Download Geometry Honors - Belvidere School District

Curriculum Guide
for
Geometry Honors
Course Number 220
Department of Mathematics
Belvidere High School
Department Philosophy
The principle reason for studying mathematics is to learn how to apply
mathematical thought to problem solving. Students are expected to develop
facility in performing the fundamental operations that are associated with
each course and to acquire a set of meaningful concepts that they can use
effectively to solve problems. The ultimate goal of the department is to
provide students with a facility to deal with mathematical functions in daily
life and for post-secondary education.
Course Overview
Geometry is a logical system of thought and proof that deals with
visualization and also links algebraic concepts to geometric figures. This
course extends understanding of theory and practice through analysis,
reasoning, deduction, the development of formal proof, and problem solving.
The goal is to help students appreciate the geometrical value of the universe
as we know it while examining the relationships in geometry and applying it
to other areas of math. This course has been designed to provide an in depth
understanding of Euclidean Geometry.
Course Description & Course Proficiency
COURSE TITLE: Geometry Honors
COURSE LENGTH: 36 weeks
COURSE # 220
PERIODS PER WK:
5
TOTAL CREDIT: 5
GRADE LEVEL: 9 10
PREREQUISITE(S): Algebra 1, teacher recommendation, and successful application for Honors Program.
Pursuant to the High School Graduation Act (NJSA 18A: 7, et. Seq.), expectations for this course of study are
outlined below
OVERVIEW:
Geometry is a logical system of thought and proof that deals with visualization and also
links algebraic concepts to geometric figures. This course extends students' understanding
of theory and practice through analysis, informal reasoning, deduction, the development of
a formal proof, problem solving and enrichment assignments.
TEXTBOOK: Geometry, Glencoe 2004
Online at:
www.geometryonline.com
User Name: geo
Password: v7tr2swagu
SUPPLEMENTARY MATERIAL:
Teacher prepared skills review and practice materials.
PROFICIENCIES: Successful completion of this course will require the student:
Learn the language of geometry.
2.
Organize the ideas of geometry into a logical system of thought and reasoning.
3.
Learn the meaning of geometric proof by deductive reasoning for the following:
a. triangles
b. parallel and perpendicular lines
c. polygons other than triangles
d. circles, angles and arcs, e. similar polygons.
4.
Apply the concepts and properties of geometric figures and integrate with the coordinate plane.
5.
Learn how to solve problems involving:
a. areas of polygons
b. measurement of angles and arcs
c. proportion and proportional line segments
d. similarity.
6.
Understand the Pythagorean Theorem, special right triangles, trigonometric ratios.
7.
Learn to construct various geometric figures.
8.
Area of plane figures.
9.
Apply coordinate geometry.
10.
Surface area and volume of solids.
11.
Complete daily homework assignments.
STUDENT ACHIEVEMENT:
STUDY STRATEGIES:
Students should review materials given in class on a daily basis to reinforce skills and concepts
presented. It is expected for students to review Algebra skills throughout course.
HOMEWORK EXPECTATIONS:
Each homework assignment will be worth up to 5 points. Late work will not be given any
credit except in the case of an extended absence. Students must make arrangements with
teacher immediately upon return to school in order to receive credit. Parents may be notified
by phone or email when student does not have assignment.
PROCEDURES FOR MAKING UP WORK:
The student is responsible for any make-up work missed from being out of class. This includes
any homework, class work, tests/quizzes, and updating of notebooks. Any work that is not made up
following above procedures will result in a grade of zero. Board policy applies .
MAJOR PROJECTS TO EXPECT:
Triangle Project 2nd marking period. Plane or Solid Figure Project 4th marking period.
MEASUREMENTS OF STUDENT ACHIEVEMENT:
The measurement of student achievement will be done through various evaluative criteria, which
will include:
1. Quizzes – Each quiz will count as 20-50 points. There will be at least one quiz given per chapter/unit. These quizzes will be announced. Except in the case of an extended absence, if a
student is absent on the day a quiz is given, the quiz must be taken on the day he returns.
2. Mini Quizzes – Each mini quiz will count as 1-20 points. Mini quizzes may or may not be announced. They will occur periodically to check immediate understanding of a skill or topic.
3. Tests – Each test will count as 100 points. Tests will be given at the end of each chapter/unit
and will be announced at least three (3) days in advance. There will be a review prior to each
test. Except in the case of an extended absence, if a student is absent on the day a test is given, the test must be taken on the day he returns.
4. Mid-term & Final Examinations- A mid-term will be given midway through the school year
and a final at the end of the year. These two grades will be worth 20% of the student’s final
grade.
PURPOSE AND METHODS OF ASSESSMENT:
Tests and quizzes are given to accurately assess each student’s skill level and understanding of
concepts.
CAREER OBJECTIVE:
To help students appreciate the geometrical value of the universe as we know it. To help
students see interrelationships between geometry and other areas of mathematics, particularly: arithmetic, algebra, trigonometry and coordinate geometry.
ADDITIONAL NOTES:
1.
Regular attendance at school is required of all students by the laws of the State of
New Jersey. Failure to attend on a regular basis may result in poor achievement
and/or loss of credit as per Board of Education Policy and as stated in the Student
Handbook.
2.
This list must be returned, signed by parent or guardian, no later than the last day in
September.
Student Signature
Parent/Guardian Signature
Teacher Signature
Parent/Guardian Email _______________________________________
Updated 05/07/2010
Course Content
Weekly Curriculum Map: Geometry Honors
Duration Content
New Jersey
of Unit
Common Core
Assessment
Resources
Standards
G-CO: 1, 2, 4, 5, 12
A-SSE: 1
A-CED: 4
G-GPE: 7
class discussion,
homework, quiz,
test
New Jersey Geometry Model
Course Content- Draft 11/6/08
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
3 wks
Point, Lines,
Planes, and
Angles
3 wks
Reasoning and
Proof
A-REI: 1
G-CO: 9, 10
A-CED: 1
class discussion,
homework, quiz,
test
3 wks
Parallel and
Perpendicular
Lines
G-CO: 1, 9, 12
G-GPE: 5
class discussion,
homework, quiz,
test
4 wks
Triangle
Congruence
G-CO: 6,7,8,10
G-SRT: 5
G-MG: 3
G-GPE: 4,5,7
class discussion,
homework, quiz,
test
2 wks
Properties and
Attributes of
Triangles
G-CO: 9, 10, 12
G-SRT: 4, 6, 8
G-C: 3
G-MG: 2,3
class discussion,
homework, quiz,
test
3 wks
Quadrilaterals
G-CO: 11
G-GPE: 5
G-MG: 3
G-SRT: 5
class discussion,
homework, quiz,
test
1 wk
Review / Testing
3wks
Similarity
Mid-term Exam
G-SRT:2
G-MG: 3
G-C: 1
G: SRT: 1, 3, 4, 5
G-CO: 2
class discussion,
homework, quiz,
test
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
G-GPE: 6
3 wks
Right Triangles
and
Trigonometry
G-SRT: 4, 6, 7, 8,
10, 11
class discussion,
homework, quiz,
test
1 wk
Transformations
G-CO: 3, 4
G-SRT 1
class discussion,
homework, quiz,
test
3wks
Areas of
Polygons and
Circles
class discussion,
homework, quiz,
test
2 wks
Surface Area
and Volume
A-SSE: 1
A-CED: 4
G-GPE:7
G-GMD: 1
G-MG: 3
G-SRT:9
S-CP:1
G-GMD: 1,2,3,4
G-MG: 1,2
4 wks
Circles
G-C:2,3,4,5
G-CO: 13
class discussion,
homework, quiz,
test
1 wk
Review/Testing
class discussion,
homework, quiz,
test
Final Exams
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Text: Geometry: Holt McDougal
Supplemental:
Geometry: Glencoe
Discovering Geometry An
Inductive Approach: Key
Curriculum Press
Unit I. Points, Lines, Planes, and Angles
State Standard Area of Concentration:
Mathematics/High School-Geometry
Unit Summary: Identify points, lines, planes, angles, and their relationships. Measure segments, angles, and
polygons using a variety of methods. Introduce the terms and symbols of geometry. Make conjectures about
vertical angles, linear pairs of angles, midpoints, and distance. Explore these properties in the coordinate plane.
