Algebra I: Unit 3 Exponential Functions, Quadratic Functions, & Polynomials

MATHEMATICS

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Course Philosophy/Description

The fundamental purpose of Algebra 1 is to formalize and extend the mathematics that students learned in the elementary and middle grades. The Standards for Mathematical Practice apply throughout each course, and, together with the New Jersey Student Learning Standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. Conceptual knowledge behind the mathematics is emphasized. Algebra I provides a formal development of the algebraic skills and concepts necessary for students to succeed in advanced courses as well as the PARCC. The course also provides opportunities for the students to enhance the skills needed to become college and career ready.

The content shall include, but not be limited to, perform set operations, use fundamental concepts of logic including Venn diagrams, describe the concept of a function, use function notation, solve real-world problems involving relations and functions, determine the domain and range of relations and functions, simplify algebraic expressions, solve linear and literal equations, solve and graph simple and compound inequalities, solve linear equations and inequalities in real-world situations, rewrite equations of a line into slope-intercept form and standard form, graph a line given any variation of information, determine the slope, x- and y- intercepts of a line given its graph, its equation or two points on the line, write an equation of a line given any variation of information, determine a line of best fit and recognize the slope as the rate of change, factor polynomial expressions, perform operations with polynomials, simplify and solve algebraic ratios and proportions, simplify and perform operations with radical and rational expressions, simplify complex fractions, solve rational equations including situations involving mixture, distance, work and interest, solve and graph absolute value equations and inequalities, graph systems of linear equations and inequalities in two and three variables and quadratic functions, and use varied solution strategies for quadratic equations and for systems of linear equations and inequalities in two and three variables.

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ESL Framework

This ESL framework was designed to be used by bilingual, dual language, ESL and general education teachers. Bilingual and dual language programs use the home language and a second language for instruction. ESL teachers and general education or bilingual teachers may use this document to collaborate on unit and lesson planning to decide who will address certain components of the SLO and language objective. ESL teachers may use the appropriate leveled language objective to build lessons for ELLs which reflects what is covered in the general education program. In this way, whether it is a pull-out or push-in model, all teachers are working on the same Student Learning Objective connected to the New Jersey Learning Student Standards. The design of language objectives are based on the alignment of the World-Class Instructional Design Assessment (WIDA) Consortium’s English Language Development (ELD) standards with the 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 3

# Student Learning Objective NJSLS Big Ideas Math Correlation Instruction: 1/29/22 – 4/8/22 Solving Exponential Equations A.CED.A.1 1 A.REI.A.1 6-5 A.REI.D.11

Geometric Sequences F.IF.A.3 6-6 2 F.BF.A.2 F.LE.A.2

Recursively Defined Sequences F.IF.A.3 6-7 F.BF.A.1a 3 F.BF.A.2 F.LE.A.2

4 Adding and Subtracting Polynomials A.APR.A.1 7-1

A.APR.A.1 5 Multiplying Polynomials 7-2

A.APR.A.1 6 Special Products of Polynomials 7-3

7 Solving Polynomial Equations in Factored Form A.APR.B.3 7-4 A.REI.B.4b

7-5 8 Factoring 푥2 + 푏푥 + 푐 A.SSE.A.2 A.SSE.B.3a

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Pacing Chart – Unit 3

2 A.SSE.A.2 9 Factoring 푎푥 + 푏푥 + 푐 A.SSE.B.3a 7-6

A.SSE.A.2 10 Factoring Special Products A.SSE.B.3a 7-7

A.SSE.A.2

11 Factoring Polynomials Completely A.SSE.B.3a 7-8

A.CED.A.2 ( ) 2 8-1 12 Graphing 푓 푥 = 푎푥 F.IF.C.7a F.BF.B.3 A.CED.A.2 2 13 Graphing 푓(푥) = 푎푥 + 푐 F.IF.C.7a 8-2 F.BF.B.3 A.CED.A.2 Graphing 푓(푥) = 푎푥2 + 푏푥 + 푐 8-3 14 F.IF.C.7a F.IF.C.9 F.BF.B.3 A.CED.A.2 2 Graphing 푓(푥) = 푎(푥 − ℎ) + 푘 F.IF.B.4 8-4 15 F.BF.B.3 F.BF.A.1

A.SSE.B.3a

Using Intercept Form A.APR.B.3 A.CED.A.2 16 F.IF.B.4 8-5 F.BF.A.1a F.IF.C.8a

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Pacing Chart – Unit 3

8-6 17 Comparing Linear, Exponential, and Quadratic F.IF.B.6 Functions F.BF.A.1a F.LE.A.3

