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MATHEMATICS
Algebra II Honors: Unit 3
Periodic Models and the Unit Circle
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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.
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ESL Framework
This ESL framework was designed to be used by bilingual, dual language, ESL and general education teachers. Bilingual and dual language programs
use the home language and a second language for instruction. ESL teachers and general education or bilingual teachers may use this document to
collaborate on unit and lesson planning to decide who will address certain components of the SLO and language objective. ESL teachers may use the
appropriate leveled language objective to build lessons for ELLs which reflects what is covered in the general education program. In this way, whether
it is a pull-out or push-in model, all teachers are working on the same Student Learning Objective connected to the New Jersey Student Learning
Standards. The design of language objectives are based on the alignment of the World-Class Instructional Design Assessment (WIDA) Consortium’s
English Language Development (ELD) standards with the New Jersey Student Learning Standards (NJSLS). WIDA’s ELD standards advance academic
language development across content areas ultimately leading to academic achievement for English learners. As English learners are progressing
through the six developmental linguistic stages, this framework will assist all teachers who work with English learners to appropriately identify the
language needed to meet the requirements of the content standard. At the same time, the language objectives recognize the cognitive demand required
to complete educational tasks. Even though listening and reading (receptive) skills differ from speaking and writing (expressive) skills across
proficiency levels the cognitive function should not be diminished. For example, an Entering Level One student only has the linguistic ability to respond
in single words in English with significant support from their home language. However, they could complete a Venn diagram with single words which
demonstrates that they understand how the elements compare and contrast with each other or they could respond with the support of their native
language with assistance from a teacher, para-professional, peer or a technology program.
http://www.state.nj.us/education/modelcurriculum/ela/ELLOverview.pdf
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Pacing Chart – Unit 3
# Student Learning Objective NJSLS Big Ideas Math
Correlation
Instruction: 8 weeks
Assessment: 1 week
1
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
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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.
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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
Using technological resources and other 21st century skills to support and enhance mathematical understanding
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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.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/
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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 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.
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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: 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.
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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 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.
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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 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.
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Self-Awareness
Self-Management
Social-Awareness
Relationship Skills
Responsible Decision-Making
Example practices that address Social-Awareness:
• Encourage students to reflect on the perspective of
others
• Assign appropriate groups
• Help students to think about social strengths
• Provide specific feedback on social skills
• Model positive social awareness through
metacognition activities
When there is a difference of opinion among
students (perhaps over solution strategies),
allow them to reflect on how they are feeling
and then share with a partner or in a small
group. It is important to be heard but also to
listen to how others feel differently in the same
situation.
Have students re-conceptualize application
problems after class discussion, by working
beyond their initial reasoning to identify
common reasoning between different
approaches to solve the same problem.
Self-Awareness
Self-Management
Social-Awareness
Relationship Skills
Responsible Decision-Making
Example practices that address Relationship Skills:
• Engage families and community members
• Model effective questioning and responding to
students
• Plan for project-based learning
• Assist students with discovering individual strengths
• Model and promote respecting differences
• Model and promote active listening
• Help students develop communication skills
• Demonstrate value for a diversity of opinions
Have students team-up at the at the end of the
unit to teach a concept to the class. At the end
of the activity students can fill out a self-
evaluation rubric to evaluate how well they
worked together. For example, the rubric can
consist of questions such as, “Did we
consistently and actively work towards our
group goals?”, “Did we willingly accept and
fulfill individual roles within the group?”, “Did
we show sensitivity to the feeling and learning
needs of others; value their knowledge, opinion,
and skills of all group members?”
Encourage students to begin a rebuttal with a
restatement of their partner’s viewpoint or
argument. If needed, provide sample stems,
such as “I understand your ideas are ___ and I
think ____because____.”
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Self-Awareness
Self-Management
Social-Awareness
Relationship Skills
Responsible Decision-Making
Example practices that address Responsible
Decision-Making:
• Support collaborative decision making for academics
and behavior
• Foster student-centered discipline
• Assist students in step-by-step conflict resolution
process
• Foster student independence
• Model fair and appropriate decision making
• Teach good citizenship
Have students participate in activities that
requires them to make a decision about a
situation and then analyze why they made that
decision. Students encounter conflicts within
themselves as well as among members of their
groups. They must compromise in order to
reach a group consensus.
Today’s students live in a digital world that
comes with many benefits — and also increased
risks. Students need to learn how to be
responsible digital citizens to protect
themselves and ensure they are not harming
others. Educators can teach digital citizenship
through social and emotional learning.
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Differentiated Instruction
Accommodate Based on Students Individual Needs: Strategies
Time/General
Extra time for assigned tasks
Adjust length of assignment
Timeline with due dates for
reports and projects
Communication system
between home and school
Provide lecture notes/outline
Processing
Extra Response time
Have students verbalize steps
Repeat, clarify or reword
directions
Mini-breaks between tasks
Provide a warning for
transitions
Partnering
Comprehension
Precise processes for balanced
math instructional model
Short manageable tasks
Brief and concrete directions
Provide immediate feedback
Small group instruction
Emphasize multi-sensory
learning
Recall
Teacher-made checklist
Use visual graphic organizers
Reference resources to
promote independence
Visual and verbal reminders
Graphic organizers
Assistive Technology
Computer/whiteboard
Tape recorder
Video Tape
Tests/Quizzes/Grading
Extended time
Study guides
Shortened tests
Read directions aloud
Behavior/Attention
Consistent daily structured
routine
Simple and clear classroom
rules
Frequent feedback
Organization
Individual daily planner
Display a written agenda
Note-taking assistance
Color code materials
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Differentiated Instruction
Accommodate Based on Content Specific Needs
Anchor charts to model strategies 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/
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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.
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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
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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 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
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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.
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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.
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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/#).
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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).
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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.
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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
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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/