MATHEMATICS...The design of language objectives are based on the alignment of the World-Class Instructional Design Assessment (WIDA) Consortium’s ... extend, and evaluate their progress

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MATHEMATICS

Algebra II: Unit 3 Exponential and Logarithmic Functions

Rational Functions Sequences and Series

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Course Philosophy/Description

Algebra II continues the students’ study of advanced algebraic concepts including functions, polynomials, rational expressions, systems of functions and inequalities, and matrices. Students will be expected to describe and translate among graphic, algebraic, numeric, tabular, and verbal representations of relations and use those representations to solve problems. Emphasis will be placed on practical applications and modeling. Students extend their knowledge and understanding by solving open-ended real-world problems and thinking critically through the use of high level tasks.

Students will be expected to demonstrate their knowledge in: utilizing essential algebraic concepts to perform calculations on polynomial expression; performing operations with complex numbers and graphing complex numbers; solving and graphing linear equations/inequalities and systems of linear equations/inequalities; solving, graphing, and interpreting the solutions of quadratic functions; solving, graphing, and analyzing solutions of polynomial functions, including complex solutions; manipulating rational expressions, solving rational equations, and graphing rational functions; solving logarithmic and exponential equations; and performing operations on matrices and solving matrix equations.

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ESL Framework This ESL framework was designed to be used by bilingual, dual language, ESL and general education teachers. Bilingual and dual language programs use the home language and a second language for instruction. ESL teachers and general education or bilingual teachers may use this document to collaborate on unit and lesson planning to decide who will address certain components of the SLO and language objective. ESL teachers may use the appropriate leveled language objective to build lessons for ELLs which reflects what is covered in the general education program. In this way, whether it is a pull-out or push-in model, all teachers are working on the same Student Learning Objective connected to the New Jersey Student Learning Standards. The design of language objectives are based on the alignment of the World-Class Instructional Design Assessment (WIDA) Consortium’s English Language Development (ELD) standards with the New Jersey Student Learning Standards (NJSLS). WIDA’s ELD standards advance academic language development across content areas ultimately leading to academic achievement for English learners. As English learners are progressing through the six developmental linguistic stages, this framework will assist all teachers who work with English learners to appropriately identify the language needed to meet the requirements of the content standard. At the same time, the language objectives recognize the cognitive demand required to complete educational tasks. Even though listening and reading (receptive) skills differ from speaking and writing (expressive) skills across proficiency levels the cognitive function should not be diminished. For example, an Entering Level One student only has the linguistic ability to respond in single words in English with significant support from their home language. However, they could complete a Venn diagram with single words which demonstrates that they understand how the elements compare and contrast with each other or they could respond with the support of their native language with assistance from a teacher, para-professional, peer or a technology program.

http://www.state.nj.us/education/modelcurriculum/ela/ELLOverview.pdf

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Pacing Chart – Unit 3 # Student Learning Objective NJSLS Big Ideas Math

Correlation

Instruction: 8 weeks Assessment: 1 week 1

Exponential Growth and Decay Functions A‐SSE.B.3c F‐IF.C.7e F‐IF.C.8b F‐LE.A.2 F‐LE.B.5

6‐1

2 The Natural Base e F‐IF.C.7e F‐LE.B.5

6‐2

3 Logarithms and Logarithmic Functions F‐IF.C.7e

F‐BF.B.4a F‐LE.A.4

6‐3

4 Transformations of Exponential and Logarithmic Functions

F‐IF.C.7e F‐BF.B.3

6‐4

5 Properties of Logarithms A‐SSE.A.2 F‐LE.A.4

6‐5

6 Solving Exponential and Logarithmic Equations A‐REI.A.1 F‐LE.A.4

6‐6

7 Modeling with Exponential and Logarithmic Functions A‐CED.A.2

F‐BF.A.1a F‐LE.A.2

6‐7

8 Inverse Variation A‐CED.A.1,

A‐CED.A.2 A‐CED.A.3

7‐1

9 Graphing Rational Functions A‐APR.D.6 F‐BF.B.3

7‐2

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Pacing Chart – Unit 3 10 Multiplying and Dividing Rational Expressions A-APR.D.6

A-APR.D.7 7‐3

11 Adding and Subtracting Rational Expressions A-APR.D.6, A-APR.D.7

7‐4

12 Solving Rational Equations A-CED.A.4

A-REI.A.1, A-REI.A.2

7‐5

13 Defining and Using Sequences and Series F-IF.A.3 8‐1

14 Analyzing Arithmetic Sequences and Series F-IF.A.3

F-BF.A.2, F-LE.A.2

8‐2

15 Analyzing Geometric Sequences and Series A-SSE.B.4

F-IF.A.3 F-BF.A.2, F-LE.A.2

8‐3

16 Finding Sums of Infinite Geometric Series A-SSE.B.4 8‐4

17 Using Recursive Rules with Sequences F-IF.A.3

F-BF.A.1a F-BF.A.2

8‐5

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Research about Teaching and Learning Mathematics Structure teaching of mathematical concepts and skills around problems to be solved (Checkly, 1997; Wood & Sellars, 1996; Wood & Sellars, 1997) Encourage students to work cooperatively with others (Johnson & Johnson, 1975; Davidson, 1990) Use group problem-solving to stimulate students to apply their mathematical thinking skills (Artzt & Armour-Thomas, 1992) Students interact in ways that support and challenge one another’s strategic thinking (Artzt, Armour-Thomas, & Curcio, 2008) Activities structured in ways allowing students to explore, explain, extend, and evaluate their progress (National Research Council, 1999) There are three critical components to effective mathematics instruction (Shellard & Moyer, 2002):

Teaching for conceptual understanding Developing children’s procedural literacy Promoting strategic competence through meaningful problem-solving investigations

Teachers should be: Demonstrating acceptance and recognition of students’ divergent ideas. Challenging students to think deeply about the problems they are solving, extending thinking beyond the solutions and algorithms required

to solve the problem Influencing learning by asking challenging and interesting questions to accelerate students’ innate inquisitiveness and foster them to

examine concepts further Projecting a positive attitude about mathematics and about students’ ability to “do” mathematics

Students should be: Actively engaging in “doing” mathematics Solving challenging problems Investigating meaningful real-world problems Making interdisciplinary connections Developing an understanding of mathematical knowledge required to “do” mathematics and connect the language of mathematical ideas

with numerical representations Sharing mathematical ideas, discussing mathematics with one another, refining and critiquing each other’s ideas and understandings Communicating in pairs, small group, or whole group presentations Using multiple representations to communicate mathematical ideas Using connections between pictures, oral language, written symbols, manipulative models, and real-world situations Using technological resources and other 21st century skills to support and enhance mathematical understanding

