MATHEMATICS - Algebra II: Unit 1 Linear Functions, Quadratic Functions Quadratic Equations and Complex Numbers
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MATHEMATICS
Algebra II: Unit 1
Linear Functions, Quadratic Functions
Quadratic Equations and Complex Numbers
1|Page
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. 2|Page
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
3|PagePacing Chart – Unit 1
# Student Learning Objective NJSLS Big Ideas Math
Correlation
1‐1
F.BF.B.3 Instruction: 8 weeks
Parent Functions and Transformations. 1‐2
1,2 Assessment: 1 week
Transformations of Linear and Absolute Value Functions.
A-CED.A.2 1‐3
Modeling with Linear Functions.
F-IF.C.9
3 F-BF.A.1a
F-LE.A.2,
S-ID.B.6a
4 A-CED.A.3 1.4
Solving Linear Systems.
A-REI.C.6
5 F-IF.C.7c 2-1
Transformations of Quadratic Functions.
F-BF.B.3
Characteristics of Quadratic Functions. F-IF.B.4 2-2
6 F-IF.C.7c
F-IF.C.9
A-APR.B.3
F-IF.B.4, 2-3
Focus of a Parabola.
7 F-IF.C.7c
G-GPE.A.2
A-CED.A.2 2-4
Modeling with Quadratic Functions
8 F-IF.B.6,
F-BF.A.1a
S-ID.B.6a
9 A-SSE.A.2 3-1
Solving Quadratic Equations
A-REI.B.4b
4|PageF-IF.C.8a
Complex Numbers N-CN.A.1 3-2
10 N-CN.A.2
N-CN.C.7
A-REI.B.4b
Completing the Square N-CN.C.7 3-3
11 A-REI.B.4b
F-IF.C.8a
Using the Quadratic Formula N-CN.C.7 3-4
12 A-REI.B.4b
Solving Nonlinear Systems A-CED.A.3 3-5
13 A-REI.C.7
A-REI.D.11
Quadratic Inequalities A-CED.A.1 A- 3-6
14 CED.A.3
5|PageResearch 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
6|PageMathematics 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 a balanced mathematical
understanding is the focal point of developing mathematical proficiency. Students should frequently work on rigorous tasks, talk about the math,
explain their thinking, justify their answer or process, build models with graphs or charts or manipulatives, and use technology.
When balanced math is used in the classroom it provides students opportunities to:
solve problems
make connections between math concepts and real-life situations
communicate mathematical ideas (orally, visually and in writing)
choose appropriate materials to solve problems
reflect and monitor their own understanding of the math concepts
practice strategies to build procedural and conceptual confidence
Teacher builds conceptual understanding by Teacher and students practice mathematics
modeling through demonstration, explicit processes together through interactive
instruction, and think alouds, as well as guiding activities, problem solving, and discussion.
students as they practice math strategies and apply (whole group or small group instruction)
problem solving strategies. (whole group or small
group instruction)
Students practice math strategies independently to
build procedural and computational fluency. Teacher
assesses learning and reteaches as necessary. (whole
7|Page
group instruction, small group instruction, or centers)Effective Pedagogical Routines/Instructional Strategies
Collaborative Problem Solving Analyze Student Work
Connect Previous Knowledge to New Learning Identify Student’s Mathematical Understanding
Identify Student’s Mathematical Misunderstandings
Making Thinking Visible
Interviews
Develop and Demonstrate Mathematical Practices
Role Playing
Inquiry-Oriented and Exploratory Approach Diagrams, Charts, Tables, and Graphs
Multiple Solution Paths and Strategies Anticipate Likely and Possible Student Responses
Collect Different Student Approaches
Use of Multiple Representations
Multiple Response Strategies
Explain the Rationale of your Math Work
Asking Assessing and Advancing Questions
Quick Writes
Revoicing
Pair/Trio Sharing
Marking
Turn and Talk
Recapping
Charting
Challenging
Gallery Walks
Pressing for Accuracy and Reasoning
Small Group and Whole Class Discussions
Maintain the Cognitive Demand
Student Modeling
8|PageEducational 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 solving systems of equations.
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
solving simple systems of linear and quadratic equations in two variables.
