Algebra I Honors Unit 2.pdf - MATHEMATICS

1 | P a g e B o a r d A p p r o v e d 0 9 . 0 8 . 2 0 2 1

MATHEMATICS

Algebra I Honors: Unit 2

Writing Linear Functions, Linear Systems, &

Exponential Functions and Sequences


Course Philosophy/Description

Algebra I Honors is a rigorous course designed to develop and extend the algebraic concepts and processes that can be used to solve a variety of real-

world and mathematical problems. The fundamental purpose of Algebra 1 is to formalize and extend the mathematics that students learned in the

elementary and middle grades. The Standards for Mathematical Practice apply throughout each course, and, together with the New Jersey Student

Learning Standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make

sense of problem situations. Conceptual knowledge behind the mathematics is emphasized. Algebra I provides a formal development of the algebraic

skills and concepts necessary for students to succeed in advanced courses as well as the PARCC. The course also provides opportunities for the

students to enhance the skills needed to become college and career ready. Students will learn and engage in applying the concepts through the use of

high-level tasks and extended projects. This course is designed for students who are highly motivated in the areas of Math and/or Science.

The content shall include, but not be limited to, perform set operations, use fundamental concepts of logic including Venn diagrams, describe the

concept of a function, use function notation, solve real-world problems involving relations and functions, determine the domain and range of

relations and functions, simplify algebraic expressions, solve linear and literal equations, solve and graph simple and compound inequalities, solve

linear equations and inequalities in real-world situations, rewrite equations of a line into slope-intercept form and standard form, graph a line given

any variation of information, determine the slope, x- and y- intercepts of a line given its graph, its equation or two points on the line, write an

equation of a line given any variation of information, determine a line of best fit and recognize the slope as the rate of change, factor polynomial

expressions, perform operations with polynomials, simplify and solve algebraic ratios and proportions, simplify and perform operations with radical

and rational expressions, simplify complex fractions, solve rational equations including situations involving mixture, distance, work and interest,

solve and graph absolute value equations and inequalities, graph systems of linear equations and inequalities in two and three variables and quadratic

functions, use varied solution strategies for quadratic equations and for systems of linear equations and inequalities in two and three variables,

perform operations on matrices, and use matrices to solve problems.

in two and three variables.


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 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


Pacing Chart – Unit 2 # Student Learning Objective NJSLS Big Ideas Math

Correlation

Instruction:

11/13/21 – 1/28/22

Assessment:

Mid-Year

Assessment

1

Writing Equations in Slope-Intercept Form A.CED.A.2

F.BF.A.1a

F.LE.A.1b

F.LE.A.2

4-1

2

Writing Equations in Point-Slope Form A.CED.A.2

F.BF.A.1a

F.LE.A.1b

F.LE.A.2

4-2

3 Writing Equations of Parallel and Perpendicular Lines A.CED.A.2

F.LE.A.2 4-3

4

Scatter Plots and Lines of Fit F.LE.B.5

S.ID.B.6a

S.ID.B.6c

S.ID.C.7

4-4

5

Analyzing Lines of Fit

F.LE.B.5

S.ID.B.6a

S.ID.B.6b

S.ID.B.6c

S.ID.C.7

S.ID.C.8

S.ID.C.9

4-5

6

Arithmetic Sequences F.IF.A.3

F.BF.A.1a

F.BF.A.2

F.LE.A.2

4-6

7 Piecewise Functions A.CED.A.2

A.REI.D.10

F.IF.C.7b 4-7


Pacing Chart – Unit 2

8 Solving Systems of Linear Equations by Graphing A.CED.A.3

A.REI.C.6 5-1

9 Solving Systems of Linear Equations by Substitution A.CED.A.3

A.REI.C.6 5-2

10 Solving Systems of Linear Equations by Elimination A.CED.A.3

A.REI.C.5

A.REI.C.6

5-3

11 Solving Special Systems of Linear Equations A.CED.A.3

A.REI.C.6 5-4

12 Solving Equations by Graphing A.CED.A.3

A.REI.D.11 5-5

13 Graphing Linear Inequalities in Two Variables A.CED.A.3

A.REI.D.12 5-6

14 Systems of Linear Inequalities A.CED.A.3

A.REI.D.12 5-7

15 Properties of Exponents N.RN.A.2 6-1

16 Radicals and Rational Exponents N.RN.A.1

N.RN.A.2 6-2

17

Exponential Functions A.CED.A.2

F.IF.B.4

F.IF.C.7e

F.IF.C.9

F.BF.A.1a

F.BF.B.3

F.LE.A.1a

F.LE.A.2

6-3

18

Exponential Growth and Decay A.SSE.B.3c

A.CED.A.2

F.IF.C.7e

F.IF.C.8b

FF.BF.A.1a

F.LE.A.1c

F.LE.A.2

6-4


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 within the Balanced Math approach

• 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


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 re-teaches 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)


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

Re-voicing

Marking

Recapping

Challenging

Pressing for Accuracy and Reasoning

Maintain the Cognitive Demand


Computer Science and Design Thinking

Standards

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

➢ Data and Analysis

• Create data visualizations from large data sets to summarize, communicate, and support different interpretations of real-

world phenomena.

Example: Students can use Google Docs and digital tools as a means of discussing and collaboratively solving complex tasks that

focus on concepts related to polynomials. https://www.mathsisfun.com/data/

• Create and refine computational models to better represent the relationships among different elements of data collected

from a phenomenon or process.

Example: Students can use digital tools to graph linear systems of inequalities that model a solution to a real world situation.

Students can then write a position statement to justify their solution, mathematical thinking and modeling with peers.

http://www.mathsisfun.com/data/graphs-index.html.