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Mathematics is not a stagnate field of textbook problems; rather, it is a dynamic way of constructing meaning about the world around us, generating knowledge and understanding about the real world every day. Students should be metaphorically rolling up their sleeves and “doing mathematics” themselves, not watching others do mathematics for them or in front of them. (Protheroe, 2007)

Balanced Mathematics Instructional Model

Balanced math consists of three different learning opportunities; guided math, shared math, and independent math. Ensuring a balance of all three approaches will build conceptual understanding, problem solving, computational fluency, and procedural fluency. Building conceptual understanding is the focal point of developing mathematical proficiency. Students should frequently work on rigorous tasks, talk about the math, explain their thinking, justify their answer or process, build models with graphs or charts or manipulatives, and use technology. When balanced math is used in the classroom it provides students opportunities to:

solve problems make connections between math concepts and real-life situations communicate mathematical ideas (orally, visually and in writing) choose appropriate materials to solve problems reflect and monitor their own understanding of the math concepts practice strategies to build procedural and conceptual confidence

Teacher builds conceptual understanding by modeling through demonstration, explicit instruction, and think alouds, as well as guiding students as they practice math strategies and apply problem solving strategies. (Whole group or small group instruction)

Students practice math strategies independently to build procedural and computational fluency. Teacher assesses learning and reteaches as necessary. (whole group instruction, small group instruction, or centers)

Teacher and students practice mathematics processes together through interactive activities, problem solving, and discussion. (whole group or small group instruction)

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Effective Pedagogical Routines/Instructional Strategies Collaborative Problem Solving

Connect Previous Knowledge to New Learning

Making Thinking Visible

Develop and Demonstrate Mathematical Practices

Inquiry-Oriented and Exploratory Approach

Multiple Solution Paths and Strategies

Use of Multiple Representations

Explain the Rationale of your Math Work

Quick Writes

Pair/Trio Sharing

Turn and Talk

Charting

Gallery Walks

Small Group and Whole Class Discussions

Student Modeling

Analyze Student Work

Identify Student’s Mathematical Understanding

Identify Student’s Mathematical Misunderstandings

Interviews

Role Playing

Diagrams, Charts, Tables, and Graphs

Anticipate Likely and Possible Student Responses

Collect Different Student Approaches

Multiple Response Strategies

Asking Assessing and Advancing Questions

Revoicing

Marking

Recapping

Challenging

Pressing for Accuracy and Reasoning

Maintain the Cognitive Demand

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Educational Technology Standards

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

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

variety of digital tools and resources. Example: Students create personal digital portfolios for coursework using Google Sites, Evernote, WordPress, Edubugs, Weebly, etc.

Communication and Collaboration Develop an innovative solution to a real world problem or issue in collaboration with peers and experts, and present ideas for

feedback through social media or in an online community. Example: Use Google Classroom for real-time communication between teachers, students, and peers to complete assignments and discuss strategies for analyzing and comparing properties of two functions when each is represented in a different way (algebraically, graphically, and numerically in tables, or by verbal descriptions).

Critical Thinking, Problem Solving, and Decision Making Evaluate the strengths and limitations of emerging technologies and their impact on educational, career, personal or social needs. Example: Students use graphing calculators and graph paper to reveal the strengths and weaknesses of technology associated with graphing trigonometric functions.

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

games). Example: Students will create a set of instructions explaining how to use the Pythagorean Identity to find sin Ɵ, cos Ɵ, tan Ɵ, given sin Ɵ, cos Ɵ, or tan Ɵ, and the quadrant of the angle.

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

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Career Ready Practices Career Ready Practices describe the career-ready skills that all educators in all content areas should seek to develop in their students. They are practices that have been linked to increase college, career, and life success. Career Ready Practices should be taught and reinforced in all career exploration and preparation programs with increasingly higher levels of complexity and expectation as a student advances through a program of study.

CRP2. Apply appropriate academic and technical skills.

Career-ready individuals readily access and use the knowledge and skills acquired through experience and education to be more productive. They make connections between abstract concepts with real-world applications, and they make correct insights about when it is appropriate to apply the use of an academic skill in a workplace situation. Example: Students will apply prior knowledge when solving real world problems. Students will make sound judgements about the use of specific tools, such as graph paper, graphing calculators and technology to deepen understanding of finding the length of an arc in a unit circle.

CRP4. Communicate clearly and effectively and with reason.

Career-ready individuals communicate thoughts, ideas, and action plans with clarity, whether using written, verbal, and/or visual methods. They communicate in the workplace with clarity and purpose to make maximum use of their own and others’ time. They are excellent writers; they master conventions, word choice, and organization, and use effective tone and presentation skills to articulate ideas. They are skilled at interacting with others; they are active listeners and speak clearly and with purpose. Career-ready individuals think about the audience for their communication and prepare accordingly to ensure the desired outcome. Example: Students will communicate precisely using clear definitions and provide carefully formulated explanations when constructing arguments. Students will communicate and defend mathematical reasoning using objects, drawings, diagrams, and/or actions. Students will ask probing questions to clarify or improve arguments.

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Career Ready Practices CRP8. Utilize critical thinking to make sense of problems and persevere in solving them.

Career-ready individuals readily recognize problems in the workplace, understand the nature of the problem, and devise effective plans to solve the problem. They are aware of problems when they occur and take action quickly to address the problem; they thoughtfully investigate the root cause of the problem prior to introducing solutions. They carefully consider the options to solve the problem. Once a solution is agreed upon, they follow through to ensure the problem is solved, whether through their own actions or the actions of others. Example: Students will understand the meaning of a problem and look for entry points to its solution. They will analyze information, make conjectures, and plan a solution pathway to solve simple rational and radical equations.

CRP12. Work productively in teams while using cultural global competence. Career-ready individuals positively contribute to every team, whether formal or informal. They apply an awareness of cultural difference to avoid barriers to productive and positive interaction. They find ways to increase the engagement and contribution of all team members. They plan and facilitate effective team meetings. Example: Students will work collaboratively in groups to solve mathematical tasks. Students will listen to or read the arguments of others and ask probing questions to clarify or improve arguments. They will be able to explain how to graph trigonometric functions.

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WIDA Proficiency Levels At the given level of English language proficiency, English language learners will process, understand, produce or use

6‐ Reaching

Specialized or technical language reflective of the content areas at grade level A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse as

required by the specified grade level Oral or written communication in English comparable to proficient English peers

5‐ Bridging

Specialized or technical language of the content areas A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse,

including stories, essays or reports Oral or written language approaching comparability to that of proficient English peers when presented with

grade level material.

