Download Triangles Unit Guide Geometry

Triangles
Unit Guide Geometry
Big Idea (Cluster):
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Prove geometric theorems.
Make geometric constructions.
Understand and apply theorems about circles.
Use coordinates to prove simple geometric theorems algebraically.
Apply geometric concepts in modeling situations.
Edited 5/5/14
Renton School District
Triangles Unit Guide (G.CO.10, G.CO.12, G.CO.13, G.C.3, G.GPE.4, G.MG.1)
Big Idea (Cluster):
 Prove geometric theorems, make geometric constructions, understand and apply theorems about circles, use coordinates to prove simple
geometric theorems algebraically, and apply geometric concepts in modeling situations.
Standard G.CO.10
Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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.
Standard G.CO.12
Make geometric constructions. 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.
Standard G.CO.13
Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Standard G.C.3
Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
Standard G.GPE.4
Use coordinates to prove simple geometric theorems algebraically.
Standard G.MG.1
Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree
trunk or a human torso as a cylinder).*
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Relevant Math Practices and Student Actions
MP1 Make sense of problems and persevere in solving them.
 Perseveres when solving problems
 Understands what the problem asks and the relationship among the
problem’s parts
 Looks for an entry point into solving the problem
 Uses multiple strategies and representations
 Explains why a solution is reasonable
MP 2 Reason abstractly and quantitatively.
 Interprets problems in context
 Uses representations to make meaning of problems
 Translates a problem from situation to equation
 Creates a situation for a given equation
 Explains connections between equation and situation
 Understands the meaning of quantities and units
 Flexibly uses properties of operations and place value
MP 3 Construct viable arguments and critique the reasoning of
others.
 Uses definitions and draws on prior mathematical knowledge when
constructing arguments
 Makes conjectures and evaluates their accuracy
 Justifies conclusions with mathematical evidence and responds to
arguments of others
 Communicates and defends mathematical reasoning using objects,
drawings, diagrams, actions, examples and counterexamples
 Asks clarifying and probing questions to improve argument
 Attends to, questions, and evaluates the reasoning and arguments of
others
MP 4 Model with mathematics.
 Applies prior mathematical knowledge to describe, analyze, and
solve problems arising in everyday life, society and workplace
 Makes assumptions and approximations to make problems easier
 Identifies quantities necessary to solve a problem and uses
representations to map their relationships
 Analyzes mapped relationships mathematically to draw conclusions
 Checks to see if an answer makes sense within the context of a
situation and improves model when necessary
MP 5 Use appropriate tools strategically.
 Selects tools strategically for visualizing, exploring, comparing,
predicting, and solving problems
 Uses technological tools and resources to pose problems, solve
problems and deepen understanding
MP 6 Attend to precision.
 Communicates mathematical thinking accurately both orally and in
writing
 Understands the meaning of mathematical symbols and vocabulary
and uses them appropriately
 Labels consistently and accurately graphs and diagrams
 Pursues a level of precision appropriate to the context of the problem
 Formulates questions and explanations carefully
MP 7 Look for and make use of structure.
Looks for, identifies, develops and generalizes patterns and
relationships
 Makes connections to prior mathematical knowledge to solve new
problems
 Breaks down complex problems into manageable parts
 Views complicated quantities both as single objects or compilations
of several objects and uses operations to make sense of problems
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
SBAC Claims 1-4 Item Specification Information
Common Core State Standard G.CO.10 will be assessed through Claim 3 “Communicating Reasoning” assessment items. In Claim 3, students are
asked to clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Common Core State
Standard G.MG.1 will be assessed through Claim 4 “Modeling and Data Analysis” assessment items. In Claim 4, students are asked to analyze
complex, real-world scenarios. Students are also asked to construct and use models to interpret and solve problems. For more information on Claim
3 and 4 assessment targets for this group of standards, read the SBAC Claim 3 and Claim 4 Assessment Item Specifications.
Vocabulary
Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical
language. Students should learn the following terms with increasing precision within the cluster.
Adjacent interior angles
Altitude
Angle bisector
Auxiliary line
Base angle
Center of gravity
Centroid
Circumcenter
Circumscribed
Concurrent
Converse
Exterior angle
Incenter
Inscribed
Isosceles triangle
Leg
Median
Midsegment
Orthocenter
Perpendicular bisector
Point of concurrency
Remote interior angles
Vertex angle
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Triangles Unit Guide (G.CO.10, G.CO.12, G.CO.13, G.C.3, G.GPE.4, G.MG.1)
Big Idea (Cluster):
 Prove geometric theorems, make geometric constructions, understand and apply theorems about circles, use coordinates to prove simple
geometric theorems algebraically, and apply geometric concepts in modeling situations.