➢ Algorithms & Programming

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

Example: Students can use and explain the advantages of using graphing calculators, GeoGebra or Desmos to graph systems of

inequalities to solve contextual problems. The discussion should also involve developing ways to improve and develop even better

algorithms. https://www.desmos.com/calculator https://www.geogebra.org/geometry?lang=en-US

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

https://www.mathsisfun.com/data/

http://www.mathsisfun.com/data/graphs-index.html

https://www.desmos.com/calculator

https://www.geogebra.org/geometry?lang=en-US

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


Career Readiness, Life Literacies and Key Skills

Career readiness, life literacies, and key skills education provides students with the necessary skills to make informed career and financial decisions, engage as

responsible community members in a digital society, and to successfully meet the challenges and opportunities in an interconnected global economy.

• Credit and Debt Management (9.1.12.CDM.8)

Compare and compute interest and compound interest and develop an amortization table using business tools.

Example: Students will use their growing understanding of exponentials functions to compare interest rates and make informed decisions about the

merits and pitfalls of different types of debt.

• Career Awareness and Planning (9.2.12.CAP.10)

Identify strategies for reducing overall costs of postsecondary education (e.g. tuition assistance, loans, grants, scholarships, and student loans.

Example: Students will compare the immediate and long term costs and benefits of different options available to pay for college and career education

after high school. Students can develop a suggested plan for hypothetical students based on their qualifications and financial circumstances.

• Technology Literacy (9.4.12.TL.1)

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

Example: Students can use and explain the advantages of using graphing calculators, GeoGebra or Desmos to graph systems of inequalities

to solve contextual problems. The discussion should also involve developing ways to improve and develop even better algorithms.

https://www.desmos.com/calculator https://www.geogebra.org/geometry?lang=en-US

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

https://www.desmos.com/calculator

https://www.geogebra.org/geometry?lang=en-US

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





Culturally Relevant Pedagogy Examples

• 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. • 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, visual

cues, graphic representations, gestures, pictures, practice and cognates. Model to students that some vocabulary has multiple

meanings. Have students create the Word Wall with their definitions and examples to foster ownership.

• Establish Inclusion: Highlight how the topic may relate or apply to students.

Example: After a brief explanation of slope, have students come up with examples of slope at home, in their neighborhood and

outside of their neighborhood. After having a volunteer list a few in each category, use the examples in class with the students.

Establishing inclusion also involves regularly grouping students with different classmates to share unique perspectives.

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

cultures.

Example: When learning different types of 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.

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

Example: Students can learn about different function types 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.



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

require them to identify variables, write

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

questions based on their understanding to ask the

teacher. This well help them to gain confidence

in working through the problems.

Set up small-group discussions that allows

students to reflect and discuss challenges or how

they solved a problem they faced. Students can

discuss their process and challenges when

writing linear and exponential functions given a

graph, table of values, or written description.

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 how to give and

receive feedback on errors/flaws in their

reasoning. As a result of these self-management

efforts, the class is able to engage in productive

and positive 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 distinguish between and

explain situations modeled with linear functions

or exponential functions.

Use real-world application problems to lead a

discussion about taking different approaches to

solving a problem and respecting the feelings

and thoughts of those that used a different

strategy.

Self-Awareness

Self-Management

Social-Awareness

✓ Relationship Skills


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 attempting to model with linear

functions, linear systems and exponential

functions, have student discuss their approach to

the problem in small groups. In doing so, they

demonstrate good relationship skills by listening

and responding to each other.

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.


Self-Awareness

Self-Management

Social-Awareness

Relationship Skills

✓ Responsible Decision-Making

Example practices that address Responsible

Decision-Making:

• Support collaborative decision making for

academics and behavior

• Foster student-centered discipline

• Assist students in step-by-step conflict resolution

process

• Foster student independence

• Model fair and appropriate decision making

• Teach good citizenship

Use a lesson to teach students a simple formula

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

identify the choice to be made, consider the

options, make a choice and do it, how did it go?)

Post the decision-making formula in the

classroom.

Routinely encourage students to use the

decision-making formula as they face a choice

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

a friend). Support students through the steps of

making a decision anytime they face a choice or

decision. Simple choices like “Which tool

should I use to explain the relationship between

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

for this problem?” are good places to start.


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


Differentiated Instruction

Accommodate Based on Content Needs: Strategies

• Anchor charts to model strategies and process

• Reference sheets that list formulas, step-by-step procedures and model strategies

• Conceptual word wall that contain definitions, translations, pictures and/or examples

• Graphic organizer to help students solve quadratic equations using different methods (such as quadratic formula, completing the

square, factoring, etc.)

• Translation dictionary

• Sentence stems to provide additional language support for ELL students

• Teacher modeling

• Highlight and label solution steps for multi-step problems in different colors

• Create an interactive notebook with students with a table of contents so they can refer to previously taught material readily

• Targeted assistance for students when summarizing and interpreting two-way frequency tables by using real world examples

• Graph paper

• Step by step directions on how to use a graphing calculator to fit functions to data and plot residuals

• Visual, verbal and algebraic models of quadratic functions

• A chart noting key features of functions from graphs and tables

• Videos to reinforce skills and thinking behind concepts

• Use real world data sets to facilitate students’ ability to compare center and spread of two sets of data


Interdisciplinary Connections

Model interdisciplinary thinking to expose students to other disciplines.