Define the term polygon and many specific types of polygons. Practice using two tools of geometry, the compass
and the straightedge.
Unit Rationale: The three building blocks of geometry are points, lines, and planes. These terms remain
undefined. A general description is used to give a sense of what is meant by point, line, and plane. Gaining an
intuitive understanding of the meaning of these terms is essential to begin a study of geometry. The language of
geometry is used to describe many real-world objects. Lines and angles are found all around us in nature.
Knowledge of the terms in this unit will help students begin to appreciate the geometrical value of the universe as
we know it.
Standards:
Congruence G-CO
Seeing Structure in Expressions A-SSE
Creating Equations A-CED
Expressing Geometric Properties with Equations G-GPE
Common
Core #:
Text:
Common Core:
G-CO 1
1-1, 1-2, 1-3, Know precise definitions of angle, circle, perpendicular line, parallel line, and line
1-4,
segment, based on the undefined notions of point, line, distance along a line, and
distance around a circular arc.
G-CO 12
1-3
Make formal geometric constructions with a variety of tools and methods.(compass
and straightedge, string, reflective devices, paper folding, dynamic geometry software,
etc.) Copying a segment; copying an angle, bisecting a segment; bisecting an angle,
and constructing perpendicular lines.
A-SSE 1
1-5
Interpret expressions that represent a quantity in terms of its context.
A-CED 4
1-5
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations.
G-GPE 7
1-6
Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
G-CO 4
1-7
Develop definitions of rotations, reflections, and translations in terms of angles,
circles, perpendicular lines, parallel lines, and line segments.
G-CO 2
1-7
Represent transformations in the plane using e.g., transparencies and geometry
software
G-CO 5
1-7
Given a geometric figure and rotation, reflection, or translations, draw the transformed
figure…Specify a sequence of transformations that will carry a given figure onto
another.
Essential Questions:
How can geometric/algebraic relationships best be
represented?
What is the importance of definitions?
What is the necessity for undefined terms?
What applications led to the development of coordinate
Enduring Understanding:
Definitions will improve mathematical reasoning by
being able to illustrate and examine geometric concepts.
Euclidean geometry begins with the three undefined
terms of point, line, and plane.
The coordinate system helps locate objects on a plane.
systems?
Identifying locations on a map or globe using latitude and
longitude in order to travel compares to coordinates in a
plane.
Instructional Focus:.
Identify, name, and draw points, lines, segments, rays, Review and apply the following Algebra 1 topics:
and planes.
 Simplify algebraic expressions
Apply basic facts about points, lines, and planes.
 Solve linear equations
Identify collinear and coplanar points and intersecting
 Simplify absolute value expressions
lines and planes in space.
 Operations with signed numbers
Use length and midpoint of a segment.
 Translate word problems into algebraic
Construct midpoints and congruent segments.
equations
Name and classify angles.
Measure and construct angles and angle bisectors.
Identify adjacent, vertical, complementary, and
supplementary angles.
Find measures of pairs of angles.
Apply formulas for perimeter, area, and circumference.
Develop and apply the formula for midpoint.
Use the Distance Formula and the Pythagorean
Theorem to find the distance between two points.
Identify reflections, rotations, and translations.
Graph transformations in the coordinate plane.
Summative Assessment
Multiple-choice test
Free-Response test
Performance Assessment
Cumulative Test
“ExamView”
Evidence of Learning
Formative Assessment:
Study Guide and Intervention Worksheets
Skills Practice and Practice Worksheets
Portfolio or Journal
Definition and Conjecture List
Quizzes
Geometry Activities:
a. Modeling Intersecting Planes
b. Midpoint of a Segment
c. Modeling the Pythagorean Theorem
d. Angle Relationships
Mini-Project Intersecting Planes
Pythagorean Puzzle
Project: Finding Treasure with Coordinates
Open-Ended Investigation: Vertical Angles Conjecture
and the Linear Pair Conjecture
Equipment needed:
Text Chapter 1
Compass
Straightedge
Protractor
Landscape of Geometry Video (Program 1 The Shape of Things & Program 3 Lines that Cross)
Unit 2. Reasoning and Proof
State Standard Area of Concentration:
Mathematics/High School-Geometry
Mathematics/High School-Algebra
Unit Summary: Use reasoning to make conjectures and prove conjectures. If a conjecture is false find
counterexamples. If a conjecture is true, verify using informal and formal proofs. Determine the truth values
of compound statements and construct truth tables. Analyze conditional statements and write related
conditionals. Introduce terms postulates and theorems. Algebraic properties of equality are applied to
geometry. Write formal and informal proofs proving segment and angle relationships.
Unit Rationale: Thinking logically is an important skill for daily living. Exploring different methods of
reasoning helps us to think through situations logically. The ideas of geometry are organized into a logical
system of thought and reasoning. Geometry uses and applies these methods to solve problems. Many different
professions rely on these reasoning skills. For example, doctors use reasoning to diagnose and treat patients
and meteorologists use patterns to make predictions. Topics in this unit will help develop clean thinking and
the ability to weigh an argument critically and impartially.
Standards:
Reasoning with Equations and Inequalities A-REI
Congruence G-CO
Seeing Structure in Expressions A-SSE
Creating Equations A-CED
Common
Core:
Text:
Cumulative Progress Indicators:
A-REI 1
2-5
Explain each step in solving a simple equation as following from the equality of
numbers asserted at the previous step, starting from the assumption that the original
equation has a solution. Construct a viable argument to justify a solution method.
