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Summary
Through a careful treatment of number theory and geometry, Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory helps readers understand serious mathematical ideas and proofs. Classroom-tested, the book draws on the authors' successful work with undergraduate students at the University of Chicago, seventh to tenth grade mathematically talented students in the University of Chicago's Young Scholars Program, and elementary public school teachers in the Seminars for Endorsement in Science and Mathematics Education (SESAME).
The first half of the book focuses on number theory, beginning with the rules of arithmetic (axioms for the integers). The authors then present all the basic ideas and applications of divisibility, primes, and modular arithmetic. They also introduce the abstract notion of a group and include numerous examples. The final topics on number theory consist of rational numbers, real numbers, and ideas about infinity.
Moving on to geometry, the text covers polygons and polyhedra, including the construction of regular polygons and regular polyhedra. It studies tessellation by looking at patterns in the plane, especially those made by regular polygons or sets of regular polygons. The text also determines the symmetry groups of these figures and patterns, demonstrating how groups arise in both geometry and number theory.
The book is suitable for pre-service or in-service training for elementary school teachers, general education mathematics or math for liberal arts undergraduate-level courses, and enrichment activities for high school students or math clubs.
Table of Contents
The Triangle Game
The Beginnings of Number Theory Setting the Table: Numbers, Sets and Functions Rules of Arithmetic A New System One's Digit Arithmetic
Axioms in Number Theory Consequences of the Rules of Arithmetic Inequalities and Order
Divisibility and Primes Divisibility Greatest Common Divisor Primes
The Division and Euclidean Algorithms The Division Algorithm The Euclidean Algorithm and the Greatest Common Divisor The Fundamental Theorem of Arithmetic
Variations on a Theme Applications of Divisibility More Algorithms
Congruences and Groups Congruences and Arithmetic of Residue Classes Groups and Other Structures
Applications of Congruences Divisibility Tests Days of the Week Check Digits
Practice Problem Solutions and Hints as well as Exercises appear at the end of each chapter.
Author Bio(s)
Editorial Reviews
All budding mathematicians should have the opportunity to savour this marvelously engaging book. The authors bring to the text an extensive background working with students and have mastered the fine art of both motivating and delighting them with mathematics. Their experience is evident on every page: creative practice problems draw the reader into the discussion, while frequent examples and detailed diagrams keep each section lively and appealing. Herrmann and Sally have carefully charted a course that takes the reader through number theory, introductory group theory, and geometry, with an emphasis on symmetries in the latter two subjects. The result is a labour of love that should inspire young minds for years to come. —Sam Vandervelde, author of Bridge to Higher Mathematics and coordinator of the Mandelbrot Competition
Number, Shape, and Symmetry accomplishes the rare feat of presenting real and deep mathematics in a clear and accessible manner. This book distills the beauty of some of the most fundamental ideas of mathematics and is a terrific resource for anyone interested in exploring these subjects. —Bridget Tenner, Associate Professor of Mathematics, DePaul University
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Mathematical Ideas plus MyMathLab/MyStatLab consists of the textbook plus an access kit for MyMathLab/MyStatLab. #xA0; Mathematical Ideascaptures the interest of non-majors who take the Liberal Arts Math course by showing how mathematics plays an important role in scenes from popular movies and television. By incorporating John Hornsby#x19;s #x1C;Math Goes to Hollywood#x1D; approach into chapter openers, margin notes, examples, exercises, and resources, this text makes it easy to weave this engaging theme into your course. #xA0; The Twelfth Editioncontinues to deliver the su... MOREperlative writing style, carefully developed examples, and extensive exercise sets that instructors have come to expect. MyMathLabcontinues to evolve with each new edition, offering expanded online exercise sets, improved instructor resources, and new section-level videos. #xA0; MyMathLabprovides a wide range of homework, tutorial, and assessment tools that make it easy to manage your course online.
Collaborative Investigation: Investigating an Interesting Property of Number Squares
Chapter 5 Test
6. The Real Numbers and Their Representations
6.1 Real Numbers, Order, and Absolute Value
6.2 Operations, Properties, and Applications of Real Numbers
6.3 Rational Numbers and Decimal Representation
6.4 Irrational Numbers and Decimal Representation
6.5 Applications of Decimals and Percents
Extension: Complex Numbers
Collaborative Investigation: Budgeting to Buy a Car
Chapter 6 Test
7. The Basic Concepts of Algebra
7.1 Linear Equations
7.2 Applications of Linear Equations
7.3 Ratio, Proportion, and Variation
7.4 Linear Inequalities
7.5 Properties of Exponents and Scientific Notation
7.6 Polynomials and Factoring
7.7 Quadratic Equations and Applications
Extension: Complex Solutions of Quadratic Equations
Collaborative Investigation: How Does Your Walking Rate Compare to That of Olympic Race-walkers?
Chapter 7 Test
8. Graphs, Functions, and Systems of Equations and Inequalities
8.1 The Rectangular Coordinate System and Circles
8.2 Lines, Slope, and Average Rate of Change
8.3 Equations of Lines and Linear Models
8.4 An Introduction to Functions: Linear Functions, Applications, and Models
8.5 Quadratic Functions, Graphs, and Models
8.6 Exponential and Logarithmic Functions, Applications, and Models
8.7 Systems of Equations
8.8 Applications of Systems
Extension: Using Matrix Row Operations to Solve Systems
8.9 Linear Inequalities, Systems, and Linear Programming
Collaborative Investigation: Tracking an Epidemic
Chapter 8 Test
9. Geometry
9.1 Points, Lines, Planes, and Angles
9.2 Curves, Polygons, and Circles
Extension: Geometric Constructions
9.3 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem
9.4 Perimeter, Area, and Circumference
9.5 Volume and Surface Area
9.6 Transformational Geometry
9.7 Non-Euclidean Geometry, Topology, and Networks
9.8 Chaos and Fractal Geometry
Collaborative Investigation: Generalizing the Angle Sum Concept
Chapter 9 Test
10. Counting Methods (formerly Chapter 11)
10.1 Counting by Systematic Listing
10.2 Using the Fundamental Counting Principle
10.3 Using Permutations and Combinations
10.4 Using Pascal's Triangle
Extension: Magic Squares
10.5 Counting Problems Involving "Not" and "Or"
Collaborative Investigation: Solving a Traveling Salesman Problem
Chapter 10 Test
11. Probability (formerly Chapter 12)
11.1 Basic Concepts
11.2 Events Involving "Not" and "Or"
11.3 Conditional Probability; Events Involving "And"
11.4 Binomial Probability
11.5 Expected Value
Extension: Estimating Probabilities by Simulation
Collaborative Investigation: Finding Empirical Values of
Chapter 11 Test
12. Statistics (formerly Chapter 13)
12.1 Visual Displays of Data
12.2 Measures of Central Tendency
12.3 Measures of Dispersion
12.4 Measures of Position
12.5 The Normal Distribution
Extension: Regression and Correlation
Collaborative Investigation: Combining Sets of Data
Chapter 12 Test
13. Personal Financial Management (formerly Chapter 14)
13.1 The Time Value of Money
13.2 Consumer Credit
13.3 Truth in Lending
13.4 The Costs and Advantages of Home Ownership
13.5 Financial Investments
Extension: Ponzi Schemes and Other Investment Frauds
Collaborative Investigation: To Buy or to Rent?
Chapter 13 Test
14. Trigonometry (formerly Ch. 10)
14.1 Angles and Their Measures
14.2 Trigonometric Functions of Angles
14.3 Trigonometric Identities
14.4 Right Triangles and Function Values
14.5 Applications of Right Triangles
14.6 The Laws of Sines and Cosines
Extension: Area Formulas for Triangles
Collaborative Investigation: Making a Point about Trigonometric Function Values
Chapter 14 Test
15. Graph Theory
15.1 Basic Concepts
15.2 Euler Circuits
15.3 Hamilton Circuits and Algorithms
15.4 Minimum Spanning Trees
Extension: Route Planning
Collaborative Investigation: Finding the Number of Edges in a Complete Graph
Chapter 15 Test
16. Voting and Apportionment
16.1 The Possibilities of Voting
16.2 The Impossibilities of Voting
16.3 The Possibilities of Apportionment
Extension: Two Additional Apportionment Methods
16.4 The Impossibilities of Apportionment
Collaborative Investigation: Favorites, An Election Exploration
Chapter 16 Test
Appendix: The Metric System
Charles Miller has taught at America River College for many years.
Vern Heeren received his bachelor's degree from Occidental College and his master's degree from the University of California, Davis, both in mathematics. He is a retired professor of mathematics from American River College where he was active in all aspects of mathematics education and curriculum development for thirty-eight years. Teaming with Charles D. Miller in 1969 to write Mathematical Ideas, the pair later collaborated on Mathematics: An Everyday Experience; John Hornsby joined as co-author of Mathematical Ideas on the later six editions. Vern enjoys the support of his wife, three sons, three daughters in-law, and eight grandchildren.
John Hornsby: When a young John Hornsby enrolled in Lousiana State University, he was uncertain whether he wanted to study mathematics education or journalism. Ultimately, he decided to become a teacher. After twenty five years in high school and university classrooms, each of his goals has been realized. His passion for teaching and mathematics manifests itself in his dedicated work with students and teachers, while his penchant for writing has, for twenty five years, been exercised in the writing of mathematics textbooks. Devotion to his family (wife Gwen and sons Chris, Jack, and Josh), numismatics (the study of coins) and record collecting keep him busy when he is not involved in teaching or writing. He is also an avid fan of baseball and music of the 1960's. Instructors, students, and the 'general public' are raving about his recent Math Goes to Hollywood presentations across the country.
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books.google.co.in - This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when... Analysis
Reviews
User reviews
This book is brimming with clarity and intuition. It develops basic Fourier analysis, and features *many* applications to other areas of mathematics. The proofs are elegant, the exercises terrific. It's one of the best books I have ever read.Read full review
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Mathematics
3102 Algebra I
Grade Level: 9-12
Credit: 1 Recommended: Successful completion of 8th grade mathematics
Algebra I uses problem situations, physical models, and appropriate techno logy
to extend algebraic thinking and
engage student reasoning. Problem solving promotes communication and fosters
connections within mathematics, to
other disciplines and to the real world . At the end of the course, students will
take the Algebra I End-of-Course
Examination as 25% of the second semester grade. In order to graduate, a student
must have a credit from Algebra I.
Concepts emphasized: Algebraic properties, scientific notation, operations (real
numbers, expressions, and
polynomials), factoring, equations and inequalities , study of basic function
characteristics and graphs (linear, quadratic , exponential), number line and coordinate plane, linear modeling ,
systems of equations , irrational
numbers, radicals, proportionality, probability and data analysis.
3108 Geometry
Grade level: 9-12
Credit: 1 Recommended: Successful completion of Algebra I
In Geometry, students will investigate and justify geometric concepts and
relationships using both inductive and
deductive reasoning. A credit in Geometry is a graduation requirement. Concepts
emphasized: structures of
geometry, lines, angles, measurement, coordinate geometry, analysis of and
relationships among of two - and three-
dimensional figures, transformational geometry, congruence, similarity and
proportional thinking, inductive and
deductive reasoning, logic, and proof.
3108 GeometryHonors
Grade Level: 9
Credit: 1
Recommended: Successful completion of Algebra I Honors or successful completion
of Algebra I and teacher
recommendation. -solving, research
involving reading/writing,
investigations and explorations, advanced use of technology, and making
connections within the discipline and to
the workplace.
In Honors Geometry, students will investigate and justify geometric concepts and
relationships using both inductive
and deductive reasoning. A credit in Geometry is a graduation requirement.
Concepts emphasized: structures of
geometry, lines, angles, measurement, coordinate geometry, analysis of and
relationships among of two- and three-
dimensional figures, transformational geometry, congruence, similarity and
proportional thinking, inductive and
deductive reasoning, logic, and proof.
3103 Algebra II
Grade Level: 9-12
Credit: 1 Recommended: Successful completion of Geometry fol lowing :103 Algebra II Honors
Grade Level: 10
Credit: 1
Recommended: Successful completion of Geometry Honors, or teacher recommendation the workplace.
following:126 Pre-Calculus Honors
Grade Level: 11
Credit: 1
Recommended: Successful completion of Algebra II Honors or teacher
recommendation
the workplace.
Pre-calculus is an advanced mathematics course using meaningful problems and
appropriate technologies to build
upon the study of functions and algebraic concepts in order to develop the
underpinnings of calculus. Concepts
emphasized: trigonometric functions and trigonometric in problem-solving;
vectors; complex numbers; algebraic
functions, their characteristics, graphs, transformations; limits; data
analysis, modeling, and predicting exponential
and logarithmic functions; applications of conic sections; recursive and
explicit sequences; series and sums.
3136 Probability and Statistics
Grade Level: 11-12
Credit: 0.5-1
Recommended: Successful completion of Algebra II or Algebra II Honors
In Statistics, students use appropriate technology to study probability,
descriptive statistics, and inferential statistics.
This course prepares students for a major/career involving research, such as
history, psychology, economics,
journalism, statistics or education. Concepts emphasized: collecting,
displaying, interpreting, and analyzing data;
surveys and experimental de sign ; drawing conclusions about a population from a
sample and predicting with data.
3129 AP Statistics
Grade Level: 11-12
Credit: 1
Recommended: Successful completion of Algebra II Honors or teacher
recommendation
The AP Statistics course introduces students to the major concepts and tools for
collecting, interpreting, analyzing
and drawing conclusions from data. Students may be granted credit or advanced
placement or both for an
introductory college statistics based on the AP Exam score. A graphing
calculator is required for study during the
course as well as for the AP Exam. This course prepares students for a
major/career involving research, such as
history, psychology, economics, journalism, statistics or education. Concepts
emphasized: data exploration;
sampling and experimentation; probability, probability distributions, and
simulation; statistical inference.
3127 AP Calculus (AB)
Grade Level: 11-12
Credit: 1
Recommended: Successful completion of Pre-Calculus or teacher recommendation
AP Calculus AB primarily focuses on the development of the concepts of limit,
differential and integral calculus. In
theory and problem-solving, students gain experience with methods and
applications. As the course emphasizes a
multi-representational approach to calculus with concepts, results, and problems
being expressed graphically, numerically , analytically, and verbally, a graphing calculator is required for
this course. In May students will take the
College Board AP Calculus AB examination; many universities grant credit based
on the score attained.
Concepts emphasized: the derivative as a limit and rate of change and its
applications; function analysis and curve
sketching; Fundamental Theorem of Calculus; the definite integral and its
applications; differential equations.
3128 AP Calculus (BC)
Grade Level: 11-12
Credit: .5
Recommended: B or higher in Calculus AP (AB)
AP Calculus BC is an extension of Calculus AB rather than an enhancement; common
topics require a similar depth
of understanding. A graphing calculator is required for this course. In May
students will take the College Board AP
Calculus BC examination; many universities grant credit based on the score
attained. Calculus BC includes concepts
of the AB course as well as additional concepts beyond those of AB course:
further applications of integrals,
parametric, polar, and vector functions; sequences, series, and polynomial
approximations ; advanced techniques of
integration.
3197 ACT Mathematics
Grade Level: 11-12
Credit: .5 elective
Recommended: Successful completion of Algebra II
Under the direction of a content teacher, students will complete practice tests
which will be utilized to inform the
teacher of instructional needs.. They will spend the semester with instruction
on the essential skills and
competencies that are indicated, review other essential ACT assessed skills,
complete practice tests and exercises,
and review basic test-taking skills.
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Im taking quite a bit of math this year and would like to know what is learned from these courses
.pre calculus is it just alg 2and trig?
Physics 1honors would pre calc help me to get a better understanding?
College algebra- is it like pre calc (taking it at the community college)
AP Statistics- I have no idea what math is in it, is it hard
Also do you think these classes will look good for UCF? Thanks in advance also going to be a junior
I had no idea what category to put this in, sorry in advance
Precalc is typically a review of trig and algebra 2.
College Algebra is typically a remedial class equivalent to algebra 2.
AP Statistics is concerned with analysis of data and various types of ways of analyzing data (for example, a way to calculate whether there is a correlation between two variables and how strong that correlation is). It also covers basic probability. It also covers statistical inference testing. For example, given a set of data (let's say, heights of randomly selected people), you will run a statistical test (and this involves looking at a normal distribution) , and determine with a 95% level of confidence that the actual average height of humans falls within the range of X to Y.
The math in AP Stat is very simple (plug into calculator). It's all about memorizing rules and equations and different situations.
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Introductory Algebra for College Students 3rd Edition
0130328391
9780130328397 experience in algebra and for those who need a review of basic algebraic concepts.The goal of the Blitzer Algebra series is to provide students with a strong foundation in Algebra. Each text is designed to develop students' critical thinking and problem-solving capabilities and prepare students for subsequent Algebra courses as well as "service" math courses. Topics are presented in an interesting and inviting format, incorporating real world sourced data and encouraging modeling and problem-solving. «Show less... Show more»
Rent Introductory Algebra for College Students 3rd Edition today, or search our site for other Blitzer
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Using and Understanding Mathematics: A Quantitative Reasoning Approach increases students' mathematical literacy so that they better understand the mathematics used in their daily lives.
Ideal for courses that emphasize quantitative reasoning, Bennett and Briggs prepare students to use math effectively to make better decisions throughout their lives. Contents are organized with that in mind, with engaging coverage in sections like Taking Control of Your Finances, Dividing the Political Pie, and a full chapter about Mathematics and the Arts.
This Fifth Edition offers new hands-on Activities for use with students in class, new ways for students to check their understanding through Quick Quizzes, and a new question type in MyMathLab that applies math to recent events in the news. In addition, the authors increase their coverage of consumer math, and provide a stronger emphasis on technology through new Using Technology features and exercises. The new Insider's Guide provides instructors with tips and ideas for effective use of the text in teaching the course.
The real-world focus turns studentsí attention to the math they will need for their daily lives, and keeps them engaged during the course. A wide range of exercises and problem types end each unit, making it easy for instructors to create assignments to fit their course goals.
Jeffrey Bennett specializes in mathematics and science education. He has taught at every level from pre-school through graduate school, including more than 50 college courses in mathematics, physics, astronomy, and education. His work on Using and Understanding Mathematics began in 1987, when he helped create a new mathematics course for the University of Colorado's core curriculum. Variations on this course, with its quantitative reasoning approach, are now taught at hundreds of colleges nationwide. In addition to his work in mathematics, Dr. Bennett (whose PhD is in astrophysics) has written leading college-level textbooks in astronomy, statistics, and the new science of astrobiology, as well as books for the general public. He also proposed and developed both the Colorado Scale Model Solar System on the University of Colorado at Boulder campus and the Voyage Scale Model Solar System, a permanent, outdoor exhibit on the National Mall in Washington, DC. He has recently begun writing science books for children, including the award-winning Max Goes to the Moon and Max Goes to Mars. When not working, he enjoys swimming as well as hiking the trails of Boulder, Colorado with his family.
William L. Briggs has been on the mathematics faculty at the University of Colorado at Denver for 22 years. He teaches numerous courses within the undergraduate and graduate curriculum, and has special interest in teaching calculus, differential equations, and mathematical modeling. He developed the quantitative reasoning course for liberal arts students at University of Colorado at Denver supported by his textbook Using and Understanding Mathematics. He has written two other tutorial monographs, The Multigrid Tutorial and The DFT: An Owner's Manual for the Discrete Fourier Transform, as well as Ants, Bikes, Clocks, a mathematical problem solving text for undergraduates. He is a University of Colorado President's Teaching Scholar, an Outstanding Teacher awardee of the Rocky Mountain Section of the MAA, and the recipient of a Fulbright Fellowship to Ireland. Bill lives with his wife, Julie, and their Gordon setter, Seamus, in Boulder, Colorado. He loves to bake bread, run trails, and rock climb in the mountains near his home.
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Tasks
Welcome to the Mindset Project for Mathematics Instruction
Mathematics INstruction using Decision Science and Engineering Tools (MINDSET) uses decision making tools from Industrial and Systems Engineering and Operations Research in a fourth-year high school mathematics curriculum. Principal performance related goals of the project are to improve upon the math students' ability to formulate and solve multi-step problems and interpret results, and to improve students' attitude toward mathematics.
Summer 2013 Teacher Workshop Registration is now closed
We are sorry if you missed out, but the workshops are filled. Two workshops are being held this year in Raleigh, NC (June 18-21 and June 24-27) and Greensboro, NC (July 23-26 and July 29-August 1).
Getting Started
Practical mathematics applications with problems and solutions are featured throughout the course materials. Educators will need a user name and password to gain access to chapters -- continue reading for details.
New to MINDSET ?
If you are not currently part of the MINDSET project then please send a request for information to the MINDSET Project Help Desk. Please describe your request and give MINDSET some information about yourself.
Site Features
The Project menu on the left provides access to the Deterministic and Probabilistic topics.
Participation by educators and feedback is critical to the success of the MINDSET Project, so this site also allows educators to:
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How is the
study of mathematics a creative venture? How can
mathematics help us understand the world around us and the
social constructs we live by? In this class we will answer these
questions by exploring a number of interesting, contemporary applications. At the same time, we will examine the nature of mathematics and mathematics research.
Topics:
Mathematics
& Social Justice
Graph Theory
& Management Science
Mathematics of
Symmetry & Secrecy
Objectives:
Strengthen our
logic & reasoning skills
Improve our
understanding of what the field of mathematics encompasses
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Course Content and Outcome Guide for MCH 130
Date:
17-NOV-2011
Posted by:
Curriculum Office
Course Number:
MCH 130
Course Title:
Machine Shop Trigonometry
Credit Hours:
2.5
Lecture hours:
0
Lecture/Lab hours:
50
Lab hours:
0
Special Fee:
$15
Course Description
Introduces the rules, methods and procedures for using trigonometry formulas that deal with both the sides and the angles of the right triangle and oblique triangle to solve for the unknown parts. Prerequisite: MCH 100. Audit available.
Addendum to Course Description
Intended Outcomes for the course
Upon successful completion of this course students will be able to:
1. Use sine, cosine, and tangent and functions to determine coordinates of a part to be machined.
Course Activities and Design
MCH 130 will be presented by means of audio-visual presentations, demonstrations, lab experiences, and research activities. The course activities and design emphasize the development of skills and knowledge outcomes prescribed by established industry standards. The identified outcomes will be achieved by means of individual and team activities.
Outcome Assessment StrategiesMathematics for the Machine Technology, by Robert D. Smith
Course Content (Themes, Concepts, Issues and Skills)
Machine Shop Math / Trigonometry consists of the following modules:
Angles - Other than linear measurement, the most common measurement with which the machinists work is the measurement of angles. Basically angular measurement is finding the size of angles.
Triangles - A plan figure is a flat, two-dimensional figure. A triangle is a closed plane figure that has three straight sides. The triangle also has three interior angles whose sum is 1800. Triangles are divided into two categories - right triangles and oblique triangles. A right triangle has a right angle as an interior angle. An oblique triangle does not have a right angle.
Trigonometry - Most of the trigonometric relationships used by machinists are based on the right triangle. The right triangle formulas we used previously dealt with the length of the sides only. Trigonometry formulas deal with both the sides and the angles.
Related Instruction
Computation
Hours: 65
Outcome:
Use sine, cosine, and tangent and functions to determine coordinates of a part to be machined.
Through direct instruction students practice Right triangle trig using Pythagorean theorem, Sin, Cosine, and Tangent functions. Some obtuse and acute triangle trig using the law of sins.
Students are given blue prints of machine parts that require the use of trigonometry to determine bolt hole patterns and how to locate x and y coordinates.
Communication
Hours: 10
Outcomes:
· Students will be able to communicate technical information to co-workers, clients, and or engineers
Technical communication skills are practiced as students need to communicate questions related to higher level math.
Students learn to ask specific questions related to their current problem by defining the case and what they need to complete the issue at hand.
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Introduction
Mathematica is a software package which is ideal for communicating scientific ideas, whether this is visualization of a concept in an intro-level course, or creating a simulation of a new idea related to research. Mathematica is used in virtually all of the world's top universities and colleges, and is commonly used in the following types of departments – Mathematical Sciences, Physical Sciences, Business and Finance, Life Sciences, Engineering, Computer Science.
Wolfram|Alpha Pro is an engine for computing answers and providing knowledge. It works by using its vast store of expert-level knowledge and algorithms to automatically answer questions, do analysis, and generate reports. Wolfram|Alpha's long-term goal is to make all systematic knowledge computable and broadly accessible.
Activating Mathematica on a PC
In the Mathematica License Agreement window, read the Mathematica License Agreement, check the box by "I accept the terms of this agreement", and click
OK to launch the program. (Image)
Reviewed: 0213 By: JB
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The student workbook includes a set of lesson review boxes accompanied by questions that provide practice for previously taught concepts and the concepts taught in the lesson. "Exploring Math Through..." sections help students understand how ordinary people use algebraic math, providing concrete examples of how math is useful in life. Students will need to supply paper to work the problems. 333 pages, softcover.
The teacher's guide includes the main concepts, lesson objectives, materials needed, teaching tips, the assignment for the day, and the reduced student pages with the correct answers supplied. Each lesson will take approximately 45-60 minutes, and is designed to be teacher-directed.
Product:
Horizons Algebra 1 Teacher's Guide
Vendor:
Alpha Omega Publications
Edition Description:
JMT080
Binding Type:
Paperback
Media Type:
Book
Minimum Grade:
8th Grade
Maximum Grade:
12th Grade
Number of Pages:
333
Weight:
0.0001 pounds
Length:
11 inches
Width:
8.5 inches
Street Date:
8/15/2012
Subject:
Math
Curriculum Name:
Horizons/Alpha Omega
Learning Style:
Auditory, Kinesthetic, Visual
Teaching Method:
Charlotte Mason, Classical, Traditional, Unit Study, Unschooling
There are currently no reviews for Horizons Algebra 1 Teacher's Guide.
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Our mission is to introduce students to the core concepts in mathematics and then lead them to more advanced inquiries and show them how to apply mathematical knowledge to different areas of scientific studies. We aims to educate outstanding talents in mathematics who can put what they have learned to use in the real world.
II. Characteristics
1. Emphasis on both teaching of fundamentals and advanced research
2. Computerized teaching
3. Teachers and students grow together
4. Balance between theories and applications
5. Current research efforts are directed toward nonlinear analysis, theories of computation, statistics and information systems
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Blog Archives
Much of Algebra I and Algebra II is spent learning how to solve polynomial equations- in particular, finding solutions / roots / zeroes of an equation. This process can be very tedious, but fortunately, there are some theorems and techniques…
I've developed a great online tool that can perform the division for you. You'll need to know at least a root to test for and will need the terms of the equation written in descending order (power). The online synthetic…
Below are some handy crib sheets. Most college level math instructors would rather students know how to apply different formulas and rules than memorize them. Creating your own crib sheet is a great study tool for tests and exams as…
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Description
This book covers mathematics of finance, linear algebra, linear programming, probability, and descriptive statistics, with an emphasis on cross-discipline principles and practices. Designed to be reader-friendly and accessible, it develops a thorough, functional understanding of mathematical... Expand concepts in preparation for their application in other areas. Each chapter concentrates on developing concepts and ideas followed immediately by developing computational skills and problem solving. Three-part coverage presents a library of elementary functions, finite mathematics, and calculus.Collapse
9th Edition. Pre-loved books for the budget-conscious consumer. With more than 50 years' experience,
[...]
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Understandable Statistics (Hardcover)
9780618949922
ISBN:
0618949925
Edition: 9 Publisher: Houghton Mifflin Company
Summary: This algebra based text is a thorough yet approachable statistics guide for students. The new edition addresses the growing importance of developing students' critical thinking and statistical literacy skills with the introduction of new features and exercises
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New Brunswick ACT would review with the student the basic concepts in algebra I such as equations and inequalities, the coordinate system and functions of multiple variables; then examine in greater depth. Other topics will then include quadratic functions and factoring, polynomials; including complex numbers a...
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This well-thought-out program, contained on 2 DVDs, would be a great help to any student. The instructor uses his extensive experience as a tutor to clearly explain the concepts chosen Trigonometry and Pre-Calculus Tutor, scroll to the concept needed, and watch! The following concepts are covered:
Complex Numbers
Exponential Functions
Logarithmic Functions
Solving Exponential and Logarithmic Equations
Angles
Finding Trig Functions Using Triangles
Finding Trig Functions Using the Unit Circle
Graphing Trig Functions
Trig Identities
This program is without bells and whistles; it is just the DVDs. No curriculum or worksheets are used. It simply explains the concept
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College Algebra Essentials CourseSmart eTextbook, 4th Edition
Description
Bob Blitzer has inspired thousands of students with his engaging approach to mathematics, making this beloved series the #1 in the market. Blitzer draws on his unique background in mathematics and behavioral science to present the full scope of mathematics with vivid applications in real-life situations. Students stay engaged because Blitzer often uses pop-culture and up-to-date references to connect math to students' lives, showing that their world is profoundly mathematical.
With the Fourth Edition, Blitzer takes student engagement to a whole new level. In addition to the multitude of exciting updates to the text and MyMathLab® course, new application-based MathTalk videos allow students to think about and understand the mathematical world in a fun, yet practical way. Assessment exercises allow instructors to assign the videos and check for understanding of the mathematical concepts presented.
Table of Contents
P. Prerequisites: Fundamental Concepts of Algebra
P.1 Algebraic Expressions, Mathematical Models, and Real Numbers
P.2 Exponents and Scientific Notation
P.3 Radicals and Rational Exponents
P.4 Polynomials
Mid-Chapter Check Point
P.5 Factoring Polynomials
P.6 Rational Expressions
SUMMARY, REVIEW, AND TEST
REVIEW EXERCISES
CHAPTER P TEST
1. Equations and Inequalities
1.1 Graphs and Graphing Utilities
1.2 Linear Equations and Rational Equations
1.3 Models and Applications
1.4 Complex Numbers
1.5 Quadratic Equations
Mid-Chapter Check Point
1.6 Other Types of Equations
1.7 Linear Inequalities and Absolute Value Inequalities
SUMMARY, REVIEW, AND TEST
REVIEW EXERCISES
CHAPTER 1 TEST
2. Functions and Graphs
2.1 Basics of Functions and Their Graphs
2.2 More on Functions and Their Graphs
2.3 Linear Functions and Slope
2.4 More on Slope
Mid-Chapter Check Point
2.5 Transformations of Functions
2.6 Combinations of Functions; Composite Functions
2.7 Inverse Functions
2.8 Distance and Midpoint Formulas; Circles
SUMMARY, REVIEW, AND TEST
REVIEW EXERCISES
CHAPTER 2 TEST
CUMULATIVE REVIEW EXERCISES (CHAPTERS 1-2)
3. Polynomial and Rational Functions
3.1 Quadratic Functions
3.2 Polynomial Functions and Their Graphs
3.3 Dividing Polynomials; Remainder and Factor Theorems
3.4 Zeros of Polynomial Functions
Mid-Chapter Check Point
3.5 Rational Functions and Their Graphs
3.6 Polynomial and Rational Inequalities
3.7 Modeling Using Variation
SUMMARY, REVIEW, AND TEST
REVIEW EXERCISES
CHAPTER 3 TEST
CUMULATIVE REVIEW EXERCISES (CHAPTERS 1-3) 410
4. Exponential and Logarithmic Functions
4.1 Exponential Functions
4.2 Logarithmic Functions
4.3 Properties of Logarithms
Mid-Chapter Check Point
4.4 Exponential and Logarithmic Equations
4.5 Exponential Growth and Decay; Modeling Data
SUMMARY, REVIEW, AND TEST
REVIEW EXERCISES
CHAPTER 4 TEST
CUMULATIVE REVIEW EXERCISES (CHAPTERS 1-4)
5. Systems of Equations and Inequalities
5.1 Systems of Linear Equations in Two Variables
5.2 Systems of Linear Equations in Three Variables
5.3 Partial Fractions
5.4 Systems of Nonlinear Equations in Two Variables
Mid-Chapter Check Point
5.5 Systems of Inequalities
5.6 Linear Programming
SUMMARY, REVIEW, AND TEST
REVIEW EXERCISES
CHAPTER 5 TEST
CUMULATIVE REVIEW EXERCISES (CHAPTERS 1-5)
Appendix: Where Did That Come From? Selected Proofs
Answers to Selected Exercises
Subject Index
Photo Credits
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This would be a problem if it was a history book like lenoat's school, as schools should be getting new history textbooks every five or so years in my opinion (unless no new edition of the book they're using has come out and there's no better textbook that they can find), as companies like to publish new editions of history books every few years to keep up-to-date with modern events and to rewrite history a bit in accordance with those events. An Algebra II book, however, isn't going to be affected by that, so the school doesn't really need to replace them that often. Sure, the problem may include something that doesn't exist, but will that affect your answers to the problem or the concept they're teaching you? If it does, then I think there might be a problem.