4‐ Expanding

Specific and some technical language of the content areas A variety of sentence lengths of varying linguistic complexity in oral discourse or multiple, related

sentences or paragraphs Oral or written language with minimal phonological, syntactic or semantic errors that may impede the

communication, but retain much of its meaning, when presented with oral or written connected discourse, with sensory, graphic or interactive support

3‐ Developing

General and some specific language of the content areas Expanded sentences in oral interaction or written paragraphs Oral or written language with phonological, syntactic or semantic errors that may impede the

communication, but retain much of its meaning, when presented with oral or written, narrative or expository descriptions with sensory, graphic or interactive support

2‐ Beginning

General language related to the content area Phrases or short sentences Oral or written language with phonological, syntactic, or semantic errors that often impede of the

communication when presented with one to multiple-step commands, directions, or a series of statements with sensory, graphic or interactive support

1‐ Entering

Pictorial or graphic representation of the language of the content areas Words, phrases or chunks of language when presented with one-step commands directions, WH-, choice or

yes/no questions, or statements with sensory, graphic or interactive support

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Culturally Relevant Pedagogy Examples

Integrate Relevant Word Problems: Contextualize equations using word problems that reference student interests and cultures. Example: When learning about trigonometric functions, problems that relate to student interests such as music, sports and art enable the students to understand and relate to the concept in a more meaningful way.

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

Run Problem Based Learning Scenarios: Encourage mathematical discourse among students by presenting problems that are relevant to them, the school and /or the community.

Example: Using a Place Based Education (PBE) model, students explore math concepts such as systems of equations while determining ways to address problems that are pertinent to their neighborhood, school or culture.

Encourage Student Leadership: Create an avenue for students to propose problem solving strategies and potential projects. Example: Students can learn to interpret functions in a context by creating problems together and deciding if the problems fit the necessary criteria. This experience will allow students to discuss and explore their current level of understanding by applying the concepts to relevant real-life experiences.

Present New Concepts Using Student Vocabulary: Use student diction to capture attention and build understanding before using academic terms. Example: Teach math vocabulary in various modalities for students to remember. Use multi-modal activities, analogies, realia, visual cues, graphic representations, gestures, pictures and cognates. Directly explain and model the idea of vocabulary words having multiple meanings. Students can create the Word Wall with their definitions and examples to foster ownership.

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SEL Competency

Examples Content Specific Activity & Approach to SEL

Self-Awareness

Self-Management Social-Awareness Relationship Skills Responsible Decision-Making

Example practices that address Self-Awareness:

• Clearly state classroom rules • Provide students with specific feedback regarding academics and behavior • Offer different ways to demonstrate understanding • Create opportunities for students to self-advocate • Check for student understanding / feelings about performance • Check for emotional wellbeing • Facilitate understanding of student strengths and challenges

Have students keep a math journal. This can create a record of their thoughts, common mistakes that they repeatedly make when solving problems, and/or record how they would approach or solve a problem.

Lead frequent discussion in class to give students an opportunity to reflect after learning new concepts. Discussion questions may include asking students: “What difficulties do you have when asked to analyze and compare properties of two functions when each is represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions).

Self-Awareness Self-Management Social-Awareness Relationship Skills Responsible Decision-Making

Example practices that address Self-Management:

• Encourage students to take pride/ownership in work and behavior • Encourage students to reflect and adapt to classroom situations • Assist students with being ready in the classroom • Assist students with managing their own emotional states

Teach students to set attainable learning goals and self-assess their progress towards those learning goals. For example, a powerful strategy that promotes academic growth is teaching instructional routines to self-assess within the Independent Phase of the Balanced Mathematics Instructional Model.

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Teach students a lesson on the proper use of equipment (such as the computers, graphing calculators and textbooks) and other resources properly. Routinely ask students who they think might be able to help them in various situations, including if they need help with a math problem or using an equipment.

Self-Awareness Self-Management

Social-Awareness Relationship Skills Responsible Decision-Making

Example practices that address Social-Awareness:

• Encourage students to reflect on the perspective of others • Assign appropriate groups • Help students to think about social strengths • Provide specific feedback on social skills • Model positive social awareness through metacognition activities

When there is a difference of opinion among students (perhaps over solution strategies), allow them to reflect on how they are feeling and then share with a partner or in a small group. It is important to be heard but also to listen to how others feel differently in the same situation.

Have students re-conceptualize application problems after class discussion, by working beyond their initial reasoning to identify common reasoning between different approaches to solve the same problem.

Self-Awareness Self-Management Social-Awareness

Relationship Skills Responsible Decision-Making

Example practices that address Relationship Skills:

• Engage families and community members • Model effective questioning and responding to students • Plan for project-based learning • Assist students with discovering individual strengths • Model and promote respecting differences • Model and promote active listening • Help students develop communication skills • Demonstrate value for a diversity of opinions

Have students team-up at the at the end of the unit to teach a concept to the class. At the end of the activity students can fill out a self-evaluation rubric to evaluate how well they worked together. For example, the rubric can consist of questions such as, “Did we consistently and actively work towards our group goals?”, “Did we willingly accept and fulfill individual roles within the group?”, “Did we show sensitivity to the feeling and learning needs of others; value their knowledge, opinion, and skills of all group members?”

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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 Self-Management Social-Awareness Relationship Skills

Responsible Decision-Making

Example practices that address Responsible Decision-Making:

• Support collaborative decision making for academics and behavior • Foster student-centered discipline • Assist students in step-by-step conflict resolution process • Foster student independence • Model fair and appropriate decision making • Teach good citizenship

Have students participate in activities that requires them to make a decision about a situation and then analyze why they made that decision. Students encounter conflicts within themselves as well as among members of their groups. They must compromise in order to reach a group consensus.

Today’s students live in a digital world that comes with many benefits — and also increased risks. Students need to learn how to be responsible digital citizens to protect themselves and ensure they are not harming others. Educators can teach digital citizenship through social and emotional learning.

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Differentiated Instruction Accommodate Based on Students Individual Needs: Strategies

Time/General

Extra time for assigned tasks

Adjust length of assignment

Timeline with due dates for reports and projects

Communication system between home and school

Provide lecture notes/outline

Processing

Extra Response time

Have students verbalize steps

Repeat, clarify or reword directions

Mini-breaks between tasks

Provide a warning for transitions

Partnering

Comprehension

Precise processes for balanced math instructional model

Short manageable tasks

Brief and concrete directions

Provide immediate feedback

Small group instruction

Emphasize multi-sensory learning

Recall

Teacher-made checklist

Use visual graphic organizers

Reference resources to promote independence

Visual and verbal reminders

Graphic organizers

Assistive Technology

Computer/whiteboard

Tape recorder

Video Tape

Tests/Quizzes/Grading

Extended time

Study guides

Shortened tests

Read directions aloud

Behavior/Attention

Consistent daily structured routine

Simple and clear classroom rules

Frequent feedback

Organization

Individual daily planner

Display a written agenda

Note-taking assistance

Color code materials

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Differentiated Instruction Accommodate Based on Content Specific Needs

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

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

Pre-teach pertinent vocabulary

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

Word wall with visual representations of mathematical terms

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

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

Record formulas, processes, and mathematical rules in reference notebooks

Graphing calculator to assist with computations and graphing of trigonometric functions

Utilize technology through interactive sites to represent nonlinear data

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

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Interdisciplinary Connections Model interdisciplinary thinking to expose students to other disciplines.