G-CO 9
2-6, 2-7
Prove theorems about lines and angles.
G-CO 10
2-7
Prove theorems about triangles.
A-CED 1
2-8
Create equations in one variable and use them to solve problems.
Essential Questions:
How can mathematical reasoning help you make
generalizations?
How do you know when you have proved something?
Enduring Understanding:
Reasoning allows us to make conjectures and prove
conjectures.
Many professions rely on reasoning and logic to solve
problems and reach valid conclusions.
Instructional Focus:
Use inductive reasoning to identify patterns and make Review the following Algebra 1 topics:
conjectures.
 Identify algebraic properties of equality
Find counterexamples to disprove conjectures.
 Solve linear equations; with focus on fractions
Identify, write, and analyze the truth value of
 Apply algebraic properties
conditional statements.
Write the inverse, converse, and contrapositive of a
conditional statement.
Apply the Law of Detachment and the Law of
Syllogism in logical reasoning.
Write and analyze biconditional statements.
Review properties of equality and use them to write
algebraic proofs.
Identify properties of equality and congruence.
Write two-column proofs.
Prove geometric theorems by using deductive
reasoning.
Write flowchart and paragraph proofs.
Analyze the truth value of conjunctions and
disjunctions.
Construct truth tables to determine the truth value of
logical statements.
Evidence of Learning
Summative Assessment
Formative Assessment:
Multiple-choice test
Portfolio or Journal
Free-Response test
Definition and Conjecture List
Performance Assessment
Quizzes
Cumulative Test
Geometry Activities:
“ExamView”
a. If-then statements
b. Matrix Logic
c. Right Angles
Mini-project: Traveling Networks
Project: Euler’s Formula for Networks
The Daffynition Game
Cooperative Problem Solving: Lunar Survival; Patterns
at the Lunar Colony
Three-Peg Puzzle
Group Activity: Number of Handshakes
Envelope Proofs
Equipment needed:
Text Chapter 2
Patty Paper
Protractor
Scissors
Rulers
Unit 3. Parallel and Perpendicular Lines
State Standard Area of Concentration:
Mathematics/High School-Geometry
Mathematics/High School-Algebra
Unit Summary: Examine the special angle relationships that occur when lines are cut by transversals.
Examine special properties that occur when those lines are parallel. Connect algebra to properties of lines.
Review solving systems of equations, slope, and writing the equation of a line from Algebra 1. Use slope to
determine whether two lines are parallel, perpendicular, or intersecting. Solve problems by writing linear
equations.
Unit Rationale: Many buildings are designed using basic shapes, lines, and planes. Examples of parallel,
perpendicular, and skew lines can be seen in real world structures. Carpenters, designers, and construction
managers must know the relationships of angles created by parallel lines and their transversals. When
examining the relationship between algebra and the geometric topics in this unit, students will see the great
impact geometry has had on some of the world’s greatest architectural achievements.
Standards:
Congruence G-CO
Expressing Geometric Properties with Equations G-GPE
Common
Core:
Text:
Cumulative Progress Indicators:
G-CO 1
3-1
Know precise definitions of angles, 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.
G-CO 9
3-2, 3-3, 3-4 Prove theorems about lines and angles. (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)
G-CO 12
3-3,3-4
Make formal geometric constructions with a variety of tools and methods (compass
and straightedge, paper folding, dynamic geometric software). 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.
G-GPE 5
3-5, 3-6
Prove the slope criteria for parallel and perpendicular lines and use them to solve
geometric problems (eg find the equation of a line parallel or perpendicular to a
given line that passes through a given point).
Essential Questions:
Enduring Understanding:
How can the relationship of angles formed by parallel There are many applications of the properties of
or perpendicular lines be used in real-world situations? parallel and perpendicular lines in the real-world. For
example, using a plumb bob or carpenter’s square,
What are some applications of linear equations?
making perspective drawings, and strings on a piano or
guitar.
The graph of a line can be used to describe the speed
(rate of change) when traveling.
There are many real-world uses for slope including
grade of a road, pitch of a roof, and incline of a ramp.
Instructional Focus:
Identify parallel, perpendicular, and skew lines.
Identify the angles formed by two lines and a
transversal.
Review and apply the following Algebra 1 topics:
 Graph linear equations
 Calculate slope
Prove and use theorems about the angles formed by
parallel lines and a transversal.
Use the angles formed by a transversal to prove two
lines are parallel.
Prove and apply theorems about perpendicular lines.
Find the slope of a line.
Use slopes to identify parallel and perpendicular lines.
Graph lines and write their equations in slope-intercept
form and point-slope form.
Classify lines as parallel, intersecting, or coinciding.
Summative Assessment
Multiple-choice test
Free-Response test
Performance Assessment
Cumulative Test
“ExamView”
Equipment needed:
Text Chapter 3
Geometer’s Sketchpad Software
Graphing Calculator
Compass
Straightedge
Protractor
Pick up sticks
Graphing Lines Game Board
Number cubes



Write equations of lines
Solve systems of equations
Translate word problems into equations
Formative Assessment:
Study Guide and Intervention Worksheets
Skills Practice and Practice Worksheets
Portfolio or Journal
Definition and Conjecture List
Quizzes
Geometry Activities:
a. Draw a Rectangular Prism
b. Graphing Lines in the Coordinate Plane
c. Non-Euclidean Geometry
Constructions: Parallel & Perpendicular Lines
Mini-Project: Avoiding Lattice Points
Open-Ended Investigation: Parallel and Perpendicular
Slope Conjecture
Project: Making a Clinometer
Cooperative Problem Solving: Geometrivia
Geometer’s Sketchpad: Parallel Lines Conjecture and
its converse
Graphing Calculator Investigation: Line Designs; Intersections of Lines
Unit 4. Triangle Congruence
State Standard Area of Concentration:
Mathematics/High School-Geometry
Unit Summary: Measure the lengths of the sides and angles of a triangle so that it can be classified and
compared to other triangles. Apply the Triangle Sum Theorem and Exterior Angle Theorem. Test for triangle
congruence and prove triangles are congruent by writing informal and formal proofs. Identify special
properties of isosceles and equilateral triangles.
Unit Rationale: Triangles can be found in nature, art, construction, and other areas of life. Triangles
have properties that make them useful in structures such as buildings and bridges. Congruent
triangles are used by surveyors by determining if one triangle can be mapped into another by means of
a sequence of rigid transformations. This unit will allow students to have a better understanding of the
purpose and importance of triangles in real-life.