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 add, subtract, and multiply complex numbers using the commutative,
associative and distributive properties.
Link: http://www.state.nj.us/education/cccs/2014/tech/
9|PageCareer 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.
10 | P a g eCareer Ready Practices
CRP8. Utilize critical thinking to make sense of problems and persevere in solving them.
Career-ready individuals readily recognize problems in the workplace, understand the nature of the problem, and devise effective plans to
solve the problem. They are aware of problems when they occur and take action quickly to address the problem; they thoughtfully
investigate the root cause of the problem prior to introducing solutions. They carefully consider the options to solve the problem. Once a
solution is agreed upon, they follow through to ensure the problem is solved, whether through their own actions or the actions of others.
Example: Students will understand the meaning of a problem and look for entry points to its solution. They will analyze information,
make conjectures, and plan a solution pathway to solve linear and quadratic equations in two variables.
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
11 | P a g eWIDA Proficiency Levels
At the given level of English language proficiency, English language learners will process, understand, produce or use
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
6‐ Reaching required by the specified grade level
Oral or written communication in English comparable to proficient English peers
Specialized or technical language of the content areas
A variety of sentence lengths of varying linguistic complexity in extended oral or written discourse,
5‐ Bridging including stories, essays or reports
Oral or written language approaching comparability to that of proficient English peers when presented with
grade level material.
Specific and some technical language of the content areas
A variety of sentence lengths of varying linguistic complexity in oral discourse or multiple, related
4‐ Expanding 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
General and some specific language of the content areas
Expanded sentences in oral interaction or written paragraphs
3‐ Developing 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
General language related to the content area
Phrases or short sentences
2‐ Beginning 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
12 | P a g e13 | P a g e
14 | P a g e
Culturally Relevant Pedagogy Examples
Integrate Relevant Word Problems: Contextualize equations using word problems that reference student interests and
cultures.
Example: When learning about building functions in two variables, 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 construct and compare linear, quadratic and exponential models 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.
15 | P a g e16 | P a g e
SEL Competency Examples Content Specific Activity & Approach
to SEL
Self-Awareness Example practices that address Self-Awareness: Encourage students to articulate their thoughts
Self-Management by restating the problem in their own words or
Social-Awareness • Clearly state classroom rules by describing to you what they know about the
Relationship Skills • Provide students with specific feedback regarding scenario and the question(s) being asked.
Responsible Decision-Making academics and behavior Acknowledge any student frustrations with the
• Offer different ways to demonstrate understanding task and remind them that frustration is normal
when working with a challenging task.
• Create opportunities for students to self-advocate
• Check for student understanding / feelings about Ask students to identify their own personal
performance interest, strengths, and weaknesses in math. For
• Check for emotional wellbeing example, have students evaluate their strengths
• Facilitate understanding of student strengths and and weaknesses when writing arithmetic and
challenges geometric sequences both recursively and with
an explicit formula.
Self-Awareness Example practices that address Self-Management: Have students brainstorm ways to motivate
Self-Management themselves and self-monitor. Use fraction bars
Social-Awareness • Encourage students to take pride/ownership in work and other linear diagrams to represent their
Relationship Skills and behavior learning targets (e.g., mood thermometers or
Responsible Decision-Making • Encourage students to reflect and adapt to classroom progress lines). Discuss your own self-
situations motivation techniques that keeps you going
when you want to give up.
• Assist students with being ready in the classroom
• Assist students with managing their own emotional Practicing how to make tough decisions can
state help your students learn how their actions affect
others. Give your students real-life application
problems in which they would have to make an
important choice. Have them write down their
answer to each situation by themselves, then
discuss their answers as a class.
17 | P a g eSelf-Awareness Example practices that address Social-Awareness: During the first week of school, work
Self-Management collaboratively with students to establish shared
Social-Awareness • Encourage students to reflect on the perspective of classroom rules and expectations and
Relationship Skills others consequences so that students can see the
Responsible Decision-Making • Assign appropriate groups impact of their own actions and behaviors on
• Help students to think about social strengths outcomes.
