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Geometry

Mathematics Dorchester County Public Schools 2017-2018

Course Code: 17045

Prerequisites: Algebra I

Credits: 1.0 Math

Resource: Discovery Education

Major Content Standard

Supporting Content Standard

Additional Content Standard

Unit 1

Foundations of Geometry

12 Days

Unit Window: September 6, 2017 – September 27, 2017

Assessment Window: September 28-29, 2017

Students will learn the roles of postulates, definitions, theorems, and precise geometric notation in the construction of more formal geometric proofs. They will apply the

Pythagorean theorem and the distance formula to calculate the perimeter and area of figures in the coordinate plane. They will use their knowledge of proportional reasoning to

identify the midpoint of a line segment and to partition line segments into equal portions. Students will use the definitions of a circle and a right angle to help create triangles

and solve for unknown distances. They will extend their understanding of congruency by learning to construct congruent segments and congruent angles using a variety of tools

and methods. They will also use coordinates to prove or disprove that segments on the coordinate plane are congruent or perpendicular.

Concept 1.1: Explore the Building Blocks of Geometry

Students discover the roles of postulates, definitions, and theorems and precise geometric notation in the construction of more formal geometric proofs. They begin their more

formal explorations in geometry by interpreting and using the axioms and undefined terms that lay the foundation of the discipline. Students explore important definitions and

postulates to prepare for the development of formal arguments to establish geometric facts. They explore evidence of vocabulary, postulates, and theorems in real-world

scenarios and mathematical diagrams to prepare for geometric modeling.

Standards covered Main Ideas Mathematical Practices Prior courses/Previous

instruction

HSG-CO.A.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.

HSG-CO.C.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.

HSG-MG.A.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).*

Lesson Objectives

• Interpret the axioms and undefined terms of

geometry.

• Articulate common axioms (postulates),

including the following: the intersection of

two lines is a point, the ruler postulate, the

segment addition postulate, and the angle

addition postulate.

Essential Question

• What are the fundamental parts of geometry,

and how can they be used to create valid

arguments?

Enduring Understandings

• Definitions are tools for geometric

investigation.

• Diagrams provide insight and evidence for

geometric conjectures.

• Proof relies on true statements and

assumptions.

MP1: Make sense of

problems and persevere in

solving them.

MP3: Construct viable

arguments and critique the

reasoning of others.

MP5: Use appropriate tools

strategically.

Students developed informal

arguments and explained

proofs of theorems. Students

also learned informal

definitions of the vocabulary

addressed in this concept

throughout elementary and

middle school.

Concept 1.2: Explore Measurements in the Coordinate Plane

Students apply the Pythagorean theorem and the distance formula to calculate the perimeter and area of figures in the coordinate plane. They apply their knowledge of

proportional reasoning to identify the midpoint of a line segment and to partition line segments into equal portions. Students use the definitions of a circle and a right angle to

help create triangles and solve for unknown distances.


instruction





arc.

HSG-GPE.B.6: Find the point on a directed line

segment between two given points that partitions the

segment in a given ratio.

HSG-GPE.B.7: Use coordinates to compute

perimeters of polygons and areas of triangles and

rectangles, e.g., using the distance formula.*

HSG-GPE.B.4: Use coordinates to prove simple

geometric theorems algebraically. For example, prove

or disprove that a figure defined by four given 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).

Lesson Objectives

• Apply the distance formula to calculate

perimeter and area in the coordinate plane.

• Derive formulas to partition segments

equally.

• Determine the position of a point relative to

another point or set of points.

Essential Question

• How can you use the structure of the

coordinate plane to calculate measurements

in geometric figures and justify properties of

figures algebraically?

Enduring Understanding

• The structure of the coordinate plane can be

used to calculate measures in geometric

figures and prove geometric theorems

algebraically.

MP1: Make sense of


solving them.





strategically.

Students became familiar

with the coordinate plane, its

use in modeling real-world

locations, and a method for

finding the distance between

two points with a mutual

coordinate. Students first

derived the distance formula

from the Pythagorean

theorem in Grade 8.

Unit 2

Geometric Transformations

12 Days

Unit Window: October 2, 2017 – October 25, 2017

Assessment Window: October 26 - 27, 2017

Students will recognize, describe, and construct reflections, rotations, and translations of points, segments, and polygons. They will also construct these transformations on the

coordinate plane. In addition, students will recognize line and rotational symmetry of figures and use transformation to create tessellations. They will build on their

understanding of congruence of two figures by expressing congruence in terms of rigid motion. They also will predict and demonstrate a sequence of translations, rotations, and

reflections to carry a given figure onto another.

Concept 2.1: Explore transformations

Students learn to recognize, describe, and construct reflections, rotations, and translations of points, segments, and polygons. They also construct these transformations on the

coordinate plane. In addition, students recognize line and rotational symmetry of figures and use transformation to create tessellations.


instruction

HSG-CO.A.4: Develop definitions of rotations,

reflections, and translations in terms of angles, circles,

perpendicular lines, parallel lines, and line segments.

