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MATHEMATICS
Geometry Honors: Unit 3
Relationships within Triangles
Quadrilaterals and Other Polygons
Similarity Right Triangles and Trigonometry
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Course Philosophy/Description
Geometry Honors is for our accelerated Math students. It stresses the ability to reason logically and to think critically, using spatial sense. In this
course, students will engage in activities that allow them to create geometric understanding. A major part of the course will be devoted to teaching
the student how to present a formal proof. Geometric properties of both two and three dimensions are emphasized as they apply to points, lines,
planes, and solids. In this course students learn to recognize and work with geometric concepts in various contexts. They build on ideas of inductive
and deductive reasoning, logic, concepts, and techniques of Euclidean plane and solid geometry and develop an understanding of mathematical
structure, method, and applications of Euclidean plane and solid geometry. Students use visualizations, spatial reasoning, and geometric modeling to
solve problems. Topics of study include points, lines, and angles; triangles; quadrilaterals and other polygons; circles; coordinate geometry; three-
dimensional solids; geometric constructions; symmetry; similarity; and the use of transformations. Students use tools of geometry to develop, verify
and prove geometric principles and relationships. Students extend their knowledge and understanding by solving open-ended real-world problems
and thinking critically through the use of high level tasks and long-term projects.
Upon successful completion of this course, students will be able to: Use and prove basic theorems involving congruence and similarity of figures;
determine how changes in dimensions affect perimeter and area of common geometric figures; apply and use the properties of proportion; perform
basic constructions with straight edge and compass; prove the Pythagorean Theorem; use the Pythagorean Theorem to determine distance and find
missing dimensions of right triangles; know and use formulas for perimeter, circumference, area, volume, lateral and surface area of common figures;
find and use measures of sides, interior and exterior angles of polygons to solve problems; use relationships between angles in polygons,
complementary, supplementary, vertical and exterior angle properties; use special angle and side relationships in special right triangles; understand,
apply, and solve problems using basic trigonometric functions; prove and use relationships in circles to solve problems; prove and use theorems
involving properties of parallel lines cut by a transversal, quadrilaterals and circles; write geometric proofs, including indirect proofs; construct and
judge validity of logical arguments; prove theorems using coordinate geometry including the midpoint of a segment and distance formula; understand
transformations in the coordinate plane; construct logical verifications to test conjectures and counterexamples; write basic mathematical arguments
in paragraph and statement-reason form; calculate basic probabilities; investigate the role of permutations and combinations in probability; and use
probability to make and analyze decisions.
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ESL Framework
This ESL framework was designed to be used by bilingual, dual language, ESL and general education teachers. Bilingual and dual language
programs use the home language and a second language for instruction. ESL teachers and general education or bilingual teachers may use this
document to collaborate on unit and lesson planning to decide who will address certain components of the SLO and language objective. ESL teachers
may use the appropriate leveled language objective to build lessons for ELLs which reflects what is covered in the general education program. In this
way, whether it is a pull-out or push-in model, all teachers are working on the same Student Learning Objective connected to the New Jersey Student
Learning Standards. The design of language objectives are based on the alignment of the World-Class Instructional Design Assessment (WIDA)
Consortium’s English Language Development (ELD) standards with the New Jersey Student Learning Standards (NJSLS). WIDA’s ELD standards
advance academic language development across content areas ultimately leading to academic achievement for English learners. As English learners
are progressing through the six developmental linguistic stages, this framework will assist all teachers who work with English learners to
appropriately identify the language needed to meet the requirements of the content standard. At the same time, the language objectives recognize the
cognitive demand required to complete educational tasks. Even though listening and reading (receptive) skills differ from speaking and writing
(expressive) skills across proficiency levels the cognitive function should not be diminished. For example, an Entering Level One student only has
the linguistic ability to respond in single words in English with significant support from their home language. However, they could complete a Venn
diagram with single words which demonstrates that they understand how the elements compare and contrast with each other or they could respond
with the support of their native language with assistance from a teacher, para-professional, peer or a technology program.
http://www.state.nj.us/education/modelcurriculum/ela/ELLOverview.pdf
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Pacing Chart – Unit 3
# Student Learning Objective NJSLS Big Ideas Math
Correlation
Instruction: 8
weeks
Assessment:
1 week
1 The Triangle Midsegment Theorem 1 G-CO.C.10
G-MG.A.1 6-4
2 Indirect Proof and Inequalities in One Triangle 2 G-CO.C.10 6-5
3 Inequalities in Two Triangles 3 G-CO.C.10 6-6
4 Angles of Polygons 4
G-CO.C.11 7-1
5 Properties of Parallelograms 5
G-CO.C.11 7-2
6 Proving That a Quadrilateral Is a Parallelogram 6 G-CO.C.11
G-SRT.B.5
G-MG.A.1
7-3
7 Properties of Special Parallelograms 7 G-CO.C.11
G-SRT.B.5
G-MG.A.1
G-MG.A.3
7-4
8 Properties of Trapezoids and Kites 8 G-CO.C.11
G-SRT.B.5
G-MG.A.1
G-MG.A.3
7-5
9 Similar Polygons 9 G-SRT.A.2
G-MG.A.3 8-1
10 Proving Triangle Similarity by AA 10
G-SRT.A.3
G-SRT.B.5 8-2
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Pacing Chart – Unit 3 11 Proving Triangle Similarity by SSS and SAS 11 G-SRT.B.4
G-SRT.B.5
G-GPE.B.5
G-MG.A.1
8-3
12 Proportionality Theorems 12 G-SRT.B.4
G-SRT.B.5
G-GPE.B.6
8-4
13 The Pythagorean Theorem 13 G-SRT.B.4
G-SRT.C.8 9-1
14 Special Right Triangles 14 G-SRT.C.8
G-MG.A.1
9-2 15 Similar Right Triangles 15 G-SRT.B.5 9-3
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Research about Teaching and Learning Mathematics Structure teaching of mathematical concepts and skills around problems to be solved (Checkly, 1997; Wood & Sellars, 1996; Wood & Sellars, 1997)
Encourage students to work cooperatively with others (Johnson & Johnson, 1975; Davidson, 1990)
Use group problem-solving to stimulate students to apply their mathematical thinking skills (Artzt & Armour-Thomas, 1992)
Students interact in ways that support and challenge one another’s strategic thinking (Artzt, Armour-Thomas, & Curcio, 2008)
Activities structured in ways allowing students to explore, explain, extend, and evaluate their progress (National Research Council, 1999)
There are three critical components to effective mathematics instruction (Shellard & Moyer, 2002):
● Teaching for conceptual understanding
● Developing children’s procedural literacy
● Promoting strategic competence through meaningful problem-solving investigations
Teachers should be:
● Demonstrating acceptance and recognition of students’ divergent ideas.
● Challenging students to think deeply about the problems they are solving, extending thinking beyond the solutions and algorithms
required to solve the problem
● Influencing learning by asking challenging and interesting questions to accelerate students’ innate inquisitiveness and foster them to
examine concepts further.
● Projecting a positive attitude about mathematics and about students’ ability to “do” mathematics
Students should be:
● Actively engaging in “doing” mathematics
● Solving challenging problems
● Investigating meaningful real-world problems
● Making interdisciplinary connections
● Developing an understanding of mathematical knowledge required to “do” mathematics and connect the language of mathematical
ideas with numerical representations
● Sharing mathematical ideas, discussing mathematics with one another, refining and critiquing each other’s ideas and understandings
● Communicating in pairs, small group, or whole group presentations
● Using multiple representations to communicate mathematical ideas
● Using connections between pictures, oral language, written symbols, manipulative models, and real-world situations
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● Using technological resources and other 21st century skills to support and enhance mathematical understanding
Mathematics is not a stagnate field of textbook problems; rather, it is a dynamic way of constructing meaning about the world around us,
generating knowledge and understanding about the real world every day. Students should be metaphorically rolling up their sleeves and “doing
mathematics” themselves, not watching others do mathematics for them or in front of them. (Protheroe, 2007)
Balanced Mathematics Instructional Model
Balanced math consists of three different learning opportunities; guided math, shared math, and independent math. Ensuring a balance of all three
approaches will build conceptual understanding, problem solving, computational fluency, and procedural fluency. Building conceptual
understanding is the focal point of developing mathematical proficiency. Students should frequently work on rigorous tasks, talk about the math,
explain their thinking, justify their answer or process, build models with graphs or charts or manipulatives, and use technology.
When balanced math is used in the classroom it provides students opportunities to:
• solve problems
• make connections between math concepts and real-life situations
• communicate mathematical ideas (orally, visually and in writing)
• choose appropriate materials to solve problems
• reflect and monitor their own understanding of the math concepts
• practice strategies to build procedural and conceptual confidence
Teacher builds conceptual understanding by
modeling through demonstration, explicit
instruction, and think alouds, as well as guiding
students as they practice math strategies and apply
problem solving strategies. (whole group or small
group instruction)
Students practice math strategies independently to
build procedural and computational fluency. Teacher
assesses learning and reteaches as necessary. (whole
group instruction, small group instruction, or centers)
Teacher and students practice mathematics
processes together through interactive
activities, problem solving, and discussion.
