Download Unit 4 Triangles - Clover Park School District

Geometry Mathematics Curriculum Guide – Unit 4 Triangles
Unit 4: Triangle Properties and Congruence
2016 – 2017
Time Frame: 16 days
Primary Focus
There are three primary parts of this unit:
A) Building off the student understanding of transformations that was developed in Unit 2, students will use rigid motion to develop proofs for triangle
congruence.
B) Students will continue to work with triangles proving theorems about triangles.
C) Develop theorems of parallelograms using theorems of lines and angles.
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.
MP4 - Model with mathematics.
MP5 - Use appropriate tools strategically.
MP6 - Attend to precision.
MP7 - Look for and make use of structure.
MP8 - Look for and express regularity in repeated reasoning.
Unit 4
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 a variety of methods to model, represent, and solve real-world problems.
Select and use appropriate tools to best model/solve problems.
Transform figures efficiently and accurately and label them appropriately.
Use patterns or structure to make sense of mathematics and connect prior
knowledge to similar situations and extend to novel situations.
Generalize the process to create a shortcut which may lead to developing rules or
creating a formula.
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 4 Triangles
2016 – 2017
Stage 1 Desired Results
Transfer Goals
Students will be able to independently use their learning to…
Determine if 2 figures are congruent by comparing corresponding parts
Prove triangles are congruent without having to compare all corresponding parts
Show the angles and sides of Isosceles and Equilateral triangles have special relationships.
Meaning Goals
UNDERSTANDINGS
ESSENTIAL QUESTIONS
Students will understand that…
How can the properties of rigid motion be used to prove that two triangles are
• The properties of transformations that are rigid motion can be used to
congruent (ASA, SAS, SSS)?
identify and prove congruence of figures in a plane.
• Constructing a viable argument using the precise vocabulary of
transformations and congruence to prove geometric theorems in a variety
of formats is important to Geometry proof.
• Triangles have special properties that allow you to use shortcuts for proving
triangles congruent.
Acquisition Goals
Students will know and will be skilled at…
 Define congruent polygons as a one-to-one relationship between the corresponding congruent parts.
 Demonstrate that two polygons are congruent by mapping one onto another using rigid transformation.
 Determine the composition of rigid transformations that map two congruent polygons onto each other.
 Determine the minimum number of transformations that are needed to map one congruent polygon onto another.
 Identify the minimum number of parts of a triangle and their relationships to each other (SAS, ASA, SSS) that can be transformed to result in congruent
triangles.
 Understand that AAA and SSA criteria do not necessarily create congruent triangles; use compass and straightedge constructions to create two noncongruent triangles that satisfy AAA and SSA.
Unit 4
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 4 Triangles
2016 – 2017
Stage 1 Established Goals: Common Core State Standards for Mathematics
Cluster: Standard(s)
Understand congruence in terms of rigid motions
G-CO.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.
G-CO.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.
G-CO.8 Explain how the criteria of triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Explanations, Examples, and Comments
What students should know prior to this unit and may need to be reviewed
Describe and apply the different types of transformations and be able to differentiate between them
Perform translations, reflections, and rotations with and without the use of technology; including reflecting
over parallel lines and reflecting over intersecting lines
Describe and perform the composition of transformations that will map a given object onto a congruent object
in the plane with or without technology
Perform all the basic constructions using a compass and straightedge
Explanations:
A rigid motion is a transformation of points in space consisting of a sequence of one or more translations,
reflections, and/or rotations. Rigid motions are assumed to preserve distances and angle measures.
Students may use geometric software to explore the effects of rigid motion on a figure(s) and theorems about
lines and angles.
Congruence of triangles
Two triangles are said to be congruent if one can be exactly superimposed on the other by a rigid motion, and
the congruence theorems specify the conditions under which this can occur. Develop the relationship between
transformations and congruency. Allow adequate time and provide hands-on activities for students to visually
and physically explore rigid motions and congruence.
Use graph paper, tracing paper or dynamic geometry software to obtain images of a given figure under
specified rigid motions. Note that size and shape are preserved.
Stage 3
MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine
which of the materials below best meet student
instructional needs. All materials listed may not be
needed.
Holt Geometry Lesson 4-3 Congruent Triangles
Holt Geometry Lesson 4-4 Triangle Congruence SSS
and SAS
Holt Geometry Lesson 4-5 Triangle Congruence ASA,
AAS, and HL
Holt Geometry Lesson 4-6 Triangle Congruence
CPCTC
Supplemental Materials
Discovering Geometry 4.4 Are There Congruence
Shortcuts?
Discovering Geometry 4.5 Are There Other
Congruence Shortcuts?
Discovering Geometry 4.6 Corresponding Parts of
Congruent Triangles
Discovering Geometry 4.7 Flowchart Thinking
Discovering Geometry 13.3 Triangle Proofs
Performance Tasks
MAP: Analyzing Congruence Proofs
MVP: Congruent Triangles
Unit 4
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 4 Triangles
2016 – 2017
Use rigid motions (translations, reflections and rotations) to determine if two figures are congruent. Compare a
given triangle and its image to verify that corresponding sides and corresponding angles are congruent.
Work backwards – given two figures that have the same size and shape, find a sequence of rigid motions that
will map one onto the other.
MVP: Congruent Triangles to the Rescue
Georgia CCGPS: Proving Two Triangles are
Congruent
Build on previous learning of transformations and congruency to develop a formal criterion for proving the
congruency of triangles. Construct pairs of triangles that satisfy the ASA, SAS or SSS congruence criteria, and
use rigid motions to verify that they satisfy the definition of congruent figures. Investigate rigid motions and
congruence both algebraically (using coordinates) and logically (using proofs).
