mathematics - paterson public schools...4 use polynomial identities to describe numerical...

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MATHEMATICS

Algebra II: Unit 2

Polynomials and Analysis of Nonlinear Functions

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&docid=AwOKz0mO1gEXQM&tbnid=prj82ZSOleBaDM:&ved=0CAUQjRw&url=http://schools.nyc.gov/SchoolPortals/10/X085/Academics/Mathematics/&ei=zKG9U5yqE4b4oAS6ooCADg&bvm=bv.70138588,d.aWw&psig=AFQjCNGWaNnv41cJZ3Bco1IS4VvAQUM2ng&ust=1405023038489225

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Course Philosophy/Description

Algebra II continues the students’ study of advanced algebraic concepts including functions, polynomials, rational expressions,

systems of functions and inequalities, and matrices. Students will be expected to describe and translate among graphic,

algebraic, numeric, tabular, and verbal representations of relations and use those representations to solve problems. Emphasis

will be placed on practical applications and modeling. Students extend their knowledge and understanding by solving open-

ended real-world problems and thinking critically through the use of high level tasks.

Students will be expected to demonstrate their knowledge in: utilizing essential algebraic concepts to perform calculations on

polynomial expression; performing operations with complex numbers and graphing complex numbers; solving and graphing

linear equations/inequalities and systems of linear equations/inequalities; solving, graphing, and interpreting the solutions of

quadratic functions; solving, graphing, and analyzing solutions of polynomial functions, including complex solutions;

manipulating rational expressions, solving rational equations, and graphing rational functions; solving logarithmic and

exponential equations; and performing operations on matrices and solving matrix equations.

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ESL Framework

This ESL framework was designed to be used by bilingual, dual language, ESL and general education teachers. Bilingual and dual language programs

use the home language and a second language for instruction. ESL teachers and general education or bilingual teachers may use this document to

collaborate on unit and lesson planning to decide who will address certain components of the SLO and language objective. ESL teachers may use the

appropriate leveled language objective to build lessons for ELLs which reflects what is covered in the general education program. In this way, whether

it is a pull-out or push-in model, all teachers are working on the same Student Learning Objective connected to the Common Core standard. The design

of language objectives are based on the alignment of the World-Class Instructional Design Assessment (WIDA) Consortium’s English Language

Development (ELD) standards with the Common Core State Standards (CCSS). WIDA’s ELD standards advance academic language development

across content areas ultimately leading to academic achievement for English learners. As English learners are progressing through the six developmental

linguistic stages, this framework will assist all teachers who work with English learners to appropriately identify the language needed to meet the

requirements of the content standard. At the same time, the language objectives recognize the cognitive demand required to complete educational tasks.

Even though listening and reading (receptive) skills differ from speaking and writing (expressive) skills across proficiency levels the cognitive function

should not be diminished. For example, an Entering Level One student only has the linguistic ability to respond in single words in English with

significant support from their home language. However, they could complete a Venn diagram with single words which demonstrates that they

understand how the elements compare and contrast with each other or they could respond with the support of their native language with assistance from

a teacher, para-professional, peer or a technology program.

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

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

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Pacing Chart – Unit 1

# Student Learning Objective NJSLS Big Ideas Math

Correlation

Instruction: 8

weeks

Assessment: 1

week

1 Apply the Remainder Theorem in order to determine the factors of a

polynomial. A.APR.B.2

4.3, 4.4

2

Use an appropriate factoring technique to factor polynomials. Explain

the relationship between zeros and factors of polynomials, and use the

zeros to construct a rough graph of the function defined by the

polynomial

A.SSE.A.2

A.APR.B.3

2.2, 3.1, 4.4, 4.5,

4.6, 4.8, 6.5

3 Graph polynomial functions from equations; identify zeros when

suitable factorizations are available; show key features and end

behavior.

F.IF.C.7c 2.1, 2.2, 2.3, 4.1,

4.7, 4.8

4 Use polynomial identities to describe numerical relationships and prove

polynomial identities.

A.APR.C.4 4.2

5 Rewrite simple rational expressions in different forms using inspection,

long division, or, for the more complicated examples, a computer

algebra system.

A.APR.D.6 4.3, 7.2, 7.3, 7.4

6

Solve simple rational and radical equations in one variable, use them to

solve problems and show how extraneous solutions may arise. Create

simple rational equations in one variable and use them to solve

problems.

A.REI.A.1

A.REI.A.2

A.CED.A.1

3.6, 5.4, 6.6, 7.1,

7.5

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Pacing Chart – Unit 1

7 For radical functions, interpret key features of graphs and tables in

terms of the quantities, and sketch graphs showing key features given a

verbal description of the relationship.

F.IF.B.4

F.IF.B.6

5.3

8 Derive the equation of a parabola given a focus and directrix. G.GPE.A.2 2.3

9 Graph logarithmic functions expressed symbolically and show key

features of the graph (including intercepts and end behavior).

F.IF.C.7e 6.1, 6.2, 6.3, 6.4

10

Find approximate solutions for f(x)=g(x), using technology to graph,

make tables of values, or find successive approximations. Include

cases where f(x) and/or g(x) are linear, polynomial, rational, absolute

value, logarithmic and exponential functions.

A.REI.D.11 3.5

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Research about Teaching and Learning Mathematics Structure teaching of mathematical concepts and skills around problems to be solved (Checkly, 1997; Wood & Sellars, 1996; Wood & Sellars, 1997)

Encourage students to work cooperatively with others (Johnson & Johnson, 1975; Davidson, 1990)

Use group problem-solving to stimulate students to apply their mathematical thinking skills (Artzt & Armour-Thomas, 1992)

Students interact in ways that support and challenge one another’s strategic thinking (Artzt, Armour-Thomas, & Curcio, 2008)

Activities structured in ways allowing students to explore, explain, extend, and evaluate their progress (National Research Council, 1999)

There are three critical components to effective mathematics instruction (Shellard & Moyer, 2002):

Teaching for balanced mathematical understanding

Developing children’s procedural literacy

Promoting strategic competence through meaningful problem-solving investigations Teachers should be:

Demonstrating acceptance and recognition of students’ divergent ideas.

Challenging students to think deeply about the problems they are solving, extending thinking beyond the solutions and algorithms required to solve the problem

Influencing learning by asking challenging and interesting questions to accelerate students’ innate inquisitiveness and foster them to examine concepts further.