Our math textbooks in our school are the first to be replaced and last is our history books. For the most part the required classes don't need history that comes up to present day. My sophomore US history class finished the year with the Vietnam War which is over 30 years old.
Even books that deal with older history need to be replaced. Vietnam is actually more recent history (especially in comparison to something like ancient Greece or Rome), and is especially something that would be updated and cause a new textbook to be needed. Authors of history books like to often produce new editions so that they can update history to be more accurate and more in-line with what is accepted to have happened in modern times. This can change a lot, which some people find surprising, but shouldn't. My AP US History textbook for this year (America: A Narrative History by George Brown Tindall and David Emory Shi) is on its 8th edition, and it was first published in 1984. History changes quickly, and the world should not be lagging behind it.
True but most history is also just that history, set in stone, fact. You are right that the opinion on that does change a little because we don't exactly know everything that went on then but I know that math like his Algebra book is changing in the same way as national organizations are always trying to find what is the best way to do math or whatever topic you are in. I have also heard many good arguments for why math textbooks are and should be updated more than history books (most from math teachers but still good reasons). The hard thing I think for most schools is the cost of updating 100 books is a couple 100 dollars a piece so you could spend 2000-3000 dollars on one class not even the entire history section but just government or just algebra I
I find that it's the opposite way around. There is rarely any piece of history that can be set in stone, whereas almost the entirety of mathematics will never change because most of the theorems and axioms taught are generally accepted as fact. This is the same reason I prefer math over history - it's far more static, so when you learn something, you almost never need to go back and revise your knowledge on it. We're always uncovering new things about history, though, and we're always changing the way we interpret it. For example, consider a theoretical textbook that has a chapter detailing US involvement in the Middle East in the late 20th century. The first edition of this book was published before the September 11th attacks, and praises the theocratic rebel leaders in the area for standing up against the Soviets, and for being leaders of a new democracy in the area. Obviously, after the September 11th attacks, such leaders were no longer seen that way. So what do the authors do? They create a new edition of the textbook detailing the history as the United States fighting the Soviet Union but failing to predict the outcome of letting the rebel groups take power in the area. Math, on the other hand, does not face such changes that often. Algebra II and Geometry are especially areas that do not need to be revised - much of those classes are based around concepts will likely never change, and the methods to solve problems involving those concepts are also unlikely to change. I'm not sure what level of math you've taken, so I'll assume at least Algebra II. Consider a problem involving the properties of logarithms. Say you're asked to simplify log(2x3 )+log(6x). Using the properties of logarithms, you would get log(12x4 ), and then 4log(12x). There really are no alternate ways to solve such a problem, so such a concept will never be change. Nor would properties of exponents, matrices, or polynomials. Math teachers often want new books because math students for some reason are very good at destroying the books. However, I've not met a math teacher that is always complaining for a new edition, just new books to replace the 8 year old books that are missing 20 pages and half a cover.
I agree. However, I do find it odd that the publisher didn't bother changing a few words when they reprinted the textbook to avoid offending 9/11 victims. Math textbooks aren't the only problem. My freshman year world history book said "Egyptian president Hosni Mubarak is working to implement democratic reforms in the country." I laughed at that.
It's the same in my district, and even within my school. In my school, the entire foreign language department has iPads. In my district's middle school, there's a Kindle for every student and they do all their textbooks through that. However, that was only something done after I got into high school (in fact, they did that last year, and I'm a junior now) because the tech department was given a huge budget that it didn't need that year.
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Numerical the intelligent application of approximation techniques to the type of problems that commonly occur in engineering and the physical sciences. Students learn why the numerical methods work, what type of errors to expect, and when an application might lead to difficulties. The authors also provide information about the availability of high-quality software for numerical approximation routines. The techniques are essentially the same as those covered in the authors' top-selling Numerical Analysis text, but in this text, full m... MOREathematical justifications are provided only if they are concise and add to the understanding of the methods. The emphasis is placed on describing each technique from an implementation standpoint, and on convincing the student that the method is reasonable both mathematically and computationally. This book emphasizes the intelligent application of approximation techniques to the type of problems that commonly occur in engineering and the physical sciences. Readers learn why the numerical methods work, what type of errors to expect, and when an application might lead to difficulties. The authors also provide information about the availability of high-quality software for numerical approximation routines. In this book, full mathematical justifications are provided only if they are concise and add to the understanding of the methods. The emphasis is placed on describing each technique from an implementation standpoint, and on convincing the reader that the method is reasonable both mathematically and computationally.
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Past Courses
Fractals, Dynamics and Chaos (2002, 2003, 2005, 2007) This was a course where students studied the geometric and analytic properties of certin fractals obtained from iterative schemes. Students used computer programs to run the iterative schemes and viewed the resulting images which showed interesting patterns coming from seemingly random processes. Students focused on the mathematical aspects of fractals and chaos.
Fourier Analysis (2002, 2003, 2004) This course introduced in an easy, intuitive form the idea of decomposing a function into frequencies and show some examples of practical applications of such decompositions. The emphasis was on hands on experience. Students played with digital images and sound in a computer classroom.
Geometry (2005)Going 'Round in Circles: Circles provide a unifying theme for a study of Euclidean and non-Euclidean geometries. Many opportunities for student investigation were possible. Extensive use of the Geometer's Sketchpad were expected. Time was devoted to this in four parts. 1) Euclidean geometry of the circle. 2) Studies in mutual tangency of congruent circles. 3) Circles on the sphere and a new geometry. 4) Circles and a model of non-Euclidean geometry (hyperbolic).
Graph Theory and Matrices (2003, 2004, 2006) Students covered special applications of linear algebra. Incidence matrices will be used to introduce matrix analysis of graphs; students studied how matrix operations are related to interesting properties of the graphs. Once matrices and matrix multiplication have been introduced, some ideas of computer graphics were explored. The students finished the week by using basic linear transformations to make their own "mini" movies with Mathematica or Matlab.
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KS3 Algebra – Formulae (MEP – Year 8 – Unit 12)
Worksheets and activities. The topic of Formulae from the Year 8 book of the Mathematics Enhancement Program (MEP). For information about these resources and an index for the whole collection please visit
Keywords: Algebra, Formulae, Formula, Substitution, Ev More…aluate, Positive, Negative, Change the Subject, Linear, Non Linear, Equation, Solve, Solution, Laws.
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Discrete Mathematics
9780198534273
ISBN:
0198534272
Publisher: Oxford University Press, Incorporated
Summary: This text is a carefully structured, coherent, and comprehensive course of discrete mathematics. The approach is traditional, deductive, and straightforward, with no unnecessary abstraction. It is self-contained including all the fundamental ideas in the field. It can be approached by anyone with basic competence in arithmetic and experience of simple algebraic manipulations. Students of computer science whose curric...ulum may not allow the study of many ancillary mathematics courses will find it particularly useful. Mathematics students seeking a first approach to courses such as graph theory, combinatorics, number theory, coding theory, combinatorial optimization, and abstract algebra will also enjoy a clear introduction to these more specialized fields. The main changes to this new edition are to present descriptions of numerous algorithms on a form close to that of a real programming language. The aim is to enable students to develop practical programs from the design of algorithms. Students of mathematics and computer science seeking an eloquent introduction to discrete mathematics will be pleased by this work
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Channels: Computer-Based Math™
Tim Garry, Head of Mathematics at the International School of Aberdeen and Textbook Author for Pearson Publishing, shares examples of how he integrates handheld devices and computers into his teaching in this presentation from The Computer-Based Math™ Education Summit 2011.
At The Computer-Based Math™ Education Summit 2011, James Nicholson, Consultant for the SMART Centre at Durham University, shares his experiences developing data visualizations to support the teaching and learning of mathematical and statistical techniques in other courses, including social science and physical science.
John Perram, Professor at the School of Mathematics and Statistics at the University of New South Wales, explains how he developed a computer-based alternative to teaching freshman mathematics in this presentation from The Computer-Based Math™ Education Summit 2011.
In this presentation from The Computer-Based Math™ Education Summit 2011, Bruce Schneider, Professor at the University of Toronto, shares how a computer-based curriculum in his large introductory statistics courses provides his students with hands-on problem solving experience.
Debra Woods, Director of the NetMath program at the University of Illinois at Urbana-Champaign, shares her experiences with re-purposing how math is taught from classroom to online in this presentation from The Computer-Based Math™ Education Summit 2011.
computerbasedmath.org is a project to build a new math curriculum with computer-based computation at its heart. In this talk from the Wolfram Technology Conference 2011, Conrad Wolfram discusses the concept, progress, and plans.
In this talk from The Computer-Based Math™ Education Summit, Sue Black presents her thoughts on the importance of math and computer science in society and explains how she hopes her non-profit organization, The <goto> foundation, will promote computing within the general public.
Tim Oates of Cambridge Assessment shares his views on the changing needs for math in society and the importance of transforming how children view the subject in this video from The Computer-Based Math™ Education Summit.
In this Computer-Based Math™ Education Summit presentation, Maggie Philbin describes how TeenTech, a program she founded to give teenagers a look at the wide range of science and technology careers, is changing views about the importance of math and STEM subjects.
Mike Ellicock shares the vision of National Numeracy, a charity focused on increasing the level of mathematical thinking and understanding across the population of the United Kingdom, in this presentation from The Computer-Based Math™ Education Summit.
In this presentation from The Computer-Based Math™ Education Summit 2011, Benjamin Koo, an Associate Professor at Tsinghua University/Ningbo Polytechnic, shares stories of how he uses physical and digital spaces to bring modern technology into his courses.
David Stern, Lecturer at Maseno University, shares his thoughts on different countries' mathematical cultures and how Computer-Based Math™ematics can move math education forward around the world in this presentation from The Computer-Based Math™ Education Summit 2011.
In this presentation from The Computer-Based Math™ Education Summit 2011, Simon Walsh, Managing Director of Maths Doctor, shows how his company is providing e-tutoring in maths and physics to students throughout the UK and discusses some of his future goals.
Rosamund Sutherland, a Professor of Education at the University of Bristol, shares details of her research on using technology to support science and math education in Rwanda during The Computer-Based Math™ Education Summit 2011.
Rupa Chilvers offers insights on how her organization, Forward 25 Careers, engages the disenfranchised with computer-based learning in this presentation from The Computer-Based Math™ Education Summit 2011.
Bruce Dickson, a retired schoolmaster, shares his personal success stories for engaging the disenfranchised with Computer-Based Math™ during this presentation from The Computer-Based Math™ Education Summit 2011.
Deborah Donnelly-McLay, an airline pilot and professor of aviation, shares her views on how to transform STEM education, especially for children and women, through Computer-Based Math™ and enhanced learning activities in this presentation from The Computer-Based Math™ Education Summit 2011.
In this video from The Computer-Based Math™ Education Summit 2011, Cristina Luminea, Founder and Managing Director of ThoughtBox, shares her thoughts about the role games, competitions, and other new modalities for learning will play in the future of STEM education.
In this video from The Computer-Based Math™ Education Summit 2011, Richard Lissaman, Deputy Programme Leader of Further Mathematics Network, shares his thoughts about the role games, competitions, and other new modalities for learning will play in the future of STEM education.
In this video from The Computer-Based Math™ Education Summit 2011, Maria Droujkova, Director of Natural Math, shares her thoughts about the role games, competitions, and other new modalities for learning will play in the future of STEM education.
In this video from The Computer-Based Math™ Education Summit 2011, Eddie Wilde, Qualifications Team Manager at Oxford Cambridge and RSA Examinations (OCR), discusses how assessment will work for Computer-Based Math™ematics.
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hands-on introduction to the tools needed for rigorous and theoretical mathematical reasoning
Successfully addressing the frustration many students experience as they make the transition from computational mathematics to advanced calculus and algebraic structures, Theorems, Corollaries, Lemmas, and Methods of Proof equips students with the tools needed to succeed while providing a firm foundation in the axiomatic structure of modern mathematics.
This essential book: * Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs * Reinforces the foundations of calculus and algebra * Explores how to use both a direct and indirect proof to prove a theorem * Presents the basic properties of real numbers * Discusses how to use mathematical induction to prove a theorem * Identifies the different types of theorems * Explains how to write a clear and understandable proof * Covers the basic structure of modern mathematics and the key components of modern mathematics
A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. In addition, the author has supplied many clear and detailed algorithms that outline these proofs.
Theorems, Corollaries, Lemmas, and Methods of Proof uniquely introduces scratch work as an indispensable part of the proof process, encouraging students to use scratch work and creative thinking as the first steps in their attempt to prove a theorem. Once their scratch work successfully demonstrates the truth of the theorem, the proof can be written in a clear and concise fashion. The basic structure of modern mathematics is discussed, and each of the key components of modern mathematics is defined. Numerous exercises are included in each chapter, covering a wide range of topics with varied levels of difficulty.
Intended as a main text for mathematics courses such as Methods of Proof, Transitions to Advanced Mathematics, and Foundations of Mathematics, the book may also be used as a supplementary textbook in junior- and senior-level courses on advanced calculus, real analysis, and modern algebra. less
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Trade in Math for 3D Game Programming and Computer Graphics (Charles River Media Game Development) for an Amazon.co.uk gift card of up to £1.78, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book Description is an excellent book for those who have some mathematical experience. Proofs of the key theorems are given, and examples for the reader to work through, with answers at the back of the book. Descriptions are concise and to the point, with diagrams provided where needed. The first 3 chapters will be a useful reference to anyone studying vectors and matrices, not just to game developers. The rest of the book moves on to more specific areas of game programming such as collision detection and illumination. Finally the book ends with 3 excellent chapters on linear, rotational and fluid physics. The author states that some prior knowledge of vectors is required, but I think the first chapter is detailed enough for people with little or no experience to work through and understand. I would highly recommend this book for anyone with some experience of algebra and trigonometry.
This book Rocks!!!. Buy this book now. This book explains math concepts in relation to game programming. Really good collection of math concepts. It even covers the physics (ie) implementing it through maths. Its sad to see people struggle to write games. The problem lies in people not learning the math concepts first before delving into programming. In this regard this book is very good. It explains everything clearly. But be warned though. The book is purely mathematical and it is like any math text book. But the concepts are easily illustrated with theorems and their proofs. All derivations are explained clearly and hence it provides clear understanding of the math concepts. Though i knew many of the things before reading this book, it serves as a single reference for the math concepts. Granted, this book dosen't cover everything, no single book can do that, but it serves as a guide and a very good one at that. You should know atleast some math like calculus, basic algebra etc to achieve full benefit from this book Overall this book is highly recommended for any 3d game programmer. Buy this book NOW if you want to learn the concepts behind those amazing games and engines.
OK, I don't work for River publishing, or whatever they are called, but I all I can say is I was immediately impressed with this book, and as you work through it the quality remains at the same high value.
Now that said if you want an easy reading book you can read in the bath then maybe this is not for you. But if you want something that is comprehensive and will be eternally useful then get this book. Any work in 3D requires you use vectors as it is the only sensible way. So you need to get fluent with the vector approach; although once you do you'll wonder how you ever managed 'the old way.' The book starts with a description of vectors, then moves on to matrices (not matrixes, please!). From these building blocks it then takes you on a journey that every 3D programmer needs to know.
As I said, you need to take it as a reasonable pace -- it is a maths book after all. But it is written in such a clear manner, and the layout is so damned good that you will enjoy reading it. Well at least I did; when you see how elegant some of the conclusions are (again thanks to vectors and matrices) you will be impressed.
Did I mention you must buy it? No? Well let me do that now. Go and buy it! I guaranteed you won't regret it. I always write the dates of when I buy books inside the front cover, habit I suppose. But looking at my copy I've just pulled it off the shelf (again!) and the date is 23/09/06 -- so after four and a half years I'm still using it. Cool or what?
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The d'Arbeloff Interactive Mathematics Project
Mathematics is the language of science and engineering. Our goal is to
help our students become fluent in it. We want them to know how to frame
questions mathematically and to recognize when and how to apply
mathematical skills and techniques to the problems they face at MIT
and in their subsequent careers.
The d'Arbeloff Interactive Mathematics Project aims to reach
its goal using several mechanisms, each of which represents
a form of interactivity.
-- We are initiating a change in the culture of classroom education
in the Mathematics Department, introducing a variety of active learning
and just-in-time teaching methods into freshman and sophomore level
lectures, as well as computer based lecture demonstrations.
-- We are tightening the connection between lecture and recitation
by moving towards a system of explicitly given problems designed to
foster group work. We are developing protocols and training procedures
to help recitation leaders use these new methods.
-- We are constructing a wide variety of computer manipulatives,
often simulating instances of general concepts in applications and
inviting active involvement by the student in controlling parameters.
These simulations will form the basis for homework assignments,
enforcing students' interaction with this material.
-- We are creating a variety of computer based courses and tutorials, incorporating
text, video, manipulatives, and corrected problems. These tutorials will
serve a number of distinct purposes, providing support for students
in mathematics classes, remediation for students in need, and reference
material to which faculty from across MIT can refer students for
re-learning mathematics material as needed.
-- We are increasing the transparency of basic Mathematics Department
courses. Transfer is a two-step process, and these measures will
make it much easier for down-stream courses to bring students back to
fluency with this material.
All the components of this project are under development.
Much of this material has been used in courses over the past two years.
We are actively engaged in a program of formative assessment of various
components of this project.
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ATH 100 (SURVEY OF MATHEMATICS) includes a variety of selected mathematical topics designed to acquaint students with examples of mathematical reasoning. The topics included in a given section or academic term are chosen by the instructor to demonstrate the beauty and power of mathematics from applied, symbolic, and abstract standpoints. MATH 100 is not intended as, and does not qualify as, a prerequisite for advanced mathematics courses.
Prerequisite: C or better (or CR) in MATH 82, MATH 83 or equivalent non-Leeward CC course, within the past two years. Equivalent courses include MATH 25 and 26 (but NOT 22, 23, 24, or 81). Other equivalencies include qualifying COMPASS placement scores (50 or greater in the algebra placement domain) and qualifying SAT scores (530 or greater in the quantitative section).
The lectures were taped in advance and will be broadcast on Oceanic cable digital channel 355. Students are expected to subscribe to cable and record each lecture. There are also internet components.
This course is NOT self-paced and CANNOT be completed entirely from home. Four proctored, on-campus chapter exams are required and must be taken during specified testing windows.
TEXTBOOK: Mathematics: Reasons, Results, and Applications, Second Preliminary Edition, by Eric Matsuoka. This book is available for purchase from the Leeward CC bookstore or can be download from this group.
CALCULATOR: A scientific calculator capable of two-variable statistical functions is REQUIRED. The following Texas Instruments models are supported (meaning help will be available): TI-30xIIB, TI-30xIIS, TI-82, TI-83, and TI-84.
Joining this course group will allow you to view the course materials (in the files sections) for the current and prior academic terms. Materials to be used in upcoming semesters may not yet be available.
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Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations.
Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle, inverse trigonometric functions (principal value only).
Analytical geometry:
Two dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.
Equation of a straight line in various forms, angle between two lines, distance of a point from a line; Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines; Centroid, orthocentre, incentre and circumcentre of a triangle.
Equation of a circle in various forms, equations of tangent, normal and chord.
Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal.
Locus Problems.
Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.
Derivatives of implicit functions, derivatives up to order two, geometrical interpretation of the derivative, tangents and normals, increasing and decreasing functions, maximum and minimum values of a function, Rolle's Theorem and Lagrange's Mean Value Theorem
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Course Description:A study of
geometric concepts and measurement using nonstandard, English, and metric
units.Coordinate geometry, inductive
and deductive reasoning, and concepts related to two- and three-dimensional
objects including similarity, congruence, and transformations are explored.
Prerequisites:Admission to
Teacher Education
Objectives
The
purpose of this course is for you to:
·Develop
mathematical content knowledge in the areas of geometry and measurement
§You will be
provided with scissors, ruler, protractor, and compass for classroom use.You will need to have your own supplies for
out-of-class use.
§You will need
access to Geometer's Sketchpad.On
campus you will find this program in the Mathematics and Computer Science
folder of the Campus Software folder.I
recommend that you purchase the student edition of GSP to install on your
personal computer.It can be ordered
from Key Curriculum Press at may be several copies for purchase in
the ASU bookstore.)
§TI-83 or TI-84
graphing calculator
PARTIAL COURSE REQUIREMENTS
·Reflective
Journal:A component of effective teaching is reflection
on practice.The NCTM Principles and
Standards for School Mathematics stress that teachers must have
opportunities to reflect on and refine instructional practice –during class and
outside class, alone and with others.As
a part of this course, you are expected to maintain a reflective journal.This journal is a log of your experiences within the course.It should include thoughts about new skills
and accomplishments that you acquire; critical incidents that occur; and your
thoughts and feelings about content (mathematics/geometry investigations) and
technology.Please see me if you are
having difficulty keeping a journal so that I can make some suggestions.
Please
write the date prior to each entry.You
may type or write in long hand (as long as it is neatly done).Do not write on the back.Keep your journal in a folder with a 3-hole
punch.These will be collected every 2
weeks for me to see the progress you are making.I will read them and make comments but no
grade will be assigned.You will receive
credit for the assignment if you submit the journal when it is due and if you
submit an entry that shows you are reflecting on your experiences.
·Electronic
Portfolio of Write-ups:Each person will develop a personal Web Page
for the course. There will be a set of at least 10 "Write-up"
projects.Each Write-up will be prepared
as an HTML document (i.e. a Web Page document) and linked to your personal web
page.
What is
a write-up?
The "write-ups" represent your synthesis
and presentation of a mathematics investigation you have done --usually under
the direction of one of the assignments. The major point is that it
convincingly communicates what you have found to be important from the
investigation. A write-up should communicate the essential material you have
synthesized from your investigation. The format could be entirely in a
word-processing document. After all, an HTML document is basically a word
processing document with links. The HTML format, however, can combine
narrative, pictures, and program applications in a dynamic document. Write-ups
should be posted to your personal Web Page. If you work with a classmate on an
investigation, you should still do your own write-up so that you explain your
thinking and what you learned.You
should also acknowledge the collaborative effort.Criteria for assessment will include correct
mathematics and how well you communicate."Solution" might be another word for "Write-up."
·Reaction
papers:You will have 4 one to two page reaction
papers to write.The papers will be your
reaction to articles you will read related to teaching and learning geometry in
the middle school. The Rubric for Reaction Papers will be used to evaluate your
writing.
·Working
Portfolio/Notebook:You should organize all materials (handouts,
classnotes, homework, readings, writings, tests) in a 3-ring binder, writing
the date on each.This notebook will be
a record of your work in the course and will also serve as a tool for
reflection.You will need this notebook
to help you prepare for tests and to help you develop your final reflective
portfolio.It will also be a valuable
resource to you when you begin teaching.
·Final
Reflective Portfolio:This assignment is designed for you to reflect over
the activities of the course to determine what kind and how much progress you have
made in your understanding of geometry concepts so that you will be able to
teach geometry meaningfully to middle school students.The directions for this assignment will be
given later.For now, you need to be
sure you are keeping a well-organized notebook.
** name is first initial plus last name, e.g. Linda
Crawford "lcrawford"
The
percentages to determine your course grade
Reflective Journal
5%
Daily class participation
5%
Other written or presented
assignments
13%
Reaction Papers
7%
Write-ups posted to
webpage
25%
Midterm exam
20%
Final exam
25%
Class Policies
§Attendance and participation are required in this
class, both for you to learn and for others to benefit from your input.Much of the learning in the course takes
place by participating, sharing, and interacting with others; this cannot take
place if you are absent so regular attendance and punctuality are
expected.Frequently, ideas that we
introduce in one class are expanded upon and developed more fully in later
classes.Thus, every class is
important.However, if you have to
miss class for good reasons, you should contact me as much in advance as
possible.Any student who is absent
more than 10% of the class time (3 class periods) may be dropped with a
WF.Excused absences will count toward
the 10%.If you are absent, you are
responsible for the assignment as well as any announcements made in class.
§Out-of-class assignments are due at the beginning of
class—place them on my desk when you arrive for class.If you are absent, your assignment is still
due so you will have to make arrangements to get it to me.Late assignments are accepted at my
discretion but will be assessed a penalty of 10% for each day (not class period)
the assignment is late.
§No eating or drinking in the classroom—this is a
policy of Allgood Hall.Bottles and cups
should be capped and put away.
§Visitors, including children, are not permitted
without my prior permission.
Academic honesty:Cheating
will not be tolerated.Although you
may collaborate on outside assignments, your write-up should be your
own.Any student who is caught
cheating will face serious consequences.You should read ASU's statement on academic honesty in the catalog.
Professional Organizations
You
are encouraged to join the following professional organizations:
·Georgia Council
of Teachers of Mathematics (GCTM) at
(this membership is free so I will see that you join this organization)
·National Council
of Teachers of Mathematics (NCTM) at
($38 student membership includes your choice of journal—Mathematics Teaching
in the Middle School is recommended for middle grades teachers.)
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MATH 190 - Real-World Mathematics: A Service-Learning Math Course (4)
Contemporary society is filled with political, economic and cultural issues that arise from mathematical ideas. This service-learning Core mathematics course will engage students in using mathematics as a tool for understanding their world with a focus on the connection between quantitative literacy and social justice.Topics covered will include financial mathematics, voting theory, data representation and statistics.
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History of Mathematics
9780130190741
ISBN:
0130190748
Pub Date: 2001 Publisher: Prentice Hall
Summary: For junior and senior level undergraduate courses, this text attempts to blend relevant mathematics and relevant history of mathematics, giving not only a description of the mathematics, but also explaining how it has been practiced through time.
May include moderately worn cover, writing, markings or slight discoloration. SKU:9780130190741-4-0-3 Orders ship the same or next business day. Expedited shipping within U.S. [more]
May include moderately worn cover, writing, markings or slight discoloration. SKU:97801301907Blending relevant mathematics and history, this book immerses readers in the full, rich detail of mathematics. It provides a description of mathematics and shows how mathemat [more]
Blending relevant mathematics and history, this book immerses readers in the full, rich detail of mathematics. It provides a description of mathematics and shows how mathematics was actually practiced throughout the millennia by p.[less]
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Welcome to our Classjump website for Intermediate Algebra! This website is a tool to help connect our classroom to home. Throughout the year, use this website to find class notes, homework assignments, important due dates and information regarding access to the online version of the textbook. These tools will be especially useful if you are absent or know that you will be gone in advance!
We only have about 55 minutes together each day, so it is extremely important that you come to class prepared and ready to engage in learning each and every day. This website will help to keep you prepared, so please check it regularly (like your Facebook!). Intermediate Algebra is a co-taught class in order to provide the best learning opportunities for students. I will be co-teaching Intermediate Algebra with Jesse Ziebarth, so please feel free to contact either of us with any questions or concerns. I look forward to an exciting and wonderful year!
Math lab serves as a support class for students enrolled in one of my sections of Intermediate Algebra w/ Statistics.
Each day, we will start off by checking and working through homework problems from the previous day. Students are expected to come to lab each day with every problem from their assignment ATTEMPTED. Students will work in their assigned groups in order to practice collaboration and teach each other before we come together as a whole class.
We will then extend the lesson from the previous day either through extra practice or some sort of activity, usually involving groupwork.
Time permitting, we will spend the end of the period previewing the lesson students will have in Intermediate Algebra later that day so they can get a jump start on learning that material.
Students will not be receiving extra homework in lab. All work will be done in class.
For any other information, such as the online version of the textbook and contact information, please look under the Intermediate Algebra with Statistics ClassJump page.
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National Math Panel: Major Topics of School Algebra
The National Mathematics Advisory Panel (NMP) Final Report and Reports of the Task Groups and Subcommittees The National Mathematics Advisory Panel conducted a systematic and rigorous review of the best available scientific evidence for the teaching and learning of mathematics and provided recommendations that lay out concrete steps to improve mathematics education, with a specific focus on preparation for learning algebra. The Panel worked in task groups and subcommittees to address areas of mathematics teaching and learning including Conceptual Knowledge and Skills, Learning Processes, Instructional Practices, Teachers and Teacher Education, and Assessment. Five task groups carried out detailed syntheses of research evidence that addressed each group's major questions and met standards of methodological quality. Three subcommittees were charged with completion of a particular advisory function for the Panel. The research findings cited in these reports underpin the mathematics practices and content included on the Doing What Works website.
Multimedia Overview
National Mathematics Advisory Panel
Watch this brief overview to learn about the purpose and findings of the National Mathematics Advisory Panel and research-based recommendations for improving mathematics instruction. Find out why it's important for schools to focus on teaching critical mathematics skills to better prepare students for entry into algebra.
(3:41 min)
Explore these recommended practices:
<<Topics of Algebra Teach the comprehensive set of major topics in algebra recommended by the National Mathematics Advisory Panel.
<<Multiple Paths Expect that all students will learn school algebra through a coherent progression of topics.
Related Links
Achieve has mapped out what students need to know and be able to do in mathematics in grades K-12, connecting the expectations throughout the grades with those for the end of high school. The American Diploma Project (ADP) benchmarks outline a progression of mathematics content for the Elementary Grades K-6 and Secondary Grades 7-12 to ensure that students master the content needed to succeed in college and careers.
The Society was founded in 1888 to further mathematical research and scholarship. AMS promotes mathematical research and its uses, strengthens mathematical education, and fosters awareness and appreciation of mathematics and its connections to other disciplines and to everyday life. Programs and services include meetings and conferences, support for Young Scholars programs and the Mathematical Moments program of the Public Awareness Office, resources for researchers and authors, and a Washington office that connects the mathematical community with the broader scientific community and with decision-makers who determine science funding. In addition, the site provides links to mathematics articles such as Professional Development of Mathematics Teachers.
