1 | P a g e

MATHEMATICS

Algebra II Honors: Unit 3

Periodic Models and the Unit Circle

2 | P a g e

Course Philosophy/Description

Algebra II Honors course is for accelerated students. This course is designed for students who exhibit high interest and

knowledge in math and science. In this course, students will extend topics introduced in Algebra I and learn to manipulate and

apply more advanced functions and algorithms. 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.

Students will be expected to demonstrate their knowledge in: utilizing essential algebraic concepts to perform calculations on

polynomial expression; performing operations with complex numbers and graphing complex numbers; solving and graphing

linear equations/inequalities and systems of linear equations/inequalities; solving, graphing, and interpreting the solutions of

quadratic functions; solving, graphing, and analyzing solutions of polynomial functions, including complex solutions;

manipulating rational expressions, solving rational equations, and graphing rational functions; solving logarithmic and

exponential equations; and performing operations on matrices and solving matrix equations.

3 | P a g e

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

4 | P a g e

Pacing Chart – Unit 3

# Student Learning Objective NJSLS Big Ideas Math

Correlation

Instruction: 8 weeks

Assessment: 1 week

1

Use the radian measure of an angle to find the length of

the arc in the unit circle subtended by the angle and find

the measure of the angle given the length of the arc.

F.TF.A.1

9.2

2

Explain how the unit circle in the coordinate plane

enables the extension of trigonometric functions to all

real numbers, interpreted as radian measures of angles

traversed counterclockwise around the unit circle.

F.TF.A.2

9.3

3

Graph trigonometric functions expressed symbolically,

showing key features of the graph, by hand in simple

cases and using technology for more complicated cases.

F.IF.C.7e

F.IF.B.4

9.4, 9.5

4

Choose trigonometric functions to model periodic

phenomena with specified amplitude, frequency, and

midline.

F.TF.B.5

9.6

5

Use the Pythagorean identity sin2(θ) + cos2(θ) = 1 to find

sin θ , cos θ , or tan θ , given sin θ , cos θ , or tan θ , and

the quadrant of the angle.

F.TF.C.8

9.7

6

Represent nonlinear (exponential and trigonometric) data

for two variables on a scatter plot, fit a function to the

data, analyze residuals (in order to informally assess fit),

and use the function to solve problems. Use given

functions or choose a function suggested by the context;

emphasize exponential and trigonometric models.

S.ID.B.6a

6.7, 9.6

7

Analyze and compare properties of two functions when

each is represented in a different way (algebraically,

graphically, numerically in tables, or by verbal

descriptions).

F.IF.C.9

9.5, 9.6

8

Construct a function that combines, using arithmetic

operations, standard function types to model a

relationship between two quantities.

F.BF.A.1b

N.Q.A.2

A.APR.B.3*

4.4, 4.5, 4.6, 4.8, 5.5,

9.6

5 | P a g e

Pacing Chart – Unit 3

9

Identify the effect on the graph of a polynomial,

exponential, logarithmic, or trigonometric function of

replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for

specific values of k (both positive and negative). Find

the value of k given the graphs and identify even and odd

functions from graphs and equations.

F.BF.B.3

4.7, 4.8, 5.3, 6.4, 7.2,

9.4, 9.5

10(+)

Find inverse functions, use them to solve equations and

verify if a function is the inverse of another. Read values

of an inverse function from a graph or table and use

restricted domain to produce invertible functions from

previously non-invertible functions.

F.BF.B.4a-d*

5.6, 6.3 It is suggested that this

SLO is taught after SLO

9.

11(+)

Use special triangles to determine the values of sine,

cosine, tangent at various point along the unit circle. Use

the unit circle to explain the symmetry and periodicity of

trigonometric functions.

F.TF.A.3

F.TF.A.4

N/A It is suggested that this

SLO is taught after SLO

2.

+ These standards/SLOs have been added designed specifically for Algebra 2 Honors to deepen and

extend student understanding and to better prepare students for Pre-Calculus.

6 | P a g e

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

Using technological resources and other 21st century skills to support and enhance mathematical understanding

7 | P a g e

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)

8 | P a g e

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

9 | P a g e

Educational Technology

Standards

8.1.12.A.1, 8.1.12.C.1, 8.1.12.F.1, 8.2.12.E.3

Technology Operations and Concepts

Create a personal digital portfolio which reflects personal and academic interests, achievements, and career aspirations by using a

variety of digital tools and resources.

Example: Students create personal digital portfolios for coursework using Google Sites, Evernote, WordPress, Edubugs, Weebly, etc.

http://www.edudemic.com/tools-for-digital-portfolios/

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: Use Google Classroom for real-time communication between teachers, students, and peers to complete assignments and

discuss strategies for analyzing and comparing properties of two functions when each is represented in a different way (algebraically,

graphically, and numerically in tables, or by verbal descriptions).

Critical Thinking, Problem Solving, and Decision Making

Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal or social needs.

Example: Students use graphing calculators and graph paper to reveal the strengths and weaknesses of technology associated with

graphing trigonometric functions.

Computational Thinking: Programming

Use a programming language to solve problems or accomplish a task (e.g., robotic functions, website designs, applications and

games).

Example: Students will create a set of instructions explaining how to use the Pythagorean Identity to find sin Ɵ, cos Ɵ, tan Ɵ, given sin Ɵ,

cos Ɵ, or tan Ɵ, and the quadrant of the angle.

Link: http://www.state.nj.us/education/cccs/2014/tech/

10 | P a g e

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 graph paper, graphing calculators and technology to deepen understanding of finding the length of an arc in a unit

circle.

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 communicate precisely using clear definitions and provide carefully formulated explanations when constructing

arguments. Students will communicate and defend mathematical reasoning using objects, drawings, diagrams, and/or actions. Students

will ask probing questions to clarify or improve arguments.

11 | P a g e

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: Students will understand the meaning of a problem and look for entry points to its solution. They will analyze information,

make conjectures, and plan a solution pathway to solve simple rational and radical equations.

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 collaboratively in groups to solve mathematical tasks. Students will listen to or read the arguments of

others and ask probing questions to clarify or improve arguments. They will be able to explain how to graph trigonometric functions.

12 | P a g e

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

13 | P a g e

14 | P a g e

15 | P a g e

Culturally Relevant Pedagogy Examples

Integrate Relevant Word Problems: Contextualize equations using word problems that reference student interests and

cultures.