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

This task could be used for assessment or for practice. It allows the students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically.

Name of Task: Bicycle Wheel Science Standard HS-PS2-2 The purpose of this task is to introduce radian measure for angles in a situation where it arises naturally. Radian measure focuses on the arc

length of a circle cut out by a given angle. Degree measure, on the other hand, focuses on the angle. If the radius of the circle were one unit, then the radian measure table would be particularly simple to fill out. Even here where the radius is not one unit, the radian angle measure is more ''natural'' for this scenario because it measures the length of a circular arc and the distance a wheel is traveling is also the length of a circular arc.

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

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

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Enrichment What is the Purpose of Enrichment?

The purpose of enrichment is to provide extended learning opportunities and challenges to students who have already mastered, or can quickly master, the basic curriculum. Enrichment gives the student more time to study concepts with greater depth, breadth, and complexity.

Enrichment also provides opportunities for students to pursue learning in their own areas of interest and strengths. Enrichment keeps advanced students engaged and supports their accelerated academic needs. Enrichment provides the most appropriate answer to the question, “What do you do when the student already knows it?”

Enrichment is… Planned and purposeful

Different, or differentiated, work – not just more work

Responsive to students’ needs and situations

A promotion of high-level thinking skills and making connections within content

The ability to apply different or multiple strategies to the content

The ability to synthesize concepts and make real world and cross-curricular connections

Elevated contextual complexity

Sometimes independent activities, sometimes direct instruction

Inquiry based or open-ended assignments and projects

Using supplementary materials in addition to the normal range of resources

Choices for students

Tiered/Multi-level activities with flexible groups (may change daily or weekly)

Enrichment is not… Just for gifted students (some gifted students may need

intervention in some areas just as some other students may need frequent enrichment)

Worksheets that are more of the same (busywork)

Random assignments, games, or puzzles not connected to the content areas or areas of student interest

Extra homework

A package that is the same for everyone

Thinking skills taught in isolation

Unstructured free time

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Assessments Required District/State Assessments

Unit Assessment NJSLA

SGO Assessments

Suggested Formative/Summative Classroom Assessments Describe Learning Vertically Identify Key Building Blocks

Make Connections (between and among key building blocks) Short/Extended Constructed Response Items

Multiple-Choice Items (where multiple answer choices may be correct) Drag and Drop Items

Use of Equation Editor Quizzes

Journal Entries/Reflections/Quick-Writes Accountable talk

Projects Portfolio

Observation Graphic Organizers/ Concept Mapping

Presentations Role Playing

Teacher-Student and Student-Student Conferencing Homework

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New Jersey Student Learning Standards 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-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 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. A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples howing how extraneous solutions may arise.

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New Jersey Student Learning Standards A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. F-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and Amplitude. F-IF.C.8b Use the properties of exponents to interpret expressions for exponential functions. F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. 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-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. 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.

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New Jersey Student Learning Standards F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. 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). F-LE.A.4 Understand the definition of a logarithm as the solution to an exponential equation. F-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.

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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: 3 (Three)

Topics: Exponential and Logarithmic Functions Rational Functions, Sequences and Series

NJSLS: A-APR.D.6,A-APR.D.7,A-CED.A.1,A-CED.A.2,A-CED.A.3,A-CED.A.4,A-REI.A.1,A-REI.A.2,A-SSE.A.2,A-SSE.B.3c,A-SSE.B.4, F-IF.C.7e,F-IF.C.8b,F-BF.A.1a,F-BF.A.2, F-BF.B.4a,F-BF.B.3,F-IF.A.3,F-LE.A.2,F-LE.A.4,F-LE.B.5 Unit Focus:

Understand exponential and logarithmic functions. Determine whether a function represents exponential growth or decay. Simplify exponential and logarithmic expressions. Solve exponential and logarithmic equations. Model exponential and logarithmic functions. Understand rational functions. Determine whether an equation represents a direct variation or an inverse variation. Describe how to graph rational functions. Add, subtract, multiply, and divide rational expressions. Solve rational equations. Understand sequences and series. Define and use sequences and series. Describe how to find sums of infinite geometric series. Analyze arithmetic and geometric sequences and series. Explain how to write recursive rules for sequences.

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New Jersey Student Learning Standard(s): A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. F-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and Amplitude. F-IF.C.8b Use the properties of exponents to interpret expressions for exponential functions. 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). F-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.

Student Learning Objective 1: Exponential Growth and Decay Functions

Modified Student Learning Objectives/Standards: MPs Evidence Statement Key/

Clarifications Skills, Strategies & Concepts Essential Understandings/

Questions (Accountable Talk)

Tasks/Activities

MP2 MP3 MP8

A.SSE.B.3c Evidence Statement • As stated in the standard. Clarification • Items must have real‐world context.

Pose the following problem to students: “If $10,000 is invested when a baby is born, do you think there will be sufficient money to pay for a college education 18 years later? Explain to them , “If b > 1, then is exponential growth. Explain why. If

What are some of the characteristics of the graph of an exponential function?

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• The equivalent form must reveal something about the real‐world context. • The domain of exponential functions limited to integers. F-LE.A.2, F-LE.B.5 Evidence Statement • Solve multi‐step contextual problems with degree of difficulty appropriate to the course by constructing linear, or exponential function models, where exponentials are limited to integer exponents. Clarification • Items must have real‐world context. • Explain the meaning of the slope and y intercept in terms of real‐world context given a linear model. • Explain the meaning of the base, the exponent and the coefficient in terms of real‐world context, given an exponential model with a domain in the integers.

0 < b < 1, then is exponential decay. Explain why. Explain compound interest. If possible, show printed ads stating compound interest rates. 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 growth and decay 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

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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.

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New Jersey Student Learning Standard(s): F-IF.C.7e

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

F-LE.B.5

Interpret the parameters in a linear or exponential function in terms of a context.

Student Learning Objective 2: The Natural Base e Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/ Clarifications

Skills, Strategies & Concepts Essential Understandings/ Questions

(Accountable Talk)

Tasks/Activities

MP4 MP7

F.LE.B.5 Evidence Statement • As stated in the standard. Clarification • Items must have real-world context. • Explain the meaning of the slope and y intercept in terms of real-world context given a linear model. • Explain the meaning of the base, the exponent and the coefficient in

Define and use the natural base e. Graph natural base functions. Solve real-life problems. Write the natural base exponential function and explore the affect r has on the graph of

Explain continuously compounded interest to the students.

What is the natural base e? Can you calculate the value of e ?

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terms of real-world context, given an exponential model with a domain in the integers.

