mathematics - paterson public schools...4 use polynomial identities to describe numerical...
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MATHEMATICS
Algebra II: Unit 2
Polynomials and Analysis of Nonlinear Functions
http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&docid=AwOKz0mO1gEXQM&tbnid=prj82ZSOleBaDM:&ved=0CAUQjRw&url=http://schools.nyc.gov/SchoolPortals/10/X085/Academics/Mathematics/&ei=zKG9U5yqE4b4oAS6ooCADg&bvm=bv.70138588,d.aWw&psig=AFQjCNGWaNnv41cJZ3Bco1IS4VvAQUM2ng&ust=1405023038489225
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Course Philosophy/Description
Algebra II continues the students’ study of advanced algebraic concepts including functions, polynomials, rational expressions,
systems of functions and inequalities, and matrices. Students will be expected to describe and translate among graphic,
algebraic, numeric, tabular, and verbal representations of relations and use those representations to solve problems. Emphasis
will be placed on practical applications and modeling. Students extend their knowledge and understanding by solving open-
ended real-world problems and thinking critically through the use of high level tasks.
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 Common Core standard. 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 Common Core State Standards (CCSS). WIDA’s ELD standards advance academic language development
across content areas ultimately leading to academic achievement for English learners. As English learners are progressing through the six developmental
linguistic stages, this framework will assist all teachers who work with English learners to appropriately identify the language needed to meet the
requirements of the content standard. At the same time, the language objectives recognize the cognitive demand required to complete educational tasks.
Even though listening and reading (receptive) skills differ from speaking and writing (expressive) skills across proficiency levels the cognitive function
should not be diminished. For example, an Entering Level One student only has the linguistic ability to respond in single words in English with
significant support from their home language. However, they could complete a Venn diagram with single words which demonstrates that they
understand how the elements compare and contrast with each other or they could respond with the support of their native language with assistance from
a teacher, para-professional, peer or a technology program.
http://www.state.nj.us/education/modelcurriculum/ela/ELLOverview.pdf
http://www.state.nj.us/education/modelcurriculum/ela/ELLOverview.pdf
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Pacing Chart – Unit 1
# Student Learning Objective NJSLS Big Ideas Math
Correlation
Instruction: 8
weeks
Assessment: 1
week
1 Apply the Remainder Theorem in order to determine the factors of a
polynomial. A.APR.B.2
4.3, 4.4
2
Use an appropriate factoring technique to factor polynomials. Explain
the relationship between zeros and factors of polynomials, and use the
zeros to construct a rough graph of the function defined by the
polynomial
A.SSE.A.2
A.APR.B.3
2.2, 3.1, 4.4, 4.5,
4.6, 4.8, 6.5
3 Graph polynomial functions from equations; identify zeros when
suitable factorizations are available; show key features and end
behavior.
F.IF.C.7c 2.1, 2.2, 2.3, 4.1,
4.7, 4.8
4 Use polynomial identities to describe numerical relationships and prove
polynomial identities.
A.APR.C.4 4.2
5 Rewrite simple rational expressions in different forms using inspection,
long division, or, for the more complicated examples, a computer
algebra system.
A.APR.D.6 4.3, 7.2, 7.3, 7.4
6
Solve simple rational and radical equations in one variable, use them to
solve problems and show how extraneous solutions may arise. Create
simple rational equations in one variable and use them to solve
problems.
A.REI.A.1
A.REI.A.2
A.CED.A.1
3.6, 5.4, 6.6, 7.1,
7.5
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Pacing Chart – Unit 1
7 For radical functions, 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.
F.IF.B.4
F.IF.B.6
5.3
8 Derive the equation of a parabola given a focus and directrix. G.GPE.A.2 2.3
9 Graph logarithmic functions expressed symbolically and show key
features of the graph (including intercepts and end behavior).
F.IF.C.7e 6.1, 6.2, 6.3, 6.4
10
Find approximate solutions for f(x)=g(x), using technology to graph,
make tables of values, or find successive approximations. Include
cases where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, logarithmic and exponential functions.
A.REI.D.11 3.5
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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 balanced mathematical 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.
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 factoring polynomials.
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 polynomial functions from equations.
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 derive the equation of a parabola given a focus and directrix.