Example: When learning about trigonometric functions, 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 such as systems of

equations 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 learn to interpret functions in a context 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.

16 | P a g e

17 | P a g e

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 record how they

would approach or solve a problem.

Lead frequent discussion in class to give

students an opportunity to reflect after learning

new concepts. Discussion questions may

include asking students: “What difficulties do

you have when asked to analyze and compare

properties of two functions when each is

represented in different ways (algebraically,

graphically, numerically in tables, or by verbal

descriptions).

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 routines 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.

18 | P a g e

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____.”

19 | P a g e

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.

20 | P a g e

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

21 | P a g e

Differentiated Instruction

Accommodate Based on Content Specific Needs

Anchor charts to model strategies for finding the length of the arc of a circle

Review Algebra concepts to ensure students have the information needed to progress in understanding

Pre-teach pertinent vocabulary

Provide reference sheets that list formulas, step-by-step procedures, theorems, and modeling of strategies

Word wall with visual representations of mathematical terms

Teacher modeling of thinking processes involved in solving, graphing, and writing equations

Introduce concepts embedded in real-life context to help students relate to the mathematics involved

Record formulas, processes, and mathematical rules in reference notebooks

Graphing calculator to assist with computations and graphing of trigonometric functions

Utilize technology through interactive sites to represent nonlinear data

www.mathopenref.com https://www.geogebra.org/

22 | P a g e

Interdisciplinary Connections

Model interdisciplinary thinking to expose students to other disciplines.

Science Connection: Name of Task: Throwing Baseballs Science Standard MS-PS2-2

This task could be used for assessment or for practice. It allows the students to compare characteristics of two quadratic functions that are

each represented differently, one as the graph of a quadratic function and one written out algebraically.

Name of Task: Bicycle Wheel Science Standard HS-PS2-2

The purpose of this task is to introduce radian measure for angles in a situation where it arises naturally. Radian measure focuses on the arc

length of a circle cut out by a given angle. Degree measure, on the other hand, focuses on the angle. If the radius of the circle were one unit,

then the radian measure table would be particularly simple to fill out. Even here where the radius is not one unit, the radian angle measure

is more ''natural'' for this scenario because it measures the length of a circular arc and the distance a wheel is traveling is also the length of a

circular arc.

Name of Task: Temperatures in degrees Fahrenheit and Celsius Science Standard HS-ESS3-5

Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our everyday lives

when we travel abroad.

23 | P a g e

Enrichment

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

24 | P a g e

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

25 | P a g e

New Jersey Student Learning Standards

F.TF.A.1:

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

F.TF.A.2:

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian

measures of angles traversed counterclockwise around the unit circle.

F.TF.A.3:

Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /6, and use the unit circle to express the values

of sine, cosines, and tangent for x, +x, and 2–x in terms of their values for x, where x is any real number.

F.TF.A.4:

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F.TF.B.5:

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

F.TF.C.8

Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the

angle.

F.IF.B.4:

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch

graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is

increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.C.7:

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated

cases.

26 | P a g e

New Jersey Student Learning Standards

F.IF.C.7e :

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and

amplitude.

F.IF.C.9:

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal

descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

S.ID.B.6:

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related

S.ID.B.6a:

Fit a function to the data (including with the use of technology); use functions fitted to data to solve problems in the context of the data. Use

given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

F.BF.A.1:

Write a function that describes a relationship between two quantities.

F.BF.A.1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body

by adding a constant function to a decaying exponential, and relate these functions to the model.

F.BF.B.3:

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the

value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing

even and odd functions from their graphs and algebraic expressions for them.

F.BF.B.4:

Find inverse functions.

F.BF.B.4a:

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2

x3 or f(x) = (x+1)/(x–1) for x ≠1. [*note: composition of functions is not introduced here]

27 | P a g e

New Jersey Student Learning Standards

F.BF.B.4b: Verify by composition that one function is the inverse of another.

F.BF.B.4c: Read values of an inverse function from a graph or a table, given that the function has an inverse.

F.BF.B.4d: Produce an invertible function from a non-invertible function by restricting the domain.

N.Q.A.2:

Define appropriate quantities for the purpose of descriptive modeling.

A.APR.B.3:

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the

polynomial.

28 | P a g e

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.

29 | P a g e

Course: Algebra II Honors Unit: 3 (Three) Topic: Periodic Models and the Unit Circle

NJSLS: F.TF.A.1, F.TF.A.2, F.TF.A.3, F.TF.A.4, F.IF.C.7, F.IF.C.7e, F.IF.B.4, F.TF.B.5, F.TF.C.8, S.ID.B.6, S.ID.B.6a, F.IF.C.9, F.BF.A.1,

F.BF.A.1b, N.Q.A.2, A.APR.B.3, F.BF.B.3, F.BF.B.4, F.BF.B.4a, F.BF.B.4a, F.BF.B.4b,F.BF.B.4c,F.BF.B.4d

Unit Focus: Extend the domain of trigonometric functions using the unit circle

Analyze functions using different representations

Interpret functions that arise in applications in terms of the context

Model periodic phenomena with trigonometric functions

Prove and apply trigonometric identities

Summarize, represent, and interpret data on two categorical and quantitative variables

Build a function that models a relationship between two quantities

Build new functions from existing functions

New Jersey Student Learning Standard(s): F.TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Student Learning Objective 1: Use the radian measure of an angle to find the length of the arc in the unit circle subtended by the angle and

find the measure of the angle given the length of the arc.

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 6

N/A Understand that radian measure of an angle as

the length of the arc on the unit circle that is

subtended by the angle

Relationship between degrees and radians.

How can you find measure of

an angle in radians and what is

its relationship to degree

measurement?

Type II, III:

Bicycle Wheel

30 | P a g e

The unit circle is a circle with radius of length

1 centered at the origin.

Find the measure of the angle given the length

of the arc.

Find the length of an arc given the measure of

the central angle.

Determine the radian measure of an angle

using the formula arc length S

radius r .

Convert between radians and degrees.

Recognize the equivalence of commonly used

angle measures given in degrees and their

radian measures in terms of .

(e.g. 360 2 ,180 ,902

, 45

4

,

603

and 30

6

)

SPED Strategies:

Pre-teach vocabulary using visual and verbal

models that are connected to real life

situations.

Model the thinking and processes behind

radian measure linking it to degree measure

and prior learning from Geometry.

Explain what radian measure is

and how it is applied to real

life situations.