Projecting a positive attitude about mathematics and about students’ ability to “do” mathematics Students should be:

Actively engaging in “doing” mathematics

Solving challenging problems

Investigating meaningful real-world problems

Making interdisciplinary connections

Developing an understanding of mathematical knowledge required to “do” mathematics and connect the language of mathematical ideas with numerical representations

Sharing mathematical ideas, discussing mathematics with one another, refining and critiquing each other’s ideas and understandings

Communicating in pairs, small group, or whole group presentations

Using multiple representations to communicate mathematical ideas

Using connections between pictures, oral language, written symbols, manipulative models, and real-world situations

Using technological resources and other 21st century skills to support and enhance mathematical understanding

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Mathematics is not a stagnate field of textbook problems; rather, it is a dynamic way of constructing meaning about the world around us,

generating knowledge and understanding about the real world every day. Students should be metaphorically rolling up their sleeves and “doing

mathematics” themselves, not watching others do mathematics for them or in front of them. (Protheroe, 2007)

Balanced Mathematics Instructional Model

Balanced math consists of three different learning opportunities; guided math, shared math, and independent math. Ensuring a balance of all three

approaches will build conceptual understanding, problem solving, computational fluency, and procedural fluency. Building conceptual

understanding is the focal point of developing mathematical proficiency. Students should frequently work on rigorous tasks, talk about the math,

explain their thinking, justify their answer or process, build models with graphs or charts or manipulatives, and use technology.

When balanced math is used in the classroom it provides students opportunities to:

solve problems

make connections between math concepts and real-life situations

communicate mathematical ideas (orally, visually and in writing)

choose appropriate materials to solve problems

reflect and monitor their own understanding of the math concepts

practice strategies to build procedural and conceptual confidence

Teacher builds conceptual understanding by

modeling through demonstration, explicit

instruction, and think alouds, as well as guiding

students as they practice math strategies and apply

problem solving strategies. (whole group or small

group instruction)

Students practice math strategies independently to

build procedural and computational fluency. Teacher

assesses learning and reteaches as necessary. (whole

group instruction, small group instruction, or centers)

Teacher and students practice mathematics

processes together through interactive

activities, problem solving, and discussion.

(whole group or small group instruction)

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Effective Pedagogical Routines/Instructional Strategies Collaborative Problem Solving

Connect Previous Knowledge to New Learning

Making Thinking Visible

Develop and Demonstrate Mathematical Practices

Inquiry-Oriented and Exploratory Approach

Multiple Solution Paths and Strategies

Use of Multiple Representations

Explain the Rationale of your Math Work

Quick Writes

Pair/Trio Sharing

Turn and Talk

Charting

Gallery Walks

Small Group and Whole Class Discussions

Student Modeling

Analyze Student Work

Identify Student’s Mathematical Understanding

Identify Student’s Mathematical Misunderstandings

Interviews

Role Playing

Diagrams, Charts, Tables, and Graphs

Anticipate Likely and Possible Student Responses

Collect Different Student Approaches

Multiple Response Strategies

Asking Assessing and Advancing Questions

Revoicing

Marking

Recapping

Challenging

Pressing for Accuracy and Reasoning

Maintain the Cognitive Demand

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Educational Technology

Standards

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

Technology Operations and Concepts

Create a personal digital portfolio which reflects personal and academic interests, achievements, and career aspirations by using a variety of digital tools and resources.

Example: Students create personal digital portfolios for coursework using Google Sites, Evernote, WordPress, Edubugs, Weebly, etc.

Communication and Collaboration

Develop an innovative solution to a real world problem or issue in collaboration with peers and experts, and present ideas for feedback through social media or in an online community.

Example: Use Google Classroom for real-time communication between teachers, students, and peers to complete assignments and

discuss strategies for factoring polynomials.

Critical Thinking, Problem Solving, and Decision Making

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

graphing polynomial functions from equations.

Computational Thinking: Programming

Use a programming language to solve problems or accomplish a task (e.g., robotic functions, website designs, applications and games).

Example: Students will create a set of instructions explaining how to derive the equation of a parabola given a focus and directrix.

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

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

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Career Ready Practices Career Ready Practices describe the career-ready skills that all educators in all content areas should seek to develop in their students. They are

practices that have been linked to increase college, career, and life success. Career Ready Practices should be taught and reinforced in all career

exploration and preparation programs with increasingly higher levels of complexity and expectation as a student advances through a program of

study.

CRP2. Apply appropriate academic and technical skills. Career-ready individuals readily access and use the knowledge and skills acquired through experience and education to be more productive.

They make connections between abstract concepts with real-world applications, and they make correct insights about when it is appropriate

to apply the use of an academic skill in a workplace situation

Example: Students will apply prior knowledge when solving real world problems. Students will make sound judgments about the use of

specific tools, such as algebra tiles, graphing calculators and technology to deepen their understanding of solving quadric equations.

CRP4. Communicate clearly and effectively and with reason. Career-ready individuals communicate thoughts, ideas, and action plans with clarity, whether using written, verbal, and/or visual methods.

They communicate in the workplace with clarity and purpose to make maximum use of their own and others’ time. They are excellent

writers; they master conventions, word choice, and organization, and use effective tone and presentation skills to articulate ideas. They are

skilled at interacting with others; they are active listeners and speak clearly and with purpose. Career-ready individuals think about the

audience for their communication and prepare accordingly to ensure the desired outcome.

Example: Students will communicate precisely using clear definitions and provide carefully formulated explanations when constructing arguments. Students will communicate and defend mathematical reasoning using objects, drawings, diagrams, and/or actions. Students

will ask probing questions to clarify or improve arguments.

CRP8. Utilize critical thinking to make sense of problems and persevere in solving them. Career-ready individuals readily recognize problems in the workplace, understand the nature of the problem, and devise effective plans to

solve the problem. They are aware of problems when they occur and take action quickly to address the problem; they thoughtfully

investigate the root cause of the problem prior to introducing solutions. They carefully consider the options to solve the problem. Once a

solution is agreed upon, they follow through to ensure the problem is solved, whether through their own actions or the actions of others.

Example: Students will understand the meaning of a problem and look for entry points to its solution. They will analyze information, make conjectures, and plan a solution pathway to solve linear and quadratic equations in two variables.

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Career Ready Practices

CRP12. Work productively in teams while using cultural global competence. Career-ready individuals positively contribute to every team, whether formal or informal. They apply an awareness of cultural difference to

avoid barriers to productive and positive interaction. They find ways to increase the engagement and contribution of all team members.

They plan and facilitate effective team meetings.

Example: Students will work collaboratively in groups to solve mathematical tasks. Students will listen to or read the arguments of others and ask probing questions to clarify or improve arguments. They will be able to explain how to perform operations with complex

numbers.

http://www.state.nj.us/education/aps/cccs/career/CareerReadyPractices.pdf

http://www.state.nj.us/education/aps/cccs/career/CareerReadyPractices.pdf

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WIDA Proficiency Levels

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

6- Reaching

Specialized or technical language reflective of the content areas at grade level

A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse as required by the specified grade level

Oral or written communication in English comparable to proficient English peers

5- Bridging

Specialized or technical language of the content areas

A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse, including stories, essays or reports

Oral or written language approaching comparability to that of proficient English peers when presented with grade level material.