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Research about Teaching and Learning Mathematics Structure teaching of mathematical concepts and skills around problems to be solved (Checkly, 1997; Wood & Sellars, 1996; Wood & Sellars, 1997) Encourage students to work cooperatively with others (Johnson & Johnson, 1975; Davidson, 1990) Use group problem-solving to stimulate students to apply their mathematical thinking skills (Artzt & Armour-Thomas, 1992) Students interact in ways that support and challenge one another’s strategic thinking (Artzt, Armour-Thomas, & Curcio, 2008) Activities structured in ways allowing students to explore, explain, extend, and evaluate their progress (National Research Council, 1999) There are three critical components to effective mathematics instruction (Shellard & Moyer, 2002): • Teaching for conceptual understanding • Developing children’s procedural literacy • Promoting strategic competence through meaningful problem-solving investigations Teachers should be: • Demonstrating acceptance and recognition of students’ divergent ideas. • Challenging students to think deeply about the problems they are solving, extending thinking beyond the solutions and algorithms required to solve the problem • Influencing learning by asking challenging and interesting questions to accelerate students’ innate inquisitiveness and foster them to examine concepts further. • Projecting a positive attitude about mathematics and about students’ ability to “do” mathematics Students should be: • Actively engaging in “doing” mathematics • Solving challenging problems • Investigating meaningful real-world problems • Making interdisciplinary connections • Developing an understanding of mathematical knowledge required to “do” mathematics and connect the language of mathematical ideas with numerical representations • Sharing mathematical ideas, discussing mathematics with one another, refining and critiquing each other’s ideas and understandings • Communicating in pairs, small group, or whole group presentations • Using multiple representations to communicate mathematical ideas • Using connections between pictures, oral language, written symbols, manipulative models, and real-world situations • 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 8 | P a g e assesses learning andBoard reteaches Approved as necessary. 08.18.21 (whole group instruction, small group instruction, or centers)

Effective Pedagogical Routines/Instructional Strategies

Collaborative Problem Solving Identify Student’s Mathematical Misunderstandings Connect Previous Knowledge to New Learning Interviews Making Thinking Visible

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 Use of Multiple Representations Collect Different Student Approaches Explain the Rationale of your Math Work Multiple Response Strategies Quick Writes

Pair/Trio Sharing Asking Assessing and Advancing Questions

Turn and Talk Revoicing

Charting Marking Gallery Walks Recapping Small Group and Whole Class Discussions

Student Modeling Challenging

Pressing for Accuracy and Reasoning Analyze Student Work

Maintain the Cognitive Demand Identify Student’s Mathematical Understanding

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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, exponential functions and geometric sequences. 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 exponential functions and geometric sequences 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 exponential functions and geometric sequences 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.1) Identify the purposes, advantages, and disadvantages of debt.

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.4)

Students will evaluate different careers and develop various plans(e.g., costs of public, private, training schools) and timetables for achieving them, including educational/training requirements, costs, loans, and debt repayments. .

Example: Students will compare different career paths in terms of costs to be trained, time needed, income potential, and personal fulfillment. Students can then discuss what they learned and how it could impact their choices for education and training after high school.

• Critical Thinking and Problem Solving (9.4.12.CT.2)

Students will demonstrate an understanding of the potential benefits of collaborating to enhance critical thinking and problem solving.

Example: Students will engage in mathematical discourse on a daily basis. This practice requires collaboration and engagement with others to explain your thought process and to understand the approach of others. Discourse serves to clarify and deepen understanding of all parties.

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 the key features of quadratic functions, incorporating problems that relate to student interests such

as music, sports and art enable the students to understand and relate to the concept in a more meaningful way.

• Everyone has a Voice: Create a classroom environment where students know that their contributions are expected and valued. Example: Norms for sharing are established that communicate a growth mindset for mathematics. All students are capable of expressing mathematical thinking and contributing to the classroom community. Students learn new ways of looking at problem solving by working with and listening to each other.

• Run Problem Based Learning Scenarios: Encourage mathematical discourse among students by presenting problems that are relevant to them, the school and /or the community. Example: Using a Place Based Education (PBE) model, students explore math concepts while determining ways to address

problems that are pertinent to their neighborhood, school or culture.

• Encourage Student Leadership: Create an avenue for students to propose problem solving strategies and potential projects. Example: Students can deepen their understanding of quadratic functions and polynomials 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: Have students keep a math journal. This can Self-Management create a record of their thoughts, common Social-Awareness • Clearly state classroom rules mistakes that they repeatedly make when Relationship Skills • Provide students with specific feedback regarding solving problems; and a place to record how Responsible Decision-Making academics and behavior they would approach or solve a problem. • Offer different ways to demonstrate understanding • Create opportunities for students to self-advocate Lead frequent discussions in class to give • Check for student understanding / feelings about students an opportunity to reflect after performance learning new concepts. Discussion questions • Check for emotional wellbeing may include asking students: “Which method • Facilitate understanding of student strengths and of solving a quadratic equation do you find challenges the easiest to implement?” or “What difficulty are you having when graphing a quadratic function?”

Self-Awareness Example practices that address Self- Teach students to set attainable learning goals ✓ Self-Management Management: and self-assess their progress towards those Social-Awareness learning goals. For example, a powerful Relationship Skills • Encourage students to take pride/ownership in strategy that promotes academic growth is Responsible Decision-Making work and behavior teaching an instructional routine to self-assess • Encourage students to reflect and adapt to within the independent phase of the Balanced classroom situations Mathematics Instructional Model. • Assist students with being ready in the classroom • Assist students with managing their own Teach students a lesson on the proper use of emotional states equipment (such as the computers, textbooks and graphing calculators) and other resources properly. Routinely ask students who they

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think might be able to help them in various situations, including if they need help with a math problem or using a technology tool.