Standards:
Congruence G-CO
Similarity, Right Triangles, and Trigonometry G-SRT
Modeling with Geometry G-MG
Expressing Geometric Properties with Equations G-GPE
Common
Core:
Text:
Cumulative Progress Indicators:
G-CO 6
4-1
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.
G-CO 7
4-1, 4-5, 4-6 Use the definition of 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.
G-CO 8
4-5, 4-6
G-CO 10
4-2, 4-3. 4-9 Prove theorems about triangles. Theorems include: measures of interior angles of
a triangle sum to 180, base angles of isosceles triangles are congruent.
G-SRT 5
4-4, 4-5, 4-6, Use congruence and similarity criteria for triangles to solve problems and prove
4-7
relationships in geometric figures.
G-MG 3
4-7, 4-8
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).
G-GPE 5
4-7
Prove the slope criteria for parallel and perpendicular lines and use them to solve
geometric problems(e.g. find the equation of line parallel or perpendicular to a
given line that passes through a given point).
G-GPE 4
4-8
Use coordinates to prove simple geometric theorems algebraically.
G-GPE 7
4-8
Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from
the definition of congruence in terms of rigid motion.
Essential Questions:
What makes shapes alike and different?
How can the study of triangles help solve problems in
architecture?
Enduring Understanding:
Congruent shapes are also similar, but similar shapes
may not be congruent.
Properties of triangles are used to design and build
bridges, towers, and other structures.
Builders of ancient Egypt used a tool called a plumb
level in building architectural wonders such as the
Egyptian pyramids using basic properties of isosceles
triangles to determine if a surface is level.
Instructional Focus:
Draw, identify, and describe transformations in the
Review and apply the following Algebra 1 topics:
coordinate plane.
 Solve linear equations
Use properties of rigid motions to determine whether
 Translate word problems into equations
figures are congruent and to prove figures congruent.
 Solve systems of linear equations in two
Classify triangles by their angle measures and side
variables by substitution
lengths.
Use triangle classification to find angle measures and
side lengths.
Find the measures of interior and exterior angles of
triangles.
Apply theorems about the interior and exterior angles
of triangles.
Use properties of congruent triangles.
Prove triangles congruent by using the definition of
congruence.
Apply SSS and SAS to construct triangles and solve
problems.
Prove triangles congruent by using SSS and SAS.
Apply ASA, AAS, and HL to construct triangles and
solve problems.
Prove triangles congruent by using ASA, AAS, and
HL.
Use CPCTC to prove parts of triangles are congruent.
Position figures in the coordinate plane for use in
coordinate proofs.(Optional)
Prove geometric concepts by using coordinate
proof.(Optional)
Prove theorems about isosceles and equilateral
triangles.
Apply properties of isosceles and equilateral triangles.
Summative Assessment
Multiple-choice test
Free-Response test
Performance Assessment
Cumulative Test
“ExamView”
Evidence of Learning
Formative Assessment:
Study Guide and Intervention Worksheets
Skills Practice and Practice Worksheets
Portfolio or Journal
Definition and Conjecture List
Quizzes
Geometry Activities:
a. Equilateral Triangles
b. Angles of Triangles
c. Congruent Triangles
d. Congruence in Right Triangles
e. Isosceles Triangles
Mini-Project: Perimeters and Unknown Values; Folding
Triangles
Graphing Calculator Investigation: Lines and Isosceles
Triangles
Open-Ended Investigation: Triangle Exterior Angle
Conjecture; Congruent Triangle Conjectures; Vertex
Angle Bisector Conjecture
The Big Triangle Puzzles
Equipment needed:
Text Chapter 4
Graph paper
Patty paper
Unit 5. Properties and Attributes of Triangles
State Standard Area of Concentration:
Mathematics/High School-Geometry
Mathematics/High School-Algebra
Unit Summary:
Investigate and apply the following theorems about triangles: Perpendicular Bisector Theorem, Angle Bisector
Theorem, Circumcenter Theorem, Centroid Theorem, and Triangle Midsegment Theorem. Define equidistant,
concurrency, and locus. Review from algebra 1 solving inequalities and compound inequalities. Explore
triangle inequalities. Review the Pythagorean Theorem and how to use it to find an unknown side length in a
right triangle. Review how to simplify square roots and radicals. Discuss Pythagorean triples and how to
identify them by using the converse of the Pythagorean theorem. Then explain how to use the Pythagorean
Inequalities Theorem to classify triangles by their angle measures. Examine the relationships between the side
lengths of a 45-45-90 and a 30-60-90 triangle by using the Pythagorean Theorem. Use these relationships to
find unknown side lengths of special right triangles.
Unit Rationale:
The triangle inequality theorem can be used to find the shortest distance when traveling. Because the
triangle is rigid it is used to add strength to structures. But the triangle is also used in mechanisms that
move by changing the length of one side. Some examples of mechanisms that take advantage of the
rigidity of the triangle while allowing one side to vary in length are dump trucks, reclining deck
chairs, and car jacks.
Standards:
Congruence G-CO
Similarity, Right Triangles, and Trigonometry G-SRT
Circles G-C
Modeling with Geometry G-MG
Common
Core:
Text:
Cumulative Progress Indicators:
G-CO 9
5-1
Prove geometric theorems about lines and angles.
G-SRT 4
5-1, 5-7
Prove theorems about triangles.
G-C 3
5-2
Construct the inscribed and circumscribed circles of a triangle and prove properties
of angles for a quadrilateral inscribed in a circle.
G-CO 12
5-2, 5-3
Make formal geometric constructions with a variety of tools and methods(compass
and straight edge, string, reflective devices, paper folding, dynamic geometry
software, etc).
G-MG 2
5-2
Apply concepts of density based on area and volume in modeling situations…
G-CO 10
5-3, 5-4,5-5, Prove theorems about triangles.
5-6
G-MG 3
5-3
Apply geometric methods to solve design problems…
G-SRT 8
5-7
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
G-SRT 6
5-8
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.
Essential Questions:
Is there a relationship between the size of angles and
the lengths of the sides of a triangle?
Why are triangles so useful in structures?
Enduring Understanding:
The triangle is used in mechanisms that move by
changing the length of one side. The lengths of the
sides of a triangle are in relationship to the size of
angles.
Because the triangle is rigid it is used to add strength
to structures.
Instructional Focus:
Prove and apply theorems about perpendicular
Review and apply the following Algebra 1 topics:
bisectors.
 Solve linear equations
Prove and apply theorems about angle bisectors.
 Solve inequalities
Prove and apply properties of perpendicular bisectors
 Simplify radical expressions
of a triangle.
 Rationalize the denominator
Prove and apply properties of angle bisectors of a
 Multiply and divide radicals
triangle.
 Add and subtract radicals
Apply properties of medians of a triangle.
Apply properties of altitudes of a triangle.
Prove and use properties of triangle midsegments.
Write indirect proofs.
Apply inequalities in one triangle.