(whole group or small group instruction)
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Effective Pedagogical Routines/Instructional Strategies
Collaborative Problem Solving
Connect Previous Knowledge to New Learning
Making Thinking Visible
Develop and Demonstrate Mathematical Practices
Inquiry-Oriented and Exploratory Approach
Multiple Solution Paths and Strategies
Use of Multiple Representations
Explain the Rationale of your Math Work
Quick Writes/
Pair/Trio Sharing
Turn and Talk
Charting/Gallery Walks
Small Group and Whole Class Discussions
Student Modeling
Analyze Student Work
Identify Student’s Mathematical Understanding
Identify Student’s Mathematical Misunderstandings
Interviews
Role Playing
Diagrams, Charts, Tables, and Graphs
Anticipate Likely and Possible Student Responses
Collect Different Student Approaches
Multiple Response Strategies
Asking Assessing and Advancing Questions
Revoicing
Marking/Recapping
Challenging
Pressing for Accuracy and Reasoning
Maintain the Cognitive Demand
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Educational Technology
Standards
8.1.12.A.5, 8.1.12.B.2, 8.1.12.C.1
➢ Technology Operations and Concepts ● Create a report from a relational database consisting of at least two tables and describe the process, and explain the report results.
Example: Students will create tables in Excel to show the angles in right triangles are related to the ratios of the side lengths.
➢ Creativity and Innovation ● Apply previous content knowledge by creating a piloting a digital learning game or tutorial.
Example: Using video software, students will use their knowledge to create a tutorial that proves all circles are similar.
➢ Communication and Collaboration ● Develop an innovative solution to a real world problem or issue in collaboration with peers and experts, and present ideas for feedback
through social media or in an online community. Example: Students will be able to explain and prove the slope criteria for parallel and perpendicular lines and use them to solve
geometric problems on Edmodo. Students can present their solutions and communicate through this platform.
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Career Ready Practices
Career Ready Practices describe the career-ready skills that all educators in all content areas should seek to develop in their students. They are
practices that have been linked to increase college, career, and life success. Career Ready Practices should be taught and reinforced in all career
exploration and preparation programs with increasingly higher levels of complexity and expectation as a student advances through a program of
study.
● CRP2. Apply appropriate academic and technical skills.
Career-ready individuals readily access and use the knowledge and skills acquired through experience and education to be more productive.
They make connections between abstract concepts with real-world applications, and they make correct insights about when it is appropriate to
apply the use of an academic skill in a workplace situation.
Example: Students will apply prior knowledge when solving real world problems. Students will make sound judgements about the use of
specific tools, such as geometric tools and geometric software, to prove the properties of angles for a quadrilateral inscribed circle and
construct inscribed and circumscribed circles of triangles.
● CRP4. Communicate clearly and effectively and with reason.
Career-ready individuals communicate thoughts, ideas, and action plans with clarity, whether using written, verbal, and/or visual methods.
They communicate in the workplace with clarity and purpose to make maximum use of their own and others’ time. They are excellent
writers; they master conventions, word choice, and organization, and use effective tone and presentation skills to articulate ideas. They are
skilled at interacting with others; they are active listeners and speak clearly and with purpose. Career-ready individuals think about the
audience for their communication and prepare accordingly to ensure the desired outcome.
Example: Students will on a daily basis communicate their reasoning behind their solution paths by making connections to the context and
the quantities, using proper vocabulary, along with decontextualizing and/or contextualizing the problem. Students will show and explain
that definitions for trigonometric ratios derive from similarity of right triangles. They will also explain the meaning behind the quantities and
units involved. Students will also ask probing questions to clarify and improve arguments.
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Career Ready Practices
● CRP8. Utilize critical thinking to make sense of problems and persevere in solving them.
Career-ready individuals readily recognize problems in the workplace, understand the nature of the problem and devise effective plans to
solve the problem. They are aware of problems when they occur and take action quickly to address the problem; they thoughtfully investigate
the root cause of the problem prior to introducing solutions. They carefully consider the options to solve the problem. Once a solution is
agreed upon, they follow through to ensure the problem is solved, whether through their own actions or the actions of others.
Example: Throughout their daily lessons, students will understand the meaning of a problem and look for entry points into solving their
problems by analyzing the relationships of the quantities, constraints and goals of the task. Plans for solution paths will be made and have
meaning. Students will self-monitor, evaluate and critique their process and progress as they are working and make changes as necessary.
● CRP12. Work productively in teams while using cultural global competence.
Career-ready individuals positively contribute to every team, whether formal or informal. They apply an awareness of cultural difference to
avoid barriers to productive and positive interaction. They find ways to increase the engagement and contribution of all team members. They
plan and facilitate effective team meetings.
Example: Students will work in collaborative and whole group settings to develop various solutions to math tasks that are presented to them.
They will work together to understand the terms of the problem, ask clarifying and challenging questions among each other, and develop
agreed upon solutions using a variety of strategies and models. Students will listen to, read and discuss arguments with each other with
respect and courtesy at all times and will be willing to assist those that may need assistance. Students will prove and explain to a peer or small
group the slope criteria for parallel and perpendicular line and use them to solve geometric problems.
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WIDA Proficiency Levels
At the given level of English language proficiency, English language learners will process, understand, produce or use
6- Reaching
• Specialized or technical language reflective of the content areas at grade level
• A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse as
required by the specified grade level
• Oral or written communication in English comparable to proficient English peers
5- Bridging
• Specialized or technical language of the content areas
• A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse,
including stories, essays or reports
• Oral or written language approaching comparability to that of proficient English peers when presented with
grade level material.
4- Expanding
• Specific and some technical language of the content areas
• A variety of sentence lengths of varying linguistic complexity in oral discourse or multiple, related
sentences or paragraphs
• Oral or written language with minimal phonological, syntactic or semantic errors that may impede the
communication, but retain much of its meaning, when presented with oral or written connected discourse,
with sensory, graphic or interactive support
3- Developing
• General and some specific language of the content areas
• Expanded sentences in oral interaction or written paragraphs
• Oral or written language with phonological, syntactic or semantic errors that may impede the
communication, but retain much of its meaning, when presented with oral or written, narrative or expository
descriptions with sensory, graphic or interactive support
2- Beginning
• General language related to the content area
• Phrases or short sentences
• Oral or written language with phonological, syntactic, or semantic errors that often impede of the
communication when presented with one to multiple-step commands, directions, or a series of statements
with sensory, graphic or interactive support
1- Entering
• Pictorial or graphic representation of the language of the content areas
• Words, phrases or chunks of language when presented with one-step commands directions, WH-, choice or
yes/no questions, or statements with sensory, graphic or interactive support
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Culturally Relevant Pedagogy Examples
• Integrate Relevant Word Problems: Contextualize equations using word problems that reference student
interests and cultures.
Example: When learning about trigonometric ratios and the Pythagorean Theorem, problems that relate to student
interests such as music, sports and art enable the students to understand and relate to the concept in a more meaningful
way.
• Everyone has a Voice: Create a classroom environment where students know that their contributions are
expected and valued.
Example: Norms for sharing are established that communicate a growth mindset for mathematics. All students are
capable of expressing mathematical thinking and contributing to the classroom community. Students learn new ways of
looking at problem solving by working with and listening to each other.
• Run Problem Based Learning Scenarios: Encourage mathematical discourse among students by presenting
problems that are relevant to them, the school and /or the community.
Example: Using a Place Based Education (PBE) model, students explore math concepts while determining ways to
address problems that are pertinent to their neighborhood, school or culture.
• Encourage Student Leadership: Create an avenue for students to propose problem solving strategies and
potential projects.
Example: Students can deepen their understanding of the slope criteria for parallel and perpendicular lines by creating
problems together and deciding if the problems fit the necessary criteria. This experience will allow students to discuss
and explore their current level of understanding by applying the concepts to relevant real-life experiences.
• Present New Concepts Using Student Vocabulary: Use student diction to capture attention and build
understanding before using academic terms. Example: Teach math vocabulary in various modalities for students to remember. Use multi-modal activities,
analogies, realia, visual cues, graphic representations, gestures, pictures and cognates. Directly explain and model the
idea of vocabulary words having multiple meanings. Students can create the Word Wall with their definitions and
examples to foster ownership.
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SEL Competency
Examples Content Specific Activity & Approach to
SEL
✓ Self-Awareness
Self-Management
Social-Awareness
Relationship Skills
Responsible Decision-Making
Example practices that address Self-
Awareness:
• Clearly state classroom rules
• Provide students with specific feedback
regarding academics and behavior
• Offer different ways to demonstrate
understanding
• Create opportunities for students to self-advocate
• Check for student understanding / feelings about
performance
• Check for emotional wellbeing
• Facilitate understanding of student strengths and
challenges
Have students keep a math journal. This can
create a record of their thoughts, common
mistakes that they repeatedly make when solving
problems and/or a place to record how they would
approach or solve a problem.
Lead frequent discussions in class to give students
an opportunity to reflect after learning new
concepts. Discussion questions may include
asking students: “What are your strength and
weaknesses in proving the properties of angles for
a quadrilateral inscribed in a circle and construct
inscribed and circumscribed circles of a triangle
using geometric tools and geometric software?”
