Algebra II Honors: Unit 2 Polynomial Functions Rational Exponents and Radical Functions

MATHEMATICS

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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 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

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Pacing Chart – Unit 2 # Student Learning Objective NJSLS Big Ideas Math Correlation 1 Graphing Polynomial Functions F-IF.B.4, 4-1 Instruction: F-IF.C.7c 11/13/21 – 1/28/22 Adding, Subtracting, and Multiplying Polynomials A-APR.A.1 4-2 2 A-APR.C.4 A-APR.C.5 Assessment: Mid-Year 3 Dividing Polynomials A-APR.B.2 4-3 A-PR.D.6 Assessment Factoring Polynomials A-SSE.A.2, A- 4-4 4 APR.B.2 A-APR.B.3 5 Solving Polynomial Equations A-APR.B.3 4-5

The Fundamental Theorem of Algebra N-CN.C.8 4-6 6 N-CN.C.9 A-APR.B.3 7 Transformations of Polynomial Functions F-IF.C.7c, F-BF.B.3 4-7

Analyzing Graphs of Polynomial Functions A-APR.B.3, 4-8 8 F-IF.B.4, F-IF.C.7c, F-BF.B.3 9 Modeling with Polynomial Functions A-CED.A.2 4-9 F-BF.A.1a

10 nth Roots and Rational Exponents N-RN.A.1, 5-1 N-RN.A.2

11 Properties of Rational Exponents and Radicals N-RN.A.2 5-2

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Pacing Chart – Unit 2 12 Graphing Radical Functions F-IF.C.7b, 5-3 F-BF.B.3

13 Solving Radical Equations and Inequalities A-REI.A.1 5-4 A-REI.A.2

14 Performing Function Operations F-BF.A.1b 5-5

15 Inverse of a Function A-CED.A.4 5-6 F-BF.B.4a

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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 Teacher and students practice mathematics modeling through demonstration, explicit processes together through interactive instruction, and think alouds, as well as guiding activities, problem solving, and discussion. students as they practice math strategies and apply (whole group or small group instruction) 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 7 | P a g e group instruction, small group instruction, or centers)

Effective Pedagogical Routines/Instructional Strategies Collaborative Problem Solving Analyze Student Work

Connect Previous Knowledge to New Learning Identify Student’s Mathematical Understanding

Identify Student’s Mathematical Misunderstandings Making Thinking Visible Interviews Develop and Demonstrate Mathematical Practices Role Playing

Inquiry-Oriented and Exploratory Approach Diagrams, Charts, Tables, and Graphs

Multiple Solution Paths and Strategies Anticipate Likely and Possible Student Responses

Collect Different Student Approaches Use of Multiple Representations Multiple Response Strategies Explain the Rationale of your Math Work Asking Assessing and Advancing Questions Quick Writes Revoicing Pair/Trio Sharing Marking Turn and Talk Recapping Charting Challenging Gallery Walks Pressing for Accuracy and Reasoning Small Group and Whole Class Discussions Maintain the Cognitive Demand Student Modeling

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Computer Science and Design Thinking Standards

8.1.12.DA.5, 8.1.12.DA.6, 8.1.12.AP.1

 Data and Analysis

 Create data visualizations from large data sets to summarize, communicate, and support different interpretations of real- world phenomena.

Example: Students can use Google Docs and digital tools as a means of discussing and collaboratively solving complex tasks that focus on concepts related to polynomials. https://www.mathsisfun.com/data/

 Create and refine computational models to better represent the relationships among different elements of data collected from a phenomenon or process.

Example: Students can use digital tools to graph linear systems of inequalities that model a solution to a real world situation. Students can then write a position statement to justify their solution, mathematical thinking and modeling with peers. http://www.mathsisfun.com/data/graphs-index.html.

 Algorithms & Programming

 Design algorithms to solve computational problems using a combination of original and existing algorithms.

Example: Students can use and explain the advantages of using graphing calculators, GeoGebra or Desmos to graph systems of inequalities to solve contextual problems. The discussion should also involve developing ways to improve and develop even better algorithms. https://www.desmos.com/calculator https://www.geogebra.org/geometry?lang=en-US

Link https://www.nj.gov/education/cccs/2020/2020%20NJSLS-CSDT.pdf

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Career Readiness, Life Literacies and Key Skills

Career readiness, life literacies, and key skills education provides students with the necessary skills to make informed career and financial decisions, engage as responsible community members in a digital society, and to successfully meet the challenges and opportunities in an interconnected global economy.

 Credit and Debt Management (9.1.12.CDM.8) Compare and compute interest and compound interest and develop an amortization table using business tools.