Self-Awareness Example practices that address Social- When there is a difference of opinion among Self-Management Awareness: students (perhaps over solution strategies), ✓ Social-Awareness allow them to reflect on how they are feeling Relationship Skills • Encourage students to reflect on the perspective of and then share with a partner or in a small Responsible Decision-Making others group. It is important to be heard but also to • Assign appropriate groups listen to how others feel differently, in the • Help students to think about social strengths same situation. • Provide specific feedback on social skills • Model positive social awareness through Have students re-conceptualize application metacognition activities problems after class discussion, by working beyond their initial reasoning to identify common reasoning between different approaches to solve the same problem. Self-Awareness Example practices that address Relationship Have students team-up at the at the end of the Self-Management Skills: unit to teach a concept to the class. At the Social-Awareness end of the activity students can fill out a self- ✓ Relationship Skills • Engage families and community members evaluation rubric to evaluate how well they Responsible Decision-Making • Model effective questioning and responding to worked together. For example, the rubric can students consist of questions such as, “Did we • Plan for project-based learning consistently and actively work towards our • Assist students with discovering individual group goals?”, “Did we willingly accept and strengths fulfill individual roles within the group?”, • Model and promote respecting differences “Did we show sensitivity to the feeling and • Model and promote active listening learning needs of others; value their • Help students develop communication skills

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• Demonstrate value for a diversity of opinions knowledge, opinion, and skills of all group members?”

Encourage students to begin a rebuttal with a restatement of their partner’s viewpoint or argument. If needed, provide sample stems, such as “I understand your ideas are ___ and I think ____because____.” Self-Awareness Example practices that address Responsible Have students participate in activities that Self-Management Decision-Making: requires them to make a decision about a Social-Awareness situation and then analyze why they made Relationship Skills • Support collaborative decision making for that decision, students encounter conflicts ✓ Responsible Decision-Making academics and behavior within themselves as well as among members • Foster student-centered discipline of their groups. They must compromise in • Assist students in step-by-step conflict resolution order to reach a group consensus. process • Foster student independence Today’s students live in a digital world that • Model fair and appropriate decision making comes with many benefits — and also • Teach good citizenship increased risks. Students need to learn how to be responsible digital citizens to protect themselves and ensure they are not harming others. Educators can teach digital citizenship through social and emotional learning.

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Differentiated Instruction Accommodate Based on Students Individual Needs: Strategies

Time/General Processing Comprehension Recall

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

• Provide lecture notes/outline • Provide a warning for • Small group instruction • Graphic organizers transitions • Emphasize multi-sensory • Partnering 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 and process • Reference sheets that list formulas, step-by-step procedures and model strategies • Conceptual word wall that contains definition, translation, pictures and/or examples • Graphic organizer to help students solve quadratic equations using different methods (such as quadratic formula, completing the square, factoring, etc.) • Translation dictionary • Sentence stems to provide additional language support for ELL students • Teacher modeling • Highlight and label solution steps for multi-step problems in different colors • Create an interactive notebook with students with a table of contents so they can refer to previously taught material readily • Exponent rules chart • Graph paper • Algebra tiles • Graphing calculator • Visual, verbal and algebraic models of quadratic functions • A chart noting key features of quadratic functions based on visual, graphical and verbal presentation • Videos to reinforce skills and thinking behind concepts • Access to tools such as tables, graphs and charts to solve problems

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Interdisciplinary Connections Model interdisciplinary thinking to expose students to other disciplines. Social Studies Connection: Population and Food Supply (6.1.12.GeoHE16.a) • Students will analyze the population and food production growth and analyze the main challenge of food shortages. They will also explain why natural resources (i.e., fossil fuels, food, and water) continue to be a source of conflict and analyze how the United States and other nations have addressed issues concerning the distribution and sustainability of natural resources and climate change.

Financial Literacy Connection: Stock Prices (9.1.12.D.1) • Students will calculate short- and long-term returns on various investments (e.g., stocks, bonds, mutual funds, IRAs, deferred pension plans, and so on).

Physical Education Connection: Springboard Dive (2.12.MSC.2) • Students will analyze the time, pathway and height of a diver.

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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 Unit Assessment NJSLA SGO Assessments

Suggested Formative/Summative Classroom Assessments Describe Learning Vertically Identify Key Building Blocks Make Connections (between and among key building blocks) Short/Extended Constructed Response Items Multiple-Choice Items (where multiple answer choices may be correct) Drag and Drop Items Use of Equation Editor Quizzes Journal Entries/Reflections/Quick-Writes Accountable talk Projects Portfolio Observation Graphic Organizers/ Concept Mapping Presentations Role Playing Teacher-Student and Student-Student Conferencing Homework

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New Jersey State Learning Standards

A.CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions and quadratic functions, and simple rational and exponential functions.

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.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.* [Focus on linear equations.]

F.IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

F.BF.A.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

F.LE.A.2: Construct linear and exponential functions - including arithmetic and geometric sequences - given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). *[Algebra 1 limitation: exponential expressions with integer exponents]

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

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

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New Jersey State Learning Standards rough graph of the function defined by the polynomial. *[Algebra 1: limit to quadratic and cubic functions in which linear and quadratic factors are available].

A.REI.B.4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

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.SSE.B.3a: Factor a quadratic expression to reveal the zeros of the function it defines.

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.C.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima. 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.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.*[Limit to linear and exponential]

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. *[Focus on exponential functions]

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New Jersey State Learning Standards

F.BF.A.1: Write a function that describes a relationship between two quantities.

F.IF.C.8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

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.LE.A.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

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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 1 Unit: 3 (Three) Topic: Exponential Functions, Quadratic Functions, & Polynomials

NJSLS: A.CED.A.1, A.REI.A.1, A.REI.D.11, F.IF.A.3, F.BF.A.2, F.LE.A.2, F.BF.A.1a, A.APR.A.1, A.APR.B.3, A.REI.B.4b, A.SSE.A.2, A.SSE.B.3a, A.CED.A.2, F.IF.C.7a, F.BF.B.3, F.IF.C.9, F.IF.B.4, F.BF.A.1, F.IF.C.8a, F.IF.B.6, F.LE.A.3 Unit Focus: • Solve exponential equations • Geometric and recursively defined sequences • Perform arithmetic operations on polynomials • Solving polynomials in factored form. • Factoring polynomials • Graphing quadratic functions. • Using intercept form • Interpret functions that arise in applications in terms of the context • Construct & compare linear, quadratic, & exponential models

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New Jersey Student Learning Standard(s): A.CED.A.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions and quadratic functions, and simple rational and exponential functions.