Social Studies Connection:

Sieves of Eratosthenes (6.2.8.HistoryCA.3.b)

• Students will learn to identify prime numbers using the Sieve of Eratosthenes. They will also study how the sieve was discovered and

about Eratosthenes, the Greek mathematician who was responsible for the sieve. See more about the incredible things that Eratosthenes

did at: http://encyclopedia.kids.net.au/page/er/Eratosthenes

Science Connection:

Cicadas Brood X (MS-LS1-4)

• Students will review and learn about cicadas found in North America that emerge from the ground every 17 years. These cicadas are

called Magicicada Septendecim. They will discuss the life cycle of an insect and the predators that an insect has. Learn more information

about cicadas at: http://bugfacts.net/cicada.php

Earth Day Project (MS-ESS3-3)

• Students will learn about recycling and Earth Day. They will discuss different ways to recycle and activities that can be done for Earth

Day. Learn more information about recycling and Earth Day at: http://www.earthday.org/ or choose a video to watch at

http://www.bing.com/videos/search?q=earth+day&qpvt=earth+day&FORM=VDRE

http://encyclopedia.kids.net.au/page/er/Eratosthenes

http://bugfacts.net/cicada.php

http://www.earthday.org/

http://www.bing.com/videos/search?q=earth+day&qpvt=earth+day&FORM=VDRE


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


Assessments

Required District/State Assessments

District Assessments

NJSLA

SGO Assessments

Suggested Formative/Summative Classroom Assessments

Describe Learning Vertically

Identify Key Building Blocks

Make Connections (between and among key building blocks)

Short/Extended Constructed Response Items

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

Drag and Drop Items

Use of Equation Editor

Quizzes

Journal Entries/Reflections/Quick-Writes

Accountable talk

Projects

Portfolio

Observation

Graphic Organizers/ Concept Mapping

Presentations

Role Playing

Teacher-Student and Student-Student Conferencing

Homework


New Jersey State Learning Standards

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 Recognize situations in which one

quantity changes at a constant rate per unit interval relative to another.

F.LE.A.1b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

F.LE.A.2: Construct linear and exponential functions - including arithmetic and geometric sequences - given a graph, a description of a

relationship, or two input-output pairs (include reading these from a table). *[Algebra 1 limitation: exponential expressions with integer

exponents]

F.LE.B.5: Interpret the parameters in a linear or exponential function in terms of a context.

S.ID.B.6a: Fit a function to the data (including the use of technology); use functions fitted to data to solve problems in the context of the data.

Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

S.ID.B.6b: Informally assess the fit of a function by plotting and analyzing residuals, including with the use of technology.

S.ID.B.6c: Fit a linear function for a scatter plot that suggests a linear association.

S.ID.C.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.C.8: Compute (using technology) and interpret the correlation coefficient of a linear fit.

S.ID.C.9: Distinguish between correlation and causation.

F.IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the

Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.



F.BF.A.2: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the


A.REI.D.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming

a curve (which could be a line). [Focus on linear equations.]

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

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.REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two

variables.

A.REI.C.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the

other produces a system with the same solutions.

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.

A.REI.D.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality),

and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

N.RN.A.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those

values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want

(51/3) 3 = 5(1/3) 3 to hold, so (51/3) 3 must equal 5

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



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

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

function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. *[Focus

on exponential functions]

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

midline, and amplitude.

F.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal

descriptions).

For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum

F.BF.B.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative);

find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include

recognizing even and odd functions from their graphs and algebraic expressions for them.

F.LE.A.1a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over

equal intervals.

A.SSE.B.3c: Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be

rewritten as (1.151/12) 12t ≈1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

F.IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in

functions such as y = (1.02)t , y = (0.97)t , y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

F.LE.A.1c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.


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.


Course: Algebra I Unit: 2 (Two) Topic: Writing Linear Functions, Linear

Systems, & Exponential Functions and

Sequences

NJSLS:

A.CED.A.2, F.BF.A.1a, F.LE.A.1b, F.LE.A.2, F.LE.B.5, S.ID.B.6a, S.ID.B.6b, S.ID.B.6c, S.ID.C.7, S.ID.C.8,

S.ID.C.9, F.IF.A.3, B.BF.A.2, A.REI.D.10, F.IF.C.7b, A.CED.A.3, A.REI.C.6, A.REI.C.5, A.REI.D.11,

A.REI.D.12, N.RN.A.1, N.RN.A.2, F.IF.B.4, F.IF.C.7e, F.IF.C.9, F.BF.B.3, F.LE.A.1a, A.SSE.B.3c, F.IF.C.8b,

F.LE.A.1c

Unit Focus:

• Writing linear equations

• Scatter plots and lines of fit

• Arithmetic sequences

• Piecewise functions

• Solving systems of linear equations and inequalities

• Properties of exponents, radicals and rational exponents

• Exponential functions and exponential growth and decay


New Jersey Student Learning Standard(s):


labels and scales.



F.LE.A.1b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

F.LE.A.2: Construct linear and exponential functions - including arithmetic and geometric sequences - given a graph, a description of a

relationship, or two input-output pairs (include reading these from a table). *[Algebra 1 limitation: exponential expressions with integer exponents]

Student Learning Objective 1: Writing equations in slope-intercept form.

Student Learning Objective 2: Writing equations in point-slope form.

Student Learning Objective 3: Writing equations of parallel and perpendicular lines.

Modified Student Learning Objectives/Standards:

M.EE.F-BF.1: Select the appropriate graphical representation (first quadrant) given a situation involving constant rate of change.

M.EE.A-CED.2–4: Solve one-step inequalities.

M.EE.F-LE.1–3: Model a simple linear function such as y = mx to show that these functions increase by equal amounts over equal intervals.

MPs Evidence Statement Key/

Clarifications

Skills, Strategies & Concepts Essential

Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 2

MP 3

MP 4

F-LE.2-1

• Construct linear and

exponential functions,

Write equations in slope-intercept form.

Use linear equations to solve real-life problems.

Given the graph of a

linear function, how can

you write an equation of

the line?

Real Life STEM

Video Task: Future

Wind Power


including arithmetic and

geometric sequences,

given a graph, a

description of a

relationship, or two input-

output pairs (include

reading these from a

table). i) Tasks are limited

to constructing linear and

exponential functions

with domains in the

integers, in simple real-

world context (not multi-

step).