Example: Students will use their growing understanding of exponentials functions to compare interest rates and make informed decisions about the merits and pitfalls of different types of debt.

 Career Awareness and Planning (9.2.12.CAP.10) Identify strategies for reducing overall costs of postsecondary education (e.g. tuition assistance, loans, grants, scholarships, and student loans.

Example: Students will compare the immediate and long term costs and benefits of different options available to pay for college and career education after high school. Students can develop a suggested plan for hypothetical students based on their qualifications and financial circumstances.

 Technology Literacy (9.4.12.TL.1)

Assess digital tools based on features of accessibility options, capacities, and utility for accomplishing a specified task.

Example: Students can use and explain the advantages of using graphing calculators, GeoGebra or Desmos to graph polynomial functions to solve contextual problems. The discussion should also involve developing ways to improve and develop even better algorithms. https://www.desmos.com/calculator https://www.geogebra.org/geometry?lang=en-US

Link https://www.nj.gov/education/cccs/2020/2020%20NJSLS-CLKS.pdf

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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 Example practices that address Self-Awareness: Students scan multistep contextual problems Self-Management that requires them to identify variables, write Social-Awareness • Clearly state classroom rules equations, create graphs, etc., and make a list of Relationship Skills • Provide students with specific feedback regarding questions based on their understanding to ask Responsible Decision-Making academics and behavior the teacher. This will help students to gain • Offer different ways to demonstrate understanding confidence in working through the problems.

• Create opportunities for students to self-advocate • Check for student understanding / feelings about Set up small-group discussions that allows performance students to reflect and discuss challenges or • Check for emotional wellbeing how they have worked through a problem. For • Facilitate understanding of student strengths and examples, when students learn factoring techniques, students can discuss which method challenges they have the most difficulty working with.

Self-Awareness Example practices that address Self-Management: Lead discussions that encourages students to  Self-Management reflect on barriers they encounter when Social-Awareness • Encourage students to take pride/ownership in work completing an assignment (e.g., finding a Relationship Skills and behavior computer, needing extra help or needing a quiet Responsible Decision-Making • Encourage students to reflect and adapt to classroom place to work) and help them think about situations solutions to overcome those barriers. • Assist students with being ready in the classroom Teach and model for students not to become • Assist students with managing their own emotional defensive or angry when errors/flaws in their states 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 Example practices that address Social-Awareness: Organize a class service project to examine and Self-Management address a community issue. Use math to  Social-Awareness • Encourage students to reflect on the perspective of examine the situations and find possible Relationship Skills others solutions. For example, students can discuss as Responsible Decision-Making • Assign appropriate groups a class or in groups how to determine whether a • Help students to think about social strengths polynomial is a repeated solution. • Provide specific feedback on social skills Use real-world application problems to lead a • Model positive social awareness through discussion about taking different approaches to metacognition activities solving a problem and respecting the feeling and thoughts of those that used a different strategy.

Self-Awareness Example practices that address Relationship Skills: Instead of simply jumping into their own Self-Management solution when asked to graph a trigonometric Social-Awareness • Engage families and community members function, have students discuss the key features  Relationship Skills • Model effective questioning and responding to and how to find those key features before Responsible Decision-Making students graphing. • Plan for project-based learning During class or group discussion, have students • Assist students with discovering individual strengths expound upon and clarify each other’s • Model and promote respecting differences questions and comments, ask follow-up • Model and promote active listening questions, and clarify their own questions and • Help students develop communication skills reasoning. • Demonstrate value for a diversity of opinions

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Self-Awareness Example practices that address Responsible Use a lesson to teach students a simple formula Self-Management Decision-Making: for making good choices. (e.g., stop, calm Social-Awareness down, identify the choice to be made, consider Relationship Skills • Support collaborative decision making for academics the options, make a choice and do it, how did it  Responsible Decision-Making and behavior go?) Post the decision-making formula in the • Foster student-centered discipline classroom. • Assist students in step-by-step conflict resolution Routinely encourage students to use the process decision-making formula as they face a choice • Foster student independence (e.g., whether to finish homework or go out • Model fair and appropriate decision making with a friend). Support students through the • Teach good citizenship 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 Processing Comprehension Recall

 Extra time for assigned tasks  Extra Response time  Precise processes for balanced  Teacher-made checklist mathematics instructional  Adjust length of assignment  Have students verbalize steps model  Use visual graphic organizers  Timeline with due dates for  Repeat, clarify or reword  Reference resources to reports and projects  Short manageable tasks directions promote independence  Communication system  Brief and concrete directions between home and school  Mini-breaks between tasks  Visual and verbal reminders  Provide immediate feedback  Provide lecture notes/outline  Provide a warning for  Graphic organizers transitions  Small group instruction