4- Expanding

Specific and some technical language of the content areas

A variety of sentence lengths of varying linguistic complexity in oral discourse or multiple, related sentences or paragraphs

Oral or written language with minimal phonological, syntactic or semantic errors that may impede the communication, but retain much of its meaning, when presented with oral or written connected discourse,

with sensory, graphic or interactive support

3- Developing

General and some specific language of the content areas

Expanded sentences in oral interaction or written paragraphs

Oral or written language with phonological, syntactic or semantic errors that may impede the communication, but retain much of its meaning, when presented with oral or written, narrative or expository

descriptions with sensory, graphic or interactive support

2- Beginning

General language related to the content area

Phrases or short sentences

Oral or written language with phonological, syntactic, or semantic errors that often impede of the communication when presented with one to multiple-step commands, directions, or a series of statements

with sensory, graphic or interactive support

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

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

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Culturally Relevant Pedagogy Examples

Integrate Relevant Word Problems: Contextualize equations using word problems that reference student interests and cultures.

Example: When learning about interpreting the structure of expressions, problems that relate to student interests such as music,

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

Everyone has a Voice: Create a classroom environment where students know that their contributions are expected and valued.

Example: Norms for sharing are established that communicate a growth mindset for mathematics. All students are capable

of expressing mathematical thinking and contributing to the classroom community. Students learn new ways of looking at

problem solving by working with and listening to each other.

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

Example: Using a Place Based Education (PBE) model, students explore math concepts such as systems of

equations while determining ways to address problems that are pertinent to their neighborhood, school or culture.

Encourage Student Leadership: Create an avenue for students to propose problem solving strategies and potential projects.

Example: Students can learn to interpret functions in a context by creating problems together and deciding if the problems

fit the necessary criteria. This experience will allow students to discuss and explore their current level of understanding by

applying the concepts to relevant real-life experiences.

Present New Concepts Using Student Vocabulary: Use student diction to capture attention and build understanding before using academic terms.

Example: Teach math vocabulary in various modalities for students to remember. Use multi-modal activities, analogies, realia,

visual cues, graphic representations, gestures, pictures and cognates. Directly explain and model the idea of vocabulary words

having multiple meanings. Students can create the Word Wall with their definitions and examples to foster ownership.

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SEL Competency

Examples Content Specific Activity & Approach

to SEL

Self-Awareness Self-Management

Social-Awareness

Relationship Skills

Responsible Decision-Making

Example practices that address Self-Awareness:

• Clearly state classroom rules

• Provide students with specific feedback regarding

academics and behavior

• Offer different ways to demonstrate understanding

• Create opportunities for students to self-advocate

• Check for student understanding / feelings about

performance

• Check for emotional wellbeing

• Facilitate understanding of student strengths and

challenges

Students scan multistep contextual problems

that requires them to identify variables, write

equations, create graphs, etc., and make a list of

questions based on their understanding to ask

the teacher. This will help students to gain

confidence in working through the problems.

Set up small-group discussions that allows

students to reflect and discuss challenges or

how they have worked through a problem. For

examples, when students learn factoring

techniques, students can discuss which method

they have the most difficulty working with.

Self-Awareness

Self-Management Social-Awareness

Relationship Skills


Example practices that address Self-Management:

• Encourage students to take pride/ownership in work

and behavior

• Encourage students to reflect and adapt to classroom

situations

• Assist students with being ready in the classroom

• Assist students with managing their own emotional

states

Lead discussions that encourages students to

reflect on barriers they encounter when

completing an assignment (e.g., finding a

computer, needing extra help or needing a quiet

place to work) and help them think about

solutions to overcome those barriers.

Teach and model for students not to become

defensive or angry when errors/flaws in their

reasoning are pointed out by their classmates

during class discussions but rather to readily

accept their peer’s correction, and continue to

contribute to the discussion. Because of these

self-management efforts, the class is able to

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continue their discussion on the definition of

functions.

Self-Awareness

Self-Management

Social-Awareness Relationship Skills


Example practices that address Social-Awareness:

• Encourage students to reflect on the perspective of

others

• Assign appropriate groups

• Help students to think about social strengths

• Provide specific feedback on social skills

• Model positive social awareness through

metacognition activities

Organize a class service project to examine and

address a community issue. Use math to

examine the situations and find possible

solutions. For example, students can discuss as

a class or in groups how to determine whether a

polynomial is a repeated solution.

Use real-world application problems to lead a

discussion about taking different approaches to

solving a problem and respecting the feeling

and thoughts of those that used a different

strategy.

Self-Awareness

Self-Management

Social-Awareness

Relationship Skills Responsible Decision-Making

Example practices that address Relationship Skills:

• Engage families and community members

• Model effective questioning and responding to

students

• Plan for project-based learning

• Assist students with discovering individual strengths

• Model and promote respecting differences

• Model and promote active listening

• Help students develop communication skills

• Demonstrate value for a diversity of opinions

Instead of simply jumping into their own

solution when asked to graph a trigonometric

function, have students discuss the key features

and how to find those key features before

graphing.

During class or group discussion, have students

expound upon and clarify each other’s

questions and comments, ask follow-up

questions, and clarify their own questions and

reasoning.

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Self-Awareness

Self-Management

Social-Awareness

Relationship Skills


Example practices that address Responsible

Decision-Making:

• Support collaborative decision making for academics

and behavior

• Foster student-centered discipline

• Assist students in step-by-step conflict resolution

process

• Foster student independence

• Model fair and appropriate decision making

• Teach good citizenship

Use a lesson to teach students a simple formula

for making good choices. (e.g., stop, calm

down, identify the choice to be made, consider

the options, make a choice and do it, how did it

go?) Post the decision-making formula in the

classroom.

Routinely encourage students to use the

decision-making formula as they face a choice

(e.g., whether to finish homework or go out

with a friend). Support students through the

steps of making a decision anytime they face a

choice or decision. Simple choices like “Which

tool should I use to explain the relationship

between domain and range?” or “Do I need a

calculator for this problem?” are good places to

start.