F-LE.2-2

• Solve multi-step

contextual problems with

degree of difficulty

appropriate to the course

by constructing linear

and/or exponential

function models, where

exponentials are limited to

integer exponents.★ i)

Prompts describe a

scenario using everyday

language. Mathematical

language such as

"function," "exponential,"

etc. is not used. ii)

Students autonomously

choose and apply

Write an equation of a line when you are given its

slope and a point on the line.

Write an equation of a line given two points on the

line.

Use linear equations to solve real-life problems.

Identify and write equations of parallel lines.

Identify and write equations of perpendicular lines.

Use parallel and perpendicular lines in real life

problems.

SPED Strategies:

Create a Google Doc/Anchor Chart/Notes that

illustrates the similarities and differences of point-

slope form and slope –intercept form using real life

examples.

Provide visual examples of equations of parallel and

perpendicular lines and ample time to practice the

thinking and algorithms associated.

Use assessing and advancing questions to help

students verbalize and move their level of thinking

and understanding to a higher level.

How can you write an

equation of a line when

you are given the slope

and a point on the line?

How can you recognize

lines that are parallel or

perpendicular?

Interactive

Explorations:

• Writing

Equations in

Slope-Intercept

Form

• Mathematical

Modeling

• Writing

Equations of

Lines

• Writing a

Formula

• Writing an

Equation

• Recognizing

Parallel Lines

• Recognizing

Perpendicular

Lines


appropriate mathematical

techniques without

prompting. For example,

in a situation of doubling,

they apply techniques of

exponential functions. iii)

For some illustrations, see

tasks at

http://illustrativemathemat

ics.org under F-LE.

ELL Support:

Teacher provides the necessary support (linguistic

and conceptual) so that students can work

independently on problems involving point-slope

form and slope–intercept form using real life

examples.

Strategies include reframing questions, filling in

background knowledge gaps and use of native

language.

Give students notes that include expectations,

common misconceptions and vocabulary (English

and native language) relevant to systems of

equations. This will facilitate independence and

increased proficiency.



F.LE.B.5: Interpret the parameters in a linear or exponential function in terms of a context.

S.ID.B.6a: Fit a function to the data (including the use of technology); use functions fitted to data to solve problems in the context of the data.

Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

S.ID.B.6b: Informally assess the fit of a function by plotting and analyzing residuals, including with the use of technology.

S.ID.B.6c: Fit a linear function for a scatter plot that suggests a linear association.

S.ID.C.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.C.8: Compute (using technology) and interpret the correlation coefficient of a linear fit.

S.ID.C.9: Distinguish between correlation and causation.

Student Learning Objective 4: Scatter plots and lines of fit.

Student Learning Objective 5: Analyzing lines of fit.

Modified Student Learning Objectives/Standards: N/A

MPs Evidence Statement Key/ Clarifications Skills, Strategies & Concepts Essential

Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 2

MP 4

MP 5

MP 6

S-ID.Int.1

• Solve multi-step contextual word problems

with degree of difficulty appropriate to the

course, requiring application of course-

level knowledge and skills articulated in S-

ID, excluding normal distributions and

limiting function fitting to linear functions

and exponential functions with domains in

the integers.

Understand the use of scatter plots

in real life situations.

Create and interpret scatter plots.

Identify correlations between data

sets.

How can you use a

scatter plot and a line of

fit to make conclusions

about data?

How can you

analytically find a line

of best fit for a scatter

plot?

Interactive

Explorations:

• Finding a Line

of Fit

• Finding a Line

of Best Fit


S-ID.Int.2

• Solve multi-step contextual word problems

with degree of difficulty appropriate to the

course, requiring application of course-

level knowledge and skills articulated in S-

ID, excluding normal distributions and

limiting function fitting to quadratic, linear

and exponential (with domains in the

integers) functions with an emphasis on

quadratic functions.

Use lines of fit to model data.

Use residuals to determine how

well lines of fit model data.

Use technology to find lines of best

fit.

Distinguish between correlation and

causation.

Solve problems using functions

fitted to data (prediction equations).

Interpret the intercepts of models in

context.

Plot residuals of linear functions.

Analyze residuals in order to

informally evaluate the fit of linear

functions.

Describe the form, strength and

direction of the relationship.

Use algebraic methods and

technology to fit a linear function to

the data.

Use the function to predict values.

How can using

technology to fit a

function to data help

students learn more

about functions?

How can the context of

a problem be used to

interpret the intercepts

of models?

What do students need

to know to accurately

plot the residuals of

linear and non-linear

functions?

What can students

determine from

analyzing residuals to

evaluate the fit of linear

and non-linear

functions?

Why would you want to

identify trends or

associations in a data

set?

Why would you want to

informally assess and


Explain the meaning of the slope

and y-intercept in context.

SPED Strategies:

Model the thinking process and

action steps involved in fitting

functions to data, plotting residuals

and assessing fit. Create a resource

by documenting these items in a

Google Doc/Anchor Chart/Graphic

Organizer.

Use contextualized data to illustrate

the essential concepts to increase

connections to prior learning and

likelihood of increased proficiency.

ELL Support:

Using Desmos or graphing

calculators, model how to fit

functions to a data set by explaining

the thinking and processes involved

at each step.

Create notes with students that

highlight the learning that has taken

place and becomes a reference for

later use. It should include steps,

identify a type of

function to fit a data

set?


thinking, key terms/concepts and

common misconceptions.

Teachers can increase

understanding and proficiency by

asking assessing and advancing

questions as students work with

their peers on conceptually based

problems requiring the fitting of

functions to data and plotting

residuals.


