Download Geometry A Syllabus - Spokane Public Schools

Geometry A Syllabus
School Year: Summer School 2015
Certificated Teacher: Luke Leifer
Desired Results
Course Title: Geometry A
Credit: __x__ one semester (.5)
Prerequisites and/or recommended preparation: Completion of Algebra 1
Estimate of hours per week engaged in learning activities:
8 hours of class work per week for 6 weeks of summer school
Instructional Materials:
All learning activity resources and folders are contained within the student online
course. Online course is accessed via login and password assigned by student’s school
(web account) or emailed directly to student upon enrollment, with the login website.
Other resources required/Resource Costs:
This course requires a MathXL for Schools account which will be provided by your course
instructor. Holt McDougal Geometry 2011 – online videos, examples, and activities.
Course Description:
Students will explore the basic concepts and methods of Euclidean Geometry while deepening their
understanding about plane and solid geometry. Course topics include reasoning and proof, line and
angle relationships, two and three dimensional figures, coordinate plane geometry, geometric
transformations, surface area and volume. Core processes include reasoning, problem solving and
communication. The Algebra End of Course (EOC) assessment given during the spring in this class is a
graduation requirement.
Enduring Understandings for Course (Performance Objectives):
We can use a variety of mathematical tools to describe our world and help solve daily problems.
We can use logic to reason out solutions to problems and prove or disprove conjectures about the world
around us.
Course Learning Goals (including WA State Standards, Common Core Standards, National Standards):
Geometry Unit 1 CCSS Standards
Unit 1: Geometric Transformations
In high school, students formalize much of the geometric exploration from middle school. In this unit, students
develop rigorous definitions of three familiar congruence transformations: reflections, translations, and rotations and
use these transformations to discover and prove geometric properties. Throughout the course, students will use
transformations as a tool to analyze and describe relationships between geometric figures.
Common Core State Standards
Congruence — G--‐CO
A. Experiment with transformations in the plane
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined
notions of point, line, distance along a line, and distance around a circular arc.
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).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto
itself.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel
lines, and line segments.
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.
B. Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on
a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are
congruent.
Common Core State Standards for Mathematical Practice
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
Geometry Unit 2 CCSS Standards
Unit 2: Angles and Lines
This unit gives students the foundational tools for developing viable geometric arguments using relationships
students studied in middle school related to lines, transversals, and special angles associated with them. Students
learn how to combine true statements within a mathematical system to deductively prove other statements.
Students should begin to see the structure of a mathematical system as they make conjectures and then prove
statements involving lines and angles.
Congruence — G--‐CO
A. Experiment with transformations in the plane
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc.
C. Prove geometric theorems
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.
Expressing Geometric Properties with Equations — G--‐GPE
B. Use coordinates to prove simple geometric theorems algebraically
5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the
equation of a line parallel or perpendicular to a given line that passes through a given point).
Common Core State Standards for Mathematical Practice
3. Construct viable arguments and critique the reasoning of others.
6. Attend to precision.
7. Look for and make use of structure.
Geometry Unit 3 CCSS Standards
Unit 3: Triangles
This unit explores basic theorems and conjectures about triangles, including the triangle inequality conjecture, the
Triangle Sum Theorem, and theorems regarding centers of a triangle. Students explored some of these
relationships in middle school but will build on their work in unit 2 with deductive reasoning and proof related to
triangles in this unit. Students make and verify conjectures related to isosceles triangles and explore physical
properties of the centroid of a triangle. In this unit, students also learn basic construction techniques and use these
as they explore triangle properties. Throughout this unit, students will use the precise definitions developed in G--‐
CO.A.1.
C. Prove geometric theorems.
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to
180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is
parallel to the third side and half the length; the medians of a triangle meet at a point.
D. Make geometric constructions
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.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Circles — G--‐C
A. Understand and apply theorems about circles
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
Modeling with Geometry — G--‐MG
A. Apply geometric concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a
human torso as a cylinder).★
Common Core State Standards for Mathematical Practice
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
8. Look for and express regularity in repeated reasoning.
Geometry Unit 4 CCSS Standards
Unit 4: Triangles Congruence
This unit builds on students’ work with transformations in unit 1 and properties of triangles in unit 3 to formalize
the definition of congruent triangles. Students reason to identify criteria for triangle congruence and use precise
notation to describe the correspondence in congruent triangles.
Congruence — G--‐CO
B. Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion
on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they
are congruent.
7. Use the definition of 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.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence
in terms of rigid motions.
Common Core State Standards for Mathematical Practice
3. Construct viable arguments and critique the reasoning of others.
6. Attend to precision.
7. Look for and make use of structure.
Geometry Unit 5 CCSS Standards
Unit 5: Similarity Transformations
This unit moves away from rigid motion and focuses on dilations and similarity. Students prove theorems involving
similarity and apply dilations and similarity to model situations in the real world.
C. Prove geometric theorems
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to
180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is
parallel to the third side and half the length; the medians of a triangle meet at a point.
Similarity, Right Triangles, and Trigonometry — G--‐SRT
A. Understand similarity in terms of similarity transformations
1. 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.