• Provide specific feedback on social skills Tell stories about famous mathematicians who
• Model positive social awareness through showed respect for each other within the
metacognition activities discipline (e.g., Blaise Pascal and Pierre de
Fermat) to demonstrate to students how to listen
or read the arguments of others, decide whether
they make sense, and ask useful questions to
clarify or improve their arguments.
Self-Awareness Example practices that address Relationship Skills: Use cooperative learning and project-based
Self-Management learning to provide students with frequent
Social-Awareness • Engage families and community members opportunities to develop and routinely practice
Relationship Skills • Model effective questioning and responding to communication and social skills. Ask students
Responsible Decision-Making students to explain their partner’s reasoning to you.
• Plan for project-based learning Frequent checking-in with students establishes
“perspective taking” as the classroom norm.
• Assist students with discovering individual strengths
• Model and promote respecting differences Implement the use of a group rubric for students
• Model and promote active listening to complete for group activities or projects.
• Help students develop communication skills Students should reflect after working together
• Demonstrate value for a diversity of opinions how well the group works together, follows the
lead of others, supports each person in the
group, provides structure, and supports ideas.
18 | P a g eSelf-Awareness Example practices that address Responsible Encourage students to justify their solution path
Self-Management Decision-Making: and to model the thinking and decision making
Social-Awareness that they used to arrive at their answers.
Relationship Skills • Support collaborative decision making for academics
Responsible Decision-Making Use rich math problems that demand group
and behavior
effort. Before proceeding, have students
• Foster student-centered discipline discuss norms for mathematical collaboration.
• Assist students in step-by-step conflict resolution Have them answer questions about what they
process like to do while working in groups. How can
• Foster student independence they act to make the group work well?
• Model fair and appropriate decision making
• Teach good citizenship
19 | P a g eDifferentiated Instruction
Accommodate Based on Students Individual Needs: Strategies
Time/General Processing Comprehension Recall
Extra time for assigned tasks Extra Response time Precise processes for balanced Teacher-made checklist
mathematics instructional
Adjust length of assignment Have students verbalize steps Use visual graphic organizers
model
Timeline with due dates for Repeat, clarify or reword Reference resources to
reports and projects Short manageable tasks
directions promote independence
Communication system Brief and concrete directions
between home and school Mini-breaks between tasks Visual and verbal reminders
Provide immediate feedback
Provide lecture notes/outline Provide a warning for Graphic organizers
transitions Small group instruction
Partnering Emphasize multi-sensory
learning
Assistive Technology Tests/Quizzes/Grading Behavior/Attention Organization
Computer/whiteboard Extended time Consistent daily structured Individual daily planner
routine
Tape recorder Study guides Display a written agenda
Simple and clear classroom
Video Tape Shortened tests rules Note-taking assistance
Read directions aloud Frequent feedback Color code materials
20 | P a g e21 | P a g e
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.
22 | P a g eInterdisciplinary Connections
Model interdisciplinary thinking to expose students to other disciplines.
Social Studies Connection: Social Studies Standard 6.2.12.B.1.a; 6.2.12.B.1.b
Name of Task: Carbon 14 Dating
The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in
archaeology and geology, among other places
Science Connection: Science Standard HS-PS-1-6
Name of Task: Course of Antibiotics
This task presents a real-world application of finite geometric series. The context can lead into several interesting follow-up questions and
projects. Many drugs only become effective after the amount in the body builds up to a certain level. This can be modeled very well with
geometric series.
23 | P a g eEnrichment
What is the purpose of Enrichment?
The purpose of enrichment is to provide extended learning opportunities and challenges to students who have already mastered, or can quickly master, the
basic curriculum. Enrichment gives the student more time to study concepts with greater depth, breadth, and complexity.
Enrichment also provides opportunities for students to pursue learning in their own areas of interest and strengths.
Enrichment keeps advanced students engaged and supports their accelerated academic needs.
Enrichment provides the most appropriate answer to the question, “What do you do when the student already knows it?”