F.IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the


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

F.BF.A.2: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the


F.LE.A.2: Construct linear and exponential functions - including arithmetic and geometric sequences - given a graph, a description of a relationship,

or two input-output pairs (include reading these from a table). *[Algebra 1 limitation: exponential expressions with integer exponents]

Student Learning Objective 6: Arithmetic sequences.


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



M.EE.F-BF.2: Determine an arithmetic sequence with whole numbers when provided a recursive rule.



Clarifications


Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 6

MP 7

F-LE.2-1

• Construct linear and



geometric sequences, given

a graph, a description of a


output pairs (include

reading these from a table).

F-LE.2-2

• Solve multi-step contextual

problems with degree of

difficulty appropriate to the

course by constructing

linear and/or exponential

function models, where


integer exponents.★

HS.D.2-8

Write the terms of arithmetic sequences.

Graph arithmetic sequences.

Write arithmetic sequences as functions.

Relate the concept of arithmetic functions to real-

life situations.

SPED Strategies:

Review the notes and resources given regarding

linear and exponential functions.

Link the task of writing the arithmetic sequences as

functions given the verbal, graphical or pictorial

depiction of the information to prior learning and

review.

Create Google Doc/Anchor Chart/Notes that

verbally and pictorially describe the characteristics

of arithmetic sequences using concrete examples.

How can you use an

arithmetic sequence to

describe a pattern?

How can functions be

used to find solutions to

real-word problems and

predict outcomes?

Interactive

Explorations:

• Describing a

Pattern



word problems with degree

of difficulty appropriate to

the course, requiring

application of course-level

knowledge and skills

articulated in F-BF.1a, F-

BF.3, ACED.1, A-SSE.3,

F-IF.B, F-IF.7, limited to

linear functions and

exponential functions with

domains in the integers.

HS.D.2-8 S


word problems with degree

of difficulty appropriate to

the course, requiring

application of course-level

knowledge and skills

articulated in F-BF.1a, F-

BF.3, ACED.1, A-SSE.3,

F-IF.B, F-IF.7, limited to

linear functions and

exponential functions with

domains in the integers

ELL Support:

Create an anchor chart/notes/graphic organizer that

verbally and pictorially describes the characteristics

of arithmetic sequences using concrete examples.

Ensure that students are provided with the

necessary language support to access these

concepts.

Provide students with the necessary support to

write simpler functions based on a context and

relate the required skills to the task of writing

arithmetic sequences.




labels and scales.

A.REI.D.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a

curve (which could be a line). [Focus on linear equations.]

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

Student Learning Objective 7: Piecewise 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.

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


Clarifications


Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 6

MP 7

A-REI.10

• Understand that the graph

of an equation in two

variables is the set of all its

solutions plotted in the

coordinate plane, often

forming a curve (which

could be a line).

Evaluate piecewise functions.

Graph and write piecewise functions.

Describe piecewise functions in terms of the real-

life circumstances.

Graph and write step functions.

How can you describe a

function that is

represented by more

than one equation?

How can piecewise

functions be used in

real-life applications?

Interactive

Explorations:

• Writing

Equations for a

Function

Performance Task:

Any Beginning


Describe piecewise functions in terms of the real-

life circumstances.

Write absolute value functions.

SPED Strategies:

Create examples of piecewise, step and absolute

value functions so that students have a reference to

use when working with and discussing the

similarities and differences of the functions.

Develop questions that encourage students to think

through their understanding of piecewise, step and

absolute value functions

ELL Support:

Provide students with notes that illustrate the

essential characteristics of piecewise, step and

absolute value functions.

Provide students with examples and non-examples

of piecewise, step and absolute value functions.

Simplify linguistic complexity, use native language

and provide real life examples to increase

likelihood of conceptual understanding., i.e. native

language explanations, word-to-word dictionary,

simplified linguistic complexity, assessing and

advancing questions.



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.REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A.REI.C.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the

other produces a system with the same solutions.

Student Learning Objective 8: Solve systems of linear equations by graphing.

Student Learning Objective 9: Solve systems of linear equations by substitution.

Student Learning Objective 10: Solve systems of linear equations by elimination.

Student Learning Objective 11: Solve special systems of linear equations.




Clarifications

Skills, Strategies & Concepts Essential Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 3

MP 6

A-CED.3-1


contextual problems that

require writing and

Understand and describe the meaning of a

system of equations in context.

Check solutions of systems of linear equations.

How can you solve a system

of linear equations?

Interactive

Explorations:

• Writing a System of

Linear Equations


analyzing systems of

linear inequalities in two

variables to find viable

solutions.

i) Tasks have hallmarks

of modeling as a

mathematical practice

(less defined tasks,

more of the modeling

cycle, etc.).

ii) Scaffolding in tasks

may range from

substantial to very little

or none.

HS.C.5.10-1

• Given an equation or

system of equations,

reason about the number

or nature of the

solutions.

Solve systems of linear equations by graphing.

Solve systems of linear equations by

substitution.

Solve systems of linear equations by

elimination.

Discuss the thinking process behind the choice

of method when solving a system of linear

equations.

Determine the number of solutions of linear

systems.

Use systems of linear equations to solve real-

life problems.

SPED Strategies:

Introduce the topic in a contextual way so that

students can visualize how the graphing of a

system of a linear equations would apply to real

life.

Have students graph the functions in different

colors to enable students to see the intersection more

clearly.

How can you use substitution

to solve a system of linear

equations?

How can you use elimination

to solve a system of linear

equations?

How do you determine

which method to use when

solving a system of

equations?

Can a system of equations

have no solution or infinitely

many solutions?

• Using a Table or

Graph to Solve a

System

• Using Substitution

to Solve Systems

• Writing and

Solving a System of

Equations

• Using Elimination

to Solve Systems

• Using Elimination

to Solve a System

• Using a Table to

Solve a System

• Writing and

Analyzing a System


Use graphing calculators to enable students to

practice the skills needed, to see the

intersections clearly and to check their work.