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Differentiated Instruction

Accommodate Based on Students Individual Needs: Strategies

Time/General

Extra time for assigned tasks

Adjust length of assignment

Timeline with due dates for reports and projects

Communication system between home and school

Provide lecture notes/outline

Processing

Extra Response time

Have students verbalize steps

Repeat, clarify or reword

directions

Mini-breaks between tasks

Provide a warning for

transitions

Partnering

Comprehension

Precise processes for balanced

mathematics instructional

model

Short manageable tasks

Brief and concrete directions

Provide immediate feedback

Small group instruction

Emphasize multi-sensory

learning

Recall

Teacher-made checklist

Use visual graphic organizers

Reference resources to

promote independence

Visual and verbal reminders

Graphic organizers

Assistive Technology

Computer/whiteboard

Tape recorder

Video Tape

Tests/Quizzes/Grading

Extended time

Study guides

Shortened tests

Read directions aloud

Behavior/Attention

Consistent daily structured

routine

Simple and clear classroom

rules

Frequent feedback

Organization

Individual daily planner

Display a written agenda

Note-taking assistance

Color code materials

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Differentiated Instruction

Accommodate Based on Content Needs: Strategies

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

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

Pre-teach pertinent vocabulary

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

Word wall with visual representations of mathematical terms

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

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

Record formulas, processes, and mathematical rules in reference notebooks

Graphing calculator to assist with computations and graphing of trigonometric functions

Utilize technology through interactive sites to represent nonlinear data

Graphic organizers to help students interpret the meaning of terms in an expression or equation in context

Translation dictionary

Sentence stems to provide additional language support for ELL students.

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Interdisciplinary Connections

Model interdisciplinary thinking to expose students to other disciplines.

Social Studies Connection: Social Studies Standard 6.1.12.D.2.a

Name of Task: Logistic Growth Model, Explicit Version This problem introduces a logistic growth model in the concrete setting of estimating the population of the U.S. The model gives a

surprisingly accurate estimate and this should be contrasted with linear and exponential models, studied in ''U.S. Population 1790-1860.

Science Connection: Science Standard K-ESS3-3

Name of Task: Combined Fuel Efficiency

The US Department of Energy keeps track of fuel efficiency for all vehicles sold in the United States. Each car has two fuel economy numbers, one measuring efficient for city driving and one for highway driving. For example, a 2012 Volkswagen Jetta gets 29.0 miles per

gallon (mpg) in the city and 39.0 mpg on the highway. Many banks have "green car loans'' where the interest rate is lowered for loans on

cars with high combined fuel economy. This number is not the average of the city and highway economy values. Rather, the combined fuel

economy (as defined by the federal Corporate Average Fuel Economy standard) for mpg in the city and mpg on the highway, is computed

as.

Name of Task: Ideal Gas Law

The goal of this task is to interpret the graph of a rational function and use the graph to approximate when the function takes a given value. The first two parts of the question focus student attention on the meaning of the function within the context of pressure and volume.

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Enrichment

What is the purpose of Enrichment?

The purpose of enrichment is to provide extended learning opportunities and challenges to students who have already mastered, or can quickly master, the

basic curriculum. Enrichment gives the student more time to study concepts with greater depth, breadth, and complexity.

Enrichment also provides opportunities for students to pursue learning in their own areas of interest and strengths.

Enrichment keeps advanced students engaged and supports their accelerated academic needs.

Enrichment provides the most appropriate answer to the question, “What do you do when the student already knows it?”

Enrichment is…

Planned and purposeful

Different, or differentiated, work – not just more work

Responsive to students’ needs and situations

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

The ability to apply different or multiple strategies to the content

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

Elevated contextual complexity

Sometimes independent activities, sometimes direct instruction

Inquiry based or open-ended assignments and projects

Using supplementary materials in addition to the normal range of resources

Choices for students

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

Enrichment is not…

Just for gifted students (some gifted students may need intervention in some areas just as some other students may need

frequent enrichment)

Worksheets that are more of the same (busywork)

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

Extra homework

A package that is the same for everyone

Thinking skills taught in isolation

Unstructured free time

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Assessments

Required District/State Assessments Unit Assessment

NJSLA

SGO Assessments

Suggested Formative/Summative Classroom Assessments Describe Learning Vertically

Identify Key Building Blocks

Make Connections (between and among key building blocks)

Short/Extended Constructed Response Items

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

Drag and Drop Items

Use of Equation Editor

Quizzes

Journal Entries/Reflections/Quick-Writes

Accountable talk

Projects

Portfolio

Observation

Graphic Organizers/ Concept Mapping

Presentations

Role Playing

Teacher-Student and Student-Student Conferencing

Homework

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New Jersey Student Learning Standards

A.APR.B.2:

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only

if (x – a) is a factor of p(x).

A.APR.B.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the

polynomial.

A.APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the difference of two squares; the sum and difference

of two cubes; the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

A.APR.D.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials

with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra

system.

F.IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch

graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is

increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of

change from a graph.

F.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated

cases.

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New Jersey Student Learning Standards

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

A.SSE.A.2:

Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of

squares that can be factored as (x2 – y2)(x2 + y2).

A.REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the

assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A.REI.A.2:

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

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

and simple rational and exponential functions.

G.GPE.A.2: Derive the equation of a parabola given a focus and directrix.

A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) =

g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.

Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

27 | P a g e

Mathematical Practices

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

28 | P a g e

Course: Algebra II Unit: 2 (Two) Topic: Polynomials and Analysis of

Nonlinear Functions

NJSLS:

A.APR.B.2, A.SSE.A.2, A.APR.B.3, F.IF.C.7c, A.APR.C.4, A.APR.D.6, A.REI.A.1, A.REI.A.2, A.CED.A.1, F.IF.B.4, F.IF.B.6, G.GPE.A.2 ,

F.IF.C.7e, A.REI.D.11

Unit Focus:

Understand the relationship between zeros and factors of polynomials

Interpret the structure of expressions

Use polynomial identities to solve problems

Analyze functions using different representations

Rewrite rational expressions

Understand solving equations as a process of reasoning and explain the reasoning

Interpret functions in terms of the context

Translate between the geometric description and the equation for a conic section

Represent and solve equations and inequalities graphically

New Jersey Student Learning Standard(s):

A.APR.B.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) =

0 if and only if (x – a) is a factor of p(x).

Student Learning Objective 1: Apply the Remainder Theorem in order to determine the factors of a polynomial.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement

Key/ Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 6

A-APR.2

Polynomial division: For a polynomial p(x) and a

number a:

p(a) = 0 if and only if (x – a) is a factor of p(x)

How can you use the

factors of a cubic

polynomial to solve a

Type II, III:

Zeroes and

factorization of a

29 | P a g e

(x – a) is a factor of p(x) if and only if p(a) = 0

The Remainder theorem says that if a polynomial

p(x) is divided by ax , then the remainder is the value of the polynomial evaluated at a.

Saying that x – a is a factor of a polynomial p(x) is

equivalent to saying that p(a) = 0, by the zero

property of multiplication.

Any polynomial of degree n can be factored into n

binomials of the form x – c, with possibly complex

values for c.

Use the Remainder Theorem to determine factors of

a polynomial.

SPED Strategies:

Provide students with background information about

dividing polynomials and connect it the material

they already know.