 Partnering  Emphasize multi-sensory learning

Assistive Technology Tests/Quizzes/Grading Behavior/Attention Organization

 Computer/whiteboard  Extended time  Consistent daily structured  Individual daily planner routine  Tape recorder  Study guides  Display a written agenda  Simple and clear classroom  Video Tape  Shortened tests rules  Note-taking assistance

 Read directions aloud  Frequent feedback  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: Name of Task: Logistic Growth Model, Explicit Version (6.2.12.HistoryCC.2.a)  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:

Name of Task: Combined Fuel Efficiency (K-ESS3-3)  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.

Name of Task: Ideal Gas Law (K-ESS3-3)  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… Enrichment is not…  Planned and purposeful  Just for gifted students (some gifted students may need  Different, or differentiated, work – not just more work intervention in some areas just as some other students may need frequent enrichment)  Responsive to students’ needs and situations  Worksheets that are more of the same (busywork)  A promotion of high-level thinking skills and making connections within content  Random assignments, games, or puzzles not connected to the content areas or areas of student interest  The ability to apply different or multiple strategies to the content  Extra homework  The ability to synthesize concepts and make real world and cross- curricular connections  A package that is the same for everyone  Elevated contextual complexity  Thinking skills taught in isolation  Sometimes independent activities, sometimes direct instruction  Unstructured free time

 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)

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Assessments Required District/State Assessments District Assessments 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.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.

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.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.

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).

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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-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-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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.7c, Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

F-IF.C.7b, Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

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New Jersey Student Learning Standards 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.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.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.

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.

N-RN.A.1, Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 =5(1/3)3 to hold, so (51/3)3 must equal 5.

N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

N-CN.C.8 Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x - 2i).

N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

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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: Polynomial Functions

Rational Exponents and Radical Functions

NJSLS: A-APR.A.1 , A-REI.A.2,A-APR.C.4,A-APR.C.5,A-APR.B.2,A-APR.B.3,A-APR.D.6,A-SSE.A.2, A-APR.B.2,A-APR.B.3,A-REI.A.1, A- CED.A.4,A-CED.A.2,F-IF.B.4,F-IF.C.7c, F-IF.C.7b,F-BF.B.3,F-BF.A.1b,F-BF.A.1a,F-BF.B.4a,N-RN.A.1,N-RN.A.2,N-RN.A.2,N-CN.C.8 N-CN.C.9

Unit Focus:

 Understand polynomial functions.  Graph the polynomial functions.  Add, subtract, multiply, divide, and factor polynomials.  Solve the polynomial equations.  Model with and analyze graphs of polynomial functions.  Understand rational exponents and radical functions.  Represent roots using rational exponents.  Describe the properties of rational exponents and radicals.  Solve radical equations and inequalities.  Perform function operations.

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New Jersey Student Learning Standard(s):

F-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Student Learning Objective 1: Graphing Polynomial Functions. Modified Student Learning Objectives/Standards: N/A MPs Evidence Statement Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Key/ Clarifications Questions (Accountable Talk) MP2 F.IF.B.4 MP5 Clarification • Identify polynomial functions and explain what is Interactive MP6 Items must have real- meant by the end behavior of a polynomial function. What are some common Explorations: world context. • Limit Graph polynomial functions using tables and show characteristics of the  Identifying key features to: them end behavior. graphs of cubic and quartic intercepts; intervals polynomial functions? Graphs of where the function is Use graphing calculators and ask students to graph Polynomial increasing, decreasing, f(x) = x3, f(x) = −x3, f(x) = x4, and f(x) = −x4. After When you know the end Functions positive, or negative; each graph, have students stand up and use their behavior of a polynomial,  Identifying x- relative maximums and arms to describe the end behavior of the function. what other information is intercepts of minimums; symmetries; needed to Polynomial end behavior. Share the polynomial model with students and ask graph the function? them to determine a good viewing window. Graphs • Items may include “Will plotting discrete determining the equation SPED Strategies: points produce an accurate of the axis of symmetry graph? of the graph of a given Model the thinking behind determining when and quadratic function. • End how to use the graphing calculator to graph “Does the average rate of behavior would be complicated polynomials. change describe 29 | P a g e

assessed informally. (e.g. the graph well? Describe what happens to Provide students with opportunities to practice the the graph ( ) = 5 .9( ) x f x thinking and processes involved in graphing What do you know about as the values of x polynomial equations by hand and using technology the graph of increase.) • Limit items to by working small groups. f(x) = 3.5x4 + 4x2 − 7x + 2 linear, quadratic, square without actually graphing root functions, piecewise- Develop a reference sheet for student use that the defined (including step includes formulas, processes and procedures and function?” functions, and absolute- sample problems to encourage proficiency and value functions), and independence. exponential functions. Exponential functions are ELL Strategies: limited to those with

domains in the integers. • Demonstrate comprehension of quadratic questions Items may include in student’s native language and/or simplified determining and/or questions with drawings and selected technical interpreting the slope/rate words concerning graphing functions symbolically of change of a linear by showing key features of the graph by hand in function from a table or simple cases and using technology for more graph complicated cases. Use technology to graph polynomial and identify the end behavior and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the polynomial.