“What is the y-intercept for the graph of y = 3ex?” 3 “What is the value of the function when x = 1?

SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated Logarithmic functions. Provide students with opportunities to practice the thinking and processes involved in graphing Logarithmic 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 natural base e quadratic 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 Logs and natural logs. Identify the end behavior x and y intercept in the figure.

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Use technology to create table of values to verify the positive zeros of the Logarithmic functions.

New Jersey Student Learning Standard(s): F-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and Amplitude. 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. F-LE.A.4 Understand the definition of a logarithm as the solution to an exponential equation. Student Learning Objective 3: Logarithms and Logarithmic Functions Modified Student Learning Objectives/Standards:



(Accountable Talk)

Tasks/Activities

MP3

N/A

Define and evaluate logarithms. Use inverse properties of logarithmic and exponential functions. Graph logarithmic functions. Define common logarithms and natural logarithms. Write a few that can be evaluated without a calculator, such as log 1000, log 0.01,

Explain continuously compounded interest

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and ln 1. Explain the properties of logs and natural logs. Show the parent graphs for logarithmic functions describe the general shape for b > 1 and 0 < b < 1. Explain that calculators have two built-in bases that can be graphed: b = 10 and b = e. Do not get into the change-of-base formula. SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated Logarithmic functions. Provide students with opportunities to practice the thinking and processes involved in graphing Logarithmic 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 natural base e quadratic 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.

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New Jersey Student Learning Standard(s): F-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and Amplitude. 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 4: Transformations of Exponential and Logarithmic Functions

Modified Student Learning Objectives/Standards: N/A MPs Evidence Statement

Key/ Clarifications Skills, Strategies & Concepts Essential Understandings/

Questions (Accountable Talk)

Tasks/Activities

F-BF.B.3 Evidence Statement • As stated in the standard. Clarification • Limit to linear and quadratic functions.

Transform graphs of exponential functions. Transform graphs of logarithmic functions. Write transformations of graphs of exponential and logarithmic functions. Discuss the Core Concept with students. Again, the issue for students is to be able to distinguish

How can you transform the graphs of exponential and logarithmic functions?

1

Use technology to graph Logs and natural logs. Identify the end behavior x and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the Logarithmic functions.

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• Even and odd functions are not assessed In Algebra I. • The experiment part of the standard is instructional only. This aspect of the standard is not assessed.

whether the transformation affects the graph horizontally or vertically. Students also need to be aware of what the parent logarithmic function looks like, as well as its domain and its range.

SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated Logarithmic functions. Provide students with opportunities to practice the thinking and processes involved in graphing Logarithmic 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.


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New Jersey Student Learning Standard(s): A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. F-LE.A.4 Understand the definition of a logarithm as the solution to an exponential equation. Student Learning Objective 5: Properties of Logarithms Modified Student Learning Objectives/Standards: xxxx


Skills, Strategies & Concepts Essential Understandings/ Questions (Accountable Talk)

Tasks/Activities

MP2 MP3

A-SSE.A.2 Evidence Statement

Use the properties of logarithms to evaluate logarithms. Use the properties of logarithms to expand or condense logarithmic expressions.

How can you use properties of exponents to derive properties of logarithms?

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• Rewrite linear, quadratic and exponential expressions. Clarification • Unlike standard A.SSE.1, items that address A.SSE.2 do not need to have real‐world context. • In Algebra I, expressions are limited to one variable expression. • Items could ask a student to identify expressions equivalent to a given expression. • Items could ask a student to rewrite an expression in one variable.

Use the change-of-base formula to evaluate logarithms. Have students estimate an answer for log6 24 before they change the base. SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated Logarithmic functions. Provide students with opportunities to practice the thinking and processes involved in graphing Logarithmic 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.


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New Jersey Student Learning Standard(s): A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. F-LE.A.4 Understand the definition of a logarithm as the solution to an exponential equation. Student Learning Objective 6: Solving Exponential and Logarithmic Equations. Modified Student Learning Objectives/Standards: N/A



(Accountable Talk)

Tasks/Activities

MP1 MP5 MP6

A-REI.A.1 Evidence Statement • Identify or justify a solution method.

Solve exponential, logarithmic equations. Solve exponential and logarithmic inequalities. Explain what it means to exponentiate each side of an equation and why a base of 2 was chosen.

How can you solve exponential and logarithmic equations?

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Clarification • Items that require solving linear equations will require the student to explain the properties used in the solution process. • Items may include solving quadratic equations.

Write the Core Concept and ask students, why b has to be positive and not equal to 1. SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated Logarithmic functions. Provide students with opportunities to practice the thinking and processes involved in graphing Logarithmic 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 natural base e 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.


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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. 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). Student Learning Objective 7: Modeling with Exponential and Logarithmic Functions. Modified Student Learning Objectives/Standards: xxxx



(Accountable Talk)

Tasks/Activities

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MP5

A.CED.A.2 Evidence Statement • As stated in the standard. Clarification • Items must have real-world context • Limit equations to two variables.

F.BF.A.1a, F-LE.A.2 Evidence Statement • Write a function based on an observed pattern in a real-world scenario. Clarification • Items must have real-world context. • Limit to linear, quadratic and exponential functions with domains in the integers. • Similar to creating a function from a scatterplot but for this standard the relationship between

Classify data sets. Write exponential functions. Use technology to find exponential and logarithmic models. The first step with any data set is to make a scatter plot to determine what the graph of the data looks like. Is there some theory or past experience that we have with the data that tells us what type of function would make the most sense? SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated exponential and logarithmic Functions Provide students with opportunities to practice the thinking and processes involved in graphing exponential and logarithmic 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 natural base e in student’s native language and/or simplified

How can you recognize polynomial, exponential, and logarithmic models?

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the two quantities is clear from the context.

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 Logs and natural logs. Identify the end behavior x and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the exponential and logarithmic Functions.

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 and quadratic functions, and simple rational and exponential functions. 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. A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Student Learning Objective 8: Inverse Variation Modified Student Learning Objectives/Standards: xxxx



Tasks/Activities

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(Accountable Talk)

MP4 MP6

A-CED.A.1, A-CED.A.2, A-CED.A.3 Evidence Statement Provide constraints based on real-world context for equations, inequalities, systems of equations and systems of inequalities. • Determine if a solution is viable based on real-world context. Clarification • Items must have real-world context. • Limit equations to those arising from linear, quadratic functions and exponential functions with domain limited to integers. • Limit inequalities to those arising from linear functions. • Rational functions are addressed in Algebra II. Items must have real-world context • Limit equations to two variables.

Inverse variation is a nice bridge to graphs of rational functions, which is the next lesson. Remind students that before they can add or subtract fractions, they must first find a common denominator. Students can confuse inverse variation with exponential decay. Contexts for inverse variation are generally restricted to the first quadrant and do not have a y-intercept. It is the decreasing function and curve of the graph that mislead students. SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated functions Provide students with opportunities to practice the thinking and processes involved in graphing 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 inverse variation quadratic questions in student’s native language and/or simplified questions

How can you recognize when two quantities vary directly or inversely? how can you distinguish between direct and inverse variation?

