Download Unit 1 - Asbury Park School District

Asbury Park
Name of Unit: Congruence, Proof & Constructions
Content Area: HS Math - Geometry
Big Idea:
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Unit #/Duration: Unit 1 / 45 days
Grade Level: 10th
Triangles are congruent if and only if pairs of corresponding sides and all corresponding angles are
congruent
Congruence can be defined in terms of rigid motions [ translations, rotations, reflections ]; Dilations do not
maintain congruence
Formal geometric constructions can be made with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software)
Geometric Proofs are comprised of theorems, postulates properties , definitions and demonstrate
understanding of geometric concepts
Essential Questions:
● How can you prove 2 triangles congruent?
● What are undefined notions in Geometry? Why do we
refer to them as undefined?
● How are transformations and functions related?
● What is “Identity Symmetry”?
● What is the relationship between a rotation and a
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School District
reflection?
Can you explain why any point on the perpendicular
bisector is equidistant from any pair of pre-image
and image point?
What is a correspondence
Why does a congruence naturally yield a
correspondence?
What is a centroid, and how does it divide a
median?
What is the “protractor postulate? And why must it
be used to prove vertical angles are congruent?
I Can Statements:
● construct an equilateral triangle
● communicate mathematical concepts clearly and
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concisely
bisect an angle
copy an angle
construct a perpendicular bisector
Define and identify two points of concurrencies
Apply the definition of supplementary and
complementary angles to determine unknown angles
Identify congruent “special angles” formed when a
transversal cuts parallel lines
review formerly learned geometry facts ex: interior
and exterior angles of a triangle, isosceles triangle,
etc.
Apply SMAT 180 ( sum of measures of angles inside a
triangle =180 degrees)to find a missing interior angle
write unknown angle proofs
draw valid conclusions based upon observations and
proven fact
write unknown angle proofs involving auxiliary lines
Describe parallel line theorems and their converses
articulate what differentiates rigid motions from
non-rigid motions
identify the parameters needed to complete any
rigid motion
Manipulate rotations by each parameter: center of
rotation, angle of rotation, and a point under the
rotation
construct the line of reflection of a figure and its
reflected image
construct the image of a figure when provided the
line of reflection
identify rotational symmetry within an individual
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figure
define translation and perform a translation by
construction
understand that any point on a line of reflection is
equidistant from any pair of pre-image and image
points in a reflection
construct a line parallel to a given line through a
point not on that line using a rotation by 180°
prove the alternate interior angles theorem using the
parallel postulate and a construction
define congruence in terms of rigid motions
state the correspondence that arises from a given
congruence
apply a sequence of rigid motions from one figure
onto another figure in order to demonstrate that the
figures are congruent
prove SAS congruence using rigid motion
transformations
complete proofs involving properties of an isosceles
triangle
prove ASA or SSS congruence using transformations
complete triangle congruency proofs requiring a
synthesis of the skills
prove properties of parallelograms
Use the properties of midsegments to solve for the
unknown value
Construct a Square and a Nine-Point Circle
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Common Core State Standards:
● G.CO.1 ; G.CO.2 ; G.CO.3 ; G.CO.4 ; G.CO.5 ; G.CO.6 ; G.CO.7 ; G.CO.8 ; G.CO.9 ; G.CO.10 ; G.CO.11 ; G.CO.12 ; G.CO.13
Pre-requisite Standards:
 8.G.1 ; 8.G.2 ; 8.G.3 ; 8.G.5
Mathematical Practices Highlighted:
● MP 3 ; MP 4 ; MP 5 ; MP 6
Interdisciplinary Connections:
History : proofs without technology
Symbol: **IC
Technology Integration: (Standards included only if students will be demonstrating knowledge/understanding/skill.)