Use contextual examples to illustrate what the

Remainder Theorem states and does when applied.

Create a reference document with students that

contain all relevant information regarding

polynomial division and Remainder Theorem to

encourage independence and increase proficiency

and confidence.

division problem involving

the polynomial?

How can you factor a

polynomial?

How can you determine

whether a polynomial has a

repeated solution?

Why is it important to

supply a zero for a

coefficient of any missing

term, when you are dividing

polynomials?

How can you determine

whether x – a is a factor of

a polynomial p(x)? Why

does this work?

How do you determine how

many zeros a polynomial

function will have?

quadratic polynomial

I

Zeroes and

factorization of a

quadratic polynomial

II

Additional Tasks:

Graphing from Roots

The Missing

Coefficient

Zeroes and

factorization

of a general

polynomial

Zeroes and

factorization

of a non-polynomial

function

30 | P a g e

ELL Strategies:

Read and write to restructure by performing

arithmetic operations on polynomial/rational

expressions in student’s native language and/or use

gestures, examples and selected technical words.

Model a polynomial function using the height of the

roller coaster as a function of time.

Let students write and explain the Remainder

Theorem and why it is useful.


A.SSE.A.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a

difference of squares that can be factored as (x2 – y2)(x2 + y2).

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

defined by the polynomial.

Student Learning Objective #2: Use an appropriate factoring technique to factor polynomials. Explain the relationship between zeros and factors

of polynomials, and use the zeros to construct a rough graph of the function defined by the polynomial.



Key/ Clarifications

Skills, Strategies & Concepts Essential

Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 7

A-SSE.2-3

Additional examples: In the equation x2 +

2x + 1 + y2 = 9, see

an opportunity to

rewrite the first three

terms as (x+1)2. See

Factors of polynomials can be used to identify

zeros to be used to develop a rough graph of the

polynomial function.

Factor polynomials.

How does using the

structure of an expression

help to simplify the

expression?

What type of symmetry

does the graph of 𝑓(𝑥) =

Type II, III:

Seeing Dots

Graphing from

Factors I

31 | P a g e

(x2 + 4)/(x2 + 3) as

((x2+3) + 1)/(x2+3),

thus recognizing an

opportunity to write

it as 1 + 1/(x2 + 3).

Tasks will not include sums and

differences of cubes.

A-SSE.2-6

Factor completely: 6cx - 3cy - 2dx + dy.

(A first iteration

might give 3c(2x-y)

+ d(-2x+y), which

could be recognized

as 3c(2x-y) - d(2x-y)

on the way to

factoring completely

as (3c-d)(2x-y).)

Tasks do not have a context.

Analyze a table of values to determine where the

polynomial is increasing and decreasing.

Use the zeros of the polynomial to create a rough

graph.

If p(a) = 0, then a is a zero of p.

If a is a zero of p, then a is an x-intercept of the

graph of y = p(x).

The values and multiplicity of the zeros of a

polynomial, along with the end behavior, can be

used to sketch a graph of the function defined by

the polynomial.

Complicated expressions can be interpreted by

viewing parts of the expression as single entities.

Structure within expressions can be identified and

used to factor or simplify the expression.

SPED Strategies:

Model the thinking and processes involved in

analyzing polynomials in algebraic and table form

to determine graph behavior.

Provide students with opportunities to practice the

thinking and processes involved in small groups.

Develop a reference sheet for student use that

includes formulas, processes and procedures and

sample problems to encourage proficiency and

independence.

𝑎(𝑥 − ℎ)2 + 𝑘 have and how can you describe this

symmetry?

How many turning points

can the graph of a

polynomial function have?

Why is it important to

supply a zero for a

coefficient of any missing

term, when you are

dividing polynomials?

How does the concept of

the zero product property

allow you to find the roots

of a quadratic function?

What information do you

need to sketch a rough

graph of a polynomial

function?

How are the zeros of a

polynomial related to its

graph?

Graphing from

Factors II

Graphing from

Factors III

Additional Tasks:

A Cubic Identity

Animal Populations

Graphing from Roots

Equivalent

Expressions

Solving a Simple Cubic

Equation

32 | P a g e

ELL Strategies:

Demonstrate orally and in writing an appropriate

factoring technique to factor expressions

completely including expressions with complex

number in student’s native language and/or use

selected technical vocabulary in phrases and short

sentences.

Describe and explain the relationship between

zeros and factors of polynomials and use zeros to

construct a rough graph of the function defined by

the polynomial using key technical vocabulary in a

series of simple sentences.

Guide students to think of a polynomial as a

product, reassure students that they will be learning

various strategies for finding those factors.

Create a reference document with the special

factoring patterns.

33 | P a g e


F.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more

complicated cases.

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

Student Learning Objective 3: Graph polynomial functions from equations; identify zeros when suitable factorizations are available; show key

features and end behavior.

Modified Student Learning Objectives/Standards:

M.EE.F-IF.1–3: Use the concept of function to solve problems.


Key/ Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 5

MP 6

F-IF.7c Factors of polynomials can be used to identify

zeros to be used to develop a rough graph of the

polynomial function.

Graph a polynomial function given its equation.

Identify zeros from the graph and using an

appropriate factoring technique.

Use technology to graph and describe key features

of the graph for complicated cases.

Key features of a graph or table may include

intercepts; intervals in which the function is

increasing, decreasing or constant; intervals in

which the function is positive, negative or zero;

symmetry; maxima; minima; end behavior;

asymptotes; domain; range and periodicity.

The graph of a trigonometric function shows

period, amplitude, midline and asymptotes.

How does the constants a,

h, and k affect the graph of

the quadratic function

𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 ?


does the graph of 𝑓(𝑥) =𝑎(𝑥 − ℎ)2 + 𝑘 have and how can you describe this

symmetry?

What are some common

characteristics of the graphs

of cubic and quartic

polynomial functions?

How can you transform the

graph of a polynomial

function?

Type II, III:

Graphs of Power

Functions

Running Time

Additional Tasks:

Identifying Graphs of

Functions

34 | P a g e

The graph of a polynomial function shows zeros

and end behavior.

A function can be represented algebraically,

graphically, numerically in tables, or by verbal

descriptions.

SPED Strategies:

Model the thinking behind determining when and

how to use the graphing calculator to graph

complicated polynomials.


thinking and processes involved in graphing

polynomial equations by hand and using

technology by working small groups.




independence.

ELL Strategies:

Demonstrate comprehension of complex questions

in student’s native language and/or simplified

questions with drawings and selected technical

words concerning graphing functions symbolically

by showing key features of the graph by hand in

simple cases and using technology for more

complicated cases.

Use technology to graph polynomial and identify

the end behavior and y intercept in the figure.


can the graph of a

polynomial function have?