Link: http://www.state.nj.us/education/cccs/2014/tech/
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 judgments about the use of
specific tools, such as algebra tiles, graphing calculators and technology to deepen their understanding of solving quadric equations.
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.
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 linear and quadratic equations in two variables.
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Career Ready Practices
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 perform operations with complex
numbers.
http://www.state.nj.us/education/aps/cccs/career/CareerReadyPractices.pdf
http://www.state.nj.us/education/aps/cccs/career/CareerReadyPractices.pdf
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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 interpreting the structure of expressions, 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
Students scan multistep contextual problems
that requires them to identify variables, write
equations, create graphs, etc., and make a list of
questions based on their understanding to ask
the teacher. This will help students to gain
confidence in working through the problems.
Set up small-group discussions that allows
students to reflect and discuss challenges or
how they have worked through a problem. For
examples, when students learn factoring
techniques, students can discuss which method
they have the most difficulty working with.
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
Lead discussions that encourages students to
reflect on barriers they encounter when
completing an assignment (e.g., finding a
computer, needing extra help or needing a quiet
place to work) and help them think about
solutions to overcome those barriers.
Teach and model for students not to become
defensive or angry when errors/flaws in their
reasoning are pointed out by their classmates
during class discussions but rather to readily
accept their peer’s correction, and continue to
contribute to the discussion. Because of these
self-management efforts, the class is able to
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continue their discussion on the definition of
functions.
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
Organize a class service project to examine and
address a community issue. Use math to
examine the situations and find possible
solutions. For example, students can discuss as
a class or in groups how to determine whether a
polynomial is a repeated solution.
Use real-world application problems to lead a
discussion about taking different approaches to
solving a problem and respecting the feeling
and thoughts of those that used a different
strategy.
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
Instead of simply jumping into their own
solution when asked to graph a trigonometric
function, have students discuss the key features
and how to find those key features before
graphing.
During class or group discussion, have students
expound upon and clarify each other’s
questions and comments, ask follow-up
questions, and clarify their own questions and
reasoning.
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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
Use a lesson to teach students a simple formula
for making good choices. (e.g., stop, calm
down, identify the choice to be made, consider
the options, make a choice and do it, how did it
go?) Post the decision-making formula in the
classroom.
Routinely encourage students to use the
decision-making formula as they face a choice
(e.g., whether to finish homework or go out
with a friend). Support students through the
steps of making a decision anytime they face a
choice or decision. Simple choices like “Which
tool should I use to explain the relationship
between domain and range?” or “Do I need a
calculator for this problem?” are good places to
start.
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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
mathematics 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 Needs: Strategies
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
Graphic organizers to help students interpret the meaning of terms in an expression or equation in context
Translation dictionary
Sentence stems to provide additional language support for ELL students.
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Interdisciplinary Connections
Model interdisciplinary thinking to expose students to other disciplines.
Social Studies Connection: Social Studies Standard 6.1.12.D.2.a
Name of Task: Logistic Growth Model, Explicit Version This problem introduces a logistic growth model in the concrete setting of estimating the population of the U.S. The model gives a
surprisingly accurate estimate and this should be contrasted with linear and exponential models, studied in ''U.S. Population 1790-1860.
Science Connection: Science Standard K-ESS3-3
Name of Task: Combined Fuel Efficiency
The US Department of Energy keeps track of fuel efficiency for all vehicles sold in the United States. Each car has two fuel economy numbers, one measuring efficient for city driving and one for highway driving. For example, a 2012 Volkswagen Jetta gets 29.0 miles per
gallon (mpg) in the city and 39.0 mpg on the highway. Many banks have "green car loans'' where the interest rate is lowered for loans on
cars with high combined fuel economy. This number is not the average of the city and highway economy values. Rather, the combined fuel
economy (as defined by the federal Corporate Average Fuel Economy standard) for mpg in the city and mpg on the highway, is computed
as.
Name of Task: Ideal Gas Law
The goal of this task is to interpret the graph of a rational function and use the graph to approximate when the function takes a given value. The first two parts of the question focus student attention on the meaning of the function within the context of pressure and volume.
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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
A.APR.B.2:
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only
if (x – a) is a factor of p(x).
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.
A.APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the difference of two squares; the sum and difference
of two cubes; the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
A.APR.D.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials
with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra
system.
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.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph.
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.