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New Jersey Student Learning Standard(s):

A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

A-APR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle Student Learning Objective 2: Adding, Subtracting, and Multiplying Polynomials Modified Student Learning Objectives/Standards: N/A MPs Evidence Statement Skills, Strategies & Concepts Essential Tasks/Activities Key/ Clarifications Understandings/ Questions (Accountable Talk) MP2 A-APR.A.1 How can you cube a MP7 Add, subtract and multiply the polynomials binomial? Interactive Use Pascal’s Triangle to expand binomials. MP8 The understand part Explorations:

of this standard is not What other observations  Cubing Binomials Write the expression (5x2 + 7x + 1) + (3x2 − 4x + assessed. can be made about the nth 2) on the board as a warm up problem, then Give  Generalizing • Polynomials must row in Pascal’s Triangle an introduction to Pascal’s Triangle if students are Patterns for be of degree one, two and the expansion of or three. unfamiliar with it. Note: The top row of the triangle Cubing a Binomial is called the 0th row. 푛 (푎 + 푏) = ?

• Students should quickly see that the third row of the triangle (1, 3, 3, 1) contains the coefficients of the expansion of (x + 1)3.

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• Students may be less able to make a connection between the entries in Pascal’s Triangle and he cube of any binomial.

There are many patterns found in Pascal’s Triangle. Assign students an independent project to research and write a report on three different patterns in Pascal’s Triangle.

If additional examples are needed, you might wait for the formal lesson and then refer back to Pascal’s Triangle. • Alternately, ask students to consider what the expansion of (a + b)3 would be when b = 2. This may help students understand the usefulness of Pascal’s Triangle.

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.

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ELL Strategies:

Demonstrate comprehension of quadratic 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. Use technology to create table of values to verify the positive zeros of the polynomial.

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).

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Student Learning Objective 3: Dividing Polynomials Modified Student Learning Objectives/Standards:

M.EE.F-IF.1–3: Use the concept of function to solve problems. MPs Evidence Statement Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Key/ Clarifications Questions (Accountable Talk) N/A Use long division to divide polynomials by other MP2 polynomials. Use synthetic division to divide How can you use the factors Interactive polynomials by binomials of the form x − k. of a cubic polynomial to Explorations: solve a division problem  Dividing Use the Remainder Theorem. involving the polynomial? polynomials

Use a completed polynomial long division problem as an example. Circle the coefficients throughout the example. Explain to students that the numeric portion is the key part of the problem.

Writing variables and exponents is extraneous. The like terms are aligned for a reason.

• Explain and model the steps of synthetic division. Compare the results with the same problem done as long division so that students see the parallel work and the same numbers. It may take several examples for students to understand why synthetic division works.

SPED Strategies:

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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:

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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).

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 4: Factoring Polynomials Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Key/ Clarifications Questions (Accountable Talk) A-SSE.A.2 How can you factor a MP3 Factor polynomials and use the Factor Theorem. polynomial? Interactive MP5 Evidence Statement • Students may think that factors must be binomials. Explorations: Rewrite linear, quadratic Were all quadratics Show them whether there is a common monomial  Factoring and exponential factorable? expressions. Clarification factor. Polynomials • Unlike standard A polynomial f (x) has a factor x − k if and only A.SSE.1, items that if f (k) = 0. What information can you address A.SSE.2 do not obtain about the graph of a need to have real-world Use technology to make a table of values to verify polynomial function context. • In Algebra I, the positive zeros and heights between the zeros. written in factored form? 36 | P a g e

expressions are limited to one variable expression. • SPED Strategies: Items could ask a student to identify expressions Model the thinking behind determining when and equivalent to a given how to use the graphing calculator to graph expression. • Items could complicated polynomials. ask a student to rewrite an expression in one Provide students with opportunities to practice the variable. thinking and processes involved in graphing polynomial equations by hand and using A.APR.B.3 technology by working small groups.