HSG-CO.A.5: Given a geometric figure and a

rotation, reflection, 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.

HSG-CO.A.3: Given a rectangle, parallelogram,

trapezoid, or regular polygon, describe the rotations

and reflections that carry it onto itself.

HSG-CO.A.2: Represent transformations in the plane

using, e.g., 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).

HSG-SRT.A.1.a: A dilation takes a line not passing

through the center of the dilation to a parallel line, and

leaves a line passing through the center unchanged.

HSG-SRT.A.1.b: The dilation of a line segment is

longer or shorter in the ratio given by the scale factor.

Lesson Objectives

• Define, identify, and draw translations,

rotations, and reflections of points, line

segments and polygons informally and with

coordinates.

• Define, identify, and show line and rotational

symmetry, including tessellations informally

and with coordinates.

• Define, identify, and draw dilations

informally and with coordinates.

•

Essential Questions

• What distinct features identify

transformations as a family of functions?

• What effects do rigid motion transformations

have on the properties of a figure?

• How are rigid motion transformations used to

determine the type and number of

symmetries of a figure?


• Transformations are functions that take

points in the plane as inputs and give other

points as outputs.

• Reflections, rotations, and translations are

rigid motion transformations that preserve

distance and angle measure.

MP1: Make sense of


solving them.





strategically.

Students have drawn,

constructed, and described

geometric figures. In Grade

7, students used compasses,

rulers, and technology to

draw geometric shapes with

given conditions. In Grade 8,

students verified

experimentally the properties

of rotations, reflections, and

translations. Students have

examined diagrams to

identify evidence and use the

evidence and definitions to

create mathematically

precise and convincing

arguments. They have

learned the definition of a

function and used function

notation.

Concept 2.2: Investigate and Apply Congruence Definitions

Students realize that congruent figures are created by a finite number of isometries; that is, students build on their understanding of congruence of two figures by expressing

congruence in terms of rigid motion. They also predict and demonstrate a sequence of translations, rotations, and reflections to carry a given figure onto another, using rigid

motions to discover and establish the SSS, SAS, and ASA triangle congruence criteria.


instruction

HSG-CO.B.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.

Lesson Objectives

• Explain congruence of two objects in terms

of rigid motion.

• Predict and create a sequence of translations,

rotations, and reflections to carry a given

figure onto another.

• Determine whether two objects are

congruent.

• Show that two triangles are congruent.

Essential Questions

• How can rigid motion transformations be

used to show that two figures are congruent?

• What is the relationship between rigid motion

transformations and triangle congruence

criteria?


• Rigid motion transformations are functional

relationships that can be used to establish the

congruence of two figures.

MP1: Make sense of


solving them.





strategically.

Students learned to

recognize, describe, and

construct reflections,

rotations, and translations of

points, segments, and

polygons. They also

constructed these

transformations on the

coordinate plane. In addition,

students learned to recognize

line and rotational symmetry

of figures and to use

transformation to create

tessellations. Students

explored congruence in terms

of rigid motions and

described the effect of rigid

motion transformations using

coordinates in a prior

course. During the previous

unit, students learned to

examine diagrams and to

identify evidence. They used

the evidence and definitions

to create mathematically

precise and convincing

arguments. Students have

used function notation to

represent geometric

transformations and

interpreted statements that

use function notation in

terms of a context.

Unit 3

Intersecting Lines

11 Days

Unit Window: October 30, 2017 – November 17, 2017

Assessment Window: November 20 - 21, 2017

Students will investigate and construct parallel and perpendicular lines. They will examine the relationship between geometric transformations and algebraic representations of

parallel and perpendicular lines in preparation for the proof of the slope criteria. Students will extend their understanding of geometric evidence and valid arguments and apply

their understanding of rigid motion transformations to formally prove theorems about angle pairs, parallel lines, and perpendicular lines.

Concept 3.1: Explore Parallel and Perpendicular Lines

Students explore and construct parallel and perpendicular lines. They examine the relationship between geometric transformations and algebraic representations of parallel and

perpendicular lines in preparation for the proof of the slope criteria.


instruction





arc.

HSG-CO.D.12: Make formal geometric constructions

with a variety of tools and methods (compass and

straightedge, string, reflective devices, paper folding,

dynamic geometric software, etc.). Copying a

segment; copying an angle; bisecting a segment;

bisecting an angle; constructing perpendicular lines,

including the perpendicular bisector of a line

segment; and constructing a line parallel to a given

line through a point not on the line.

HSG-GPE.B.5: Prove the slope criteria for parallel

and perpendicular lines and use them to solve

geometric problem (e.g., find the equation of a line

parallel or perpendicular to a given line that passes

through a given point).

Lesson Objectives

• Construct parallel and perpendicular lines

manually and with technology.

• Construct a line through a point that is

perpendicular or parallel to a given line.

• Write equations for lines parallel and

perpendicular to a given line.

• Construct angle bisectors and perpendicular

bisectors.

Essential Question

• How can you use geometric transformations

and algebraic to determine whether two lines

are parallel or perpendicular?


• Geometric transformations can be used to

construct parallel and perpendicular lines.