CBMS is an umbrella organization consisting of seventeen professional societies, all of which have as one of their primary objectives the increase or diffusion of knowledge in one or more of the mathematical sciences. Its purpose is to promote understanding and cooperation among these national organizations so that they work together and support each other in their efforts to promote research, improve education, and expand the uses of mathematics. The group serves as a point of representation for the mathematical sciences to government agencies, other professional societies, and private foundations. Other activities include convening forums for the discussion of issues of broad concern to the mathematical sciences community such as the National Mathematics Advisory Panel Forum held in October 2008.
The mission of the MAA is to advance the mathematical sciences, especially at the collegiate level. Core interests of the group include: 1) supporting mathematical education and learning by encouraging effective curriculum, teaching, and assessment at all levels; 2) supporting research and scholarship; 3) providing professional development that fosters scholarship, professional growth, and cooperation among teachers, other professionals, and students; 4) influencing institutional and public policy through advocacy for the importance, uses, and needs of the mathematical sciences; and 5) promoting the general understanding and appreciation of mathematics. MAA encourages students of all ages, particularly those from underrepresented groups, to pursue activities and careers in the mathematical sciences.
The U.S. Department of Education Mathematics and Science Partnerships program funds professional development activities intended to increase the academic achievement of students in mathematics and science by enhancing the content knowledge and teaching skills of classroom teachers. Partnerships between high-need school districts and the science, technology, engineering, and mathematics (STEM) faculty in institutions of higher education are at the core of these improvement efforts. Other partners may include state education agencies, public charter schools or other public schools, businesses, and nonprofit or for-profit organizations concerned with mathematics and science education.
MSRI's mission is the advancement and communication of fundamental knowledge in mathematics and the mathematical sciences, to the development of human capital for the growth and use of such knowledge, and to the cultivation in the larger society of awareness and appreciation of the beauty, power, and importance of mathematical ideas and ways of understanding the world. From its beginning in 1982, the Institute has been primarily funded by the NSF with additional support from other government agencies, private foundations, and academic and corporate sponsors. This site provides links to workshops, streaming video, and other algebra-related resources, including workshop lectures by Hung-Hsi Wu on the topics of The Mathematics K-12 Teachers Need to Know and Preparing Teachers to Teach Algebra.
NAGB was created by Congress in 1988 to formulate policy for the National Assessment of Educational Progress (NAEP). The Governing Board monitors external contracts; prepares and recommends procedures for reporting and disseminating NAEP results; reviews and recommends test content for NAEP; and recommends policies to guide other NAEP activities. Among the Governing Board's responsibilities are developing objectives and test specifications and designing the assessment methodology for NAEP. The Mathematics Framework for the 2009 National Assessment of Educational Progress describes a design for the main NAEP assessments at the national, state, and district levels, including mathematics content and types of assessment questions.
NCSM is a mathematics leadership organization for educational leaders, which provides professional learning opportunities necessary to support and sustain improved student achievement. NCSM envisions a professional and diverse learning community of educational leaders that ensures every student in every classroom has access to effective mathematics teachers, relevant curricula, culturally responsive pedagogy, and current technology.
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Opa Locka GeEach lesson uses what was learned before. If you don't understand one of the foundational steps, you will get lost later on. Kelvin uses multiple techniques to help students understand the basics and does a thorough review so those building blocks stay fresh.
...Sample testing will be provided to evaluate student readiness to take this test. The math portion of the ACT test is 60 questions to be done within 60 minutes. The subjects and % of each included are; Pre Algebra (23%)/ Elementary Algebra (17%) = 40%
The next section is Intermediate Algebra (1...
...Pre algebra deals with the basic concepts needed in order to advance to algebra. Some of the basic concepts found in prealgebra are the following:
linear equations
Probability
Number sense
Geometry
Solving for x
In order to understand algebra, a student must be able to figure out equations tha...
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Financial literacy for students
Why is the world descending into a financial meltdown? Lack of financial math knowledge on the part of consumers and financial institutions has a lot to do with it. Here's a resource that aims to improve the situation....
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View of /branches/gage_dev/pg/doc/MathObjects/MathObjectsAnswerCheckers.pod
1 =head1 MathObjects-based Answer Checkers
2 3 MathObjects are designed to be used in two ways. First, you can use it
4 within your perl code when writing problems as a means of making it
5 easier to handle formulas, and in particular, to be able to use a single
6 object to produce numeric values, TeX output and answer strings from a
7 single formula entry. This avoids having to type a function three
8 different ways (which makes maintaining a problem much harder). Since
9 MathObjects also included vector and complex arthimetic, it is easier to
10 work with these types of values as well.
11 12 Secondly using MathObjects improves the processing of student input.
13 This is accomplished through special answer checkers that are part of
14 the Parser package (rather than the traditional WeBWorK answer
15 checkers). Each of these checkers has error checking customized to the
16 type of input expected from the student and can provide helpful feedback
17 if the syntax of the student's entry is incorrect.
18 19 Checkers are available for each of the types of values that the parser
20 can produce (numbers, complex numbers, infinities, points, vectors,
21 intervals, unions, formulas, lists of numbers, lists of points, lists of
22 intervals, lists of formulas returning numbers, lists of formulas
23 returning points, and so on).
24 25 To use one of these checkers, simply call the ->cmp method of the
26 object that represents the correct answer. For example:
27 28 $n = Real(sqrt(2));
29 ANS($n->cmp);
30 31 will produce an answer checker that matches the square root of two.
32 Similarly,
33 34 ANS(Vector(1,2,3)->cmp);
35 36 matches the vector <1,2,3> (or any computation that produces it, e.g.,
37 i+2j+3k, or <4,4,4>-<3,2,1>), while
38 39 ANS(Interval("(-inf,3]")->cmp);
40 41 matches the given interval. Other examples include:
42 43 ANS(Infinity->cmp);
44 ANS(String('NONE')->cmp);
45 ANS(Union("(-inf,$a) U ($a,inf)")->cmp);
46 47 and so on.
48 49 Formulas are handled in the same way:
50 51 ANS(Formula("x+1")->cmp);
52 53 $a = random(-5,5,1); $b = random(-5,5,1); $x = random(-5,5,1);
54 $f = Formula("x^2 + $a x + $b")->reduce;
55 ANS($f->cmp);
56 ANS($f->eval(x=>$x)->cmp);
57 58 $x = Formula('x');
59 ANS((1+$a*$x)->cmp);
60 61 Context("Vector")->variables->are(t=>'Real');
62 $v = Formula("<t,t^2,t^3>"); $t = random(-5,5,1);
63 ANS($v->cmp);
64 ANS($v->eval(t=>$t)->cmp);
65 66 and so on.
67 68 Lists of items can be checked as easily:
69 70 ANS(List(1,-1,0)->cmp);
71 ANS(List(Point($a,$b),Point($a,-$b))->cmp);
72 ANS(List(Vector(1,0,0),Vector(0,1,1))->cmp);
73 ANS(Compute("(-inf,2),(4,5)")->cmp); # easy way to get list of intervals
74 ANS(Formula("x, x+1, x^2-1")->cmp);
75 ANS(Formula("<x,2x>,<x,-2x>,<0,x>")->cmp);
76 ANS(List('NONE')->cmp);
77 78 and so on. The last example may seem strange, as you could have used
79 ANS(String('NONE')->cmp), but there is a reason for using this type
80 of construction. You might be asking for one or more numbers (or
81 points, or whatever) or the word 'NONE' of there are no numbers (or
82 points). If you used String('NONE')->cmp, the student would get an
83 error message about a type mismatch if he entered a list of numbers,
84 but with List('NONE')->cmp, he will get appropriate error messages for
85 the wrong entries in the list.
86 87 It is often appropriate to use the list checker in this way even when
88 the correct answer is a single value, if the student might type a list
89 of answers.
90 91 On the other hand, using the list checker has its disadvantages. For
92 example, if you use
93 94 ANS(Interval("(-inf,3]")->cmp);
95 96 and the student enters (-inf,3), she will get a message indicating
97 that the type of interval is incorrect, while that would not be the
98 case if
99 100 ANS(List(Interval("(-inf,3]"))->cmp);
101 102 were used. (This is because the student doesn't know how many
103 intervals there are, so saying that the type of interval is wrong
104 would inform her that there is only one.)
105 106 The rule of thumb is: the individual checkers can give more detailed
107 information about what is wrong with the student's answer; the list
108 checker allows a wider range of answers to be given without giving
109 away how many answers there are. If the student knows there's only
110 one, use the individual checker; if there may or may not be more than
111 one, use the list checker.
112 113 Note that you can form lists of formulas as well. The following all
114 produce the same answer checker:
115 116 ANS(List(Formula("x+1"),Formula("x-1"))->cmp);
117 118 ANS(Formula("x+1,x-1")->cmp); # easier
119 120 $f = Formula("x+1"); $g = Formula("x-1");
121 ANS(List($f,$g)->cmp);
122 123 $x = Formula('x');
124 ANS(List($x+1,$x-1)->cmp);
125 126 See the files in webwork2/doc/parser/problems for more
127 examples of using the parser's answer checkers.
128 129 =head2 Controlling the Details of the Answer Checkers
130 131 The action of the answer checkers can be modified by passing flags to
132 the cmp() method. For example:
133 134 ANS(Real(pi)->cmp(showTypeWarnings=>0));
135 136 will prevent the answer checker from reporting errors due to the
137 student entering in the wrong type of answer (say a vector rather than
138 a number).
139 140 =head3 Flags common to all answer checkers
141 142 There are a number of flags common to all the checkers:
143 144 =over
145 146 =item S<C<< showTypeWarnings=>1 or 0 >>>
147 148 show/don't show messages about student
149 answers not being of the right type.
150 (default: 1)
151 152 =item S<C<< showEqualErrors=>1 or 0 >>>
153 154 show/don't show messages produced by
155 trying to compare the professor and
156 student values for equality, e.g.,
157 conversion errors between types.
158 (default: 1)
159 160 =item S<C<< ignoreStrings=>1 or 0 >>>
161 162 show/don't show type mismatch errors
163 produced by strings (so that 'NONE' will
164 not cause a type mismatch in a checker
165 looking for a list of numbers, for example).
166 (default: 1)
167 168 =back
169 170 In addition to these, the individual types have their own flags:
171 172 =head3 Flags for Real()->cmp
173 174 =over
175 176 =item S<C<< ignoreInfinity=>1 or 0 >>>
177 178 Don't report type mismatches if the
179 student enters an infinity.
180 (default: 1)
181 182 =back
183 184 =head3 Flags for String()->cmp
185 186 =over
187 188 =item S<C<< typeMatch=>value >>>
189 190 Specifies the type of object that
191 the student should be allowed to enter
192 (in addition the string).
193 (default: 'Value::Real')
194 195 =back
196 197 =head3 Flags for Point()->cmp
198 199 =over
200 201 =item S<C<< showDimensionHints=>1 or 0 >>>
202 203 show/don't show messages about the
204 wrong number of coordinates.
205 (default: 1)
206 207 =item S<C<< showCoordinateHints=>1 or 0 >>>
208 209 show/don't show message about
210 which coordinates are right.
211 (default: 1)
212 213 =back
214 215 =head3 Flags for Vector()->cmp
216 217 =over
218 219 =item S<C<< showDimensionHints=>1 or 0 >>>
220 221 show/don't show messages about the
222 wrong number of coordinates.
223 (default: 1)
224 225 =item S<C<< showCoordinateHints=>1 or 0 >>>
226 227 show/don't show message about
228 which coordinates are right.
229 (default: 1)
230 231 =item S<C<< promotePoints=>1 or 0 >>>
232 233 do/don't allow the student to
234 enter a point rather than a vector.
235 (default: 1)
236 237 =item S<C<< parallel=>1 or 0 >>>
238 239 Mark the answer as correct if it
240 is parallel to the professor's answer.
241 Note that a value of 1 forces
242 showCoordinateHints to be 0.
243 (default: 0)
244 245 =item S<C<< sameDirection=>1 or 0 >>>
246 247 During a parallel check, mark the
248 answer as correct only if it is in
249 the same (not the opposite)
250 direction as the professor's answer.
251 (default: 0)
252 253 =back
254 255 =head3 Flags for Matrix()->cmp
256 257 =over
258 259 =item S<C<< showDimensionHints=>1 or 0 >>>
260 261 show/don't show messages about the
262 wrong number of coordinates.
263 (default: 1)
264 265 =back
266 267 The default for showEqualErrors is set to 0 for Matrices, since
268 these errors usually are dimension errors, and that is handled
269 separately (and after the equality check).
270 271 =head3 Flags for Interval()->cmp
272 273 =over
274 275 =item S<C<< showEndpointHints=>1 or 0 >>>
276 277 do/don't show messages about which
278 endpoints are correct.
279 (default: 1)
280 281 =item S<C<< showEndTypeHints=>1 or 0 >>>
282 283 do/don't show messages about
284 whether the open/closed status of
285 the enpoints are correct (only
286 shown when the endpoints themselves
287 are correct).
288 (default: 1)
289 290 =back
291 292 =head3 Flags for Union()->cmp and List()->cmp
293 294 all the flags from the Real()->cmp, plus:
295 296 =over
297 298 =item S<C<< showHints=>1 or 0 >>>
299 300 do/don't show messages about which
301 entries are incorrect.
302 (default: $showPartialCorrectAnswers)
303 304 =item S<C<< showLengthHints=>1 or 0 >>>
305 306 do/don't show messages about having the
307 correct number of entries (only shown
308 when all the student answers are
309 correct but there are more needed, or
310 all the correct answsers are among the
311 ones given, but some extras were given).
312 (default: $showPartialCorrectAnswers)
313 314 =item S<C<< partialCredit=>1 or 0 >>>
315 316 do/don't give partial credit for when
317 some answers are right, but not all.
318 (default: $showPartialCorrectAnswers)
319 (currently the default is 0 since WW
320 can't handle partial credit properly).
321 322 =item S<C<< ordered=>1 or 0 >>>
323 324 give credit only if the student answers
325 are in the same order as the
326 professor's answers.
327 (default: 0)
328 329 =item S<C<< entry_type=>'a (name)' >>>
330 331 The string to use in error messages
332 about type mismatches.
333 (default: dynamically determined from list)
334 335 =item S<C<< list_type=>'a (name)' >>>
336 337 The string to use in error messages
338 about numbers of entries in the list.
339 (default: dynamically determined from list)
340 341 =item S<C<< typeMatch=>value >>>
342 343 Specifies the type of object that
344 the student should be allowed to enter
345 in the list (determines what
346 constitutes a type mismatch error).
347 (default: dynamically determined from list)
348 349 =item S<C<< requireParenMatch=>1 or 0 >>>
350 351 Do/don't require the parentheses in the
352 student's answer to match those in the
353 professor's answer exactly.
354 (default: 1)
355 356 =item S<C<< removeParens=>1 or 0 >>>
357 358 Do/don't remove the parentheses from the
359 professor's list as part of the correct
360 answer string. This is so that if you
361 use List() to create the list (which
362 doesn't allow you to control the parens
363 directly), you can still get a list
364 with no parentheses.
365 (default: 0 for List() and 1 for Formula())
366 367 =back
368 369 =head3 Flags for Formula()->cmp
370 371 The flags for formulas are dependent on the type of the result of
372 the formula. If the result is a list or union, it gets the flags
373 for that type above, otherwise it gets that flags of the Real
374 type above.
375 376 More flags need to be added in order to allow more control over the
377 answer checkers to give the full flexibility of the traditional
378 WeBWorK answer checkers. Note that some things, like whether trig
379 functions are allowed in the answer, are controlled through the
380 Context() rather than the answer checker itself. For example,
381 382 Context()->functions->undefine('sin','cos','tan');
383 384 would remove those three functions from use. (One would need to remove
385 cot, sec, csc, arcsin, asin, etc., to do this properly; there could be
386 a function call to do this.)
387 388 Similarly, which arithmetic operations are available is controlled
389 through Context()->operations.
390 391 The tolerances used in comparing numbers are part of the Context as
392 well. You can set these via:
393 394 Context()->flags->set(
395 tolerance => .0001, # the relative or absolute tolerance
396 tolType => 'relative', # or 'absolute'
397 zeroLevel => 1E-14, # when to use zeroLevelTol
398 zeroLevelTol => 1E-12, # smaller than this matches zero
399 # when one of the two is less
400 # than zeroLevel
401 limits => [-2,2], # limits for variables in formulas
402 num_points => 5, # the number of test points
403 );
404 405 [These need to be handled better.]
406 407 Note that for testing formulas, you can override the limits and
408 num_points settings by setting these fields of the formula itself:
409 410 $f = Formula("sqrt(x-10)");
411 $f->{limits} = [10,12];
412 413 $f = Formula("log(xy)");
414 $f->{limits} = [[.1,2],[.1,2]]; # x and y limits
415 416 You can also specify the test points explicitly:
417 418 $f = Formula("sqrt(x-10)");
419 $f->{test_points} = [[11],[11.5],[12]];
420 421 $f = Formula("log(xy)");
422 $f->{test_points} = [[.1,.1],[.1,.5],[.1,.75],
423 [.5,.1],[.5,.5],[.5,.75]];
424 425 [There still needs to be a means of handling the tolerances similarly,
426 and through the ->cmp() call itself.]
427
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Learning through guided discovery - MathOverflow most recent 30 from through guided discoveryThéophile Cantelobre2013-01-23T05:09:18Z2013-05-05T11:18:37Z
<p>I have been working through Kenneth P. Bogart's "Combinatorics Through Guided Discovery". You can download it from this page: <a href=" rel="nofollow">
<p>I've found that it is a great way to learn and makes me think about the concepts as if I were discovering them. I think that a lot of people will find benefit in working through such a book.</p>
<p>I've looked for books in the same spirit as this one: learning through guided discovery, but my searches haven't been fruitful.</p>
<p>Does anyone know of any such books? </p>
by Rodrigo A. Pérez for Learning through guided discoveryRodrigo A. Pérez2013-01-23T05:54:09Z2013-01-23T05:54:09Z<p>You may be interested in learning about the Moore Method. The idea is to "encourages students to solve problems using their own skills of critical analysis and creativity" without relying on textbooks. <a href=" rel="nofollow">HERE</a> you can find some references.</p>
by Daniel Moskovich for Learning through guided discoveryDaniel Moskovich2013-01-23T06:27:58Z2013-01-23T06:27:58Z<p><a href=" rel="nofollow">Linear Algebra Problem Book</a> by P.R. Halmos is written very much in this spirit: learning through guided discovery. I use it for my "Advanced Investigations in Linear Algebra" course.</p>
by Aakumadula for Learning through guided discoveryAakumadula2013-01-23T06:28:46Z2013-01-23T06:28:46Z<p>My favourite is Alexandre Kirillov's "Elements of the Theory of Representations" Grundlehren der Mathematischen Wissenschaften, Springer, vol 220. A lot of representation theory is worked out through examples and exercises. </p>
by Carl Najafi for Learning through guided discoveryCarl Najafi2013-01-23T07:47:07Z2013-01-23T07:47:07Z<p>Similar to his "Linear Algebra Problem Book", Halmos also wrote "A Hilbert Space Problem Book". I have only skimmed it but it seems as good as LAPB which I remember liking a lot.</p>
by Thomas Sauvaget for Learning through guided discoveryThomas Sauvaget2013-01-26T15:03:15Z2013-01-26T15:03:15Z<p>The book <em>Abel's Theorem in Problems and Solutions</em> by Alekseev & Arnold is a great one to learn about group theory and complex analysis (see excerpts <a href=" rel="nofollow">here</a>)</p>
<p>Also, have a look at the following related MO questions: <a href=" rel="nofollow">12709</a>, <a href=" rel="nofollow">28158</a> and <a href=" rel="nofollow">56314</a>.</p>
by Jon Bannon for Learning through guided discoveryJon Bannon2013-01-26T18:51:12Z2013-01-26T18:51:12Z<p>This guided discovery approach goes by other names, as well. One such name is "Inquiry Based Learning" or IBL. A list of guided discovery problems is often referred to as an "IBL script". Many such scripts are available from the Journal of Inquiry Based Learning in Mathematics (JIBLM): <a href=" rel="nofollow">
by Elden Elmanto for Learning through guided discoveryElden Elmanto2013-01-26T21:37:29Z2013-01-26T21:37:29Z<p>I have not read (or, in this case, worked) through the book, but Jeffrey Strom's ``Modern Classical Homotopy Theory" guides the reader through the proofs of all the theorems stated in the book (as opposed to proving them himself). To my very limited knowledge, this is the first "IBL-type" book in algebraic topology.</p>
<p>This is the book:</p>
<p><a href=" rel="nofollow">
by cams for Learning through guided discoverycams2013-02-08T05:57:21Z2013-02-08T05:57:21Z<p>I'm impressed with two books by Dr R. P. Burn that seem to be in the spirit of your question:- </p>
<ol>
<li>Groups: a path to geometry, CUP, 1985, 0-521-30037-1</li>
<li>A pathway to number theory, CUP, 2nd ed., 1997, 978-0-521-57540-9</li>
</ol>
<p>Each consists of an ordered sequence of problems (answers provided):- </p>
<blockquote>
<p>... to enable students to participate in the formulation of central mathematical ideas <em>before</em> a formal treatment (which, suitably introduced, they may well be able to provide themselves) </p>
</blockquote>
<p><em>Source:</em> a preface to <em>A pathway to number theory)</em></p>
<p>They are aimed at advanced high school, or early undergraduate level students. The sequence starts by getting the reader to initially explore special cases and then work towards a generalisation, usually a theorem. The books include references to selected standard texts that are recommended to be read concurrently.</p>
by Amir Asghari for Learning through guided discoveryAmir Asghari2013-05-04T21:27:10Z2013-05-04T21:27:10Z<p>You may find this one interesting: Number Theory Through Inquiry (MAA textbooks). I have used it three times. First time, which I strictly followed the method, we just coverd the first four chapters. Second time, I have relaxed myself a bit and we covered the first six chapters. Last time (current term), I have used all the teaching methods I know (including modified Moore method), we are nearly covering all chapters! </p>
<p>You may also find this paper interesting: "Moore and Less" (PRIMUS,22(7):509-524, 2012) where I told the story of using a very modified Moore method in a Multivariable Calculus Course. </p>
by jim-hefferon for Learning through guided discoveryjim-hefferon2013-05-05T11:18:37Z2013-05-05T11:18:37Z<p>I've just put up such a text for an Introduction to Proofs course, <a href=" rel="nofollow">here</a>. It is Free, including LaTeX source. (I've only taught out of it one time so no doubt there are typos, places that could use refinement, etc.)</p>
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Basic Mathematics, CourseSmart eTextbook
Description
Basic Mathematics, by Goetz, Smith, and Tobey, is your students' on-ramp to success in mathematics! Providing generous levels of support and interactivity throughout their text, the authors help students experience many small successes, one concept at a time. Students take an active role while using this text–they participate and learn by making decisions, solving exercises, or answering questions as they read. This interactive structure allows students to get up to speed at their own pace, while developing the skills necessary to succeed in future mathematics courses. To deepen the interactive nature of the book, Twitter® is used throughout the text, with the authors providing a tweet for every exercise set of every section, giving students timely hints and suggestions to help with specific exercises. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
Table of Contents
1. Whole Numbers
1.1 Understanding Whole Numbers
1.2 Adding Whole Numbers
1.3 Subtracting Whole numbers
1.4 Multiplying Whole Numbers
1.5 Dividing Whole Numbers
1.6 Exponents, Groupings, and the Order of Operations
1.7 Properties of Whole Numbers
1.8 The Greatest Common Factor and Least Common Multiple
1.9 Applications with Whole Numbers
Chapter 1 Chapter Organizer
Chapter 1 Review Exercises
Chapter 1 Practice Test
2. Fractions
2.1 Visualizing Fractions
2.2 Multiplying Fractions
2.3 Dividing Fractions
2.4 Adding and Subtracting Fractions
2.5 Fractions and the Order of Operations
2.6 Mixed Numbers
Chapter 2 Chapter Organizer
Chapter 2 Review Exercises
Chapter 2 Practice Test
3. Decimals
3.1 Understanding Decimal Numbers
3.2 Adding and Subtracting Decimal Numbers
3.3 Multiplying Decimal Numbers
3.4 Dividing Decimal Numbers
Chapter 3 Chapter Organizer
Chapter 3 Review Exercises
Chapter 3 Practice Test
4. Ratios, Rates, and Proportions
4.1 Ratios and Rates
4.2 Writing and Solving Proportions
4.3 Applications of Ratios, Rates and Proportions
Chapter 4 Chapter Organizer
Chapter 4 Review Exercises
Chapter 4 Practice Test
5. Percents
5.1 Percents, Fractions, and Decimals
5.2 Use Proportions to Solve Percent Exercises
5.3 Use Equations to Solve Percent Exercises
Chapter 5 Chapter Organizer
Chapter 5 Review Exercises
Chapter 5 Practice Test
6. Units of Measure
6.1 U.S. System Units of Measure
6.2 Metric System Units of Measure
6.3 Converting Between the U.S. System and the Metric System
Chapter 6 Chapter Organizer
Chapter 6 Review Exercises
Chapter 6 Practice Test
7. Geometry
7.1 Angles
7.2 Polygons
7.3 Perimeter and Area
7.4 Circles
7.5 Volume
7.6 Square Roots and the Pythagorean Theorem
7.7 Similarity
Chapter 7 Chapter Organizer
Chapter 7 Review Exercises
Chapter 7 Practice Test
8. Statistics
8.1 Reading Graphs
8.2 Mean, Median and Mode
Chapter 8 Chapter Organizer
Chapter 8 Review Exercises
Chapter 8 Practice Test
9. Signed Numbers
9.1 Understanding Signed Numbers
9.2 Adding and Subtracting Signed Numbers
9.3 Multiplying and Dividing Signed Numbers
9.4 The Order of Operations and Signed Numbers
Chapter 9 Chapter Organizer
Chapter 9 Review Exercises
Chapter 9 Practice Test
10. Introduction to Algebra
10.1 Introduction to Variables
10.2 Operations with Variable Expressions
10.3 Solving One-Step Equations
10.4 Solving Multi-Step Equations
Chapter 10 Chapter Organizer
Chapter 10 Review Exercises
Chapter 10 Practice Test
Appendices
A. Additional Practice and Review
Section 1.2 Extra Practice, Addition Facts
Section 1.3 Extra Practice, Subtraction Facts
Section 1.4 Extra Practice, Multiplication Facts
Mid Chapter Review, Chapter 1
Mid Chapter Review, Chapter 2
Mid Chapter Review, Chapter 9
B. Tables
Basic Facts for Addition
Basic Facts for Multiplication
Square Roots
U.S. and Metric Measurements and Convers
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Kimberton Algebra comes putting intervals together to build chords, how major, minor, augmented, and diminished chords are formed. And there are also inversions and 7th chords. Finally comes which chords appear in the different keys, which chords in a key will be major or minor or diminished or augmented and why, and how it changes depending on whether it's in major or minor key
...I aim to help the student understand what the theorems and axioms mean physically, so the student will better understand the concept theoretically and ultimately feel comfortable solving a geometric proof. The first pre-algebra topic to focus on is knowing and understanding types of numbers, suc...
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The Mathematics Department at Saint Vincent College views
mathematics in several ways. First, we see the role of mathematics as a
"universal language". In this role mathematics enables mankind
to engage in various activities such as engineering, chemistry, physics,
computing, and economics (to mention just a few) with a much deeper
understanding than would otherwise be possible. Indeed, mathematics facilitates
the quantification of various results, notions, and concepts, allowing them to
be studied, revised, and refined to such an extent that, in many cases, they
become excellent descriptors of human behavior and natural phenomena. It
is also in this sense that mathematics is seen as a tool which allows for
profound development, not only in the mathematical arena, but in the natural
and social sciences as well. Indeed, in the words of Galileo Galilei:
"Mathematics is the
language with which God has written the universe."
The role of mathematics as a language or tool is
perhaps the most common perception of the discipline. We, in the
Mathematics Department, perceive a second role of mathematics as being an
object of beauty, essentially an art form; something to be shared and
appreciated by all - not just by a select few. It is our hope that
students will come to appreciate the unique way in which mathematics often
starts with a few basic definitions and premises and then unfolds into a tremendous
body of knowledge, often yielding many new and exciting insights and
problem-solving techniques. Thus it is our hope that every one of our
students will come to enjoy the actual excitement and fun that can be
experienced in the rigorous study of mathematicws.
The ability to enjoy the study of mathematics comes with
a certain degree of mathematical maturity which is achieved only after working
through many problems and examples. As a result, the members of the
Mathematics Department strongly encourage their students to carefully complete
all homework assignments in order to facilitate the development of their own
personal maturity. It is important to note that mathematics is not a
spectator sport; in order to learn mathematics, one must actually do mathematics.
We understand that in the normal course of events students may encounter difficulties
in their studies of mathematics. To address this the faculty holds numerous
office hours each week and students who need extra help are invited and encouraged
to attend. In addition, the department has many student tutors who are
quite capable in helping students complete their assignments and understand
key techniques and crucial concepts. Click here for a list of our student
tutors and their schedules.
It is often the case that the profit derived from a
venture is proportional to the amount invested. For this reason we urge
our students to approach their studies of mathematics wholeheartedly.
Thus, in order to derive the maximum benefit from math courses (or any course
for that matter) the student should:
a) attend each class and participate frequently
b) complete each and every homework assignment
c) seek additional help as necessary.
We believe that if this prescription is followed, then students will
develop their own mathematical maturity and come to enjoy and understand the
beauty of mathematics.
We end with a note concerning our take on the use of
technology in the classroom. It is impossible to neglect or deny the
effect of technology on modern life. Technology is here and it is here to
stay. As a result, graphing calculators are used in many courses numbered
above the MA 100 level. The mathematics faculty is currently using the
Texas Instruments model # TI-85 or TI-86 and recommends that each student
taking Calculus I or above obtain one. (Check with your instructor to see
if a graphing calculator is required for your particular mathematics
course.) In addition the Computer Algebra System of Mathematica
will be used in one of our sections of freshman Calculus this fall.
Although technology is frequently used to eliminate much of the tedious work
encountered in mathematics, the faculty believes that technology is not a
substitute for a firm understanding of basic definitions and concepts. As
a result we strongly encourage our students to use technology only as a tool
for understanding; that is, as a tool to help enforce or discover fundamental
mathematical principles and notions.
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Myungsuk Chung
In this paper, we shall use the software, Scientific Workplace, to
demonstrate how we can interact with functions which are defined
in Maple. We also use examples from Calculus to illustrate how we can
teach live mathematics and communicate with others more effectively
and efficiently without learning the syntax.
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Linear Independence
In this lesson our instructor talks about linear independence. He discusses definition, meaning, and procedure for determining if a given list of vectors is linear dependence. He ends the lesson with two complete example problems.