What is the Unit Circle and

why do you need it?

What Exactly is a

Radian?

31 | P a g e

Create a reference document with all necessary

terms, formulas, processes and sample

problems to encourage proficiency and

independence.

Provide students with opportunities to practice

skills and concepts involved in radian measure

using contextually based problems.

ELL Strategies:

Demonstrate comprehension of complex word

problems in the student’s native language

and/or problems with Visuals and selected

technical words by answering questions in

writing using the radian measure of an angle to

find the length of the arc in the unit circle and

finding the measure of the angle given the

length of the arc.

Use PAIRED VERBAL FLUENCY (PVF):

Let student take turns speaking, uninterrupted

for a specified period of time talking about

previous knowledge of a “circle”.

Provide students with a visual aid when

defining radian measure. Use “string

wrapping” to show that the length of the circle

can be wrapped around the circle.

32 | P a g e

New Jersey Student Learning Standard(s): F.TF.A.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as

radian measures of angles traversed counterclockwise around the unit circle.

Student Learning Objective 2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real

numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

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 6

N/A Find the measure of the angle given the length of

the arc.

Find the length of an arc given the measure of

the central angle.

Convert between radians and degrees.

Use the unit circle to evaluate sine, cosine and

tangent of standard reference angles.

Identify, label and be able to use a unit circle to

solve problems.

Define an angle in standard form as an angle

drawn on a plane that has its vertex at the origin

and its initial side along the positive x-axis.

Define the sine, cosine, tangent, cosecant, secant

and cotangent functions using the unit circle.

How can you use the unit

circle to define the

trigonometric functions of any

angle?

How does the unit circle let

you extend trigonometric

functions to all real numbers?

Type II, III:

Trig Functions and

the Unit Circle

Additional Tasks:

Properties of

Trigonometric

Functions

Trigonometric

functions

for arbitrary angles

(radians)

What exactly is a

radian?

33 | P a g e

Identify the domain and range of the

trigonometric functions based on their

definitions in terms of the unit circle.

Determine the output values of trigonometric

functions for input values whose reference

angles have measures of

30 ;45 ;606 4 3

, without using a

calculator or table.

SPED Strategies:

Model the thinking and processes behind the

Unit Circle, radians, degree measure and real life

applications.

Create a reference document with all necessary

terms, formulas, processes and sample problems

to encourage proficiency and independence.

Provide students with opportunities to practice

using skills developed and concepts involved

working in small groups with contextually based

problems.

ELL Strategies:

Explain orally and in writing how the unit circle

in the coordinate plane enables the extension of

trigonometric functions to all real numbers and

use the Pythagorean identity to find sin θ, cos θ,

or tan θ, given sin θ, cos θ, or tan θ, and the

quadrant of the angle using key, technical

vocabulary in simple sentences.

34 | P a g e

Write a mnemonic that will help them remember

the unit circle properties.

Have students recite the measure of the sides of

triangles 30-60-90, and 45-45-90 to help them

understand the measure of the unit circle.

New Jersey Student Learning Standard(s): F.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more

complicated cases.

F.IF.C.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period,

midline, and amplitude.

F.IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and

sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is

increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Student Learning Objective 3: Graph trigonometric functions expressed symbolically, showing key features of the graph, by hand in simple

cases and using technology for more complicated cases.

Modified Student Learning Objectives/Standards: M.EE.F-IF.1–3: Use the concept of function to solve problems.

M.EE.F-IF.4–6: Construct graphs that represent linear functions with different rates of change and interpret which is faster/slower, higher/lower, etc.

MPs Evidence Statement

Key/ Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

35 | P a g e

MP 1

MP 4

MP 5

MP 6

MP 7

F-IF.7e-2

About half of tasks

involve logarithmic

functions, while the

other half involves

trigonometric

functions.

F-IF.4-2

For an exponential,

polynomial,

trigonometric, or

logarithmic function

that models a

relationship between

two quantities,

interpret key features

of graphs and tables in

terms of the

quantities, and sketch

graphs showing key

features given a verbal

description of the

relationship. Key

features include:

intercepts; intervals

where the function is

increasing,

decreasing, positive,

or negative; relative

maximums and

minimums; end

behavior; symmetries;

and periodicity.

Relationship between the unit circle in the

coordinate plane and graph of trigonometric

functions.

Graph trigonometric functions, showing period,

midline, and amplitude.

Key features of a graph or table may include

intercepts; intervals in which the function is

increasing, decreasing or constant; intervals in

which the function is positive, negative or zero;

symmetry; maxima; minima; and end behavior.

Given a verbal description of a relationship that

can be modeled by a function, a table or graph

can be constructed and used to interpret key

features of that function.

SPED Strategies:

Model the relationship between the Unit Circle

in the coordinate plane and the graphs of

trigonometric functions highlighting and

defining key features.

Provide students with a reference document that

illustrates verbally and pictorially the important

characteristics of these functions and their

graphical representation.

Demonstrate how to use technology to solve

more complicated trigonometric functions and

provide students with the opportunity to practice

with peers.

How can you describe the

shape of a graph?

How can you relate the shape

of a graph to the meaning of

the relationship it represents?

How would you determine the

appropriate domain for a

function describing a real-

world situation?

How do exponential functions

model real-world problems

and their solutions?

How do logarithmic functions

model real-world problems

and their solutions?

How can you transform the

graphs of exponential and

logarithmic functions and

when?

How are exponential functions

and logarithmic functions

related?

Type II, III:

Logistic Growth

Model, Abstract

Version

Logistic Growth

Model, Explicit

Version

Telling a Story With

Graphs

Warming and

Cooling

Exponential Kiss

Additional Tasks:

Identifying

Exponential

Functions

Lake Sonoma

Model Airplane

Acrobatics

Playing Catch

The Aquarium

The Story of a Flight

36 | P a g e

New Jersey Student Learning Standard(s): F.TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Student Learning Objective 4: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and

midline.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement

Key/ Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 4

N/A Periodic functions may model real-world

scenarios.

Use characteristics of real world phenomena to

select a trigonometric model.

What are the characteristics of

real-life problems that can be

modeled by trigonometric

functions?

Type II, III:

Foxes and Rabbits 2

Foxes and Rabbits 3

As the Wheel Turns

See illustrations for F-

IF.4 at:

http://illustrativemathe

matics.org/illustration

s/649

http://illustrativemathe

matics.org/illustration

s/637

http://illustrativemathe

matics.org/illustration

s/639

Key features may also

include

discontinuities.