Student Learning Objective 1: Solve exponential equations. 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. 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/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions (Accountable Talk) How can you solve an exponential MP 1 HS.D.2-5 Introduce solving exponential equations equation graphically? IFL “Solving MP 3 graphically by using graphing calculators so Problems Using MP 5 • A-CED is the primary students can develop and understanding of How is solving exponential equations Linear and MP 7 content; other listed how exponential graphs behave. different when you have like or unlike Exponential content elements may bases? What do you need to think Models.” *Also be involved in tasks as Review the Property of Equality for about? addressed in IFL well. Exponential Equations as a way to introduce unit is F.LE.B.5. (F.IF.A.3 is not HS.D.2-6 30 | P a g e Board Approved 08.18.21

solving exponential equations when the What are some applications where you addressed in IFL A-CED is the primary bases are the same. may choose to solve exponential Unit) content; other listed content equations graphically? elements may be involved Review the properties of exponents before Interactive in tasks as well. solving exponential equations with unlike Explorations: bases to facilitate students understanding. • Solving Solve exponential equations with unlike Exponential bases including bases where 0 > b < 1. Equations Graphically SPED Strategies: • The Number Use real-life examples with visual models to of Solutions of demonstrate how to solve exponential an equations with like and unlike bases. Be Exponential sure to provide students with time for Equation guided practice on the use of the graphing calculator when solving exponential

equations.

Provide graphic organizers/notes/Google Docs/Anchor Charts that include helpful hints and strategies to employ when solving exponential equations. Students should be able to use this as a resource to increase confidence, proficiency and understanding.

ELL Support: If students are receiving mathematics instruction in Spanish, use the Khan

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Academy video from Khan Academy en Español to provide students with a native language explanation of exponential equations. Khan Academy Video

Create a graphic organizers/notes/Google Docs/Anchor Charts with students that include helpful hints and strategies to employ when solving exponential equations. Students should be able to use this as a resource to increase confidence, proficiency and understanding.

New Jersey Student Learning Standard(s): A F.IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. F.BF.A.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★ F.LE.A.2: Construct linear and exponential functions - including arithmetic and geometric sequences - given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). *[Algebra 1 limitation: exponential expressions with integer exponents] F.BF.A.1a: Determine an explicit expression, a recursive process, or steps for calculation from a context.

Student Learning Objective 2: Geometric sequences. Student Learning Objective 3: Recursively defined sequences.

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Modified Student Learning Objectives/Standards: M.EE.F-IF.1–3: Use the concept of function F-IF.2. Use function notation, evaluate functions for inputs in their domains, and to solve problems. M.EE.F-BF.1: Select the appropriate graphical representation (first quadrant) given a situation involving constant rate of change. M.EE.F-BF.2: Determine an arithmetic sequence with whole numbers when provided a recursive rule. M.EE.F-LE.1–3: Model a simple linear function such as y = mx to show that these functions increase by equal amounts over equal intervals. MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions (Accountable Talk) Develop a conceptual understanding of How can you use a Interactive MP 1 F-LE.2-1 what geometric sequences model in real geometric sequence to Explorations: MP 3 • Construct linear and exponential life. This will enable students to learn how describe a pattern? • Describing MP 5 functions, including arithmetic to identify geometric sequences and to Calculator and geometric sequences, given MP 7 understand the concept of the common How can you define a Patterns a graph, a description of a ratio. sequence recursively? relationship, or two input-output • Folding a Sheet of Paper pairs (include reading these from Learn how to use the information from What are some of the things • Describing a a table). i) Tasks are limited to identifying a geometric sequence and a that we need to think about Pattern constructing linear and stepping off point to extending geometric to determine patterns in a exponential functions with • Using a sequences. sequence? domains in the integers, in Recursive simple real-world context (not Equation multi-step). Graph geometric sequences. F-LE.2-2 Performance Task: • Solve multi-step contextual Write geometric sequences as functions The New Car problems with degree of and find the nth term in a geometric difficulty appropriate to the sequence from real life situations to course by constructing linear solidify the conceptual understanding and and/or exponential function provide a context for understanding the models, where exponentials are processes. limited to integer exponents.★ i)

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Prompts describe a scenario Write terms of recursively defined using everyday language. sequences. Mathematical language such as Write recursive rules for sequences. "function," "exponential," etc. is not used. ii) Students Translate between explicit and recursive autonomously choose and apply rules. appropriate mathematical

techniques without prompting. For example, in a situation of Write recursive rules for special sequences doubling, they apply techniques such as the Fibonacci sequence. of exponential functions. iii) For some illustrations, see tasks at SPED Strategies: http://illustrativemathematics.org Work through the explorations with under F-LE. sequences so students develop a conceptual understanding of how sequences look and behave.

Provide graphic organizers/notes/Google Docs/Anchor Charts that include helpful hints and strategies to employ when identifying, extending, graphing and writing geometric sequences. Students should be able to use this as a resource to increase confidence, proficiency and understanding.