Enrichment is… Enrichment is not…
Planned and purposeful Just for gifted students (some gifted students may need
Different, or differentiated, work – not just more work intervention in some areas just as some other students may need
frequent enrichment)
Responsive to students’ needs and situations
Worksheets that are more of the same (busywork)
A promotion of high-level thinking skills and making connections
within content Random assignments, games, or puzzles not connected to the
content areas or areas of student interest
The ability to apply different or multiple strategies to the content
Extra homework
The ability to synthesize concepts and make real world and cross-
curricular connections. A package that is the same for everyone
Elevated contextual complexity Thinking skills taught in isolation
Sometimes independent activities, sometimes direct instruction Unstructured free time
Inquiry based or open-ended assignments and projects
Using supplementary materials in addition to the normal range
of resources.
Choices for students
Tiered/Multi-level activities with flexible groups (may change
daily or weekly)
24 | P a g eAssessments
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
25 | P a g eNew Jersey Student Learning Standards A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 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.B.4b Solve quadratic equations by inspection. 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.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically 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) 26 | P a g e
New Jersey Student Learning Standards A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval F-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 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.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. 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 27 | P a g e
New Jersey Student Learning Standards F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F-IF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. G-GPE.A.2 Derive the equation of a parabola given a focus and directrix. N-CN.A.1 Know there is a complex number i such that i2 = -1, and every complex number as the form a + bi with a and b real. N-CN.A.2 Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions. S-ID.B.6a Fit a function to the data; 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. 28 | P a g e
New Jersey Student Learning Standards 29 | 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.
Course: Algebra II Unit: 1 (One) Topics: Linear Functions, Quadratic Functions
Quadratic Equations and Complex Numbers
30 | P a g eNJSLS:
A-CED.A.1, A-CED.A.2, A-CED.A.3, A-SSE.A.2, A-REI.B.4b, A-REI.C.6, A-REI.C.7, A-REI.D.11, A-APR.B.3, F-IF.B.4, F-IF.B.6,
F-IF.C.7c, F-IF.C.9, F-BF.A.1a, F.BF.B.3, F-LE.A.2, F-IF.C.8a, G-GPE.A.2, N-CN.A.1, N-CN.A.2, N-CN.C.7, S-ID.B.6a
Unit Focus:
Identify parent functions and transformations.
Describe transformations of parent functions.
Model with linear functions and solve linear systems.
Describe transformations of quadratic functions.
Identify characteristics of quadratic functions.
Write equations of parabolas.
Model with quadratic functions.
Perform operations with complex numbers.
Solve quadratic equations by completing the square.
Describe how to use the quadratic formula.
Solve nonlinear systems and quadratic inequalities.
New Jersey Student Learning Standard(s):
31 | P a g eF.BF.B.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the
value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing
even and odd functions from their graphs and algebraic expressions for them
Student Learning Objective 1: Parent Functions and Transformations.
Student Learning Objective 2: Transformations of Linear and Absolute Value Functions.
Modified Student Learning Objectives/Standards: N/A
.
Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities
MPs
Clarifications Questions
(Accountable Talk)
F‐BF.3‐2
MP2 Identify the effect on the graph of How do the graphs of Type II, III:
MP4 replacing f(x) by f(x) + k, k f(x), f(kx), Explain the term parent function if it is y = f(x) + k, y = f (x − h),
MP6 and f(x + k) for specific values of k unfamiliar. and y = −f(x) compare to Exploring Sinusoidal
MP7 (both positive and negative); find the the graph of the parent Functions
value of k given the graphs, limiting Write the Core Concept, which shows the function
the function types to polynomial, graphs of linear functions transformed. f ? Building a quadratic
exponential, logarithmic, and function from f(x)=x 2
Make the connection to the results found in
trigonometric functions. i.) What are the characteristics
translating the absolute value function in the Building an Explicit
of some of the basic parent
explorations. Quadratic Function by
Experimenting with cases functions?
and illustrating an Composition
explanation are not
• MP7 Look For and Make Use of Structure:
Is it possible to use more
assessed here. Use the notation g(x) = f(x) + (−3) and then Identifying Quadratic
than one transformation on
F‐BF.3‐3 Recognize a function? Functions (Standard Form)
even and odd functions
substitute for f(x) to help students see the
from their graphs and vertical shift of 3 units. Identifying Quadratic
algebraic expressions for Functions (Vertex Form)
them, limiting the
32 | P a g efunction types to • MP7: Use the notation h(x) = f(x − (−2)) Medieval Archer
polynomial, exponential, and then evaluate the function for an input
logarithmic, and Transforming the graph of
of x + 2.
trigonometric functions. a function
Experimenting with The result is a horizontal translation of 2 units.
cases and illustrating an Additional tasks:
explanation are not • Have students translate f up 4 units to see
assessed here. F‐BF.3‐5 that the result is the same as the horizontal shift Identifying Even and Odd
Illustrating an Functions
in. Have students compare y = − ∣ x ∣ with the
explanation is not
parent function and look at the table of values.
assessed here.