Cluster: Standard(s)
Prove geometric theorems
G-CO.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
Explanations, Examples, and Comments
What students should know prior to this unit and may need to be reviewed
Understand and apply the definitions of: interior angles, isosceles triangles, parts of isosceles
triangles, mid-segment of a triangle, medians of a triangle.
Stage 3
MATERIALS BY STANDARD(S):
Theorems Included in these standards:
Triangle sum theorem
Isosceles triangle theorem
Mid-segment of a triangle is parallel to a side and half the length
Medians of a triangle are concurrent
Holt Geometry Lesson 4-2 Angle Relationships in Triangles
Holt Geometry Lesson 5-2 Triangle Bisectors
Holt Geometry Lesson 5-3 Medians &Altitudes of Triangles
Holt Geometry Lesson 5-4 Triangle Mid-segment
Holt Geometry Lesson 4-8 Isosceles and Equilateral Triangles
Teacher should use assessment data to determine which of the
materials below best meet student instructional needs. All
materials listed may not be needed.
Supplemental Material
Discovering Geometry 3.4 Constructing Angle Bisectors
Discovering Geometry 3.7 Constructing Points of Concurrency
Discovering Geometry 3.8 The Centroid
Performance Tasks
MVP: It’s All In Your Head
MVP: Conjectures and Proofs
MVP: Centers of Triangles
Georgia CCGPS: Triangle Proofs
Unit 4
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 4 Triangles
2016 – 2017
Prove geometric theorems
G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
The focus of this standard is to use lines and angles to prove theorems about parallelograms.
What students should know prior to this unit and may need to be reviewed
Understand and apply the definitions of: parallelograms, diagonals, opposite sides, opposite
angles, consecutive angles, rectangles
Theorems Included in this standards:
Opposite sides of a parallelogram are congruent
Opposite angles of a parallelogram are congruent
Diagonals of a parallelogram bisect each other
Rectangles are parallelograms with congruent diagonals
Georgia Analytical Geometry EOCT Study Guide; pgs 42-46 provides an overview of content.
MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which of the
materials below best meet student instructional needs. All materials
listed may not be needed.
Holt Geometry Lesson 6-1 Properties and Attributes of
Polygons
Holt Geometry Lesson 6-2 Properties of Parallelograms
Holt Geometry Lesson 6-3 Conditions of Parallelograms
Holt Geometry Lesson 6-4 Properties of Special Parallelograms
Holt Geometry Lesson 6-5 Conditions for Special
Parallelograms
Holt Geometry Lesson 6-6 Kites and Trapezoids
Supplemental Materials
Parallelograms and other Quadrilaterals
Discovering Geometry 1.6 Special Quadrilaterals
Discovering Geometry 5.3 Kites and Trapezoid Properties
Discovering Geometry 5.5 Properties of Parallelograms
Discovering Geometry 5.6 Properties of Special Parallelograms
Discovering Geometry 5.7 Proving Quadrilateral Properties
Discovering Geometry 13.4 Quadrilateral Proofs
Other Materials
EngageNY Geometry Module 1 Lesson 22-27 Teacher Student
MAP: Square
Georgia CCGPS: Constructing Diagonals
Georgia CCGPS: Proving Quadrilaterals in the Coordinate
Plane
MVP: Parallelogram Conjectures and Proof
Unit 4
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 4 Triangles
Unit 4
Clover Park School District 2016-2017
2016 – 2017
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Geometry Mathematics Curriculum Guide – Unit 4 Triangles
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
Understand congruence in terms of rigid
motions
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.
Semester Cumulative
Unit 4
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 4 Triangles
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 Sections of the Unit (See NOTES section for more detail for Extended Geometry.)
Parallelograms
Holt Geometry Chapter 6 Lesson 2 & 3
- Emphasize problems that reinforce algebra, such as pg.395 #9-12 and #21-24.
- Formal language in proofs is less important. Supply students with the language as shown in the
Theorems sections on pgs. 398 & 399
- Exit Tasks, see pg.407 or 439
Triangle Properties:
Holt Geometry Chapter 4 Lesson 3
Holt Geometry Chapter 4 Lesson 8
Holt Geometry Chapter 5 Lesson 3 (Could move to circle unit.)
- Supply students with the language as shown in Theorems sections on pgs. 252-254, 273-275, and
314.
Triangle Congruence
EngageNY Geometry Module 1, Lesson 22, 24, 29, 30
Holt Geometry Chapter 4 Lesson 4-6
- Again, supply students with the language as shown in Theorems sections on pgs. 242, 243, 252-255.
Common Assessment (with theorem language supplied for them)
Semester Cumulative
Daily Lesson Components
Learning Target
Warm-up
Activities
 Whole Group:
 Small Group/Guided/Collaborative/Independent:
 Whole Group:
Checking for Understanding (before, during and after):
Assessments
Unit 4
Clover Park School District 2016-2017
NOTES:
Triangle centers can be included in this unit or in
the circles unit because they can relate to
standards in either unit.
The sections of this unit can be rearranged to
building philosophy.
Extended Geometry classes should select four or
more of these Discovering Geometry lessons to
support student success in the core curriculum.
Parallelograms and other Quadrilaterals
Discovering Geometry 1.6 Special Quadrilaterals
Discovering Geometry 3.4 Constructing Angle
Bisectors
Discovering Geometry 3.7 Constructing Points of
Concurrency
Discovering Geometry 5.3 Kites and Trapezoid
Properties
Discovering Geometry 5.5 Properties of
Parallelograms
Discovering Geometry 5.6 Properties of Special
Parallelograms
Discovering Geometry 5.7 Proving Quadrilateral
Properties
Discovering Geometry 13.4 Quadrilateral Proofs
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Geometry Mathematics Curriculum Guide – Unit 4 Triangles
Unit 4
Clover Park School District 2016-2017
2016 – 2017
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