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POMPTON LAKES SCHOOL DISTRICT INTRO TO COLLEGE MATHEMATICS June 2012 COURSE OF STUDY Submitted By The Mathematic Department Dr. Paul Amoroso, Superintendent Mr. Vincent Przybylinski, Principal Mr. Anthony Mattera, Vice Principal Frances J. Macdonald District Mathematics Supervisor K-12 BOARD MEMBERS Mr. Jose A. Arroyo, Mrs. Catherine Brolsma, Mr. Shawn Dougherty, Mrs. Nancy Lohse-Schwartz, Mr. Garry Luciani, Mr. Carl Padula, Mr. Tom Salus, Mrs. Stephanie Shaw, Mr. Timothy Troast, Jr. Unit Overview Content Area: Mathematics Unit 1 Unit Title: Number Sense, Computation, and Reasoning Target Course/Grade Level: College Math/12th Unit Summary: In this unit students will explore number sense and reasoning through patterns, graphs and estimation. In addition students develop their computation skills through order of operations, sequences and simplifying expressions. Primary interdisciplinary connections: Trends in Science and Business Education 21st century themes: Modeling real situations i.e. Stock Market, Scientific Data Unit Rationale: Students need number sense and problem solving skills to analyze mathematical real world situations. Learning Targets Standards 4:1 Number and Numerical Operations: Students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways. Content Statements: Number sense is used to analyze graphs, use order of operations and complete arithmetic and geometric sequences. CPI # Cumulative Progress Indicator (CPI) 4:1.12A Extend understanding of the number system to all real numbers 4:1.12B Use of operations to real numbers and algebraic procedures 4:1.12C Use and application of estimation and problem solving Unit Essential Questions How do you analyze graphs? How do we apply number sense? Unit Enduring Understandings Describe trends from graphs, figure out patterns Use the appropriate operations necessary for problem solving Unit Learning Targets Students will ... Analyze graphs and patterns in sequences of numbers Apply number sense to problem solving Evidence of Learning Summative Assessment: 2 days In groups students compare and contrast their interpretations of a given problem. Equipment needed: overhead transparencies, copies of notes, Smartboard, Mobi, TI 84 graphing calculators Teacher Resources: Math for Your World textbook, Internet for real life data Formative Assessments Tests and quizzes Explanations of examples Discussions Daily homework assignments Notes Individual practice Lesson Plans Lesson Lesson 1 Patterns and Estimation Timeframe 3 days Lesson 2 Problem Solving and Analyzing Graphs 3 days Lesson 3 Number Theory and Order of Operations 4 days Lesson 4 Rational and Irrational Numbers and Sequences 3 days Lesson 5 Exponents and Scientific Notation 2 days Teacher Notes: Curriculum Development Resources Click the links below to access additional resources used to design this unit: www.mathxlforschool.com Unit Overview Content Area: Trigonometry Unit 2 Unit Title: Introduction to trigonometry Target Course/Grade Level: College Math Grade/12th Unit Summary: In this unit, students will explore four quadrant trigonometry. In particular graphing the six trigonometric functions, learning the values of each of the 6 trigonometric functions and then using the common identities that will be discussed further in calculus. Primary interdisciplinary connections: Physics and how the trigonometric functions determine things such as a force. 21st century themes: Polynomial time algorithm and the supercomputer Unit Rationale: Trigonometric functions and their graphs must be analyzed in order to be able to solve limits, derivatives and integrals of trigonometric functions. Learning Targets Standard 4-2: All students will develop special sense and the ability to use geometric properties relationships and measurement to model describe and analyze phenomena Standard 4.3: All students will represent and analyze relationships among variable quantities and solve problems involving functions and algebraic concepts and processes. Standard 4.5: All students will use mathematical processes of problem solving communication connections reasoning representations and technology to solve problems and communicate mathematical ideas. Content Statements: The unit circle will be used to develop the concepts of trigonometric functions. This in turn will be used to analyze limits, derivatives and integrals using the concepts of trigonometry. CPI # Cumulative Progress Indicator (CPI) 4.2.12.C.1 Use coordinate geometry to represent circles 4.2.12.C.2 Show position of angles in standard position in the coordinate plane 4.2.12.C.3 Find an equation of a circle given its center and radius and given an equation of the circle in standard form find its center and radius 4.3.12.B.1 Understand the relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables and graphs. 4.3.12.B.2 Analyze and explain the general properties and behavior of functions or relations using algebraic and graphing techniques. 4.3.12.B.3 Understand and perform transformations on commonly used functions. 4.3.12.B.4 Understand and compare the properties of classes of functions including exponential, polynomial, rational and trigonometric functions. 4.5.A.1 Learn mathematics through problem solving inquiry and discovery. 