How can you compare

properties of two functions

if they are represented in

different ways?

How do different forms of

a function help you to

identify key features?

How do you determine

which type of function best

models a given situation?

35 | P a g e

Use technology to create table of values to verify

the positive zeros of the polynomial.


A.APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the difference of two squares; the sum and

difference of two cubes; the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

Student Learning Objective 4: Use polynomial identities to describe numerical relationships and prove polynomial identities.



Key/ Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 3

MP 7

A.APR.C.4 Polynomial identities can be used to describe

numerical relationships.

Show that the polynomial identity (x2 + y2)2 = (x2 –

y2)2 + (2xy)2 can be used to generate Pythagorean

triples.

Prove polynomial identities.

SPED Strategies:

Ground the new learning in a real life context to

help students internalize the concept and develop

understanding.




independence.

ELL Strategies:

How can you cube a

binomial?

How does cubing binomials

enhance the understanding

of polynomial identities?

How are polynomials used

to represent situations?

Type II, III:

Trina's Triangles

Additional Tasks:

The Power of

Algebra—Finding

Pythagorean Triples

36 | P a g e

Explore Pascal’s Triangle to display the patterns in

the expansion of (a+b)n.

Create in a notebook a list of polynomial identities

to help students recognize different polynomial

identities


A.APR.D.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are

polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer

algebra system.

Student Learning Objective 5: Rewrite simple rational expressions in different forms using inspection, long division, or, for the more complicated

examples, a computer algebra system.


MPs Evidence Statement Key/

Clarifications


Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

A-APR.6

Examples will be simple enough to

allow inspection or

long division.

Simple rational expressions are limited

to numerators and

denominators that

have degree at most 2.

Rational expressions can be written in different

forms.

Write a(x)/b(x) in the form q(x) + r(x)/b(x), where

a(x), b(x), q(x), and r(x) are polynomials with the

degree of r(x) less than the degree of b(x).

Use inspection, factoring and long division to

rewrite rational expressions.

Use technology to rewrite rational expressions for

more complicated cases.

SPED Strategies:

How can you use the

factors of a cubic

polynomial to solve a

division problem involving

the polynomial?

What are some of the

characteristics of the graph

of a rational function?

How can you determine the

excluded values in a

product or quotient of two

rational expressions?

Type II, III:

Combined Fuel

Efficiency

Egyptian Fractions II

37 | P a g e

Connect the rewriting of polynomial expressions to

previous learning about equivalent expressions.

Model how to use inspection, factoring and long

division to rewrite rational expressions and develop

a reference document that uses verbal and pictorial

models.


thinking and processes involved in rewriting

polynomials by inspection, factoring and long

division.

Model the use of technology to rewrite more

complicated cases and encourage them to practice

this skill while working small groups.




independence.

ELL Strategies:

Read in order to rewrite simple rational expressions

in different forms in student’s native language

and/or use gestures, examples and selected,

technical words.

Connect rational numbers and rational functions by

asking student to define rational numbers.

Highlight and circle each factor in the denominator

of a rational expression. Guide students to set each

factor equal to zero

How are rational functions

used to represent real

world situations?

38 | P a g e


A.REI.A.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the

assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

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

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

functions, and simple rational and exponential functions.

Student Learning Objective 6: Solve simple rational and radical equations in one variable, use them to solve problems and show how extraneous

solutions may arise. Create simple rational equations in one variable and use them to solve problems.


M.EE.A-CED.1: Create an equation involving one operation with one variable, and use it to solve a real-world problem.


Clarifications


Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 3

MP 4

MP 6

A-REI.2

Simple rational equations are limited

to numerators and

denominators that

have degree at most 2.

Inverse relationships exist between roots and

powers.

Extraneous solutions do not result in true

statements.

Simple rational and radical equations can have

extraneous solutions.

Use the inverse relationship between roots and

powers when solving radical equations.

Identify any extraneous solutions.

What is the significance of

being able to identify

extraneous solutions?

What do you use to justify

your reasoning when

solving an equation?

How do you determine if

an equation is solved

properly?

How do you determine

and justify if a solution to

an equation is correct?

Type II, III:

Paying the rent

Radical Equations

Additional Tasks:

An Extraneous

Solution

Products and

Reciprocals

39 | P a g e

Solve simple rational equations in one variable

(degree of numerators and denominator is not

greater than 2).

Write simple rational equations in one variable and

use the rational equation to solve problems.

Equations are solved as a process of reasoning

using properties of operations and equality, which

can justify each step of the process.

A solution to an equation can be checked, by

substituting in that value for the variable and

simplifying to see if the equation holds true.

Equations and inequalities can be created to

represent and solve real-world and mathematical

problems.

Solutions are viable or not in different situations

depending upon the constraints of the given

context.

SPED Strategies:


solving simple rational and radical equations

explaining the significance of extraneous roots by

using real life examples to illustrate.




independence.

Why are properties of real

numbers important when

solving equations?

Give an example of a

simple rational or radical

equation that has an

extraneous solution and

explain why it is an

extraneous solution.

Who wins the Race?

40 | P a g e


thinking and processes involved in solving simple

rational and radical equations including those with

extraneous roots by working in small groups.

ELL Strategies:

Explain orally and in writing how to solve simple

equations in one variable and use them to solve

problems, justify each step in the process in

student’s native language and/or use gestures,

examples and selected, technical words.

Build on past knowledge by explaining that

simplifying rational expressions is similar to

simplifying fractions.

Model a real-life problem: The income function

and population, divide and graph the result.

41 | P a g e


F.IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and

sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is

increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the

rate of change from a graph.

Student Learning Objective 7: For radical functions, interpret key features of graphs and tables in terms of the quantities, and sketch graphs

showing key features given a verbal description of the relationship.


M.EE.F-IF.4–6: Construct graphs that represent linear functions with different rates of change and interpret which is faster/slower, higher/lower,

etc.


Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 4

MP 5

MP 6

MP 7

F-IF.4-5

For an exponential, polynomial,

trigonometric, or

logarithmic function

that models a

relationship between

two quantities,

interpret key features

of graphs and tables in

terms of the quantities,

and sketch graphs

showing key features

given a verbal

description of the

relationship. Key

features include:

A radical function is any function that contains a

variable inside a root.

Interpret key features of radical functions from

graphs and tables in the context of the problem.

Sketch graphs of radical functions given a verbal

description of the relationship between the

quantities.

Identify intercepts and intervals where function is

increasing/decreasing.

Determine the practical domain of a radical

function.


does the graph of 𝑓(𝑥) =𝑎(𝑥 − ℎ)2 + 𝑘 have and how can you describe this

symmetry?

What are some of the

characteristics of the graphs

of cubic and quartic

polynomial functions?


can a graph of a polynomial

function have and why is

this important?