Self-Awareness
✓ Self-Management
Social-Awareness
Relationship Skills
Responsible Decision-Making
Example practices that address Self-
Management:
• Encourage students to take pride/ownership in
work and behavior
• Encourage students to reflect and adapt to
classroom situations
• Assist students with being ready in the classroom
• Assist students with managing their own
emotional states
Teach students to set attainable learning goals and
self-assess their progress towards those learning
goals. For example, a powerful strategy that
promotes academic growth is teaching
instructional routine to self-assess within the
Independent Phase of the Balanced Mathematics
Instructional Model.
Teach students a lesson on the proper use of
equipment (such as the computers, graphing
calculators and textbooks) and other resources
properly. Routinely ask students who they think
might be able to help them in various situations,
including if they need help with a math problem or
using an equipment.
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Self-Awareness
Self-Management
✓ Social-Awareness
Relationship Skills
Responsible Decision-Making
Example practices that address Social-
Awareness:
• Encourage students to reflect on the perspective
of others
• Assign appropriate groups
• Help students to think about social strengths
• Provide specific feedback on social skills
• Model positive social awareness through
metacognition activities
When there is a difference of opinion among
students (perhaps over solution strategies), allow
them to reflect on how they are feeling and then
share with a partner or in a small group. It is
important to be heard but also to listen to how
others feel differently, in the same situation.
Have students re-conceptualize application
problems after class discussion, by working
beyond their initial reasoning to identify common
reasoning between different approaches to solve
the same problem.
Self-Awareness
Self-Management
Social-Awareness
✓ Relationship Skills
Responsible Decision-Making
Example practices that address Relationship
Skills:
• Engage families and community members
• Model effective questioning and responding to
students
• Plan for project-based learning
• Assist students with discovering individual
strengths
• Model and promote respecting differences
• Model and promote active listening
• Help students develop communication skills
• Demonstrate value for a diversity of opinions
Have students team-up at the at the end of the unit
to teach a concept to the class. At the end of the
activity students can fill out a self-evaluation
rubric to evaluate how well they worked together.
For example, the rubric can consist of questions
such as, “Did we consistently and actively work
towards our group goals?”, “Did we willingly
accept and fulfill individual roles within the
group?”, “Did we show sensitivity to the feeling
and learning needs of others; value their
knowledge, opinion, and skills of all group
members?”
Encourage students to begin a rebuttal with a
restatement of their partner’s viewpoint or
argument. If needed, provide sample stems, such
as “I understand your ideas are ___ and I think
____because____.”
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Self-Awareness
Self-Management
Social-Awareness
Relationship Skills
✓ Responsible Decision-Making
Example practices that address Responsible
Decision-Making:
• Support collaborative decision making for
academics and behavior
• Foster student-centered discipline
• Assist students in step-by-step conflict resolution
process
• Foster student independence
• Model fair and appropriate decision making
• Teach good citizenship
Have students participate in activities that requires
them to make a decision about a situation and then
analyze why they made that decision. Students
encounter conflicts within themselves as well as
among members of their groups. They must
compromise in order to reach a group consensus.
Today’s students live in a digital world that comes
with many benefits — and also increased risks.
Students need to learn how to be responsible
digital citizens to protect themselves and ensure
they are not harming others. Educators can teach
digital citizenship through social and emotional
learning.
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Differentiated Instruction
Accommodate Based on Students Individual Needs: Strategies
Time/General
● Extra time for assigned tasks
● Adjust length of assignment
● Timeline with due dates for
reports and projects
● Communication system
between home and school
● Provide lecture notes/outline
Processing
● Extra Response time
● Have students verbalize steps
● Repeat, clarify or reword
directions
● Mini-breaks between tasks
● Provide a warning for
transitions
● Partnering
Comprehension
● Precise processes for balanced
math instructional model
● Short manageable tasks
● Brief and concrete directions
● Provide immediate feedback
● Small group instruction
● Emphasize multi-sensory
learning
Recall
● Teacher-made checklist
● Use visual graphic organizers
● Reference resources to promote
independence
● Visual and verbal reminders
● Graphic organizers
Assistive Technology
● Computer/whiteboard
● Tape recorder
● Video Tape
Tests/Quizzes/Grading
● Extended time
● Study guides
● Shortened tests
● Read directions aloud
Behavior/Attention
● Consistent daily structured
routine
● Simple and clear classroom
rules
● Frequent feedback
Organization
● Individual daily planner
● Display a written agenda
● Note-taking assistance
● Color code materials
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Differentiated Instruction
Accommodate Based on Content Specific Needs
● Anchor charts to model strategies and use of formulas
● Review Algebra concepts to ensure students have the information needed to progress in understanding
● Pre-teach vocabulary using visual models that are connected to real-life situations
● Reference sheets that list formulas
● Teacher modeling of thinking processes involved in constructing a two column or paragraph proof
● Area models to represent the Pythagorean Theorem
● Record formulas, theorems, and postulates in reference notebooks
● Use string to demonstrate arc length
● Use graph paper to represent the coordinate plane
● Word wall with visual representations of geometric terms
● Calculator to assist with computations
● Have students use a dynamic geometry software package to measure the distances and the lengths of arcs
● Utilize technology through interactive sites to explore Plane Geometry, Constructions, and Coordinate Geometry.
www.mathopenref.com https://www.geogebra.org/
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Interdisciplinary Connections
Model interdisciplinary thinking to expose students to other disciplines.
Geography Connection:
Task Name: Neglecting the Curvature of the Earth (RH.9-10.7)
● This task takes into consideration the curvature of the earth to find the distance between two locations.
Task name: How Far Is the Horizon (RH.9-10.7)
● The purpose of this modeling task is to have students use mathematics to answer a question in a real-world context using mathematical
tools that should be very familiar to them. The task gets at particular aspects of the modeling process, namely, it requires them to make
reasonable assumptions and find information that is not provided in the task statement.
Architecture and Construction Career Cluster Connection:
Task name: Access Ramp (9.3.ST-SM-2)
● You have been commissioned to design an access ramp, which complies with the Americans with Disabilities Act (ADA) requirements, for
an entry that is 3 feet above ground level. Your client has asked you to design the ramp and to determine costs, using local pricing, for two
types of ramps, wooden and concrete.
Enrichment
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What is the Purpose of Enrichment?
• The purpose of enrichment is to provide extended learning opportunities and challenges to students who have already mastered, or can quickly master, the
basic curriculum. Enrichment gives the student more time to study concepts with greater depth, breadth, and complexity.
• Enrichment also provides opportunities for students to pursue learning in their own areas of interest and strengths.
• Enrichment keeps advanced students engaged and supports their accelerated academic needs.
• Enrichment provides the most appropriate answer to the question, “What do you do when the student already knows it?”
Enrichment is…
• Planned and purposeful
• Different, or differentiated, work – not just more work
• Responsive to students’ needs and situations
• A promotion of high-level thinking skills and making connections
within content
• The ability to apply different or multiple strategies to the content
• The ability to synthesize concepts and make real world and cross-
curricular connections
• Elevated contextual complexity
• Sometimes independent activities, sometimes direct instruction
• Inquiry based or open-ended assignments and projects
• Using supplementary materials in addition to the normal range
of resources
• Choices for students
• Tiered/Multi-level activities with flexible groups (may change
daily or weekly)
Enrichment is not…
• Just for gifted students (some gifted students may need
intervention in some areas just as some other students may need
frequent enrichment)
• Worksheets that are more of the same (busywork)
• Random assignments, games, or puzzles not connected to the
content areas or areas of student interest
• Extra homework
• A package that is the same for everyone
• Thinking skills taught in isolation
• Unstructured free time
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Assessments
Required District/State Assessments Unit Assessment
NJSLA
SGO Assessments
Suggested Formative/Summative Classroom Assessments Describe Learning Vertically
Identify Key Building Blocks
Make Connections (between and among key building blocks)
Short/Extended Constructed Response Items
Multiple-Choice Items (where multiple answer choices may be correct)
Drag and Drop Items
Use of Equation Editor
Quizzes
Journal Entries/Reflections/Quick-Writes
Accountable talk
Projects
Portfolio
Observation
Graphic Organizers/ Concept Mapping
Presentations
Role Playing
Teacher-Student and Student-Student Conferencing
Homework
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New Jersey State Learning Standards
G-CO.C.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.
G-CO.C.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.
G-GPE.B.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.
G-GPE.B.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-SRT.B.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.
G-SRT.B.5: 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.
G-SRT.B.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-SRT.A.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.
G-SRT.A.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-MG.A.1: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
25 | Page
G-MG.A.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).
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
26 | Page
Geometry
Unit: 3 (Three ) Topic: Relationships within Triangles
Quadrilaterals and Other Polygons
Similarity Right Triangles and trigonometry
NJSLS: _
G-CO.C.10, G-CO.C.11, G-MG.A.1, G-MG.A.3, G-GPE.B.5, G-GPE.B.6, G-SRT.A.2, G-SRT.A.3, G-SRT.B.4, G-
SRT.B.5, G-SRT.C.8
Unit Focus:
● Understand and classifying all type of triangles and their properties.
