Honors Project (Must Complete All)

MATHEMATICS

Pre-Calculus Honors: Unit 4 Concepts in Probability and Statistics, Limits and Introduction to Calculus

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Course Philosophy/Description

The high school Pre-calculus course covers mathematical topics ranging from Basics of Functions to Limits of Functions. It provides opportunities to the students to expand their knowledge base and understanding of mathematics in general. The overarching goal of the course is to build a solid foundation for the students who choose Mathematics, Engineering, Sciences, or Business as their college major and/or career options. The major topics in the course such as, Polynomials, Exponents, Trigonometry, Logarithms, Complex numbers, Series/sequences, and Limits help generate students’ inquiries about the mathematical nature, complexities, and applications of these topics in real-life situations. Students not only acquire new knowledge, but also deepen their topical and overall understanding of the content for future transfer to new situations or other disciplines.

Assessment results from this course may be used for the purpose of placements into Calculus/AP Calculus, Statistics, Physics/AP Physics, or other higher level courses.

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ESL Framework

This ESL framework was designed to be used by bilingual, dual language, ESL and general education teachers. Bilingual and dual language programs use the home language and a second language for instruction. ESL teachers and general education or bilingual teachers may use this document to collaborate on unit and lesson planning to decide who will address certain components of the SLO and language objective. ESL teachers may use the appropriate leveled language objective to build lessons for ELLs which reflects what is covered in the general education program. In this way, whether it is a pull-out or push-in model, all teachers are working on the same Student Learning Objective connected to the Common Core standard. The design of language objectives are based on the alignment of the World-Class Instructional Design Assessment (WIDA) Consortium’s English Language Development (ELD) standards with the Common Core State Standards (CCSS). WIDA’s ELD standards advance academic language development across content areas ultimately leading to academic achievement for English learners. As English learners are progressing through the six developmental linguistic stages, this framework will assist all teachers who work with English learners to appropriately identify the language needed to meet the requirements of the content standard. At the same time, the language objectives recognize the cognitive demand required to complete educational tasks. Even though listening and reading (receptive) skills differ from speaking and writing (expressive) skills across proficiency levels the cognitive function should not be diminished. For example, an Entering Level One student only has the linguistic ability to respond in single words in English with significant support from their home language. However, they could complete a Venn diagram with single words which demonstrates that they understand how the elements compare and contrast with each other or they could respond with the support of their home language (L1) with assistance from a teacher, para-professional, peer or a technology program.

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

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Pacing Chart – Unit 4

# Student Learning Objective NJSLS Textbook/WebAssign WebAssign Instructor Resources • A.REI.C.8 Write matrices and determine their dimensions. • Perform elementary row operations on matrices. A.REI.C.9 7.4 1 • Use matrices and Gaussian or Gauss-Jordan elimination to 7.5 solve systems of equations. 7.6 • Determine if two matrices are equal. 7.7 • Add and subtract matrices, multiply matrices by a scalar. 7.8 • Multiply two matrices. • Use matrix operations to model and solve real-life problems. • Verify two matrices are inverses. • Find the inverse of a matrix • S.CP.A.2 9.5 Use the Binomial Theorem to calculate binomial coefficients and write binomial expansions. S.CP.B.7 9.6 2 • Use Pascal’s Triangle to calculate binomial coefficients. S.CP.B.9 9.7 • Use counting principles to solve more complicated counting problems. • 12.1 Use the definition of a limit to estimate limits. • Determine whether limits of functions exists F.IF.A.3 12.2 3 • Use properties of limits to evaluate limits. F.IF.B.4 • Evaluate one-sided limits. • 12.3 4 Approximate limits of functions graphically and numerically. • Use a tangent line to approximate the slope of a graph at a F.LE.A.1b 12.4 point F.IF.C.7 • Use the limit definition of slope to find exact slopes of graphs. • Limits at Infinity