ELL Strategies:

Demonstrate comprehension of complex

questions in the student’s native language and/or

simplified questions with drawings and selected

technical words concerning graphing functions

symbolically by showing key features of the

graph by hand in simple cases and using

technology for more complicated cases.

Explore using a graphic calculator what the

parent functions y = sin and y = cos looks like.

Use jigsaw with the first four identities assigning

one identity to each group of students. Give

students time to think, reason and verbalize

identities.

Words-Tables-

Graphs

37 | P a g e

Identify amplitude, frequency and midline

appropriate for the model.

In order to model a periodic phenomenon, you

need to know the amplitude, frequency or

period, and midline.

SPED Strategies:

Model how to visualize trigonometric functions

in real life scenarios.

Encourage students to work in small groups to

practice the application of trigonometric

functions to contextually based problems.

ELL Strategies:

Demonstrate understanding of oral explanations

and written word problems in the student’s

native language and/or problems with Visuals

and selected technical words of trigonometric

functions by choosing the correct function to

model periodic phenomena with specified

amplitude, frequency and midline.

Make a table that shows the time in 15 minute

increments from noon until 6 pm. Ask students

to graph the number of minutes’ vs the time and

display an example of periodic function.

Make a comparison and contrast chart of the sine

and cosine functions.

What are the key features of a

trigonometric function?

What kinds of phenomena can

be modeled by trigonometric

functions?

Give an example.

What information about a

situation do you need in order

to model it with a

trigonometric function?

What do the amplitude,

frequency, and midline of a

trigonometric function tell

you about the situation it

models?

How are period and frequency

related?

Hours of Daylight 1

38 | P a g e

New Jersey Student Learning Standard(s): F.TF.C.8: Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the

quadrant of the angle.

Student Learning Objective 5: Use the Pythagorean identity sin2(θ) + cos2(θ) = 1 to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ)

and the quadrant of the angle.

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

MP 7

F-TF.8-2

The "prove" part of

standard F-TF.8 is not

assessed here.

Prove the Pythagorean identity:

sin2(θ) + cos2(θ) = 1.

Use the Pythagorean identity to find sin(θ),

cos(θ), or tan(θ) when given sin(θ), cos(θ), or

tan(θ) and the quadrant of the angle.

SPED Strategies:

Ground the discussion of this theoretical

understanding using a real life example to make

it clearer and more relevant for students.

Create a reference document with students that

includes terms, formulas and illustrations which

can serve as a tool to facilitate understanding,

problem solving and independence.

ELL Strategies:

Verbalize the Pythagorean Theorem: “In a right

triangle, the sum of the squares of the lengths of

the legs equals the square of the length of the

hypotenuse.”

How can you prove the

Pythagorean identity?

How can you find sin(θ),

cos(θ), or tan(θ) using the

Pythagorean identity?

Type II, III:

Calculations with sine

and cosine

Finding Trig Values

Trigonometric Ratios

and the Pythagorean

Theorem

39 | P a g e

Create a reference document with all necessary

terms, formulas, processes and sample

problems.

Provide visuals as a point of reference.

New Jersey Student Learning Standard(s): S.ID.B.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

S.ID.B.6a: Fit a function to the data (including with the use of technology); use functions fitted to data to solve problems in the context of the

data. Uses given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Student Learning Objective 6: Represent nonlinear (exponential and trigonometric) data for two variables on a scatter plot, fit a function to

the data, analyze residuals (in order to informally assess fit), and use the function to solve problems. Uses given functions or choose a function

suggested by the context; emphasize exponential and trigonometric models.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement

Key/ Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 2

MP 4

MP 5

MP 6

S-ID.6a-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 S-

ID.6a, excluding

normal distributions

and limiting function

Fit exponential and trigonometric functions to

data using technology.

Solve problems using functions fitted to data

(prediction equations).

Interpret the intercepts of models in context.

Plot residuals of non-linear functions.

Analyze residuals in order to informally

evaluate the fit of exponential and

trigonometric functions.

How can you use a quadratic

function to model a real-life

situation?

Why would you want to

identify trends or associations

in a data set?

Why would you want to

informally assess and identify

a type of function to fit a data

set?

Type II, III:

Olympic Men's 100-

meter dash

Ball Drop

Snakes

The Wave

40 | P a g e

fitting to exponential

functions.

S-ID.6a-2

Solve multi-step

contextual word

problems with degree

of difficulty

appropriate to the

course, requiring

application of course-

level knowledge and

skills articulated in S-

ID.6a, excluding

normal distributions

and limiting function

fitting to

trigonometric

functions.

Example:

Measure the wrist and neck size of each person

in your class and make a scatterplot. Find the

least squares regression line. Calculate and

interpret the correlation coefficient for this

linear regression model. Graph the residuals

and evaluate the fit of the linear equations.

SPED Strategies:

Review the concepts related to and

characteristics of scatter plots with students.

Link this prior learning to nonlinear functions.

Provide contextualized problems to illustrate

the concepts clearly and give students

opportunities to practice the skills.

Create a reference document with students that

highlight the key aspects of representing

nonlinear data via scatter plots.

ELL Strategies:

Model an exponential function using “baby

born invested problem” and let students graph

the function, and define terms of growth and

decay.

Explore “Scatter plot” by letting students label

a graph diagram. Pair students and let them

discuss possible form of the graph, display a

scatter plot, and continue the discussion.

41 | P a g e

New Jersey Student Learning Standard(s): F.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal

descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Student Learning Objective 7: Analyze and compare properties of two functions when each is represented in a different way (algebraically,

graphically, numerically in tables, or by verbal descriptions).

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement

Key/ Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 3

MP 5

MP 6

MP 8

F-IF.9-2

Function types are

limited to

polynomial,

exponential,

logarithmic, and

trigonometric

functions.

Tasks may or may

not have a real-world

context.

Compare key attributes of functions each

represented in a different way (i.e. zeros, end

behavior, periodicity, asymptotes).

A function can be represented algebraically,

graphically, numerically in tables, or by verbal

descriptions.

SPED Strategies:

Review the different ways that functions can be

represented with students (algebraically,

graphically, numerically in tables, or by verbal

descriptions).

Model how to identify properties of a function

such as end behavior and highlight how it can

be seen in all of the representations.