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• Systems are limited to systems of equations with two equations and two unknowns.

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 functions. Identify the end behavior x and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the exponential and logarithmic Functions.

New Jersey Student Learning Standard(s): A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. F-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Student Learning Objective 9: Graphing Rational Functions Modified Student Learning Objectives/Standards: N/A



(Accountable Talk)

Tasks/Activities

MP1 MP2 MP7

F-BF.B.3

Evidence Statement

What are some of the characteristics of the graph of a rational function?

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• As stated in the standard Clarification • Limit to linear and quadratic functions. • Even and odd functions are not assessed In Algebra I. • The experiment part of the standard is instructional only. This aspect of the standard is not assessed.

Graph simple rational functions. Translate simple rational functions. Graph other rational functions.

Students should know, what information is known, and how should they calculate an average cost? Students should learn how to find vertical and horizontal asymptote. Give partners time to discuss this matter. Do not rush in. SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated functions Provide students with opportunities to practice the thinking and processes involved in graphing rational 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 rational functions in student’s native language and/or

What value(s) make the denominator equal to 0?”

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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 functions. Identify the end behavior x and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the rational functions exponential.

New Jersey Student Learning Standard(s): A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Student Learning Objective #10: Multiplying and Dividing Rational Expressions Modified Student Learning Objectives/Standards: N/A



(Accountable Talk)

Tasks/Activities

Simplify rational expressions.

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MP6

N/A

Multiply and divide rational expressions. In this lesson, students will add and subtract rational expressions, some with like denominators and others without like denominators. Because the rational expressions themselves could be a single term or a polynomial, it is necessary to work slowly through the various cases. Remind students that this is not the Cross Products Property! They are multiplying rational expressions. Students can confuse multiplying rational expressions that are monomials and those that are binomials. Keep reminding students that when they simplify first before performing the multiplication, they must be dividing out common factors. SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated functions Provide students with opportunities to practice the thinking and processes involved in graphing rational functions by hand and using technology by working small groups. Develop a reference sheet for student use that includes formulas, processes and procedures

How can you determine the excluded values in a product or quotient of two rational expressions? Is it possible for the product or quotient of two rational expressions to have no excluded values? Explain your reasoning. If it is possible, give an example.

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and sample problems to encourage proficiency and independence. ELL Strategies: Demonstrate comprehension of rational 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.


New Jersey Student Learning Standard(s): A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Student Learning Objective #11: Adding and Subtracting Rational Expressions

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(Accountable Talk)

Tasks/Activities

MP2

N/A

Add or subtract rational expressions. Rewrite rational expressions and graph the related function. Simplify complex fractions. Go over LCM one more time.

SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated functions Provide students with opportunities to practice the thinking and processes involved in graphing rational 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 rational 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

How can you determine the domain of the sum or difference of two rational expressions?

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and using technology for more complicated cases.

Use technology to graph rational functions. Identify the end behavior x and y intercept in the figure. Use technology to create table of values to verify the positive zeros of the rational functions exponential.

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. A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples howing how extraneous solutions may arise.

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Student Learning Objective #12: Solving Rational Equations Modified Student Learning Objectives/Standards: xxxxxx



(Accountable Talk)

Tasks/Activities

MP1

A-CED.A.4

Evidence Statement • As stated in the standard. Clarification • Items must have real-world context. • Limit quadratic formulas to those that do not contain a linear term.

Solve rational equations by cross multiplying. Solve rational equations by using the least common denominator. Use inverses of functions. Have students graph the function and its inverse to see symmetry about y = x.

SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to graph complicated functions Provide students with opportunities to practice the thinking and processes involved in graphing rational 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:

How can you solve a rational equation? Why is it important to check solutions?

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Demonstrate comprehension of rational 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.


New Jersey Student Learning Standard(s): F-IF.A.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Student Learning Objective #13: Defining and Using Sequences and Series Modified Student Learning Objectives/Standards: N/A



(Accountable Talk)

Tasks/Activities

F.IF.A.3

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MP4 MP6

Evidence Statement • Identify a given sequence as arithmetic or geometric • Match a given sequence to a given algebraic representation for the sequence. • Given a recursive or explicit rule, evaluate the expression to find the value for a specified term in the sequence • Create an explicit function rule for a sequence. Clarification • Limit sequences to simple arithmetic or geometric sequences.

Use sequence notation to write terms of sequences. Write a rule for the nth term of a sequence. Sum the terms of a sequence to obtain a series and use summation notation. A sequence is an ordered list of numbers. A finite sequence is a function that has a limited number of terms and whose domain is the finite set {1, 2, 3, . . . , n}. The values in the range are called the terms of the sequence. When the terms of a sequence are added together, the resulting expression is a series. A series can be finite or infinite. If time permits, derive the three formulas. This may indeed help students remember the formulas SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to work with sequence. Provide students with opportunities to practice the thinking and processes involved in graphing rational 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:

How can you write a rule for the nth term of a sequence? what they should do when they do not recognize a pattern and therefore cannot figure out the rule?

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Demonstrate comprehension understanding of sequence 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 sequence by hand in simple cases and using technology for more complicated cases.

Use technology to graph functions. Identify the end sequence and its sum.

New Jersey Student Learning Standard(s):

F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. F-BF.A.2 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

F-LE.A.2 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

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Student Learning Objective #14: Analyzing Arithmetic Sequences and Series

Modified Student Learning Objectives/Standards:xxxx


Skills, Strategies & Concepts Essential Understandings/ Questions (Accountable Talk)

Tasks/Activities

MP1

MP4

F.IF.A.3

Evidence Statement

• Identify a given sequence as arithmetic or geometric.

• Match a given sequence to a given algebraic representation for the sequence.

• Given a recursive or explicit rule, evaluate the expression to find the value for a specified term in the sequence.

• Create an explicit function rule for a sequence.

Clarification

Identify arithmetic sequences. Write rules for arithmetic sequences. Find sums of finite arithmetic series.

In an arithmetic sequence, the difference of consecutive terms is constant. This constant difference is called the common difference and is denoted by d.

Students need to be careful as they find the common difference. In finding the common difference, subtract each term from the term that follows it, meaning .

SPED Strategies: Review the differences between geometric and arithmetic sequences by giving students examples and illustrating the characteristics that distinguish them. Pre-teach the vocabulary and provide verbal and pictorial descriptions. (i.e. recursive and explicit formulas).

How can you recognize an arithmetic sequence from its graph? “How will knowing the number of common differences between a7 and a26 help?

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• Limit sequences to simple arithmetic or geometric sequences.