Create a Google Doc that lists and describes the

different types of solution methods.

ELL Support:

Use a contextual example of linear systems

highlighting where they intersect to ground

student understanding of this new concept.

Ensure that new vocabulary is explained

thoroughly using visual cues, native language

and by modifying the linguistic complexity of

explanations.

Create a Google Doc that lists and describes the




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.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.

A.REI.D.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and

graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes

Student Learning Objective 12: Solving equations by graphing.

Student Learning Objective 13: Graphing linear inequalities in two variables.

Student Learning Objective 14: Systems of linear inequalities.



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 2

MP 4

MP 1

MP 5

MP 6

MP 7

A-CED.3-1


contextual problems that

require writing and

analyzing systems of

linear inequalities in two

variables to find viable

solutions.

iii) Tasks have hallmarks

of modeling as a

mathematical practice

(less defined tasks,

more of the modeling

cycle, etc.).

iv) Scaffolding in tasks

may range from

substantial to very little

or none.

A-REI.11-1

• The "explain" part of

standard A-REI.11 is

not assessed here. For

this aspect of the

standard, see Sub-

Claim C.

A-REI.12

Make the connection that graphing systems is

another way to solve in addition to algebraic

methods.

Discuss the idea of why graphing systems may be

advantageous in a context.

Solve linear equations by graphing.

Solve absolute value equations by graphing.

Use graphing to solve linear systems in real-life

situations.

Examine graphs of linear inequalities and discuss

the key features and how they relate to the context

of the problem.

Check the solutions of linear inequalities.

Graph linear inequalities in two variables, verify

accuracy and explain it.

Use linear inequalities and systems of inequalities

to solve real-life problems.

Examine graphs of systems of linear inequalities

and discuss the key features and how they relate to

the context of the problem.

How can you use a system

of linear equations to

solve an equation with

variables on both sides?

How is the process of

solving a system of

absolute value equations

by graphing similar to

and/or different from

solving a system of linear

equations?

When would you use

graphing as a method of

solving systems?

How can you graph a

linear inequality in two

variables?

How can you graph a system

of linear inequalities?

Real Life STEM Video

Task: Setting Fishery

Limits

Performance Task: Prize

Patrol

Interactive Explorations:

• Solving an Equation

by Graphing

• Solving Equations

Algebraically and

Graphically

• Writing a Linear

Inequality in One

Variables

• Writing a Linear

Inequality in Two

Variables

• Graphing Linear

Inequalities

• Graphing a System

of Linear Inequalities


• Graph the solutions to a

linear inequality in two

variables as a half-plane

(excluding the

boundary in the case of

a strict inequality), and

graph the solution set to

a system of linear

inequalities in two

variables as the

intersection of the

corresponding half-

planes.

Graph systems of linear inequalities, verify

accuracy and explain it.

Write systems of inequalities.

SPED Strategies:

Review the Google Doc notes and resources on

systems of equations and add graphing as another

way of solving systems.

Explicitly describe situations where graphing

would be the preferred method. Have students

share their thinking and connect it to a real life

situation.

Create a model for students to visualize the graph

of a system of inequalities. Provide memorable

connections that help students remember how to

interpret the graph and know when it is accurate.

ELL Support:

Use a contextual example of graphing linear

systems and absolute value highlighting where

they intersect to ground student understanding of

this new concept.

Create a model for students to visualize the graph

of a system of inequalities. Provide memorable

connections that help students remember how to

interpret the graph and know when it is accurate.


Ensure that new vocabulary is explained

thoroughly using visual cues, native language and

by modifying the linguistic complexity of

explanations.

Add graphing systems as another solution method

to the Google Doc that lists and describes the


AdNew Jersey Student Learning Standard(s):

N.RN.A.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those

values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want

(51/3) 3 = 5(1/3) 3 to hold, so (51/3) 3 must equal 5

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

Student Learning Objective 15: Properties of exponents.

Student Learning Objective 16: Radicals and rational exponents.


M. EE.N-RN.1: Determine the value of a quantity that is squared or cubed.


Clarifications


Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 2

MP 4

N/A

Review the properties of exponents and connect

them to real life examples to help students internalize

How can you write

general rules involving

properties of exponents?

Interactive

Explorations:


and apply the properties in problem solving

situations.

Use zero and negative exponents to evaluate

expressions.

Use the properties of exponents to simplify

expressions.

Draw a connection between the properties of

exponents and finding the nth root.

Evaluate expressions with rational exponents and

explain the rationale for the answer in terms of the

properties of exponents.

functions in context.

Identify the different parts of the expression and

explain their meaning within the context of a

problem.

Decompose expressions and make sense of the

multiple factors and terms by explaining the meaning

of the individual parts.

Rewrite algebraic expressions in different equivalent

forms such as factoring or combining like terms.

The properties of

exponents can be used to

develop equivalent forms

of the same expression

How can you

Write and evaluate the

nth root of a number?

• Writing Rules for

the Properties of

Exponents

• Finding Cube

Roots

• Estimating nth

Roots


SPED Strategies:

Connect properties of exponents to standards

8.EE.A.1 and 8.EE.A.2 from Grade 8 to review and

reinforce previously taught concepts.

Review the meaning of the parts of an expression by

linking it to a real-life context of high interest.

Create a Google Doc with students that highlights

the properties of exponents and provide ample time

for students to use the document when completing

problems to build confidence.

ELL Support:

Create a Google Doc anchor chart with students that

illustrates the properties of exponents. The

document should have visual and linguistic supports.

Use think-alouds as a support that allows students to

discuss how they are processing information.