Standard G.CO.10
Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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.
See High School Flip Book pages 151-152 for explanations and examples of this standard.
Learning Objectives
 Students will be able to prove and apply triangle sum theorem and third angle sum theorem.
 Students will be able to prove and apply isosceles triangle theorem and its converse.
 Students will be able to prove and apply medians are concurrent at the triangle centroid (or center of mass).
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
Common Core State Standard G.CO.10 will be assessed through Claim
3 “Communicating Reasoning” assessment items. In Claim 3, students
are asked to clearly and precisely construct viable arguments to support
their own reasoning and to critique the reasoning of others. For more
information on Claim 3 learning targets for this group of standards,
read the SBAC Claim 3 Assessment Item Specifications.
8.G.5 Understand congruence and similarity using physical models,
transparencies, or geometry software. Use informal arguments to establish
facts about the angle sum and exterior angle of triangles, about the angles
created when parallel lines are cut by a transversal, and the angle-angle
criterion for similarity of triangles.
Connections to Curricular Materials
Additional Resources/Technology Resources
Discovering Geometry sections 3.7, 3.8, 4.1, 4.2, 4.3, 5.4, and 13.3
address this standard.
Content missing that will need to be supplemented includes:
 The transformational approach to proving these triangle
theorems.
Mathematics Vision Project Secondary Mathematics II Module 5 unit
(lesson and tasks, OSPI reviewed OER)
Engage NY High School Geometry Module 1 – Congruence, Proof and
Constructions (OSPI reviewed OER)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
Geometry Common Core district site HCPSS Unit 2 (tasks, videos, objectives)
5
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Triangles Unit Guide (G.CO.10, G.CO.12, G.CO.13, G.C.3, G.GPE.4, G.MG.1)
Big Idea (Cluster):
 Prove geometric theorems, make geometric constructions, understand and apply theorems about circles, use coordinates to prove simple
geometric theorems algebraically, and apply geometric concepts in modeling situations.
Standard G.CO.12
Make geometric constructions. 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.
See High School Flip Book pages 154-156 for explanations and examples of this standard.
Learning Objectives
 Students will be able to use constructions from previous units to explore basic theorems and conjectures about triangles.
SBAC Claims 1-4 Item Specification Information
Although G.CO.12 is not specifically identified within the assessment
claims for SBAC, the content should be taught and assessed at the
classroom level.
Connections to Prior Learning
7.G.2 Draw (freehand, with ruler and protractor, and with technology)
geometric shapes with given conditions. Focus on constructing triangles
from three measures of angles or sides, noticing when the conditions
determine a unique triangle, more than one triangle, or no triangle.
Connections to Curricular Materials
Additional Resources/Technology Resources
Content missing that will need to be supplemented includes:

Use constructions from previous units to explore basic theorems
and conjectures about triangles.
Mathematics Vision Project Secondary Mathematics I Module 6 unit
(lesson and tasks, OSPI reviewed OER)
Engage NY High School Geometry Module 1 – Congruence, Proof and
Constructions (OSPI reviewed OER)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
Geometry Common Core district site HCPSS Unit 1 (tasks, videos, objectives)
Geometry Common Core district site HCPSS Unit 2 (tasks, videos, objectives)
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Triangles Unit Guide (G.CO.10, G.CO.12, G.CO.13, G.C.3, G.GPE.4, G.MG.1)
Big Idea (Cluster):
 Prove geometric theorems, make geometric constructions, understand and apply theorems about circles, use coordinates to prove simple
geometric theorems algebraically, and apply geometric concepts in modeling situations.
Standard G.CO.13
Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
See High School Flip Book page 157 for explanations and examples of this standard.
Learning Objectives
 Students will be able to formally construct using a variety of tools and methods: construct an equilateral triangle inscribed in a circle.
SBAC Claims 1-4 Item Specification Information
Although G.CO.13 is not specifically identified within the assessment
claims for SBAC, the content should be taught and assessed at the
classroom level.
Connections to Curricular Materials
Discovering Geometry section 3.7 addresses this standard loosely.
Content missing that will need to be supplemented includes:
 How to construct an equilateral triangle
Connections to Prior Learning
7.G.2 Draw (freehand, with ruler and protractor, and with technology)
geometric shapes with given conditions. Focus on constructing triangles
from three measures of angles or sides, noticing when the conditions
determine a unique triangle, more than one triangle, or no triangle.