Provide graphic organizers/notes/Google Docs/Anchor Charts that include helpful hints and strategies to employ when learning, practicing, sand studying the

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characteristics of recursive sequences. Students should be able to use this as a resource to increase confidence, proficiency and understanding.

ELL Support:

Work with students in a guided format to explore the concepts of geometric and recursive sequences. Create visual supports that help students to reinforce learning and reinforce the concepts when students are working independently.

Encourage discussion of the topic through the use of the explorations of sequences so that students develop an understanding of how sequences look and behave.

Ensure that language supports are provided such as access to word-to-word dictionary, notes, anchor charts, linguistically simpler explanations and visual models.

New Jersey Student Learning Standard(s):

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Student Learning Objective 4: Adding and subtracting polynomials Student Learning Objective 5: Multiplying polynomials Student Learning Objective 6: Special products of polynomials

Modified Student Learning Objectives/Standards: N/A MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions (Accountable Talk) A-APR.1-1 Develop an understanding of the degree of Interactive MP 1 • Add, subtract, and polynomials and relate it to previous learning What rules govern how Explorations: MP 3 multiply polynomials. about combining like terms. polynomials can be added and • Adding MP 5 i) The "understand" part subtracted? Polynomials MP 7 of the standard is not Classify polynomials and write them in • Subtracting assessed here; it is standard form. What is the process and the Polynomials assessed under Sub-claim thinking involved in • Multiplying Polynomials Using C. Use the horizontal and vertical format for multiplying and dividing Algebra Tiles adding and subtracting polynomials. polynomials? • Multiplying

Binomials Using Multiply binomials and trinomials and What are the patterns in the Algebra Tiles connect the process to the distributive following special products? • Finding a Sum and 2 property. (푎 + 푏)(푎 − 푏), (푎 + 푏) , Difference Pattern 2 (푎 − 푏) • Finding the Square Solve real-life problems that use the addition, of a Binomial subtraction, and multiplication of Pattern polynomials.

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Use algebra tiles to help students understand the square of a binomial pattern and the equations that represent it.

SPED Strategies: Provide students with ample opportunity to work with polynomials in context and determine the appropriate approach to solving the problem.

Create a Google Doc graphic organizer to help students keep operation of polynomials organized.

ELL Support: Create anchor chart/notes with students that document the thinking and processes behind adding, subtracting, and multiplying polynomials.

Relate multiplication of polynomials to students by using real-life problems such as area problems to help students understand the concept on a different level.

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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. *[Algebra 1: limit to quadratic and cubic functions in which linear and quadratic factors are available]. A.REI.B.4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. 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.SSE.B.3a: Factor a quadratic expression to reveal the zeros of the function it defines. A.REI.B.4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. 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.SSE.B.3a: Factor a quadratic expression to reveal the zeros of the function it defines.

Student Learning Objective 7: Solving polynomials in factored form Student Learning Objective 8: Factoring 푥2 + 푏푥 + 푐 Student Learning Objective 9: Factoring 푎푥2 + 푏푥 + 푐 Student Learning Objective 10: Factoring special products Student Learning Objective 11: Factoring polynomials completely

Modified Student Learning Objectives/Standards: N/A MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions (Accountable Talk)

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Use the Zero Product Property. How can you solve Interactive MP 2 A-APR.3-1 polynomial equations by Explorations: MP 6 • Identify zeros of quadratic Factor polynomials using the greatest finding the roots or zeros? • Matching common factor (GCF). MP 7 and cubic polynomials in Equivalent Forms

which linear and quadratic How can you use algebra tiles of an Equation factors are available, and Solve polynomial equations by factoring. to factor the trinomial • Writing a use the zeros to construct a Solve multi-variable formulas or literal 푥2 + 푏푥 + 푐 into the product Conjecture rough graph of the function equations, for a specific variable. of two binomials? • Special defined by the polynomial. Properties of 0 i) For example, find the Factor polynomials with coefficient = 1 and How can you use algebra tiles zeros of (x-2)(x2 -9). ii) and 1. Sketching graphs is limited terms b and c are either positive or negative. to factor the trinomial • Finding Binomial 2 to quadratics. iii) For cubic 푎푥 + 푏푥 + 푐 into the product Factors polynomials, at least one Factor polynomials with coefficient >1 and of two binomials? • Factoring Special linear factor must be terms b and c are either positive or negative. Products provided or one of the linear How can you recognize and • Writing a factors must be a GCF Factor the difference of two squares. factor special products? A A-REI.4a-1 Product of Linear

• Solve quadratic equations in Factors Factor perfect square trinomials. one variable. a) Use the • Matching How are graphs of equations method of completing the Standard and square to transform any Solve real life problems involving factoring and inequalities similar and Factored Forms quadratic equation in x into to ground the concept in contexts that different? an equation of the form (x – facilitate understanding. Real Life STEM p)2 = q that has the same How do you determine if a Video Task: Birds solutions. i) The derivation SPED Strategies: given point is a viable solution Dropping Food part of the standard is not Create Google Doc with students that to a system of equations or assessed here; it is assessed provides support when learning to factor. It inequalities, both on a graph Performance Task: under Sub-Claim C. and using the equations? should include the thinking process behind • The View A-SSE.2-4 the algorithm. Matters • Use the structure of a numerical expression or 39 | P a g e Board Approved 08.18.21

polynomial expression in Model how to approach different polynomial • Flight Path of a one variable to rewrite it, in types when factoring. Bird a case where two or more rewriting steps are required. Use real world situations that can be i) Example: Factor modeled with polynomials to create a deeper completely: 풙 ퟐ - 1 + (풙 − level of understanding. ퟏ) ퟐ . (A first iteration

might give (x+1)(x-1) + (풙 Ask students assessing and advancing − ퟏ) ퟐ , which could be rewritten as (x-1)(x+1+x1) questions to elicit their level of on the way to factorizing understanding and propel thinking forward. completely as 2x(x-1). Or the student might first ELL Support: expand, as 풙 ퟐ - 1 + 풙 ퟐ - Create a document with students that clearly 2x + 1, rewriting as 2풙 ퟐ - depict visual and verbal models for factoring 2x, then factorizing as 2x(x- using appropriate language level or native 1).) ii) Tasks do not have a language. real-world context.