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Research about Teaching and Learning Mathematics Structure teaching of mathematical concepts and skills around problems to be solved (Checkly, 1997; Wood & Sellars, 1996; Wood & Sellars, 1997) Encourage students to work cooperatively with others (Johnson & Johnson, 1975; Davidson, 1990) Use group problem-solving to stimulate students to apply their mathematical thinking skills (Artzt & Armour-Thomas, 1992) Students interact in ways that support and challenge one another’s strategic thinking (Artzt, Armour-Thomas, & Curcio, 2008) Activities structured in ways allowing students to explore, explain, extend, and evaluate their progress (National Research Council, 1999) There are three critical components to effective mathematics instruction (Shellard & Moyer, 2002): • Teaching for conceptual understanding • Developing children’s procedural literacy • Promoting strategic competence through meaningful problem-solving investigations Teachers should be: • Demonstrating acceptance and recognition of students’ divergent ideas. • Challenging students to think deeply about the problems they are solving, extending thinking beyond the solutions and algorithms required to solve the problem • Influencing learning by asking challenging and interesting questions to accelerate students’ innate inquisitiveness and foster them to examine concepts further. • Projecting a positive attitude about mathematics and about students’ ability to “do” mathematics Students should be: • Actively engaging in “doing” mathematics • Solving challenging problems • Investigating meaningful real-world problems • Making interdisciplinary connections • Developing an understanding of mathematical knowledge required to “do” mathematics and connect the language of mathematical ideas with numerical representations • Sharing mathematical ideas, discussing mathematics with one another, refining and critiquing each other’s ideas and understandings • Communicating in pairs, small group, or whole group presentations • Using multiple representations to communicate mathematical ideas • Using connections between pictures, oral language, written symbols, manipulative models, and real-world situations • Using technological resources and other 21st century skills to support and enhance mathematical understanding 5 | Page

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.

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Effective Pedagogical Routines/Instructional Strategies

Collaborative Problem Solving Analyze Student Work

Connect Previous Knowledge to New Learning Identify Student’s Mathematical Understanding

Making Thinking Visible Identify Student’s Mathematical Misunderstandings

Develop and Demonstrate Mathematical Practices Interviews

Inquiry-Oriented and Exploratory Approach Role Playing

Multiple Solution Paths and Strategies Diagrams, Charts, Tables, and Graphs

Use of Multiple Representations Anticipate Likely and Possible Student Responses

Explain the Rationale of your Math Work Collect Different Student Approaches

Quick Writes Multiple Response Strategies

Pair/Trio Sharing Asking Assessing and Advancing Questions

Turn and Talk Revoicing

Charting Marking

Gallery Walks Recapping

Small Group and Whole Class Discussions Challenging

Student Modeling Pressing for Accuracy and Reasoning

Maintain the Cognitive Demand

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Computer Science and Design Thinking Standards

8.2.12.ED.3, 8.2.12.ITH.2, 8.1.12.CS.4, 8.1.12.DA.1  Engineering Design • Evaluate several models of the same type of product and make recommendations for a new design based on a cost benefit analysis. Example: Students will be able to use mathematical modeling to discuss and compare products to maximize profit and minimize cost.

 Interaction of Technology and Humans • Propose an innovation to meet future demands supported by analysis of the potential costs, benefits, trade-offs, and risks related to the use of the innovation. Example: Changes caused by the introduction and use of new technology can range from gradual to rapid and from subtle to obvious, and can change over time. Students will observe these changes and recognize the varying data from society to society as a result of differences in economic status, politics and culture.

 Computing Systems • Develop guidelines that convey systemic troubleshooting strategies that others can use to identify and fix errors. Example: Students will be able to successfully troubleshoot complex problems that involve multiple approaches including research, analysis, reflection, interaction with peers, and drawing on past experiences.

 Data and Analysis • Create interactive data visualizations using software tools to help others better understand real world phenomena, including climate change. Example: Individuals select digital tools and design automated processes to collect, transform, generalize, simplify, and present large data sets in different ways to influence how other people interpret and understand underlying information.

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Career Readiness, Life Literacies and Key Skills

Career readiness, life literacies, and key skills education provides students with the necessary skills to make informed career and financial decisions, engage as responsible community members in a digital society, and to successfully meet the challenges and opportunities in an interconnected global economy. .

• Civil Financial Responsibility (9.1.12.CFR.4) Students will demonstrate an understanding of the interrelationships among attitudes, assumptions, and patterns of behavior regarding money, saving, investing, and work across cultures.

The potential for building and using personal wealth includes responsibility to the broader community and an understanding of the legal rights and responsibilities of being a good citizen. Example: Students will use technology and content specific material to determine financial responsibilities across various professions and lifestyles.

• Credit Debt Management (9.1.12.CDM.4) Students will identify issues associated with student loan debt, requirements for repayment, and consequences of failure to repay student loan debt. Example: Throughout this curriculum, students work to improve of debt management, the benefits and consequences of various debt types.