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New Jersey Student Learning Standards
F.IF.C.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
A.SSE.A.2:
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of
squares that can be factored as (x2 – y2)(x2 + y2).
A.REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the
assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.A.2:
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A.CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions,
and simple rational and exponential functions.
G.GPE.A.2: Derive the equation of a parabola given a focus and directrix.
A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) =
g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.
Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
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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.
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Course: Algebra II Unit: 2 (Two) Topic: Polynomials and Analysis of
Nonlinear Functions
NJSLS:
A.APR.B.2, A.SSE.A.2, A.APR.B.3, F.IF.C.7c, A.APR.C.4, A.APR.D.6, A.REI.A.1, A.REI.A.2, A.CED.A.1, F.IF.B.4, F.IF.B.6, G.GPE.A.2 ,
F.IF.C.7e, A.REI.D.11
Unit Focus:
Understand the relationship between zeros and factors of polynomials
Interpret the structure of expressions
Use polynomial identities to solve problems
Analyze functions using different representations
Rewrite rational expressions
Understand solving equations as a process of reasoning and explain the reasoning
Interpret functions in terms of the context
Translate between the geometric description and the equation for a conic section
Represent and solve equations and inequalities graphically
New Jersey Student Learning Standard(s):
A.APR.B.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) =
0 if and only if (x – a) is a factor of p(x).
Student Learning Objective 1: Apply the Remainder Theorem in order to determine the factors of a polynomial.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement
Key/ Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 6
A-APR.2
Polynomial division: For a polynomial p(x) and a
number a:
p(a) = 0 if and only if (x – a) is a factor of p(x)
How can you use the
factors of a cubic
polynomial to solve a
Type II, III:
Zeroes and
factorization of a
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(x – a) is a factor of p(x) if and only if p(a) = 0
The Remainder theorem says that if a polynomial
p(x) is divided by ax , then the remainder is the value of the polynomial evaluated at a.
Saying that x – a is a factor of a polynomial p(x) is
equivalent to saying that p(a) = 0, by the zero
property of multiplication.
Any polynomial of degree n can be factored into n
binomials of the form x – c, with possibly complex
values for c.
Use the Remainder Theorem to determine factors of
a polynomial.
SPED Strategies:
Provide students with background information about
dividing polynomials and connect it the material
they already know.
Use contextual examples to illustrate what the
Remainder Theorem states and does when applied.
Create a reference document with students that
contain all relevant information regarding
polynomial division and Remainder Theorem to
encourage independence and increase proficiency
and confidence.
division problem involving
the polynomial?
How can you factor a
polynomial?
How can you determine
whether a polynomial has a
repeated solution?
Why is it important to
supply a zero for a
coefficient of any missing
term, when you are dividing
polynomials?
How can you determine
whether x – a is a factor of
a polynomial p(x)? Why
does this work?
How do you determine how
many zeros a polynomial
function will have?
quadratic polynomial
I
Zeroes and
factorization of a
quadratic polynomial
II
Additional Tasks:
Graphing from Roots
The Missing
Coefficient
Zeroes and
factorization
of a general
polynomial
Zeroes and
factorization
of a non-polynomial
function
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ELL Strategies:
Read and write to restructure by performing
arithmetic operations on polynomial/rational
expressions in student’s native language and/or use
gestures, examples and selected technical words.
Model a polynomial function using the height of the
roller coaster as a function of time.
Let students write and explain the Remainder
Theorem and why it is useful.
New Jersey Student Learning Standard(s):
A.SSE.A.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2 – y2)(x2 + y2).
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 #2: Use an appropriate factoring technique to factor polynomials. Explain the relationship between zeros and factors
of polynomials, and use the zeros to construct a rough graph of the function defined by the polynomial.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement
Key/ Clarifications
Skills, Strategies & Concepts Essential
Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 7
A-SSE.2-3
Additional examples: In the equation x2 +
2x + 1 + y2 = 9, see
an opportunity to
rewrite the first three
terms as (x+1)2. See
Factors of polynomials can be used to identify
zeros to be used to develop a rough graph of the
polynomial function.
Factor polynomials.
How does using the
structure of an expression
help to simplify the
expression?
What type of symmetry
does the graph of 𝑓(𝑥) =
Type II, III:
Seeing Dots
Graphing from
Factors I
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(x2 + 4)/(x2 + 3) as
((x2+3) + 1)/(x2+3),
thus recognizing an
opportunity to write
it as 1 + 1/(x2 + 3).