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.
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
B. Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures.
Expressing Geometric Properties with Equations — G--‐GPE
B. Use coordinates to prove simple geometric theorems algebraically
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Modeling with Geometry — G--‐MG
A. Apply geometric concepts in modeling situations
3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or
minimize cost; working with typographic grid systems based on ratios).★
Common Core State Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
3. Construct viable arguments and critique the reasoning of others.
8. Look for and express regularity in repeated reasoning.
Evidence of Assessment
Performance Tasks:
Units are arranged so that students work through MathXL to master many of the objectives. Additional
exercises are provided to develop application of newly learned concepts. Each unit concludes with both
a skills-based assessment and one with applications.
Other Evidence (self-assessments, observations, work samples, quizzes, tests and so on):


Work in Math XL for school
Quizzes and Unit tests in Math XL for school
Types of Learning Activities
Each unit follows the same basic pattern:
1.
Work in MathXL for school
2.
Target Activities (including additional work if needed to meet standard)
3.
Unit Assessments
Direct Instruction
Indirect Instruction
_x___Structured
Overview
____Mini
presentation
__x__Drill & Practice
____Demonstrations
__x__Other (List)
__x__Problembased
___Case Studies
____Inquiry
____Reflective
Practice
____Project
____Paper
____Concept
Mapping
____Other (List)
MathXL
Experiential
Learning
____ Virt. Field
Trip
____Experiments
____Simulations
____Games
____Field
Observ.
___Role-playing
____Model Bldg.
____Surveys
____Other (List)
Independent Study
____Essays
_x__Self-paced
computer
__x__Journals
____Learning Logs
____Reports
____Directed
Study
____Research
Projects
____Other (List)
Interactive
Instruction
_x___Discussion
____Debates
___Role Playing
____Panels
____Peer Partner
Learning
____Project team
____Laboratory
Groups
____Think, Pair,
Share
____Cooperative
Learning
____Tutorial Groups
____Interviewing
____Conferencing
____Other (List)
Other:
Learning Activities
These learning activities are aligned with the successful completion of the course learning goals and
progress towards these learning activities will be reported monthly on a progress report.
1st Semester Geometry
Unit 1 Geometric Transformations
In high school, students formalize much of the geometric exploration from middle school. In this unit, students
develop rigorous definitions of three familiar congruence transformations: reflections, translations, and rotations
and use these transformations to discover and prove geometric properties. Throughout the course, students will
use transformations as a tool to analyze and describe relationships between geometric figures.
Lessons
1. Points, Lines, and Planes
2. Angles
3. Parallel and Perpendicular Lines
4. Reflections
5. Translations
6.
7.
8.
9.
Rotations
Compositions of Transformations
Congruence and Transformations
Dilations
Unit 2 Angles and Lines
This unit gives students the foundational tools for developing viable geometric arguments using relationships
students studied in middle school related to lines, transversals, and special angles associated with them. Students
learn how to combine true statements within a mathematical system to deductively prove other statements.
Students should begin to see the structure of a mathematical system as they make conjectures and then prove
statements involving lines and angles.
Lessons
1. Pairs of Angles
2. Lines and Angles
3. Parallel Lines and Transversals
4. Proving Lines Parallel
5. Perpendicular Lines
6. Slopes of Lines
7. Lines in the Coordinate Plane
Unit 3 Triangles
This unit explores basic theorems and conjectures about triangles, including the triangle inequality conjecture, the
Triangle Sum Theorem, and theorems regarding centers of a triangle. Students explored some of these relationships
in middle school but will build on their work in unit 2 with deductive reasoning and proof related to triangles in this
unit. Students make and verify conjectures related to isosceles triangles and explore physical properties of the
centroid of a triangle. In this unit, students also learn basic construction techniques and use these as they explore
triangle properties. Throughout this unit, students will use the precise definitions developed in G--‐CO.A.1.
Lessons
1. Classifying Triangles
2. Angle Relationships in Triangles
3. Isosceles & Equilateral Triangles
4. Perpendicular and Angle Bisectors
5. Bisectors of Triangles
6. Median of Triangles
Unit 4 Triangle Congruence
This unit builds on students’ work with transformations in unit 1 and properties of triangles in unit 3 to formalize
the definition of congruent triangles. Students reason to identify criteria for triangle congruence and use precise
notation to describe the correspondence in congruent triangles
Lessons
1. Transformations and Congruence
2. Congruence and Triangles
3. Triangle Congruence Postulates
4. Proving Triangle Congruence Through
Transformations
5. Using Congruence Criteria
Unit 5 Similarity Transformations
This unit moves away from rigid motion and focuses on dilations and similarity. Students prove theorems involving
similarity and apply dilations and similarity to model situations in the real world.
Lessons
1. Ratios and Proportions
2. Similarity and Transformations
3. Ratios in Similar Polygons
4. Triangle Similarity
5.
6.
7.
8.
Pythagorean Theorem and Similarity
Triangle Similarity AA,SSS, and SAS
Properties of Similar Triangles
Using Proportional Relationships