Provide students with ample opportunity to practice

using the properties of exponents and to verbalize

their thinking.




labels and scales.

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

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

is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. *[Focus on

exponential functions]

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

midline, and amplitude.

F.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal

descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum



F.LE.A.1a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over

equal intervals.

F.LE.A.2: Construct linear and exponential functions - including arithmetic and geometric sequences - given a graph, a description of a relationship,

or two input-output pairs (include reading these from a table). *[Algebra 1 limitation: exponential expressions with integer exponents]

F.IF.C.8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in

functions such as y = (1.02)t , y = (0.97)t , y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

F.LE.A.1c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

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.


A.SSE.B.3c: Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten

as (1.151/12) 12t ≈1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Student Learning Objective 17: Exponential functions.

Student Learning Objective 18: Exponential growth and decay.


M.EE.A-SSE.3. Solve simple algebraic equations with one variable using multiplication and division.





Clarifications


Understandings/

Questions

(Accountable Talk)

Tasks/Activities

MP 1

MP 2

MP 4

MP 7

A-SSE.3c-1

• Choose and produce an

equivalent form of an

expression to reveal and

explain properties of the

quantity represented by the

expression, where


integer exponents.

• Use the properties of

exponents to transform

expressions for

exponential functions.

Identify and evaluate exponential functions.

Graph exponential functions and describe the key

features in a real life problem.

Compare exponential functions and draw

conclusions.

Use exponential functions to model real life

exponential growth and decay.

Use properties of exponents (such as power of a

power, product of powers, power of a product, and

rational exponents, etc.) to write an equivalent form

of an exponential function to reveal and explain

Discuss some of the key

features of an

exponential graph in the

context of the problem.

What are some of the

characteristics of

exponential growth and

exponential decay?

How do exponential

growth and exponential

decay compare?

Interactive

Explorations:

• Exploring an

Exponential

Function

• Predicting a

Future Event

• Describing a

Decay Pattern


F-LE.2-1

Construct linear and



geometric sequences, given a

graph, a description of a


output pairs (include reading

these from a table).

F-LE.2-2

Solve multi-step contextual

problems with degree of

difficulty appropriate to the

course by constructing linear

and/or exponential function

models, where exponentials

are limited to integer

exponents.★

specific information about its approximate rate of

growth or decay.

SPED Strategies:

Review the properties of exponents and provide clear

examples to remind students of this prior learning.

Develop mnemonic devices with students to help

them remember the properties of exponents and the

circumstances under which the rules apply.

Create a Google Doc/Anchor Chart/Notes recapping

the properties of exponents with students for their

reference when problem solving.

Provide students with opportunities to relate the

mathematical depiction of exponential functions to

real life scenarios that can be modeled as exponential

functions.

ELL Support:

Create an anchor chart/graphic organizer that lists

the properties of exponents and the charateristics of

exponential functions with a clear example for each

using native language or simpler English

terminology to engage all students.

Provide students with opportunities to relate the

mathematical depiction of exponential functions to

How can you

differentiate between

exponential growth and

exponential decay?

IFL “Solving

Problems Using

Linear and

Exponential

Models.” *Also

addressed in IFL

unit is F.LE.B.5.

(F.IF.A.3 is not

addressed in IFL

Unit


real life scenarios that can be modeled as exponential

functions.


Honors Projects (Choose a minimum of 3)

Project 1 Project 2 Project 3 Project 4

Wearing Through the Pipes

Essential Question:

What are the characteristics of a problem

that determine which method of solving

is most efficient?

Skills:

1. Identify and define variables

representing essential features of the

model.

2. Model real world situations by

creating a system of linear equations.

3. Interpret the solution(s) in context.

4. Systems of equations can be solved

exactly (algebraically) and approximately

(graphically).

Growing by Leaps and

Bounds

Essential Question:

How do I interpret exponential

functions in context?

Skills:

1. Graphing equations on

coordinate axes

with labels and scales.

2. Interpreting expressions that

represent a quantity in terms of

its context.

3. Determining constraints.

4. Modeling with exponential

functions.

5. Interpreting functions and its

key features.

6. Analyzing sequences as

functions.

Paula’s Peaches

Essential Questions:

How do we use quadratic

functions to represent

contextual situations?

How do we solve quadratic

equations?

How do we interpret quadratic

functions in context?

Skills:

1. Factoring

2. Solving quadratic equations.

3. Interpreting functions in

context.

Table Tiles

Essential Questions:

1. How do you generalize using

numerical, geometrical or

algebraic structure?

2. How do you describe and

explain findings clearly and

effectively?

Skills:

Building a function to represent

patterns algebraically.


Integrated Evidence Statements

F-IF.A.Int.1: Understand the concept of a function and use function notation.

• Tasks require students to use function notation, evaluate functions for inputs in their domains, and interpret statements that use function

notation in terms of a real-world context • About a quarter of tasks involve functions defined recursively on a domain in the integers.

F-Int.1-1: Given a verbal description of a linear or quadratic functional dependence, write an expression for the function and demonstrate various

knowledge and skills articulated in the Functions category in relation to this function.

• Given a verbal description of a functional dependence, the student would be required to write an expression for the function and then, e.g.,

identify a natural domain for the function given the situation; use a graphing tool to graph several input-output pairs; select applicable features

of the function, such as linear, increasing, decreasing, quadratic, nonlinear; and find an input value leading to a given output value. o e.g., a functional dependence might be described as follows: "The area of a square is a function of the length of its diagonal." The

student would be asked to create an expression such as f(x) = (1/2) x 2 for this function. The natural domain for the function would be

the positive real numbers. The function is increasing and nonlinear. And so on. o e.g., a functional dependence might be described as follows: "The slope of the line passing through the points (1, 3) and (7, y) is a

function of y." The student would be asked to create an expression such as s(y) = (3- y)/(1-7) for this function. The natural domain for

this function would be the real numbers. The function is increasing and linear. And so on.