Tasks will not include sums and
differences of cubes.
A-SSE.2-6
Factor completely: 6cx - 3cy - 2dx + dy.
(A first iteration
might give 3c(2x-y)
+ d(-2x+y), which
could be recognized
as 3c(2x-y) - d(2x-y)
on the way to
factoring completely
as (3c-d)(2x-y).)
Tasks do not have a context.
Analyze a table of values to determine where the
polynomial is increasing and decreasing.
Use the zeros of the polynomial to create a rough
graph.
If p(a) = 0, then a is a zero of p.
If a is a zero of p, then a is an x-intercept of the
graph of y = p(x).
The values and multiplicity of the zeros of a
polynomial, along with the end behavior, can be
used to sketch a graph of the function defined by
the polynomial.
Complicated expressions can be interpreted by
viewing parts of the expression as single entities.
Structure within expressions can be identified and
used to factor or simplify the expression.
SPED Strategies:
Model the thinking and processes involved in
analyzing polynomials in algebraic and table form
to determine graph behavior.
Provide students with opportunities to practice the
thinking and processes involved in small groups.
Develop a reference sheet for student use that
includes formulas, processes and procedures and
sample problems to encourage proficiency and
independence.
𝑎(𝑥 − ℎ)2 + 𝑘 have and how can you describe this
symmetry?
How many turning points
can the graph of a
polynomial function have?
Why is it important to
supply a zero for a
coefficient of any missing
term, when you are
dividing polynomials?
How does the concept of
the zero product property
allow you to find the roots
of a quadratic function?
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?
Graphing from
Factors II
Graphing from
Factors III
Additional Tasks:
A Cubic Identity
Animal Populations
Graphing from Roots
Equivalent
Expressions
Solving a Simple Cubic
Equation
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ELL Strategies:
Demonstrate orally and in writing an appropriate
factoring technique to factor expressions
completely including expressions with complex
number in student’s native language and/or use
selected technical vocabulary in phrases and short
sentences.
Describe and explain the relationship between
zeros and factors of polynomials and use zeros to
construct a rough graph of the function defined by
the polynomial using key technical vocabulary in a
series of simple sentences.
Guide students to think of a polynomial as a
product, reassure students that they will be learning
various strategies for finding those factors.
Create a reference document with the special
factoring patterns.
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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.7c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Student Learning Objective 3: Graph polynomial functions from equations; identify zeros when suitable factorizations are available; show key
features and end behavior.
Modified Student Learning Objectives/Standards:
M.EE.F-IF.1–3: Use the concept of function to solve problems.
MPs Evidence Statement
Key/ Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 1
MP 5
MP 6
F-IF.7c Factors of polynomials can be used to identify
zeros to be used to develop a rough graph of the
polynomial function.
Graph a polynomial function given its equation.
Identify zeros from the graph and using an
appropriate factoring technique.
Use technology to graph and describe key features
of the graph for complicated cases.
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; end behavior;
asymptotes; domain; range and periodicity.
The graph of a trigonometric function shows
period, amplitude, midline and asymptotes.
How does the constants a,
h, and k affect the graph of
the quadratic function
𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 ?
What type of symmetry
does the graph of 𝑓(𝑥) =𝑎(𝑥 − ℎ)2 + 𝑘 have and how can you describe this
symmetry?
What are some common
characteristics of the graphs
of cubic and quartic
polynomial functions?
How can you transform the
graph of a polynomial
function?
Type II, III:
Graphs of Power
Functions
Running Time
Additional Tasks:
Identifying Graphs of
Functions
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The graph of a polynomial function shows zeros
and end behavior.
A function can be represented algebraically,
graphically, numerically in tables, or by verbal
descriptions.
SPED Strategies:
Model the thinking behind determining when and
how to use the graphing calculator to graph
complicated polynomials.
Provide students with opportunities to practice the
thinking and processes involved in graphing
polynomial equations by hand and using
technology by working small groups.
Develop a reference sheet for student use that
includes formulas, processes and procedures and
sample problems to encourage proficiency and
independence.
ELL Strategies:
Demonstrate comprehension of complex questions
in 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.
Use technology to graph polynomial and identify
the end behavior and y intercept in the figure.