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Linear Independence
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Course Overview
Calculus I is an introduction to differential and integral calculus: the study of change. The course is designed for students working on a degree in science, mathematics, computer science, and those planning on certain types of graduate work. Others are welcome. The prerequisites are motivation and a good working knowledge of high school algebra and trigonometry. Those needing extra background work should consider MATH 130, Calculus I With Review. Calculus emphasizes skills, theory, and applications. Calculus opens doors to higher mathematics, science, and technology.
Text and Calculator
The text is Stewart*, Calculus: Early Transcendentals, 6th ed., chapters 1-5. Also required are the trade book A Tour of the Calculus by David Berlinski and a graphing calculator. If you already have a calculator that you know how to use, that will be fine. If you are purchasing one, buy the TI-84 Plus. Instructions for its use are on line and also will be given in class.
Grading
Your grade will be based on 4 tests and a final exam, 1000 points total. The grading scale is A: 900 - 1000 points, B: 800 - 899, etc., with +'s and -'s in the top and bottom 20 points.
Test 1
100 points
Tests 2 – 4
200 points
each
Final Exam
300 points
Total
1000 points
Policies and Due DatesTests must be taken during the scheduled time unless you have a valid excuse cleared with me ahead of time. Make-ups must take place by class time on Friday of test week. Grade penalties will be imposed for infringements. This includes Test 4 which is the day before Thanksgiving break begins. Tests are on Tuesdays: 9/1, 9/29, 10/27, 11/24. The final exam is:
Homework should consume about 8 hours per week outside of class. Homework is in three parts: review, current exercises, anticipate — past, present, future. All three are important for successfully mastering the material. Part 1: review recent work and catch up on problems you could not do previously. Part 2: problems are grouped on the syllabus by type. Within each group, do problems until you have mastered the technique. You need not do all the problems the first night, but should do most problems before the test. Part 3: read the section for the next class. This prepares the ground for planting new ideas, helping you make the most of class time. Homework is not collected: I trust you to keep current and to ask timely questions.
Answers to odd-numbered problems are in the Appendix. Problems marked with a graph symbol usually require graphical or calculator answers, Read the first day's lecture Class, College, and Life online, and frequently review the Study Tips. Use the publisher's
*Note: The Stewart text comes in three packages. Choose according to how many semesters of calculus you want in the book, and be sure to get the version labeled Early Transcendentals, 6th edition:
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Pescadero Trigonometry
...Calculus brings a new concept for students, approaching infinity. It needs a strong foundation but the concepts carry through the course that is basically differentiation and integration. After learning the basic skills, application becomes very important
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The Advanced Algebra Tutor: Learning By Example DVD Series teaches students through step-by-step example problems that progressively become more difficult. This DVD covers arithmetic sequences and series in algebra, including what an arithmetic sequence and series is and why it is important in algebra. Grades 9-College. 24 minutes on DVD.
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Mathematics Department
Developmental Mathematics:
Courses are provided to assist students who need to brush up their mathematics skills before they take their college level mathematics, science and other courses. Incoming student take a mathematics placement test as part of the admissions procedure that will tell them which developmental mathematics course to start with, or if
they are ready to go into a college level mathematics class. Counselors will also help students choose the correct mathematics class according to both readiness and curriculum.
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Discovering Math: Advanced Probability DVD This program introduces and develops concepts of probability, such as discrete and continuous variables, and dependent and independent events. It also discusses various methods of determining probabilities, as well as their applications.
What are the Odds? DVD Professor Jeff Rosenthal breaks down the probability of twenty-five fantastic events in an entertaining blend of mathematics, science, and popular culture.
6 - 12
DVD
$59.95
Math: Money and Time DVD "When are we going to use math anyway?" If you've heard that before, here is a video with irrefutable evidence that we use math every day—whether counting change, changing an appointment, or balancing our checkbooks.
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whats the diff between calc based and algerbra
whats the diff between calc based and algerbra
Ok, so i just finished my first semester ever of calculus. I have never ever made it this far in math in my life period. im way excited. But im still new to calc and not sure what everything means or is. Physics 1 at my school is what they call calc based. I dont understand what they mean. apparently there is a algerbra based physics as well.
isnt physics physics? i mean i know that f=ma. but how does that involve calc? basically im looking for a single problem that can be solved algebraically and or with calc so i can see the difference. if that makes sense.
i know that freq=1/tone and via basic math tone=1/freq but who did we get those basic equations? is that where the calc comes in?
An algebra based physics is just that there is no calculus; all problems can be solved using algebra. Trigonometry would also come into play in algebra based physics too.
Calculus based physics can get deeper into physics concepts where now you can derive the formulas that are only stated and used in a algebra physics course. Calculus gives you some powerful methods to solve more real world physics problems.
Algebra based Physics: Given some random facts that are taken at face value, solve some simple physics problems.
Calculus based Physics: Given some general laws of physics, realize the random facts aren't random, at all; that there's a very good reason those formulas can't help but be true.
Algebra based: Learn Kepler's 2nd Law and Kepler's 3rd Law.
Calculus based: It's the same damn law, dammit!
(Okay, actually it doesn't take calculus to realize the latter, but the difference between algebra based physics classes and calculus based physics classes is usually a lot more than just using calculus once in a while in a calculus based class. It's just a lot more in depth.)No its more like starting with F=ma applying it to a problem by setting up the appropriate calculus integral with applicable constraints along the way and then solving and reducing it to one of the algebra based equations that you then solve to get the answer.
An example of a constraint is the string of a pendulum constrains the bob to moving along a circular arc or the inclined plane constrains the object to fall along the incline...
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ID: 189 | Video: None | Audio: Low | Animation: Low
Improve your algebra, calculus and trigonometry skills
This free online training explores complex numbers and equations, polynomial equations, conics, advanced trigonometry, differentiation, antiderivatives, and vectors in 2- and 3-space. This course is both an ideal study-aid for students to improve their skills in their spare time or for anyone interested in exploring the world of mathematics.
Upon completion of this course you will understand nominal, discrete and continuous data in frequency tables. You will know the product rule and you will be able to easily perform calculations of differentiation. You will gain a good knowledge of antiderivatives including Hyperbolic functions, partial fractions, linear substitution, odd and even powers. You will understand the relationship between graphs and their antiderivatives. You will be able to calculate complex numbers using addition, subtraction, multiplication and division. You will be familiar with DeMoivre's Theorom, polar form multiplication and division. This course will teach you how to calculate conics, degrees, radians, inverse circular functions, polynomial equations and much more.
Comments & Reviews
Ahmed Sheikha - Kenya
Donna Mcneill - United States of America
2012-05-30 10:05:49
Course Module: Reciprocal function graphs Course Topic: Graphs of Reciprocal Functions Comment: I like the fact that this course feeds you small sections of the topic, which allows you to take things in one bite at a time. The online interactive examples, really engages you in the course. I'm having an awesome time absorbing the topics.
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There are times when a student needs specific help with a given homework problem. So they will look for somewhere that provides Algebra 1 answers. These can come from tutors or any source that is easily accessible and has the information readily available. It doesn't matter the nature of the assignment or what topic it involves. There are resources available to fit every question and homework problem. With on line options they will provide a given outline of options. That way the student can quickly search for the one that meets his or her needs.
And for the person who is in need of a specific answer to a given problem these web sites are an invaluable source. It helps to take the stress out of seeking additional help with a certain problem without having to wade through a lot of items that don't relate. Algebra 1 answers therefore are as easy to find as any other math solutions needed. Plus they will have every possible solution that meets a student's needs. Also they will provide at times examples to a given problem. They will be provided in a form that is compatible with the lesson being used by the student. Which then allows the student to apply it better to his or her assignment.
Most web sites who specialize in such information do more than provide Algebra 1 answers. They normally offer a comprehensive list of answers that can fit many different forms of math. And that will allow the student to go back to the same sight on more than one occasion as he or she moves from one math course to another. Anyone who is in search of help will thus want to keep this option as one they may wish to explore. For it will help to making the process of learning math easier. Such knowledge will always be a blessing for those who choose to use the data.
Like so many aids, the Algebra 1 answers will be provided in a way that is intended to help the student master this subject. To have a means for help outside the classroom that will give an extra sense of freedom about studying alone without concern answers can't be found.
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More About
This Book
showing quadrilateral geometric shapes, and still others showing ways of charting statistics, measuring vectors, and more. Here is an imaginative new approach to mathematics, a great classroom supplement, a useful homework helper for middle school and high school students, and a reference book that belongs in every school library.
Includes alphabetically arranged terms of the basic vocabulary of mathematics along with definitions for each
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Practical Problems in Mathematics for Weld workbook, offers a concise review of the mathematic principles used in the welding shop. Each unit begins with a review of the basic procedures used in standard operations, and builds to feature more advanced formulas and procedures. Special enhancements of this new edition include updates on present-day shop practices to give students an accurate overview of the welding field. ALSO AVAILABLE INSTRUCTOR SUPPLEMENTS CALL CUSTOMER SUPPORT TO ORDER Instructor's Guide, ISBN: 0-8273-6707-4
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ECE 3022: Mathematics for Upper Elementary Grades
A comprehensive understanding of NCTM standards and the Ohio academic content standards, pedagogy, assessment procedures, and materials for the teaching of mathematics to upper elementary grade students. Candidates will become familiar with number, number sense and operations; measurement; geometry and spatial sense; patterns, functions and algebra; data analysis and probability; and mathematical process, which includes problem solving, reasoning and proof, communication, connections, and representation.... more »
Credits:2
Overall Rating:0 Stars
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Thanks, enjoy the course! Come back and let us know how you like it by writing a review.
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Writing Algebraic Story found a Dr. Seuss book that demonstrates the idea of linear equations. I started the lesson by reading the book to the students and creating linear equations to follow the storyline. Then we created our own, following a model I put together.
By downloading this file, you will get the Dr. Seuss book title, the lesson plan I followed, the notes I gave to the students, the examples I showed them, and a few examples that my own students made.
It took me a few months to find this book, but it demonstrates linear equations (using y=mx+b) perfectly! The students might be a bit shocked to find out they are about to read a children's story, but they will enjoy it. Who doesn't like being read to? :)
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
7343.12
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Education college math courses for teachers are now available in several sites. Graduate and post graduate courses can be taken on the web. Formal education and free programs may be studied.
Coursework
Practically every mathematics branch may be studied. This includes trigonometry, algebra, geometry, calculus and advanced computations. Each of these subjects have different levels (geometry I, algebra I, II and so on). They are also very detailed. There are introductory subjects such as fractions, decimals, whole number reviews and so on. An algebra course will cover topics like bilinear forms, linear transformations, vector spaces and polynomials.
There are also subjects that will teach you about the number system. Teachers will also have to study theorems, graphs and mathematical sciences. Mechanics of deformable solids, fluid mechanics, particle mechanics and computer algebra are also taught in these classes. There are also courses about statistics, probability theory and Stochastic processes.
More advanced concepts include manifolds, algebraic topology and general topology. The latter is also known as point set topology. This area includes topics about dimension theory, open spaces, separation axioms and continuous functions.
Free Courses
These can be taken by teachers too. While they have a lot of information, not all of them earn credit. Online universities offer credit earning courses. The price of each course varies. States have different rules regarding the credit hours needed. You have to check your universality regarding credit transfer policies and rules.This will vary from university to university.
Format
Websites have different set ups. Most of these programs are self-paced, although some of them assign instructors to their students. The majority of these sites divide the lessons into modules. Some courses only take a few hours to finish. But since these courses are self paced, students can decide to study full or part time. This will determine the length of each course.
Other Information
Online classes have lectures, assignments and video guides. Just like other online classes, there are discussion forums, boards and email. Some of them now use Skype and other means to enhance communication.
Some of these courses prepare teachers for the state exams. These subjects will cover a lot of topics.
Different study materials are available. There are several lessons available. Study materials include electronic notes, lectures, question sheets, problem sets and tests.
Online college math courses for teachers have to be accredited. Free programs do not earn credit. By taking one of these classes, you will be better prepared for the job.
Online courses for teachers classroom can get you ready to get a certification or degree. In the past you had to take courses in a traditional class. But the Internet has made it possible to take some of the coursework on the Internet.
Coursework Overview
Teacher education colleges are run by teachers and other qualified experts. Subjects include studying the meaning of school administration, office management and record keeping, Key Issues in educational management and characteristics of good head teachers.
Among the subjects you have to study are school and community, management of co-curricular activities, organizational structure, school discipline and human relations. Aspiring students also have the option of going through decision making, types of administrators and administrative functions in education.
Other Areas of Study
Universities with classes for teachers include subjects such as administrative functions in education, nature and scope of educational management, supervision and inspection, meaning and scope of school organization. Basic concepts that have to be studied are nature, aims, objectives and principles of school Administration. You also have to learn about the difference between administration, supervision & management.
Features and Format
These colleges offer more than just classroom management lessons. Teachers can take up courses on all aspects of educational leadership, physical education and its foundations. You also have to study curriculum and elementary teaching. An aspiring teacher can take up a teacher study program.
They can also take a major in other related courses. You can study for a degree or a certificate. There are graduate and undergraduate programs for students. Some of these classes have live and online classes. You have to undergo fieldwork and Internet based coursework.
Other Information
Issues in education are almost always covered in these courses. Among these subjects are training vs. development, productivity vs. human relations and efficiency and effectiveness. There are also subjects on management of school libraries, using the internet cyber bullying, management of school time-table and service training. You will also study challenges in school administration, common weaknesses of teachers, workload and common problems.
Would-be teachers also have to study guiding principles for schools, how to place emphasis on co-curricular activities and organization of education. Courses on school discipline include old and new concepts of discipline, factors that affect discipline and importance of decision making.
Online courses for teachers classroom is becoming a standard feature in universities and colleges. After completing these web based courses, you will be ready for the state teacher exam.
Online continuing education courses for social workers allow a professional to complete these requirements without going to school. Online options are available in many websites and colleges that have CEU courses.
Coursework Overview
Colleges with social work degrees and their online counterparts arrange their subjects in different sections. Divisions vary but there are several topics which are almost always covered. These include studies in human relations, social psychology basic and advanced concepts.
You will find topics about becoming a helper, ethics, responsibilities and the law. Other subjects are about case management, social work in rural areas and drug abuse and correctional services. Other available subjects are about child welfare services, social work in health care, mental health, administration and research.
License and CEUs
These topics are often reviewed in CEU courses. These programs are available only for social workers who have a valid license. This can be obtained by completing a master's or bachelor's degree in social work. These CEU courses have to be taken every three years to renew the license. After the CEUs are completed, the social worker will submit them to the state.
Requirements
Social work classes are
required to complete a specific number of CE units; the number varies by state. Some do not require you to submit evidence of CEU. However, audits are performed on a random basis. Proof of CEUs have to be submitted. For this reason, continuing education records have to be kept for four years.
Format and Features
A course in ethics is mandatory in most states as is pain management. Other topics that may be needed are common errors in workplace social interactions, stress and frustration, social issues in the workplace and in general. Continuing education classes include learning about the self, schemas, inference and emotion.
Other Information
There are also courses that focus on nonverbal communication, attributing the causes of behavior and research methods. The latter is often studied because methods are always changing. After completing the required courses, you will earn the credits needed to complete the program. If continuing education is not completed, they will be in violation of the state's health code. This can result in probation or license suspension.
Online continuing education courses for social workers work in different ways, depending on the state. To renew licenses, a certain number of CEU hours have to be completed. You need to consult the state licensing board for more information.
Online ESL courses for teachers differ by state. However, they share some common features like videos, audio and other multimedia features. Schedules are also flexible, allowing teachers to study during the day or night.
Coursework Overview
Those who want to instruct ESL must have an education major in college. These courses focus on teaching skills. This is necessary for those who want to operate classes proficiently. Other courses include intercultural communication and applied linguistics. Students also discover methods for teaching conversational English. These courses explain how to speak the language at work.
You will also learn how to assign meaningful homework and plan classes. Specific courses teach you how to work with adults or K-12 students who are studying English. Teachers also learn word meanings, pronunciation and grammar rules.
Formats of these virtual schools vary. There are courses designed for educators who want to work full time. Others are for those who want to teach part time.
Additional Subjects
These universities with education majors also teach about instructional techniques. This is necessary for instructing students with different cultural backgrounds. ESL teachers must have a bachelor's or master's degree.
There are also courses specific for teaching classes. At the same time they are taught how to run computer software tools. More advanced courses focus on the other tasks of a teacher. This includes being a community resource.
Continuing Education
This is required in many states. There are ESL educators with a bachelor's degree who can get a master's degree. This can lead to a better job and pay. Formal continuing education is necessary for their professional development. These courses help ensure their teaching methods are current. Many of these courses are can now be studied on the web.
Other Information
These subjects emphasize teaching methods and cultures. These classes may include a semester of educating. Clinical training is also required in many courses. All ESL educators must also have a license. The requirements differ depending on the state.
Admission requirements will vary depending on whether it is a public or private schools. In other states, you have to take an adult education license. They also study about job placements and the places where you can find work. There are many other courses that you can study on the web.
Online ESL courses for teachers are now being taken by several educators. The convenience that they afford is something that many are now discovering.
Free online courses for teachers aide offer several topics that aspirants and professionals can take. Teachers' aide programs and free courses give you the opportunity to learn the skills necessary to become successful. You will discover the core skills required for the job.
Coursework Overview
Major courses are speaking effectively with parents. Teachers also discover how to develop a good working relationship with coworkers. There are also topics on work team effectiveness and development. As a teacher's assistant, you will find out how to help pupils access the curriculum and develop their numeracy and literacy skills.
These assistants also learn how to help students with their social, emotional and behavioral needs. Aides must also learn about health, hygiene, safety and security. As a student you just take up subjects concerning education and its implications and how to organize the environment.
Other Tasks of Teachers Assistants
A teachers' assistant must also study how to use integrate communication and information technology in the classroom. An assistant must also be aware of how to help students with physical and / or sensory impairments.
Free educational sites for teachers assistants explore different methods for helping students with interaction, learning, cognition, interaction and communication problems. They also have to support students with multilingual and bilingual abilities.
Other Subjects in Free Courses
Their activities are not limited to the classroom. They must also help students with their health and well-being. Teachers assistants must also help when it comes to maintaining the safety of the environment for students.
These free resources also explain what it takes to promote the emotional development of a student. Additional topics include evaluation of learning activities, observing and reporting pupil performance. As an assistant they must also keep stock of pupil records.
Format
These courses use video, audio and forums to help students. Using these resources, you will find it easier to learn how to help during learning activities and maintain good relationships with other students.
Other Information
You will also find out about management of pupil behavior, helping literacy and numeracy activities and working with other people. There are also subjects that focus on supporting your colleagues, planning and evaluating learning activities. Among the other subjects that have to be studied are dealing with behavior problems, communicating with pupils, praise and encouragement.
Free online courses for teachers aide also teach you how to observe changes in students. An integral part of the course is guiding pupil behavior.
If you are a military family always moving from one deployment to the next and your kids need help with their studies; your kids are home-schooled; your young ones are
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being tutored at home or you want to give your kids lessons for advanced studies; your best option is to enroll them in online courses designed for kids of various ages.
There are online courses for kindergarten through sixth grade if your kids are within this age bracket and they are usually assigned their own online instructor.
This will give you the assurance that your kids are being given the best learning tools and assistance even if they are not enrolled in a regular school.
What Courses Can Your Kids Take Online?
For lessons specific to online courses for kindergarten through sixth grade, these are broken down per level to allow the students to take courses appropriate for their age as well as for their level of knowledge.
The core courses however are basically the same for each grade level, it's the extent of the extent and coverage of the courses that vary from one level to the next.
Depending on the online school that you have selected, the curriculum may be categorized differently. For instance, Language Arts may be categorized as Language Arts and Reading while History may be presented as Social Studies.
How are the Courses Presented?
Since the courses are designed for younger students, these have to be presented in a way that they will not get easily bored with the lessons as well as explained in a manner that they can easily comprehend what is being taught to them.
Audio-visual learning aids, interactive practice tests and animated graphics to emphasize certain areas covered by the lessons are the most common methods of how each course is presented online.
You can enroll your kids for a full course or you can also choose specific areas that you want them to focus on. When choosing courses, it is important to check the child's capacity in term of course load so they can work on their courses more effectively.
If you enroll them in more than two courses at a time, they might feel overwhelmed and pressured and thus greatly affect the quality of their online education.
Entrance tests may also be required by the online school prior to admission to check how much the child already knows. This will help the school in determining what grade level they are most suitable forFor licensed and practicing teachers who wish to advance in their careers, a Master's Degree could be the answer. With a Master's Degree, more doors will be opened for you with regards to your career advancement.
Understandably, going back to school may not be your priority for two reasons: time and budget. Graduate school is expensive, maybe even more expensive than taking your undergraduate courses.
For this reason, not a lot of professionals consider getting a higher education. However, with education now also offered online, teachers and other professionals now have a better chance of continuing their education and getting a Master's Degree.
Inexpensive Online Courses for Educators
To find inexpensive online courses for educators, you should first check those that are established specifically to provide online courses alone but whose courses, modules and instructors are affiliated with some of the top universities worldwide.
You may discover that online courses from campuses that have put their presence on the internet are basically priced the same whereas courses offered by websites thru their affiliate schools are much more affordable.
There are also online schools that are actually non-profit organizations which offer more affordable graduate courses with the help of financiers, donors and other supporters.
On the other hand, you can also ask the online university or college if they offer discounts and under what terms and conditions can you avail of these.
Your other options for tuition reduction are applying for a full or partial scholarship and applying for other financial aid options provided by the online school.
Graduate Courses for Teachers
You won't lack in online graduate courses to take once you have decided to earn your Master's Degree as there are quite a lot of offerings on the internet for M.S., M.A. and M.Ed. courses for teachers.
Among the Master of Science courses that you can take are:
- Curriculum and Instruction
- K-12 Special Education
- Educational Leadership
For Master of Arts courses, the following are some of your choices:
- Mathematics Education
- Science Education
- Science Ed. in Chemistry
- Science Ed. in Physics
- English Language Learning
There are also Science Ed. courses in Biological Sciences as well as Science Ed. in Geosciences.
For Master of Education, there are also several online courses to choose from such as Instructional Design and M. Ed. in Learning and Technology.
Bear in mind that the courses listed here have their own areas of study and they may or may not have several domains of study per course.
You should read the course overview and course guide to get more information about them and also to give you an idea if these are what you want to take.
Teaching is a challenging task. Teaching gifted students come with a different set of challenges that only the well-trained and knowledgeable teachers would be able to tackle. Plus, there are certain certifications and degrees that are required to be able to pursue a career in teaching gifted students, who definitely have special needs.
The Gifted, Special and Fragile
Gifted students are classified as such because with the high IQ that they have, they easily get bored with regular flow of classroom discussions. Their minds are advanced and would need a curriculum and a system of teaching that will keep them interested in learning so they continue to do so.
Like special education teachers, teachers who deal with the gifted are required with a special training. Such training will help them know and understand the unique behavioral patters and cognitive abilities of such children. Their skills in ensuring that learning is promoted are further enhanced. That is so they will be able to develop a teaching system that is suitable to the kind of kids they deal with.
Get an Education
The key towards pursuing a career in teaching gifted students is having proper education. Some states may not require an education degree but would ask you to pass licensure exams. There are online courses that will help you through this process.
Teachers of gifted students use different techniques in order to promote learning. At times they are required with individualized instructions that suit their behavior towards learning in general. All these and more are being taught in school.
When choosing a course to become a teacher for the gifted, choose one that will suit your requirements and needs. Make sure that it is worthy of your time, effort, and money. Make sure that it is valuable in a sense that it will be credited for earning your degree or finishing your certification for licensing.
There are a number of courses available for teachers. Each course are directed towards a specific career goal that you may have set previously.
While you are being cautious choosing your course, make sure that you choose an accredited online school as well. You just cannot ignore the possible existence of Internet frauds, which may work towards stealing away either your money or your identity away from you.
As with many other aspects of our modern lives, technology is an important building block in taking a career as a teacher for the gifted. So make sure that you are not left behind.
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A Concise Introduction to Matlab is a simple, concise book designed to cover all the major capabilities of MATLAB that are useful for beginning students. Thorough coverage of Function handles, Anonymous functions, and Subfunctions. In addition, key applications including plotting, programming, statistics and model building are also all covered.
MATLAB is presently a globally available standard computational tool for engineers and scientists. The terminology, syntax, and the use of the programming language are well defined and the organization of the material makes it easy to locate information and navigate through the textbook.
1 An Overview of MATLAB
2 Numeric, Cell, and Structure Arrays
3 Functions and Files
4 Decision-Making Programs
5 Advanced Plotting and Model Building
6 Statistics, Probability, and Interpolation
7 Numerical Methods for Calculus and Differential Equations
8 Symbolic Processing
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Math
At Earl Warren, students enroll in Pre-Algebra, Algebra, or Geometry. In Pre-Algebra, students prepare for Algebra by studying topics such as number sense, algebra & functions, measurement & geometry, statistics, probability & data analysis, and mathematical reasoning. In Algebra, students focus on mathematical concepts such as solving equations, solving systems of equations, using quadratic equations, working with rational expressions and equations, and solving multi-step word problems. Geometry students learn how to use theorems, logical arguments, and proofs to derive and solve geometric problems.
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for Elementary School Teachers
Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be ...Show synopsisFuture elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be called upon to teach. This text uniquely balances "what" they will teach (concepts and content) with "how" to teach (processes and communication). As a result, students using "Mathematics for Elementary School Teachers" leave the course knowing more than basic math skills; they develop a deep understanding of concepts that enables them to effectively teach others. This Fourth Edition features an increased focus on the 'big ideas' of mathematics, as well as the individual skills upon which those ideas are built
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Schaum's Outline of Geometry ideal review for your geometry course More than 40 million students have trusted Schaum's Outlines for their expert knowledge and helpful solved problems. Written by a renowned expert in this field, Schaum's Outline of Geometrycovers what you need to know for your course and, more important, your exams. Step-by-step, the author walks you through coming up with solutions to exercises in this topic. Outline format supplies a concise guide to the standard college course in geometry 712 problems solved step-by-step Clear, concise explanations o... MOREf all geometry concepts Easily-understood review of basic geometry principles Hundreds of practice problems with step-by-step solutions Supports all the major textbooks for the introductory geometry course
Tough Test Questions? Missed Lectures? Not Enough Time?
Fortunately, there's Schaum's.
665 fully solved problems
Concise explanations of all geometry concepts
Support for all major textbooks for geometrycourses
Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time--and get your best test scores!
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Mathematics and Statistics Colleges
A program with a general synthesis of mathematics and statistics or a specialization which draws from mathematics and statistics. Includes instruction in calculus, linear algebra, numerical analysis and partial differential equations, discrete mathematics, probability theory, statistics, computing, and other
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The Mathematics department at Bishop Allen is committed to helping each student realize their mathematical potential. Department members encourage all those enrolled in courses of mathematics to complete and review their work daily, seeking assistance immediately when necessary. To facilitate this learning process, extra help sessions are available for any student in need of clarification in mathematics. Math teachers will be available on selected dates before or after school if you get stumped or confused!
The department offers enriched mathematics classes beginning in grade 9. The academic curriculum is enhanced and augmented by Advanced Placement resource material. In their senior year, students may write the Advanced Placement exam in Calculus.
Students are invited to participate in various Canadian mathematics competitions prepared by the University of Waterloo:
The Pascal, Cayley and Fermat contests (prepared for students
in grades 9, 10, and 11, respectively):
The aim of the contests is to provide an opportunity
for students to have fun and to develop their mathematical
problem solving ability.
The top five contestants in Canada receive Gold Medals
and students scoring in the top 25% of Canadian competitors
will receive a Certificate of Distinction.
The Euclid contest is designed primarily for students in their final year of secondary school:
The aim of the competition is to provide students with an opportunity to develop their mathematical problem solving ability.
Each of the top five competitors in Canada will receive a Gold Medal and will be awarded a Centre for Education in Mathematics and Computing cash prize of $500. Each of the next top ten competitors will receive a Centre for Education in Mathematics and Computing book prize. Each student scoring in the top 25% of Canadian competitors will receive a Certificate of Distinction.
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Academics
Mathematics
In science, there are no such things as unrelated courses. One finds that a mathematical form that represents competition in an ecology course stands a good chance of being the same one that represents a chemical reaction or a knight's move on a chess board. This is one of the beauties of mathematics and, at the same time, one of its powers. Mathematics is a field that (by its own structure) uses forms to investigate itself. This power of introspection will become more and more manifest as you move up the course ladder. Indeed, learning mathematics at Marlboro is learning to spot the reflections of one course in another and to see that there is a great unity and beauty to the subject.
The mathematics curriculum at Marlboro has a two-fold purpose:
Power: to exercise the ability to formalize and abstract, and to develop a facility in specializing abstractions in order to attack applied problems.
Introspection: to appreciate the inherent beauty of mathematical enquiry, and to investigate and to understand the intertwined relationships between the three major branches of mathematics.
Mathematics courses at Marlboro are designed to serve both of these goals, though some are skewed strongly in favor of one or the other.
My personal research is in the areas of combinatorics and group theory, especially in questions motivated by the design of experiments. These areas are very accessible to study at the undergraduate level, and expose students to both the problem solving and theory building sides of mathematics.
Areas of Interest for Plan-level Work:
Students are encouraged to pursue whatever topic demands their attention from across the many sub-fields of mathematics. Recent examples include chaos theory, combinatorics, complex analysis, cryptography, formal languages, game theory, graph theory, group theory, history of mathematics, mathematical modeling and statistics. Interdisciplinary Plans are welcomed.