HS-Int.2: Solve multi-step mathematical problems with degree of difficulty appropriate to the course that requires analyzing quadratic functions

and/or writing and solving quadratic equations.

• Tasks do not have a real-world context. • Exact answers may be required or decimal approximations may be given. Students might choose to take advantage of the graphing utility to

find approximate answers or clarify the situation at hand. For rational solutions, exact values are required. For irrational solutions, exact or

decimal approximations may be required. Simplifying or rewriting radicals is not required. Some examples: Given the function f(x) = 𝑥 2 + x, find all values of k such that f(3 - k) = f(3). (Exact answers are required.)


Integrated Evidence Statements

Find a value of c so that the equation 2𝑥 2 - cx + 1 = 0 has a double root. Give an answer accurate to the tenths place.

HS.C.12.1: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures about functions. Content

scope: F-IF.8a

• Tasks involve using algebra to prove properties of given functions. For example, prove algebraically that the function h(t) = t(t-1) has

minimum value 14 ; prove algebraically that the graph of g(x) = x2 - x + 14 is symmetric about the line x = 12 ; prove that x2+ 1 is never less

than -2x.

• Scaffolding is provided to ensure tasks have appropriate level of difficulty. (For example, the prompt could show the graphs of x2+1 and -2x

on the same set of axes, and say, "From the graph, it looks as if x2+ 1 is never less than -2x. In this task, you will use algebra to prove it." And

so on, perhaps with additional hints or scaffolding.)

• Tasks may have a mathematical or real-world context.

HS.D.2-6: 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.12, A-REI.11-1, limited to linear and quadratic equations

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


level knowledge and skills articulated in F-BF.1a, F-BF.3, A-CED.1, A-SSE.3, F-IF.B, F-IF.7, limited to linear functions and exponential

functions with domains in the integers.

• F-BF.1a is the primary content; other listed content elements may be involved in tasks as well.


level knowledge and skills articulated in F-BF.1a, F-BF.3, A-CED.1, A-SSE.3, F-IF.B, F-IF.7, limited to linear and quadratic functions.

• F-BF.1a is the primary content; other listed content elements may be involved in tasks as well.


Unit 2 Vocabulary

• Algorithm

• Area Model

• Arithmetic Sequence

• Array

• Associative Property of Multiplication

• Causation

• Common Denominator

• Common Difference

• Common Factor

• Common Multiple

• Common Ratio

• Commutative Property of

Multiplication

• Compare

• Compatible Numbers

• Composite Number

• Compound Interest

• Correlation

• Correlation Coefficient

• Decompose

• Denominator

• Distributive Property

• Divisibility Rules

• Divisor

• Dividend

• Decompose

• Denominator

• Distributive Property

• Divisibility Rules

• Divisor

• Dividend

• Elimination

• Equation

• Equivalent Fractions

• Estimate

• Expanded Form

• Explicit Rule

• Exponential Decay

• Exponential Equations

• Exponential Function

• Exponential Growth

• Exponential Growth Function

• Expression

• Extrapolation

• Factoring

• Factors

• Factor Pairs

• Geometric Sequence

• Graphing

• Graph of a Linear Inequality

• Graph of a System of Linear

Inequalities

• Greatest Common Factor

• Half-planes

• Identity Property of Multiplication

• Index

• Interpolation


• Interpret

• Linear Inequality in Two Variables

• Linear Model

• Linear Regression

• Line of Best Fit

• Line of Fit

• Mixed Number

• Model/ Visual Model

• Multiples Multiplicative Identity

Property of 1

• Negative Exponents

• nth Root of a

• Numerator

• Parallel Lines

• Partial Product

• Partial Quotient

• Piecewise function

• Place Value

• Perpendicular Lines

• Exponential Decay Function

• Point-Slope Form

• Power of a Power Property

• Power of a Product Property

• Prime Number

• Product

• Product of Powers Property

• Property of Equality for Exponential

Equations

• Quotient

• Quotient of Powers Property

• Radical

• Radical expression

• Rational Exponents

• Reasonableness

• Recursive Rule

• Related Facts

• Residual

• Remainder

• Scatter Plot

• Rule

• Inverse Operations

• Sequence

• Simplest Form

• Simplify

• Solution of a Linear Inequality in Two

Variables

• Solution of a System of Linear

Equations

• Solution of a System of Linear

Inequalities

• Step function

• Substitution

• System of Linear Equations

• System of Linear Inequalities

• Term

• Unit Fraction

• Unlike Denominators

• Unlike Numerators

• Variable

• Zero Exponent

• Zero Property of Multiplication


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

Achieve the Core http://achievethecore.org

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

Inside Mathematics www.insidemathematics.org

Learn Zillion https://learnzillion.com

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

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

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

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

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

Desmos https://www.desmos.com/

Geogebra http://www.geogebra.org/

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

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

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

https://www.khanacademy.org/

http://achievethecore.org/

https://www.illustrativemathematics.org/

http://www.insidemathematics.org/

https://learnzillion.com/

http://nlvm.usu.edu/en/nav/vlibrary.html

https://www.bigideasmath.com/

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

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

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

https://www.desmos.com/

http://www.geogebra.org/

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

https://parcc.pearson.com/

https://www.aleks.com/


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/

http://www.sixflags.com/

http://www.momath.org/

https://en.wikipedia.org/wiki/Interactive

https://en.wikipedia.org/wiki/Science_museum

https://en.wikipedia.org/wiki/Liberty_State_Park

https://en.wikipedia.org/wiki/IMAX_Dome

https://en.wikipedia.org/wiki/Hoberman_sphere

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

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