● Proving Triangle Similarity by AA, SSS, and SAS.
● Understand all type of parallelograms and their properties.
● Solve real-life problems involving polygons.
● The Pythagorean Theorem and special triangles.
27 | Page
New Jersey Student Learning Standard(s):
G-CO.C.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.
G-MG.A.1: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Student Learning Objective 1: The Triangle Midsegment Theorem.
Modified Student Learning Objectives/Standards:
M. EE.G-MG.1–3. Use properties of geometric shapes to describe real-life objects.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 3
G-CO.C
● About 75% of tasks
align to G.CO.9 or
G.CO.10. ● Theorems include but
are not limited to the
examples listed in
standards G-CO.9,
10,11. HS.C.14.1
● Construct,
autonomously, chains
of reasoning that will
justify or refute
geometric propositions
or conjectures. Content
Use midsegments of triangles in the coordinate
plane.
Use the Triangle Midsegment Theorem to find
distances. Students will be familiar with
midsegments and the midsegment triangle from
the explorations.
State the Triangle Midsegment Theorem. Note:
This theorem is sometimes presented later in the
course after similar triangles have been
introduced.
Logically, it fits with the other special segments
in a triangle.
How are the midsegments of
a triangle related to the sides
of the triangle?
If you found the segment
midway between Pear Street
and Plum Street and parallel
to both, what would its length
be?
How do you solve the
triangle Midsegment
Theorem?
How do you know to which
side of a triangle a
Type II, III:
Classifying Triangles
Congruent angles in
isosceles triangles
Midpoints of Triangle
Sides
Eratosthenes and the
circumference of the
earth
How far is the horizon
28 | Page
scope: G-CO.9,
G-CO.10
A midsegment triangle has side lengths of 3
inches, 4 inches, and 5 inches.
Sketch the original triangle. What are the
lengths of its sides?
SPED Strategies:
Pre-teach vocabulary using visual and verbal
models that are connected to real life situations
and ensure that students include these
definitions their reference notebook.
Review algebra concepts as needed to ensure
that students have the information they need to
progress in understanding.
Model verbally and visually how to use algebra
to prove geometry theorems using contextual
problems.
ELL Strategies:
Demonstrate comprehension of how to use
coordinates to prove simple geometric theorems
algebraically by explaining the process in the
student’s native language and/or use gestures,
examples and selected technical words.
Posters/charts are a great support for ELLs to
reference. A chart can contain certain words
from a unit, definitions, helpful hints, etc. This
support provides consistency because the
Midsegment is parallel?
What are the 3 Midsegments
of a triangle?
What are the key features of
Midsegments of a triangle?
A Paper Clip
29 | Page
information on a chart/poster is always in the
same place.
Students can use bilingual (paper or electronic)
dictionaries to clearly define words that they do
not know.
Explain the process of simple geometric
theorems algebraically using key vocabulary in
the student’s native language and/simple
sentences.
Utilize Manipulatives and develop hands-on
activities.
Have students visualize actual models of shapes
and create their own
Utilize Pictures/illustrations and have student
write meaning in their Math Journals.
30 | Page
New Jersey Student Learning Standard(s):
G-CO.C.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.
Student Learning Objective 2: Indirect Proof and Inequalities in One Triangle
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 2
MP 5
G-CO.C
● About 75% of tasks
align to G.CO.9 or
G.CO.10. ● Theorems include but
are not limited to the
examples listed in
standards G-CO.9,
10,11. HS.C.14.1
● Construct,
autonomously, chains
of reasoning that will
justify or refute
geometric propositions
or conjectures. Content
scope: G-CO.9,
G-CO.10
Write indirect proofs. List sides and angles of a
triangle in order by size. Use the Triangle
Inequality Theorem to find possible side lengths
of triangles.
Scenario is suggested as a way of introducing
indirect reasoning. You may have your own
favorite example. It is helpful to have a
contextual, non-geometric example to discuss
before explaining the steps of an indirect proof.
If students have worked through the
explorations, the example and theorems will
make sense.
Pose Example 3 and have students work with
their partners to solve. Use Popsicle Sticks for
students to share their results.
Have students answer Questions 2 and 3, and
How are the sides related to
the angles of a triangle? How
are any two sides of a
triangle related to the third
side?
What would it sound like if
you combined both theorems
into one statement?
What are the inequalities in
one triangle?
What are the 3 ways to prove
triangles similar?
How do you prove triangle
inequalities?
What do you assume in an
Type II, III:
Classifying Triangles
Congruent angles in
isosceles triangles
Midpoints of Triangle
Sides
31 | Page
then share and discuss as a class.
SPED Strategies:
Review how to calculate slope with students.
Model verbally and visually how to do Indirect
Proof and Inequalities in One Triangle
Encourage students to add this concept to their
reference notebook by providing notes or
guiding notetaking.
ELL Strategies:
Explain orally and in writing work on Indirect
Proof and Inequalities in One Triangle in the
student’s native language and/or use gestures,
examples and selected technical words.
Explain orally and in writing the criteria to proof
of geometric problems in the student’s native
language and/or use selected technical
vocabulary in phrases and short sentences and
simple sentences.
Utilize Manipulatives and develop hands-on
activities.
Provide students with translation dictionary.
Have students work with partners, small groups.
indirect proof?
32 | Page
New Jersey Student Learning Standards (s):
G-CO.C.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.
Student Learning Objective 3: Inequalities in Two Triangles.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 3
MP 4
MP 6
G-CO.C
● About 75% of tasks
align to G.CO.9 or
G.CO.10. ● Theorems include but
are not limited to the
examples listed in
standards G-CO.9,
10,11. HS.C.14.1
● Construct,
autonomously, chains
of reasoning that will
justify or refute
geometric propositions
or conjectures. Content
scope: G-CO.9,
G-CO.10
Compare measures in triangles. Solve real-life
problems using the Hinge Theorem.
Write the Hinge Theorem and its converse. Both
theorems are wordy but readable. Have students
annotate the theorems with marked diagrams
that convey the meaning of the theorems.
In proving the Converse of the Hinge Theorem
(Example 3), state the given information. The
congruent sides are marked on the diagram, but
the given information that AC > DF is not. It is
Important that this is written. The contradiction
relates back to this piece of given information.
SPED Strategies:
Use a contextual problem to model the thinking,
processes and procedures involved in
What are the inequalities in
two triangles?
What are the inequalities in
one triangle?
What is the SAS Inequality
Theorem?
Can three segments with
lengths 8 ,15 and 6 make a
triangle?
Type II, III:
Classifying Triangles
Congruent angles in
isosceles triangles
Midpoints of Triangle
Sides
33 | Page
Inequalities in Two Triangles.
Encourage students to add this concept to their
reference notebook by providing notes or
guiding note taking.
Provide students with hands on opportunities to
explore and extend their understanding by
working in small groups.
ELL Strategies:
Use oral and writing explanations in the
student’s native language and/or using gestures,
graphs and selected technical words, in order to
explain how to find the point on a directed line
segment
Explain orally and in writing how to work with
Inequalities in Two Triangles in the student’s
native language and/or use gestures, examples
and selected technical words and using technical
vocabulary in phrases and short simple
sentences.
Utilize manipulatives and develop hands-on
activities.
Have students visualize actual models of shapes
and create their own.
Provide students with translation dictionary.
34 | Page
Have students work with partners, small groups.
Develop interactive games and activities to
promote retention.
New Jersey Student Learning Standard(s):
G-CO.C.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.
Student Learning Objective 4: Angles of Polygons.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement
Key/ Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 3
G-CO.C
● About 75% of tasks
align to G.CO.9 or
G.CO.10. ● Theorems include but
are not limited to the
examples listed in
standards G-CO.9,
10,11. HS.C.14.1
● Construct,
autonomously, chains
of reasoning that will
justify or refute
geometric
propositions or
Use the interior and exterior angle measures of
polygons.
Theorem 7.1 is a statement of the results found
by students in the first exploration. Note that it is
written in the factored form. Connect this to the
number of triangles formed when the diagonals
from one vertex are drawn. Note that the
theorem is stated for convex n-gons. While the
formula is valid for concave polygons, reflex
angles have not been defined.
Have students work independently on Example 1
and then compare with partners.
Students become adept at remembering the list of
What is the sum of the measures
of the interior angles of a
polygon?
Is it possible for a polygon to be
equilateral and not equiangular?
Equiangular and not equilateral?
Both? Neither?
How do you work out the angles
of an irregular polygon?
What are the angles of a polygon?
What are the angles of a 10 sided
polygon?
Type II, III:
Congruence of
parallelograms
Is this a
parallelogram
Midpoints of the
Sides of a
Parallelogram
Parallelograms
and Translations
35 | Page
conjectures. Content
scope: G-CO.9,
G-CO.10
polygon sums and may recall the number of sides
in the polygon without solving.
Students can discover that the measures of the
exterior angles of a polygon sum to 360° using
the method shown.
SPED Strategies:
Pre-teach vocabulary using visual and verbal
models that are connected to real life situations
and ensure that students include these definitions
their reference notebook.
Review angles of polygons and find the general
formula of sum of interior polygons. Bridge into
trigonometric ratios and explain how they are
used in real world to solve problems.