F.LE.A.2

Evidence Statement

• Solve multi‐step contextual problems with degree of difficulty appropriate to the course by constructing linear or exponential function models, where exponentials are limited to integer exponents.

Clarification

• Items must have real‐world context.

• Items do not reveal the type of function that should be constructed.

Model the thinking and procedure involved in writing geometric and arithmetic sequences in recursive and explicit form. ELL Strategies: After listening to an oral explanation in the student’s native language, demonstrate comprehension of arithmetic and geometric sequences both recursively and with an explicit formula and/or an explanation which uses drawings and selected technical words. Interpret orally and in writing the parameters in a linear or exponential function in terms of a context the parameters in a linear or exponential function in terms of a context in the student’s native language and/or use gestures, pictures and selected, technical words. Explain the classic tale “The Story of Gauss” that helps students to make sense of arithmetic sequences.

Model an example and do “Think Aloud” in pair reasoning: How do I get from a1 to a19 ?

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New Jersey Student Learning Standard(s):

A-SSE.B.4

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

F-IF.A.3

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

F-BF.A.2

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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

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Student Learning Objective #15: Analyzing Geometric Sequences and Series

Modified Student Learning Objectives/Standards:xxxx


Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP1

F.IF.A.3

Evidence Statement

• Identify a given sequence as arithmetic or geometric.

• Match a given sequence to a given algebraic representation for the sequence.

• Given a recursive or explicit rule, evaluate the

Identify geometric sequences. Write rules for geometric sequences. Find sums of finite geometric series.

Students need to be careful as they find the common ratio.

Discuss why the nth term, , makes sense. The 12th term is found by multiplying the 1st term by the common ratio 11 times.

SPED Strategies:

How can you recognize a geometric sequence from its graph?

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expression to find the value for a specified term in the sequence.

• Create an explicit function rule for a sequence.

Clarification

• Limit sequences to simple arithmetic or geometric sequences.

F.LE.A.2

Evidence Statement

• Solve multi-step contextual problems with degree of difficulty appropriate to the course by constructing linear or exponential function models, where exponentials are limited to integer exponents.

Clarification

• Items must have real-world context.

Pre-teach vocabulary and provide verbal and pictorial descriptions to maximize understanding and interest. Introduce the concept imbedded in a real-life context to help students relate to and internalize the mathematics involved. Model the thinking and processes involved in solving problems involving finite geometric series. ELL Strategies: Read and write the formula for geometric series and use the formula to solve the problems in the student’s native language and/or use gestures, examples and selected, technical words. Build on past knowledge when is needed. Encourage students to organize what they know about arithmetic and geometric sequences in a Chart, including formulas.

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• Items do not reveal the type of function that should be constructed

New Jersey Student Learning Standard(s): A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Student Learning Objective #16: Finding Sums of Infinite Geometric Series



Tasks/Activities

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(Accountable Talk)

MP2 MP3 MP5

N/A

Find partial sums of infinite geometric series. Find sums of infinite geometric series. The sum of an infinite geometric series with first term a1 and common ratio r is given by

, provided ∣r ∣ < 1. If ∣r ∣ ≥ 1, then the series has no sum. Write the Core Concept. Give students time to consider that although there are infinitely many terms, the series has a finite sum when ∣r ∣ < 1. It is much easier to think about the case of ∣r ∣ > 1 not having a finite sum

SPED Strategies: Pre-teach vocabulary and provide verbal and pictorial descriptions to maximize understanding and interest. Introduce the concept imbedded in a real-life context to help students relate to and internalize the mathematics involved. Model the thinking and processes involved in solving problems involving finite geometric series. Provide students with a graphic organizer/reference sheet/Google Doc that

How can you find the sum of an infinite geometric series?

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highlights the thinking and procedure involved solving problems involving finite geometric series.

ELL Strategies: Read and write the formula for the sum of a finite geometric series and use the formula to solve the problems in the student’s native language and/or use gestures, examples and selected, technical words. Build on past knowledge by reminding students that a ratio is a comparison of the two quantities and can be written as a fraction.

Encourage students to organize what they know about arithmetic and geometric sequences in a Chart, including formulas.

New Jersey Student Learning Standard(s): F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. F-BF.A.2

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Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Student Learning Objective #17: Using Recursive Rules with Sequences Modified Student Learning Objectives/Standards: N/A



(Accountable Talk)

Tasks/Activities

MP1 MP5

F-IF.A.3 Evidence Statement • Identify a given sequence as arithmetic or geometric • Match a given sequence to a given algebraic representation for the sequence. • Given a recursive or explicit rule, evaluate the expression to find the value for a specified term in the sequence • Create an explicit function rule for a sequence. Clarification • Limit sequences to simple arithmetic or geometric sequences.

F.BF.A.1a Evidence Statement

Evaluate recursive rules for sequences. Write recursive rules for sequences. Translate between recursive and explicit rules for sequences. Use recursive rules to solve real-life problems. A recursive rule gives the beginning term(s) of a sequence and a recursive equation that tells how is related to one or more preceding terms.

SPED Strategies: Model the thinking behind determining when and how to use the graphing calculator to work with sequence. Provide students with opportunities to practice the thinking and processes involved in graphing rational functions by hand and using technology by working small groups.

How can you define a sequence recursively?

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• Write a function based on an observed pattern in a real‐world scenario. Clarification • Items must have real‐world context. • Limit to linear, quadratic and exponential functions with domains in the integers. • Similar to creating a function from a scatterplot but for this standard the relationship between the two quantities is clear from the context.

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 understanding of sequence 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 sequence by hand in simple cases and using technology for more complicated cases.

Use technology to graph functions. Identify the end of sequence and its sum.

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.

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F-BF.Int.2: Find inverse functions to solve contextual problems. Solve an equation of the form for a simple function f that has an inverse and write an expression for the inverse. For example, or for .

For example, see http://illustrativemathematics.org/illustrations/234. As another example, given a function C(L) = 750 2 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.7.1: Base explanations/reasoning on the relationship between zeros and factors of polynomials. Content Scope: A-APR.B

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

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HS.C.9.2: Express reasoning about transformations of functions. Content scope: F-BF.3, which may involve polynomial, exponential, logarithmic or trigonometric functions. Tasks also may involve even and odd functions.

HS.C.11.1: Express reasoning about trigonometric functions and the unit circle. Content scope: F-TF.2, F-TF.8 For example, students might explain why the angles and have the same cosine value; or use the unit circle to prove that sin2( ) +

cos2( ) = 1; or compute the tangent of the angle in the first quadrant having sine equal to . HS.C.18.4: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures about polynomials, rational expressions, or rational exponents. Content scope: N-RN, A-APR.(2, 3, 4, 6)

HS.C.CCR: Solve multi-step mathematical problems requiring extended chains of reasoning and drawing on a synthesis of the knowledge and skills articulated across: 7-RP.A.3, 7-NS.A.3, 7-EE.B.3, 8-EE.C.7B, 8-EE.C.8c, N-RN.A.2, A-SSE.A.1b, A-REI.A.1, A-REI.B.3, A-REI.B.4b, F-IF.A.2, F-IF.C.7a, F-IF.C.7e, G-SRT.B.5 and G-SRT.C.7.

