mathematics - paterson.k12.nj.uspaterson.k12.nj.us/11_curriculum/math/math... · 4 show and explain...

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MATHEMATICS

Geometry: Unit 3

Relationships within Triangles

Quadrilaterals and Other Polygons

Similarity Right Triangles and Trigonometry

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Course Philosophy/Description

Geometry stresses the ability to reason logically and to think critically, using spatial sense. 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.

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; and write basic mathematical

arguments in paragraph and statement-reason form.

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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

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


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


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


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/

http://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



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):





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





Student Learning Objective 2: Indirect Proof and Inequalities in One Triangle

Modified Student Learning Objectives/Standards: N/A


Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 5

G-CO.C


align to G.CO.9 or



examples listed in

standards G-CO.9,

10,11. HS.C.14.1

● Construct,



justify or refute



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:


Congruent angles in

isosceles triangles


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.


activities.

Provide students with translation dictionary.

Have students work with partners, small groups.

indirect proof?

32 | Page

New Jersey Student Learning Standards (s):


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.



Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 3

MP 4

MP 6

G-CO.C


align to G.CO.9 or



examples listed in

standards G-CO.9,

10,11. HS.C.14.1

● Construct,



justify or refute



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


two triangles?


one triangle?

What is the SAS Inequality

Theorem?

Can three segments with

lengths 8 ,15 and 6 make a

triangle?

Type II, III:


Congruent angles in

isosceles triangles


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.


34 | Page


Develop interactive games and activities to

promote retention.


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.


MPs Evidence Statement

Key/ Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 3

G-CO.C


align to G.CO.9 or



examples listed in

standards G-CO.9,

10,11. HS.C.14.1

● Construct,



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:



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.


activities.

Model structure and clarify unfamiliar syntax.

What are the angles of a seven

sided polygon?

36 | Page



Have students visualize and create their own

conversion charts, and diagrams with

measurement labels.


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.



Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 6

MP 8

G-CO.C


align to G.CO.9 or



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



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.


activities. Model structure and clarify

unfamiliar syntax.


Have students work with partners, small

groups.

Have students visualize and create their own

conversion charts, and diagrams with

measurement labels.


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?


properties of a parallelogram

problem?

38 | Page








Student Learning Objective 6: Proving That a Quadrilateral is a Parallelogram.




Clarifications


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,


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


What are the 6 ways to prove a


How do you prove a

parallelogram?

Type II, III:

Congruence of

parallelograms



of a Parallelogram

Parallelograms and

Translations

39 | Page


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.



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


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










Student Learning Objective 7: Properties of Special Parallelograms.




Clarifications


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.


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


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



Rectangle


thirds

Tangent Line to Two

Circles

42 | Page










Student Learning Objective 8: Properties of Trapezoids and Kites.




Clarifications


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?


trapezoids and kites?

What are 3 properties of a

trapezoid?

What are the 5 properties of

a kite?

Type II, III:

Congruence of

parallelograms



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:


models that are connected to real life



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


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



Rectangle


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.


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





Student Learning Objective 9: Similar Polygons.




Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 6

HS.C.14.5

● Construct,


reasoning that will

justify or refute


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


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.



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.



the relationships between chords and angles by



G-SRT.A.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.




Student Learning Objective 10: Proving Triangle Similarity by AA



Clarifications


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


reasoning that will

justify or refute



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?


triangles similar?

How do you prove AA

similarity?

What is AA triangle

similarity?


triangles congruent?

Similar triangles



Rectangle


thirds



Rectangle


thirds

Tangent Line to Two

Circles

48 | Page

needs.



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.


activities.







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


Student Learning Objective 11: Proving Triangle Similarity by SSS and SAS




Clarifications


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,


reasoning that will

justify or refute



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:



Rectangle


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






Encourage students to verbalize their thinking

using assessing and advancing questions. Use

student responses to tailor instructional

strategies and meet student needs.



the relationships between by working in small

groups.

ELL Strategies:


triangle in the student’s native language and/or

use gestures, examples and selected technical

words in phrases and short simple sentences.


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


tree


tree Version 2

How thick is a soda

can Variation_I

51 | Page







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.



Clarifications


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?


proportionality theorem?

What is the proportionality

theorem?

Type II, III:

Finding triangle

coordinates



Pythagorean Theorem

Tangent

Line to Two Circles

Pythagorean Theorem


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.



Encourage students to verbalize their

thinking using assessing and advancing

questions. Use student responses to tailor

instructional strategies and meet student

needs.



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


What is the converse of basic


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


Utilize manipulatives and develop hands-

on activities.





Student Learning Objective 13: The Pythagorean Theorem.



Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 5

MP 6

HS.C.14.6


chains of reasoning that

will justify or refute

geometric propositions or


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:



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:


related to figures inscribed within and

circumscribed by circles using visual and verbal










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



sentences.


activities.




Student Learning Objective 14: Special Right Triangles.




Clarifications


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:



circumscribed by circles using visual and












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


tree


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


ELL Strategies:

Explain orally or in writing the Special Right

Triangles in the student’s native language



sentences.


activities.





Student Learning Objective 15: Similar Right Triangles.



Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 6

HS.C.14.6


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



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?


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.


thirds

Tangent Line to Two

Circles

59 | Page

SPED Strategies:



circumscribed by circles using visual and












the relationships between figures by working

in small groups.

60 | Page

ELL Strategies:


Similar Right Triangles in the student’s native


selected technical words in phrases and short

simple sentences.


activities.

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.

https://www.illustrativemathematics.org/

62 | Page


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.


GPE.4.


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


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

64 | Page

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

65 | Page

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/

https://www.desmos.com/

http://www.geogebra.org/

http://www.cpalms.org/Public/ToolkitGradeLevelGroup/Toolkit?id=14

http://www.corestandards.org/

http://www.nctm.org/

https://www.khanacademy.org/

http://achievethecore.org/

https://www.illustrativemathematics.org/

http://www.insidemathematics.org/

https://learnzillion.com/resources/75114-math

http://nlvm.usu.edu/en/nav/vlibrary.html

http://nrich.maths.org/

https://www.youcubed.org/week-of-inspirational-math/

http://illuminations.nctm.org/Lessons-Activities.aspx

http://www.ck12.org/

http://map.mathshell.org/tasks.php?collection=9&unit=HE06

http://www.shmoop.com/common-core-standards/math.html

http://www.ccsstoolbox.org/

66 | Page

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/

http://www.sixflags.com/

http://www.momath.org/

https://en.wikipedia.org/wiki/Interactive

https://en.wikipedia.org/wiki/Science_museum

https://en.wikipedia.org/wiki/Liberty_State_Park

https://en.wikipedia.org/wiki/IMAX_Dome

https://en.wikipedia.org/wiki/Hoberman_sphere

http://lsc.org/plan-your-visit/

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