Use relevant contextual examples to

reinforce and extend understanding of polynomials and factoring.

Ensure that language supports are provided such as access to word-to-word dictionary, notes, anchor charts, linguistically simpler explanations and visual models.

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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.IF.C.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima. 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.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.*[Limit to linear and exponential] 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. *[Focus on exponential functions] F.BF.A.1: Write a function that describes a relationship between two quantities. Student Learning Objective 12: Graphing 푓(푥) = 푎푥2 Student Learning Objective 13: Graphing 푓(푥) = 푎푥2 + 푐 Student Learning Objective 14: Graphing 푓(푥) = 푎푥2 + 푏푥 + 푐 Student Learning Objective 15: Graphing 푓(푥) = 푎(푥 − ℎ)2 + 푘 Modified Student Learning Objectives/Standards: M.EE.A-CED.2–4: Solve one-step inequalities. 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. M.EE.F-BF.1: Select the appropriate graphical representation (first quadrant) given a situation involving constant rate of change. MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions 41 | P a g e Board Approved 08.18.21

(Accountable Talk) MP 4 Identify characteristics of quadratic functions to What are some of the IFL Set of Related MP 6 F-BF.3-4 develop an understanding of how the graphs characteristics of the graph of Tasks: “Developing • Identify the effect on the look and what changes in the form can tell us. a function of the form an Understanding of ( ) 2 graph of a quadratic 푓 푥 = 푎푥 ? Quadratics”

function of replacing f(x) Sketch graphs of parent functions and the by f(x) + k, kf(x), f(kx), How does the value of c affect changes that occur as the value of the lead ( ) 2 Interactive and f(x + k) for specific the graph of 푓 푥 = 푎푥 + 푐? coefficient changes. Explorations: values of k (both positive How can you find the vertex • Graphing and negative); find the value of k given the Sketch graphs of parent functions and the of the graph Quadratic 2 graphs. Experiment with changes that occur as the value of the constant 푓(푥) = 푎푥 + 푏푥 + 푐? Functions

cases using technology. i) changes. • Graphing 풚 = Illustrating an explanation How can you describe the ퟐ is not assessed here. graph of a 푓(푥) = 푎(푥 − 풂풙 + 풄 Explanations are assessed Graph quadratic functions in the form of ℎ)2 + 푘 ? • Finding x- 2 under Sub-claim C. 푓(푥) = 푎푥 + 푏푥 + 푐 and find the maximum intercepts of and minimum values of the function. graphs What are the relative HS.C.9.1 Identify odd and even functions. • Comparing x- • Express reasoning about maximums and minimums of the function? What does that intercepts with transformations of functions. Develop an understanding of vertex form of a tell us? the Vertex Content scope: F-BF.3, quadratic equation and how it impacts graphing. • Finding x- limited to linear and quadratic functions. Tasks Key features of the graph of a intercepts Use real life situations that quadratic equations will not involve ideas of quadratic function–vertex, • Deductive even or odd functions. can model to make the process of graphing average rate of change, and reasoning them more relevant. symmetry–can be analyzed • Graphing 풂(풙 −

HS.D.2-6 and interpreted within a 풉)ퟐ when h>0 SPED Strategies: • Solve multi-step contextual problem • Graphing 풂(풙 − word problems with degree Fortify students’ understanding of quadratic context and can be used to 풉)ퟐ when h<0 of difficulty appropriate to functions by reviewing and creating a document solve problems. the course, requiring

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application of course-level that highlights the key features of a quadratic knowledge and skills function and how to notice them in a graph. articulated in A-CED, N- Q.2, A-SSE.3, A-REI.6, A- Provide opportunities to practice sketching REI.12, A-REI.11-1, limited graphs based on verbal models of quadratic to linear and quadratic functions. equations. i) A-CED is the

primary content; other listed content elements may be Model the thinking that is needed when creating involved in tasks as well. sketches from verbal models. F-IF.7a-2 • Graph functions expressed ELL Support: symbolically and show key Create a document that describes the key features of the graph, by features of a quadratic and provides helpful hand in simple cases and hints on how to notice them in a graph using using technology for more visual/pictorial representations, simplified complicated cases.★ a) language or native language as appropriate. Graph quadratic functions and show intercepts, Provide students with the ongoing support and maxima, and minima. guidance they need to increase conceptual understanding of quadratics and simultaneously build language proficiency. This can be done by planning effective questions, filling in the background knowledge gaps and providing language supports.