4.5.A.2 Solve problems that arise in mathematics and in other contexts 4.5.B.1 Use communication to organize and clarify their mathematical thinking 4.5.B.4 Use the language of mathematics to express mathematical ideas precisely Summative Assessment: 20 days Tests and quizzes Equipment needed: overhead, handouts, textbook, graphing calculators and Smartboard Teacher Resources: www.mathworldwolfram.com Formative Assessments Tests and quizzes Homework Unit circle activities Lesson Plans Lesson Unit circle and angles Timeframe 4 days Equation of circles and domain and range 2 days Trig values of the 6 functions 3 days Trig functions and graphs 7 days Trig identities 4 days Teacher Notes: Teachers are to be sure that students have the basic understanding of trigonometric analysis in order to assure that students will be able to do the concepts needed in a introductory calculus course Curriculum Development Resources www.mathworldwolfram.com Unit Overview Content Area: Mathematics Unit 3 Unit Title: Functions Target Course/Grade Level: College Math/12th Unit Summary: Functions and how they will be a factor in the discussion of integral and differential calculus. Domain and range of a derivative graph cannot be discussed, for example, without first discussing the use of domain and range of a function. Primary interdisciplinary connections: Students will be able to distinguish between the use of domain and range in real life applications. Further the use of trigonometry and calculus applications can de done to further enrich the interdisciplinary instruction of the course. 21st century themes: How time affects the displacement, velocity and acceleration of an object. Unit Rationale: While functional analysis is a popular unit with all mathematics levels, it is often introduced in Pre-calculus and then expanded upon when a student takes Calculus for the first time. For example, how time affects an object in free fall or the motion of a particle. The content and skills acquired in this unit will help students to tangibly address the concepts that are presented in the calculus curriculum. Learning Targets Standard 4.3: All students will represent and analyze relationships among variable quantities and solve problems involving functions and algebraic concepts and processes. Standard 4.5: All students will use mathematical processes of problem solving communication connections reasoning representations and technology to solve problems and communicate mathematical ideas. Content Statements: Functional investigations among various graphical analysis as well using equations in order to understand the picture of the graph in order to further investigate limits and differential calculus in the future curriculum that is studied. CPI # Cumulative Progress Indicator (CPI) Understand the relations and functions and select, convert flexibly among, and use various 4.3.12.B.1 representations for them, including equations or inequalities, tables and graphs. 4.3.12.B.2 Analyze and explain the general properties and behavior of functions or relations using algebraic and graphing techniques. 4.3.12.B.3 Understand and perform transformations on commonly used functions. 4.3.12.B.4 Understand and compare the properties of classes of functions including exponential, polynomial, rational and trigonometric functions. 4.5.A.1 Learn mathematics through problem solving inquiry and discovery. 4.5.A.2 Solve problems that arise in mathematics and in other contexts 4.5.B.1 Use communication to organize and clarify their mathematical thinking 4.5.B.4 Use the language of mathematics to express mathematical ideas precisely Unit Essential Questions Unit Enduring Understandings Will students be able to solve the domain and the By analyzing the “x” and “y” values of the graphs range of an equation and a graph? and equations What procedure is used to find the domain and Denominators of equations cannot be zero range of an equation that has a variable in the denominator? Unit Learning Targets Students will ... Be able to identify functions using the vertical line test Be able to find domain and range of a graph Be able to find the domain and range of an equation Be able to find values of functions Be able to combine function (composite functional analysis) Be able to find the inverse of a function Be able to graph functions Evidence of Learning Summative Assessment: 15 days Tests and quizzes Equipment needed: Overhead projector, graphing calculators and Smartboard Teacher Resources: Calculus textbook, www.mathworldwolfram.com Formative Assessments Discussions about the functional analysis of a graph and equation Tests and quizzes homework Lesson Plans Lesson Relations and functions Timeframe 4 days Domain and range of equations and graphs 4 days Composite functions and inverses 4 days Graphing functions 3 days Teacher Notes: Teachers are to be sure that students have the basic understanding of functional analysis in order to assure that students will be able to do the concepts needed in a introductory calculus course. Curriculum Development Resources www.mathworldwolfram.com Unit Overview Content Area: Mathematics Unit 4 Unit Title: Geometry and Measurement Target Course/Grade Level: College Math/12th Unit Summary: In this unit students use geometry and measurement to explore applications of length, area, perimeter and volume. Primary interdisciplinary connections: Science and Business Education 21st century themes: Modeling real situations; i.e. tiling a floor, fencing a yard Unit Rationale: Students need geometry to understand real world situations involving architecture, construction, and design. Learning Targets Standards 4:2 Geometry and Measurement: Students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe, and analyze phenomena. Content Statements: Geometry and measurement are used in spatial sense and