ELL Strategies:
Identify and explain orally and in writing the
angles of polygons in the student’s native
language and/or use gestures, examples and
selected technical words, phrases and short
simple sentences.
Utilize Manipulatives and develop hands-on
activities.
Model structure and clarify unfamiliar syntax.
What are the angles of a seven
sided polygon?
36 | Page
Provide students with translation dictionary.
Have students work with partners, small groups.
Have students visualize and create their own
conversion charts, and diagrams with
measurement labels.
New Jersey Student Learning Standard(s):
G-CO.C.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.
Student Learning Objective 5: Properties of Parallelograms.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 6
MP 8
G-CO.C
● About 75% of tasks
align to G.CO.9 or
G.CO.10. ● Theorems include but
are not limited to the
examples listed in
standards G-CO.9,
10,11. HS.C.14.1
● Construct,
autonomously, chains of
reasoning that will
Use properties to find side lengths and
angles of parallelograms. Use
parallelograms in the coordinate plane.
The Midpoint Formula is used in Example
4. Alternately, determine the equations
of the diagonals and find the point of
intersection. In Example 5, the name of the
parallelogram is given, which determines
the order of the vertices. Ask, “In general,
given three coordinates of a parallelogram,
is there more than one possible location for
the fourth vertex? Explain.”
What are the properties of
parallelograms?
If you know the measure of one
angle of a parallelogram, how
can you find the measure of the
two adjacent angles?
What do you know about the
diagonals of a parallelogram?
What is the given information
in the proof of the
Type II, III:
Congruence of
parallelograms
Is this a parallelogram
Midpoints of the Sides
of a Parallelogram
Parallelograms and
Translations
37 | Page
justify or refute
geometric propositions
or conjectures. Content
scope: G-CO.9,
G-CO.10
SPED Strategies:
Use a contextual problem to model the
thinking, processes and procedures involved
in Properties of Parallelograms.
Encourage students to add this concept to
their reference notebook by providing notes
or guiding note taking.
Provide students with hands on
opportunities to explore and extend their
understanding by working in small groups.
ELL Strategies:
Identify and explain orally and in writing
the angles of polygons in the student’s
native language and/or use gestures,
examples and selected technical words,
phrases and short simple sentences.
Utilize Manipulatives and develop hands-on
activities. Model structure and clarify
unfamiliar syntax.
Provide students with translation dictionary.
Have students work with partners, small
groups.
Have students visualize and create their own
conversion charts, and diagrams with
measurement labels.
Provide students with translation dictionary.
Develop interactive games and activities to
promote retention
parallelogram Diagonals
Theorem and what
do we need to prove?
In Example 2 ask, Does the
measure of ∠ADC always equal
110°?
What are the 7 properties of a
parallelogram?
Which has all properties of a
parallelogram?
What are the 5 properties of a
parallelogram?
How do you solve the
properties of a parallelogram
problem?
38 | Page
New Jersey Student Learning Standard(s):
G-CO.C.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.
G-SRT.B.5: 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.
G-MG.A.1: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Student Learning Objective 6: Proving That a Quadrilateral is a Parallelogram.
Modified Student Learning Objectives/Standards:
M. EE.G-MG.1–3. Use properties of geometric shapes to describe real-life objects.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 6
MP 7
G-CO.C
● About 75% of tasks align
to G.CO.9 or G.CO.10. ● Theorems include but are
not limited to the
examples listed in
standards G-CO.9, 10,11. HS.C.14.1
● Construct,
autonomously, chains of
reasoning that will justify
or refute geometric
propositions or
Identify and verify parallelograms. Show
that a quadrilateral is a parallelogram in the
coordinate plane.
Sketch quadrilateral ABCD that appears to
be a parallelogram. “What information
would you need in order to know that ABCD
is a parallelogram?” The opposite sides are
parallel. If students start to list additional
properties, remind them of the definition.
Explain to students that in this lesson they
will look back at the theorems of the last
How can you prove that a
quadrilateral is a parallelogram?
How do you prove if a
quadrilateral is a parallelogram?
What are the 6 ways to prove a
quadrilateral is a parallelogram?
How do you prove a
parallelogram?
Type II, III:
Congruence of
parallelograms
Is this a parallelogram
Midpoints of the Sides
of a Parallelogram
Parallelograms and
Translations
39 | Page
conjectures. Content
scope: G-CO.9,
G-CO.10
lesson to see whether the converse of each is
true.
Discuss the amusement park ride to be sure
students know how it operates. Listen for
student justification of why A and B are
parallel to the ground.
SPED Strategies:
Pre-teach vocabulary using visual and
verbal models that are connected to real life
situations and ensure that students include
these definitions their reference notebook.
Model how to derive the Prove That a
Quadrilateral is a Parallelogram Ensure that
students include this information in their
reference notebook.
Provide students with hands on
opportunities to explore and extend their
understanding by working in small groups,
see the application to real life
ELL Strategies:
Explain orally and in writing the equation of
a quadrilateral in the student’s native
language and/or use gestures, examples and
selected technical words, and short simple
sentences.
Review vocabulary words and provide
visual examples for students to reference.
Bank Shot
Extensions Bisections
and Dissections in a
Rectangle
Folding a square into
thirds
Tangent Line to Two
Circles
How many cells are in
the human body
How thick is a soda
can Variation I
The Lighthouse
Problem
Toilet Roll
40 | Page
New Jersey Student Learning Standard(s):
G-CO.C.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.
G-SRT.B.5: 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.
G-MG.A.1: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-MG.A.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).
Student Learning Objective 7: Properties of Special Parallelograms.
Modified Student Learning Objectives/Standards:
M. EE.G-MG.1–3. Use properties of geometric shapes to describe real-life objects.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 1
MP 2
MP 3
MP 6
G-CO.C
● Theorems include but are
not limited to the
examples listed in
standards G-CO.9, 10,
11.
Use properties of special parallelograms.
Use properties of diagonals of special
parallelograms. Use coordinate geometry to
identify special types of parallelograms.
Relationships between various quadrilaterals
can be shown in a Venn diagram or a
flowchart.
What are the properties of
the diagonals of rectangles,
rhombuses, and squares?
What do you know about
the diagonals of a rhombus
from the exploration?
Type II, III:
Congruence of
parallelograms
Is this a
parallelogram
41 | Page
G-SRT.5
Use congruence and
similarity criteria for
triangles to solve
problems and to prove
relationships in
geometric figures.
i) For example, find a
missing angle or side in a
triangle
Example 1 can be difficult for students, as
they often answer with their eyes versus
reasoning with the given information.
SPED Strategies:
Review knowledge of special parallelograms
(i.e. equation, definition) and model how to
graph all special parallelograms.
Provide students with hands on opportunities
to explore and extend their understanding by
working in small groups to graph special
parallelograms.
ELL Strategies:
Explain orally and in writing the sequence of
steps to work with special parallelograms in
the student’s native language and/or use
gestures, selected technical words and in
phrases and short simple sentences.
Have students work with partners, small
groups.
Are the diagonals of a
rectangle (that is not a
square) perpendicular?
What are the special
properties of a
parallelogram?
What are the 10 properties
of a parallelogram?
What is special about the
diagonals in a
parallelogram?
What are the 3 special
parallelograms?
Midpoints of the
Sides of a
Parallelogram
Parallelograms and
Translations
How many leaves on
a tree Version 2
How thick is a soda
can Variation I
How thick is a soda
can Variation II
Solar Eclipse
Ice Cream Cone
Satellite
Extensions Bisections
and Dissections in a
Rectangle
Folding a square into
thirds
Tangent Line to Two
Circles
42 | Page
New Jersey Student Learning Standard(s):
G-CO.C.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.
G-SRT.B.5: 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.
G-MG.A.1: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-MG.A.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).
Student Learning Objective 8: Properties of Trapezoids and Kites.
Modified Student Learning Objectives/Standards:
M. EE.G-MG.1–3. Use properties of geometric shapes to describe real-life objects.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 2
G-CO.C
● Theorems include but are
not limited to the
examples listed in
standards G-CO.9, 10,11.
G-SRT.5
Use congruence and
Use properties of trapezoids. Use the
Trapezoid Midsegment Theorem to find
distances. Use properties of kites. Identify
quadrilaterals.
What is a trapezoid and what properties does
a trapezoid have? From this assessing
question, you’ll find that students have prior
knowledge about trapezoids. The theorems
about trapezoids will make sense to students.
What are some properties of
trapezoids and kites?
What are the properties of
trapezoids and kites?
What are 3 properties of a
trapezoid?
What are the 5 properties of
a kite?
Type II, III:
Congruence of
parallelograms
Is this a parallelogram
Midpoints of the Sides
of a Parallelogram
43 | Page
similarity criteria for
triangles to solve
problems and to prove
relationships in
geometric figures.
i) For example, find a missing angle or side in a triangle
Have partners select a theorem and discuss
how to prove it. Students may use different
strategies. Allow ample work time before
asking selected students to share their proofs.
The flowchart shows how the special
quadrilaterals are related. Any quadrilateral
takes the properties and attributes of the
quadrilateral above it and adds an additional
property. It also shows that special
quadrilaterals are parallelograms, trapezoids,
or kites.