Apply inequalities in two triangles.
Use the Pythagorean Theorem and its converse to solve
problems.
Use Pythagorean inequalities to classify triangles.
Justify and apply properties of 45-45-90 triangles.
Justify and apply properties of 30-60-90 triangles.
Summative Assessment
Multiple-choice test
Free-Response test
Performance Assessment
Cumulative Test
“ExamView”
Equipment needed:
Text Chapter 5
Graph paper
Linguine
Formative Assessment:
Study Guide and Intervention Worksheets
Portfolio or Journal
Definition and Conjecture List
Geometry activity: Inequalities for Sides and Angles of
Triangles
Project: Triangles at Work; Napoleon’s Theorem
Unit 6 Quadrilaterals
State Standard Area of Concentration:
Mathematics/High School-Geometry
Mathematics/High School-Algebra
Unit Summary:
Investigate the interior and exterior angles of polygons. Apply the properties of regular polygons to solve realworld problems. Several different geometric shapes are examples of quadrilaterals. These shapes each have
individual characteristics. Recognize and apply the properties of parallelograms. Recognize and apply the
properties of parallelograms extending to rectangles, rhombi, and squares. Explore properties of quadrilaterals
that do not have both pairs of opposite side parallel, such as trapezoids and kites.
Unit Rationale:
Polygons are seen in nature, in everyday objects, in art, and in architecture. Examining some of these objects
while looking for patterns and developing conjectures about polygons can be an important opportunity for
critical thinking. Quadrilaterals played an important part in the history of the parallel postulate. Properties of
parallelograms can be examined when using a pantograph to copy a drawing or using a parallel rule to plot a
course on a navigation chart. Because of many characteristics of special quadrilaterals, discoveries have been
made that helped civilizations advance such as the role of trapezoids in arches, mechanisms that use
quadrilateral linkages, and squaring the frame of a window or frame of a house.
Standards:
Congruence G-CO
Expressing Geometric Properties with Equations G-GPE
Modeling with Geometry G-MG
Similarity, Right Triangles, and Trigonometry G-SRT
Common
Core:
Text:
Cumulative Progress Indicators:
G-CO 11
6-1, 6-2, 6-3, Prove theorems about parallelograms. Theorems include: opposite sides are
6-4, 6-5
congruent, opposite angles are congruent, the diagonals of a parallelogram bisect
each other, and conversely, rectangles are parallelograms with congruent diagonals.
G-GPE 5
6-2, 6-3
Use coordinates to prove simple geometric theorems algebraically. For example:
prove or disprove that a figure define by four given points in the coordinate plane
is a rectangle.
G-MG 3
6-3
Apply geometric methods to solve design problems…
G- SRT 5
6-5
Use congruence and similarity criteria for triangles to solve problems and prove
relationships in geometric figures.
Essential Questions:
How can knowledge of polygons be used in solving
real-world situations?
How can coordinates be used to describe and analyze
geometric objects?
Enduring Understanding:
The properties of quadrilaterals are essential to success
in engineering, architecture, or design.
An object’s location on a plane or in space can be
described quantitatively. The position of any point on
a surface can be specified by tow numbers.
Computations with these numbers allow us to describe
and measure geometric objects.
Instructional Focus:
Classify polygons based on their sides and angles.
Find and use the measures of interior and exterior
angles of polygons.
Prove and apply properties of parallelograms.
Review and apply the following Algebra 1 topics:
 Use Laws of Exponents to simplify
 Factor polynomials
 Multiply polynomials
Use properties of parallelograms to solve problems.
 Multiply and divide rational expressions
Prove that a given quadrilateral is a parallelogram.
 Solve quadratic equations by factoring
Prove and apply properties of rectangles, rhombuses,
and squares.
Use properties of rectangles, rhombuses, and squares
to solve problems.
Prove that a given quadrilateral is a rectangle,
rhombus, or square.
Use properties of kites to solve problems.
Use properties of trapezoids to solve problems.
Evidence of Learning
Summative Assessment
Formative Assessment:
Multiple-choice test
Study Guide and Intervention Worksheets
Free-Response test
Skills Practice and Practice Worksheets
Performance Assessment
Portfolio or Journal
Cumulative Test
Definition and Conjecture List
“ExamView”
Quizzes
Geometry Activities:
a. Sum of the Exterior Angles of a Polygon
b. Properties of Parallelograms
c. Testing for a Parallelogram
d. Kites
e. Construct Median of a Trapezoid
f. Linear Equations
Mini-Project: Quadrilateral Sort; Square Search
Constructions: Rectangle and Rhombus
Graphing Calculator Investigation: Drawing Regular
Polygons
Geometer’s Sketchpad Project: Star Polygons
Project: Quadrilateral Linkages; Building an Arch
Cooperative Problem Solving: The Geometry Scavenger
Hunt
The Big Quadrilateral Puzzles
Equipment needed:
Text Chapter 6
Compass
Straightedge
Protractor
Pipe cleaners
Straws
Patty Paper
Geoboard/Geobands
Unit 7 Similarity
State Standard Area of Concentration:
Mathematics/High School-Geometry
Mathematics/High School-Algebra
Unit Summary:
Extend knowledge of ratios and proportions to similar figures. Solve problems using cross products of
proportions. Justify conditions for triangle similarity. Apply similarity to solve real-world problems.
Unit Rationale: Similar polygons and their properties can be used to model and analyze many real-
world situations. Similar figures are used to read maps accurately, work with blueprints, make
movies, or even adapt recipes. Solving similarity proportions is a useful skill for working in the fields
of chemistry, physics, and medicine.
Standards:
Similarity, Right Triangles, and Trigonometry G-SRT
Modeling with Geometry G-MG
Circles G-C
Expressing Geometric Properties with Equations G-GPE
Congruence G-CO
Common
Core:
Text:
Cumulative Progress Indicators:
G-SRT 2
7-1, 7-3, 7-4 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.
G- MG 3
7-1
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).
G- C 1
7-2
Prove that all circles are similar.
G SRT 1
7-2, 7-6
Verify experimentally the properties of dilations given by a center and a scale
factor:
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 enter unchanged,
b) The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
G-SRT 3
7-3
Use the properties of similarity transformations to establish the AA criterion for
two triangles to be similar.
G-SRT 4
7-3, 7-4
Prove theorems about triangles. Theorems include: a line parallel to one side of a
triangle divides the other two proportionally, and conversely
G-SRT 5
7-3, 7-4, 7-5 Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G- CO 2
7-6
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., translations versus horizontal stretch).
G-GPE 6
7-6
Find the point on a directed line segment between two given points that partitions
the segment in a given ratio.
Essential Questions:
How are similarity and congruence related?
How is similarity important in industry?
Enduring Understanding:
Congruent shapes are also similar, but similar shapes
may not be congruent.