How many turning points
can the graph of a
polynomial function have?
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?
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Use technology to create table of values to verify
the positive zeros of the polynomial.
New Jersey Student Learning Standard(s):
A.APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the difference of two squares; the sum and
difference of two cubes; the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
Student Learning Objective 4: Use polynomial identities to describe numerical relationships and prove polynomial identities.
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 7
A.APR.C.4 Polynomial identities can be used to describe
numerical relationships.
Show that the polynomial identity (x2 + y2)2 = (x2 –
y2)2 + (2xy)2 can be used to generate Pythagorean
triples.
Prove polynomial identities.
SPED Strategies:
Ground the new learning in a real life context to
help students internalize the concept and develop
understanding.
Develop a reference sheet for student use that
includes formulas, processes and procedures and
sample problems to encourage proficiency and
independence.
ELL Strategies:
How can you cube a
binomial?
How does cubing binomials
enhance the understanding
of polynomial identities?
How are polynomials used
to represent situations?
Type II, III:
Trina's Triangles
Additional Tasks:
The Power of
Algebra—Finding
Pythagorean Triples
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Explore Pascal’s Triangle to display the patterns in
the expansion of (a+b)n.
Create in a notebook a list of polynomial identities
to help students recognize different polynomial
identities
New Jersey Student Learning Standard(s):
A.APR.D.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are
polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer
algebra system.
Student Learning Objective 5: Rewrite simple rational expressions in different forms using inspection, long division, or, for the more complicated
examples, a computer algebra system.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential
Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 1
A-APR.6
Examples will be simple enough to
allow inspection or
long division.
Simple rational expressions are limited
to numerators and
denominators that
have degree at most 2.
Rational expressions can be written in different
forms.
Write a(x)/b(x) in the form q(x) + r(x)/b(x), where
a(x), b(x), q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of b(x).
Use inspection, factoring and long division to
rewrite rational expressions.
Use technology to rewrite rational expressions for
more complicated cases.
SPED Strategies:
How can you use the
factors of a cubic
polynomial to solve a
division problem involving
the polynomial?
What are some of the
characteristics of the graph
of a rational function?
How can you determine the
excluded values in a
product or quotient of two
rational expressions?
Type II, III:
Combined Fuel
Efficiency
Egyptian Fractions II
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Connect the rewriting of polynomial expressions to
previous learning about equivalent expressions.
Model how to use inspection, factoring and long
division to rewrite rational expressions and develop
a reference document that uses verbal and pictorial
models.
Provide students with opportunities to practice the
thinking and processes involved in rewriting
polynomials by inspection, factoring and long
division.
Model the use of technology to rewrite more
complicated cases and encourage them to practice
this skill while working small groups.
Develop a reference sheet for student use that
includes formulas, processes and procedures and
sample problems to encourage proficiency and
independence.
ELL Strategies:
Read in order to rewrite simple rational expressions
in different forms in student’s native language
and/or use gestures, examples and selected,
technical words.
Connect rational numbers and rational functions by
asking student to define rational numbers.
Highlight and circle each factor in the denominator
of a rational expression. Guide students to set each
factor equal to zero
How are rational functions
used to represent real
world situations?
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New Jersey Student Learning Standard(s):
A.REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the
assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.A.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A.CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic
functions, and simple rational and exponential functions.
Student Learning Objective 6: Solve simple rational and radical equations in one variable, use them to solve problems and show how extraneous
solutions may arise. Create simple rational equations in one variable and use them to solve problems.
Modified Student Learning Objectives/Standards:
M.EE.A-CED.1: Create an equation involving one operation with one variable, and use it to solve a real-world problem.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential
Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 2
MP 3
MP 4
MP 6
A-REI.2
Simple rational equations are limited
to numerators and
denominators that
have degree at most 2.
Inverse relationships exist between roots and
powers.
Extraneous solutions do not result in true
statements.
Simple rational and radical equations can have
extraneous solutions.
Use the inverse relationship between roots and
powers when solving radical equations.
Identify any extraneous solutions.
What is the significance of
being able to identify
extraneous solutions?
What do you use to justify
your reasoning when
solving an equation?
How do you determine if
an equation is solved
properly?
How do you determine
and justify if a solution to
an equation is correct?