Create a reference document that includes

verbal and pictorial representations of the

properties of functions and how they can be

identified in all representations.

How do you compare the

properties of two functions

when they are represented in

different forms?

Why is it important to

analyze and compare the

properties of functions when

they are represented in

different ways?

How can you compare

properties of two functions if

they are represented in

different ways?

How do different forms of a

function help you to identify

key features?

How do you determine which

type of function best models

a given situation?

Type II, III:

Throwing Baseballs

Comparing Multiple

Representations of

Fractions

42 | P a g e

ELL Strategies:

Compare and contrast orally and in writing the

properties of two functions when each is

represented in a different form in the student’s

native language and/or use gestures, examples

and selected technical words.

Create a list of steps that students will use to

interpret functions.

Brainstorm two functions by comparing the

most significant point of solving those

equations using different methods. Encourage

students to write their statements on a visual

display

How can a given function be

represented graphically,

within a table, by an

equation, and in the real-

world?

What connections can be

made between various

functions and various

representations of functions?

43 | P a g e

New Jersey Student Learning Standard(s): F.BF.A.1: Write a function that describes a relationship between two quantities.

F.BF.A.1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a

cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

N.Q.A.2: Define appropriate quantities for the purpose of descriptive modeling.

A.APR.B.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function

defined by the polynomial.

Student Learning Objective 8: Construct a function that combines, using arithmetic operations, standard function types to model a

relationship between two quantities.

Modified Student Learning Objectives/Standards:

EE.F-BF.1. Select the appropriate graphical representation (first quadrant) given a situation involving constant rate of change.

EE.N-Q.1–3. Express quantities to the appropriate precision of measurement.

MPs Evidence Statement

Key/ Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 4

MP 7

F-BF.1b-1

Represent arithmetic

combinations of

standard function

types algebraically.

Tasks may or may not

have a context.

For example, given

f(x) = 𝑒𝑥and g(x) = 5,

write an expression

for h(x) = 2f(-3x) +

g(x).

Functions of various types can be combined to

model real world situations.

Use arithmetic operations to combine

functions of varying types in order to model

relationships between quantities.

SPED Strategies:

Pre-teach vocabulary using visual and verbal

models that are connected to real life

situations.

Model how combining functions using

arithmetic operations develops more accurate

function models for real life situations.

How can you use the graphs

of two functions to sketch the

graph of an arithmetic

combination of two functions

and why is this important?

What data would you need to

write a function to model a

given situation?

What information do you need

to sketch a rough graph of a

polynomial function?

How are the zeros of a

polynomial related to its graph?

IFL Sets of Related

Lessons “Building

Polynomial

Functions”

Type II, III:

1,000 is half of 2,000

Sum of Functions

Solving a simple

cubic equation

Graphing from

Factors 1

44 | P a g e

Create a reference document such as a Google

Doc or anchor chart that illustrates and

explains the features and importance of

combined functions.

ELL Strategies:

After listening to an oral explanation and

reading the directions, construct and explain,

in writing, a function that combines standard

function types using arithmetic operations in

the student’s native language and/or use

gestures, examples and selected technical

words.

Model and draw conclusions about the “White

Rhino Problems”. Describe fun facts and

share.

Practice reading aloud and write function

notations, then interpreted operations on

functions.

Polynomial functions can be

written as a product of two or

more linear factors.

The value of a polynomial

function is equal to zero if

and only if at least one factor

of the polynomial is equal to

zero. Therefore, a polynomial

function will have the same

x-intercepts as its factor

functions.

The product of two or more

polynomial functions is a

polynomial function. The

product function will have the

same x-intercepts as the

original functions because the

original functions are factors

of the polynomial.

Two or more functions can be

added or subtracted using

their algebraic representations

by combining like terms.

Two or more functions can be

multiplied using the algebraic

representation by applying

the distributive property and

combining like terms.

45 | P a g e

Two or more polynomial

functions can be added using

their graphs or tables of

values because given two

functions f(x1) and g(x1) and

a specific x-value, x1, the

point (x1, (f(x1) + g(x1)) will

be on the graph and in the

table of the sum f(x) + g(x).

(This is true for subtraction

and multiplication as well.)

The degree of the sum of two

polynomial functions is

dependent upon the degree of

the addends. When two

polynomial functions are

added using their algebraic

representations by combining

like terms, the coefficient of

the highest order terms may

change, but the exponent will

not. Therefore, if the degree

of the addends is unequal, the

sum will have the degree of

the addend with the higher

degree. If the degree of the

addends is equal, the degree

of the sum is less than or

equal to the degree of the

addends.

46 | P a g e

New Jersey Student Learning Standard(s): F.BF.B.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative);

find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include

recognizing even and odd functions from their graphs and algebraic expressions for them.

Student Learning Objective 9: Identify the effect on the graph of a polynomial, exponential, logarithmic, or trigonometric function of replacing

f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative). Find the value of k given the graphs and identify even

and odd functions from graphs and equations.

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

MP 7

MP 8

F-BF.3-2

Identify the effect on

the graph of replacing

f(x) by f(x) + k, k f(x),

f(kx), and f(x + k) for

specific values of k

(both positive and

negative); find the

value of k given the

graphs, limiting the

function types to

polynomial,

exponential,

logarithmic, and

trigonometric

functions. i.)

Experimenting with

cases and illustrating

an explanation are not

assessed here.

Function notation representation of

transformations

Perform transformations on graphs of

polynomial, exponential, logarithmic, or

trigonometric functions.

Identify the effect on the graph of replacing f(x)

by:

o f(x) + k;

o k f(x);

o f(kx);

o and f(x + k) for specific values

of k (both positive and negative).

Identify the effect on the graph of combinations

of transformations.

Given the graph, find the value of k.

What are some of the

characteristics of some of the

basic parent functions?

How do the graphs of y =f(x)+k,

y=f(x)-h, and y= -f(x) compare

to the graph of the parent

function f?

How do the constants a, h, and k

affect the graph of the quadratic

function 𝑔(x)=a[(x-h)] 2+k?

How can you transform the

graph of a polynomial function?

How can you transform the

graphs of exponential and

logarithmic functions?

Type II, III:

Exploring Sinusoidal

Functions

Building a quadratic

function from f(x)=x

2

Building a General

Quadratic Function

Building an Explicit

Quadratic Function

by

Composition

Identifying

Quadratic Functions

(Standard Form)

47 | P a g e

F-BF.3-3

Recognize even and

odd functions from

their graphs and

algebraic expressions

for them, limiting the

function types to

polynomial,

exponential,

logarithmic, and

trigonometric

functions.