• Planning and Budgeting (9.1.12.PB.1, 9.1.12.PB.2) Students will explain the difference between saving and investing. Students will prioritize financial decisions by considering alternatives and possible consequences. Example: Students will learn about living within their means, identifying and relating financial planning and budgeting throughout various real-life application problems.

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WIDA 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 sentences or paragraphs • Oral or written language with minimal phonological, syntactic or semantic errors that may impede the 4- Expanding 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 • 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 3- Developing expository descriptions with sensory, graphic or interactive support

• General language related to the content area • Phrases or short sentences • Oral or written language with phonological, syntactic, or semantic errors that often impede of the communication when presented with one to multiple-step commands, directions, or a series of statements 2- Beginning with sensory, graphic or interactive support

1- Entering • Pictorial or graphic representation of the language of the content areas, Words, phrases or chunks of language when presented with one-step commands directions, WH-, choice or yes/no questions, or statements with sensory, graphic or interactive support

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Culturally Relevant Pedagogy Examples • Integrate Relevant Word Problems: Contextualize equations using word problems that reference student interests and cultures. Example: Create and use word problems that include student interests, current events, and/or relevance to real-world situations in order to make problems relatable to students when adding to, taking from, putting together, taking apart, and comparing. Using content that students can relate to adds meaning, value, and connection.

• Use Learning Stations: Provide a range of material by setting up learning stations. Example: Reinforce understanding of concepts and skills by promoting learning through student interests and modalities, experiences and/or prior knowledge. Encourage the students to make choices in content based upon their strengths, needs, values and experiences. Providing students with choice boards will give them a sense of ownership to their learning and understanding.

• 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, 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. Work with students to create a variety of sorting and match games of vocabulary words in this unit. Students can work in teams or individually to play these games for approximately 10-15 minutes each week. This will give students a different way of becoming familiar with the vocabulary rather than just looking up the words or writing the definition down.

• 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.

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SEL Competency Examples Content Specific Activity & Approach to SEL

Example practices that address Self-Awareness: Work with students to collaboratively develop the  Self-Awareness rules for working in the math classroom. Establish Self-Management • Clearly state classroom rules a charter for how ideas are respectfully shared. Social-Awareness • Provide students with specific feedback regarding Relationship Skills academics and behavior Responsible Decision-Making Establish a safe space for the sharing of divergent • Offer different ways to demonstrate thinking on problem solving. 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 Self-Awareness Example practices that address Self- Work with students to help them to prepare for  Self-Management Management: classroom expectations, to identify their challenges Social-Awareness and develop or seek assistance with strategies to Relationship Skills • Encourage students to take pride/ownership in address challenges. Responsible Decision-Making work and behavior

• Encourage students to reflect and adapt to classroom situations

• Assist students with being ready in the classroom

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SEL Competency Examples Content Specific Activity & Approach to SEL

• Assist students with managing their own emotional states Self-Awareness Example practices that address Social- Model the ways that students can work effectively Self-Management Awareness: as a group by fostering social awareness when  Social-Awareness students are working collaboratively as problem Relationship Skills • Encourage students to reflect on the perspective solvers. Responsible Decision-Making 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 Self-Awareness Example practices that address Relationship Engage students in tasks that are relevant to real Self-Management Skills: life, require communication among stakeholders, Social-Awareness and have multiple solution paths.  Relationship Skills • Engage families and community members Responsible Decision-Making • Model effective questioning and responding to students

• Plan for project-based learning

• Assist students with discovering individual strengths

• Model and promote respecting differences

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SEL Competency Examples Content Specific Activity & Approach to SEL

• Model and promote active listening

• Help students develop communication skills

• Demonstrate value for a diversity of opinions Self-Awareness Example practices that address Responsible Provide students with a classroom environment that Self-Management Decision-Making: fosters independence and provides students with Social-Awareness the opportunity to resolve conflicts productively. Relationship Skills • Support collaborative decision making for  Responsible Decision-Making 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

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Differentiated Instruction Accommodate Based on Students Individual Needs: Strategies

Time/General Processing Comprehension Recall

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

• Provide lecture notes/outline • Provide a warning for • Small group instruction • Graphic organizers transitions • Emphasize multi-sensory • Reading partners learning