Clarification Develop a reference sheet for student use that • The understand part of includes formulas, processes and procedures and this standard is not sample problems to encourage proficiency and assessed. • Polynomials independence. must be of degree one, two or three. ELL Strategies:

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New Jersey Student Learning Standard(s):

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 5: Solving Polynomial Equations Modified Student Learning Objectives/Standards: N/A MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Tasks/Activities Clarifications Understandings/ Questions (Accountable Talk) A.APR.B.3 MP1 Find solutions of polynomial equations and zeros of How can you determine Interactive Evidence Statement polynomial functions. Use the Rational Root whether a polynomial Explorations: equation has a repeated Theorem and the Irrational Conjugates Theorem.  Cubic Equations • As stated in the standard. solution?

and Repeated Give an example of a cubic with only one solution. Clarification What is the maximum Solutions Use the word multiplicity to describe the repeated possible number of real  Quartic Equations roots. Discuss the difference between a graph • Cubic polynomials may zeros that the function and Repeated be used if one linear factor crossing at a root versus touching the axis at a root. can have? Solutions and an easily factorable

quadratic factor are Explain why synthetic division is more efficient for

provided. finding the roots than substituting the possible roots • Zeros of cubic into the equation. Explain that from the Rational polynomials must be Root Theorem a list of possible roots can be integers. • Construction of a rough graph is limited to written, but the actual roots still need to be found. the graph of a quadratic polynomial. Explain why the Irrational Conjugates Theorem should make sense.

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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:

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New Jersey Student Learning Standard(s):

N-CN.C.8 Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x - 2i).

N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

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 6: The Fundamental Theorem of Algebra

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Tasks/Activities Clarifications Understandings/ Questions (Accountable Talk) MP1 A.APR.B.3 Use the Fundamental Theorem of Algebra. How can you determine whether a polynomial Interactive Find conjugate pairs of complex zeros of Evidence Statement equation has imaginary Explorations: polynomial functions. solutions?  Cubic Equations • As stated in the Use Descartes’s Rule of Signs. standard. Does every polynomial and Imaginary Make sure students to understand, equation have at least one Solutions Clarification If f is a polynomial function with real coefficients, real solution?  Quartic Equations

and a + bi is an imaginary zero of f, then a − bi is and Imaginary • Cubic polynomials Is it possible, polynomial Solutions may be used if one also a zero of f and also remind them that if the degree 4 have three real

linear factor and an polynomial has three solutions does not mean they roots and one imaginary easily factorable are all real solutions. root?

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quadratic factor are provided. • Zeros of cubic polynomials must be SPED Strategies: integers. • Construction of a Model the thinking behind determining when and rough graph is limited how to use the graphing calculator to graph to the graph of a complicated polynomials. quadratic polynomial. 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:

Use technology to graph polynomial and identify the end behavior and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the polynomial.

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New Jersey Student Learning Standard(s): F-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 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 7: Transformations of Polynomial Functions 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/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions (Accountable Talk)

F.BF.B.3 Describe transformations of polynomial functions. How can you transform the Interactive MP5 Write transformations of polynomial functions. graph of a polynomial Explorations: Evidence Statement function?  Transforming the Draw the table and fill in the second column, f(x) • As stated in the standard. Notation. Have partners fill in the first column. Do you recognize that Graph of a Cubic Write the examples and have students describe the transformations of Function Clarification transformation. polynomials will follow the  Transforming the patterns found when they Graph of a Quartic • Limit to linear and Pose a problem then have students work with worked with Function quadratic functions. • partners to describe the transformations and to transformations of

Even and odd functions graph the original and transformed functions. quadratics.. are not assessed In Algebra I. • The SPED Strategies: experiment part of the standard is instructional Model the thinking behind determining when and only. This aspect of the how to use the graphing calculator to graph standard is not assessed. complicated polynomials.

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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 quadratic 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. Use technology to create table of values to verify the positive zeros of the polynomial.

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New Jersey Student Learning Standard(s): 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. 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.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 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 8: Analyzing Graphs of Polynomial Functions Modified Student Learning Objectives/Standards: N/A MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions (Accountable Talk) A-APR.B.3 Real Life STEM MP3 Use x-intercepts to graph polynomial functions. Video Task: MP5 Evidence Statement Use the Location Principle to identify zeros of How many turning points can Quonset Huts the graph of a polynomial polynomial functions. function have? Find turning points and identify local maximums • As stated in the Interactive standard. and local minimums of graphs of polynomial How many turning points can Explorations: functions. Identify even and odd functions. the graph of a polynomial  Approximating Clarification function have? Turning Points

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• Limit to linear and Make sure students to know, If f is a polynomial quadratic functions. • function, and a and b are two real numbers such Even and odd that f (a) < 0 and f (b) > 0, then f has at least one Is it possible to sketch the real zero between a and b. graph of a cubic polynomial functions are not function that has no turning assessed In Algebra I. Students should know, the graph of every points? Justify your answer • The experiment part polynomial function of degree n has at most n − 1 of the standard is turning points and if a polynomial function of instructional only. This degree n has n distinct real zeros, then its graph aspect of the standard has exactly n − 1 turning points. is not assessed.