Experimenting with

cases and illustrating

an explanation are not

assessed here.

F-BF.3-5

Illustrating an

explanation is not

assessed here.

Illustrate an explanation of the effects on

polynomial, exponential, logarithmic, or

trigonometric graphs using technology.

SPED Strategies:

Model how the function notation of

transformations correlates to changes in the

values and graph of a function.

Provide students with a reference document that

illustrates verbally and pictorially the features

of a function and how they are changed due to

transformation.

ELL Strategies:

Demonstrate comprehension of the effects on

the graph of replacing f(x) by f(x) + k, k f(x),

f(kx), and f(x + k) for specific values of k, by

illustrating an explanation using technology and

finding the value of k given the graphs in the

student’s native language and/or use gestures,

examples and selected technical words.

Practice sketching the graph of a parent’s

function and their transformation.

Verbalize observations made when students use

graphic calculator to display functions

transformation.

Identifying

Quadratic Functions

(Vertex Form)

Medieval Archer

Transforming the

graph of a function

Additional tasks:

Identifying Even and

Odd Functions

48 | P a g e

New Jersey Student Learning Standard(s): F.BF.B.4: Find inverse functions.

F.BF.B.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example,

f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠1. [*note: composition of functions is not introduced here]

F.BF.B.4b: Verify by composition that one function is the inverse of another.

F.BF.B.4c: Read values of an inverse function from a graph or a table, given that the function has an inverse.

F.BF.B.4d: Produce an invertible function from a non-invertible function by restricting the domain.

Student Learning Objective 10: Find inverse functions, use them to solve equations and verify if a function is the inverse of another. Read

values of an inverse function from a graph or table and use restricted domain to produce invertible functions from previously non-invertible

functions.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence

Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 6

MP 8

N/A For a function f(x) that has an inverse, the domain/input for

f(x) is the inverse function’s range/output and that the

range/output for f(x) is the inverse function’s domain/input.

Use function notation to represent the inverse of a function –

f-1(x).

Transform an equation in order to isolate the independent

variable, recognizing that the domain/input for f(x) is the

inverse function’s range/output and that the range/output for

f(x) is the inverse function’s domain/input.

Determine what restrictions to a domain are necessary to

produce invertible functions from previously non-invertible

functions.

How can you sketch the graph

of the inverse of a function and

how does it enhance

understanding?

How do you describe a

transformation of a given

polynomial and its inverse

function?

What is an inverse of a

function?

Type II, III:

Temperature

Conversions

Temperatures in

degrees

Fahrenheit and

Celsius

Additional Tasks:

Invertible or Not

US Households

49 | P a g e

SPED Strategies:

Pre-teach vocabulary using visual and verbal models that are

connected to real life situations.

Model the relationship between a function and its inverse

verbally, algebraically and graphically using contextualized

examples.

Provide students with the opportunity to identify situations

when the inverse of a function is needed and how to use

function notation to represent the inverse by working on real

life models with peers in small groups.

ELL Strategies: After listening to an oral explanation and reading the

directions in the student’s native language, and/or using

drawings and selected technical words, demonstrate

comprehension of the inverse function for a simple function

that has an inverse and write an expression for it.

Guide a discussion by letting students state their

interpretation of inverse functions with the goal of creating a

correct definition of an inverse function.

Use a reflector to sketch the graph of the inverse function to

help students to make sense of reflections.

50 | P a g e

New Jersey Student Learning Standard(s):

F.TF.A.3: Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /6, and use the unit circle to express

the values of sine, cosines, and tangent for x, +x, and 2–x in terms of their values for x, where x is any real number.

F.TF.A.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Student Learning Objective 11: Use special triangles to determine the values of sine, cosine, tangent at various point along the unit circle.

Use the unit circle to explain the symmetry and periodicity of trigonometric functions.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence

Statement Key/

Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 6

MP 8

N/A Understand, explore, and apply the Unit Circle

Use special triangles to determine the values of sine,

cosine and tangent, and the unit circle

Explain symmetry and periodicity using the unit

circle

SPED Strategies:

Pre-teach vocabulary using visual and verbal models

that are connected to real life situations. Create

artifacts of the learning that can be used as reference

documents.

Have students construct unit circles to discover their

properties including radians. Annotate this learning

experience by asking thought provoking and

clarifying questions.

How do you evaluate trigonometric

functions for given values, periods, and

intervals?

How trigonometric functions relate to

the unit circle?

How do we model “Real world”

scenarios to trigonometric functions?

Equilateral

Triangles and

Trigonometric

Functions

Properties of

Trigonometric

Functions

Special Triangles

1

Special Triangles

2

51 | P a g e

Use software that models tides and periodicity, real

world examples such as Ferris wheels videos

showing motion with graphing simultaneously

(several websites offer this visual)

http://www.mathdemos.org/mathdemos/sinusoidapp

/sinusoidapp.html

http://demonstrations.wolfram.com/TrigonometricFi

ttingAndInterpolation/

ELL Strategies: Use the modeling software mentioned in the SPED

Strategies as a means of helping students to

visualize real-world applications of trigonometric

function. Be certain to share technical terms and

give students ample time to explore and clarify their

understanding.

Guide a discussion by letting students state their

interpretation of trigonometric functions with the

goal being increased understanding of trigonometric

functions and academic language.

Have students construct unit circles to discover their

properties including radians. Annotate this learning

experience by creating a document that includes

important terms, has visual examples and enhances

students’ understanding and ability to discuss the

topic with peers.

52 | P a g e

Honors Projects (Must complete all)

Project 1 Project 2 Project 3

Trig Project!!!

Essential Question(s):

Why do we need radian measure?

How can sine, cosine, and tangent

functions be defined using the unit

circle?

Skills: Emphasize all the mathematical practice

standards as you address the standards in

this project. F-TF.5 would provide the

opportunity to link mathematics to

everyday life, work, and decision making.

Exponential and Logarithmic Equations

Project

Essential Question: How do exponential and logarithmic equations

allow for extrapolation and/or interpolation in a

given context?

Skills: Select and accurately model a context in which

exponential and/or logarithmic equations apply.

Unit Circle Project

Essential Question: How do the principles of the Unit Circle

apply to technology, design and art?