Starting Points (Basic and Introductory Courses)
TOPICS IN ALGEBRA, TRIGONOMETRY & PRE-CALCULUS (NSC556) Introductory | Credits: Variable
ASPECTS OF GEOMETRY (NSC563) Introductory | Credits: Variable
CALCULUS (NSC515)
A one semester course covering differential and integral calculus and their applications. This course provides a general background for more advanced study in mathematics and science. Prerequisite: Topics in Algebra, Trigonometry and Pre-Calculus or equivalent Introductory | Credits: 4
DISCRETE MATHEMATICS (NSC562)
Discrete math is the study of mathematical objects on which there is no natural notion of continuity. Examples include the integers, networks, permutations and search trees. After an introduction to the tools needed to study the subject, the emphasis will be on you doing mathematics. Series of problems will lead gradually to proofs of major theorems in various areas of the discipline. This course is recommended for those intending to do advanced work in math or computer science. Prerequisite: None Introductory | Credits: 4
STATISTICS (NSC123)
We look at three main topics: the collection and presentation of data, the probability theory behind statistical methods and the analysis of data. Statistical tests covered include the t-test, linear regression, ANOVA and chi-squared. The open source statistical computing package R is introduced and used throughout the class. Prerequisite: Topics in Algebra, Trigonometry and Pre-Calculus or equivalent Introductory | Credits: 4
A WHIRLWIND TOUR OF MATHEMATICS (NSC577)
Do you want a thorough understanding of the most important and deep theorems in every branch of mathematics? Do you want to achieve this in a three credit course from a standing start? Good luck with that—you won't manage it in this course. Instead, we'll look at six to ten topics, chosen for their accessibility and beauty, and drawn from a broad range of sub-disciplines of math. The interests of students in the class will drive the exact choice of topics. Possibilities include: irrational and imaginary numbers, the infinite, chaos and fractals, Fermat's Last Theorem, the Platonic solids, the fourth dimension, the combinatorial explosion, P vs. NP, the Four Color Theorem, non-Euclidean geometry, logical paradoxes and many others. No prior mathematical experience is expected. Prerequisite: None Introductory | Credits: 3
GROUP THEORY & RUBIK'S CUBE (NSC203)
ThisPUZZLED? (NSC541)
This course will give students a chance to test and develop their puzzle-solving ingenuity. We'll attack a series of puzzles, going from Lewis Carroll's logic problems via the classic "recreational math" puzzles of Lucas, Loyd and Dudeney to modern crazes such as the sudoku. Pass/Fail grading. Prerequisite: None Introductory | Credits: 2
Pursuing Interests (Intermediate and Thematic Courses)
CALCULUS II (NSC212)
We build on the theory and techniques developed in Calculus (NSC515). Topics include techniques and applications of integration, epsilon/delta definitions, power series, parametric equations and differential equations. Prerequisite: Calculus or permission of the instructor Intermediate | Credits: 4
EXCURSIONS IN CALCULUS (NSC576)
A deeper study and appreciation of the ideas of calculus, focusing on intuitive understanding of key concepts as well as the contemporary and historical role of the calculus in mathematics and science as conveyed by its wealth of important and beautiful applications. Prerequisite: Calculus or equivalent, or concurrent enrollment in Calculus Introductory | Credits: 2
GAME THEORY (CDS31)
While game theory is a field of mathematics, little math will be used in this course. The intent of this course is to provide students with a formal method for looking at the interactions of individuals (either firms, states, people or bacteria) and explaining or prediction the outcome. Optimally, students will learn or find applications for game theory in their respective fields. Prerequisite: None Introductory | Credits: 3
DIFFERENTIAL EQUATIONS (NSC384)
Learn how to solve differential equations in closed form and by approximation when closed form is impossible or impractical. Applications will cover topics from fields of interest of students enrolled in the course. We will follow a text, but the largest part of the course will concentrate on computer simulations and solutions. Prerequisite: Concurrent enrollment in Calculus II Intermediate | Credits: 4
LINEAR ALGEBRA (NSC164)
Next to Calculus, this is the most important math course offered. It is important for its remarkable demonstration of abstraction and idealization on the one hand, and for its applications to many branches of math and science on the other. Whereas Calculus introduces undergraduates to a large warehouse of constantly used mathematical items, Linear Algebra has the power to use and manipulate those items. Matrices, vector spaces and transformations are studied extensively (most work is done in the n-dimensional real case). Prerequisite: Calculus Intermediate | Credits: 4
NUMBER THEORY (NSC514)
Numbers have been a source of fascination since ancient times. We investigate some of the more intriguing properties numbers can have, and study the work of some of the great mathematicians, including Euclid, Fermat and Gauss. We also look at cryptography—a modern application of number theory. Prerequisite: Topics in Algebra, Trigonometry and Pre-Calculus or equivalent Intermediate | Credits: 4
STATISTICS WORKSHOP (NSC574)
A follow-up to Statistics (NSC123) in which students will acquire and hone the statistical skills needed for their work on Plan. Course content will be driven by the interests and requirements of those taking the class. Prerequisite: Statistics (NSC123) or permission of the instructor Intermediate | Credits: Variable
MULTIVARIABLE CALCULUS (NSC603)
An extension of the ideas from Calculus and Calculus II to multivariable and vector functions. Topics covered include the geometry of 3-dimensional space, partial derivatives, multiple integrals and higher dimensional analogues of the fundamental theorem of calculus. Prerequisite: Calculus II or equivalent Intermediate | Credits: 4
PROBABILITY (NSC604)
Probabilities pop up every day like "There's a 30 percent chance of rain" or "The probability of being dealt a full house in stud poker is approximately 0.00144." Our main goal for the class will be developing various tools to calculate probabilities. Topics include axioms of probability, counting techniques, conditional probability, discrete and continuous random variables, special discrete and continuous distributions and joint distributions. Prerequisite: Calculus I Intermediate | Credits: 4
DYNAMICAL SYSTEMS AND CHAOS THEORY (NSC605)
Our goal for this class will be to study dynamical systems. To study dynamical systems, we will be using the software Mathematica (no initial computer programming knowledge is required). We will not only learn some of the theory behind dynamical systems, but we will also experiment on the computer. We will look at simple dynamical systems, use graphical analysis to help describe the behavior of a system, explore symbolic dynamics, examine fractals and look at the Mandelbrot set and Julia sets. Prerequisite: Linear Algebra or instructor's approval Intermediate | Credits: 4
ADVANCED CALCULUS (NSC207)
The geometry of Euclidean space and generalizations to n-dimensions. The study of differentiation and integration for functions of several variables, coordinate changes and the geometry of maps between spaces. Double and triple integration. Prerequisite: Permission of instructor Advanced | Credits: 4
COMPLEX VARIABLES (NSC590)
An introduction to functions of one complex variable. We look at geometry and transformations in the complex plane and then extend the notions of calculus of functions of real variables to this setting. Prerequisite: Calculus II Advanced | Credits: 4
REAL ANALYSIS (NSC336)
Real analysis is the study of the real numbers and functions of real numbers. After looking in some detail at the underpinnings of the real number system we'll consider sequences, continuity, differentiation and integration. This course will contain very few theorems that you haven't seen and used in Calculus. However, our interest here will be on their proofs rather than their applications. Prerequisites: Calculus and permission of instructor Advanced | Credits: 4
WRITING MATH (NSC534)
In this class you will study the writing and presentation of mathematics. All skills needed for writing Plan-level math will be discussed, from the overall flow of a paper to the use of the typesetting package LaTeX, via writing proofs well and choosing good examples. You will write short papers, based on material in your other math classes, that we will read and discuss as a group. May be repeated for credit. Prerequisite: Permission of instructor and concurrent math class Advanced | Credits: 1
Good Foundation for Plan
All students of mathematics must search for a basic understanding in each of the three major areas: algebra, geometry and analysis. To accomplish this in one undergraduate lifetime, it is important to get some core courses—Discrete Math, Calculus, Calculus II and Linear Algebra—completed as soon as possible. Doing this will widen your range of options as you look for a topic on which to focus. When pursuing an interdisciplinary Plan, there is less urgency to complete these courses, since the "introspection" side of the Plan will turn towards investigating the relationships between mathematics and the chosen field.
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This website is all about making an informed choice, choosing a math
course that is right for you. However, this Self-Assessment site does
not apply to everyone needing to take a math class. Should you go through the process?
Let's find out.
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Algebra Functions
Up until now, you've learned everything there is to know about equations and expressions! Now we are going to change our focus to algebra functions! You will be introduced to functions in Algebra 1, but you will see a lot more of them as you continue your math journey into Algebra 2!
Functions are a lot like equations, but there are a few little things that make them unique! You'll see!
Below is your table of contents for the Functions Unit. Click on the lesson that interests you, or follow them in order for a complete study of functions!
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Monroe Township Schools
Monroe Township Schools
Curriculum Management System
Algebra 1 A/B
Grade 9
July 2010
* For adoption by all regular education programs Board Approved:
as specified and for adoption or adaptation by
all Special Education Programs in accordance
with Board of Education Policy # 2220.
Table of Contents
Monroe Township Schools Administration and Board of Education Members Page 3
Acknowledgments Page 4
District Vision, Mission, and Goals Page 5
Introduction/Philosophy/Educational Goals Pages 5-6
Core Curriculum Content Standards Page 7
Scope and Sequence Pages 8-11
Algebra I Core Content Overview Pages 12-14
Goals/Essential Questions/Objectives/Instructional Tools/Activities Pages 15-67
Benchmarks Page 68
2
MONROE TOWNSHIP SCHOOL DISTRICT
ADMINISTRATION
Dr. Kenneth Hamilton, Superintendent
Mr. Jeff Gorman, Assistant Superintendent
Ms. Sharon M. Biggs, Administrative Assistant to the District Superintendent
BOARD OF EDUCATION
Mr. Lew Kaufman, President
Mr. Marvin I. Braverman, Vice President
Mr. Ken Chiarella
Mr. Mark Klein
Ms. Kathy Kolupanowich
Mr. John Leary
Ms. Kathy Leonard
Mr. Louis C. Masters
Mr. Ira Tessler
JAMESBURG REPRESENTATIVE
Ms. Patrice Faraone
Student Board Members
Ms. Reena Dholakia
Mr. Jonathan Kim
3
Acknowledgments
The following individuals are acknowledged for their assistance in the preparation of this Curriculum
Management System:
Writers Names: Jaclyn E. Varacallo
Technology Staff: Al Pulsinelli
Reggie Washington
Secretarial Staff: Debby Gialanella
Gail Nemeth
4
Monroe Township Schools
Vision, Mission, and Goals
Vision Statement
The Monroe Township Board of Education commits itself to all children by preparing them to reach
their full potential and to function in a global society through a preeminent education.
Mission Statement
The Monroe Public Schools in collaboration with the members of the community shall ensure that all
children receive an exemplary education by well trained committed staff in a safe and orderly
environment.
Goals
Raise achievement for all students paying particular attention to disparities between subgroups.
Systematically collect, analyze, and evaluate available data to inform all decisions.
Improve business efficiencies where possible to reduce overall operating costs.
Provide support programs for students across the continuum of academic achievement with an
emphasis on those who are in the middle.
Provide early interventions for all students who are at risk of not reaching their full potential.
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INTRODUCTION, PHILOSOPHY OF EDUCATION, AND EDUCATIONAL GOALS
Philosophy
Monroe Township Schools are committed to providing all students with a quality education resulting in life-long learners who can
succeed in a global society. The mathematics program, grades K-12, is predicted on that belief and is guided by the following six
principals as stated by the National Council of Teachers of Mathematics (NCTM) in the Principles and Standards for School
Mathematics, 2000. First, a mathematics education requires equity. All students will be given worthwhile opportunities and strong
support to meet high mathematical expectations. Second, a coherent mathematics curriculum will effectively organize, integrate, and
articulate important mathematical ideas across the grades. Third, effective mathematics teaching requires the following: a) knowing
and understanding mathematics, students as learners, and pedagogical strategies, b) having a challenging and supportive classroom
environment and c) continually reflecting on and refining instructional practice. Fourth, students must learn mathematics with
understanding. A student's prior experiences and knowledge will actively build new knowledge. Fifth, assessment should support the
learning of important mathematics and provide useful information to both teachers and students. Lastly, technology enhances
mathematics learning, supports effective mathematics teaching, and influences what mathematics is taught.
As students begin their mathematics education in Monroe Township, classroom instruction will reflect the best thinking of the
day. Children will engage in a wide variety of learning activities designed to develop their ability to reason and solve complex problems.
Calculators, computers, manipulatives, technology, and the Internet will be used as tools to enhance learning and assist in problem
solving. Group work, projects, literature, and interdisciplinary activities will make mathematics more meaningful and aid understanding.
Classroom instruction will be designed to meet the learning needs of all children and will reflect a variety of learning styles.
In this changing world those who have a good understanding of mathematics will have many opportunities and doors open to
them throughout their lives. Mathematics is not for the select few but rather is for everyone. Monroe township Schools are committed
to providing all students with the opportunity and the support necessary to learn significant mathematics with depth and understanding.
This curriculum guide is designed to be a resource for staff members and to provide guidance in the planning, delivery, and assessment
of mathematics instruction.
Educational Goals
Algebra I is the first course of the college preparatory sequence. It is designed to provide an in-depth analysis of the real world
system and introduce process of algebra. Topics included are: data analysis, roots and powers, simplify mathematical expressions,
linear equations, graphing linear equations, theoretical and experimental probability, linear inequalities, systems of equations and
inequalities, polynomial equations, quadratic functions, graphing quadratic functions, mathematical models, functions, matrices, and
solve rational equations. The A/B curriculum is designed to teach and remediate with the same instructor so as to aid students in
meeting all the standards and requirements to pass the End of Course Algebra 1 Exam.
6
New Jersey State Department of Education
Core Curriculum Content Standards
A note about Common Core State Standards for Mathematics
The Common Core State Standards for Mathematics were adopted by the state of New Jersey in 2010. The standards referenced in
this curriculum guide refer to these new standards and may be found in the Curriculum folder on the district servers. A complete copy
of the new Common Core State Standards for Mathematics and the end of year algebra 1 test content standards may also be found at:
7
Algebra 1 A/B
Scope and Sequence
Quarter I
Big Idea I: Representation and Modeling with Variables Big Idea II: Equivalence
I. Variables in Algebra I. Absolute Value
a. Writing and Evaluating Variable Expressions II. Graphing and Comparing Real Numbers on a Number Line
b. Evaluating Simple Interest III. Addition and Subtraction of Real Numbers
II. Expressions Containing Exponents IV. Multiplication and Division of Real Numbers
III. Order of Operations V. Distributive Property
IV. Equations and Inequalities
a. Checking and Solving Equations
b. Checking Solutions of Inequalities
V. Translating Verbal Phrases to use in Algebraic Models
a. Translating verbal phrases into Algebra
b. Using verbal models
VI. Functions
a. Input-Output tables
b. Domain and Range
Big Idea III: Connections and Data Analysis Big Idea IV: Equivalence/ Representation & Modeling with
Variables
I. Construct and Interpret Data Displays
a. Line Graph I. One-Step Equations
b. Bar Graph II. Multi-Step
c. Box and Whisker Plots a. Combining like terms
d. Stem and Leaf Plots b. Distribution
II. Probability and Odds c. Multiplying by reciprocals
a. Experimental vs. Theoretical d. Variables on Both Sides
b. Combinations and Permutations e. Rational Coefficients
i. Using a Graphing Calculator f. Reciprocal Property and Cross Products
III. Measures of Central Tendency III. Using Linear Equations for Problem Solving
IV. Rates, Ratios, Proportions, Percents a. Translating verbal models
b. Drawing a diagram
c. Using tables to solve
d. Using graphs to solve
IV. Transforming Formulas
Course Quarterly Benchmark Assessment: (Higher level 5-10 questions,
45 minutes)
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Quarter II
Big Idea V: Representation & Modeling with Big Idea VI: Linearity
Variables/Linearity
I. Slope-Intercept Form
I. Plotting Cartesian Coordinates II. Point- Slope Form
II. Scatterplots III. Writing an Equation
a. Graphing Data a. Given two points
III. Graphing Linear Equations b. Given a point and slope
a. Using Input-Output Table c. Given a point and a line parallel
b. Using Intercepts d. Given a point and a line perpendicular
c. Using Slope and y-intercept IV. Converting to Standard Form
d. Horizontal and Vertical Lines V. Reintroducing Scatterplots and Predicting with Linear Models
e. Using a Graphing Calculator a. Graphing Data
IV. Solving Linear Equations Using Graphs b. Calculate Line of Best Fit by Hand
a. Graphical Check for a Solution c. Calculate Line of Best Fit with Graphing Calculator
b. Solving an Equation Using a Graph VI. Graphing Absolute Value Equations
c. Approximating Solutions Using a Graph a. Using Input-Output Table
V. Functions vs. Relations b. Using Vertex and Slope
a. Using a graph to determine c. Using a Graphing Calculator
b. Using a table to determine
c. Vertical Line Test
Big Idea VII: Linearity Big Idea VIII: Linearity
I. Solving and Graphing Inequalities in One Variable I. Solving Linear Systems
a. One Step a. Checking Validity of Solutions
b. Multi Step i. Substituting in values
c. Compound ii. Using a Graphing Calculator
d. Absolute Value b. Determining the Number of Solutions
II. Graphing Linear Inequalities in Two Variable c. By Graphing
a. Checking Solutions d. By Substitution
b. Using a Graphing Calculator e. By Elimination (Linear Combination)
II. Solving Systems of Linear Inequalities
a. Graphing by Hand
b. Using a Graphing Calculator
III. Applications of Linear Systems
Course Quarterly Benchmark Assessment: (Higher level 5-10 questions)
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Quarter III
Big Idea IX: Non-linear Relationships Big Idea X: Representation and Modeling with Variables
I. Properties of Exponents I. Radicals
a. Multiplication a. Simplification
b. Power of Power b. Multiplication
c. Power of Product c. Division
d. Zero and Negative Exponents d. Rationalizing Denominators
e. Division e. Addition and Subtraction of Rational Expressions
II. Scientific Notation II. Solving Radical Equations
a. Converting from Expanded Form to Scientific Notation III. Evaluating a Discriminant
b. Converting from Scientific Notation to Expanded Form IV. Distance Formula (Pythagorean Theorem)
c. Computations with Scientific Notation V. Graphing a Quadratic Function
III. Exponential Graphs a. Determine the Vertex and Axis of Symmetry
a. Growth and Decay Functions and their Graphs b. Using an Input-Output Table
i. Growth and Decay Factor c. Using a Graphing Calculator
ii. Interpreting Using Graphing Calculator d. Identify Domain and Range
b. Determining Domain and Range Using a Graph VI. Solving Quadratic Equations using the Quadratic Formula
c. Compound Interest VII. Application of the Discriminant
Course Quarterly Benchmark Assessment: (Higher level 5-10 questions)
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Quarter IV
Big Idea XI: Representations and Modeling with Variables Big Idea XII: Nonlinear Relationships
VIII. Polynomial Functions XI. Direct and Inverse Variation
a. Naming a. Using a Model to Solve Application Problems
b. Addition/Subtraction i. Using a Graphing Calculator
c. Multiplication XII. Simplifying Rational Expressions
d. Solving in Factored Form a. By Factoring
IX. Solving Quadratic Equations by Factoring b. By Using Greatest Common Factor
a. With a Leading Coefficient of 1 c. Finding Values Where a Rational Expression is Undefined
b. With a Leading Coefficient other than 1 d. Using Addition and Subtraction
c. With a Greatest Common Factor e. Using Multiplication and Division
d. Special Products XIII.Solving Rational Equations
e. Grouping
X. Finding Zeros/Intercepts of an Quadratic Equation
a. By Solving Quadratic Equations
b. Graphically
c. Using a Graphing Calculator
Big Idea XIII: Connections and Extensions
XIV. Operations with Radical Expressions (Chapter 12) Course Quarterly Benchmark Assessment: (Higher level 5-10 questions)
XV. Pythagorean Theorem and its Converse (Chapter 12)
XVI. Identifying Patterns (External Resources – HSPA Review Packet)
XVII. Application Problems
11
Algebra I Core Content Overview
O1.B1 Using variables in different ways.
Big Idea I: L1.a Representing linear functions in multiple ways.
Representation and Modeling L1.b Analyzing linear functions.
L1.d Using linear models.
O1.a Reasoning with real numbers.
Big Idea II: O1.b Using ratios, rates, and proportions.
D1.b Comparing data using summary statistics.
Connections and Data D1.c Evaluating data-based reports in the media.
Analysis D2.a Using counting principles.
D2.b Determining probability.
O1.a Reasoning with real numbers.
L1.a Representing linear functions in multiple ways.
L1.b Analyzing linear functions.
Big Idea III:
L1.d Using linear models.
Equivalence L2.a Solving linear equation and inequalities.
L2.e Modeling with single variable linear equations, one or two variable inequalities, or
systems of equations.
O1.B1 Using variables in different ways.
Big Idea IV: L1.a Representing linear functions in multiple ways.
Representation and Modeling L1.b Analyzing linear functions.
L1.d Using linear models.
L1.a Representing linear functions in multiple ways.
L1.b Analyzing linear functions.
Big Idea V:
L1.d Using linear models.
Linearity L2.c Graphing linear functions involving absolute value.
L2.e Modeling with single variable linear equations, one or two variable inequalities, or
systems of equations.
12
Algebra I Core Content Overview
L2.a Solving linear equation and inequalities.
L2.b Solving equations involving absolute value.
Big Idea VI:
L2.c Graphing linear inequalities.
Linearity L2.e Modeling with single variable linear equations, one or two variable inequalities, or
systems of equations.
L1.b Analyzing linear functions.
L1.d Using linear models.
Big Idea VII: L2.c Graphing linear functions involving absolute value.
Linearity L1.d Using linear models.
L2.e Modeling with single variable linear equations, one or two variable inequalities, or
systems of equations.
O1.c Using numerical exponential expressions.
Big Idea VIII:
O2.a Using algebraic exponential expressions.
Relationships N2.B1 Solving simple exponential equations.
O1.d Using numerical radical expressions.
O2.d Using algebraic radical expressions.
Big Idea IX: O2.b Operating with polynomial expressions.
Relationships N1.a Representing quadratic functions in multiple ways.
N1.c Using quadratic models.
N2.b Solving quadratic equations.
O2.b Operating with polynomial expressions.
Big Idea X: O2.c Factoring polynomial expressions.
Representation and Modeling N1.b Distinguishing between function types.
N1.c Using quadratic models.
N2.b Solving quadratic equations.
13
Algebra I Core Content Overview
O1.b Using ratios, rates, and proportions.
Big Idea XI:
L2.e Modeling with single variable linear equations, one or two variable inequalities, or
Relationships systems of equations.
O1.d Using numerical radical expressions.
Big Idea XII: O2.d Using algebraic radical expressions.
Connections and Extensions L2.e Modeling with single variable linear equations, one or two variable inequalities, or
systems of equations.
14
BIG IDEA I 6
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
Variables can be used to describe Equations are used to describe patterns. Write and evaluate a variable expression.
1.1
number relationships. Operations are used to represent verbal models. Evaluate Simple Interest.
Symbols can be manipulated using different operations to
Exponents are tools to model model and communicate relationships. Evaluate and write expressions containing
1.2
patterns. exponents
Order of operations is a standardized Sample Conceptual Understandings Use order of operations to evaluate
1.3
method to evaluate expressions. One room in Jean's apartment is a square measuring 12.2 algebraic expressions with and without a
feet along the base of each wall. How many square feet calculator.
Verbal sentences can be translated of wall-to-wall carpet does Jean need to carpet the room? Check solutions to equations and
into mathematical sentences. inequalities.
Mathematical sentences represent Use verbal and algebraic models to
1.4
Make a table for the powers of 8. Describe any patterns.
verbal sentences. represent real-life situations.
Solutions allow number sentences to You are shopping for a mountain bike. A store sells two
make a true statement. different models. The model that has steel wheel rims
Problem solving can be achieved costs $220. The model with aluminum wheel rims costs Explain modeling using algebraic
through a system of verbal models $480. You have a summer job for 12 weeks. You save expressions.
1.5
labelsalgebraic $20 per week, which would allow you to buy the model
modelsolvingand a solution with the steel rims. You want to know how much more
check.
15
Functions are one-to-one and onto. money you would have to save each week to be able to Identify a function.
Functions can be represented in buy the model with the aluminum wheel rims. Functions can be described using an
multiple ways to model real-life o Write a verbal model and an algebraic model for input-output table, verbal description, in
situations. how much more money you would have to save symbols, and a graph.
Domain is the set of all input values each week. Describe the relationship between the
that go into a function. This results in o Use mental math to solve the equation. What domain and range of a function.
the range – the set of all output does the solution represent?
values.
If you place one marble in a measuring cup that contains
200 milliliters of water, the measure on the cup indicates
1.7
that there is a one millimeter increase in volume. How
much does the volume increase when you place from 1 to
10 marble in the measuring cup?
o Write an equation to represent the function.
o Compute an input-output table for the function
with the domain 0,1,2,3,4,5,6,7,8,9,10.
o Describe the domains and range of the function
whose values are shown in the table.
o Graph the data in the table. Use this graph to
graph the function Concept Activity: Finding Patterns (Chapter 1 Resource Books, p.56)
Chapter 1 Project: Watch It Disappear (Chapter 1 Resource Books, p.117)
11.3 Graphing Calculator Activity (Chapter 11 Resource Books, p.40)
Tiered Activity Example Big Idea #1: Tiered Example
16 You are making candles to sell at your school's art festival. You melt paraffin wax in a cubic container. Each edge is 6 inches in length.
The container is one-half full. Design a cubic candle mold that will hold all of the melted wax. Draw a diagram of the mold. Explain
why your mold will hold all of the melted wax. (McDougal-Littell: Algebra 1, pg. 14)
1.1 Real-Life Applications: Freshman Class Officer Duties (Chapter 1 Resource Books, p.21)
1.5 Real-Life Applications: Taiwan Vacation (Chapter 1 1: Alternative Assessment and Math Journal (Chapter 1 Resource Books, p.115)
Resources
Additional
McDougal-Littell: Algebra 1 2004
McDougal-Littell: Algebra 1 Chapter Resource Books
17
BIG IDEA II: Equivalence 8
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
Absolute value of a number is the Real numbers are communication tools that express Graph and compare real numbers
distance of a value from zero on the important ideas. using a number line.
2.1
number line. Addition and subtraction of real numbers are directly Find the absolute value of a number.
Real numbers are all values that are related to one another. Find the opposite value of a number.
found on a number line. Multiplication and division of real numbers are directly
The sum of two positive integers is related to one another. Add real numbers using a number line
positive. or addition rules.
The sum of two negative integers is Sample Conceptual Understandings
2.2
negative. A star's brightness as it appears to a person on Earth is
The sum of a positive integer and a measured by its apparent magnitude. A bright start has
negative integer can be positive, negative, a lesser apparent magnitude than a dim star.
or zero. Star Magnitude
To subtract two quantities, add the Canopus -0.72 Subtract real numbers using the
2.3
opposite. The result is the difference of Altair 0.77 subtraction rule.
the two quantities.
Sirius -1.46
When multiplying, if the signs of two Multiply real numbers using properties
Vega 0.03
factors are the same, the product will be of multiplication.
2.5
positive. If the signs of two factors are o Which star looks the brightest?
different, the product will be negative. o Which star looks the dimmest?
18
The distributive property is used when a o Which star looks dimmer than Altair? Use the distributive property to
factor is multiplied by a polynomial and multiply a factor and a polynomial.
the factor must be distributed to each In a game that decides the high school football
term in a polynomial. championship, your team needs to gain 14 years to score
2.6
Like terms in an expression have the same a touchdown and win. Your team's final four plays result
variable raised to the same power. in a 9-yard gain, a 5-yard loss, a 4-yard gain, a 5-yard
Constant terms are terms without a gain as time runs out. Use a number line to model the
variable. gains and losses and explain whether your team won.
The product of a nonzero number and its Divide real numbers.
reciprocal is 1. You and a friend decide to leave a 15% tip for restaurant
To divide, multiply dividend by the service. You compute the tip as , where
reciprocal of the divisor. represents the cost of the meal. Your friend claims that
Division by zero is undefined. an easier way to mentally compute the tip is to calculate
10 % of the cost of the meal plus one half of 10% of the
2.7
cost of the meal.
o Write an equation that represents your friend's
method of computing the tip.
o Simplify the equation.
o Will both methods give the same results? Explain 2.3 Visual Approach Lesson Opener (Chapter 2 Resource Books, p.42 Big Idea #2: Tiered Example
19Assessment Models
Open-Ended Assessment:
2.2 Real-Life Applications: Stockholders (Chapter 2 Resource Books, p.36)
2.5 Real-Life Applications: Hot-Air Balloons (Chapter 2 Resource Books, p.7820
BIG IDEA III: Connections and Data Analysis can odds and probability help to analyze information to interpret data?
How can you use data displays in the real world?
Describe the relationship between mean, median, mode, and outliers.
Suggested Blocks for Instruction: 10
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
Data is information, fact, or numbers that Graphs represent data in an organized manner to help Use a table to organize data into
describe something. analyze information. meaningful groupings.
1.6
Tables and graphs are used to organize Mean, median, and mode are measures of central Make and interpret a bar graph.
data. tendency of a data set. Make and interpret a line graph.
Probability of an event is the likelihood Probability is used to analyze information and Find the probability of an event and
that the event will occur. interpret significance. determine its likelihood.
Ratios are used to make inferences about large Find the odds of an event.
2.8
The odds of an event are the ratio of the population using small samples. Calculate theoretical probability of an
number of favorable outcomes divided by Percents are used to analyze and compare data from event.
the number of unfavorable outcomes. graphs. Calculate experimental probability of an
Experimental Probability uses Unit rates are factors that help to model and scale event.
A unit rate is a rate per one given unit. proportions to desired quantities. Use rates and ratios to model and solve
real-life problems.
3.8
Sample Conceptual Understandings Use percents to solve real-life
The table shows the number of commercial television problems.
A stem and leaf plot is used to organize stations for different years. Make a line graph of the Make and use a stem-and-leaf plot to
data. data. Discuss what the line graph shows. put data in order.
6.6
Find the mean, median, and mode of
data.
21
A Box and whisker plot is a data display Draw a box-and-whisker plot to
that divides a set of data into four parts. organize data.
The median separates the set into two Read and interpret a box-and-whisker
halves (50%). Suppose you randomly choose a marble from a bag plot.
The first quartile is the median of the holding 11 green, 4 blue, and 5 yellow marbles. Use
lower half (25%) and the third quartile is probability and odds to express how likely it is that you
the median of the upper half (75%) of the choose a yellow marble. If you find one (probability or
data. If a measure of position is shared odds) easier to understand or more useful than the
between two data entries, the average is other, explain why.
6.7
taken to represent that position. In that You are conducting a survey on the use of air-plane
case, those two averaged data entries are phones. You survey 320 adults and find that 288 of
included in calculating the quartiles. them never made a phone call from an airplane. If you
surveyed 3500 adults, how many of them would you
predict have made a phone call from an airplane?
Explain.
If someone said that the mean age of everyone in your
algebra class is about 16 ½ years old, do you think the
age of the teacher was included in the calculation?
Explain Cooperative Learning Activity (Chapter 1 Resource Books, p.90)
Activity Lesson Opener (Chapter 6 Resource Books, p.94 Example
22 Interdisciplinary Application (Chapter 2 Resource Books, p.120)
Real-Life Application: Skyscrapers(Chapter 3 Resource Books, p.118)
Real Life Application: Good Health and Test Scores (Chapter 6 Resource Books, p.102 should Alternative Assessment and Math Journal Multistep Problem (#2 only)(Chapter 6 Resource Books, p.112)
Resources
Additional
McDougal-Littell: Algebra 1 2004
McDougal-Littell: Algebra 1 Chapter Resource Books
23
BIG IDEA IV equations useful in everyday life?
How is an equation that has no solution different than an equation that is an identity?
How can drawing diagrams, using a table, and using a graph can be useful problem solving tools?
How are formulas similar and different to equations?
Why are ratios useful for architectural design?
How do percentages relate to you?
Suggested Blocks for Instruction: 14
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
A linear equation is when the Equations model patterns that occur in real life Solve linear equations using addition and
variable is raised to the first power problems and are used to solve for unknown subtraction.
and does not occur in a quantities.
denominator, inside a square root Diagrams help to model problems and draw
symbol, or inside an absolute value conclusions.
symbol. A graph and its equation are in an interdependent
3.1
Inverse operations are operations relationship.
that undo each other, which help Formulas are direct representations of real life
to isolate the variable on one side applications that help to solve for an unknown
of the equation. quantity.
The goal of solving a linear
equation is isolating the variable on
one side of the equation.