Function notation representation of
transformations Perform transformations on
graphs of polynomial, exponential, logarithmic,
or trigonometric functions. Identify the effect
on the graph of replacing f(x) by: o f(x) + k; o k
f(x); o f(kx); o and f(x + k) for specific values of k
(both positive and negative). Identify the effect
on the graph of combinations of
transformations. Given the graph, find the value
of k.
SPED Strategies:
Model how the function notation of
transformations correlates to changes in the
values and graph of a function. Provide students
with a reference document that illustrates
verbally and pictorially the features of a
function and how they are changed due to
transformation.
ELL Strategies:
33 | P a g eDemonstrate comprehension of the effects 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, by
illustrating an explanation using technology and
finding the value of k given the graphs in L1
and/or use gestures, examples and selected
technical words. Practice sketching the graph of
a parent’s function and their transformation.
Verbalize observations made when students use
graphic calculator to display functions
transformation.
34 | P a g eNew Jersey Student Learning Standard(s):
A-CED.A.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
F-IF.C.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.A.1a
Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-LE.A.2
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
S-ID.B.6a
Fit a function to the data; 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.
Student Learning Objective 3: Modeling with Linear Functions.
Modified Student Learning Objectives/Standards:
Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities
MPs
Clarifications Questions
(Accountable Talk)
MP1 A-CED.A.2
M6 Items must have real-world How can you use a linear
MP7 context • Limit equations to function to model and
two variables. analyze a real-life situation?
35 | P a g eUsing a Given Input and Output to Build a Model.
F-IF.C.9 Identify the input and output values. Convert the
Function types should be data to two coordinate pairs. Find the slope.
limited to linear, quadratic ,
square root, cube root, SPED Strategies:
piecewise‐defined (including
step functions and absolute Review the solving linear Functions
value functions), and
exponential functions. Pre-teach the vocabulary and provide verbal and
Exponential functions are pictorial descriptions
limited to those with domains
in the integers. • Items may
Provide students with a graphic
or may not have real world
organizer/reference sheet/Google Doc that
context.
highlights the thinking and procedure involved in
F‐BF.A.1a, F-LE.A.2 writing geometric and arithmetic sequences in
And S‐ID.B.6a recursive and explicit form.
Evidence Statement
• Write a function based on ELL Strategies:
an observed pattern in a real‐
world scenario. Write the linear model. Use the model to make a
Clarification prediction by evaluating the function at a given x-
• Items must have real‐world value. Use the model to identify an x-value that
context. results in a given y-value. Write equations of
• Limit to linear, quadratic
linear functions using points and slopes. Find lines
and exponential functions
of fit and lines of best fit. Ask probing questions
with domains in the integers.
• Similar to creating a to gauge students’ recollection of what
function from a scatterplot information is necessary to write a linear function
but for this standard the and the two forms.
relationship between the two
• Write the Core Concept. Be sure to check that
quantities is clear from the
students recall the slope formula. Discuss linear
context.
36 | P a g eregression and correlation (positive, negative,
none)
with students.
New Jersey Student Learning Standard(s):
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.
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.
Student Learning Objective 4: Solving Linear Systems.
Modified Student Learning Objectives/Standards:
Xxxxxx
Evidence Statement Key/ Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities
MPs
Clarifications Questions
(Accountable Talk)
A.APR.B.3
MP1 Evidence Statement • Visualize solutions of systems of linear How can you determine the
MP2 Provide constraints equations in three variables. Solve systems number of solutions of a linear
MP3 based on real‐world of linear equations in three variables system?
context for equations, algebraically. Solve real-life problems
37 | P a g einequalities, systems of SPED Strategies:
equations and systems Link the concept of solving a system of
of inequalities. • equations with one linear and one quadratic
Determine if a solution equation to solving a system of linear
is viable based on real‐ equations.
world context.