MP1: Make sense of


solving them.





strategically.

Students verified


of rigid motion

transformations and


motion transformations on

two-dimensional figures

using coordinates. Students

defined and identified rigid

motion transformations of

points and line segments,

investigated transformations

of segments and angles, and

identified and described

combinations of rigid

transformations.

Concept 3.2: Prove Theorems about Lines and Angles

Students extend their understanding of geometric evidence and valid arguments and apply their understanding of rigid motion transformations to formally prove theorems about

angle pairs, parallel lines, and perpendicular lines.


instruction







HSG-GPE.B.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).

HSG-CO.C.9: Prove theorems about lines and angles.







Lesson Objectives

• Use a variety of methods to prove theorems

about angle pairs, parallel lines, and

perpendicular lines.

• Apply theorems about angle pairs, parallel

lines, and perpendicular lines.

Essential Question

• What are some important theorems about

lines and angles, and how can these theorems

be proved?


• Geometric transformations can be used to

prove theorems about parallel and

perpendicular lines.

MP1: Make sense of


solving them.





strategically.

Students verified


of rigid motion

transformations and



two-dimensional figures

using coordinates. Students

defined and identified rigid

motion transformations of

points and line segments,

investigated transformations

of segments and angles, and

identified and described

combinations of rigid

transformations. Students

explored and constructed

parallel and perpendicular

lines and examined the

relationship between

geometric transformations

and algebraic representations

of parallel and perpendicular

lines in preparation for the

proof of the slope criteria.

Unit 4

Triangle Geometry

11 Days

Unit Window: November 27, 2017 – December 18, 2017

Assessment Window: December 19 -20, 2017

(MID – TERM will cover Units 1 – 4)

Students will prove theorems about triangles including the triangle sum theorem, exterior angle theorem, and isosceles and equilateral triangles theorems. They will build on

their understanding of congruence transformations to prove triangle congruence theorems. Students will learn to construct various figures, including equilateral triangles,

perpendicular bisectors, and angle bisectors using a compass and a straight edge. They will use the converse of the perpendicular bisector theorem to justify their constructions.

Concept 4.1: Prove Congruence Theorems

Students prove theorems about triangles, including the triangle sum theorem, exterior angle theorem, and isosceles and equilateral triangles theorems. They extend their

understanding of congruence transformations to prove triangle congruence theorems.


instruction

HSG-CO.B.8: Explain how the criteria for triangle

congruence (ASA, SAS, and SSS) follow from the

definition of congruence in terms of rigid motions.

HSG-CO.C.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.

HSG-SRT.B.5: Use congruence and similarity criteria

for triangles to solve problems and to prove

relationships in geometric figures.

Lesson Objectives

• Prove theorems about triangle angle

measures.

• Use theorems about triangle congruence and

angle measures to solve problems.

Essential Question

• Which triangle congruence statements and

relationships in triangles are valid for all

triangles?


• Claims about properties of triangles and

triangle congruence must be proved to be

valid before the claims can be accepted as

theorems.

• Comparing corresponding parts of two

triangles can establish whether or not the

triangles are congruent.

MP1: Make sense of


solving them.





strategically.

Students created triangles

from three measures of

angles or sides, making

conclusions about which

conditions determined a

unique triangle, more than

one triangle, and no triangle.

Students also used informal

arguments to establish facts

about the angle sum as well

as the exterior angle of

triangles. Students used rigid

motions to transform figures

and used the definition of

congruence in terms of rigid

motions to show two

triangles congruent given

that corresponding pairs of

angles and sides were

congruent.

Concept 4.2: Prove Congruence Theorems

Students make geometric constructions. They learn to construct various figures, including equilateral triangles, perpendicular bisectors, and angle bisectors using a compass and

a straight edge. Students also use the converse of the perpendicular bisector theorem to justify their constructions.


instruction








HSG-CO.D.12: Make formal geometric constructions

with a variety of tools and methods (compass and

straightedge, string, reflective devices, paper folding,

dynamic geometric software, etc.). Copying a

segment; copying an angle; bisecting a segment;

bisecting an angle; constructing perpendicular lines,

including the perpendicular bisector of a line

segment; and constructing a line parallel to a given

line through a point not on the line.

Lesson Objectives

• Prove that a construction is valid.

• Prove theorems about triangle congruence.

Essential Question

• What properties of geometric objectives are

revealed through geometric constructions?


• Properties of geometric objects can be

analyzed and verified through geometric

constructions.

MP1: Make sense of


solving them.





strategically.

Students used rulers and

protractors to draw geometric

shapes given certain

constraints. They learned

precise definitions of circles

and explored congruence

criteria for triangles.

Students then used

congruence criteria to prove

congruency of triangles.

They also developed the

definition of midpoint in and

out of the coordinate plane.

Unit 5

Similarity

14 Days

Unit Window: January 2, 2018 – February 2, 2018 (Mid-term will be administered during this window.)