Assistive Technology Tests/Quizzes/Grading Behavior/Attention Organization

• Computer/whiteboard • Extended time • Consistent daily structured • Individual daily planner routine • Tape recorder • Study guides • Display a written agenda • Simple and clear classroom • • • Video Tape Shortened tests rules Note-taking assistance • • Read directions aloud • Frequent feedback Color code materials

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Differentiated Instruction

Accommodate Based on Students Individual Needs: Strategies

• Anchor charts to model strategies. • Review Algebra concepts to ensure students have the information needed to progress in understanding. • Pre-teach pertinent vocabulary. • Provide reference sheets that list formulas, step-by-step procedures, theorems, and modeling of strategies. • Word wall with visual representations of mathematical terms. • Teacher modeling of thinking processes involved in solving, graphing, and writing equations. • Introduce concepts embedded in real-life context to help students relate to the mathematics involved. • Record formulas, processes, and mathematical rules in reference notebooks. • Graphing calculator to assist with computations and graphing of trigonometric functions. • Utilize technology through interactive sites to represent nonlinear data. • Graphic organizers to help students interpret the meaning of terms in an expression or equation in context. • Translation dictionary. • Sentence stems to provide additional language support for ELL students.

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Interdisciplinary Connections Model interdisciplinary thinking to expose students to other disciplines. Social Studies Connection: Name of Task: Security System at Quail Creek NJSLS: 6.2.12.History.CC.5.a Ten homes in the Quail Creek area were identified by a surveyor; you were told that 4 of the homes have security system. 1- If you are to pick 3 homes randomly: a. What is the probability that all 3 homes have security system? b. What is the probability that only 1 has a security system? 2- If 40% of the homes constructed in the Quail creek area include a security system. Three homes are selected at random: a. What is the probability all three of the selected homes have a security system? b. What is the probability none of the selected homes have a security system? c. What is the probability that at least one has a security system? 3- Create probability distribution tables for part 1 and part 2

Science Connection: Name of Task: Rabbits Population NJSLS: HS-LS4-2; HS-LS4-3 The population of rabbits over a 2 year period in a certain county is given below 1- Draw a scatter plot of the data 2- Find a logistic regression model for the data. Find the limit of that model as time approaches infinity 3- What can you conclude about the limit of the rabbit population growth in the county? 4- Provide a reasonable explanation for the population growth limit.

Name of Task: Rock Toss NJSLS: HS-PS2-2, HS-PS2-3 A rock is thrown straight up from level ground. The velocity of the rock at any time t (sec) is v(t)=48-32t ft/sec 1- Graph the velocity function. 2- At what time does the rock reach its maximum height? 3- Find how far the rock has traveled at its maximum height. 4- Graph the pathway followed by the rock 5- How far away from the launching point would the rock land? * Tasks can be found within the additional task folders

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Enrichment

What is the purpose of Enrichment?

• The purpose of enrichment is to provide extended learning opportunities and challenges to students who have already mastered, or can quickly master, the basic curriculum. Enrichment gives the student more time to study concepts with greater depth, breadth, and complexity. • Enrichment also provides opportunities for students to pursue learning in their own areas of interest and strengths. • Enrichment keeps advanced students engaged and supports their accelerated academic needs. • Enrichment provides the most appropriate answer to the question, “What do you do when the student already knows it?”

Enrichment is… Enrichment is not… • Planned and purposeful • Just for gifted students (some gifted students may need • Different, or differentiated, work – not just more work intervention in some areas just as some other students may need • Responsive to students’ needs and situations frequent enrichment) • A promotion of high-level thinking skills and making connections • Worksheets that are more of the same (busywork) within content • Random assignments, games, or puzzles not connected to the • The ability to apply different or multiple strategies to the content content areas or areas of student interest • The ability to synthesize concepts and make real world and cross- curricular connections. • Extra homework • Elevated contextual complexity • A package that is the same for everyone • Sometimes independent activities, sometimes direct instruction • Thinking skills taught in isolation • Inquiry based or open ended assignments and projects • Unstructured free time • Using supplementary materials in addition to the normal range of resources. • Choices for students • Tiered/Multi-level activities with Flexible groups (may change daily or

weekly)

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Assessments Required District/State Assessments SGO Baseline Assessment SGO Post Assessment

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 WebAssign Problem Sets WebAssign Instructor Help Homework

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New Jersey Student Learning Standards

S.CP.A.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S.CP.B.7: Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

S.CP.B.9: Use permutations and combinations to compute probabilities of compound events and solve problems.

