Download Geometry Mathematics Curriculum Guide

Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles
Unit 3: Lines & Angles
2016 – 2017
Time Frame: 8 Days
Primary Focus
There are 2 main parts to this unit:
a) Students will build on their work in the previous unit on coordinate geometry and further develop their understanding of slope
b) Students will use slope to develop theorems about parallel perpendicular lines
Common Core State Standards for Mathematical Practice
Standards for Mathematical Practice
MP1 - Make sense of problems and persevere in solving them.
MP2 - Reason abstractly and quantitatively.
MP3 - Construct viable arguments and critique the reasoning of
others.
Unit 3
How It Applies to this Topic…
Analyze given information to develop possible strategies for solving the problem.
Make connections between the abstract theorems and their real-world applications.
Justify (orally and in written form) the argument by deductive reasoning, including
how it fits in the context from which the problem arose.
Use observations and prior knowledge (stated assumptions, definitions, and previous
established results) to make conjectures and construct arguments.
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles
2016 – 2017
Stage 1 Desired Results
Transfer Goals
Students will be able to independently use their learning to…
Prove properties of two- and three-dimensional geometric figures when solving increasingly complex problems involving geometric figures and their
measurements.
UNDERSTANDINGS
Students will understand that…
Meaning Goals
ESSENTIAL QUESTIONS
The basic properties of parallel and perpendicular lines, their respective slopes,
and the properties of the angles formed when parallel lines are intersected by a
transversal are important when proving related theorems and solving both
mathematical and practical problems.
•
•
•
How do you summarize and explain basic theorems?
How do you know that two lines are parallel?
How do you know that two lines perpendicular to each other?
Acquisition Goals
Students will know and will be skilled at…
Determining the coordinates of a point described geometrically.
Explaining and performing basic compass and straightedge constructions related to parallel and perpendicular lines.
Proving that if two parallel lines are cut by a transversal, and then alternate-interior angles are equal.
Determining the equation of a line in the coordinate plane described geometrically from: (two points, parallel line through a given point, and perpendicular line
through a given point).
Proving and applying theorems about parallel and perpendicular lines.
Proving and applying theorems about angles, including angles that arise from parallel lines intersected by a transversal.
Summarizing and explaining basic theorems.
Unit 3
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles
2016 – 2017
Stage 1 Established Goals: Common Core State Standards for Mathematics
Note on Proofs for this unit:
Students may use geometric simulations (computer software or graphing calculator) to explore theorems about lines and angles.
Use inductive and deductive reasoning, students will solve problems, proofs, and real world situations involving parallel lines, perpendicular lines, and the angles
relationships formed by those lines
Proofs in high school geometry should not be restricted to the two-column format. Most proofs at the college level are done in paragraph form, with the writer
explaining and defending a conjecture. In many cases, the two-column format can hinder the student from making sense of the geometry by paying too much
attention to format rather than mathematical reasoning. That being said, as it relates to high school math, formal proofs have been deemphasized in the
common core geometry progression. Providing references for students will help them utilize and apply precise language more efficiently while also keeping
proofs from disproportionately taking up too much class time.
Vocabulary: hypothesis, transitive property, symmetric property, reflexive property, law of syllogism, law of detachment, counterexample, converse
Cluster: Standard(s)
Use coordinates to prove simple geometric theorems algebraically
G.GPE.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).
Explanations, Examples, and Comments
Lines can be horizontal, vertical, or neither.
Stage 3
MATERIALS BY STANDARD(S):
Students may use a variety of different methods to construct a parallel or perpendicular line to a given line and
calculate the slopes to compare the relationships.
Teacher should use assessment data to determine
which of the materials below best meet student
instructional needs. All materials listed may not be
needed.
Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra I involving systems of
equations having no solution or infinitely many solutions
 Review the concept of slope as the rate of change of the y-coordinate with respect to the x-coordinate for
a point moving along a line, and derive the slope formula.
 Use similar triangles to show that every non-vertical line has a constant slope.
 Review the point-slope, slope-intercept and standard forms for equations of lines.
 Investigate pairs of lines that are known to be parallel or perpendicular to each other and discover that
their slopes are either equal or have a product of –1, respectively.
Unit 3
Clover Park School District 2016-2017
Holt Geometry Lesson 3-6 Lines in the Coordinate
Plane
Supplemental Materials
Discovering Geometry Using your Algebra Skills 3:
p167 Slopes of Parallel and Perpendicular Lines
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Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles
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Pay special attention to the slope of a line and its applications in analyzing properties of lines.
Allow adequate time for students to become familiar with slopes and equations of lines and methods of
computing them.
Connect to G.GPE.4,6 and 7 Have student given two points, find the following: slope, distance, and
midpoint
2016 – 2017
Performance Tasks:
MVP Task: Slippery Slopes
Georgia CCGPS: Discovery Task: Slopes of Special
Pairs of Lines
Georgia Coordinate Algebra EOCT Study Guide Unit 6; pgs 166 – 179 provides an overview of content.
Cluster: Standard(s)
Make geometric constructions
G.CO.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.
Explanations, Examples, and Comments
Stage 3
MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which
Focus on: constructing perpendicular lines, including the perpendicular bisector of a line segment; and
of the materials below best meet student instructional
constructing a line parallel to a given line through a point not on the line.
needs. All materials listed may not be needed.