Assessment Window: February 5 - 6, 2018

Students will deepen their knowledge of dilations of figures by identifying a dilation as a nonrigid transformation. They will experimentally verify the properties of dilations as

functions. Students will establish the definition of a similarity transformation as a composition of rigid motion transformations followed by a dilation. They will use triangle

similarity criteria to help them prove theorems about triangles. Students will also use their understanding of similarity and congruence of triangles to solve real-world and

mathematical problems. They will build on what they have learned about similarity and apply it to three-dimensional figures. Students will apply properties of similar figures in

three dimensions to solve problems, including problems involving surface area and volume.

Concept 5.1: Explore Similarity and Dilation

Students broaden their understanding of dilations of figures by identifying a dilation as a non-rigid transformation. They use their previous work with transformations to

experimentally verify the properties of dilations as functions. Students establish the definition of a similarity transformation as a composition of rigid motion transformations

followed by a dilation. They use and expand their knowledge of similarity to decide if two given figures are similar, and they justify their reasoning by examining all

corresponding pairs of angles and sides.


instruction

HSG-SRT.A.1: Verify experimentally the properties

of dilations given by a center and a scale factor.

HSG-SRT.A.1.a: A dilation takes a line not passing

through the center of the dilation to a parallel line, and

leaves a line passing through the center unchanged.

HSG-SRT.A.1.b: The dilation of a line segment is

longer or shorter in the ratio given by the scale factor.

HSG-SRT.A.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 proportionality of all corresponding pairs of sides.

Lesson Objectives

• Define, identify, and draw dilations

informally. Define similarity in terms of

dilations, and explain why dilations do not

preserve congruence unless the scale factor is

1.

• Explain the relationship between similarity

and proportionality.

Essential Question

• What are similarity transformations, and how

do they affect the measurements of figures?


• A dilation transformation is a non-rigid

transformation that preserves angle measure

but changes the lengths of segments

proportionally by a scale factor of 𝑟.

• A similarity transformation is a composition

of rigid motion transformations and a dilation

transformation.





strategically.

MP7: Look for and make use

of structure.

Students investigated the

concept of scale drawings of

geometric figures and solved

problems involving actual

lengths and areas, in addition

to producing scale drawings

for a given scale factor.

Students learned about

proportional relationships, in

particular what a point with

coordinates (x,y) means in

terms of a specific situation,

allowing them to analyze

relationships when points

(0,0) and (1,r) are points on

the graph, and r represents

the unit rate. Students

translated, reflected, rotated,

and dilated two-dimensional

figures and described their

effects on figures using

coordinates. They used

informal arguments to

establish facts about the AA

criteria for similarity of

triangles. Students have

previously explored and

described rigid motion

transformations and used the

definition of congruence in

terms of rigid motions to

determine whether two

figures are congruent.

Concept 5.2: Prove Similarity Theorems

Students apply their prior knowledge of transformations and the properties of similar figures to establish triangle similarity criteria. They use triangle similarity criteria to help

them prove theorems about triangles. Students also use their understanding of similarity and congruence of triangles to solve real-world and mathematical problems as they

continue in their development to prepare and solve design problems.


instruction








HSG-SRT.A.3: Use the properties of similarity

transformations to establish the AA criterion for two

triangles to be similar.

HSG-SRT.B.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.




HSG-MG.A.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).*

Lesson Objectives

• Prove figures are similar using similarity

transformations.

• Prove theorems about similar figures.

• Explain the criteria necessary to show that

two triangles are similar.

• Prove and apply the triangle midsegment

theorem.

• Solve mathematical problems by apply

congruence and similarity criteria.

Essential Question

• How can similarity be verified and used to

prove properties of geometric figures?


• Similarity can be used to prove properties of

triangles.





strategically.


of structure.

Students analyzed

proportional relationships

between quantities and

applied proportional

relationships to solve

problems involving scale

drawings. In addition,

students were introduced to

transformations and

investigated the properties of

translations, reflections,

rotations, and dilations. They

predicted the effect of rigid


given figures and used the

definition of congruence in

terms of rigid motions to

determine whether two

figures are congruent.

Subsequently, students

applied their understanding

of congruence to prove

theorems about lines and

angles. They then used these

transformations to develop a

definition of similar figures.

This progression of work

with proportional reasoning

will help students as they

solve problems and develop

proofs involving side lengths

of similar figures.

Concept 5.3: Apply Similarity Theorems

Students build on what they have learned about similarity and apply it to three-dimensional figures. They develop an informal definition of similarity in three dimensions and

use it to determine whether three-dimensional figures are similar. Students also apply properties of similar figures in three dimensions to solve problems, including problems

involving surface area and volume.


instruction

HSG-SRT.A.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 proportionality of all corresponding pairs of sides.




Lesson Objectives

• Define and understand similarity in three

dimensions.

• Solve real-world problems by applying

congruence and similarity criteria.

Essential Question

• What relationships exist between different

measurements for similar solids?


• The effect of a dilation on the measurements

of a geometric solid can be represented as a

linear, quadratic, or cubic function.





strategically.


of structure.

