Download Introduction to College Mathematics

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