Dividing by a number is the Solve linear equations using multiplication
3.2
equivalent to multiplying by its Sample Conceptual Understandings and division.
reciprocal. The table shows the number of Digital Versatile Disc
To solve a multi-step equation, first (DVD) players sold in the first ten month after their Use two or more transformations to solve
simplify both sides of the equation release in 1997. an equation.
3.3
and then use inverse operations to Combine like terms in an equation.
isolate the variable.
24
Translate word problems (verbal models)
into equations.
An identity is an equation that is Collect variables on both sides of the
true for all values of the variable. equation.
3.4
Some linear equations have no
solution.
Graphs are the visualization of Draw diagrams to problem solve.
equations. Use graphs and tables to gather and/or
3.5
check answers.
o For each month, write a sales equation
Round-off error is a consequence relating cumulative and monthly sales. Let Find exact and approximate solutions of
3.6
of rounded solutions. represent the number of players sold that equations that contain decimals.
month.
A formula is an algebraic equation o Solve your sales equations to fill in the Solve a formula for one of its variables.
that relates two or more real-life monthly sales column. Rewrite an equation in function form.
3.7
quantities. o Suppose a DVD player manufacturer started
Function form is when a variable is an advertising campaign in September. Use
isolated on one side of the formula. your table to Judge the campaign's effect on
A proportion is an equation that sales. Write a brief report explaining whether Use the reciprocal property to solve
11.1
states two ratios are equal. the campaign was successful. proportions for unknown quantities.
Use the cross product property to solve
Write and solve an equation to find your average proportions for unknown quantities.
Percents can be described using speed on a trip from St. Louis to Dallas. You drove Use equations to solve problems involving
percentages, decimals, or ratios. miles in 10 ½ hours. percents.
Two student volunteers are stuffing envelopes for a
local food pantry. The mailing will be sent to 560
possible contributors. Luis can stuff 160 envelopes per
hour and Mei can stuff 120 envelopes per hour.
o Working alone, what fraction of the job can
11.2
Luis complete in one hour? In hours? Write
the fraction in lowest terms.
o Working alone, what fraction of the job can
Luis complete in hours?
o Write an expression for the fraction of the job
that Luis and Mei can complete in hours if
they work together.
o To find how long it will take Luis and Mei to
complete the job if they work together, you
25
can set the expression you wrote in part (c)
equal to 1 and solve for . Explain why this will
work.
o How long will it take Luis and Mei to complete
the job if they work together? Check your
solution.
Train A leaves the downtown station for the other end
of the line at 55 mi/h. Train B leaves the other end of
the line on a parallel route and heads downtown at 65
mi/h.
o Use the graph to tell how many minutes it will
be before the trains pass one another.
o Write and solve an equation to check your
answer.
Suppose another town has 15,860 people aged 25
years or older that 7581 of these people have
completed at least 4 years of college. Explain how you
can find out whether the number of college graduates
in that town is typical for a town of that size.
You are shopping and find a coat that is on sale for
26
30% off. It is regularly prices at $80. Your friend tells
you that she saw the same coat that she saw the same
coat for $80 in another store, but it was 20% off plus
an additional 10% off. Will you save by going to the
other store? Explain 3.1 Activity Lesson Opener (Chapter 3 Resource Books, p.13)
3.3 Application Lesson Opener(Chapter 3 Resource Books, p.37)
11.1 Application Lesson Opener(Chapter 11 Resource Books, p.12)
Tiered Learning ActivityAssessment Models
Open-Ended Assessment:
3.1 Real-Life Application: College Football Stadiums (Chapter 3 Resource Books, p.20)
3.2 Interdisciplinary Application: Pony Express(Chapter 3 Resource Books, p.32)
3.4 Real-Life Application: Recycling (Chapter 3 Resource Books, p.62)
3.5 Real-Life Application: Tunnels(Chapter 3 Resource Books, p.76)
3.6 Interdisciplinary Application: Magnification(Chapter 3 Resource Books, p.105)
11.2 Interdisciplinary Application: Markup and Cost(Chapter 11 Resource Books, p.34Sample Conceptual Understanding section. (Synthesis, Analysis, Evaluation)
27 3: Project: Ice Rescue (Chapter 1 Resource Books, p.130)
Chapter 11 Project: Miniature Room (Chapter 1128
BIG IDEA V scatterplot useful in making predictions?
How is a line a useful tool for interpreting data?
Describe an occupation in which slope plays an important role.
Suggested Blocks for Instruction: 10
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
A coordinate plane is formed by two real Scatterplot enable analysis of patterns and the Plot points in a coordinate plane.
number lines that intersect at the relationship between two quantities by yielding a visual Draw a scatter plot and make
origin, . representation of data. predictions.
Each point in the plane corresponds to an Real life situations can be modeled using an equation.
4.1
ordered pair . Equations can be used to describe real life situations to
A scatterplot is a graph of ordered data form predictions.
pairs on a coordinate plane that allow
analysis between two quantities. Sample Conceptual Understandings
A point is on the graph of an equation if it The table below shows the number of rolls developed Graph a linear equation using a table or
satisfies the statement when the values for the United States media at the Winter Olympics. a list of values.
4.2
are substituted in. Graph horizontal and vertical lines.
Describe the situation presented using
a graph of the data.
The -intercept is the value of o Construct a scatter plot of the data. Find the intercepts of a graph of a
when . Describe the pattern of the number of rolls linear equation.
4.3
The -intercept is the value of of film developed for the Winter Olympics Use intercepts to make a quick graph of
when . from 1984 to 1998. a linear equation.
Draw appropriate scales.
29
The ratio "rise to run" describes the o Predict the number of rolls of film that will Find the slope of a line using two of its
steepness of a slope. be developed for the Winter Olympics in the points.
The slope of a non-vertical line is the year 2002. Explain how you made your Interpret slope using real life contexts.
number of units the line rises or falls for prediction.
4.4
each unit of horizontal change from left Use a table of values to graph the equation:
to right. .
A vertical slope is undefined. Your school drama club is putting on a play next
Rate of change compares two different month. By selling tickets for the play, the club hopes
quantities that are changing. to raise $600 for the drama fund for new costumes,
Slope intercept is of the form scripts, and scenery for future plays. Let represent Graph a linear equation in slope-
where is the slope and is the - the number of adult tickets they sell at $8 each, and intercept form.
intercept. let represent the number of student tickets they Graph and interpret equations in slope-
4.6
sell at $5 each. intercept form that model real life
o Write a linear function to model the situations.
situation. Identify parallel lines.
o Graph the linear function. Solve a linear equation graphically.
4.7
o What is the -intercept? What does it Use a graphing calculator to
represent in this situation? approximate a solution.
o What are three possible number of adult
A relation is any set of ordered pairs. Identify when a relation is a function
and student tickets to sell that will make the
A relation is a function of the horizontal graphically and looking at sets of
drama club reach its goal?
axis variable if and only id no vertical line ordered pairs.
Draw a ramp and label its rise and run. Explain what
passes through two or more points on the
is meant by the slope of the ramp.
graph.
The volume of blood pumped from your heart
is called function notation.
each minute varies directly with your pulse rate .
Each time your heart beats, it pumps approximately
liter of blood.
o Find an equation that relates and .
4.8
o Take your pulse and find out how much
blood your heart pumps per minute.
Graph the situation: You start from home and drive
55 miles per hour for 3 hours, where is your
distance from home.
21st Century Skills
30
Critical Thinking and Problem Solving Critical Thinking and Problem Solving Communication and Collaboration
Media Literacy Media Literacy ICT Literacy
Technology Based Activities Technology Based Activities
Learning Activities
4.1 Graphing Calculator Activity(Chapter 4 Resource Books, p.15)
4.1 Activity Lesson Opener (Chapter 4 Resource Books, p.12)
4.7 Graphing Calculator Lesson Opener (Chapter 4 Resource Books, p.98)
4.8 Application Lesson Opener (Chapter 4 Resource Books, p.111)
Tiered Learning Activity Big Idea V: Tiered 4.1 Interdisciplinary Application: Mammals (Chapter 4 Resource Books, p.22)
4.4 Interdisciplinary Application: Minimum Wage(Chapter 4 Resource Books, p.62)
4.5 Real-Life Application: Gasoline Prices (Chapter 4 Resource Books, p.74)
4.6 Interdisciplinary Application: Mount Everest (Chapter 4 4: Alternative Assessment and Math Journal(Chapter 4 Resource Books, p.128)
Chapter 4 Project: Carnival Time(Chapter 4 Resource Books, p.131)
31
Resources
Additional
McDougal-Littell: Algebra 1 2004
McDougal-Littell: Algebra 1 Chapter Resource Books
32
BIG IDEA VI: Linearity linear model used to approximate a real life situation?
Explain how to use a linear model to make predictions from given data.
Describe the differences between parallel and perpendicular lines.
How do the different forms of linear functions and the concept of slope help solve real world situations?
Suggested Blocks for Instruction: 14
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
Slope-intercept form of the equation of Equations describe the relationship between a dependent Use the slope-intercept form to
the line is: . and an independent variable. write an equation of a line.
A linear model is a linear function that Best-fit line represents the relationship between two Write an equation of a line from a
is used to model a real-life situation. variables. graph.
5.1
When a linear model is used to Point-intercept form, point-slope form, and standard form Model a real life situation with a
approximate a situation, slope models are interdependently related. linear function.
the rate of change and -intercept is
the initial amount or the fixed amount. Sample Conceptual Understandings
Parallel lines have the same slope. A rental company charges a flat fee of $30 and an additional Use slope and any point on a line to
"If and only if" is a bi-conditional $.25 per mile to rent a moving van. Write an equation to write an equation of the line.
5.2
statement that means if model the total charge (in dollars) in terms of , the Use a linear model to make
. number of miles driven. predictions about a real-life
The cost of a taxi ride is an initial fee plus $1.50 for each situation.
The slope of a line can be found using mile. Your fare for 9 miles is $15.50. Write an equation that Write an equation of a line given
two points on the line. models the total cost of a taxi ride in terms of the number two points on the line.
The product of a number and its of miles . How much is the initial fee?
multiplicative inverse, its reciprocal, is Write an equation in slope-intercept form of the line that
5.3
equal to -1. passes through the points: .
Perpendicular lines are two lines that A mountain climber is scaling a 300-foot cliff at a constant
intersect at a angle. rate. The climber starts at the bottom at 12:00 PM by 12:30
Perpendicular lines have slopes that are PM, the climber has moved 62 feet up the cliff. Write an
opposite reciprocals.
33
The best-fitting line is a line that models equation that gives the distance (in feet) remaining in the Find a linear equation that
the trend through a set of data points. climb in terms of the time (in hours). What is the slope of approximates a set of data points
Correlation is a number satisfying the line? At what time will the mountain climber reach the manually.
that indicates the strength top of the cliff? Find a linear equation that
of the best fit line. Write the equation in standard form of the line that passes approximates a set of data points
5.4
Positive correlation is data that has a through the given point and has the given using a graphing calculator.
trend line with a positive slope. slope: Determine whether there is a
Negative correlation is data that has a Graph using an input output table. positive or negative or no
trend line with a negative slope. Describe the graph. correlation in a set of data.
No correlation is data that cannot be
modeled by a trend line.
The point-slope form of the equation of Use the point-slope form to write an
the non-vertical line that passes equation of a line.
5.5
through a given point with a Use the point-slope form to model a
slope of is: . real life situation.
The standard form of the equation is Write a linear equation in standard
form.
5.6
Standard form linear equations can be Use the standard form of an
useful for modeling situations involving equation to model real-life
a combination of items. situations.
The vertex of an absolute value Graph absolute value equations
6.4 Extension
equation, is the point using an input-output table.
Graph absolute value equations
using a vertex and slope.
Graph absolute value equations
using a graphing calculator.
External Resources required
21st Century Skills
Critical Thinking and Problem Solving Critical Thinking and Problem Solving Communication and Collaboration
Media Literacy Media Literacy ICT Literacy
Technology Based Activities Technology Based Activities
Learning Activities
34
Algebra: Real-Life Investigations in a Lab Setting – Leah P. McCoy,
(Reprinted from Barbara Moses, ed., Algebraic Thinking,
Grades K-12: Readings from NCTM's School-Based Journals and Other Publications (Reston, Va.: National Council of Teachers of Mathematics, 2000), pp.
202-5.
5.1 Graphing Calculator Lesson Opener(Chapter 5 Resource Books, p.12)
5.2 Activity Lesson Opener(Chapter 5 Resource Books, p.24)
5.4 Application Lesson Opener(Chapter 5 Resource Books, p.52)
5.4 Cooperative Learning Activity (Chapter 5 Resource Books, p.60)
Tiered Learning Activity 5.1 Interdisciplinary Application: Break-Even Analysis (Chapter 5 Resource Books, p.19)
5.2 Real-Life Application: Sports Participation (Chapter 5 Resource Books, p.32)
5.3 Interdisciplinary Application: Bald Eagles(Chapter 5 Resource Books, p.46)
5.5 Interdisciplinary Application: Advertising(Chapter 5 Resource Books, p.73)
5.6 Real-Life Application: Saving Money(Chapter 5 Resource Books, p.9035
BIG IDEA VII 14
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
The inequality symbol is reversed when Linear inequalities describe a range of possible Write linear inequalities.
both sides of an inequality are multiplied solutions to a situation. Graph linear inequalities in one variable.
or divided by a negative number. Solve one-step linear inequalities.
The graph of a linear inequality in one
variable is the set of points on a number
6.1
line that represent all solutions of the
Sample Conceptual Understandings
inequality. After two games of bowling, Brenda has a total
score of 475. To win the tournament, she
Solid dot represents inclusion of the
needs a total score of 684 or higher. Let
point and an empty circle represents the
represent the score she needs for her third
exclusion of the point.
game to win the tournament. Write an
means is more than . Solve multi-step linear inequalities.
inequality for . What is the lowest score she
means is less than . Use linear inequalities to model and solve real-
6.2
can get for her third game and win the
means is at least . tournament?
life problems.
means is at most . Write an inequality for the values of
A compound inequality consists of two Write, solve, and graph compound inequalities.
inequalities connected by "and" or "or". Model a real life situation with a compound
6.3
inequality.
36
Solve absolute-value equations.
Absolute value are grouping symbols. Solve absolute value inequalities.
6.4
o "Less thAND"
o "greatOR" On your basketball team, the starting players'
scoring averages are between 8 and 22 points
per game. Write an absolute value inequality
An ordered pair, is a solution of a describing the scoring averages for the players. Graph a linear inequality in two variables.
linear inequality if the inequality is true
You have $12 to spend on fruit for a meeting. Check solutions of a linear inequality.
when the values of and are
Grapes cost $1 per pound and peaches cost Model a real-life situation using a linear
6.5
substituted into the inequality. inequality in two variables.
$1.50 per pound. Let represent the number
of pounds of grapes you can buy. Write and
graph an inequality to model the amounts of
grapes and peaches you can buy 6.1 Application Lesson Opener(Chapter 6 Resource Books, p.12)
6.2 Visual Approach Lesson Opener(Chapter 6 Resource Books, p.24)
6.3 Activity Lesson Opener(Chapter 6 Resource Books, p.36)
Tiered Activity Example Example
37 6.1 Real-Life Application: Golf(Chapter 6 Resource Books, p.19)
6.2 Interdisciplinary Application: People in Flight(Chapter 6 Resource Books, p.31)
6.3 Real-Life Application: The Value and Cost of Education(Chapter 6 Resource Books, p.45)
6.4 Real-Life Application: Compact Disc (CD) Players (Chapter 6 Resource Books, p.60)
6.5 Interdisciplinary Application: Japan (Chapter 6 Resource Books, p.73 6 Project: Dinosaur Activity
Resources
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38
BIG IDEA VIII 10
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
A solution of a system of linear A solution of a system of linear equations models a Solve a system of linear equations by
equations in two variables is an ordered unique outcome for two real-life situations. graphing on a coordinate plane.
pair, , that satisfies each equation Systems of linear equations model real life situations to Solve a system of linear equations by
in the system. make predictions given certain conditions. graphing on a graphing calculator.
7.1
A solution of a linear system is the Systems of linear inequalities model all possible Check the intersection point to verify it is a
intersection point of the two lines. outcomes for two or more real-life situations. solution of the system.
Model a real-life problem using a linear
Sample Conceptual Understandings system.
When using substitution, you will get the You do 4 loads of laundry each week at a launderettte Use substitution to solve a linear system.
same solution whether you solve for where each load costs $1.25. You could buy a washing
7.2
first or first. machine that costs $400. Washing 4 loads at home will
You should begin by solving for the cost about $1 per week for electricity. How many loads
variable that is easier to isolate. of laundry must you do in order for the costs to be
Linear combination of two equations is equal? Use linear combinations to solve a system
an equation obtained by adding one of of linear equations.
7.3
You exercised on a treadmill for 1.5 hours. You ran at a
the equations (or a multiple of one of rate of 5 miles per hour, then you sprinted at a rate of 6
the equations) to the other equation.
39
Graphing is a useful method for miles per hour. If the treadmill monitor says that you Choose the best method to solve a system
approximating a solution, checking the ran and sprinted 7 miles, how long did you run at each of linear equations.
reasonableness of a solution, and speed?
providing a visual model. You have a necklace and a matching bracelet with 2
Substitution is a useful method when types of beads. There are 30 small beads and 6 large
one of the variables has a coefficient of 1 beads on the necklace. The bracelet has 10 small beads
7.4
or -1. and 2 large beads. The necklace weighs 3.6 grams and
Linear combination is a useful method the bracelet weighs 1.2 grams. If the chain has no
when none of the variables has a significant weight, can you find the weight of one large
coefficient of 1 or -1. bead? Explain.
To avoid fractional or decimal A monthly magazine is hiring reporters to cover school
coefficients, multiply the equation by a events and local events. In each magazine, the
constant first before solving. managing editor wants at least 4 reporters covering
A solution to a system of linear local news and at least 1 reporter covering school news. Identify linear systems as having one
equations is the intersection of the two The budget allows for not more than 9 different solution, no solution, or infinitely many
lines. reporters' articles to be in one magazine. Graph the solutions.
A solution to a system of linear region that shows the possible combinations of local
equations that are parallel, has no and school events covered in the magazine.
7.5
intersection, thus has no solution.
A solution to a system of linear
equations that turns out to be the same
line, has infinite intersections, thus has
infinitely many solutions.
Two or more linear inequalities form a Solving a system of linear inequalities by
system of linear inequalities. graphing using a coordinate plane.
A solution of a system of linear Solving a system of linear inequalities by
inequalities is an ordered pair that is a graphing using a graphing calculator.
solution of each inequality in the system. Use a system of linear inequalities to
The graph of a system of linear model a real-life situation.
inequalities is the graph of all solutions
that satisfy the system.
7.6
A solid line is used when the inequality is
composed of the symbols: .
A dotted line is used when the inequality
is composed of the symbols: .
40 7.4 Cooperative Learning Activity (Chapter 7 Resource Books, p.60)
7.5 Graphing Calculator Lesson Opener (Chapter 7 Resource Books, p.66 7.1 Real-Life Application: Newspaper Routes(Chapter 7 Resource Books, p.21)
Assessment Models
7.2 Interdisciplinary Application: Amphibians(Chapter 7 Resource Books, p.34)
7.3 Real-Life Application: The Juan Fernandez Islands (Chapter 7 Resource Books, p.46)
7.3 Math and History Application (Chapter 7 Resource Books, p.47)
7.4 Interdisciplinary Application: Brass Instruments(Chapter 7 Resource Books, p.61)
7.5 Interdisciplinary Application: Four Corners in Allegheny National Forest (Chapter 7 7: Alternative Assessment and Math Journal(Chapter 7 Resource Books, p.102)
Chapter 7 Project: Going Up (Chapter 7 Resource Books, p.105)
41
Resources
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42
BIG IDEA IX why it is essential to have a like base in order to use any of the exponential properties.
Describe a real-life situation that might require using exponents.
Explain why scientific notation may be particularly useful in certain occupations.
How does exponential growth and decay apply to you and your future?
Suggested Blocks for Instruction: 12
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
To multiply powers having the same Exponential functions model percentages of Use properties of exponents to multiply
base, add the exponents. change over time. exponential expressions.
To find the power of a power, multiply Exponential functions model real-life Use powers to model real-life problems.
8.1
the exponents. applications and are used to make predictions.
To find a power of product, find the Scientific notation is an efficient method of
power of each factor and multiply. representing and calculating very large and very
A non-zero number to the zero power is small numbers. Evaluate powers that have zero and
1. negative exponents.
8.2
; Sample Conceptual Understandings Graph exponential functions using an input-
Exponential function is of the form: You are offered a job that pays dollars or output table.
. dollars for hours of work. Assuming you must
Quotient of powers property states to work at least 2 hours, which method of payment Use the division properties of exponents to
divide powers having the same base, would you choose? Explain your reasoning. evaluate powers and simplify expressions.
subtract exponents. Sketch the graphs of and . Use the division properties of exponents to
8.3
Power of a quotient property states to How are the graphs related? find a probability.
find the power of the quotient, find the The racing shells (boats) used in rowing
power of the numerator and the power competition usually have 1,2,4, or 8 rowers. Top
of the denominator and divide.
43
Scientific notation is of the form speeds for racing shells in the Olympic 2000- Use scientific notation to represent
where and is an integer. meter races can be modeled by numbers.
where is the speed in Rewrite from scientific notation into decimal
kilometers per hour and is the number of form.
rowers. Use the model to estimate the ratio of Rewrite from decimal form into scientific
8.4
the speed of an 8-rower shell to the speed of a notation.
2-rower shell. Computing with scientific notation by hand.
The distance between the ninth "planet" Computing with scientific notation by hand
Pluto and the sun is of kilometers. using a calculator.
Light travels at a speed of about Use scientific notation to describe real-life
kilometers per second. How long does it take situations.
Exponential growth is when a light to travel from the Sun to Pluto. Write and use models for exponential
quantity grows by the same percent The population of 30 mice is released in a growth.
in each unit of time and is of the wildlife region. The population doubles each Graph models for exponential growth.
form: year for 4 years. What is the population after 4
years.
8.5
Exponential growth models have a Each year in the month of March, the NCAA
variable used as an exponent. Their basketball tournament is held to determine the
value will eventually change much national champion. At the start of the
more rapidly than those of linear tournament there are 64 teams, and after each
models. round, one half of the remaining teams are
eliminated.
Exponential growth is when a Write and use models for exponential decay.
o Write an exponential decay model
quantity decreases by the same Graph a model for exponential decay.
showing the number of teams left in
percent in each unit of time and is of the tournament after round .
the form: o How many teams remain after 3
rounds? 4 rounds?
A quantity that decreases by a factor
less than 1 can be modeled by an
8.6
exponential equation that
represents exponential decay.
21st Century Skills
44 8.4 Application Lesson Opener(Chapter 8 Resource Books, p.55)
8.4 Cooperative Learning Activity(Chapter 8 Resource Books, p.63)
8.5 Application Lesson Opener (Chapter 8 Resource Books, p.70)
8.6 Cooperative Learning Activity(Chapter 8 Resource Books, p.94 8.1 Real Life Application: Telephone Numbers (Chapter 8 Resource Books, p.21)
Assessment Models
8.2 Interdisciplinary Application: Carbon 14 Dating (Chapter 8 Resource Books, p.35)
8.3 Real Life Application: Internet Usage (Chapter 8 Resource Books, p.49)
8.4 Interdisciplinary Application: Sahara Desert (Chapter 8 Resource Books, p.64)
8.5 Real Life Application: Investing for College (Chapter 8 Resource Books, p.79)
8.6 Real Life Application: Record Albums (Chapter 8 Resource Books, p.95 8: Alternative Assessment and Math Journal(Chapter 8 Resource Books, p.107)
Chapter 8 Project: City Growth (Chapter 8 Resource Books, p.109)
45
Resources
Additional McDougal-Littell: Algebra 1 2004
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46
BIG IDEA X perfect squares and square roots related?
Explain how quadratic equations can be used to model real-life situations.
How are the coefficients of a quadratic equation and its graph related?
How is the quadratic formula more useful in solving quadratic equations than solving using radicals to solve?
Describe the relationship between the quadratic formula, the discriminant, and the number of solutions.
Suggested Blocks for Instruction: 24
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
If , the is the square root of . Quadratic equations are used in physics to model paths Evaluate and approximate square
All positive real numbers have two square of objects through the air (with little air resistance). roots.
roots: a positive and negative square The quadratic formula, the discriminant, and the Solve quadratic equation by finding
root. The positive square root is called the number of solutions are in an interdependent square roots.
principle square root. relationship.
The number or expression inside a radical
symbol is the radicand. Sample Conceptual Understandings
The square root of a negative number is The sales (in millions of dollars) of computer software
undefined. in the United States from 1990 to 1995 can be modeled
9.1
Numbers whose square roots are integers by , where is the number of
or quotients of integers are called perfect years since 1990. Use this model to estimate the year
squares. in which sales of computer software will be $7200
million.
47
An irrational number is a number that Find the area of the given figure.
cannot be written as the quotient of two
integers.
A radical expression involves square
roots.
means .
A quadratic equation is an equation that
can be written in the following standard Suppose a table-tennis ball is hit in such a way that its
form: path can be modeled by , where
is the leading coefficient. is the height in meters above the table and is the
When the quadratic equation is of time in seconds.
the form . o Estimate the maximum height reached by the
o If , then has two table-tennis ball.
solutions: . o About how many seconds did it take for the
o If , then has one table tennis ball to reach its maximum height
solution: after its initial bounce?
o If , then has no real o About how many seconds did it take for the
solution. table-tennis ball to travel from the initial
bounce to land on the other side of the net?
The product property states that the The number of recreational vehicles (RVs) sold in the Use properties of radicals to simplify
square root of a product equals the Unites States from 1985 to 1991 can be modeled by radicals.
product of the square roots of the , where represents the Use quadratic equations to model
factors. number of vehicles sold (in thousands) and real-life problems.
where and represents the number of years since 1985.
o Sketch a graph of the model for positive values
9.2
The quotient property states that the
of and .
square root of a quotient equals the
o Use the graph to estimate a positive root of the
quotient of the square roots of the
equation .
numerator and denominator.
o According to the model, in what year will the
where and number of RVs sold in the Unites States drop to
0?
48
You can predict the shape of the graph o Do you think the prediction is realistic? What Sketch the graph of a quadratic
of , if it opens up or factors might explain a decrease or an increase function using a coordinate plane.
down, and its general position by in the number of sales of recreational vehicles? Sketch the graph of a quadratic
examining A falcon dives toward a pigeon on the ground. When function using a graphing calculator.
The shape of a quadratic function is the falcon is at a height of 1000 feet, the pigeon sees Use quadratic models in real-life
called a parabola. the falcon, which is diving at 220 feet per second. situations.
The vertex is the highest point of a Estimate the time the pigeon has to escape.
parabola that opens up or the lowest You see a firefighter aim a fire hose from 4 feet above
9.3
point of a parabola that opens down. the ground at a window that is 26 feet above the
The vertex of the equation ground. The equation
can be found: models the path of the water when equals the height
Vertex = in feet. Estimate, to the nearest whole number, the
possible horizontal distances (in feet) between the
The line that passes through the vertex firefighter and the building.
that divides the parabola into two
symmetric parts is called the axis of
symmetry: .
The quadratic formula, , Use the quadratic formula to solve a
quadratic equation.
9.5
can be used to solve for the roots of any
quadratic equation in standard form
.
The discriminant, , is a part of
the quadratic formula that helps to
determine the number of roots or
solutions in a quadratic equation.
o If , the equation has
9.6
two solutions.
o If , the equation has
one solution.
o If , the equation has no
real solutions49
Learning Relationships
BIG IDEA XI: Activities
Curriculum Management System
Algebra 1 A/B : Grade 9
9.3 Graphing Calculator Lesson Opener (Chapter 9 Resource Books, p.37)
9.3 Graphing Calculator Activity (Chapter 9 Resource Books, p.40)
9.4 Visual Approach Lesson Opener (Chapter 9 Resource Books, p.55)
9.6 Activity Lesson Opener (Chapter 9 Resource Books, p.85 9.1 Interdisciplinary Application: Right Circular Cylinder (Chapter 9 Resource Books, p.20)
9.2 Interdisciplinary Application: Centripetal Acceleration (Chapter 9 Resource Books, p.32)
Assessment Models
9.3 Real Life Application: Ballet Recital(Chapter 9 Resource Books, p.40)
9.4 Interdisciplinary Application: Air Pollution (Chapter 9 Resource Books, p.65)
9.5 Interdisciplinary Application: Current in Electric Circuit (Chapter 9 Resource Books, p.79)
9.6 Real Life Application: Factory Sales (Chapter 9 9: Alternative Assessment and Math Journal(Chapter 9 Resource Books, p.133)
Chapter 9 Project: Light Square(Chapter 9 Resource Books, p.135)
Parachute Jump
Resources
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McDougal-Littell: Algebra 1 2004
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50
BIG IDEA XI10.2
o Degree = 0 constant long as the length of the house.
o Degree = 1 linear
51 A polynomial named by the number of
terms is as follows: o Write an expression for the area of the land surrounding
o one term is called a monomial the house.
o two terms is called a binomial o If feet, what is the area of the house? What is the
o three terms is called a trinomial area of the entire property?
FOIL is a double distributing method of An investment of dollars that gains percent of its Multiply two polynomials.
multiplication for two binomials value in one year is worth at the end of that Use polynomial multiplication in real-
: year. An investment that loses percent of its value in life situations. Multiplying vertically follows the
o If the investment gains percent the first year
traditional (multi-digit) x (multi-digit)
and loses percent the second year, what is
number pattern.
the increase or decrease in the value of the
10.2
Multiplying horizontally follows the
investment?
distributive pattern where each term in
You sell hot dogs for $1.00 each at your concession
the polynomial on the left is distributed
stand at a baseball park and have about 200 customers.
to each term in the polynomial to the
You want to increase the price of a hot dog. You
right.
estimate that you will lose three sales for every $.10
The box method is a visual method that
increase. The following equation models your hot dog
organizes the multiplication of a
sales revenue , where is the number of $.10
(polynomial) x (polynomial)
increases.
. Concession stand revenue model:
o To find your revenue from hot dog sales, you
multiply the price of each hot dog sold by the
number of hot dogs sold. In the formula above,
52
Sum and difference pattern: what does represent? What does Use special product patterns for the
10.3 represent? product of a sum and a difference, and
Square of a binomial pattern: o How many times would you have to raise the for the square of a binomial.
price by $.10 to reduce your revenue to zero?
Make a graph to help find your answer.
o Decide how high you should raise the price to
A polynomial is in factored form if it is Solve a polynomial equation in factored
make the most money. Explain how you got
written as the product of two or more form.
your answer.
linear factors. Relate factors and x-intercepts.
10.4
Consider a circle whose radius is greater than 9 and
Zero product property states that if
whose area is given by . Use
then or .
factoring to find an expression for the radius of the
The zeros of a polynomial are the circle.
x - intercepts of the graph.
Deciding whether a trinomial can be Solve by factoring, finding the square roots, or by using Factor a quadratic expression of the
factored with the use of the form: .