Students explored

proportional relationships

and applied them to solve

problems involving scale

drawings. They also

investigated properties of

transformations, including

translations, reflections,

rotations, and dilations, and

used these transformations to

develop definitions of

congruent and similar figures

in two dimensions. In

addition, students solved

problems involving the

volume of various three-

dimensional figures. Students

have distinguished between

situations that can be

modeled with linear

functions and with

exponential functions.

Unit 6

Trigonometry

12 Days

Unit Window: February 7, 2018 – March 2, 2018

Assessment Window: March 5 – 6, 2018

Students will acquire an understanding of how the ratios of sides in similar right triangles lead to the definitions of trigonometric ratios of angles. They will also learn the

relationship between the sine and cosine of complementary angles and apply this relationship to solve problems. Students will state and prove the law of sines and the law of

cosines. They will apply the two laws to solve problems, investigating when to use the law of sines and when to use the law of cosines to solve a triangle. Real-world scenarios

will be used to develop these laws, challenging students to apply geometric methods to satisfy physical constraints and to choose a level of accuracy appropriate to limitations

on measurement.

Concept 6.1: Investigate Similar Right Triangles

Students develop an understanding of how the ratios of sides in similar right triangles lead to the definitions of trigonometric ratios of angles.


instruction

HSG-SRT.D.9: Derive the formula A = 1/2 ab sin(C)

for the area of a triangle by drawing an auxiliary line

from a vertex perpendicular to the opposite side. (+)

HSG-SRT.C.7: Explain and use the relationship

between the sine and cosine of complementary angles.

HSG-SRT.C.6: Understand that by similarity, side

ratios in right triangles are properties of the angles in

the triangle, leading to definitions of trigonometric

ratios for acute angles.

Lesson Objectives

• Use the ratios of sides and similar right

triangles to find the missing sides.

Essential Question

• How can you solve problems that involve

right triangles?


• Ratios related the sides and angles of right

triangles and can be used to solve problems

involving right triangles.

MP4: Model with

mathematics.


of structure.

MP8: Look for and express

regularity in repeated

reasoning.

Students were introduced to

the concept of dilation of a

figure by a scale factor. They

learned to define, determine,

and explain similarity

through the use of

transformations. They proved

theorems about triangles,

including the Triangle Sum

theorem and the Isosceles

Triangle theorem, and

determined similarity and

congruence of triangles and

other figures by applying the

properties of similarity

transformations, as well as

congruence and similarity

criteria. Students have also

used coordinates to compute

perimeters and areas of

polygons in the coordinate

plane.

Concept 6.2 Investigate Right Triangle Trigonometry

Students develop an understanding of how the ratios of sides in similar right triangles lead to the definitions of trigonometric ratios of angles, and they use these ratios with the

Pythagorean theorem to solve real-world problems. They also learn the relationship between the sine and cosine of complementary angles and apply this relationship to solve

problems.


instruction

HSG-SRT.C.7: Explain and use the relationship

between the sine and cosine of complementary angles.

HSG-SRT.C.6: Understand that by similarity, side

ratios in right triangles are properties of the angles in

the triangle, leading to definitions of trigonometric

ratios for acute angles.

HSG-SRT-C.8: Use trigonometric ratios and the

Pythagorean Theorem to solve right triangles in

applied problems.

Lesson Objectives

• Understand that angle measure determines

ratio of side length in all right triangles.

• Define sine, cosine, and tangent ratios for

acute angles.

• Use trigonometry to calculate the area of

triangles.

• Apply trigonometric ratios and the

Pythagorean Theorem to solve real-world

problems.

Essential Question

• How can you solve problems that involve

right triangles?


• Trigonometric ratios related the sides and

angles of right triangles and can be used to

solve problems involving right triangles.

MP4: Model with

mathematics.


of structure.



reasoning.

Students were introduced to

the concept of dilation of a

figure by a scale factor. They

learned to define, determine,

and explain similarity

through the use of

transformations. They proved

theorems about triangles,

including the Triangle Sum

theorem and the Isosceles

Triangle theorem, and

determined similarity and

congruence of triangles and

other figures by applying the

properties of similarity

transformations, as well as


criteria. Students have also

used coordinates to compute


polygons in the coordinate

plane.

Unit 7

The Geometry of Circles

15 Days

Unit Window: March 7, 2018 – April 9, 2018

Assessment Window: April 10 - 12, 2018

Students will investigate relationships among central angles, arc measure, inscribed angles, and circumscribed angles. They will identify and describe relationships between

inscribed angles, radii, and chords. Students also will prove that all circles are similar. Students will work on a limit argument that proves the formulas for circumference and

area of a circle. They will also develop formulas for arc length and area of a sector and define the radian measure of an angle. Students will derive the equation of the circle in

the coordinate plane. Given the equation of a circle, they will be able to complete the square to determine the center and radius of the circle and graph the circle in the coordinate

plane. Students will apply their understanding of the equation of a circle to determine whether a given point lies on the circle.

Concept 7.1: Investigate Circles and Parts of Circles

Students explore relationships between central angles, arc measure, inscribed angles, and circumscribed angles and identify and describe relationships between inscribed angles,

radii, and chords. In addition, they construct tangents from a point outside the circle. Students also prove that all circles are similar.


instruction

HSG-C.A.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.