Please talk about even and odd functions.

A.APR.B.3 Explain to your partners everything you know

about the graph of the function. Have students Evidence Statement verify their conjectures by graphing the function

on a calculator. • As stated in the

standard. SPED Strategies:

Clarification Model the thinking behind determining when

and how to use the graphing calculator to graph • Cubic polynomials complicated polynomials. may be used if one

linear factor and an Provide students with opportunities to practice easily factorable the thinking and processes involved in graphing quadratic factor are polynomial equations by hand and using provided. technology by working small groups. • Zeros of cubic

polynomials must be Develop a reference sheet for student use that integers. • includes formulas, processes and procedures and Construction of a sample problems to encourage proficiency and rough graph is limited independence. 45 | P a g e

to the graph of a quadratic polynomial. ELL Strategies:

Use technology to graph polynomial and identify the end behavior and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the polynomial.

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New Jersey Student Learning Standard(s):

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.

Student Learning Objective 9: Modeling with Polynomial Functions 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/ Skills, Strategies & Concepts Essential Tasks/Activities Clarifications Understandings/ Questions (Accountable Talk) A-CED.A.2 MP4 Write polynomial functions for sets of points. How can you find a Interactive MP6 Evidence Statement Write polynomial functions using finite differences. polynomial model for Explorations: real-life data? Use technology to find models for data sets.  Modeling Real-Life • As stated in the standard.

Data Discuss limitations of finding an exact model for Clarification real-life applications. Use technology to find Performance Task: reasonable models. Pay attention to the units • Items must have real- For the Birds-Wildlife world context • Limit Have students to summarize what they understand Management equations to two variables. about graphs of polynomial functions and modeling

polynomial functions, and what helped them learn F.BF.A.1a these concepts and skills.

Evidence Statement SPED Strategies: 47 | P a g e

• Write a function based on Model the thinking behind determining when and an observed pattern in a how to use the graphing calculator to graph real-world scenario. complicated polynomials.

Clarification Provide students with opportunities to practice the thinking and processes involved in graphing • Items must have real- polynomial equations by hand and using world context. • Limit to technology by working small groups. linear, quadratic and exponential functions with Develop a reference sheet for student use that domains in the integers. includes formulas, processes and procedures and sample problems to encourage proficiency and • Similar to creating a independence. function from a scatterplot

but for this standard the ELL Strategies: relationship between the

two quantities is clear from Demonstrate comprehension of quadratic questions the context. 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. Use technology to create table of values to verify the positive zeros of the polynomial.

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New Jersey Student Learning Standard(s):

N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents Student Learning Objective 10: nth Roots and Rational Exponents Modified Student Learning Objectives/Standards: M.EE.N-RN.1 Determine the value of a quantity that is squared or cubed.

Evidence Statement Key/ Skills, Strategies & Concepts Essential Tasks/Activities MPs Clarifications Understandings/ Questions (Accountable Talk) How can you use a rational MP6 N/A Find nth roots of numbers. exponent to represent a Interactive Evaluate expressions with rational exponents. power involving a radical? Explorations:

Solve equations using nth roots.  Exploring the

Definition of a Please be care full of common mistakes among Rational Exponent students. For example

Students sometimes think that to solve a  Writing

problem involving taking the cube Expressions in root of an expression, you divide by 3. Rational Exponent Form  Writing Expressions in Radical Form

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New Jersey Student Learning Standard(s):

N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents Student Learning Objective 11: Properties of Rational Exponents and Radicals Modified Student Learning Objectives/Standards: M.EE.N-RN.1 Determine the value of a quantity that is squared or cubed.

MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Tasks/Activities Clarifications Understandings/ Questions (Accountable Talk)

MP6 N/A Use properties of rational exponents to Is it true, adding and Interactive MP7 simplify expressions with rational exponents. subtracting like radicals is Explorations: the same as adding and  Reviewing Use properties of radicals to simplify and write subtracting like terms or radical expressions in simplest form. like fractions? Properties of Exponents Explain that two of the properties can be How can you use  Simplifying 1 expressed using radical notation when 푚 = properties of exponents to Expressions with 푛 simplify products and n is an integer greater than 1. Rational Exponents quotients of radicals?  Simplifying Simplifying the expressions takes time for Products and students. Do not rush. Quotients of There is often more than one correct approach. Radicals. When students have had sufficient time to work the problem.