Artists use similarity and proportionality to give
paintings an illusion of depth.
Engineers us similar triangles when designing
buildings.
The mathematics of similarity and perspective are key
to making realistic movie images.
Instructional Focus:
Identify similar polygons.
Review and apply the following Algebra 1 topics:
Apply properties of similar polygons to solve
 Solve ratio and proportion problems
problems.
 Solve linear equations
Draw and describe similarity transformations in the
 Translate word problems into equations
coordinate plane.
Use properties of similarity transformations to
determine whether polygons are similar and to prove
circles are similar.
Prove certain triangles are similar by using AA, SSS,
and SAS.
Use triangle similarity to solve problems.
Use properties of similar triangles to find segment
lengths.
Apply proportionality and triangle angle bisector
theorems.
Use ratios to make indirect measurements.
Use scale drawings to solve problems.
Apply similarity properties in the coordinate plane.
Use coordinate proof to prove figures similar.
Divide a directed line segment into partitions.
Evidence of Learning
Summative Assessment
Formative Assessment:
Multiple-choice test
Study Guide and Intervention Worksheets
Free-Response test
Skills Practice and Practice Worksheets
Performance Assessment
Portfolio or Journal
Cumulative Test
Definition and Conjecture List
“ExamView”
Quizzes
Geometry Activities:
a. Similar Triangles
b. Sierpinski’s Triangle
Mini-Project: Measuring Height
Project: Making a Mural; The Shadow Knows; Why
Elephants Have Big Ears
Cooperative Problem Solving: Similarity in Space
Open-Ended Investigation: Similarity in an Eclipse;
Scale Factor; Fractals
Equipment needed:
Text Chapter 7
Video: M! Project Math: Similarity
Protractor, Ruler Compass & Straightedge
Isometric dot paper
Unit 8 Right Triangles and Trigonometry
State Standard Area of Concentration:
Mathematics/High School-Geometry
Mathematics/High School-Algebra
Unit Summary: Solve problems using the similarity relationships of right triangles. Apply trigonometric
ratios to real-world situations.
Unit Rationale:
Trigonometry was developed by astronomers who wished to map the stars. Since measurements of stars and
planets are inherently hard to make by direct methods, indirect measurement paved the way for study in this
field. Properties and applications of right triangle trigonometry are used to calculate these distances that are
difficult or impossible to measure directly as seen also in architecture, engineering, navigation, and surveying.
Standards:
Similarity, Right Triangles, and Trigonometry G-SRT
Common
Core:
Text:
G-SRT 4
Cumulative Progress Indicators:
Prove Theorems about triangles. Theorems include: the Pythagorean Theorem
proved using triangle similarity.
G-SRT 6
8-1, 8-2
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.
G-SRT 7
8-2
Explain and use the relationship between the sine and cosine of complementary
angles.
G-SRT 8
8-3, 8-4
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
G-SRT 10
(optional)
8-5
Prove the Laws of Sines and Cosines and use them to solve problems.
G-SRT 11
(optional)
8-5
Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and non-right triangles.
Essential Questions:
What is the historical impact of knowledge of right
triangles and its properties?
How can you find distance between two objects when
you cannot use direct measurement?
Enduring Understanding:
The knowledge of right triangles and its properties
allows us to lay the foundations of buildings
accurately. The “rope stretchers” of Egypt used the
properties of the 3-4-5 triangle to build the pyramids.
Measures of distances can be found indirectly by
writing and solving proportions and using
trigonometry.
Instructional Focus:
Use geometric mean to find segment lengths in right
Review and apply the following Algebra 1 topics:
triangles.
 Solve linear equations
Apply similarity relationships in right triangles to solve
 Translate word problems into equations
problems.
 Simplify radicals and apply Laws of Radicals
Find the sine, cosine, and tangent of an acute angle.
 Solve equations with radicals
Use trigonometric ratios to find side lengths in right
 Solve 2nd degree equations
triangles and to solve real-world problems.
Use the relationship between the sine and cosine of
complementary angles.
Use trigonometric ratios to find angle measures in right
triangles and to solve real-world problems.
Solve problems involving angles of elevation and
depression.
Use Law of Sines & Law of Cosines to solve
triangles.(optional)
Summative Assessment
Multiple-choice test
Free-Response test
Performance Assessment
Cumulative Test
“ExamView”
Evidence of Learning
Formative Assessment:
Study Guide and Intervention Worksheets
Skills Practice and Practice Worksheets
Portfolio or Journal
Definition and Conjecture List
Quizzes
Geometry Activities:
a. The Pythagorean Theorem
b. Trigonometric Ratios
c. Trigonometry
Mini-Project: The Pythagorean Theorem
Project: Creating a Geometry Flip Book; Indirect Measurement
Geometry Software Investigation: Right Triangles
Formed by the Altitude, A Pythagorean Fractal, The
Ambiguous Case of the Law of Sines
Graphing Calculator Investigation: The Height Reached
by a Ladder
Cooperative Problem Solving: Pythagoras in Space
Open-Ended Investigation: Pythagorean Proposition,
Converse of the Pythagorean Theorem, 30-60-90 Right
Triangle Conjecture, Pythagorean Triples
Equipment needed:
Text Chapter 8
Video: M! Project Math The Theorem of Pythagoras
Patty paper
Rulers
Scissors
Unit 9. Transformations
State Standard Area of Concentration:
Mathematics/High School-Geometry
Unit Summary:
Explore different types of transformations: reflections, translations, rotations, and dilations. Identify, draw, and
recognize figures that have been transformed.
Unit Rationale:
Geometry is not only the study of figures; it is also the study of the movement of figures. If you move all the
points of a geometric figure according to set rules, you can create a new geometric figure. A transformation
that preserves size and shape is called an isometry. Three types of isometries are translation, rotation, and
reflection. Symmetry is an integral part of nature and of the arts and crafts of cultures worldwide. Symmetry
can be found in art, architecture, crafts, poetry, music, dance, chemistry, biology, and mathematics. These
geometric procedures and characteristics make objects more visually pleasing. Tessellations can be created
when applying the principles of symmetry and isometries.
Standards:
Congruence G-CO
Similarity, Right Triangles, and Trigonometry G-SRT
Common
Core:
Text:
Cumulative Progress Indicators:
G-CO 3
9-5
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
G-CO 4
9-1, 9-2, 9-3, Given a geometric figure and a rotation, refection, or translation, draw the
9-4
transformed figure using, eg. Graph paper, tracing paper, or geometry software.
Specify a sequence of transformations that will carry a given figure onto another.
G-SRT 1
9-7
Verify experimentally the properties of dilations given by a center and a scale
factor:
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.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
Essential Questions:
How can transformations be described
mathematically?
What are the types of transformations seen in realworld situations?
Instructional Focus:
Identify and draw reflections.