Lesson Objectives

• Define and describe circles.

• Classify lines and segments as chords,

secants, tangents, radii, and diameters.

• Discover relationships between inscribed,

circumscribed, and central angles, radii, and

chords, and show that they are true for any

circle.

Essential Question

• How are the parts of a circle related to the

circle and to each other?


• Within circles, segments, lines, angles, and

arcs create special relationships.





strategically.



reasoning.

Students learned the

formulas for area and

circumference of a circle and

used them to solve problems.

They used informal

reasoning to establish the

relationship between the

circumference and area of a

circle. In addition, students

explored and verified dilation

properties and used the

definition of similarity

transformations to determine

whether or not two polygons

were similar. Students have

proved theorems, including

vertical angles theorem,

triangle sum theorem, and

exterior angle theorem for

triangles. They have made

formal constructions

including constructing the

perpendicular bisector of an

angle. Additionally, students

have applied the distance

formula and the Pythagorean

theorem in a variety of

settings.

Concept 7.2: Investigate and Apply Area and Circumference Formulas

Students develop a limit argument that proves the formulas for circumference and area of a circle and use the formulas to solve real-world problems involving area and

circumference. They also develop formulas for arc length and area of a sector and define the radian measure of an angle.


instruction

HSG-GMD.A.1: Given 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.

HSG-C.B.5: Derive using similarity the fact that the

length of the arc intercepted by an angle is

proportional to the radius, and define the radian

measure of the angle as the constant of

proportionality; derive the formula for the area of a

sector.

Lesson Objectives

• Develop and informally prove the formulas

for area and circumference of a circle.

• Solve problems involving area and

circumference.

• Use proportion to develop the formulas for

arc length and area of a sector.

• Solve problems involving arc length and area

of a sector.

Essential Question

• How do measurements of a circle related to

one another?


• Circle formulas can be derived using

similarity transformations and proportional

reasoning.





strategically.



reasoning.

Students informally derived

the formula for

circumference and area of a

circle. In geometry, they

formally defined the terms

rays, angles, circles, and line

segments. Students have

made formal constructions

with a variety of tools and

methods including paper

folding and compass and

straightedge.

Concept 7.3: Investigate and Interpret Circle Equations

Students derive the equation of the circle in the coordinate plane. Given the equation of a circle, they are able to complete the square to determine the center and radius of the

circle and to graph the circle in the coordinate plane. Students apply their understanding of the equation of a circle to determine whether a given point lies on the circle.


instruction

HSG-GPE.A.1: Derive the equation of a circle of

given center and radius using the Pythagorean

Theorem; complete the square to find the center and

radius of a circle given by an equation.

HSG-C.A.1: Prove that all circle are similar.







Lesson Objectives

• Derive the equation of a circle in a coordinate

plane.

• Prove that all circles are similar.

• Find the equation of a circle that goes

through particular coordinates.

• Identify the center, radius, and points on the

circle using an equation.

Essential Question

• How can you represent a circle in the

coordinate plane algebraically?


• A circle in the coordinate plane can be

modeled algebraically if its center and radius

are known.





strategically.



reasoning.

Students used the

Pythagorean theorem to

determine unknown lengths

in real-world situations. They

applied the Pythagorean

theorem to find the distance

between two points in the

coordinate plane. Students

wrote quadratic functions in

different but equivalent

forms by completing the

square to reveal and explain

different properties of the

function.

Unit 8 (Old Unit 10)

Three Dimensional Figures

9 Days

Unit Window: April 13, 2018 – April 30, 2018

Assessment Window: May 1 – 2, 2018

Students will expand upon their understanding of cross sections to include two-dimensional slices of real-world three-dimensional objects and will identify three-dimensional

objects generated by rotation of two-dimensional objects. They will use transformations of two-dimensional geometric shapes to describe and model real-world three-

dimensional objects. Students will explore the cross sections and characteristics of figures through informal arguments for volume of cylinders, spheres, and other solid figures.

They will use geometric shapes to describe and model real-world objects and apply their formulas for volume and surface area to solve problems.

Concept (8.1)10.1: Investigate Cross Sections and Rotations

Students extend their understanding of cross sections to include two-dimensional slices of real-world three-dimensional objects and identify three-dimensional objects generated

by rotation of two-dimensional objects. They use transformations of two-dimensional geometric shapes to describe and model real-world three-dimensional objects. Students

apply their understanding of cross sections to solve design problems.


instruction

HSG-GMD.B.4: Identify the shapes of two-

dimensional cross-sections of three-dimensional

objects, and identify three-dimensional objects

generated by rotations of two-dimensional objects.








ratios).*

Lesson Objectives

• Visualize and model cross-sections and

rotations of three-dimensional figures.

• Visualize and describe three-dimensional

objects created by rotations of two-

dimensional figures.

• Visualize and represent spheres, cylinders,

cones, pyramids and prisms as an

accumulations of cross-sections of two-

dimensional 2-D figures.