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SPED Strategies:

Model the thinking behind determining when and how to use the graphing calculator to graph complicated Exponential functions.

Provide students with opportunities to practice the thinking and processes involved in graphing exponential functions 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 exponential functions 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 exponential functions and identify the end behavior and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the exponential functions. 51 | P a g e

New Jersey Student Learning Standard(s):

F-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

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 12: Graphing Radical Functions 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/ Skills, Strategies & Concepts Essential Tasks/Activities Clarifications Understandings/ Questions (Accountable Talk) MP6 F.BF.A.1a Evidence Statement Graph radical functions. How can you identify the Interactive Write transformations of radical functions. domain and range of a Explorations: • Write a function based on radical function? Graph parabolas and circles.  Identifying Graphs an observed pattern in a real-

world scenario. What does the graph of of Radical Describe the transformations of the parent Functions function and why they are equivalent. 푦 = √푥 look like when Clarification you turn it 90°  Identifying Graphs

SPED Strategies: counterclockwise? of Transformations • Items must have real-world

context. • Limit to linear, Model the thinking behind determining when quadratic and exponential and how to use the graphing calculator to functions with domains in graph complicated Radical Functions. the integers. • Similar to

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creating a function from a Provide students with opportunities to practice scatterplot but for this the thinking and processes involved in standard the relationship graphing radical functions by hand and using between the two quantities is technology by working small groups. clear from the context. Develop a reference sheet for student use that F.BF.B.3 includes formulas, processes and procedures and sample problems to encourage proficiency Evidence Statement and independence.

• As stated in the standard. ELL Strategies:

Clarification Demonstrate comprehension of radical functions in student’s native language and/or • Limit to linear and quadratic simplified questions with drawings and functions. • Even and odd selected technical words concerning graphing functions are not assessed In functions symbolically by showing key Algebra I. • The experiment part features of the graph by hand in simple cases of the standard is instructional and using technology for more complicated only. This aspect of the standard cases. is not assessed. Use technology to graph radical functions and identify the end behavior and x and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the exponential functions.

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New Jersey Student Learning Standard(s):

A-REI.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Student Learning Objective 13: Solving Radical Equations and Inequalities Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Tasks/Activities Clarifications Understandings/ Questions (Accountable Talk)

MP7 A.REI.A.1 Solve equations containing radicals and How can you solve a Interactive rational exponents. Solve radical inequalities radical equation? Explorations: Evidence Statement • Identify  Solving Radical or justify a solution method. Ask students how they would solve 2x − 5 = Clarification • Items that Equations require solving linear equations 7. Then ask how they would solve 2√푥 − will require the student to 5 =7 . The steps are the same. explain the properties used in the solution process. • Items SPED Strategies: may include solving quadratic equations. Model the thinking behind determining when and how to use the graphing calculator to graph complicated radical equations and inequalities. Provide students with opportunities to practice the thinking and processes involved in graphing radical 54 | P a g e

equations and inequalities 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 radical equations and inequalities 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 radical equations and Inequalities polynomial and identify the end behavior and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the radical equations and inequalities.

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New Jersey Student Learning Standard(s):

Student Learning Objective 14: Performing Function Operations

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Tasks/Activities Clarifications Understandings/ Questions (Accountable Talk) How can you use the Real Life STEM Video MP5 N/A Students should know the notation: (f + g)(x) graphs of two functions to Task: The Heartbeat MP6 means they want to add two functions named f sketch the graph of an Hypothesis and g. The notation f(x) + g(x) shows the actual arithmetic combination of

functions being added together. Discuss the the two functions? domain and range of the sum, difference, Interactive product, and quotient of two functions. What are the domain and Explorations: range of (f + g)(x)?  Graphing the Sum SPED Strategies: of Two Functions

Model the thinking behind determining when and how to use the graphing calculator to graph complicated equations. Provide students with opportunities to practice the thinking and processes involved in graphing equations by hand and using technology by working small groups.

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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.

ELL Strategies:

Demonstrate comprehension of function operations 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 polynomials and identify the end behavior and y intercept in the figure.

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New Jersey Student Learning Standard(s):

A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

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.