Model the thinking and processes necessary
to decide on a solution path and solve a
Clarification
system with one linear equation and one
• Items must have real‐
quadratic equation accurately.
world context. •
Systems are limited to Provide students with reference sheets/notes
systems of equations to encourage confidence and independence.
with two equations and
two unknowns. ELL Strategies:
A.REI.C.6 Demonstrate understanding of solving
systems of linear and quadratic equations;
Clarification then explain orally how to solve the
equations in two variables in the student’s
• Items may have real‐ native language and/or use gestures,
world context. • Items equations and selected, technical words.
do not require student
to use a particular Create an outline that allows students to
method. • Systems are organize and follow information that they
are receiving. Outlines can be blank or
to be provided for
partially filled in to vary difficulty.
students when
assessing this standard Use a graphic calculator to solve a linear
system of equations to help students
understand what various type of solutions
might look like.
38 | P a g eProvide students with construction paper,
scissors and tape to build models of linear
systems that have one, infinitely many
solutions and no solutions.
New Jersey Student Learning Standard(s):
F-IF.C.7c
Graph polynomial functions, identifying zeros when suitable factorizations are available, And showing end behavior.
F.BF.B.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative);
find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Include recognizing even and odd functions from their graphs and algebraic expressions for them
Student Learning Objective 5: Transformations of Quadratic Functions
Modified Student Learning Objectives/Standards: xxxxxx
Evidence Statement Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities
MPs
Key/ Clarifications Questions
(Accountable Talk)
Describe transformations of quadratic
MP3 F.BF.B.3 How do the constants a, h, and
functions. Write transformations of quadratic
k affect the graph of the
functions.
quadratic function
Limit to linear and SPED Strategies:
quadratic functions. g(x) = a(x − h)2 + k?
The experiment part of Model how the function notation of
the standard is transformations correlates to changes in the
instructional only. This values and graph of a function.
39 | P a g easpect of the standard is “How can you tell when two
not assessed. Provide students with a reference document ordered pairs are a reflection
that illustrates verbally and pictorially the in the x-axis?”
features of a function and how they are “How can you tell when two
changed due to transformation. functions are a reflection in
the x-axis?”
ELL Strategies:
Write the Core Concept. Discuss f(x) = x2 as
the squaring function.
This allows you to talk about h affecting the
graph in a horizontal direction because the
squaring has not been done yet. The x-value
is being determined. The value of k affects
the graph in a vertical direction because the
squaring has happened. The y-value was
determined by the squaring function, and
now k is being added to or subtracted from it.
In deductive reasoning, you start with two or
more statements that you know or assume to
be true. From these, you deduce or infer the
truth of another statement.
Demonstrate comprehension of the effects 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, by
illustrating an explanation using technology
and finding the value of k given the graphs in
L1 and/or use gestures, examples and
selected technical words.
Practice sketching the graph of a parent’s
function and their transformation.
40 | P a g eVerbalize observations made when students
use graphic calculator to display functions
transformation.
New Jersey Student Learning Standard(s):
F-IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch
graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.C.7c
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
F-IF.C.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.
A-APR.B.3
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the
polynomial.
Student Learning Objective 6: Characteristics of Quadratic Functions.
Modified Student Learning Objectives/Standards: xxxxx
Evidence Statement Key/ Essential Understandings/ Tasks/Activities
MPs
Clarifications Skills, Strategies & Concepts Questions
(Accountable Talk)
41 | P a g eF.IF.C.9 Explore properties of parabolas.
What type of symmetry does
MP3 Function types should be Find maximum and minimum values of the graph of
limited to linear, quadratic , quadratic functions. Graph quadratic functions f(x) = a(x − h)2 + k have and
MP7 square root, cube root, using x-intercepts. Solve real-life problems. how can you describe this
piecewise-defined (including symmetry?
step functions and absolute Students should be able to state the vertex in
addition to describing the symmetry. Turn and Talk: “If you know
value functions), and the vertex of a parabola, can
exponential functions. What the students should know about the you graph the parabola?