• Approximate real-world objectives as

combinations of geometric shapes to make

scale models and explore properties.

Essential Question

• How can you visualize and represent three-

dimensional objects?


• Geometry and spatial sense offer ways to

visualize, to interpret, and to reflect on our

physical environment.

• Visualization, de-composition, and re-

composition of three-dimensional objects

help us explain, describe, and measure

structures in our world.

MP2: Reason abstractly and

quantitatively.

MP4: Model with

mathematics.


strategically.

Students solved problems,

including scale drawing and

reproducing scale drawings.

They explored two-

dimensional slices of right

rectangular prisms and right

rectangular pyramids and

solved problems involving

area, volume and surface

area. Students have

represented rigid motion

transformations in the plane

and predicted the effect of a

given rigid motion on a given

shape. They have explored

transformations in the

coordinate plane and used

coordinates to prove simple

geometric theorems.

8.2 (10.2) Develop and Apply Volume Formulas

Students explore the cross sections and characteristics of figures through informal arguments for volume of cylinders, spheres, and other solid figures. They use geometric

shapes to describe and model real-world objects and apply their formulas for volume and surface area to solve problems.


instruction

HSG-GMD.A.1: Give an informal argument for the





HSG-GMD.A.3: Use volume formulas for cylinders,

pyramids, cones, and spheres to solve problems.*

Lesson Objectives

• Develop volume formulas using informal

arguments about sums of areas.

• Solve problems involving the area and

volume of spheres, cones, cylinders, and

pyramids.

• Use 3-D figures to model and solve real

world problems involving area and volume.

Essential Question

• How can decomposition and recomposition

of three-dimensional objects be used to

develop and explain volume formulas?


• Volume involves the sum of areas.

• Visualization, decomposition, and

recomposition of three-dimensional objects

help us explain, describe, and measures



quantitatively.

MP4: Model with

mathematics.


strategically.

Students used nets consisting

of rectangles and triangles to

represent three-dimensional

figures to determine surface

area. They applied their

knowledge of formulas for

volume and surface area in

prisms, cones, cylinders, and

spheres to solve real-world

and mathematical problems.

They also calculated the

volume of complex figures

by adding volumes. Students

described and identified the

shapes of two-dimensional

cross sections of three-

dimensional objects.

Unit 9 (Old Unit 8)

Triangles and Circles

4 Days

Unit Window: May 3, 2018 – May 9, 2018

Assessment Window – May 10 – 11, 2018

Students will discover points of concurrency of a triangle and prove that the medians of a triangle meet at a point of concurrency. They will use points of concurrency to

construct circumscribed and inscribed triangles.

Concept 9.1 (8.1): Investigate Concurrency in Triangle

Students explore points of concurrency of a triangle and prove that the medians of a triangle meet at a point of concurrency. They use points of concurrency to construct

circumscribed and inscribed triangles.


instruction

HSG-C.A.3: Construct the inscribed and

circumscribed circles of a triangle, and prove

properties of angles for a quadrilateral inscribed in a

circle.








Lesson Objectives

• Identify and describe the relationship

between the centroid and the medians of a

triangle, the orthocenter and the altitudes of a

triangle, the vertices of a triangle and the

circumcenter, and the sides of a triangle and

the incenter.

• Understand the construction of inscribed and

circumscribed circles of triangles, and

understand their relationship to the points of

concurrency.

Essential Question

• What are the points of concurrency of a

triangle, and how are they used to solve

problems involving triangles?


• There are different centers of a triangle, and

the lines, segments, or rays used to find these

centers are always concurrent.

• The different centers of a triangle can be used

to find the center of gravity of a triangle and

to circumscribe or inscribe a triangle.

MP1: Make sense of


solving them.





of structure.

Students made formal

constructions using a variety

of tools and methods of

triangles, angles, angle

bisectors, midpoints, and

perpendicular lines and

bisectors. They proved

theorems about lines and

angles and applied the

distance formula. Students

established the AA criteria

for two triangles to be similar

and used the similarity

criteria to prove relationships

in geometric figures.

Unit 10 (Old Unit 9)

Quadrilaterals and Other Polygons

10 Days

Unit Window: May 14, 2018 – May 31, 2018

Assessment Window: June 1 – 4, 2018

Students will demonstrate how to use constructions to inscribe a polygon in a circle and explore and prove properties of angles in inscribed polygons. They will prove theorems

about parallelograms. Students will extend properties of parallelograms to other quadrilaterals and develop a classification of quadrilaterals. They will use coordinates to prove

geometric theorems involving quadrilaterals algebraically.

Concept 10.1 (9.1): Construct and Explore Polygons

Students understand how to use constructions to inscribe a polygon in a circle and explore and prove properties of angles in inscribed polygons.


instruction

HSG-CO.D.13: Construct an equilateral triangle, a

square, and a regular hexagon inscribed in a circle.

HSG-C.A.3: Construct the inscribed and

circumscribed circles of a triangle, and prove

properties of angles for a quadrilateral inscribed in a

circle.