Identify and draw translations.
Identify and draw rotations.
Apply theorems about isometries.
Identify and draw compositions of transformations,
such as glide reflections.
Identify and describe symmetry in geometric figures.
Understand how solids can be produced by rotating a
two dimensional figure through space.
Enduring Understanding:
Transformations can be represented and verified using
coordinate geometry.
Reflections, translations, rotations, and dilations can
all be seen in real-world situations.
Use transformations to draw tessellations.(Optional)
Identify regular and semiregular tessellations and
figures that will tessellate.(Optional)
Identify and draw dilations.
Evidence of Learning
Summative Assessment
Formative Assessment:
Multiple-choice test
Study Guide and Intervention Worksheets
Free-Response test
Skills Practice and Practice Worksheets
Performance Assessment
Portfolio or Journal
Cumulative Test
Definition and Conjecture List
“ExamView”
Quizzes
Geometry Activities:
a. Transformations
b. Reflections and Translations
c. Tessellations of Regular Polygons
Mini-Project: Graphing and Translations; Geomirror
Graphing Calculator Investigation: Transforming Triangles
Cooperative Problem Solving: Poolroom Math; Miniature Golf Math
Geometer’s Sketchpad Project: Tessellating with the
Conway Criterion
Constructions: Reflections in a Line
Equipment needed:
Text Chapter 9
Patty Paper
Dot paper
Straightedge
Coordinate Grids
Geomirror
Pattern Blocks
Video: Landscape of Geometry Program 8: The Range of Change
Unit 10 Areas of Polygons and Circles
State Standard Area of Concentration:
Mathematics/High School-Geometry
Unit Summary:
Find the areas of parallelograms, rhombi, trapezoids, and triangles. Identify the apothem of a regular polygon
and use that measure to find the areas of regular polygons. Find the areas of irregular figures, circles, sectors,
and segments of circles. Develop and apply area formulas for circles, polygons, and composite figures.
Unit Rationale:
People in many occupations work with areas. Carpenters calculate areas to order materials for construction,
painters calculate the area of surfaces to be painted, decorators need to know areas when installing materials in
homes, and gardeners may use area to find the maximum area given perimeter. The area of a figure is
measured by the number of squares of a unit length that can be arranged to completely fill that area. The
fundamental idea in developing area formulas is the Area Addition Postulate: the area of a region is equal to
the sum of the areas of the region’s nonoverlapping parts. Historically, methods of measuring the area of
people’s property was needed in order for governments to tax land. The Babylonians and Egyptians developed
some of the earliest mathematics, partly to keep track of land and finances.
Standards:
Expressing Geometric Properties with Equations G-GPE
Geometric Measurement and Dimension G-GMD
Modeling with Geometry G-MG
Congruence G-GO
Circles G-C
Common
Core:
Text:
Cumulative Progress Indicators:
A-SSE 1
10-1
Interpret expressions that represent a quantity in terms of its context.
A-CED 4
10-1
Rearrange formulas to highlight a quantity of interest, using the same reasoning as
in solving equations.
G-GPE 7
10-4, 10-5
Use coordinates to compute areas of triangles and rectangles.
G-GMD 1
10-2
Give an informal argument for the formula for area of a circle.
G-MG 3
10-3
Apply geometric methods to solve design problems.
G-SRT 9
10-3
Derive the formula A=1/2 absin(C) for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular to the opposite side.
S-CP 1
10-6
Describe events as subsets of sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or”, “and”, “not”
Essential Questions:
What is the advantage of deriving formulas for area?
How can area formulas be used in real-world
situations?
Enduring Understanding:
Deriving formulas for area helps strengthen
understanding of spatial relationships.
Calculating area and perimeter can be seen in real life
situations such as finding the amount of grass seed to
needed to cover various shaped regions, finding the
area of a banner, or finding area of figures in
construction.
Instructional Focus:
Develop and apply the formulas for the area of
Review and apply the following Algebra 1 topics:
triangles and special quadrilaterals.
 Solve linear and 2nd degree equations
Solve problems involving perimeters and areas of
 Translate word problems into equations
triangles and special quadrilateral.
 Use formulas
Use the Area Addition Postulate to find the area of
 Solve and simplify radicals
composite figures.
 Solve ratio and proportion problems
Use composite figures to estimate the areas of irregular
shapes.
Find the perimeters and areas of figures in a coordinate
plane.
Describe the effect on perimeter and area when one or
more dimensions of a figure are changed.
Apply the relationship between perimeter and area in
problem solving.
Calculate geometric probabilities.
Evidence of Learning
Summative Assessment
Formative Assessment:
Multiple-choice test
Study Guide and Intervention Worksheets
Free-Response test
Skills Practice and Practice Worksheets
Performance Assessment
Portfolio or Journal
Cumulative Test
Definition and Conjecture List
“ExamView”
Quizzes
Geometry Activities:
a. Area of a Parallelogram
b. Area of a Triangle
c. Area of a Trapezoid
d. Area of a Circle
e. Area of a Regular Polygon
Constructions: Regular Polygon
Mini-Project: Areas of Circular Regions
Open-Ended Investigation: Maximizing Area, Area vs
Perimeter
Project: Quilt Making
Cooperative Problem Solving: Discovering New Area
Formulas
Geometer’s Sketchpad: Calculating Area in Ancient
Egypt
Graphing Calculator Investigation: Maximizing Area
Equipment needed:
Text Chapter 10
Grid Paper
Straightedge
Scissors
Tape
Compass
Unit 11. Surface Area and Volume
State Standard Area of Concentration:
Mathematics/High School-Geometry
Unit Summary:
Students represent 3D figures using nets. Create representations of three-dimensional figures. The basics types
of geometric figures are described and their characteristics are discussed. Find the Lateral Area, Surface Area,
and Volume of prisms, cylinders, pyramids, and cones. Identify the parts of a sphere and find its surface area
and volume. Apply formulas for volume to real-world figures.
Unit Rationale:
Solids have three dimensions: length, width, and height. Solids occur in nature such as viruses, oranges,
crystals, the earth itself, or man-made objects such as books, buildings, baseballs, soup cans, or even ice cream
cones. At the molecular level, 3D geometry plays a very important role in a number of common substances
such as carbon or even water. Many solid objects have shapes that can be easily described using common
geometric terms. The volume of a figure is the measure of the amount of space that a figure encloses. Volume
is measured in cubic units. Solids can be created from different views of the figure to investigate its volume.
The formula for the volume of a rectangular prism(V=Bh) is the starting point for developing the volume
formulas for other three-dimensional figures. Another important part of developing volume formulas is
Cavalieri’s Principle, which says that two 3D figures with the same height and same cross-sectional area at
every level have the same volume.