Type II, III:
Paying the rent
Radical Equations
Additional Tasks:
An Extraneous
Solution
Products and
Reciprocals
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Solve simple rational equations in one variable
(degree of numerators and denominator is not
greater than 2).
Write simple rational equations in one variable and
use the rational equation to solve problems.
Equations are solved as a process of reasoning
using properties of operations and equality, which
can justify each step of the process.
A solution to an equation can be checked, by
substituting in that value for the variable and
simplifying to see if the equation holds true.
Equations and inequalities can be created to
represent and solve real-world and mathematical
problems.
Solutions are viable or not in different situations
depending upon the constraints of the given
context.
SPED Strategies:
Model the thinking and processes involved in
solving simple rational and radical equations
explaining the significance of extraneous roots by
using real life examples to illustrate.
Develop a reference sheet for student use that
includes formulas, processes and procedures and
sample problems to encourage proficiency and
independence.
Why are properties of real
numbers important when
solving equations?
Give an example of a
simple rational or radical
equation that has an
extraneous solution and
explain why it is an
extraneous solution.
Who wins the Race?
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Provide students with opportunities to practice the
thinking and processes involved in solving simple
rational and radical equations including those with
extraneous roots by working in small groups.
ELL Strategies:
Explain orally and in writing how to solve simple
equations in one variable and use them to solve
problems, justify each step in the process in
student’s native language and/or use gestures,
examples and selected, technical words.
Build on past knowledge by explaining that
simplifying rational expressions is similar to
simplifying fractions.
Model a real-life problem: The income function
and population, divide and graph the result.
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New Jersey Student Learning Standard(s):
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.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the
rate of change from a graph.
Student Learning Objective 7: For radical functions, 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.
Modified Student Learning Objectives/Standards:
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
MP 1
MP 4
MP 5
MP 6
MP 7
F-IF.4-5
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:
A radical function is any function that contains a
variable inside a root.
Interpret key features of radical functions from
graphs and tables in the context of the problem.
Sketch graphs of radical functions given a verbal
description of the relationship between the
quantities.
Identify intercepts and intervals where function is
increasing/decreasing.
Determine the practical domain of a radical
function.
What type of symmetry
does the graph of 𝑓(𝑥) =𝑎(𝑥 − ℎ)2 + 𝑘 have and how can you describe this
symmetry?
What are some of the
characteristics of the graphs
of cubic and quartic
polynomial functions?
How many turning points
can a graph of a polynomial
function have and why is
this important?
Type II, III:
Containers
Mathemafish
Population
Model air plane
acrobatics
The High School Gym
Words - Tables -
Graphs
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intercepts; intervals
where the function is
increasing, decreasing,
positive, or negative;
relative maximums
and minimums; end
behavior; symmetries;
and periodicity.
See illustrations for F-IF.4
at
o http://illustrativemathematics.org
Key features may also include
discontinuities.
F-IF.6-2
Calculate and interpret the average rate of
change of a function
(presented
symbolically or as a
table) over a specified
interval with functions
limited to polynomial,
exponential,
logarithmic and
trigonometric
functions.
Tasks have a real-world context.
Determine key features including intercepts;
intervals where the function is increasing,
decreasing, positive, or negative; relative maxima
and minima; symmetries; end behavior.
SPED Strategies:
Pre-teach vocabulary using visual and verbal
models that are connected to real life situations.
Model the thinking and processes involved in the
understanding of radical functions including
interpreting key features from graphs and tables
and sketching the key features of graphs from a
verbal description of the relationship.
Develop a reference sheet for student use that
includes formulas, processes and procedures and
sample problems to encourage proficiency and
independence.
Provide students with opportunities to practice the
thinking and processes involved in the
understanding of radical functions including the
interpretation of key features from graphs and
tables and sketching the key features of graphs
from a verbal description of the relationship by
working in small groups.
ELL Strategies:
Listen and read in order to interpret orally and in
writing the key features in graphs and tables and
given a verbal description of the relationship,
sketch graphs showing the key features in
How can you use a radical
function to model a real life
situation?
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 would you determine
the appropriate domain for a
function describing a real-
world situation?
Given a function that
describes a real-world
situation, what can the
average rate of change of
the function tell you?
How do the parts of a graph
of a function related to its
real-world context?
http://illustrativemathematics.org/http://illustrativemathematics.org/
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Tasks must include the interpret part of the
evidence statement.