Skills: Create a Unit Circle model with accurate

calculations from a real world context in the

field of technology, design or art.

53 | P a g e

Integrated Evidence Statements

A.Int.1: Solve equations that require seeing structure in expressions.

Tasks do not have a context.

Equations simplify considerably after appropriate algebraic manipulations are performed. For example, x4-17x2+16 = 0, 23x = 7(22x) + 22x , x -

√x = 3√x

Tasks should be course level appropriate.

F-BF.Int.2: Find inverse functions to solve contextual problems. Solve an equation of the form 𝒇(𝒙) = 𝒄 for a simple function f that has an

inverse and write an expression for the inverse. For example, 𝒇(𝒙) = 𝟐𝒙𝟑 or 𝒇(𝒙) =𝒙+𝟏

𝒙−𝟏 for 𝒙 ≠ 𝟏.

For example, see http://illustrativemathematics.org/illustrations/234.

As another example, given a function C(L) = 750𝐿2 for the cost C(L) of planting seeds in a square field of edge length L, write a function for

the edge length L(C) of a square field that can be planted for a given amount of money C; graph the function, labeling the axes.

This is an integrated evidence statement because it adds solving contextual problems to standard F-BF.4a.

F-Int.1-2: Given a verbal description of a polynomial, exponential, trigonometric, or logarithmic functional dependence, write an expression

for the function and demonstrate various knowledge and skills articulated in the Functions category in relation to this function.

Given a verbal description of a functional dependence, the student would be required to write an expression for the function and then, e.g.,

identify a natural domain for the function given the situation; use a graphing tool to graph several input-output pairs; select applicable

features of the function, such as linear, increasing, decreasing, quadratic, periodic, nonlinear; and find an input value leading to a given output

value.

F-Int.3: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level

knowledge and skills articulated in F-TF.5, F-IF.B, F-IF.7, limited to trigonometric functions.

F-TF.5 is the primary content and at least one of the other listed content elements will be involved in tasks as well.

HS-Int.3-3: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-

level knowledge and skills articulated in F-LE, A-CED.1, A-SSE.3, F-IF.B, F-IF.7★

54 | P a g e

Integrated Evidence Statements

F-LE.A, Construct and compare linear, quadratic, and exponential models and solve problems, is the primary content and at least one of the

other listed content elements will be involved in tasks as well.

HS.C.7.1: Base explanations/reasoning on the relationship between zeros and factors of polynomials. Content Scope: A-APR.B

HS.C.8.3: Construct, autonomously, chains of reasoning that will justify or refute algebraic propositions or conjectures. Content Scope: A-

APR

HS.C.9.2: Express reasoning about transformations of functions. Content scope: F-BF.3, which may involve polynomial, exponential,

logarithmic or trigonometric functions. Tasks also may involve even and odd functions.

HS.C.11.1: Express reasoning about trigonometric functions and the unit circle. Content scope: F-TF.2, F-TF.8

For example, students might explain why the angles 151𝜋

3 and

𝜋

3 have the same cosine value; or use the unit circle to prove that sin2(

3𝜋

4) +

cos2(3𝜋

4) = 1; or compute the tangent of the angle in the first quadrant having sine equal to

1

3.

HS.C.18.4: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures about polynomials, rational

expressions, or rational exponents. Content scope: N-RN, A-APR.(2, 3, 4, 6)

HS.C.CCR: Solve multi-step mathematical problems requiring extended chains of reasoning and drawing on a synthesis of the knowledge

and skills articulated across: 7-RP.A.3, 7-NS.A.3, 7-EE.B.3, 8-EE.C.7B, 8-EE.C.8c, N-RN.A.2, A-SSE.A.1b, A-REI.A.1, A-REI.B.3, A-

REI.B.4b, F-IF.A.2, F-IF.C.7a, F-IF.C.7e, G-SRT.B.5 and G-SRT.C.7.

Tasks will draw on securely held content from previous grades and courses, including down to Grade 7, but that are at the Algebra

II/Mathematics III level of rigor.

Tasks will synthesize multiple aspects of the content listed in the evidence statement text, but need not be comprehensive.

Tasks should address at least A-SSE.A.1b, A-REI.A.1, and F-IF.A.2 and either F-IF.C.7a or F-IF.C.7e (excluding trigonometric and

logarithmic functions). Tasks should also draw upon additional content listed for grades 7 and 8 and from the remaining standards in the

Evidence Statement Text.

55 | P a g e

Integrated Evidence Statements

HS.D.2-4: Solve multi-step contextual problems with degree of difficulty appropriate to the course that require writing an expression for an

inverse function, as articulated in F.BF.4a.

Refer to F-BF.41 for some of the content knowledge relevant to these tasks.

HS.D.2-7: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-

level knowledge and skills articulated in A-CED, N-Q.2, A-SSE.3, A-REI.6, A-REI.7, A-REI.12, A-REI.11-2.

A-CED is the primary content; other listed content elements may be involved in tasks as well.

HS.D.2-10: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-

level knowledge and skills articulated in F-BF.A, F-BF.3, F-IF.3, A-CED.1, A-SSE.3, F-IF.B, F-IF.7.

F-BF.A is the primary content; other listed content elements may be involved in tasks as well.

HS.D.2-13: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-

level knowledge and skills articulated in S-ID and S-IC.

If the content is only S-ID, the task must include Algebra 2 / Math 3 content (S-ID.4 or S-ID.6)

Longer tasks may require some or all of the steps of the modeling cycle (CCSSM, pp. 72, 73); for example, see ITN Appendix F, "Karnataka"

task (Section A "Illustrations of innovative task characteristics," subsection 7 "Modeling/Application," subsection f "Full Models"). As in the

Karnataka example, algebra and function skills may be used.

Predictions should not extrapolate far beyond the set of data provided.

Line of best fit is always based on the equation of the least squares regression line either provided or calculated through the use of

technology. Tasks may involve linear, exponential, or quadratic regressions. If the linear regression is in the task, the task must be written to

allow students to choose the regression.

To investigate associations, students may be asked to evaluate scatterplots that may be provided or created using technology. Evaluation

includes shape, direction, strength, presence of outliers, and gaps.

Analysis of residuals may include the identification of a pattern in a residual plot as an indication of a poor fit.

Models may assess key features of the graph of the fitted model.

Tasks that involve S-IC.2 might ask the students to look at the results of a simulation and decide how plausible the observed value is with

respect to the simulation. For an example, see question 7 on the calculator section of the online practice test

(http://practice.parcc.testnav.com/#).

