted arc major arc measure of arc minor arc point of tangency secant semicircle tangent

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

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

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

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

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

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

Blooms = Analyze

2/29/­3/11

SOL G.4†

The student will construct and justify the constructions of:

†) A tangent line from a point outside a given circle to the circle.

Blooms = Create

Greene County Public Schools Geometry Curriculum & Pacing Guide 2015­2016 Quarter 4

March 14, 2016 – May 27, 2016

Time/Dates

SOL/Strand Objective/Content/Essential Knowledge and Skills/Cognitive Level

Vertical Alignment

Vocabulary Cross­Curricular

and Advanced Geometry Extensions

3/14­4/5

Textbook 11­1

pp. 452/3 11­3 11­4 11­5

SOL

G.10/G.11/G.14

Polygons Circles

Tessellations Similarity

G.10 The student will solve real­world problems involving angles of polygons. Solve real­world problems involving

the measures of interior and exterior angles of polygons.

Identify tessellations in art, construction, and nature.

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

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

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

G.11 The student will use angles, arcs, chords, tangents, and secants to: c) Find arc lengths and areas of sectors in circles

6.10abc a) define π; b) solve practical problems with circumference and area of a circle; c) solve practical problems involving area and perimeter when given the radius or diameter.

arc length area of a sector concentric circles regular

tessellations tessellations

Advanced Geometry ∙Area of the segment of a circle ∙Area of a regular polygon ∙Area of a portion of a sector Archery and Darts Concentric Circles and Geometric Probability Pie Charts & Spinners Sectors and Geometric Probability Art and Quilting Semiregular tessellations

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

Calculate the area of a sector and the length of an arc of a circle, using proportions.

Solve real­world problems associated with circles, using properties of angles, lines, and arcs.

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

Blooms = Evaluate G.14 The student will use similar geometric figures in two­dimensions to: a) compare ratios between side lengths, perimeters and areas. Blooms = Analyze

4/6­4/22

Textbook 12­1 12­2 12­3 12­4 12­5 12­6 12­7

SOL

G.13/G.14 Solids

G.13 The student will use formulas for surface area and volume of three­dimensional objects to solve real­world problems. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

SOL 7.5abc a) describe volume and surface area of cylinders; b) solve practical problems involving volume and surface area of rectangular prisms and cylinders; c) describe how changes in

area of a base cone cylinder faces hemisphere lateral area polyhedron prism pyramid

Architecture & Design ∙Isometric Drawings ∙Orthographic Projections Advanced Geometry Platonic Solids Construction Composite Solids

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 real­world problems, involving total surface area and volume of cylinders, prisms, pyramids, cones, and spheres as well as combinations of three­dimensional figures.

Calculators may be used to find decimal approximations for results.

Blooms = Analyze G.14 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

measured attribute affects volume and surface area. SOL 8.7ab a) investigate and solve practical problems involving volume and surface area of prisms, cylinders, cones and pyramids; b) describe how changes in measured attribute affects volume and surface area. SOL 8.9 Construct a 3­dimensional model given top or bottom, side, and front view.

similar solids slant height sphere surface area volume

Surface Area and Volume & Weight or a Cinderblock (Textbook page 745) Language (Etymology) While math might not immediately seem to have a strong relationship to English plane and solid geometry provides a

very important

vocabulary lesson.

Many shape names ­­

from triangle to

quadrilateral to

pentagon ­­ provide

root words that can be

applied to other words.

Students can learn

complex words like

"trifurcated," which has

the same root word

"tri," meaning three.

Other words ­­ like the

"lateral" in quadrilateral

­­ provide other

applicable vocabulary.

For an activity,

students might first

learn about these roots

words and then, with

no previous knowledge,

be asked to guess the

definitions of words like

“bilateral” and

“periscope.”

Science Changing volume over time as solvent enters and leaves funnel at differing rates. (Textbook page 756) History Cone shaped bricks in ancient Peru. (Textbook page 756)

d) Solve real­world problems about similar geometric objects.

Blooms = Analyze

4/25­5/6

Textbook

7­1 7­2 7­3 7­4 8­7

SOL G.3

The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include:

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

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

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

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

Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, dilation, or translation.

Blooms = Apply

SOL 7.8 represent transformations (reflections, dilations, rotations, and translations) of polygons in the coordinate plane by graphing SOL 8.8ab a) apply transformations to plane figures; b) identify applications of transformations

dilation enlargement image preimage reduction reflection rotation line symmetry point symmetry transformations

Earth Science Snowflakes & Rotational Symmetry Art & Design frieze pattern Music frieze pattern of notes on a staff (Textbook page 444) Chemistry Molecular Geometry relates to polygons, polyhedra and reflections. An example of reflection is found on page 409 of the textbook.