F-IF.6-7
Estimate the rate of change from a graph.
Tasks have a real-world context.
Tasks may involve polynomial,
exponential,
logarithmic, and
trigonometric
functions.
student’s native language and/or use gestures,
examples and selected, technical words.
After estimating, and calculating the average rate
of change of a function presented symbolically, in
a table, or graphically over a specified interval;
interpret the answer in writing in student’s native
language and/or use gestures, examples and
selected technical words.
Introduce the new topic by creating partner
discussions of the equation of the top half of a
parabola with horizontal line of symmetry: y =
sqrt(0-x); then make connection with the radical
equation graph.
Using a graphic calculator, graph the parent
function of radical and later display several
transformations.
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New Jersey Student Learning Standard(s):
G.GPE.A.2: Derive the equation of a parabola given a focus and directrix.
Student Learning Objective 8: Derive the equation of a parabola given a focus and directrix.
Modified Student Learning Objectives/Standards: N/A
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 4
G.GPE.A.2 Any point on a parabola is equidistant between
the focus and the directrix.
Use the distance formula to write an equation of
a parabola when the focus and directrix are
given.
Derive equation of a parabola.
Graph a parabola.
Determine the characteristics of a parabola based
on its equation.
Determine the equation of a parabola using
certain characteristics.
SPED Strategies:
Pre-teach vocabulary using visual and verbal
models that are connected to real life situations.
Model the thinking and processes involved in
understanding the relationship and significance
between parabolas and their respective focus and
directrix by grounding it in a real life context
such as satellite dishes.
What is the focus of a
parabola and what is its
significance?
Given the focus and directrix
of a parabola, how do we find
the equation of the parabola?
How is the process of writing
equations for parabolas,
ellipses and hyperbolas
similar/different?
How do you write the
equation of a parabola given
its focus and directrix?
Type II, III:
Defining Parabolas
Geometrically
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Develop a reference sheet for student use that
includes formulas, processes and procedures and
sample problems to encourage proficiency and
independence.
Provide students with opportunities to practice
the thinking by working with a partner or in
small groups.
ELL Strategies:
After deriving the equation of a parabola (given
a focus and directrix) explain in student’s native
language and/or use gestures, examples and
selected technical words.
Analyze satellite dishes and spotlights to discuss
focus, vertex and directrix of the parabola.
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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.
Student Learning Objective 9: Graph logarithmic functions expressed symbolically and show key features of the graph (including intercepts and
end behavior).
Modified Student Learning Objectives/Standards:
M.EE.F-IF.1–3: Use the concept of function to solve problems.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential
Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 1
MP 2
MP 4
MP 6
F-IF.7e-1
F-IF.7e-2
About half of tasks involve logarithmic
functions, while the
other half involves
trigonometric functions.
Graph logarithmic functions having base 2, 10 or e,
using technology for more complicated cases.
Show intercepts and end behavior of logarithmic
functions.
Identify whether the exponential function is a
growth or decay function from its graph.
SPED Strategies:
Pre-teach vocabulary using visual and verbal
models that are connected to real life situations.
Model the thinking and processes involved in
graphing exponential and logarithmic functions by
grounding it in a real life context.
Model the thinking behind determining when and
how to use the graphing calculator to graph
complicated exponential and logarithmic functions.
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:
Identifying graphs of
functions
Additional Tasks:
Exponential Kiss
Graphs of Power
Functions
Identifying
Exponential Functions
Logistic Growth
Model, Explicit
Version
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Illustrate the relationship between logarithmic and
exponential functions deliberately and provide
students with a reference sheet to encourage
proficiency and independence.
ELL Strategies:
Explore logarithm function by giving students a set
of notecards with the term log, b, y, =, and x
written on separate cards. Ask students to form
equations using their card.
Model the graph of the energy magnitude M of an
earthquake and let students research how to use log
to represent the energy magnitude of the
earthquake.
New Jersey Student Learning Standard(s):
A.REI.D.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. *
Student Learning Objective 10: Find approximate solutions for f(x)=g(x), using technology to graph, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, logarithmic and exponential functions.
Modified Student Learning Objectives/Standards:
M.EE.A-REI.10–12: Interpret the meaning of a point on the graph of a line. For example, on a graph of pizza purchases, trace the graph to a point
and tell the number of pizzas purchased and the total cost of the pizzas.