A.REI.C.8: Represent a system of linear equations as a single matrix equation in a vector variable.

A.REI.C.9: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

F.IF.A.3: Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

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

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

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

2. Reason abstractly and quantitatively.

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

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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Course: Pre-Calculus Unit: 4 Topic: Analytic Geometry and Statistics

NJSLS:

A.REI.C.8, A.REI.C.9, A.CP.A.2, S.CP.B.7, S.CP.B.9, F.IF.A.3, A.IF.A.4, F.LE.A.1B, F.IF.C.7 Unit Focus:

• Use matrices and determine their dimensions. • Perform elementary row operations on matrices. • Use matrices and Gaussian or Gauss-Jordan elimination to solve systems of equations. • Determine if two matrices are equal. • Add, subtract and multiply matrices. • Use matrix operations to model and solve real-life problems. • Verify two matrices are inverses. • Find the inverse of a matrix. • Use the counting principles and the probability rules to analyze the Binomial Theorem and the binomial probability distribution. • Introduce parametric and polar forms for writing and graphing equations. • Find the limits of functions either by using its definition, by approximating it graphically and numerically or by evaluating one-sided ones. • Use a tangent line to approximate the slope of a graph at a point. • Find the tangent lines of a function.

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New Jersey Student Learning Standard(s):

A.REI.C.8: Represent a system of linear equations as a single matrix equation in a vector variable. A.REI.C.9: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Student Learning Objective 1: Write matrices and determine their dimensions, perform elementary row operations, use Gaussian and Gauss- Jordan elimination to solve systems of equations, determine equal matrices, add subtract and multiply matrices by a scalar, multiply two matrices, use matrix operations to model real-life problems., verify matrix inverses.

MPs Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Questions

Identify the number of rows and What are the properties of matrix? WebAssign: MP1 columns within a matrix. Problem Set: 7.4, 7.5, 7.6, 7.7, 7.8 MP2 How are matrices added and/or multiplied? MP5 Modify a matrix through individual Tasks: MP6 operations of adding, multiplying, How is the inverse of a matrix found? Fun Maze Activity MP7 and switching rows. How are matrix operations performed on a Matrix Real World Problems Modify a matrix through a graphics calculator?

combination of adding, multiplying, *Unit 4 Tasks can be found in the How are simultaneous equations solved with and switching rows. Matrix math? Unit 4 Task Folder

Correctly notate a matrix. How are Augmented Matrices put in row reduced echelon form (rref)? Identify equal and inverse matrices What is the inverse Matrix procedure for solving simultaneous equations?

What is Linear Programming and how is it used to maximize or minimize a company's profits and costs respectively? 26 | Page

New Jersey Student Learning Standard(s):

S.CP.A.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities and use this characterization to determine if they are independent.

S.CP.B.7: Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

S.CP.B.9: Use permutations and combinations to compute probabilities of compound events and solve problems.

Student Learning Objective 2: Use the counting principles and the probability rules to analyze the Binomial Theorem and the binomial probability distribution. MPs Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Questions Calculate basic combinations, How can the counting principle be used to WebAssign: permutations, and factorials. find sample spaces and probabilities? Problem Set: 9.5, 9.6, 9.7 MP 2 MP 5 Calculate binomial probability What is the difference between permutations Tasks: distributions. and combinations? Straight Poker

Use the Binomial Theorem to What methods can be used to find binomial Security System at Quail Creek expand binomials. coefficients? Is It Fair? Analyzing Pascal’s Triangle for its How do you use binomial coefficients to values and symmetries. write binomial expansions? *Unit 4 Tasks can be found in the Unit 4 Task Folder The Binomial Theorem can be used to expand polynomials and to determine the probability of an event.

The Binomial Theorem can be applied easily using tools like Pascal’s Triangle.