SPED Strategies:
Pre-teach vocabulary using visual and verbal
models that are connected to real life
situations and ensure that students include
these definitions their reference notebook.
Model how the relationships between
Trapezoids and Kites and calculate the
measure of unknown values. Ensure that
students include this information in their
reference notebook
Provide students with hands on opportunities
to explore and extend their understanding of
the relationships between Trapezoids and
Kites by working in small groups.
ELL Strategies:
After listening to oral explanation and reading
Can a kite be a trapezoid?
Parallelograms and
Translations
How many leaves on a
tree Version 2
How thick is a soda
can Variation I
How thick is a soda
can Variation II
Solar Eclipse
Ice Cream Cone
Satellite
Extensions Bisections
and Dissections in a
Rectangle
Folding a square into
thirds
Tangent_ Line to Two
Circles
44 | Page
the directions, demonstrate comprehension by
identifying and describing relationships
among Trapezoids and Kites in the student’s
native language and/or using gestures and
selected technical words.
Identify and describe relationships among
inscribed angles, radii, and chords; use these
relationships to solve problems.
New Jersey Student Learning Standards (s):
G-SRT.A.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.
G-MG.A.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).
Student Learning Objective 9: Similar Polygons.
Modified Student Learning Objectives/Standards:
M. EE.G-MG.1–3. Use properties of geometric shapes to describe real-life objects.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 2
MP 6
HS.C.14.5
● Construct,
autonomously, chains of
reasoning that will
justify or refute
geometric propositions
Use similarity statements. Find corresponding
lengths in similar polygons. Find perimeters
and areas of similar polygons. Decide whether
polygons are similar.
Compare a congruency transformation
How are similar polygons
related?
How can polygons be similar?
How do you prove a polygon is
similar?
Type II, III:
Ice Cream Cone
Satellite
45 | Page
or conjectures. Content
scope: G-SRT.A
(isometry) and a similarity transformation.
After comparisons are discussed, ask, “Are
congruent figures similar? Explain.”
Listen for correct reasoning.
Have partners work independently on Example
1. Have partners share solutions. Circulate and
listen to students’ conversations. The
statement of proportionality may be unfamiliar
language.
Have partners work independently on
Examples 2 and 3 and then compare how the
proportions were set up.
SPED Strategies: Pre-teach vocabulary, concepts and formulas
related to polygons, area of polygons using
visual and verbal models that are connected to
real life situations and ensure that students
include these definitions their reference
notebook.
Provide students with the formula needed to
convert degrees to radians and radians to
degrees and explain why it facilitates
understanding and is necessary.
Model how to use the concepts and formulas
to solve real life problems.
Provide students with hands on opportunities
to explore and extend their understanding of
How do you write a proportion
for similar polygons?
How do you prove shapes are
similar?
Congruent and Similar
Triangles
Similar Quadrilaterals
Similar Triangles
46 | Page
the relationships between polygons by working
in small groups.
ELL Strategies:
Describe orally and in writing the polygons in
the student’s native language and/or use
drawings, examples and selected technical
words in phrases and short simple sentences.
Provide students with hands on opportunities
to explore and extend their understanding of
the relationships between chords and angles by
working in small groups.
New Jersey Student Learning Standards (s):
G-SRT.A.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G-SRT.B.5: 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.
Student Learning Objective 10: Proving Triangle Similarity by AA
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 3
MP 4
HS.C.14.5
● Construct,
Use the Angle-Angle Similarity Theorem.
Solve real-life problems. State the Angle-
What can you conclude about
two triangles when you know
Type II, III:
47 | Page
autonomously, chains of
reasoning that will
justify or refute
geometric propositions
or conjectures. Content
scope: G-SRT.A
Angle Similarity Theorem. How can you
prove this theorem?
Students will know that the third angles are
congruent, and hopefully they will think
about transformations as a way to prove the
theorem. Pose Example 1, and have partners
solve independently.
You may consider having different groups
of students construct a subset of the
problems and then share results with the
whole class. For each pair of triangles,
students not only construct the triangles
from the given side lengths, they must also
measure all six angles. In constructing the
triangles, students should be careful to make
sure that the given angle is between the two
given sides.
SPED Strategies:
Pre-teach vocabulary, concepts and
formulas related to Proving Triangle
Similarity
Model the thinking process and procedures
needed to work proficiently.
Encourage students to verbalize their
thinking using assessing and advancing
reasoning. Use student responses to tailor
instructional strategies and meet student
that two pairs of corresponding
angles are congruent?
What are the 3 ways to prove
triangles similar?
How do you prove AA
similarity?
What is AA triangle
similarity?
What are the 5 ways to prove
triangles congruent?
Similar triangles
Extensions Bisections
and Dissections in a
Rectangle
Folding a square into
thirds
Extensions Bisections
and Dissections in a
Rectangle
Folding a square into
thirds
Tangent Line to Two
Circles
48 | Page
needs.
Provide students with hands on
opportunities to explore and extend their
understanding of the relationships between
figures by circles by working in small
groups.
ELL Strategies:
Explain orally or in writing the properties of
Triangles in the student’s native language
and/or use gestures, examples and selected
technical words in phrases and short simple
sentences.
Utilize manipulatives and develop hands-on
activities.
New Jersey Student Learning Standards (s):
G-SRT.B.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.
G-SRT.B.5: 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.
G-GPE.B.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.
49 | Page
G-MG.A.1: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Student Learning Objective 11: Proving Triangle Similarity by SSS and SAS
Modified Student Learning Objectives/Standards:
M. EE.G-MG.1–3. Use properties of geometric shapes to describe real-life objects.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 3
MP 7
G-SRT.5
For example, find a missing
angle or side in a triangle.
HS.C.14.6
● Construct,
autonomously, chains of
reasoning that will
justify or refute
geometric propositions
or conjectures. Content
scope: G-SRT.B
Use the Side-Side-Side Similarity Theorem.
Use the Side-Angle-Side Similarity Theorem.
Prove slope criteria using similar triangles.
State the SSS Similarity Theorem. Distinguish
the differences from the SSS Congruence
Theorem. To help pair corresponding sides,
use colored pencils and language such as
“shortest side to shortest side,” and so on.
Have partners read the proof of the SSS
Similarity Theorem. “What are the key steps in
the proof narrative presented?” Listen for:
drawing the auxiliary line so that the
overlapping triangles are similar, showing
smaller triangles are congruent, and using
corresponding parts and the AA Similarity
Theorem to show given triangles are similar.
Summarize the ways in which you can prove
two triangles are similar.
What are two ways to use
corresponding sides of two
triangles to determine that
the triangles are similar?
How do you prove the SAS
Similarity Theorem?
What are the 3 ways to
prove triangles similar?
What are the 5 ways to
prove triangles congruent?
How do I prove my SSS and
SAS?
Type II, III:
Extensions Bisections
and Dissections in a
Rectangle
Folding a square into
thirds
Tangent Line to Two
Circles
Bank Shot
Pythagorean Theorem
Joining two midpoints
of sides of a triangle
Equal Area Triangles
on the Same Base I
Equal Area Triangles
50 | Page
SPED Strategies:
Pre-teach vocabulary, concepts and formulas
related to figures by using visual and verbal
models that are connected to real life situations
and ensure that students include these
definitions their reference notebook.
Model the thinking process and procedures
needed to work proficiently.
Encourage students to verbalize their thinking
using assessing and advancing questions. Use
student responses to tailor instructional
strategies and meet student needs.
Provide students with hands on opportunities
to explore and extend their understanding of
the relationships between by working in small
groups.
ELL Strategies:
Explain orally or in writing the properties of
triangle in the student’s native language and/or
use gestures, examples and selected technical
words in phrases and short simple sentences.
Utilize manipulatives and develop hands-on
activities.
on the Same Base II
Parallel Lines in the
Coordinate Plane
Slope Criterion for
Perpendicular Lines
Triangles inscribed in
a circle
When are two lines
perpendicular
How many leaves on a
tree
How many leaves on a
tree Version 2
How thick is a soda
can Variation_I
51 | Page
New Jersey Student Learning Standards (s):
G-SRT.B.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.
G-SRT.B.5: 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.
G-GPE.B.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Student Learning Objective 12: Proportionality Theorems.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 3
MP 5
HS.C.14.6
Construct, autonomously,
chains of reasoning that will
justify or refute geometric
propositions or conjectures.
Content scope: G-SRT.B
Use the Triangle Proportionality Theorem
and its converse. Use other proportionality
theorems.
State the Triangle Proportionality Theorem
and its converse. State the contrapositive of
each theorem, and have students discuss
with their partners whether these
statements are valid.
Using lined paper, have students trace any
three of the lines on the paper. Use
a straightedge to draw two lines that
intersect the three parallel lines. Use this
model to discuss the Three Parallel Lines
Theorem.
What proportionality
relationships exist in a triangle
intersected by an angle bisector
or by a line parallel to one of
the sides?
What is an equivalent
proportion that results from this
construction?
How do you solve the
proportionality theorem?
What is the proportionality
theorem?