MPs Evidence Statement Key/
Clarifications
Skills, Strategies & Concepts Essential
Understandings/
Questions
(Accountable Talk)
Tasks/Activities
MP 1
MP 5
A-REI.11-2
Find the solutions of where the graphs of the equations
Solutions to complex systems of nonlinear
functions can be approximated graphically.
Why are the x-coordinates
of the points where the
graphs of the equations
Type II, III:
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y= f(x) and y= g(x) intersect,
e.g. using technology to
graph the functions, make
tables of values or find
successive approximations.
Include cases where f(x)
and/or g(x) are linear,
quadratic, polynomial,
rational, absolute value,
exponential, and/or
logarithmic functions. ★
The "explain" part of standard A-REI.11 is not
assessed here.
Find the solution to f(x)=g(x) approximately,
e.g., using technology to graph the functions;
include cases where f(x) and/or g(x) are linear,
polynomial, rational, absolute value,
exponential, and logarithmic functions.
Find the solution to f(x)=g(x) approximately,
e.g., using technology to make tables of values,
or find successive approximations; include
cases where f(x) and/or g(x) are linear,
polynomial, rational, absolute value,
exponential, and logarithmic functions.
Solving a system of equations algebraically
yields an exact solution; solving by graphing
or by comparing tables of values yields an
approximate solution.
The x-coordinates of the points where the
graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x)
= g(x).
SPED Strategies:
Illustrate the thinking and processes needed to
find approximate solutions for f(x)=g(x), using
technology to graph, make tables of values, or
find successive approximations using real life
examples.
Develop a reference sheet for students to
utilize when working independently and in
small groups that illustrates the thinking and
processes needed to find approximate solutions
y = f(x) and y = g(x)
intersect equal to the
solutions of the equation
f(x) = g(x)?
Why does graphing or
using a table give
approximate solutions?
In what situations would
you want an exact solution
rather than an approximate
solution or vice versa?
Introduction to
Polynomials - College
Fund
Ideal Gas Law
Population and Food
Supply
Two Squares are Equal
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for f(x)=g(x), using technology to graph, make
tables of values, or find successive
approximations.
Model how to use the graphing calculator to
find approximate solutions for f(x)=g(x), to
graph, to make tables of values, or to find
successive approximations.
ELL Strategies:
After finding approximate solutions for the
intersections of functions, explain orally why
the x-coordinates are the solutions of the
equation f(x) = g(x) in student’s native
language and/or use drawings, and selected
technical words.
Model: writing and discussing the falling
object, by using a ball, then, graph and display
in a table the results of the activity.
Using a graphing calculator, graph a variety of
equations as log, rational, radical, exponential
and ask students to compare and contrast the
shape, range and domain of each graph.
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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
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★
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.5.4: Given an equation or system of equations, reason about the number or nature of the solutions. Content Scope: A-REI.2.
Simple rational equations are limited to numerators and denominators that have degree at most 2.
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Integrated Evidence Statements
HS.C.5.11: Given an equation or system of equations, reason about the number or nature of the solutions. Content Scope: A-REI.11,
involving any of the function types measured in the standards.
For example, students might be asked how many positive solutions there are to the equation ex = x+2 or the equation ex = x+1, explaining how they know. The student might use technology strategically to plot both sides of the equation without prompting.
HS.C.6.2: Base explanations/reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted
in the coordinate plane. Content Scope: A-REI.D
HS.C.6.4: Base explanations/reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted
in the coordinate plane. Content Scope: G-GPE.2
HS.C.7.1: Base explanations/reasoning on the relationship between zeros and factors of polynomials. Content Scope: A-APR.B
HS.C.8.2: Construct, autonomously, chains of reasoning that will justify or refute algebraic propositions or conjectures. Content Scope:
A-APR.4
HS.C.8.3: Construct, autonomously, chains of reasoning that will justify or refute algebraic propositions or conjectures. Content Scope:
A-APR
HS.C.16.3: Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of
solutions (if any). Tasks are limited to simple rational or radical equations. Content scope: A-REI.1
Simple rational equations are limited to numerators and denominators that have degree at most 2.
A rational or radical function may be paired with a linear function. A rational function may not be paired with a radical function.
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)
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Integrated Evidence Statements 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 remainin