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8.1 educational Technology: all students will use digital tools to access, manage, evaluate and synthesize
information in order to solve problems individually and collaboratively and to communicate knowledge
Symbol: ***TI [ Use of TI-84’s and an Epson Brite Board are used daily in the math classroom, with the use of internet
websites to enrich the learning experience when appropriate ]
NOTE: Use of Geometer’s Sketch pad can be used for constructions & Geometric Shape features on Epson Briteboard should be used along with use of interactive pens for kinesthetic learners
Texts
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Primary Text:
Eureka math 10 [ Engage NY Geometry]
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Secondary/Supplemental Texts:
Pearson Geometry
HMH Geometry
Suggested Instructional Activities/Strategies
Topic A – Basic constructions:
Lesson 1: Construct an Equilateral Triangle * **TI
Lesson 2: Construct an Equilateral Triangle II ***TI
Lesson 3: Copy and Bisect an Angle **TI
Lesson 4: Construct a Perpendicular Bisector ***TI
Lesson 5: Points of Concurrencies
Topic B - Unknown angles:
Lesson 6: Solve for Unknown Angles—Angles and Lines at a Point
Lesson 7: Solve for Unknown Angles—Transversals ***TI
Lesson 8: Solve for Unknown Angles—Angles in a Triangle
Lesson 9: Unknown Angle Proofs—Writing Proofs
Lesson 10: Unknown Angle Proofs—Proofs with Constructions
Lesson 11: Unknown Angle Proofs—Proofs of Known Facts **IC
Topic C – Transformations / Rigid motions :
Lesson 12: Transformations—The Next Level **TI
Lesson 13: Rotations ***TI
Lesson 14: Reflections ***TI
Lesson 15: Rotations, Reflections, and Symmetry ***TI
Lesson 16: Translations
Lesson 17: Characterize Points on a Perpendicular Bisector
Lesson 18: Looking More Carefully at Parallel Lines
Lesson 19: Construct and Apply a Sequence of Rigid Motions ***TI
Lesson 20: Applications of Congruence in Terms of Rigid Motions
Lesson 21: Correspondence and Transformations ***TI
Mid-module assessment
Topic D - Congruence :
Lesson 22: Congruence Criteria for Triangles—SAS
Lesson 23: Base Angles of Isosceles Triangles
Lesson 24: Congruence Criteria for Triangles—ASA and SSS
Lesson 25: Lesson 25: Congruence Criteria for Triangles—SAA and HL
Lesson 26: Triangle Congruency Proofs—Part I
Lesson 27: Triangle Congruency Proofs—Part II
Topic E – Proving properties of geometric figures:
Lesson 28: Properties of Parallelograms
Lesson 29: Special Lines in Triangles
Lesson 30: Special Lines in Triangles
Topic F - Advanced constructions:
Lesson 31: Construct a Square and a Nine-Point Circle ***TI
Lesson 32: Construct a Nine-Point Circle
***TI
Topic G – Axiomatic Systems:
Lesson 33: Review of the Assumptions
Lesson 34: Review of the Assumptions
End-of-module assessment
Teacher Resources
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Compass
Protractors
Straight edge
Patty paper or tracing paper
Geometer’s Sketchpad or Geoebra
Document camera
Epson Brite-board ( interactive)
Communicators
Vocabulary
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Domain Specific Academic Vocabulary (Tier 3)
Isometry
Transformations
Translation
Rotation
Reflection
Congruence
Image
Pre-image
Chord
Midsegment
Median
Centroid
Inscribed
Formative Assessments:
● Do Now’s ( class openers)
● Class work
● Homework
● Exit Tickets
● Reflections
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General Academic Vocabulary (Tier 2)
Rectangle
Circle
Triangle
Bisect
Construct
Prove
Rigid
Motion
Viable argument
Assessments
Summative Assessment:
● Mid-module assessment
● End-of-module assessment
● Teacher-prepared quizzes
Differentiation/Scaffolding
(for example ELL, students who are classified, struggling learners, etc.)
Visual
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Auditory
Kinesthetic
Language Development
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Khan Academy and other math websites used presenting images of topics discussed ; use of
communicators and templates to assist with marking up triangles to determine congruence;
Geogebra ; Geometer’s Sketch pad ; Visual presentation of completed student constructions
for comparison ; Use different colored markers/ pencils to mark congruent angles and sides
Reading directions aloud and close-reading activities; khan Academy lessons
Constructions ; patty paper folding , tracing to perform physical rotations of shapes
Vocabulary rich integration in the classroom through use of diagrams and other graphic
organizers to help process concepts and increase recall ; Historical reference to math
conjecture and proof
Appendix 1
(graphic organizers, rubrics, websites, activities, manipulatives, sample assessments, etc.)
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Kutasoftware.com
Mathisfun.com
Coolmath.com
KhanAcademy.com
video, Angles and Trim
http://youtu.be/EBP3I8O9gIM
video clip (Sherlock Holmes, Master of Deduction Does it Again,
https://www.youtube.com/watch?v=o30UY_flFgM discuss the connection between Holmes’s process of
identifying the attacker and the deduction used in geometry.
Historical connection Eratosthenes:
https://youtu.be/wnElDaV4esg
Geogebra
Geometer’s Sketch pad
Mid-module assessment & rubric p. 175-188 Eureka Math
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End-of-module Assessment p. 280 – 297 Eureka Math
Graphic organizer of geometric facts & properties p. 268-270 Eureka Math
Copied angle and bisector rubric p. 35
Prior geometry key facts chart p. 58, 59, 60
Reference chart ( properties of real numbers & properties of equality )
Appendix 2
(Quad D Exemplar Lesson Plan)
http://map.mathshell.org/lessons.php?unit=9315&collection=8 [ Evaluating conditions for congruency]