Type II, III:

Containers

Mathemafish

Population

Model air plane

acrobatics

The High School Gym

Words - Tables -

Graphs

42 | P a g e

intercepts; intervals

where the function is

increasing, decreasing,

positive, or negative;

relative maximums

and minimums; end

behavior; symmetries;

and periodicity.

See illustrations for F-IF.4

at

o http://illustrativemathematics.org

Key features may also include

discontinuities.

F-IF.6-2

Calculate and interpret the average rate of

change of a function

(presented

symbolically or as a

table) over a specified

interval with functions

limited to polynomial,

exponential,

logarithmic and

trigonometric

functions.

Tasks have a real-world context.

Determine key features including intercepts;

intervals where the function is increasing,

decreasing, positive, or negative; relative maxima

and minima; symmetries; end behavior.

SPED Strategies:

Pre-teach vocabulary using visual and verbal

models that are connected to real life situations.

Model the thinking and processes involved in the

understanding of radical functions including

interpreting key features from graphs and tables

and sketching the key features of graphs from a

verbal description of the relationship.




independence.


thinking and processes involved in the

understanding of radical functions including the

interpretation of key features from graphs and

tables and sketching the key features of graphs

from a verbal description of the relationship by

working in small groups.

ELL Strategies:

Listen and read in order to interpret orally and in

writing the key features in graphs and tables and

given a verbal description of the relationship,

sketch graphs showing the key features in

How can you use a radical

function to model a real life

situation?

How can you describe the

shape of a graph?

How can you relate the

shape of a graph to the

meaning of the relationship

it represents?

How would you determine

the appropriate domain for a

function describing a real-

world situation?

How would you determine

the appropriate domain for a

function describing a real-

world situation?

Given a function that

describes a real-world

situation, what can the

average rate of change of

the function tell you?

How do the parts of a graph

of a function related to its

real-world context?

http://illustrativemathematics.org/http://illustrativemathematics.org/

43 | P a g e

Tasks must include the interpret part of the

evidence statement.

F-IF.6-7

Estimate the rate of change from a graph.

Tasks have a real-world context.

Tasks may involve polynomial,

exponential,

logarithmic, and

trigonometric

functions.

student’s native language and/or use gestures,

examples and selected, technical words.

After estimating, and calculating the average rate

of change of a function presented symbolically, in

a table, or graphically over a specified interval;

interpret the answer in writing in student’s native

language and/or use gestures, examples and

selected technical words.

Introduce the new topic by creating partner

discussions of the equation of the top half of a

parabola with horizontal line of symmetry: y =

sqrt(0-x); then make connection with the radical

equation graph.

Using a graphic calculator, graph the parent

function of radical and later display several

transformations.

44 | P a g e


G.GPE.A.2: Derive the equation of a parabola given a focus and directrix.

Student Learning Objective 8: Derive the equation of a parabola given a focus and directrix.



Clarifications


Questions

(Accountable Talk)

Tasks/Activities

MP 4

G.GPE.A.2 Any point on a parabola is equidistant between

the focus and the directrix.

Use the distance formula to write an equation of

a parabola when the focus and directrix are

given.

Derive equation of a parabola.

Graph a parabola.

Determine the characteristics of a parabola based

on its equation.

Determine the equation of a parabola using

certain characteristics.

SPED Strategies:




understanding the relationship and significance

between parabolas and their respective focus and

directrix by grounding it in a real life context

such as satellite dishes.

What is the focus of a

parabola and what is its

significance?

Given the focus and directrix

of a parabola, how do we find

the equation of the parabola?

How is the process of writing

equations for parabolas,

ellipses and hyperbolas

similar/different?

How do you write the

equation of a parabola given

its focus and directrix?

Type II, III:

Defining Parabolas

Geometrically

45 | P a g e




independence.

Provide students with opportunities to practice

the thinking by working with a partner or in

small groups.

ELL Strategies:

After deriving the equation of a parabola (given

a focus and directrix) explain in student’s native

language and/or use gestures, examples and

selected technical words.

Analyze satellite dishes and spotlights to discuss

focus, vertex and directrix of the parabola.

46 | P a g e


F.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more

complicated cases.

F.IF.C.7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period,

midline, and amplitude.

Student Learning Objective 9: Graph logarithmic functions expressed symbolically and show key features of the graph (including intercepts and

end behavior).


M.EE.F-IF.1–3: Use the concept of function to solve problems.


Clarifications


Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 2

MP 4

MP 6

F-IF.7e-1

F-IF.7e-2

About half of tasks involve logarithmic

functions, while the

other half involves

trigonometric functions.

Graph logarithmic functions having base 2, 10 or e,

using technology for more complicated cases.

Show intercepts and end behavior of logarithmic

functions.

Identify whether the exponential function is a

growth or decay function from its graph.

SPED Strategies:




graphing exponential and logarithmic functions by

grounding it in a real life context.

Model the thinking behind determining when and

how to use the graphing calculator to graph

complicated exponential and logarithmic functions.

How do exponential

functions model real-world

problems and their

solutions?

How do logarithmic

functions model real-world

problems and their

solutions?

How can you transform the

graphs of exponential and

logarithmic functions and

when?

How are exponential

functions and logarithmic

functions related?

Type II, III:

Identifying graphs of

functions

Additional Tasks:

Exponential Kiss

Graphs of Power

Functions

Identifying

Exponential Functions

Logistic Growth

Model, Explicit

Version

47 | P a g e

Illustrate the relationship between logarithmic and

exponential functions deliberately and provide

students with a reference sheet to encourage

proficiency and independence.

ELL Strategies:

Explore logarithm function by giving students a set

of notecards with the term log, b, y, =, and x

written on separate cards. Ask students to form

equations using their card.

Model the graph of the energy magnitude M of an

earthquake and let students research how to use log

to represent the energy magnitude of the

earthquake.


A.REI.D.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the

equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive

approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. *

Student Learning Objective 10: Find approximate solutions for f(x)=g(x), using technology to graph, make tables of values, or find successive

approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, logarithmic and exponential functions.


M.EE.A-REI.10–12: Interpret the meaning of a point on the graph of a line. For example, on a graph of pizza purchases, trace the graph to a point

and tell the number of pizzas purchased and the total cost of the pizzas.


Clarifications


Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 5

A-REI.11-2

Find the solutions of where the graphs of the equations

Solutions to complex systems of nonlinear

functions can be approximated graphically.

Why are the x-coordinates

of the points where the

graphs of the equations

Type II, III:

48 | P a g e

y= f(x) and y= g(x) intersect,

e.g. using technology to

graph the functions, make

tables of values or find

successive approximations.