Exponential functions are function f(x) = 1/ 2 x2 − 2x − 2?” Partners Explain.”
limited to those with domains should identify the function as a quadratic that
in the integers. • Items may or Advancing Question:
opens upward and has a y-intercept of −2.
may not have real world Students may recall the formula for the x- “If you knew the vertex and
context. coordinate of the vertex from their Algebra 1 one additional point on the
A.APR.B.3 class. graph, would
that be enough to graph the
Cubic polynomials may be • The table of values should reveal the parabola? Explain.”
used if one linear factor and an symmetry of the y-values centered at x = 2.
easily factorable quadratic SPED Strategies:
factor are provided. • Zeros of Model the thinking behind determining when
cubic polynomials must be and how to use the graphing calculator to
integers. • Construction of a graph complicated polynomials.
rough graph is limited to the
graph of a quadratic Provide students with opportunities to practice
the thinking and processes involved in
polynomial. graphing polynomial equations by hand and
using technology by working small groups.
Develop a reference sheet for student use that
includes formulas, processes and procedures
42 | P a g eand sample problems to encourage proficiency
and 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.
Use technology to create table of values to
verify
43 | P a g eNew Jersey Student Learning Standard(s):
F-IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch
graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.C.7c
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
G-GPE.A.2
Derive the equation of a parabola given a focus and directrix.
Student Learning Objective 7: Focus of a Parabola.
Modified Student Learning Objectives/Standards:
M.EE.F-IF.4–6: Construct graphs that represent linear functions with different rates of change and interpret which is faster/slower,
higher/lower, etc.
Evidence Statement Essential Understandings/ Tasks/Activities
MPs
Key/ Clarifications Skills, Strategies & Concepts Questions
(Accountable Talk)
MP2 G-GPE.A.2 Explore the focus and the directrix of a What is the focus of a
MP4 parabola. Write equations of parabolas. parabola?
MP7 Understand or Solve real-life problems.
complete a derivation What information will you need?”
SPED Strategies: To write the equation of a parabola
of the equation of a with its vertex at the origin,
circle of given center
and radius using the Pre-teach vocabulary using visual and What is the vertex of the parabola?”
Pythagorean Theorem; verbal models that are connected to real life
complete the square to situations and ensure that students include
these definitions their reference notebook.
find the center and
Model how to derive the equation of a circle
44 | P a g eradius of a circle given given the center and radius using the
by an equation. Pythagorean Theorem Ensure that students
include this information in their reference
Clarifications notebook. Model how to use the equation of
a circle to determine the radius and center.
i) Tasks must go Provide students with hands on
beyond simply finding opportunities to explore and extend their
understanding by working in small groups,
the center and radius of
see the application to real life
a circle.
ELL Strategies:
Display an image of a satellite dish. Have
students share their knowledge of satellite
dishes and how they work. The essential
piece is that the satellite dish is an antenna
that receives electromagnetic signals from
an orbiting satellite. The shape of the
satellite dish is parabolic.
Explain orally and in writing the equation of
a circle of given center and radius in the
student’s native language and/or use
gestures, examples and selected technical
words, and short simple sentences.
45 | P a g eNew Jersey Student Learning Standard(s):
A-CED.A.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
F-IF.B.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval
F-BF.A.1a
Determine an explicit expression, a recursive process, or steps for calculation from a context.
S-ID.B.6a
Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
Student Learning Objective 7: Modeling with Quadratic Functions.
Modified Student Learning Objectives/Standards: xxxx
.
Evidence Statement Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities
MPs
Key/ Clarifications Questions
(Accountable Talk)
MP4 How can you use a quadratic
Write equations of quadratic functions using
MP6 F-BF.A.1a function to model a real-life
vertices, points, and x-intercepts.
situation?
Evidence Statement • Write quadratic equations to model data sets.
This standard is not Students should be familiar with identifying Turn and Talk: Have students
assessed as a standalone a linear function from a table of values read the example and view the
standard. where the constant rate of change was found graph.
in the first differences. Introduce the idea
F.IF.B.6 that when second differences are equal, the What information is known in
data is quadratic. the example, and what
Clarification SPED Strategies:
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