Lesson Objectives

• Construct an equilateral triangle, a square,

and a regular hexagon in a circle.

• Discover relationships between angles of a

quadrilateral inscribed in a circle.

• Verify constructions both manually and with

technology tools.

• Calculate the area of regular polygons.

Essential Question

• How are geometric constructions used to

inscribe a figure in a circle, and what

properties of geometric figures are evidenced

in the construction?


• Constructions are based on the properties of

geometric figures.




MP4: Model with

mathematics.


of structure.

Students drew and explored

geometric shapes with given

conditions. They explored

and solved problems

involving vertical,

supplementary,

complementary, and adjacent

angles. They learned precise

definitions and developed an

understanding of congruence

in terms of rigid motions.

Students stated and proved

theorems involving angles.

They made formal

constructions with a variety

of tools and methods,

including copying angles,

constructing perpendicular

bisectors, bisecting a

segment, and bisecting an

angle.

Concept 10.2 (9.2): Prove and Apply Theorems about Quadrilaterals

Students prove theorems about parallelograms. They extend properties of parallelograms to other quadrilaterals and develop a classification of quadrilaterals. Students use

coordinates to prove geometric theorems involving quadrilaterals algebraically.


instruction

HSG-CO.C.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.







Lesson Objectives

• Prove theorems about parallelograms,

rectangles, kites, and trapezoids.

• Apply theorems about quadrilaterals to solve

real-world and mathematical problems.

Essential Question

• How can relationships among the sides,

angles, and diagonals of a quadrilateral be

used to analyze, describe, and classify

quadrilaterals?


• Quadrilaterals can be analyzed, described,

and classified by their attributes.




MP4: Model with

mathematics.


of structure.

Students proved theorems

about lines and angles,

including theorems related to

angles formed when a

transversal crosses parallel

lines and the slope criteria.

Students have used the

distance formula to compute


polygons, including triangles

and rectangles. They also

proved and applied the

criteria for triangle

congruence. Students applied


criteria to solve problems

involving triangles.

Concept 10.1: Investigate Cross Sections and Rotations

Students extend their understanding of cross sections to include two-dimensional slices of real-world three-dimensional objects and identify three-dimensional objects generated

by rotation of two-dimensional objects. They use transformations of two-dimensional geometric shapes to describe and model real-world three-dimensional objects. Students

apply their understanding of cross sections to solve design problems.


instruction

HSG-GMD.B.4: Identify the shapes of two-

dimensional cross-sections of three-dimensional

objects, and identify three-dimensional objects

generated by rotations of two-dimensional objects.








ratios).*

Lesson Objectives

• Visualize and model cross-sections and

rotations of three-dimensional figures.

• Visualize and describe three-dimensional

objects created by rotations of two-

dimensional figures.

• Visualize and represent spheres, cylinders,

cones, pyramids and prisms as an

accumulations of cross-sections of two-

dimensional 2-D figures.

• Approximate real-world objectives as

combinations of geometric shapes to make

scale models and explore properties.

Essential Question

• How can you visualize and represent three-

dimensional objects?


• Geometry and spatial sense offer ways to

visualize, to interpret, and to reflect on our

physical environment.

• Visualization, de-composition, and re-

composition of three-dimensional objects

help us explain, describe, and measure



quantitatively.

MP4: Model with

mathematics.


strategically.

Students solved problems,

including scale drawing and

reproducing scale drawings.

They explored two-

dimensional slices of right

rectangular prisms and right

rectangular pyramids and

solved problems involving

area, volume and surface

area. Students have

represented rigid motion

transformations in the plane

and predicted the effect of a

given rigid motion on a given

shape. They have explored

transformations in the

coordinate plane and used

coordinates to prove simple

geometric theorems.

10.2 Develop and Apply Volume Formulas

Students explore the cross sections and characteristics of figures through informal arguments for volume of cylinders, spheres, and other solid figures. They use geometric

shapes to describe and model real-world objects and apply their formulas for volume and surface area to solve problems.


instruction

HSG-GMD.A.1: Give an informal argument for the





HSG-GMD.A.3: Use volume formulas for cylinders,

pyramids, cones, and spheres to solve problems.*

Lesson Objectives

• Develop volume formulas using informal

arguments about sums of areas.

• Solve problems involving the area and

volume of spheres, cones, cylinders, and

pyramids.

• Use 3-D figures to model and solve real

world problems involving area and volume.

Essential Question

• How can decomposition and recomposition

of three-dimensional objects be used to

develop and explain volume formulas?


• Volume involves the sum of areas.

• Visualization, decomposition, and

recomposition of three-dimensional objects

help us explain, describe, and measures



quantitatively.

MP4: Model with

mathematics.


strategically.

Students used nets consisting

of rectangles and triangles to

represent three-dimensional

figures to determine surface

area. They applied their

knowledge of formulas for

volume and surface area in

prisms, cones, cylinders, and

spheres to solve real-world

and mathematical problems.

They also calculated the

volume of complex figures

by adding volumes. Students

described and identified the

shapes of two-dimensional

cross sections of three-

dimensional objects.

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