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New Jersey Student Learning Standard(s):

F.IF.A.3: Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Student Learning Objective 3: Use the definition of a limit to estimate limits. Determine whether limits of functions exist. Use properties of limits to evaluate limits. Evaluate one-sided limits. MPs Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities

Questions

Find limit at infinity. What is the limit of a function and how can WebAssign: MP 2 a limit be used to determine the continuity Problem Set: 12.1, 12.2 MP 5 Computing integral. of a function? Tasks: Find left and right limit at a point. Rock Toss

Rabbit Population

*Unit 4 Tasks can be found in the Unit 4 Task Folder

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New Jersey Student Learning Standard(s):

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

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

Student Learning Objective 4: Approximate limits of functions graphically and numerically. Use a tangent line to approximate the slope of a graph at a point. Use the limit definition of slope to find exact slopes of graphs. MPs Skills, Strategies & Concepts Essential Understandings/ Tasks/Activities Questions Define a limit. What characteristics does a function need to WebAssign: MP 2 have for a limit to exist? Problem Set: 12.3, 12.4 MP 5 Find the value of a limit graphically and as a table, What does it mean to have a limit equal Tasks: including limits at infinity. infinity? Free Fall on Another Planet

Find the limit algebraically. Why does the slope of a secant line change *Unit 4 Tasks can be found in as x approaches a point? the Unit 4 Task Folder Find one sided limits graphically and algebraically. What is rate of change and how can functions and graphs help model it? Determine the x-values at which a function is Is a function continuous at a particular value continuous/discontinuous. of x?

Is a function considered a continuous function?

The average rate of change between two points on a function can be written as a function.

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Honors Project (must complete all)

Project 1 Project 2 Project 3 Tangent Lines to Sine Curves Distribution of Number of Hits in Overbook Flights Baseball Games

Essential Question: Essential Question: Essential Question: • What is the difference between average and • How can we differentiate between discrete infinitesimal changes and how does that • How can modeling predict the future? and continuous random variables? relate to tangent lines? • To what extent does our world exhibit • What determines if an experiment is • How do you calculate the slope of a curve at binomial and geometric phenomena? binomial? a point and express it in terms of a limit? • How can the effect of multiple vectors be Skills: determined? Skills: • Determine if a probability experiment is a • Analyze the methods can be used to find binomial experiment? Skills: binomial coefficients?

• Find the slope of a line tangent to a function at a given point using limits. • Find the rate of change and understand how functions and graphs could help model it?

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Unit 4 Vocabulary

• derivative • one-sided limit • direct substitution • orientation of a curve • dividing out technique • parameter • Gaussian • plane curve • Gauss-Jordan • rationalizing technique • indeterminate form • rectangular for • inverse matrices • row operations • limacon • secant line • limit • slope of a graph • matrix • tangent line • matrix operations

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References & Suggested Instructional Websites

• IXL High School Standards: https://www.ixl.com/standards/new-jersey/math/high-school • Digital Mathematics Word Wall: http://www.mathwords.com/index_adv_alg_precal.htm • Extra Notes for Pre-Calculus Content: https://sites.google.com/a/evergreenps.org/ms-griffin-s-math-classes/updates • Review Documents for Pre-Calculus: https://sites.google.com/site/dgrahamcalculus/trigpre-calculus/trig-pre-calculus-worksheets • Pre-Calculus IXL Topics and Resources: https://www.ixl.com/math/precalculus • Classroom Challenges to Support Teachers in Formative Assessments: http://map.mathshell.org/materials/lessons.php?gradeid=24 • Binomial Theorem Resources: https://www.khanacademy.org/math/algebra2/polynomial-functions/binomial-theorem/v/coefficient-in- binomial-expansion • Statistics Education Web (STEW). http://www.amstat.org/education/STEW/ • The Data and Story Library (DASL). http://lib.stat.cmu.edu/DASL/ • WebAssign Instructor Resources • WebAssign Student Resources

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Field Trip Ideas SIX FLAGS GREAT ADVENTURE: This educational event includes workbooks and special science and math related shows throughout the day. Your students will leave with a better understanding of real world applications of the material they have learned in the classroom. Each student will have the opportunity to experience different rides and attractions linking mathematical and scientific concepts to what they are experiencing. www.sixflags.com

MUSEUM of MATHEMATICS: Mathematics illuminates the patterns that abound in our world. The National Museum of Mathematics strives to enhance public understanding and perception of mathematics. Its dynamic exhibits and programs stimulate inquiry, spark curiosity, and reveal the wonders of mathematics. The Museum’s activities lead a broad and diverse audience to understand the evolving, creative, human, and aesthetic nature of mathematics.

www.momath.org

LIBERTY SCIENCE CENTER: An interactive science museum and learning center located in Liberty State Park. The center, which first opened in 1993 as New Jersey's first major state science museum, has science exhibits, the largest IMAX Dome theater in the United States, numerous educational resources, and the original Hoberman sphere.

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

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