New Jersey Student Learning Standard(s):

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A.SSE.B.3a: Factor a quadratic expression to reveal the zeros of the function it defines. 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. *[Algebra 1: limit to quadratic and cubic functions in which linear and quadratic factors are available]. 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. *[Focus on exponential functions] F.BF.A.1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F.IF.C.8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Student Learning Objective 16: Using intercept form

Modified Student Learning Objectives/Standards: M.EE.A-CED.2–4: Solve one-step inequalities. 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. M.EE.A-SSE.3: Solve simple algebraic equations with one variable using multiplication and division. MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions (Accountable Talk) Use intercept form to find the zeros of a Interactive MP 1 A-SSE.3a, function. What are some of the Explorations: MP 2 A-SSE.3b characteristics of the graph • Using Zeros to MP 4 • The equivalent form must Use intercept form to determine the domain 푓(푥) = 푎(푥 − 푝)(푥 − 푞)? Write Functions MP 7 reveal the zeroes of the and range of a function. Alternate, equivalent forms of function. a quadratic expression may

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Write quadratic and cubic functions using the reveal specific attributes of the • Tasks require students to characteristics of the graph or given function that it defines make the connection information by using knowledge of vertex and between the equivalent intercept form. Can students simplify

forms of the expression. expressions and recognize equivalent forms of the same Write expressions in equivalent forms by expression? factoring to find the zeros of a quadratic function and explain the meaning of the zeros. What strategies can be applied Given a quadratic function explain the to create equivalent meaning of the zeros of the function. That is if expressions that lead to f(x) = (x – c) (x – a) then f(a) = 0 and f(c) = 0. accuracy and efficiency Given a quadratic expression, explain the What strategies can be used to meaning of the zeros graphically. That is for an expression (x – a) (x – c), a and c correspond factor a quadratic equation? to the x-intercepts (if a and c are real). What is the most efficient way Write expressions in equivalent forms by to transform expressions for completing the square to convey the vertex expressions for exponential form, to find the maximum or minimum value functions? of a quadratic function, and to explain the meaning of the vertex.

SPED Strategies: Illustrate to students how equivalent forms of a quadratic expression make it easier to see the key features of the function.

Create notes/Google Doc/Anchor Chart with students to illustrate explicitly what the

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different forms reveal. Ex: factoring and zeros, vertex form and the coordinates of the vertex (maxima and minima).

ELL Support: Demonstrate how graphing quadratic and cubic functions makes it easier to see the key features of the function by using language that meets students’ learning needs.

Illustrate how the different forms of the quadratic highlight key features differently and discuss when you might use them. Document this learning on an anchor chart or in notes so that students can refer to it when working independently.

New Jersey Student Learning Standard(s): 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.

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

F.LE.A.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Student Learning Objective 17: Comparing linear, exponential, and quadratic functions. Modified Student Learning Objectives/Standards: M.EE.F-BF.1: Select the appropriate graphical representation (first quadrant) given a situation involving constant rate of change. M.EE.F-LE.1–3: Model a simple linear function such as y = mx to show that these functions increase by equal amounts over equal intervals. MPs Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Clarifications Questions (Accountable Talk) Compare a linear, exponential, and quadratic How can you compare the IFL “Creating and MP 2 F-IF.6-6a functions by having students look at and growth rates of linear, Interpreting MP 4 • Estimate the rate of change discuss the graphs. exponential, and quadratic Functions.” from a graph of linear functions? functions and quadratic Compare a linear, exponential, and quadratic (F.LE.B.5 is not functions.★ i) Tasks have a functions by having students look at and What data would you need to addressed in IFL real-world context. discuss the ordered pairs. write a linear, basic quadratic, Unit)

or basic exponential function? F-IF.6-6b Real Life STEM Compare a linear, exponential, and quadratic • Estimate the rate of change Video Task: functions by having students look at and from a graph of linear Comparing Growth functions, quadratic discuss the tables. Models functions, square root functions, cube root Create a Google Doc highlighting the patterns Performance Task: functions, piecewise-defined and characteristics of linear, exponential, and Asteroid Aim functions (including step quadratic functions with students and functions and absolute value

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functions), and/or encourage students to refer to the document Interactive exponential functions with when working independently. Explorations: domains in the integers.★ i) Comparing Speeds Tasks have a real-world Write functions to model given data. context. Compare functions using average rates of change.

SPED Strategies: Model the thinking and provide students with multiple examples of how to compare linear, exponential, and quadratic functions.

Create a Google Doc highlighting the patterns and characteristics of linear, exponential, and quadratic functions with students and encourage students to refer to the document when working independently.

ELL Support: Use contexts of high interest to students to explore the comparison of linear, exponential, and quadratic functions.

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Ensure that students are provided with appropriate language supports such as native language instruction, access to notes, dictionaries and effective questioning so that they can further their conceptual understanding.

Integrated Evidence Statements F-IF.A.Int.1: Understand the concept of a function and use function notation. • Tasks require students to use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a real-world context. 49 | P a g e Board Approved 08.18.21

Integrated Evidence Statements

• About a quarter of tasks involve functions defined recursively on a domain in the integers.

F-INT.1-1: Given a verbal description of a linear or quadratic 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, nonlinear; and find an input value leading to a given output value. • e.g., a functional dependence might be described as follows: "The area of a square is a function of the length of its diagonal." The student would be asked to create an expression such as f(x) = (1/2) x2 for this function. The natural domain for the function would be the positive real numbers. The function is increasing and nonlinear. And so on. • E.g., a functional dependence might be described as follows: "The slope of the line passing through the points (1, 3) and (7, y) is a function of y." The student would be asked to create an expression such as s(y) = (3-y)/(1-7) for this function. The natural domain for this function would be the real numbers. The function is increasing and linear. And so on.