Tasks will draw on securely held content from previous grades and courses, including down to Grade 7, but that are at the Algebra II/Mathematics III level of rigor.

Tasks will synthesize multiple aspects of the content listed in the evidence statement text, but need not be comprehensive. Tasks should address at least A-SSE.A.1b, A-REI.A.1, and F-IF.A.2 and either F-IF.C.7a or F-IF.C.7e (excluding trigonometric and

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

HS.D.2-4: Solve multi-step contextual problems with degree of difficulty appropriate to the course that require writing an expression for an inverse function, as articulated in F.BF.4a.

Refer to F-BF.41 for some of the content knowledge relevant to these tasks. HS.D.2-7: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in A-CED, N-Q.2, A-SSE.3, A-REI.6, A-REI.7, A-REI.12, A-REI.11-2.

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

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HS.D.2-10: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in F-BF.A, F-BF.3, F-IF.3, A-CED.1, A-SSE.3, F-IF.B, F-IF.7.

F-BF.A is the primary content; other listed content elements may be involved in tasks as well. HS.D.2-13: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in S-ID and S-IC.

If the content is only S-ID, the task must include Algebra 2 / Math 3 content (S-ID.4 or S-ID.6) Longer tasks may require some or all of the steps of the modeling cycle (CCSSM, pp. 72, 73); for example, see ITN Appendix F,

"Karnataka" task (Section A "Illustrations of innovative task characteristics," subsection 7 "Modeling/Application," subsection f "Full Models"). As in the Karnataka example, algebra and function skills may be used.

Predictions should not extrapolate far beyond the set of data provided. Line of best fit is always based on the equation of the least squares regression line either provided or calculated through the use of

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

To investigate associations, students may be asked to evaluate scatterplots that may be provided or created using technology. Evaluation includes shape, direction, strength, presence of outliers, and gaps.

Analysis of residuals may include the identification of a pattern in a residual plot as an indication of a poor fit. Models may assess key features of the graph of the fitted model. Tasks that involve S-IC.2 might ask the students to look at the results of a simulation and decide how plausible the observed value is with

respect to the simulation. For an example, see question 7 on the calculator section of the online practice test (http://practice.parcc.testnav.com/#).

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

Tasks may involve finding a value at a given percentile based on a normal distribution. HS.D.3-5: Decisions from data: Identify relevant data in a data source, analyze it, and draw reasonable conclusions from it. Content scope: Knowledge and skills articulated in Algebra 2.

Tasks may result in an evaluation or recommendation. The purpose of tasks is not to provide a setting for the student to demonstrate breadth in data analysis skills (such as box-and-whisker

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

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HS.D.3-6: Full models: Identify variables in a situation, select those that represent essential features, formulate a mathematical representation of the situation using those variables, analyze the representation and perform operations to obtain a result, interpret the result in terms of the original situation, validate the result by comparing it to the situation, and either improve the model or briefly report the conclusions. Content scope: Knowledge and skills articulated in the Standards in grades 6-8, Algebra 1 and Math 1 (excluding statistics)

Task prompts describe a scenario using everyday language. Mathematical language such as "function," "equation," etc. is not used. Tasks require the student to make simplifying assumptions autonomously in order to formulate a mathematical model. For example, the

student might autonomously make a simplifying assumption that every tree in a forest has the same trunk diameter, or that water temperature is a linear function of ocean depth.

Tasks may require the student to create a quantity of interest in the situation being described (N-Q.2). For example, in a situation involving population and land area, the student might decide autonomously that population density is a key variable, and then choose to work with persons per square mile. In a situation involving data, the student might autonomously decide that a measure of center is a key variable in a situation, and then choose to work with the mean.

Tasks may involve choosing a level of accuracy appropriate to limitations of measurement or limitations of data when reporting quantities (N-Q.3, first introduced in AI/M1).

HS.D.CCR: Solve problems using modeling: Identify variables in a situation, select those that represent essential features, formulate a mathematical representation of the situation using those variables, analyze the representation and perform operations to obtain a result, interpret the result in terms of the original situation, validate the result by comparing it to the situation, and either improve the model or briefly report the conclusions. Content scope: Knowledge and skills articulated in the Standards as described in previous courses and grades, with a particular emphasis on 7- RP, 8 – EE, 8 – F, N-Q, A-CED, A-REI, F-BF, G-MG, Modeling, and S-ID

Tasks will draw on securely held content from previous grades and courses, include down to Grade 7, but that are at the Algebra II/Mathematics III level of rigor.

Task prompts describe a scenario using everyday language. Mathematical language such as "function," "equation," etc. is not used. Tasks require the student to make simplifying assumptions autonomously in order to formulate a mathematical model. For example, the

student might make a simplifying assumption autonomously that every tree in a forest has the same trunk diameter, or that water temperature is a linear function of ocean depth.

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

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Algebra II Vocabulary

Number and Quantity

Algebra Functions Statistics and Probability

Complex number Conjugate Determinant

Binomial Theorem Complete the square

Absolute value function Asymptote

Logarithmic function

2-way frequency table Addition Rule

Independent Inter-quartile range

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References & Suggested Instructional Websites Internet4Classrooms www.internet4classrooms.com

Desmos https://www.desmos.com/

Math Open Reference www.mathopenref.com

Fundamental theorem of Algebra Identity matrix Imaginary number Initial point Moduli Parallelogram rule Polar form Quadratic equation Polynomial Rational exponent Real number Rectangular form Scalar multiplication of Matrices Terminal point Vector Velocity Zero matrix

Exponential function Geometric series Logarithmic Function Maximum Minimum Pascal’s Triangle Remainder Theorem

Amplitude Arc Arithmetic sequenceConstant function Cosine Decreasing intervals Domain End behavior Exponential decay Exponential function Exponential growth Fibonacci sequence Function notation Geometric sequence Increasing intervals Intercepts Invertible function

Trigonometric function Midline Negative intervals Period Periodicity Positive intervals Radian measure Range Rate of change Recursive process Relative maximum Relative minimum Sine Step function Symmetries Tangent

Arithmetic sequence Box plot Causation Combinations Complements Conditional probability Conditional relative frequency Correlation Correlation coefficient Dot plot Experiment Fibonacci sequence Frequency table Geometric sequence Histogram

Joint relative frequency Margin of error Marginal relative frequency Multiplication Rule Observational studies Outlier Permutations Recursive process Relative frequency Residuals Sample survey Simulation models Standard deviation Subsets Theoretical probability Unions

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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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