geometric modeling to describe and interpret our physical environment. CPI # Cumulative Progress Indicator (CPI) 4:2.12A Use geometric models to represent real world situations and to solve problems using those models Understand and use appropriate tools and units and measurement 4:2.12D Unit Essential Questions How do you find perimeter, area and volume? How do we measure length and weight? Unit Enduring Understandings Find area, perimeter and volume to solve real word problems Use the appropriate units of measurement Unit Learning Targets Students will ... Find area, perimeter and volume of various shapes Use appropriate units of measurement for lengths and weights Evidence of Learning Summative Assessment: 2 days In groups students will design a room and figure out the cost by applying geometric knowledge. Equipment needed: overhead transparencies, copies of notes, Smartboard, Mobi, TI 84 graphing calculators Teacher Resources: Math for Your World textbook, Internet for real life data Formative Assessments Tests and quizzes Explanations of examples Discussions Daily homework assignments Individual practice Notes Lesson Plans Lesson Lesson 1 Measuring Length, Weight, and Temperature Lesson 2 Area, Perimeter and Volume Lesson 3 Applications of Area, Perimeter and Volume Timeframe 2 days 3 days 4 days Teacher Notes: Curriculum Development Resources: Click the links below to access additional resources used to design this unit: www.mathxlforschool.com Unit Overview Content Area: Mathematics Unit 5 Unit Title: Statistics and Probability Target Course/Grade Level: College Math/12th Unit Summary: In this unit students learn the fundamentals of statistics and probability to analyze data. Primary interdisciplinary connections: Trends in Science and Business Education 21st century themes: Modeling real situations; i.e. Stock Market, Scientific Data Unit Rationale: Students need an understanding of statistics and probability for problem solving and analyzing real world data. Learning Targets Standards 4.4 Data Analysis and Probability: Students will develop an understanding of the concepts and techniques of data analysis, probability, and discrete mathematics. Content Statements: Data analysis and probability are used to model situations, solve problems and draw appropriate inferences from data. CPI # Cumulative Progress Indicator (CPI) 4:4.12A Analyze data using technology and use statistical terminology to describe conclusions 4:4.12B Estimate probabilities and make predictions based on experimental and theoretical probabilities 4:4.12C Calculate combinations and justify solutions to counting problems Unit Essential Questions How do you analyze data? How do we apply counting principles? Unit Enduring Understandings Describe trends from data Use the appropriate counting principles for probability and problem solving Unit Learning Targets Students will ... Analyze data and apply statistical measures Apply probability to problem solving Evidence of Learning Summative Assessment: 2 days In groups students will perform a survey and collect real life data. They will use appropriate statistical procedure to analyze and make inferences from the data. Equipment needed: overhead transparencies, copies of notes, Smartboard, Mobi, TI 84 graphing calculators Teacher Resources: Math for Your World textbook, Internet for real life data Unit Overview Content Area: Mathematics Unit 6 Unit Title: Personal Finance Target Course/Grade Level: College Math/12th Unit Summary: In this unit students apply mathematical skills to sales tax, income tax, interest and investments. Primary interdisciplinary connections: Trends in Business Education 21st century themes: Modeling real situations i.e. Stock Market, Buying a home Unit Rationale: Students need to know how to handle personal finance. Learning Targets Standards 4.5 Mathematical Processes: Students will use communication, connections, reasoning, representations and technology to solve problems and communicate mathematical ideas. Content Statements: Mathematical skills are needed to make educated financial decisions. CPI # 4:5.12C Cumulative Progress Indicator (CPI) Use connections among mathematical ideas to explain concepts Recognize that mathematics is used in a variety of contexts outside of mathematics class Apply mathematics in practical situations Unit Essential Questions How do you pay for a home, car, etc? How do you make good investments? How do you handle loans and interest? Unit Enduring Understandings Compare and contrast options for purchases and loans Use the appropriate mathematical skills to handle personal finance Unit Learning Targets Students will ... Use mathematical skills to understand personal finance and investments Evidence of Learning Summative Assessment: 2 days In groups students will come up with a financial plan and payments for a home and a car. Equipment needed: overhead transparencies, copies of notes, Smartboard, Mobi, TI 84 graphing calculators Teacher Resources: Math for Your World textbook, Internet for real life data Formative Assessments Tests and quizzes Discussions Individual practice Explanations of examples Daily homework assignments Notes Lesson Plans Lesson Lesson 1 Percents, Discounts, Sales Tax and Income Tax Timeframe 4 days Lesson 2 Interest and Investments 4 days Lesson 3 Financing and Leasing a Car 2 days Lesson 4 Costs of Renting and Buying a Home 4 days Lesson 5 Credit Cards 2 days Teacher Notes: Curriculum Development Resources Click the links below to access additional resources used to design this unit: www.mathxlforschool.com