Type II, III:
Finding triangle
coordinates
Joining two midpoints
of sides of a triangle
Pythagorean Theorem
Tangent
Line to Two Circles
Pythagorean Theorem
Folding a square into
52 | Page
Pose Example 3. Use whiteboards and give
partners time to solve the problem.
Circulate and observe methods of solution.
Solicit solutions from multiple students,
assuring that more than one method will be
demonstrated. Discuss.
SPED Strategies:
Pre-teach vocabulary, concepts and
formulas related to figures inscribed within
and circumscribed by circles using visual
and verbal models that are connected to
real life situations and ensure that students
include these definitions their reference
notebook.
Model the thinking process and procedures
needed to work proficiently.
Encourage students to verbalize their
thinking using assessing and advancing
questions. Use student responses to tailor
instructional strategies and meet student
needs.
Provide students with hands on
opportunities to explore and extend their
understanding of the relationships between
figures inscribed within and circumscribed
by circles by working in small groups.
ELL Strategies:
What is the name of basic
proportionality theorem?
What is the converse of basic
proportionality theorem?
thirds
Scaling a Triangle in
the Coordinate Plane
53 | Page
Explain orally or in writing the
Proportionality Theorems in the student’s
native language and/or use gestures,
examples and selected technical words in
phrases and short simple sentences.
Utilize manipulatives and develop hands-
on activities.
New Jersey Student Learning Standards (s):
G-SRT.B.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.
G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Student Learning Objective 13: The Pythagorean Theorem.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 5
MP 6
HS.C.14.6
Construct, autonomously,
chains of reasoning that
will justify or refute
geometric propositions or
conjectures. Content
scope: G-SRT.B
Use the Pythagorean Theorem. Use the
Converse of the Pythagorean Theorem.
Classify triangles.
Have students pair up and follow the protocol
described on page T-462. Ask students to share
what they recall about the Pythagorean
Theorem from middle school. Students should
How can you prove the
Pythagorean Theorem?
Is the converse of every true
statement also true?
How do I use similar triangles
to prove the Pythagorean
Type II, III:
Joining two midpoints
of sides of a triangle
Pythagorean Theorem
The Pilot
54 | Page
HS.C.15.14
Present solutions to multi-
step problems in the form
of valid chains of
reasoning, using symbols
such as equals signs
appropriately (for
example, rubrics award
less than full credit for the
presence of nonsense
statements such as 1 + 4 =
5 + 7 = 12, even if the
final answer is correct), or
identify or describe errors
in solutions to multi-step
problems and present
corrected solutions.
Content scope: G-SRT.C
be precise in their summaries.
Pythagorean triples and their multiples are a
handy list to remember. Suggest to students that
they memorize the four key triples in bold.
Prove the Pythagorean Theorem using
similarity and the geometric means.
SPED Strategies:
Pre-teach vocabulary, concepts and formulas
related to figures inscribed within and
circumscribed by circles using visual and verbal
models that are connected to real life situations
and ensure that students include these
definitions their reference notebook.
Model the thinking process and procedures
needed to work proficiently.
Encourage students to verbalize their thinking
using assessing and advancing questions. Use
student responses to tailor instructional
strategies and meet student needs.
Provide students with hands on opportunities to
explore and extend their understanding of the
relationships between figures inscribed within
and circumscribed by circles by working in
small groups.
ELL Strategies:
theorem?
What strategies can I use to
determine missing side lengths and
areas of similar figures? How does coordinate geometry
allow you to find relationships in
triangles?
How do you solve problems that
involve measurements of
triangles?
Constructing Special
Angles
Neglecting the
Curvature of the Earth
Shortest line segment
rom a point P to a line
L
55 | Page
Explain orally or in writing the Pythagorean
Theorem in the student’s native language
and/or use gestures, examples and selected
technical words in phrases and short simple
sentences.
Utilize manipulatives and develop hands-on
activities.
New Jersey Student Learning Standards (s):
G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-MG.A.1: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Student Learning Objective 14: Special Right Triangles.
Modified Student Learning Objectives/Standards:
M. EE.G-MG.1–3. Use properties of geometric shapes to describe real-life objects.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 2
HS.C.15.14
Present solutions to multi-
step problems in the form of
valid chains of reasoning,
using symbols such as
equals signs appropriately
(for example, rubrics award
Find side lengths in special right triangles.
Solve real-life problems involving special right
triangles.
Discuss with students that the shorter leg is
also referred to as the side opposite 30° and
the longer leg is the side opposite 60°.
What is the relationship
among the side lengths of
45°- 45°- 90° triangles?
30°- 60°- 90° triangles?
What does it mean to "solve"
a triangle?
Type II, III:
Hexagonal Pattern of
Beehives
How many cells are in
56 | Page
less than full credit for the
presence of nonsense
statements such as 1 + 4 = 5
+ 7 = 12, even if the final
answer is correct), or
identify or describe errors in
solutions to multi-step
problems and present
corrected solutions. Content
scope: G-SRT.C
Solve right triangles (determine all angle
measures and all side lengths) using
trigonometric ratios and the Pythagorean
Theorem.
Solving a Right Triangle: To solve a right
triangle means to find all unknown side
lengths and angle measures. You can solve a
right triangle when you know either of the
following:
• Two side lengths
One side length and the measure of one acute
angle.
SPED Strategies:
Pre-teach vocabulary, concepts and formulas
related to figures inscribed within and
circumscribed by circles using visual and
verbal models that are connected to real life
situations and ensure that students include
these definitions their reference notebook.
Model the thinking process and procedures
needed to work proficiently.
Encourage students to verbalize their thinking
using assessing and advancing questions. Use
student responses to tailor instructional
strategies and meet student needs.
Provide students with hands on opportunities
to explore and extend their understanding of
the relationships between figures inscribed
within and circumscribed by circles by
What is the relationship
between the sine and cosine
of complementary angles?
How can right triangles be
used to solve real world
problems?
How do you memorize
special right triangles?
What 3 numbers can form a
right triangle?
What is a unique right
triangle?
the human body
How many leaves on a
tree
How many leaves on a
tree Version 2
How thick is a soda
can Variation I
How thick is a soda
can Variation II
Solar Eclipse
Ask the Pilot
Constructing Special
Angles
Neglecting the
Curvature of the Earth
57 | Page
working in small groups.
ELL Strategies:
Explain orally or in writing the Special Right
Triangles in the student’s native language
and/or use gestures, examples and selected
technical words in phrases and short simple
sentences.
Utilize manipulatives and develop hands-on
activities.
New Jersey Student Learning Standards (s):
G-SRT.B.5: 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.
Student Learning Objective 15: Similar Right Triangles.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 6
HS.C.14.6
Construct, autonomously,
chains of reasoning that will
justify or refute geometric
propositions or conjectures.
Content scope: G-SRT.B
Identify similar triangles. Solve real-life
problems involving similar triangles. Use
geometric means.
Use a colored pencil to shade the angles of the
triangles. Drawing the three triangles as non-
overlapping, and in the same orientation, is
helpful for students.
How are altitudes and
geometric means of
right triangles related?
What relationships among
sides and other segments in a
triangle are always true?
Type II, III:
Bank Shot
Extensions Bisections
and Dissections in a
Rectangle
58 | Page
Students can get lost in the notation and words
of these theorems. In Theorem 9.8, help
students see that the shortest side of the
original right triangle is also the hypotenuse
of the smallest right triangle, so that side is
written twice in the proportion. Similarly, the
altitude of the original triangle is the longer leg
of the smallest triangle, which leads to the
relationship given in Theorem 9.7.
Ask students to identify, aloud or on a paper,
the most significant point (or part) in the
lesson that aided their learning.
Use the similarity criteria to establish a new
and important relationship found in similar
triangles, such as the side splitting theorem
and the angle bisector theorem.
Side Splitting Theorem: A line parallel to one
side of a triangle divides the other two
proportionally.
Students are able to handle the easy
relationships of comparing pieces to pieces,
but when the ratio of the full length side is
used in a ratio they often compare the wrong
things. Many examples should be done to
work out this confusion.
What must be true for
triangles to be similar?
What are the 3 ways to prove
triangles similar?
What are the rules for similar
triangles?
How do you write a similarity
statement for a right triangle?
How does the use of
congruency and similarity
concepts allow us to model
relationships between
geometric figures?
Similarity unlocks many new
powerful relationships such as
geometric mean and special
right triangles. Two angles
locks in the side
proportionality of a triangle.
Folding a square into
thirds
Tangent Line to Two
Circles
59 | Page
SPED Strategies:
Pre-teach vocabulary, concepts and formulas
related to figures inscribed within and
circumscribed by circles using visual and
verbal models that are connected to real life
situations and ensure that students include
these definitions their reference notebook.
Model the thinking process and procedures
needed to work proficiently.
Encourage students to verbalize their thinking
using assessing and advancing questions. Use
student responses to tailor instructional
strategies and meet student needs.
Provide students with hands on opportunities
to explore and extend their understanding of
the relationships between figures by working
in small groups.
60 | Page
ELL Strategies:
Explain orally or in writing the properties of
Similar Right Triangles in the student’s native
language and/or use gestures, examples and
selected technical words in phrases and short
simple sentences.