10.5
the quadratic formula: .
discriminant.
An object is propelled from the ground with an initial Solve quadratic equations by factoring.
A polynomial equation must be set equal upward velocity of 224 feet per second. Using the
to zero in order to solve for the zeros. vertical motion equation , will the object
A polynomial expression is a sum of reach a height of 784 feet? If it does, how long will it Factor a quadratic expression of the
10.6
terms. take the object to reach that height? Solve by factoring. form: .
A polynomial equation is an equation Using the vertical motion equation , you Solve quadratic equations by factoring.
made up of a sum of terms. toss a tennis ball from a height of 96 feet with an initial
Difference of two squares pattern velocity of 16 feet per second. How long will it take for Use special product patterns to factor
the tennis ball to reach the ground? quadrcoefficients.
10.8535455
BIG IDEA XI: Representations1
0
1
1
1
0
1
1
0
2
.
.
.o Degree = 0 constant long as the length of the house.
o Degree = 1 linearo Write an expression for the area of the land surrounding
56
A polynomial named by the number of the house.
terms is as follows: o If feet, what is the area of the house? What is the
o one term is called a monomial area of the entire property?
10.1 continued
o two terms is called a binomial An investment of dollars that gains percent of its
o three terms is called a trinomial value in one year is worth at the end of that
year. An investment that loses percent of its value ino If the investment gains percent the first year
FOIL is a double distributing method of and loses percent the second year, what is Multiply two polynomials.
multiplication for two binomials the increase or decrease in the value of the Use polynomial multiplication in real-
: investment? life situations.
You sell hot dogs for $1.00 each at your concession
stand at a baseball park and have about 200 customers.
You want to increase the price of a hot dog. You
estimate that you will lose three sales for every $.10
increase. The following equation models your hot dog
Multiplying vertically follows the sales revenue , where is the number of $.10
traditional (multi-digit) x (multi-digit) increases.
Concession stand revenue model:
number pattern.
10.2
Multiplying horizontally follows the o To find your revenue from hot dog sales, you
distributive pattern where each term in multiply the price of each hot dog sold by the
the polynomial on the left is distributed number of hot dogs sold. In the formula above,
to each term in the polynomial to the what does represent? What does
right. represent?
The box method is a visual method that o How many times would you have to raise the
organizes the multiplication of a price by $.10 to reduce your revenue to zero?
(polynomial) x (polynomial) Make a graph to help find your answer.
. o Decide how high you should raise the price to
make the most money. Explain how you got
your answer.
Consider a circle whose radius is greater than 9 and
whose area is given by . Use
57
Sum and difference pattern: factoring to find an expression for the radius of the Use special product patterns for the
10.3 circle. product of a sum and a difference, and
Square of a binomial pattern: for the square of a binomial.
Solve by factoring, finding the square roots, or by using
the quadratic formula: .
An object is propelled from the ground with an initial
A polynomial is in factored form if it is Solve a polynomial equation in factored
upward velocity of 224 feet per second. Using the
written as the product of two or more form.
vertical motion equation , will the object
linear factors. Relate factors and x-intercepts.
10.4
reach a height of 784 feet? If it does, how long will it
Zero product property states that if
take the object to reach that height? Solve by factoring.
then or .
Using the vertical motion equation , you
The zeros of a polynomial are the
toss a tennis ball from a height of 96 feet with an initial
x - intercepts of the graph.
velocity of 16 feet per second. How long will it take for
Deciding whether a trinomial can be Factor a quadratic expression of the
the tennis ball to reach the ground?
factored with the use of the form: .
10.5
discriminant. Solve quadratic equations by factoring.
A polynomial equation must be set equal
to zero in order to solve for the zeros.
A polynomial expression is a sum of Factor a quadratic expression of the
10.6
terms. form: .
A polynomial equation is an equation Solve quadratic equations by factoring.
made up of a sum of terms.
Difference of two squares pattern Use special product patterns to factor
quadr10.8
coefficients.21st Century Skills
585960
BIG IDEA XII a real life situation that has a model that varies inversely and directly.
Explain the difference between a rational expression and a fraction.
Suggested Blocks for Instruction: 14
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
Direct variation is when the Inverse and direct variation model real-life patterns to make Use direct and inverse variation.
variable and vary directly if predictions. Use direct and inverse variations to model
for a constant ; or Rational expressions are used to model real-life situations real-life situations.
and can be used to make predictions.
Inverse variation is when the
variable and vary inversely if Sample Conceptual Understandings
for a constant ; or Decide if the data in the table show direct or inverse
. variation. Write an equation that relates the variables.
is the constant of variation.
1 3 5 10 0.5
11.3
5 15 25 50 2.5
You are designing a game for a school carnival. Players
will drop a coin into a basin of water, trying to hit a target
on the bottom. The water is kept moving randomly, so
the coin is equally likely to land anywhere. You use a
rectangular basin twice as long as it is wide. You place
the blue rectangular target an equal distance from each
end.
o Express the two dimensions of the target in
61
A rational number is a number terms of the variables and . Simplify a rational expression.
that can be written as the o Write a model that gives the probability that the
quotient of two integers. coin will land on the target.
A fraction whose numerator, Simplify:
denominator, or both Find an expression for the perimeter of the rectangle:
numerator and denominator are
nonzero polynomials is a
rational expression.
11.4
A rational expression is
simplified if its numerator and
denominator have no factors in
common (other than ). After 50 times at bat, a major league baseball player has
Simplifying fractions: a batting average of 0.160. How many consecutive hits
Let be nonzero numbers. must the player get to raise his batting average to 0.250?
=
To multiply rational expressions, Multiply and divide rational expressions.
let be nonzero Use rational expressions as real-life models.
polynomials, multiply the
numerators and denominators:
To divide rational expressions,
let be nonzero
polynomials, multiply by the
11.5
reciprocal of the divisor:
62
To add with a like denominator, Add and subtract rational expressions that
add the numerators and keep have like denominators.
the denominators the same: Add and subtract rational expressions that
have unlike denominators.
To subtract with a like
denominator, subtract the
11.6
numerators and keep the
denominators the same:
The least common denominator
(LCD) that you use is the least
common multiple of the original
denominators.
A rational equation is an Solve rational equations.
equation that contains rational
expressions.
11.8
Cross multiplication can only be
used when each side of the
equation is a single fraction 11.3 Graphing Calculator Activity (Chapter 11 Resource Books, p.40)
Tiered Activity Example Big Idea XII: Tiered Example
63 oral
reports, booklets, or other formats of measurement used by the teacher.
Open-Ended Assessment:
Assessment Models
11.3 Real Life Application: Light Bulbs (Chapter 11 Resource Books, p.49)
11.4 Interdisciplinary Application: Social Studies (Chapter 11 Resource Books, p.63)
11.5 Interdisciplinary Application: Health (Chapter 11 Resource Books, p.76)
11.6 Real Life Application: Television (Chapter 11 Resource Books, p.89)
11.8 Interdisciplinary Application: Medicine and Children (Chapter 11 topics.(Synthesis,
Analysis� Chapter 11: Alternative Assessment and Math Journal (Chapter 11 Resource Books, p.127)
Resources
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McDougal-Littell: Algebra 1 2004
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64
BIG IDEA XIII: Connections and Extensions how drawing a diagram, using a table, and using a graph can help with problem solving in the real world?
Suggested Blocks for Instruction: 10
KNOW UNDERSTAND DO
Students will know that: Students will understand that: Students will be able to:
The distributive property is used Drawing a diagram, using a table, and using a graph Add, subtract, multiply, and divide radical
to simplify the sums and model real life situations to help demonstrate expressions.
differences of radical expressions reasoning and model the situation. Use radical expressions in real life situations.
when the expressions have the
12.2
same radicand.
The expressions and
are conjugates.
65
The Pythagorean Theorem states: Sample Conceptual Understandings Use the Pythagorean Theorem and its converse.
if a triangle is a right triangle, At Barton High School, 45 students are taking Use the Pythagorean Theorem and its converse in
then the sum of the squares of real-life problems.
Japanese. The number has been increasing at a rate
the lengths of the legs and of 3 students per year. The number of students
equals the square of the length of taking German is 108 and has been decreasing at a
the hypotenuse rate of 4 students per year. At these rates, when will
the number of students taking Japanese equal the
Converse of the Pythagorean number taking German? Write and solve an
Theorem states: If a triangle has equation to answer the question. Check your
side lengths and such answer with a table or graph.
that, , then the
triangle is a right triangle. Find the area:
12.5
You are surveying a triangular-shaped piece of land.
You have measured and recorded two lengths on a
plot plan. What is the length of the property along
the street?
66 12.5 Visual Approach Lesson Opener (Chapter 12 Resource Books, p.67Assessment Models
oral reports, booklets, or other formats of measurement used by the teacher.
Open-Ended Assessment:
12.2 Real Life Application: Plywood (Chapter 12 Resource Books, p.33)
12.5 Real Life Application: Kites (Chapter 1267
Algebra 1 A/B
COURSE BENCHMARKS
1. The student will be able to understand that equations are used to describe patterns, operations are used to represent verbal models and symbols can
be manipulated using different operations to model and communicate relationships.
2. The student will be able to understand that real numbers are communication tools that express important ideas with addition and subtraction of real
numbers are directly related to one another and multiplication and division of real numbers are directly related to one another.
3. The student will be able to understand that graphs are used to represent data in an organized manner to help analyze information using the mean,
median, and mode as measures of center of a data set. Percents and probability are used to analyze information and interpret significance and ratios
are used to make inferences about large population using small samples. Unit rates are factors that help to model and scale proportions to desired
quantities.
4. The student will be able to understand that equations model patterns that occur in real life problems and are used to solve for unknown quantities and
formulas are direct representations of real life applications that help to solve for an unknown quantity. Diagrams help to model problems and draw
conclusions. A graph and its equation are in an interdependent relationship.
5. The student will be able to understand that scatterplots enable analysis of patterns and the relationship between two quantities by yielding a visual
representation of data. Real life situations can be modeled using an equation. Equations can be used to describe real life situations to form predictions.
6. The student will be able to understand that inverse and direct variation model real-life patterns to make predictions. Equations describe the
relationship between a dependent and an independent variable. Best-fit line represents the relationship between two variables. Point-intercept form,
point-slope form, and standard form are interdependently related.
7. The student will be able to understand that linear inequalities describe a range of possible solutions to a situation.
8. The student will be able to understand that a solution of a system of linear equations models a unique outcome for two real-life situations. Systems of
linear equations model real life situations to make predictions given certain conditions. Systems of linear inequalities model all possible outcomes for
two or more real-life situations.
9. The student will be able to understand that exponential functions model percentages of change over time. Exponential functions model real-life
applications and are used to make predictions. Scientific notation is an efficient method of representing and calculating large numbers.
10. The student will be able to understand that quadratic equations are used in physics to model paths of objects through the air (with little air resistance).
The quadratic formula, the discriminant, and the number of solutions are in an interdependent relationship.
11. The student will be able to understand that the factors and x-intercepts of a polynomial are directly related and multiplying polynomials and factoring
polynomials are reverse processes of each other.
12. The student will be able to understand that inverse and direct variation model real-life patterns to make predictions and rational expressions are used
to model real-life situations and can be used to make predictions.
13. The student will be able to understand that drawing a diagram, using a table, and using a graph, model real life situations to help demonstrate reasoning
and model the situation.
68
BIG IDEA I: Tiered Assignment
[back to Big Idea #I]69
2. If they qualified for the group rate, how many "student" Chens were present?
3. Use the data from Exercise 1 to determine how much could have been saved if the Chens arrived
after 3:00 PM.
4. Use the data from Exercise 2 to determine how much could have been saved if the Chens arrived
after70Write a Verbal Model: ______________________________________________________________
Algebraic
Let x = number of adults and y = number of children.
Evaluate
71
2. If they qualified for the group rate, how many "student" Chens were present?
Write a Verbal Model: ______________________________________________________________
Algebraic
Let x = number of adults and y = number of children.
Evaluate
3. Use the data from Exercise 1 to determine how much could have been saved if the Chens arrived
after 3:00 PM.
Write a Verbal Model for after 3:00 PM: _________________________________________________
Algebraic for 3:00 PM
Let x = number of adults and y = number of children.
Evaluate for 3:00 PM
Calculate Savings
72
4. Use the data from Exercise 2 to determine how much could have been saved if the Chens arrived
after 3:00 PM.
Let x = number of adults and y = number of children.
Evaluate for 3:00 PM
Calculate SavingsWhat is the ratio of NT to U.S. Dollars?____________________________________________________
Convert your answer to Exercise 1 to U.S. Dollars
73 opportunityVerbal
Algebraic
+ =
Let x = number of adults and y = number of children.
Evaluate
+ =
74
2. If they qualified for the group rate, how many "student" Chens were present?
Verbal
Algebraic
+ =
Let x = number of adults and y = number of children.
Evaluate
+ =
3. Use the data from Exercise 1 to determine how much could have been saved if the Chens arrived
after 3:00 PM.
Answer from Exercise 1 1 and if they arrived after 3:00 PM.
75
4. Use the data from Exercise 2 to determine how much could have been saved if the Chens arrived
after 3:00 PM.
Answer from Exercise 2 2 and if they arrived afterWrite the exchange rate for NT to U.S. dollars as a fraction:___________________________________
Answer in NT from Exercise 1: __________________________________________________________
Use the exchange rate to convert from NT to U.S dollars:
Answer in U.S. Dollars _________________________________________________________________
76
BIG IDEA II: Tiered Assignment
[back to Big Idea #II] (When would
you have made the greatest profit from selling your share?)
77
4. Suppose you do not sell your share and watch the market for another fiveday period. The results are: loses 3
cents, gains 5 cents, gains 7 cents, loses 2 cents, and gains 9 cents. Find the net profit or loss for this five-day
period.
5. Using your answer from Exercise 2, find the value of your share after the ten-day period.
6. After the ten-day period, did you make a profit or suffer a loss? How much?
78 List the profit79 Find What was the starting value of your stock?______________________________________________
c. How much of a profit/loss did your stock take?
80 Write a number that represents the number of the profit or loss for each day.
a. Day 1:__________________
b. Day 2:__________________
c. Day 3: __________________
d. Day 4: __________________
e. Day 5: __________________
d. Find your net profit or loss for this five-day period by adding together your profits and losses over the five
days.
2. You paid $8.54 for your share. After the five-day period, how much is your share worth?
a. How much money did you pay for the share? ____________________________________________
b. Did you make or lose money in part d)? _________________________________________________
c. Add together your starting amount and your answer from part d).
81
3. As you look back over the five-day period, when would have been the best time for you to sell? List the "profit" Find How much of a profit/loss did your stock take?
i. What was the starting value of your stock? __________________________________________
ii. What is the ending value of your stock? ____________________________________________
iii. Find the difference of your starting and ending value of the stock. Is it a gain or loss? Why?
82
BIG IDEA III: Tiered Assignment
[Back to Big Idea III]2. Find the probability that a female from the 18–20 age group is not a registered voter.
3. Find the probability that a female registered voter chosen at random is 25 to 44 years old.
4. Find the probability that a female registered voter chosen at random is not 21 to 24 years old.
5. Find the odds of randomly choosing a female registered voter that is 65 years and older.
6. Find the odds of randomly choosing a female registered voter that is 25 to 64 years old.
7. Find the odds of randomly choosing a female ages 21 to 24 years old that is not a registered voter.
83a. Number of Females that are registered to vote in 18-20 age group: _____________________________________
b. Number of Females 18 to 20 years old : ____________________________________________________________
c. Number of Females that are NOT registered to vote in 18-20 age group: _________________________________
d84
4. Find the probability that a female registered voter chosen at random is not 21 to 24 years old.
5. Find the odds of randomly choosing a female registered voter that is 65 years and older.
favorable outcomes
odds
unfavorable outcomes
6. Find the odds of randomly choosing a female registered voter that is 25 to 64 years old.
7. Find the odds of randomly choosing a female ages 21 to 24 years old that is not a registered voter.
85e. Number of Females that are registered to vote in 18-20 age group: _____________________________________
f. Number of Females 18 to 20 years old : ____________________________________________________________
g. Number of Females that are NOT registered to vote in 18-20 age group: _________________________________
ha. Number of Females that are in the 25 – 44 age group: _____________________________________
b. Number of Females total: ____________________________________________________________
c. Divide a) and b) to find the probability that a female is aged 25 to 44: ______________________
86
4. Find the probability that a female registered voter chosen at random is not 21 to 24 years old.
a. Number of Females that are in the 21 – 24 age group: _____________________________________
b. Number of Females that are NOT in the 21 – 24 age group: _____________________________________
c. Number of Females total: ____________________________________________________________
d. Divide b) and c) to find the probability of a female not aged 21 to 24 years old:______________________
5. Find the odds of randomly choosing a female registered voter that is 65 years and older.
favorable outcomes
odds
unfavorable outcomes
a. Number of Females that are 65 years or older age group: _____________________________________________
b. Number of Females that are NOT in the 65 years or older age group: ____________________________________
c. The odds of randomly choosing a female registered voter that is 65 years or older: _________________________
6. Find the odds of randomly choosing a female registered voter that is 25 to 64 years old.
favorable outcomes
odds
unfavorable outcomes
a. Number of Females that are in the 25 to 64 age group that is a registered voter: _________________________
b. Number of Females that are NOT registered to vote in the 25 to 64 years age group: _______________________
c. The odds of randomly choosing a female registered voter that is 25 to 64 years age: ________________________
7. Find the odds of randomly choosing a female ages 21 to 24 years old that is NOT a registered voter.
favorable outcomes
odds
unfavorable outcomes
d. Number of Females that are in the 21 to 24 age group that are not registered to vote: _____________________
e. Number of Females that are in the 21 to 24 age group that are registered to vote: _________________________
f. The odds of randomly choosing a female ages 21 to 24 years old that is NOT a registered voter:
_____________________________________________________________________________________________
87
BIG IDEA IV: Tiered Assignment
[Back to Big Idea IV]
88
BIG IDEA V: Tiered Assignment
[Back to Big Idea V]
89
BIG IDEA VI: Tiered Assignment
[Back to Big Idea VI]
90
BIG IDEA VII: Tiered Assignment
[Back to Big Idea VII]
91
BIG IDEA VIII: Tiered Assignment
[Back to Big Idea VIII]
92
BIG IDEA IX: Tiered Assignment
[Back to Big Idea IX]
93
BIG IDEA X: Tiered Assignment
[Back to Big Idea X]
94
BIG IDEA XI: Tiered Assignment
[Back to Big Idea XI]
95
BIG IDEA XII: Tiered Assignment
[Back to Big Idea XII]966. Write an algebraic expression to find the percent of physicians who are pediatric doctors in the
United States.
7. Use your algebraic expression to find the percent of physicians who were pediatric doctors in
1999.
97 obtain1990: t =
1993: t =
1996: t =
981990: t =
1993: t =
1996: t = 1990: t =
1993: t =
1996: t =
99Number of physicians in 1999 =
Number of orthopedic surgeons in 1999=
6. Write an algebraic expression to find the percent of physicians who are pediatric doctors in the
United States.
7. Use your algebraic expression to find the percent of physicians who were pediatric doctors in
1999.
Number of physicians in 1999 = ___________________________________________________
Number of pediatric doctors in 1999=_______________________________________________
100620.9 7.9t
1990: t =___________ D
1 0.01t
620.9 7.9t
1993: t =___________ D
1 0.01t
620.9 7.9t
1996: t =___________ D
1 0.01t
10114.7 0.25t
1990: t =___________ D
1 0.01t
14.7 0.25t
1993: t =___________ D
1 0.01t
14.7 0.25t
1996: t =___________ D
1 0.01t26.9 0.48t
1990: t =___________ D
1 0.03t
26.9 0.48t
1993: t =___________ D
1 0.03t
26.9 0.48t
1996: t =___________ D
1 0.03t
102Let x represent the number of orthopedic surgeons and let y represent the number of
physicians.
5. Use your algebraic expression to find the percent of physicians who were orthopedic surgeons
in 1999.
a. Number of physicians in 1999 =_________________________________________________
620.9 7.9t
1999: t =___________ D
1 0.01t
b. Number of orthopedic surgeons in 1999=_________________________________________
14.7 0.25t
1999: t =___________ D
1 0.01t
c. Calculate the percentage using your answer from #4.
103
6. Write an algebraic expression to find the percent of physicians who are pediatric doctors in the
United States.
Number of Pediatric Doctors
Total Number of Physicians
100
Let z represent the number of pediatric doctors and let y represent the number of physicians.
7. Use your algebraic expression to find the percent of physicians who were pediatric doctors in
1999.
d. Number of physicians in 1999 =________________________________________________
620.9 7.9t
1999: t =___________ D
1 0.01t
e. Number of pediatric doctors in 1999=___________________________________________
26.9 0.48t
1999: t =___________ D
1 0.03t
f. Calculate the percentage using your answer from #6.
104
BIG IDEA XIII: Tiered Assignment
[Back to Big Idea XIII]
105
|
Vector Addition
Vectors are added by placing the tail of one vector on the
head of the preceding vector; the sum of the vectors is
obtained by joining the first tail to the last head.
The analytical method of adding vectors makes use of the
laws of cosines and sines.
A vector is completely specified by giving its magnitude or
direction, or by giving its components in a coordinate
system.
Vectors may be added by adding the respective
components.
Two vectors are equal if their respective components are
equal.
The dot product of two vectors is a scalar; it is equal to
the product of the magnitudes of the vectors and cosine the
angle between them.
Vector Addition offers a complete review of your AP course, strategies to give you the edge on test day, and plenty of practice with AP-style test questions. It includes full length practice exams modeled on the real test and all the terms and concepts you need to know.
This book includes a review of all the topics tested including vectors, kinematics, fluid mechanics, optics and nuclear physics. Additionally, the book includes two full length tests made complete with descriptive solutions, and quick study tables for Physics B formulas and equations.
|
This course is a first course in algebra. Specific topics include polynomials, factoring of polynomials, rational expressions, ratios and proportions, solving linear and quadratic equations, elementary graphing in two dimensions, elementary systems of linear equations, square roots, and various application problems. The course is comparable to a first year algebra course in local K-12 schools.
|
Introduction
These notes are intended to offer a few basic points
about studying, particularly for mathematical, statistical, and other
quantitative subjects. Much of the advice given is common sense, and will be
unsurprising. But it is important for you to think about your approach to
studying. Of course youíve studied before, and have been highly successful, but
there are some differences required in approaching university-level study.
Planning Your Study
Creating time and
space for study
You worked hard to get here and our courses are
demanding, so you have already studied hard and will continue to do so. But
donít work so hard that you donít enjoy your time at LSE. You should have
plenty of free time for leisure activities. Try to spend a few hoursí work
based around each lecture, making sure you comprehend the notes, reading the
text, working through the examples. You should also spend a few more hours each
week, for each course, on any problems assigned for classes. (This is a very
important part of the learning process.) In working through the problems, it
will often be necessary to refer to your lecture notes, so your comprehension
of the lectures will be enhanced too. The exact amounts of time you spend will
vary from week to week or from course to course, but the most important thing
is to keep a regular program of work going.
The type of environment you best work in will depend
on you. You should try to use a location that you are comfortable in (but not
so comfortable that you can doze off). Some like to study with music playing in
the background, whereas others prefer complete silence. Try to reach agreement
with your friends or family about your study environment in your accommodation.
During the day, perhaps when you have a block of time between lectures or
classes, you might like to use a quiet spot in a library, or you could make use
of the Mathematics and Statistics studentsí study room (room A504). To and from
LSE, journeys by train, tube or bus (if they are not too packed) provide an
opportunity for reading and reflection.
Planning study
across the year
It is a good idea to have a fairly regular pattern to
your work. You could, for instance, set aside the same block of time each day
for some work. To try to cram your work into short periods of time as the exams
come close is a bad approach. It will be difficult and stressful. In many
subjects, particularly mathematical and statistical subjects, cramming simply
wonít work because lectures build upon each other. It is a good idea to keep up
with the lectures by doing a little work, often.
Managing a study
session
Try to have some realistic goals for each study
period. Often, this is easy: you could set aside a slot during which you will
review a particular lecture, read a specific chapter of a book, or complete a
certain problem sheet. Having achieved your goal, you can then relax with a
clear conscience. Try to organise your studying in such a way that you feel in
control of it.†
It is probably
most effective to work for reasonably short, concentrated blocks of time, of,
say 30 to 45 minutes, in between which you take short breaks. Research has
shown that concentration can only be maintained at a high level for discrete
blocks of time like this.
When you sit down to work, get stuck in. Donít spend
half an hour arranging your differently coloured pens (useful as these are).
Give yourself small goals and rewards: ĎIíll do this one problem, then Iíll
have a cup of coffee,í for example.†
Some students find making a checklist at the start of the session
useful. Tick off items as you achieve them.
Active Learning
Becoming an active learner
Being an active learner means taking a
pro-active, interrogative approach to your study. Above all, it means making
your learning your responsibility. The role of teaching staff at university is
not just to transmit information: it is to enable you to grapple with key
concepts and ideas, and to apply these. This canít be achieved through passive
learning, where you simply try to remember what lecturers have said, agree with
it, and commit it to memory. Until you know that you understand the key
concepts, you should ask questions, do some more reading, talk to your friends,
do whatever it takes until you are sure that you are on top of it.
Active learning in a study
session
When you are studying, you should regularly review
what you have learned. Itís very easy to get on with some reading and slip into
Ďauto-pilotí, losing concentration. Stop and ask yourself just exactly what
youíve learned in the past few minutes.
Try to take some initiative with reading materials. It
is a good idea to read more than you have to, if you can. Having a wider
perspective or deeper level of knowledge than is minimally required certainly
helps with understanding the material of the course, and it keeps you
interested. There is a huge amount you could try to read on any given topic,
and you could go much deeper than the lectures, so itís important to be
realistic in what you try to achieve: donít be too hard on yourself. Consulting
texts other than the core texts can also be very useful: a slightly different
perspective on a given topic might help ideas slot into place.
As you study (for example as you review your lecture
notes) you should make additional notes, either on fresh paper, or on your
lecture notes, or your text. Highlighter pens are useful, and itís good to use
different colours for different things.
†
Actively using textbooks and
hand-outs
Mathematics and statistics textbooks require a special
form of interaction with the reader: active rather than passive. Itís too easy
simply to agree with a mathematics or statistics book, without actually
understanding it, or without being able to apply the concept you have been
reading about. You should work through any calculations in the text by
yourself. And you should certainly attempt some of the textbook problems that
relate to your recent lectures and classes.†
When doing this, though, avoid the temptation of referring to the
answers at the back of the book (if there are any): wait until you think you
have solved the problem before doing this.
If you have hand-outs in advance of lectures, itís
worth skimming them to get some idea of whatís coming up.
Formal Teaching
Lectures
There are many different styles of lectures. Some will
consist of a lecturer writing or speaking continuously. Taking notes in such
lectures is a valuable skill, and one which requires practice. Try to note down
the main points, and certainly definitions, theorems, proofs, examples, and
references to any relevant reading in the textbooks. There is no need to write
down everything the lecturer says. It can sometimes be difficult in such a
lecture to concentrate enough to understand the material: note taking may be
taking up too much of your concentration. But donít panic if this is the case.
You can work on the details after the lectures, and follow it up with questions
to fellow students, class teachers and the lecturer. (For specific information
on note taking, see the book by Northedge cited at the end of these notes.)
†
Some lecturers will rely heavily on lecture notes they
have handed out, or on textbooks. Here, too, note taking will be important, but
will probably be easier. Hand-outs in such lectures are likely to take the form
of summary notes, outlining the main ideas. You should aim to augment these
hand-outs with any additional useful points the lecturer† makes, and also with important details
(examples or proofs, for instance) omitted from the hand-outs.†
The most important thing to realise about university
lectures, which you have probably already noticed, is that the rate of delivery
of new ideas is much faster than in high school or sixth-form college. A lot of
work is required by you, outside of lectures, to make sure you understand the
material.
Classes
In quantitative subjects, classes are immensely
important. Subjects like mathematics and statistics are only really mastered by
working through lots of problems. Classes are most useful if you have attempted
the assigned work: even if you canít complete the problems, you should at least
try them, and locate exactly where it is you have difficulty.
(The most common problem might be that you simply
didnít know where to begin, and weíll discuss this further later.) Class
teachers want to know what problems youíve been having. They donít particularly
want to know the answersóthey already know them! Donít be afraid to learn from
your mistakes, and donít worry too much about the class grades: these are just
for information, for you and the teacher. They do not contribute to final
assessment. Thereís no point in handing in to your class teacher a perfect set
of answers that you obtained from someone else. Unless youíve grappled with the
problems yourself, you wonít have learned anything. Even when you see a
solution presented in class, it wonít have much value if you havenít thought
about the problem for yourself.
You should ask questions in classes. You may be too
daunted by the size of lecture groups to do so in lectures, but classes are the
right sort of size for raising questions and having discussion.
Problem Solving
Problem solving is a mixture of frustration and
satisfaction. It is the most important skill you develop in studying
mathematics and statistics, and will stay with you for the rest of your life,
even if specific techniques fade from your memory.
Types of problem
Different types of problem are asked in quantitative
subjects. Here are a few of the most common types.
∑routine, Ďdrillí problems, involving straightforward (or not so
straightforward) application of a technique. These may be technically
difficult, but at least you know how to approach them.
∑modelling (or applications) problems, requiring a translation of a descriptive problem
(such as a statistical one) into mathematical language before solving using
standard techniques. The translation can be very hard.
∑proof problems, requiring the use of formal definitions and proof techniques. It is
not always clear how to approach these, and students often comment that they
simply donít know where to start.
Approaching problems
There is no recipe for successful problem solving, but
Polyaís general four-point approach is useful[1]:
1.Understand
the problem
2.Devise
a plan
3.Implement
the plan
4.Look
back/check
The first step is hugely important and often not taken
seriously enough. You need to know exactly what it is you need to establish.
This is hopeless if you donít know what key concepts mean, exactly. That is,
definitions are extremely important. This is particularly so for proof
problems. Suppose you were asked to show that a given sequence of numbers has a
limit. Unless you know exactly what is meant by a limit, thereís no way you can
even begin to solve this problem.
Once you know what you need to show, do not be afraid
to try something (steps 2 and 3 of Polyaís scheme).† If that fails, then thatís OK: try something else. Or, try to
solve a special case or a simpler version of the problem, in the hope that you
can then get a feel for the problem and generalise to the required level. These
are the ways in which real mathematics and statistics is often done.
The checking part of Polyaís approach is sometimes
easy to do. For example, if you were asked to solve a system of linear equations,
then to carry out the check you could simply put the supposed answers into each
equation, verifying that each equation holds. Sometimes checking isnít so
simple, but you should always look back over your solution to make sure that it
at least makes sense to you on a second look.
You should spend a lot of time on problems. Donít
worry about taking more time than you would have in an exam. Problem solving,
and the speed of problem solving, improves with experience, so by the time the
exam has approached, you will hopefully be proficient enough in solving
problems to cope well.