Student Learning Objective 15: Inverse of a Function

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Tasks/Activities Clarifications Understandings/ Questions (Accountable Talk)

MP2 Explore inverses of functions and verify How can you sketch the Interactive A-CED.A.4 inverses of nonlinear functions. graph of the inverse of Explorations: Solve real-life problems using inverse a function?  Graphing Evidence Statement functions. Functions and • As stated in the standard. Take time explore 푓(푥)2푥 + 3 푎푛푑 푔(푥) = Their Inverses 푥−3  Sketching Graphs 2 Clarification −1 of Inverse Show them 푓(푓 ) = 푥 Functions • Items must have real-world context. SPED Strategies: Performance Task: • Limit quadratic formulas to those Model the thinking behind determining when Turning the Tables that do not contain a linear term. and how to use the graphing calculator to graph complicated equations and its inverse. Provide students with opportunities to practice

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the thinking and processes involved in graphing 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 inverse of a function 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 equations and their inverse. They need to know the x and y intercept in the figure. Use technology to create table of values of both, equations and their inverse.

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Honors Projects (Must complete all projects)

Project 1 Project 2 Project 3 It’s a Matter of Interpretation Health Indicators Project Satellite Dish Dilemma

Essential Question: Essential Question: Essential Question: How does the interpretation of a quadratic function How do statistics and data facilitate the How does the efficacy of a satellite dish facilitate informed decision making in a real life context? informed decision making process? change based on the dimensions?

Skills: Skills: Skills: 1. Analyze and interpret a quadratic function. 1. Accurately calculate and display data.1. 1. Calculate the focus, and depth of a parabolic curve given the directrix. 2. Predict outcomes based on the behavior and features of 2. Analyze data to make informed the parabolic curve. 2. 2. Analyze the features of different assertions . parabolic curves. 3. Make decisions based on the data and support those decisions with evidence. 3. Make decisions based on analysis and support the decisions made.

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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 풇(풙) = 풄 for a simple function f that has 풙+ퟏ an inverse and write an expression for the inverse. For example, 풇(풙) = ퟐ풙ퟑ or 풇(풙) = for 풙 ≠ ퟏ. 풙−ퟏ  For example, see http://illustrativemathematics.org  As another example, given a function C(L) = 750퐿2 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. 61 | P a g e

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 remaining standards in the Evidence Statement Text.

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.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.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.

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Integrated Evidence Statements  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 Algebra Functions Statistics and Probability and Quantity Complex number Binomial Theorem Absolute value Logarithmic 2-way frequency Independent Conjugate Complete the square function function table Inter-quartile range Determinant Exponential Asymptote Trigonometric Addition Rule Joint relative Fundamental function Geometric Amplitude function Arithmetic sequence frequency theorem of Algebra series Arc Midline Box plot Margin of error Identity matrix Logarithmic Arithmetic sequence Negative intervals Causation Marginal relative Imaginary number Function Constant function Period Combinations frequency Initial point Maximum Cosine Periodicity Complements Multiplication Rule Moduli Minimum Decreasing intervals Positive intervals Conditional Observational

Parallelogram rule Pascal’s Triangle Domain Radian measure probability studies Polar form Remainder Theorem End behavior Range Conditional relative Outlier Quadratic equation Exponential decay Rate of change frequency Permutations Polynomial Exponential Recursive process Correlation Recursive process Rational exponent function Relative maximum Correlation Relative frequency Real number Exponential growth Relative minimum coefficient Residuals Rectangular form Fibonacci sequence Sine Dot plot Sample survey Scalar Function notation Step function Experiment Simulation models

multiplication of Geometric sequence Symmetries Fibonacci sequence Standard deviation Matrices Increasing intervals Tangent Frequency table Subsets Terminal point Intercepts Geometric sequence Theoretical Vector Invertible function Histogram probability

Velocity Unions Zero matrix

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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/

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Field Trip Ideas SIX FLAGS GREAT ADVENTURE: This educational event includes workbooks and special science and math related shows throughout the day. Your students will leave with a better understanding of real world applications of the material they have learned in the classroom. Each student will have the opportunity to experience different rides and attractions linking mathematical and scientific concepts to what they are experiencing. www.sixflags.com

MUSEUM of MATHEMATICS: Mathematics illuminates the patterns that abound in our world. The National Museum of Mathematics strives to enhance public understanding and perception of mathematics. Its dynamic exhibits and programs stimulate inquiry, spark curiosity, and reveal the wonders of mathematics. The Museum’s activities lead a broad and diverse audience to understand the evolving, creative, human, and aesthetic nature of mathematics. www.momath.org

LIBERTY SCIENCE CENTER - An interactive science museum and learning center located in Liberty State Park. The center, which first opened in 1993 as New Jersey's first major state science museum, has science exhibits, the largest IMAX Dome theater in the United States, numerous educational resources, and the original Hoberman sphere.

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

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