Standards:
Geometric Measurement and Dimension G-GMD
Modeling with Geometry G-MG
Common
Core:
Text:
Cumulative Progress Indicators:
G-GMD 1
11-2, 11-3
Give an informal argument for the formulas for volume of a cylinder, pyramid, and
cone.
G-GMD 2
11-2
Give an informal argument using Cavalieri’s principle for the formulas for the
volume of a sphere and other solid figures.
G-GMD 3
11-2, 11-3,
11-4
Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
G-GMD 4
11-1
Identify the shapes of two-dimensional cross-sections of three-dimensional objects,
and identify three dimensional objects generated by rotations of two dimensional
objects.
G-MG 1
11-2
Use geometric shapes, their measures, and their properties to describe objects…
G-MG 2
11-2
Apply concepts of density based on area and volume in modeling situations.
G-MG 3
11-1
Apply geometric methods to solve design problems.
Essential Questions:
How are 1, 2, and 3 dimensional shapes related?
How can 3-dimensional objects be represented in 2dimensions?
How is volume important in a real world context?
What type of shapes should be used to increase or
decrease the surface area for a given volume?
Enduring Understanding:
Shape can be seen from different perspectives.
Nets, perspective drawings, and projections are ways
of representing 3D figures in two dimensions.
It is important to know what the capacity of a given
figure is.
A cube is the rectangular prism with the minimum
surface area for a given volume, and with the
maximum volume for a given surface area. The
cylinder with the minimum surface area for a
given volume, and with the maximum volume for
a given surface area will be the one in which the
height is equal to its diameter.
Instructional Focus:
Classify three-dimensional figures according to their
Review and apply the following Algebra 1 topics:
properties.
 Solve linear equations
Use nets and cross sections to analyze three Use formulas
dimensional figures.
 Simplify ratios
Learn and apply the formula for the volume of a prism.
Learn and apply the formula for the volume of a
cylinder.
Learn and apply the formula for the volume of a
pyramid.
Learn and apply the formula for the volume of a cone.
Learn and apply the formula for the volume of a
sphere.
Learn and apply the formula for the surface area of a
sphere.
Evidence of Learning
Summative Assessment
Formative Assessment:
Multiple-choice test
Study Guide and Intervention Worksheets
Free-Response test
Skills Practice and Practice Worksheets
Performance Assessment
Portfolio or Journal
Cumulative Test
Definition and Conjecture List
“ExamView”
Quizzes
Geometry Activities:
a. Surface Area of Cylinders and Cones
b. Surface Area of a Sphere
c. Locus and Spheres
d. Volume of a Rectangular Prism
e. Investigating the Volume of a Pyramid
Constructions: Intersection of Loci, Constructing the
Platonic Solids
Mini-Project: Cone Patterns; Volume of Cylinders
Open-Ended Investigation: Euler’s Formula for Solids
Project: The Five Platonic Solids; The World’s Largest
Pyramid
Cooperative Problem Solving: Once Upon A Time
Graphing Calculator Investigation: A Maximum Volume Box
Equipment needed:
Text Chapter 11
Isometric dot paper
Straightedge
Compass
Polystyrene ball
Scissors, Tape, Glue
Centimeter cubes
Card stock
Ruler
Rice
Unit 12 Circle
State Standard Area of Concentration:
Mathematics/High School-Geometry
Mathematics/High School-Algebra
Unit Summary:
Identify parts of a circle. Examine the special relationship of angles, arcs, and segments intersecting a circle.
Find arc and angle measures and the measures of segments in a circle. Solve problems involving
circumference. Explore special properties of circles.
Unit Rationale:
Circles are geometric shapes common to everyday life such as wheels on a vehicle. Circular wheels gave rise
to circular gears, which helped bring on the industrial revolution. Potter’s wheels, clocks, and windmills which
are all based on applications of wheels, represent great advances in civilization. Archaeologists use properties
of circles to study our past when finding artifacts that are circular in design. The Tangent Conjecture can be
seen in many applications related to circular motion such as satellites and circular orbits. Circle theorems are
used in the study of the laws of motion, exploring cells in biology, creating images in art and marketing,
calculating distances to the horizon, and many other areas of real life.
Standards:
Circles G-C
Geometric Measurement and Dimension G-GMD
Common
Core:
Text:
Cumulative Progress Indicators:
G-C 2
12-1, 12-2,
12-4, 12-5,
12-6
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 angels, the radius of a circle is
perpendicular to the tangent where the radius intersects the circle.
G-C 3
12-4
Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.
G-C 4
12-1
Construct a tangent line from a point outside a given circle to the circle.
G-C 5
12-3
Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius.
G-CO 13
12-4
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a
circle.
Essential Questions:
What geometric relationships can be found in circles?
How can these relationships help describe real-world
situations?
Enduring Understanding:
Geometric relationships in circles involve angles, arcs,
and segments.
Some real-world examples are determining the length
of the line of sight on the earth’s surface, identifying
the coverage of an overlapping circular pattern of
irrigation, and finding the area or arc length of a slice
of pie.
Instructional Focus:.
Identify tangents, secants, and chords.
Use properties of tangents to solve problems.
Apply properties of arcs.
Apply properties of chords.
Find the area of sectors.
Review and apply the following Algebra 1 topics:
 Solve linear equations
 Translate word problems into equations
 Solve 2nd degree equations
Find arc lengths.
 Solve inequalities
Find the measure of an inscribed angle.
 Solve radical equations
Use inscribed angles and their properties to solve
 Simplify radical expressions
problems.
 Simplify ratios
Find the measures of angles formed by lines that
intersect circles.
Use angle measures to solve problems.
Find the lengths of segments formed by lines that
intersect circles.
Use the length of segments in circles to solve
problems.
Write equations and graph circles in the coordinate
plane.(Optional)
Evidence of Learning
Summative Assessment
Formative Assessment:
Multiple-choice test
Study Guide and Intervention Worksheets
Free-Response test
Skills Practice and Practice Worksheets
Performance Assessment
Portfolio or Journal
Cumulative Test
Definition and Conjecture List
“ExamView”
Quizzes
Geometry Activities:
a. Circumference Ratio
b. Congruent Chords and Distance
c. Measure of Inscribed Angles
d. Inscribed Angles
e. Inscribed and Circumscribed Triangles
f. Pi Day
Constructions: Construct a Circle to Inscribe a Triangle;
Construct a Tangent; Inscribe a Circle in a Triangle
Mini-Project: Locating the Center of a Circle
Open-Ended Investigation: Elton Notle’s ceramic plate;
Locate crater’s center; Tangent Conjectures
Project: Racetrack Geometry
Cooperative Problem Solving: Designing a Theater for
Galileo
Geometer’s Sketchpad Project: Turning Wheels
Graphing Calculator Investigation: Graphing Circles and
Tangents
Equipment needed:
Text Chapter 12
Video: M! Project Math: The Story of Pi