Additional Resources for Constructions
Construction Resource
Constructions
Parallel Lines: Holt Geometry Lesson 3-3 & 3-3 Lab
Perpendicular Lines: Holt Geometry Lesson 3-4 & 34 Lab
Supplemental Material:
Discovering Geometry 3.2 Constructing
Perpendicular Bisectors
Discovering Geometry 3.3 Constructing
Perpendiculars to a Line
Discovering Geometry 3.5 Constructing Parallel
Lines
Georgia Analytical Geometry EOCT Study Guide;
pgs 47-59 provides additional resources for
Constructions
Unit 3
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles
2016 – 2017
Cluster: Standard(s)
Prove geometric theorems
G.CO.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.
Explanations, Examples, and Comments
What students should know prior to this unit and may need to be reviewed
Understand and apply the definitions of: linear pair of angles, parallel lines and angles formed by
transversals, corresponding angles, corresponding sides, perpendicular lines, perpendicular bisector, and
equidistance
Stage 3
MATERIALS BY STANDARD(S):
Theorems included in this standard:
Vertical angle theorem
Alternate interior angle theorem
Corresponding angle theorem
Perpendicular bisector theorem
Holt Geometry Lesson 3-1 Lines and Angles
Holt Geometry Lesson 3-2 Angles Formed by Parallel
Line and Transversals
Holt Geometry Lesson 3-3 Proving Lines Parallel
Holt Geometry Lesson 3-4 Perpendicular Lines
Georgia Analytical Geometry EOCT Study Guide; pgs 42-46 provides an overview of content.
Supplemental Materials
Discovering Geometry 2.5 Angle Relationships
Discovering Geometry 2.6 Special Angles on Parallel Lines
Discovering Geometry 3.5 Constructing Parallel Lines
Theorems
Vertical angles are congruent -Geometry Holt Unit 2: Lesson 7
Alternate interior angles -Geometry Holt Unit 3: Lesson 2
Points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s
endpoints.-Geometry Holt Unit 5: Lesson 1
Unit 3
Clover Park School District 2016-2017
Teacher should use assessment data to determine which
of the materials below best meet student instructional
needs. All materials listed may not be needed.
Other Resources
GSE Coordinate Algebra: Unit 6: Connecting Algebra
and Geometry Through Coordinates: Slopes of
Special Pairs of Lines
MVP: Parallelism Preserved and Protected
EngageNY Geometry Module 1: Lessons 6-11
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Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles
2016 – 2017
Stage 2 - Evidence
Evaluative Criteria/Assessment Level
Descriptors (ALDs):
Claim 1 Clusters:
Sample Assessment Evidence
Concepts and Procedures
NONE
Claim 2 Clusters:
Problem Solving
NONE
Claim 3 Clusters:
Prove geometric theorems
Communicating Reasoning
Level 3 students should be able to use stated assumptions, definitions, and previously established results and
examples to test and support their reasoning or to identify, explain, and repair the flaw in an argument. Students
should be able to break an argument into cases to determine when the argument does or does not hold.
Level 4 students should be able to use stated assumptions, definitions, and previously established results to
support their reasoning or repair and explain the flaw in an argument. They should be able to construct a chain of
logic to justify or refute a proposition or conjecture and to determine the conditions under which an argument
does or does not apply.
Go here for Sample SBAC items
Go here for more information about the Achievement Level Descriptors for Mathematics
Common Assessment
See Sample Assessments for Geometry Units.
Unit 3
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 3 Lines & Angles
2016 – 2017
Stage 3 – Learning Plan: Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
LEARNING ACTIVITIES:
Suggested Sequence of the Unit (See NOTES section for more detail for Extended Geometry.)
Day 1: Holt Geometry Chapter 3 Lesson 6
- Exit Task, see pg 201 or 205
Day 2: Georgia Coordinate Algebra: Unit 6: Connecting Algebra and Geometry Through Coordinates: Slopes of
Special Pairs of Lines (Discovery Task)
Days 3-4: Holt Geometry Chapter 2 Lessons 1,5,6,7
- Provide EngageNY Geo Module 1 Teacher Guide pgs 58-60 to students for their reference. The
emphasis is not on memorization, but on logical use of given theorems.
- Holt 2-2 was removed because it does not align with Common Core. Only Converse is needed, and it
was taught in 8th Grade. You may review the term converse.
- Exit Tasks, see pgs103,127,132,133
Days 5-7: EngageNY Geometry Module 1, Lesson 6,7, and 10 (use to develop Theorems) Teacher - Student
- See note above about providing reference pages to students.
- Exit Tickets provided.
Common Assessment (Includes use of reference pages noted above.)
Daily Lesson Components
Learning Target
Opening Activities
Activities
 Whole Group:
 Small Group/Guided/Collaborative/Independent:
 Whole Group:
Checking for Understanding (before, during and after):
Assessments
Unit 3
Clover Park School District 2016-2017
NOTES:
Scaffolding that may be needed:
 Slope and Linear Function Review from Alg. 1
 Review from 8th grade identifying angle pairs
formed by parallel lines cut by a transversal
Students should get supportive scaffolds with
simple proofs, both two-column and paragraph.
Proofs have been de-emphasized in progression
documents.
Extended Geometry classes should select two or
three of these Discovering Geometry lessons to
support student success in the core curriculum.
Discovering Geometry 2.5 Angle Relationships
Discovering Geometry 2.6 Special Angles on
Parallel Lines
Discovering Geometry 3.5 Constructing Parallel
Lines
Portions of Holt Geometry Lessons 3-1 through
3-4 may also help students.
Other Resources
MVP: Parallelism Preserved and Protected
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