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics I Module 6 unit
(lesson and tasks, OSPI reviewed OER)
Engage NY High School Geometry Module 1 – Congruence, Proof and
Constructions (OSPI reviewed OER)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Mathematics Assessment Project tasks
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Triangles Unit Guide (G.CO.10, G.CO.12, G.CO.13, G.C.3, G.GPE.4, G.MG.1)
Big Idea (Cluster):
 Prove geometric theorems, make geometric constructions, understand and apply theorems about circles, use coordinates to prove simple
geometric theorems algebraically, and apply geometric concepts in modeling situations.
Standard G.C.3
Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
See High School Flip Book pages 175-176 for explanations and examples of this standard.
Learning Objectives
 Student will be able to formally construct using a variety of tools and methods: incenter, circumcenter and orthocenter.
SBAC Claims 1-4 Item Specification Information
Although G.C.3 is not specifically identified within the assessment
claims for SBAC, the content should be taught and assessed at the
classroom level.
Connections to Curricular Materials
Discovering Geometry section 3.7 addresses this standard.
Connections to Prior Learning
7.G.2 Draw (freehand, with ruler and protractor, and with technology)
geometric shapes with given conditions. Focus on constructing triangles
from three measures of angles or sides, noticing when the conditions
determine a unique triangle, more than one triangle, or no triangle.
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics II Module 7 unit
(lesson and tasks, OSPI reviewed OER)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
Missouri DOE Model Curriculum Unit Working with Circles
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Mathematics Assessment Project tasks
www.betterlesson.com
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Triangles Unit Guide (G.CO.10, G.CO.12, G.CO.13, G.C.3, G.GPE.4, G.MG.1)
Big Idea (Cluster):
 Prove geometric theorems, make geometric constructions, understand and apply theorems about circles, use coordinates to prove simple
geometric theorems algebraically, and apply geometric concepts in modeling situations.
Standard G.GPE.4
Use coordinates to prove simple geometric theorems algebraically.
See High School Flip Book pages 182-183 for explanations and examples of this standard.
Learning Objectives
 Students will be able to use coordinates to classify triangles by angle and sides.
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
Although G.GPE.4 is not specifically identified within the assessment
claims for SBAC, the content should be taught and assessed at the
classroom level.
Connections to Curricular Materials
Content missing that will need to be supplemented includes:

Use coordinates to classify triangles by angle and sides
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics II Module 8 unit
(lesson and tasks, OSPI reviewed OER)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
Mathematics Assessment Project tasks
www.betterlesson.com
9
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Triangles Unit Guide (G.CO.10, G.CO.12, G.CO.13, G.C.3, G.GPE.4, G.MG.1)
Big Idea (Cluster):
 Prove geometric theorems, make geometric constructions, understand and apply theorems about circles, use coordinates to prove simple
geometric theorems algebraically, and apply geometric concepts in modeling situations.
Standard G.MG.1
Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree
trunk or a human torso as a cylinder).*
See High School Flip Book pages 197-198 for explanations and examples of this standard.
Learning Objectives
 Students will be able to apply constructing centroid, circumcenter and incenter to solve real-world modeling situations.
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
Common Core State Standard G.MG.1 will be assessed through Claim
4 “Modeling and Data Analysis” assessment items. In Claim 4,
students are asked to analyze complex, real-world scenarios. Students
are also asked to construct and use models to interpret and solve
problems. For more information on Claim 4 learning targets for this
group of standards, read the SBAC Claim 4 Assessment Item
Specifications.
Connections to Curricular Materials
Discovering Geometry section 3.7 and 3.8 addresses this standard
loosely.
Content missing that will need to be supplemented includes:
 True modeling problems.
Additional Resources/Technology Resources
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
Modeling Progression of Learning document (Bill McCallum)
Examples: hospital equidistant to three major freeways, hanging a
triangular wind chime, build a fire station equidistant to three local
schools, etc.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Hard copies of these lessons can be found at http://staff.rentonschools.us/renton/secondary-math
The following resources were used to create this curriculum guide: SBAC Claim 3 Assessment Item Specifications, SBAC Claim 4 Assessment Item
Specifications, High School Modeling Progression of Learning document and High School Common Core State Standards Flip Book compiled by
Trish Hart.
A further unpacking of the Geometry Common Core State Standards by Hung-Hsi Wu (2013) titled “Teaching Geometry in Grade 8 and High School
According to the Common Core Standards” can be found at http://math.berkeley.edu/~wu/CCSS-Geometry.pdf.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.