Utilize manipulatives and develop hands-on
activities.
Honors Projects (must complete all)
Project 1 Project 2 Project 3
Construction – Mystery
Quadrilateral G.CO.A.1, G.CO.D.12 and G.CO.D.13
Essential Question:
What does the inscription of a quadrilateral
inside of a circle tell us about the inherent
qualities of a circle and a quadrilateral?
Skills:
1. 1. Accurately calculate and draw the figures.
2. 2. Analyze the relationship and draw
conclusions about geometric principles.
Making a Mural G.SRT.A.1
Essential Question:
How are the mathematical principles
involved in scale applied in real life
scenarios?
Skills:
1. Accurately calculate measurements
using scale factor.
2. Demonstrate the connection
between art and mathematics.
What Can I Learn from My
Shadow? G.SRT.C.8
Essential Question:
How can trigonometric ratios be used to
solve problems involving indirect
measurement?
Skills:
1.Accurately calculate values using
trigonometry. Develop a more complete understanding
of trigonometric principles.
61 | Page
Integrated Evidence Statements
G-Int.1: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-
level knowledge and skills articulated in G-MG and G-GPE.7.
● MG is the primary content.
● See examples at https://www.illustrativemathematics.org/ for G-MG.
G-C.2: Identify and describe relationships among inscribed angles, radii, and chords and apply these concepts in problem solving
situations.
● Include the relationship between central, inscribed, and circumscribed angles: inscribed angles on a diameter are right angles; the radius
of a circle is perpendicular to the tangent where the radius intersects the circle.
● This does not include angles and segment relationships with tangents and secants. Tasks will not assess angle relationships formed
outside the circle using secants and tangents.
● Tasks may involve the degree measure of an arc.
G-C.B: Find arc lengths and areas of sectors of circles. i.) Tasks involve computing arc lengths or areas of sectors given the radius and
the angle subtended; or vice versa.
G-GPE.1-1: Complete the square to find the center and radius of a circle given by an equation. i.) The "derive" part of standard G-
GPE.1 is not assessed here.
G-GPE.1-2 Understand or complete a derivation of the equation of a circle of given center and radius using the Pythagorean Theorem;
complete the square to find the center and radius of a circle given by an equation.
● Tasks must go beyond simply finding the center and radius of a circle.
G-GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
● Trigonometric ratios include sine, cosine, and tangent only.
62 | Page
Integrated Evidence Statements
G-SRT.7-2: Use the relationship between the sine and cosine of complementary angles.
● The "explain" part of standard G-SRT.7 is not assessed here; See Sub-Claim C for this aspect of the standard.
G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
● The task may have a real world or mathematical context. For rational solutions, exact values are required. For irrational solutions, exact
or decimal approximations may be required. Simplifying or rewriting radicals is not required; however, students will not be penalized if
they simplify the radicals correctly.
HS.C.13.1 Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions. Content scope: G-
GPE.6, G-GPE.7.
HS.C.13.2 Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions. Content scope: G-
GPE.4.
HS.C.13.3 Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions. Content scope: G-
GPE.5. HS.C.15.14: Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs
appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1 + 4 = 5 + 7 = 12, even if
the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions. Content
scope: G-SRT.C
HS.C.18.2 Use a combination of algebraic and geometric reasoning to construct, autonomously, chains of reasoning that will justify or
refute propositions or conjectures about geometric figures. Content scope: Algebra content from Algebra 1 course; geometry content from
the Geometry course.
● For the Geometry course, we are reaching back to Algebra 1 to help students synthesize across the two subjects.
HS.D.1-2: Solve multi-step contextual problems with degree of difficulty appropriate to the course, requiring application of knowledge
and skills articulated in 6.G, 7.G, and/or 8.G.
63 | Page
Integrated Evidence Statements
HS.D.2-1 Solve multi-step contextual problems with degree of difficulty appropriate to the course involving perimeter, area, or volume
that require solving a quadratic equation.
● Tasks do not cue students to the type of equation or specific solution method involved in the task.
o For example: An artist wants to build a right-triangular frame in which one of the legs exceeds the other in length by 1 unit, and
in which the hypotenuse exceeds the longer leg in length by 1 unit. Use algebra to show that there is one and only one such right
triangle, and determine its side lengths.
HS.D.2-2: Solve multi-step contextual problems with degree of difficulty appropriate to the course involving perimeter, area, or volume
that require finding an approximate solution to a polynomial equation using numerical/graphical means.
● Tasks may have a real world or mathematical context.
● Tasks may involve coordinates (G-GPE.7).
● Refer to A-REI.11 for some of the content knowledge from the previous course relevant to these tasks.
● Cubic polynomials are limited to polynomials in which linear and quadratic factors are available.
● To make the tasks involve strategic use of tools (MP.5), calculation and graphing aids are available but tasks do not prompt the student
to use them
HS.D.2-11: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of
course-level knowledge and skills articulated in G-SRT.8, involving right triangles in an applied setting.
● Tasks may, or may not, require the student to autonomously make an assumption or simplification in order to apply techniques of right
triangles.
o For example, a configuration of three buildings might form a triangle that is nearly, but not quite, a right triangle; then, a good
approximate result can be obtained if the student autonomously approximates the triangle as a right triangle.
HS.D.3-2a: Micro-models: Autonomously apply a technique from pure mathematics to a real-world situation in which the technique
yields valuable results even though it is obviously not applicable in a strict mathematical sense (e.g., profitably applying proportional
relationships to a phenomenon that is obviously nonlinear or statistical in nature). Content Scope: Knowledge and skills articulated in the
Geometry Type I, Sub-Claim A Evidence Statements.
HS.D.3-4a: Reasoned estimates: Use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an
unknown quantity. Content Scope: Knowledge and skills articulated in the Geometry Type I, Sub-Claim A Evidence Statements
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Unit 3 Vocabulary
● Adjacent
● Arc
● Arc length
● Area
● Center
● Center
● Central angle
● Chords
● Circle
● Circumference
● Circumscribed angle
● Complementary angles
● Completing the square
● Construct
● Converse
● Coordinate plane
● Coordinates
● Cosine
● Diameter
● Dimension
● Directrix
● Distance formula
● Equation of a circle
● Equivalent slopes
● Focus
● Heron’s Formula
● Hypotenuse
● Inscribed angle
● Inscribed Angles of a Circle Theorem
● Inscribed Quadrilateral Theorem
● Inscribed Right Triangle Theorem
● Intercepted
● Intersects
● Length
● Length of an arc
● Line segment
● Parallel lines
● Perimeter
● Perpendicular lines
● Point on the circle
● Polygon
● Proportional relationship
● Pythagorean Theorem
● Quadrilateral
● Radian
● Radii
● Radius
● Ratio
● Rectangle
● Right angle
● Right triangle
● Secant
● Similar Circles Theorem
● Similarity
● Sine
● Sine and Cosine of Complementary
Angles
● Slope criteria
● Subtend
● Tangent
● Theorem
● Trigonometric ratio
● Vertex
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References & Suggested Instructional Websites
Desmos https://www.desmos.com/
GeoGebra http://www.geogebra.org/
Cpams http://www.cpalms.org/Public/ToolkitGradeLevelGroup/Toolkit?id=14
Common Core Standards www.corestandards.org
National Council of Teacher of Mathematics www.nctm.org
Khan Academy https://www.khanacademy.org
Achieve the Core http://achievethecore.org
Illustrative Mathematics https://www.illustrativemathematics.org/
Inside Mathematics www.insidemathematics.org
Learn Zillion https://learnzillion.com/resources/75114-math
National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html
NRICH http://nrich.maths.org
YouCubed https://www.youcubed.org/week-of-inspirational-math/
NCTM Illuminations http://illuminations.nctm.org/Lessons-Activities.aspx (choose grade level and connect to search lessons)
CK-12 www.ck12.org
Mathematics Assessment Project http://map.mathshell.org/tasks.php?collection=9&unit=HE06
Shmoop http://www.shmoop.com/common-core-standards/math.html
Mathematics Common Core Toolbox http://www.ccsstoolbox.org/
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Field Trip Ideas SIX FLAGS GREAT ADVENTURE-This educational event includes workbooks and special science and math related shows throughout the
day. Your students will leave with a better understanding of real world applications of the material they have learned in the classroom. Each
student will have the opportunity to experience different rides and attractions linking mathematical and scientific concepts to what they are
experiencing.
www.sixflags.com
MUSEUM of MATHEMATICS- Mathematics illuminates the patterns that abound in our world. The National Museum of Mathematics strives
to enhance public understanding and perception of mathematics. Its dynamic exhibits and programs stimulate inquiry, spark curiosity, and reveal
the wonders of mathematics. The Museum’s activities lead a broad and diverse audience to understand the evolving, creative, human, and
aesthetic nature of mathematics.
www.momath.org
LIBERTY SCIENCE CENTER - An interactive science museum and learning center located in Liberty State Park. The center, which first
opened in 1993 as New Jersey's first major state science museum, has science exhibits, the largest IMAX Dome theater in the United States,
numerous educational resources, and the original Hoberman sphere.
http://lsc.org/plan-your-visit/