56 | P a g e

Integrated Evidence Statements

Tasks that involve S-ID.4, may require finding the area associated with a z-score using technology. Use of a z-score table will not be

required.

Tasks may involve finding a value at a given percentile based on a normal distribution.

HS.D.3-5: Decisions from data: Identify relevant data in a data source, analyze it, and draw reasonable conclusions from it. Content scope:

Knowledge and skills articulated in Algebra 2.

Tasks may result in an evaluation or recommendation.

The purpose of tasks is not to provide a setting for the student to demonstrate breadth in data analysis skills (such as box-and-whisker plots

and the like). Rather, the purpose is for the student to draw conclusions in a realistic setting using elementary techniques.

HS.D.3-6: Full models: Identify variables in a situation, select those that represent essential features, formulate a mathematical

representation of the situation using those variables, analyze the representation and perform operations to obtain a result, interpret the

result in terms of the original situation, validate the result by comparing it to the situation, and either improve the model or briefly report

the conclusions. Content scope: Knowledge and skills articulated in the Standards in grades 6-8, Algebra 1 and Math 1 (excluding statistics)

Task prompts describe a scenario using everyday language. Mathematical language such as "function," "equation," etc. is not used.

Tasks require the student to make simplifying assumptions autonomously in order to formulate a mathematical model. For example, the

student might autonomously make a simplifying assumption that every tree in a forest has the same trunk diameter, or that water temperature

is a linear function of ocean depth.

Tasks may require the student to create a quantity of interest in the situation being described (N-Q.2). For example, in a situation involving

population and land area, the student might decide autonomously that population density is a key variable, and then choose to work with

persons per square mile. In a situation involving data, the student might autonomously decide that a measure of center is a key variable in a

situation, and then choose to work with the mean.

Tasks may involve choosing a level of accuracy appropriate to limitations of measurement or limitations of data when reporting quantities

(N-Q.3, first introduced in AI/M1).

57 | P a g e

Integrated Evidence Statements

HS.D.CCR: Solve problems using modeling: Identify variables in a situation, select those that represent essential features, formulate a

mathematical representation of the situation using those variables, analyze the representation and perform operations to obtain a result,

interpret the result in terms of the original situation, validate the result by comparing it to the situation, and either improve the model or

briefly report the conclusions. Content scope: Knowledge and skills articulated in the Standards as described in previous courses and

grades, with a particular emphasis on 7- RP, 8 – EE, 8 – F, N-Q, A-CED, A-REI, F-BF, G-MG, Modeling, and S-ID

Tasks will draw on securely held content from previous grades and courses, include down to Grade 7, but that are at the Algebra

II/Mathematics III level of rigor.

Task prompts describe a scenario using everyday language. Mathematical language such as "function," "equation," etc. is not used.

Tasks require the student to make simplifying assumptions autonomously in order to formulate a mathematical model. For example, the

student might make a simplifying assumption autonomously that every tree in a forest has the same trunk diameter, or that water temperature

is a linear function of ocean depth.

Tasks may require the student to create a quantity of interest in the situation being described.

58 | P a g e

Algebra II Vocabulary

Number

and Quantity Algebra Functions Statistics and Probability

Complex number

Conjugate

Determinant

Fundamental theorem

of Algebra

Identity matrix

Imaginary number

Initial point

Moduli

Parallelogram rule

Polar form

Quadratic equation

Polynomial

Rational exponent

Real number

Rectangular form

Scalar multiplication

of Matrices

Terminal point

Vector

Velocity

Zero matrix

Binomial Theorem

Complete the square

Exponential function

Geometric series

Logarithmic Function

Maximum

Minimum

Pascal’s Triangle

Remainder Theorem

Absolute value

function

Asymptote

Amplitude

Arc

Arithmetic sequence

Constant function

Cosine

Decreasing intervals

Domain

End behavior

Exponential decay

Exponential function

Exponential growth

Fibonacci sequence

Function notation

Geometric sequence

Increasing intervals

Intercepts

Invertible function

Logarithmic function

Trigonometric

function

Midline

Negative intervals

Period

Periodicity

Positive intervals

Radian measure

Range

Rate of change

Recursive process

Relative maximum

Relative minimum

Sine

Step function

Symmetries

Tangent

2-way frequency table

Addition Rule

Arithmetic sequence

Box plot

Causation

Combinations

Complements

Conditional

probability

Conditional relative

frequency

Correlation

Correlation

coefficient

Dot plot

Experiment

Fibonacci sequence

Frequency table

Geometric sequence

Histogram

Independent

Inter-quartile range

Joint relative

frequency

Margin of error

Marginal relative

frequency

Multiplication Rule

Observational studies

Outlier

Permutations

Recursive process

Relative frequency

Residuals

Sample survey

Simulation models

Standard deviation

Subsets

Theoretical

probability

Unions

59 | P a g e

References & Suggested Instructional Websites

Internet4Classrooms www.internet4classrooms.com

Desmos https://www.desmos.com/

Math Open Reference www.mathopenref.com

National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/index.html

Georgia Department of Education https://www.georgiastandards.org/Georgia-Standards/Pages/Math-9-12.aspx

Illustrative Mathematics www.illustrativemathematics.org/

Khan Academy https://www.khanacademy.org/math/algebra-home/algebra2

Math Planet http://www.mathplanet.com/education/algebra-2

IXL Learning https://www.ixl.com/math/algebra-2

Math Is Fun Advanced http://www.mathsisfun.com/algebra/index-2.html

Partnership for Assessment of Readiness for College and Careers https://parcc.pearson.com/practice-tests/math/

Mathematics Assessment Project http://map.mathshell.org/materials/lessons.php?gradeid=24

Achieve the Core http://www.achieve.org/ccss-cte-classroom-tasks

NYLearns http://www.nylearns.org/module/Standards/Tools/Browse?linkStandardId=0&standardId=97817

Learning Progression Framework K-12 http://www.nciea.org/publications/Math_LPF_KH11.pdf

PARCC Mathematics Evidence Tables. https://parcc-assessment.org/mathematics/

Smarter Balanced Assessment Consortium. http://www.smarterbalanced.org/

Statistics Education Web (STEW). http://www.amstat.org/education/STEW/

McGraw-Hill ALEKS https://www.aleks.com/

60 | P a g e

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/