Include cases where f(x)

and/or g(x) are linear,

quadratic, polynomial,

rational, absolute value,

exponential, and/or

logarithmic functions. ★

The "explain" part of standard A-REI.11 is not

assessed here.

Find the solution to f(x)=g(x) approximately,

e.g., using technology to graph the functions;

include cases where f(x) and/or g(x) are linear,

polynomial, rational, absolute value,

exponential, and logarithmic functions.

Find the solution to f(x)=g(x) approximately,

e.g., using technology to make tables of values,

or find successive approximations; include

cases where f(x) and/or g(x) are linear,

polynomial, rational, absolute value,

exponential, and logarithmic functions.

Solving a system of equations algebraically

yields an exact solution; solving by graphing

or by comparing tables of values yields an

approximate solution.

The x-coordinates of the points where the

graphs of the equations y = f(x) and y = g(x)

intersect are the solutions of the equation f(x)

= g(x).

SPED Strategies:

Illustrate the thinking and processes needed to

find approximate solutions for f(x)=g(x), using

technology to graph, make tables of values, or

find successive approximations using real life

examples.

Develop a reference sheet for students to

utilize when working independently and in

small groups that illustrates the thinking and

processes needed to find approximate solutions

y = f(x) and y = g(x)

intersect equal to the

solutions of the equation

f(x) = g(x)?

Why does graphing or

using a table give

approximate solutions?

In what situations would

you want an exact solution

rather than an approximate

solution or vice versa?

Introduction to

Polynomials - College

Fund

Ideal Gas Law

Population and Food

Supply

Two Squares are Equal

49 | P a g e

for f(x)=g(x), using technology to graph, make

tables of values, or find successive

approximations.

Model how to use the graphing calculator to

find approximate solutions for f(x)=g(x), to

graph, to make tables of values, or to find

successive approximations.

ELL Strategies:

After finding approximate solutions for the

intersections of functions, explain orally why

the x-coordinates are the solutions of the

equation f(x) = g(x) in student’s native

language and/or use drawings, and selected

technical words.

Model: writing and discussing the falling

object, by using a ball, then, graph and display

in a table the results of the activity.

Using a graphing calculator, graph a variety of

equations as log, rational, radical, exponential

and ask students to compare and contrast the

shape, range and domain of each graph.

50 | P a g e

Integrated Evidence Statements A.Int.1: Solve equations that require seeing structure in expressions

Tasks do not have a context.

Equations simplify considerably after appropriate algebraic manipulations are performed. For example, x4-17x2+16 = 0, 23x = 7(22x) + 22x , x - √x = 3√x

Tasks should be course level appropriate.

F-BF.Int.2: Find inverse functions to solve contextual problems. Solve an equation of the form 𝒇(𝒙) = 𝒄 for a simple function f that has

an inverse and write an expression for the inverse. For example, 𝒇(𝒙) = 𝟐𝒙𝟑 or 𝒇(𝒙) =𝒙+𝟏

𝒙−𝟏 for 𝒙 ≠ 𝟏.

For example, see http://illustrativemathematics.org

As another example, given a function C(L) = 750𝐿2 for the cost C(L) of planting seeds in a square field of edge length L, write a function for the edge length L(C) of a square field that can be planted for a given amount of money C; graph the function, labeling the axes.

This is an integrated evidence statement because it adds solving contextual problems to standard F-BF.4a.

F-Int.1-2: Given a verbal description of a polynomial, exponential, trigonometric, or logarithmic functional dependence, write an

expression for the function and demonstrate various knowledge and skills articulated in the Functions category in relation to this function.

Given a verbal description of a functional dependence, the student would be required to write an expression for the function and then, e.g., identify a natural domain for the function given the situation; use a graphing tool to graph several input-output pairs; select applicable

features of the function, such as linear, increasing, decreasing, quadratic, periodic, nonlinear; and find an input value leading to a given

output value.

F-Int.3: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-

level knowledge and skills articulated in F-TF.5, F-IF.B, F-IF.7, limited to trigonometric functions.

F-TF.5 is the primary content and at least one of the other listed content elements will be involved in tasks as well.

HS-Int.3-3: Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of

course-level knowledge and skills articulated in F-LE, A-CED.1, A-SSE.3, F-IF.B, F-IF.7★

F-LE.A, Construct and compare linear, quadratic, and exponential models and solve problems, is the primary content and at least one of the other listed content elements will be involved in tasks as well.

HS.C.5.4: Given an equation or system of equations, reason about the number or nature of the solutions. Content Scope: A-REI.2.

Simple rational equations are limited to numerators and denominators that have degree at most 2.

51 | P a g e

Integrated Evidence Statements

HS.C.5.11: Given an equation or system of equations, reason about the number or nature of the solutions. Content Scope: A-REI.11,

involving any of the function types measured in the standards.

For example, students might be asked how many positive solutions there are to the equation ex = x+2 or the equation ex = x+1, explaining how they know. The student might use technology strategically to plot both sides of the equation without prompting.

HS.C.6.2: Base explanations/reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted

in the coordinate plane. Content Scope: A-REI.D

HS.C.6.4: Base explanations/reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted

in the coordinate plane. Content Scope: G-GPE.2

HS.C.7.1: Base explanations/reasoning on the relationship between zeros and factors of polynomials. Content Scope: A-APR.B

HS.C.8.2: Construct, autonomously, chains of reasoning that will justify or refute algebraic propositions or conjectures. Content Scope:

A-APR.4

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

A-APR

HS.C.16.3: Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of

solutions (if any). Tasks are limited to simple rational or radical equations. Content scope: A-REI.1

Simple rational equations are limited to numerators and denominators that have degree at most 2.

A rational or radical function may be paired with a linear function. A rational function may not be paired with a radical function.

HS.C.18.4: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures about polynomials,

rational expressions, or rational exponents. Content scope: N-RN, A-APR.(2, 3, 4, 6)

52 | P a g e

Integrated Evidence Statements HS.C.CCR: Solve multi-step mathematical problems requiring extended chains of reasoning and drawing on a synthesis of the knowledge

and skills articulated across: 7-RP.A.3, 7-NS.A.3, 7-EE.B.3, 8-EE.C.7B, 8-EE.C.8c, N-RN.A.2, A-SSE.A.1b, A-REI.A.1, A-REI.B.3, A-

REI.B.4b, F-IF.A.2, F-IF.C.7a, F-IF.C.7e, G-SRT.B.5 and G-SRT.C.7.

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

Tasks will synthesize multiple aspects of the content listed in the evidence statement text, but need not be comprehensive.

Tasks should address at least A-SSE.A.1b, A-REI.A.1, and F-IF.A.2 and either F-IF.C.7a or F-IF.C.7e (excluding trigonometric and logarithmic functions). Tasks should also draw upon additional content listed for grades 7 and 8 and from the remainin

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