HS-INT.1: Solve multi-step contextual problems with degree of difficulty appropriate to the course by constructing quadratic function models and/or writing and solving quadratic equations • A scenario might be described and illustrated with graphics (or even with animations in some cases). • Solutions may be given in the form of decimal approximations. For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required; however, students will not be penalized if they simplify the radicals correctly. • Some examples: - A company sells steel rods that are painted gold. The steel rods are cylindrical in shape and 6 cm long. Gold paint costs $0.15 per square inch. Find the maximum diameter of a steel rod if the cost of painting a single steel rod must be $0.20 or less. You may answer in units of centimeters or inches. Give an answer accurate to the nearest hundredth of a unit. - As an employee at the Gizmo Company, you must decide how much to charge for a gizmo. Assume that if the price of a single gizmo is set at P dollars, then the company will sell 1000 - 0.2P gizmos per year. Write an expression for the amount of money the company will take in each year if the price of a single gizmo is set at P dollars. What price should the company set in order to take in as much money as possible each year? How much money will the company make per year in this case? How many gizmos will the company sell per year? (Students might use graphical and/or algebraic methods to solve the problem.) - At t = 0, a car driving on a straight road at a constant speed passes a

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Integrated Evidence Statements telephone pole. From then on, the car's distance from the telephone pole is given by C(t) = 30t, where t is in seconds and C is in meters. Also at t = 0, a motorcycle pulls out onto the road, driving in the same direction, initially 90 m ahead of the car. From then on, the motorcycle's distance from the telephone pole is given by M(t) = 90 + 2.5 푡2, where t is in seconds and M is in meters. At what time t does the car catch up to the motorcycle? Find the answer by setting C and M equal. How far are the car and the motorcycle from the telephone pole when this happens? (Students might use graphical and/or algebraic methods to solve the problem.)

HS-INT.2: Solve multi-step mathematical problems with degree of difficulty appropriate to the course that requires analyzing quadratic functions and/or writing and solving quadratic equations. • Tasks do not have a real-world context.

• Exact answers may be required or decimal approximations may be given. Students might choose to take advantage of the graphing utility to find approximate answers or clarify the situation at hand. For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required. Some examples: o Given the function f(x) = 푥2 + x, find all values of k such that f(3 - k) = f(3). (Exact answers are required.) o Find a value of c so that the equation 2푥2 - cx + 1 = 0 has a double root. Give an answer accurate to the tenths place

HS-INT.3-1: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in F-LE, A-CED.1, A-SSE.3, F-IF.B, F-IF.7, limited to linear functions and exponential functions with domains in the integers.★ • 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-INT.3-2: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in F-LE, A-CED.1, A-SSE.3, F-IF.B, F-IF.7, limited to linear, quadratic, and exponential functions.★ • 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. For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required; however, students will not be penalized if they simplify the radicals correctly.

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Unit 3 Vocabulary

• Average Rate of Change • Exponential Equations • Intercept Form • Axis of Symmetry • Exponential Function • Leading Coefficient • Binomial • Exponential Growth • Linear Function • Common Ratio • Exponential Growth Function • Maximum Value • Complex Solutions • Expressions • Minimum Value • Compound Interest • Factored Completely • Monomial • Decreasing • Factored Form • Negative Exponents • Degree of a Monomial • Factoring by Grouping • nth Root of a • Degree of a Polynomial • Factor Pairs • Odd Function • Difference of Two Squares Pattern • FOIL Method • Parabola • Discriminant • Form • Parallel Lines • Distributive Property • Geometric Sequence • Partial Product

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• Divisibility Rules • Graphing • Partial Quotient • Elimination • Graph of a Linear Inequality • Parent Quadratic Function • Equation • Graph of a System of Linear • Perfect Square Trinomial Pattern • Estimate Inequalities • Perpendicular Lines • Even Function • Greatest Common Factor • Point-Slope Form • Explicit Rule • Identity Property of Multiplication • Polynomial • Exponential Decay • Increasing • Power of a Power Property • Exponential Decay Function • Index • Power of a Product Property • Prime Number • Roots • System of Linear Equations • Product • Rule • System of Linear Inequalities • Product of Powers Property • Sequence • Term • Property of Equality for Exponential • Solution of a Linear Inequality in Two • Translation Equations Variables • Transformation • Quadratic Equation • Solution of a System of Linear • Trinomial • Quadratic Function Equations • Variable • Quotient • Solution of a System of Linear • Vertex • Quotient of Powers Property Inequalities • Vertex Form • Rational Exponents • Solving by Inspection • Vertical Shrink • Reasonableness • Standard Form • Vertical Stretch • Recursive Rule • Step function • Zero Exponent • Related Facts • Substitution • Zero of a Function • Repeated Roots • Sum and Difference Pattern • Zero-Product Property

53 | P a g e Board Approved 08.18.21

References & Suggested Instructional Websites Khan Academy https://www.khanacademy.org

Achieve the Core http://achievethecore.org

Illustrative Mathematics https://www.illustrativemathematics.org/

Inside Mathematics www.insidemathematics.org

Learn Zillion https://learnzillion.com

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

Big Ideas Math https://www.bigideasmath.com/

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

NCTM Illuminations https://illuminations.nctm.org/Search.aspx?view=search&type=ls&gr=9-12

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

Desmos https://www.desmos.com/

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References & Suggested Instructional Websites Geogebra http://www.geogebra.org/

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

Partnership for Assessment of Readiness for College and Careers https://parcc.pearson.com/#

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

55 | P a g e Board Approved 08.18.21

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/

56 | P a g e Board Approved 08.18.21