Getting Help
Help from your teachers
Your teachers are there to help you. Remember to try
to be an active learner. You should make full use of classes for asking questions.
Your class teacher or lecturer will be available to see you during his or her
office hour, and these opportunities are worth taking up.† Staff are happy to help, but it will lead to
a better discussion during an office hour if you can focus in on what exactly
is causing you problems. To approach a teacher and say
ĎI havenít understood the last
three lectures. Can you explain them to me?í
is pretty pointless. In an office hour, he or she is
not going to be able to re-iterate the past three lectures in a way you will be
able to understand (if you didnít already understand them at the regular
speed). So you would need to be more specific, as in, for example,
ĎOK, so I know how to
differentiate, in that I can use the product rule and so on, but I have
difficulty when I try to understand what the derivative actually means. For
example, in question 5 of exercise sheet 3, which is all about differentiation,
I have to work out by how much a function changes if I increase the variable a
little bit. Whatís the connection with the derivative? I donít get it.í
This is a great question, which the teacher can
happily deal with. Itís also one which he or she will be able to know they have
satisfactorily answered for you, and that makes them happy too!
†
Help from your friends
Your fellow students in a course might be having
difficulties too, but these may be different difficulties. Together you might
be able to overcome the various difficulties by working in a small group. You
can discuss key ideas and concepts, and work through problems together. This is
often useful, provided everybody contributes, but bear in mind that in the end,
you will be on your own in the exam.
Exams
Exam revision
Exam revision should be revision: you should not find
yourself learning things for the first time! (See the discouraging words about
Ďcrammingí earlier.)
It is a good idea to plan your revision, and to stick
at least roughly to a timetable. Make full use of the vacations, not just
Easter, but Christmas too.
When revising your courses, it is a good idea to
summarise your lecture notes, and work through problems again. Focus in on
areas you are having difficulty with, and talk to fellow students and academic
staff about these. A particularly useful resource will be the recent exam
papers, and it is perhaps a good idea to leave them until exam revision rather
than to attempt them earlier in the session.
Some exams (such as those in pure mathematics) require
you to reproduce some Ďbookworkí, meaning the statements of definitions from
the lectures, and the proofs of key results. Proofs are enormously difficult to
memorise. The only way successfully to be able to handle bookwork is to
understand the definitions and proofs, and not simply to memorise them.
(Precise notations used in proofs are not very important. The key thing to be
remembered is the approach used.) You will, in any case, need to understand the
key ideas and concepts in order to handle the bulk of the exam questions, which
will be testing that you can solve unseen problems.
Exam technique
First of all, know when and where the exam is: donít
rely on friends for this information!
The most useful piece of advice for exams is: donít
panic. Try to relax and take control of your exam. The exam you sit will quite
probably contain problems that are unlike others you have seen before, though
there will usually also be some less surprising questions. This is part of the
nature of exams, and you should not let yourself panic about it.
†
When you start the exam, make sure you understand the
rubric. Then, itís a good idea to skim the whole paper quickly just to size it
up. You should answer questions in the order you want to. Why not bag a few
easy questions for starters, to build up your confidence?
Donít get bogged down with small bits of questions
that might be worth only a few marks: you should move on and return to these
later if you have time.
Remember that in many cases what examiners are testing
is that you know how to solve a problem: that is, that you know what technique
to use, and how it works. Some credit will be awarded for correct approaches,
even if you mess up the subsequent calculations. If you donít have time to
finish a problem, but have enough time to explain how you would have finished
it, then it might be worth doing so: some credit may be given for this.
Further Information
Much of what I have written here draws on my personal
experience, from
observing my fellow students as an undergraduate, and from my experience as a
lecturer and personal tutor. There are people at LSE who know much more than I
do about study skills generally. Dr Liz Barnett, the Teaching and Learning
Development Officer, has organised a series of lectures and practical workshops
on study skills. The slides for these sessions are available on the Outlook
Public Folders, and you should check the web-site
This has good discussions of time-management, note
taking, and revision. It also covers essay writing extensively, which is of
less relevance to mathematics and statistics but may be useful for some of your
other courses. A version of this book has been written for science students,
and may be more appropriate:
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It is recommended that students taking this course have successfully
complete Math 2255 or Math 3312.At
a minimum, students should have an avid interest in mathematics, its
foundation, and logic.
This is a rigorous development of plane geometry starting with Book I of The
Elements of Euclid and continuing through the subject as it evolved during the
19th century.Depending
on the interest of the instructor and the students, topics may include a deeper
study of projective planes, non-Euclidean geometries, early developments in
algebraic geometry, or the use of modern algebra and/or analysis in geometry.
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COURSE DESCRIPTION
Finally make sense of two fundamental branches of mathematics with this accessible and thorough two-course set delivered by ideal experts and professors.
First, imContinue learning with Algebra I, which is designed to meet the concerns of both students and their parents. These 36 accessible lectures make the concepts of first-year algebra—including variables, order of operations, and functions—easy to understand
1
of
2:
Algebra I Professor
Professor James A. Sellers,
The Pennsylvania State University Ph.D., The Pennsylvania State University Algebra I is an entirely new course designed to meet the concerns of both students and their parents. These 36 accessible lectures make the concepts of first-year algebra—including variables, order of operations, and functions—easy to grasp Lecture Titles
36
Lectures
30
minutes/lecture
1.
An Introduction to the Course
Professor Sellers introduces the general topics and themes for the course, describing his approach and recommending a strategy for making the best use of the lessons and supplementary workbook. Warm up with some simple problems that demonstrate signed numbers and operations.
19.
Factoring Trinomials
Begin to find solutions for quadratic equations, starting with the FOIL technique in reverse to find the binomial factors of a quadratic trinomial (a binomial expression consists of two terms, a trinomial of three). Professor Sellers explains the tricks of factoring such expressions, which is a process almost like solving a mystery.
2.
Order of Operations
The order in which you do simple operations of arithmetic can make a big difference. Learn how to solve problems that combine adding, subtracting, multiplying, and dividing, as well as raising numbers to various powers. These same concepts also apply when you need to simplify algebraic expressions, making it critical to master them now.
20.
Quadratic Equations—Factoring
In some circumstances, quadratic expressions are given in a special form that allows them to be factored quickly. Focus on two such forms: perfect square trinomials and differences of two squares. Learning to recognize these cases makes factoring easy.
3.
Percents, Decimals, and Fractions
Continue your study of math fundamentals by exploring various procedures for converting between percents, decimals, and fractions. Professor Sellers notes that it helps to see these procedures as ways of presenting the same information in different forms.
21.
Quadratic Equations—The Quadratic Formula
For those cases that defy simple factoring, the quadratic formula provides a powerful technique for solving quadratic equations. Discover that this formidable-looking expression is not as difficult as it appears and is well worth committing to memory. Also learn how to determine if a quadratic equation has no solutions.
4.
Variables and Algebraic Expressions
Advance to the next level of problem solving by using variables as the building blocks to create algebraic expressions, which are combinations of mathematical symbols that might include numbers, variables, and operation symbols. Also learn some tricks for translating the language of problems (phrases in English) into the language of math (algebraic expressions).
22.
Quadratic Equations—Completing the Square
After learning the definition of a function, investigate an additional approach to solving quadratic equations: completing the square. This technique is very useful when rewriting the equation of a quadratic function in such a way that the graph of the function is easily sketched.
5.
Operations and Expressions
Discover that by following basic rules on how to treat coefficients and exponents, you can reduce very complicated algebraic expressions to much simpler ones. You start by using the commutative property of multiplication to rearrange the terms of an expression, making combining them relatively easy.
23.
Representations of Quadratic Functions
Drawing on your experience solving quadratic functions, analyze the parabolic shapes produced by such functions when represented on a graph. Use your algebraic skills to determine the parabola's vertex, its x and y intercepts, and whether it opens in an upward "cup" or downward in a "cap."
6.
Principles of Graphing in 2 Dimensions
Using graph paper and pencil, begin your exploration of the coordinate plane, also known as the Cartesian plane. Learn how to plot points in the four quadrants of the plane, how to choose a scale for labeling the x and y axes, and how to graph a linear equation.
24.
Quadratic Equations in the Real World
Quadratic functions often arise in real-world settings. Explore a number of problems, including calculating the maximum height of a rocket and determining how long an object dropped from a tree takes to reach the ground. Learn that in finding a solution, graphing can often help.
7.
Solving Linear Equations, Part 1
In this lesson, work through simple one- and two-step linear equations, learning how to isolate the variable by different operations. Professor Sellers also presents a word problem involving a two-step equation and gives tips for how to solve it.
25.
The Pythagorean Theorem
Because it involves terms raised to the second power, the famous Pythagorean theorem, a2 + b2 = c2, is actually a quadratic equation. Discover how techniques you have previously learned for analyzing quadratic functions can be used for solving problems involving right triangles.
8.
Solving Linear Equations, Part 2
Investigating more complicated examples of linear equations, learn that linear equations fall into three categories. First, the equation might have exactly one solution. Second, it might have no solutions at all. Third, it might be an identity, which means every number is a solution.
26.
Polynomials of Higher Degree
Most of the expressions you've studied in the course so far have been polynomials. Learn what characterizes a polynomial and how to recognize polynomials in both algebraic functions and in graphical form. Professor Sellers defines several terms, including the degree of an equation, the leading coefficient, and the domain.
9.
Slope of a Line
Explore the concept of slope, which for a given straight line is its rate of change, defined as the rise over run. Learn the formula for calculating slope with coordinates only, and what it means to have a positive, negative, and undefined slope.
27.
Operations and Polynomials
Much of what you've learned about linear and quadratic expressions applies to adding, subtracting, multiplying, and dividing polynomials. Discover how the FOIL operation can be extended to multiplying large polynomials, and a version of long division works for dividing one polynomial by another.
10.
Graphing Linear Equations, Part 1
Use what you've learned about slope to graph linear equations in the slope-intercept form, y = mx + b, where m is the slope, and b is the y intercept. Experiment with examples in which you calculate the equation from a graph and from a table of pairs of points.
28.
Rational Expressions, Part 1
When one polynomial is divided by another, the result is called a rational function because it is the ratio of two polynomials. These functions play an important role in algebra. Learn how to add and subtract rational functions by first finding their common divisor.
11.
Graphing Linear Equations, Part 2
A more versatile approach to writing the equation of a line is the point-slope form, in which only two points are required, and neither needs to intercept the y axis. Work through several examples and become comfortable determining the equation using the line and the line using the equation
29.
Rational Expressions, Part 2
Continuing your exploration of rational expressions, try your hand at multiplying and dividing them. The key to solving these complicated-looking equations is to proceed one step at a time. Close the lesson with a problem that brings together all you've learned about rational functions.
12.
Parallel and Perpendicular Lines
Apply what you've discovered about equations of lines to two very special types of lines: parallel and perpendicular. Learn how to tell if lines are parallel or perpendicular from their equations alone, without having to see the lines themselves. Also try your hand at word problems that feature both types of lines.
30.
Graphing Rational Functions, Part 1
Examine the distinctive graphs formed by rational functions, which may form vertical or horizontal curves that aren't even connected on a graph. Learn to identify the intercepts and the vertical and horizontal asymptotes of these fascinating curves.
13.
Solving Word Problems with Linear Equations
Linear equations reflect the behavior of real-life phenomena. Practice evaluating tables of numbers to determine if they can be represented as linear equations. Conclude with an example about the yearly growth of a tree. Does it increase in size at a linear rate?
31.
Graphing Rational Functions, Part 2
Sketch the graphs of several rational functions by first calculating the vertical and horizontal asymptotes, the x and y intercepts, and then plotting several points in the function. In the final exercise, you must simplify the expression in order to extract the needed information.
14.
Linear Equations for Real-World Data
Investigating more real-world applications of linear equations, derive the formula for converting degrees Celsius to Fahrenheit; determine the boiling point of water in Denver, Colorado; and calculate the speed of a rising balloon and the time for an elevator to descend to the ground floor.
32.
Radical Expressions
Anytime you see a root symbol—for example, the symbol for a square root—then you're dealing with what mathematicians call a radical. Learn how to simplify radical expressions and perform operations on them, such as multiplication, division, addition, and subtraction, as well as combinations of these operations.
15.
Systems of Linear Equations, Part 1
When two lines intersect, they form a system of linear equations. Discover two methods for finding a solution to such a system: by graphing and by substitution. Then try out a real-world example, involving a farmer who wants to plant different crops in different proportions.
33.
Solving Radical Equations
Discover how to solve equations that contain radical expressions. A key step is isolating the radical term and then squaring both sides. As always, it's important to check the solution by plugging it into the equation to see if it makes sense. This is especially true with radical equations, which can sometimes yield extraneous, or invalid, solutions.
16.
Systems of Linear Equations, Part 2
Expand your tools for solving systems of linear equations by exploring the method of solving by elimination. This technique allows you to eliminate one variable by performing addition, subtraction, or multiplication on both sides of an equation, allowing a straightforward solution for the remaining variable.
34.
Graphing Radical Functions
In previous lessons, you moved from linear, quadratic, and rational functions to the graphs that display them. Now do the same with radical functions. For these, it's important to pay attention to the domain of the functions to ensure that negative values are not introduced beneath the root symbol.
17.
Linear Inequalities
Shift gears to consider linear inequalities, which are mathematical expressions featuring a less than sign or a greater than sign instead of an equal sign. Discover that these kinds of problems have some very interesting twists, and they come up frequently in business applications.
35.
Sequences and Pattern Recognition, Part 1
Pattern recognition is an important and fascinating mathematical skill. Investigate two types of number patterns: geometric sequences and arithmetic sequences. Learn how to analyze such patterns and work out a formula that predicts any term in the sequence
18.
An Introduction to Quadratic Polynomials
Transition to a more complex type of algebraic expression, which incorporates squared terms and is therefore known as quadratic. Learn how to use the FOIL method (first, outer, inner, last) to multiply linear terms to get a quadratic expression.
36.
Sequences and Pattern Recognition, Part 2
Conclude the course by examining more types of number sequences, discovering how rich and enjoyable the mathematics of pattern recognition can be. As in previous lessons, employ your reasoning skills and growing command of algebra to find order—and beauty—where once all was a confusion of numbers.
Course Lecture Titles
36
Lectures
30
minutes/lecture
1.
A Preview of Calculus
Calculus is the mathematics of change, a field with many important applications in science, engineering, medicine, business, and other disciplines. Begin by surveying the goals of the course. Then get your feet wet by investigating the classic tangent line problem, which illustrates the concept of limits.
19.
The Area Problem and the Definite Integral
One of the classic problems of integral calculus is finding areas bounded by curves. This was solved for simple curves by the ancient Greeks. See how a more powerful method was later developed that produces a number called the definite integral, and learn the relevant notation.
2.
Review—Graphs, Models, and Functions
In the first of two review lectures on precalculus, examine graphs of equations and properties such as symmetry and intercepts. Also explore the use of equations to model real life and begin your study of functions, which Professor Edwards calls the most important concept in mathematics.
20.
The Fundamental Theorem of Calculus, Part 1
The two essential ideas of this course—derivatives and integrals—are connected by the fundamental theorem of calculus, one of the most important theorems in mathematics. Get an intuitive grasp of this deep relationship by working several problems and surveying a proof.
3.
Review—Functions and Trigonometry
Continue your review of precalculus by looking at different types of functions and how they can be identified by their distinctive shapes when graphed. Then review trigonometric functions, using both the right triangle definition as well as the unit circle definition, which measures angles in radians rather than degrees.
21.
The Fundamental Theorem of Calculus, Part 2
Try examples using the second fundamental theorem of calculus, which allows you to let the upper limit of integration be a variable. In the process, explore more relationships between differentiation and integration, and discover how they are almost inverses of each other.
4.
Finding Limits
Jump into real calculus by going deeper into the concept of limits introduced in Lecture 1. Learn the informal, working definition of limits and how to determine a limit in three different ways: numerically, graphically, and analytically. Also discover how to recognize when a given function does not have a limit.
22.
Integration by Substitution
Investigate a straightforward technique for finding antiderivatives, called integration by substitution. Based on the chain rule, it enables you to convert a difficult problem into one that's easier to solve by using the variable u to represent a more complicated expression.
5.
An Introduction to Continuity
Broadly speaking, a function is continuous if there is no interruption in the curve when its graph is drawn. Explore the three conditions that must be met for continuity—along with applications of associated ideas, such as the greatest integer function and the intermediate value theorem.
23.
Numerical Integration
When calculating a definite integral, the first step of finding the antiderivative can be difficult or even impossible. Learn the trapezoid rule, one of several techniques that yield a close approximation to the definite integral. Then do a problem involving a plot of land bounded by a river.
6.
Infinite Limits and Limits at Infinity
Infinite limits describe the behavior of functions that increase or decrease without bound, in which the asymptote is the specific value that the function approaches without ever reaching it. Learn how to analyze these functions, and try some examples from relativity theory and biology.
24.
Natural Logarithmic Function—Differentiation
Review the properties of logarithms in base 10. Then see how the so-called natural base for logarithms, e, has important uses in calculus and is one of the most significant numbers in mathematics. Learn how such natural logarithms help to simplify derivative calculations.
7.
The Derivative and the Tangent Line Problem
Building on what you have learned about limits and continuity, investigate derivatives, which are the foundation of differential calculus. Develop the formula for defining a derivative, and survey the history of the concept and its different forms of notation.
25.
Natural Logarithmic Function—Integration
Continue your investigation of logarithms by looking at some of the consequences of the integral formula developed in the previous lecture. Next, change gears and review inverse functions at the precalculus level, preparing the way for a deeper exploration of the subject in coming lectures.
8.
Basic Differentiation Rules
Practice several techniques that make finding derivatives relatively easy: the power rule, the constant multiple rule, sum and difference rules, plus a shortcut to use when sine and cosine functions are involved. Then see how derivatives are the key to determining the rate of change in problems involving objects in motion.
26.
Exponential Function
The inverse of the natural logarithmic function is the exponential function, perhaps the most important function in all of calculus. Discover that this function has an amazing property: It is its own derivative! Also see the connection between the exponential function and the bell-shaped curve in probability.
9.
Product and Quotient Rules
Learn the formulas for finding derivatives of products and quotients of functions. Then use the quotient rule to derive formulas for the trigonometric functions not covered in the previous lecture. Also investigate higher-order derivatives, differential equations, and horizontal tangents.
27.
Bases other than e
Extend the use of the logarithmic and exponential functions to bases other than e, exploiting this approach to solve a problem in radioactive decay. Also learn to find the derivatives of such functions, and see how e emerges in other mathematical contexts, including the formula for continuous compound interest.
10.
The Chain Rule
Discover one of the most useful of the differentiation rules, the chain rule, which allows you to find the derivative of a composite of two functions. Explore different examples of this technique, including a problem from physics that involves the motion of a pendulum.
28.
Inverse Trigonometric Functions
Turn to the last set of functions you will need in your study of calculus, inverse trigonometric functions. Practice using some of the formulas for differentiating these functions. Then do an entertaining problem involving how fast the rotating light on a police car sweeps across a wall and whether you can evade it.
11.
Implicit Differentiation and Related Rates
Conquer the final strategy for finding derivatives: implicit differentiation, used when it's difficult to solve a function for y. Apply this rule to problems in related rates—for example, the rate at which a camera must move to track the space shuttle at a specified time after launch.
29.
Area of a Region between 2 Curves
Revisit the area problem and discover how to find the area of a region bounded by two curves. First imagine that the region is divided into representative rectangles. Then add up an infinite number of these rectangles, which corresponds to a definite integral.
12.
Extrema on an Interval
Having covered the rules for finding derivatives, embark on the first of five lectures dealing with applications of these techniques. Derivatives can be used to find the absolute maximum and minimum values of functions, known as extrema, a vital tool for analyzing many real-life situations.
30.
Volume—The Disk Method
Learn how to calculate the volume of a solid of revolution—an object that is symmetrical around its axis of rotation. As in the area problem in the previous lecture, you imagine adding up an infinite number of slices—in this case, of disks rather than rectangles—which yields a definite integral.
13.
Increasing and Decreasing Functions
Use the first derivative to determine where graphs are increasing or decreasing. Next, investigate Rolle's theorem and the mean value theorem, one of whose consequences is that during a car trip, your actual speed must correspond to your average speed during at least one point of your journey.
31.
Volume—The Shell Method
Apply the shell method for measuring volumes, comparing it with the disk method on the same shape. Then find the volume of a doughnut-shaped object called a torus, along with the volume for a figure called Gabriel's Horn, which is infinitely long but has finite volume.
14.
Concavity and Points of Inflection
What does the second derivative reveal about a graph? It describes how the curve bends—whether it is concave upward or downward. You determine concavity much as you found the intervals where a graph was increasing or decreasing, except this time you use the second derivative.
32.
Applications—Arc Length and Surface Area
Investigate two applications of calculus that are at the heart of engineering: measuring arc length and surface area. One of your problems is to determine the length of a cable hung between two towers, a shape known as a catenary. Then examine a peculiar paradox of Gabriel's Horn.
15.
Curve Sketching and Linear Approximations
By using calculus, you can be certain that you have discovered all the properties of the graph of a function. After learning how this is done, focus on the tangent line to a graph, which is a convenient approximation for values of the function that lie close to the point of tangency.
33.
Basic Integration Rules
Review integration formulas studied so far, and see how to apply them in various examples. Then explore cases in which a calculator gives different answers from the ones obtained by hand calculation, learning why this occurs. Finally, Professor Edwards gives advice on how to succeed in introductory calculus.
16.
Applications—Optimization Problems, Part 1
Attack real-life problems in optimization, which requires finding the relative extrema of different functions by differentiation. Calculate the optimum size for a box, and the largest area that can be enclosed by a circle and a square made from a given length of wire.
34.
Other Techniques of Integration
Closing your study of integration techniques, explore a powerful method for finding antiderivatives: integration by parts, which is based on the product rule for derivatives. Use this technique to calculate area and volume. Then focus on integrals involving products of trigonometric functions.
17.
Applications—Optimization Problems, Part 2
Conclude your investigation of differential calculus with additional problems in optimization. For success with such word problems, Professor Edwards stresses the importance of first framing the problem with precalculus, reducing the equation to one independent variable, and then using calculus to find and verify the answer.
35.
Differential Equations and Slope Fields
Explore slope fields as a method for getting a picture of possible solutions to a differential equation without having to solve it, examining several problems of the type that appear on the Advanced Placement exam. Also look at a solution technique for differential equations called separation of variables.
18.
Antiderivatives and Basic Integration Rules
Up until now, you've calculated a derivative based on a given function. Discover how to reverse the procedure and determine the function based on the derivative. This approach is known as obtaining the antiderivative, or integration. Also learn the notation for integration.
36.
Applications of Differential Equations
Use your calculus skills in three applications of differential equations: first, calculate the radioactive decay of a quantity of plutonium; second, determine the initial population of a colony of fruit flies; and third, solve one of Professor Edwards's favorite problems by using Newton's law of cooling to predict the cooling time for a cup of coffee.
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Project maths is bloody great, it's so much easier than having to learn off methods of doing stuff, you actually understand all the probability stuff. The problem here is that everyone resists any sort of change.
My biggest problem is that anyone who does project Maths now, and wants to do Maths in college hasn't a hope. Any other problem is fairly irrelevant in comparison.
What I do like is the whole idea of understanding, and the fact that, because it's so much easier than before (from what I can see so far), I actually have a chance of getting a C or D in Maths now.
Each and every one of the twenty-five extra points are also quite beautiful.
In terms of getting points it's fantastic, but if you want to be able to do Maths after school it's absolutely hopeless, but I think the old course was the same because students were rarely thought to understand the topics.
Pretty sure project maths has the exact same calculus in it as the old maths course..
Then the course has been taught wrong for the past few years. It's just worse now because project Maths seems to be easier. Either way Maths lecturers haven't too much praise for the Leaving Cert, to say the least.Not a fan of project maths at all. I'm liking the idea of 25 extra points but starting in fifth year is just a joke! Imo its not proper maths now with all the explaining and stuff. I personally don't think this was properly thought through, the book we are using wasn't even out until the middle of October! My teacher hardly understands it and is convinced that we will be moving back to the old course very soon. Anyone else confused by the little boxes we have to write in?I have always found that the best engineers, programmers and scientists are generally not the ones who got A1's in exams but the ones who can interface between the non-techies, understand what they want, and implement it.
In the real world, you will need to be able to talk to people who waffle on forever about what they want but can't describe it in any meaningful technical sense. So Project Maths sounds to me more like the real world than the fantasy academic world.
I did a Maths degree with pass Maths for the LC, as did a few of my college buddies - and outperformed a lot of my honours Maths classmates. So there must be something wrong with the old LC.
Edit: On second thoughts the major problem with LC Maths - at least in my day - were:
1. Bad teachers
2. Bad textbooks with examples for the easiest 1 or 2 problems and then 40 much harder problems with new concepts thrown in and not explained.
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Metadata
Name:
Solving Linear Equations and Inequalities: Further Techniques in Equation Solving
ID:
m21992
Language:
English
(en)
Summary:
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules ((Reference)) and ((Reference))).
Objectives of this module: be comfortable with combining techniques in equation solving, be able to recognize identities and contradictions.
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Authors
With the emphasis the Common Core State Standards (CCSS) places on modeling, Modeling With Mathematics: A Bridge to Algebra II (Bridge 2e) addresses these modeling requirements while helping prepare students for success in Algebra II. Intended for students who have taken Algebra I and Geometry but who are not yet ready for Algebra II, this program helps solidify their understanding by providing a different kind of learning experience. With Bridge 2e students model real-world applications with a functions approach netting a deeper grasp of the important concepts necessary for success in Algebra II and on the forthcoming Common Core assessments.
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Description The purpose of this course is the build a solid mathematical framework through solidifying your understanding of algebra. While there are many mechanical processes that need to be mastered, we will also try to learn how to utilize your knowledge of those processes in contextual situations. The brain is like any other muscle and must be stretched and exercised and developed through many different techniques!
Expectations My only expectation is very simple: commit yourself to being a successful student. Prepare for each class by completing homework, seeking help for concepts you do not understand, and bringing needed materials. Take advantage of your time in class to learn from me and your classmates and work hard from bell to bell. Do your part to help build a respectful and relaxed learning atmosphere and have fun while learning. The last expectation is that you must laugh each day in class. I want this class to be a great learning experience both academically and socially, and one of the best ways to develop class unity is to laugh together. Don't worry; I will supply most of the humor both intentionally and unintentionally.
What's Happening
Welcome back to another year at West....We are implementing some new and great ideas and you will help us shape many of these practices. We are excited with the direction of our Math department and hope that it will lead each of you to great success.
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MATH 100: Intermediate Algebra. 3 hours. Prerequisite: Prealgebra or beginning algebra in high school or college. The traditional topics of intermediate algebra through quadratic equations and functions.
MATH 101: Fundamental Mathematical Concepts I. 3 hours. Prerequisite: One year of high school algebra or MATH 100. Development of the number systems — whole numbers through real numbers. Problem solving strategies, functions, elementary logic, and set theory are included.
MATH 109: College Algebra. 3 hours. Prerequisite: MATH 100 or one year of high school algebra and one year of high school geometry. A study of functions and graphs, solutions of equations and inequalities and the properties of polynomial, rational, exponential, and logarithmic functions.
MATH 110: Trigonometry. 3 hours. Prerequisite: MATH 109 or two years of high school algebra and one year of high school geometry. The study of trigonometric, logarithmic, and exponential functions and their applications.
MATH 141. Applied Logic. 1 hour. This course is designed to help students learn to apply the tools of logic to concrete situations, such as those posed on LSAT and GMAT tests. The course will include a discussion of propositional logic, propositional equivalences, rules of inference and common fallacies. Students are strongly encouraged to take PHIL 100: Introduction to Logic and Critical Thinking either prior to or concurrently with this course.
MATH 203: Mathematics and Inquiry. 3 hours. In this course, students will develop quantitative and abstract reasoning abilities necessary to solve complex problems. Literacy in mathematics is developed, with concepts and skills from such areas as algebra, trigonometry, calculus, probability, statistics and computer science. This course will address critical thinking and problem-solving skills, not simply numerical manipulations related to a single subdiscipline of mathematics. Emphasis will be placed on defining and setting up problems; understanding the steps required to solve various types of problems; understanding the factual information and quantitative abilities required for problem solving; and understanding how necessary information can be obtained from text material, resource individuals and computer resources.
MATH 227: Introduction to Statistics. 3 hours. Prerequisite: One year of high school algebra. A course to acquaint the student with the basic ideas and language of statistics including such topics as: descriptive statistics, correlation and regression, basic experimental design, elementary probability, binomial and normal distributions, estimation and test of hypotheses, and analysis of variance.
MATH 228: Applied Statistics in Exercise and Sport Science. 3 hours. Prerequisite: One year of high school algebra. The course contains the mathematical basis for statistics including descriptive measures, probability and hypothesis testing. Some applications in exercise science will include tests, ANOVA, correlation and regression. Same as EXSP 228. Credit will not be given for both MATH 227 and MATH/EXSP 228.
MATH 230: Business Calculus. 3 hours. Prerequisite: Two years of high school algebra. Topics from differential and integral calculus with an emphasis on business applications. This class cannot be used as a prerequisite for MATH 232.
MATH 231: Calculus I. 4 hours. Prerequisite: Two years of high school algebra and one semester of high school trigonometry. A study of the fundamental principles of analytic geometry and calculus with emphasis on differentiation.
MATH 232: Calculus II. 4 hours. Prerequisite: MATH 231. It is recommended that students receive a grade of C or better in MATH 231 to be successful in this course. Continuation of Calculus I including techniques of integration and infinite series.
MATH 233: Calculus III. 4 hours. Prerequisite: MATH 232. It is recommended that students receive a grade of C or better in MATH 231 to be successful in this course. Functions of two variables, partial differentiation, applications of multiple integrals to areas and volumes, line and surface integrals, vectors.
MATH 234: Introduction to Mathematical Proof. 3 hours. Prerequisite: MATH 231. Recommend prerequisite: MATH 232. A careful introduction to the process of constructing mathematical arguments, covering the basic ideas of logic, sets, functions and relations. A substantial amount of time will be devoted to looking at important forms of mathematical argument such as direct proof, proof by contradiction, proof by contrapositive and proof by cases. Applications from set theory, abstract algebra, or analysis may be covered at the discretion of the instructor.
MATH 241: Discrete Mathematics. 3 hours. Prerequisite: two years of high school algebra. This course includes propositional logic, induction and recursion, number theory, set theory, relations and functions, graphs and trees, and permutations and combinations. Same as CSCI 241.
MATH 326: Probability Theory. 3 hours. Prerequisite: MATH 232. It is recommended that students receive a grade of C or better in MATH 232 to be successful in this course. This course includes an introduction to probability theory, discrete and continuous random variables, mathematical expectation, and multivariate distributions.
MATH 327: Mathematical Statistics. 3 hours. Prerequisite: MATH 326. It is recommended that students receive a grade of C or better in MATH 326 to be successful in this course. This course takes the material from MATH 326 into the applications side of statistics including functions of random variables, sampling distributions, estimations, and hypothesis testing.
MATH 330: Geometry. 3 hours. Prerequisite: MATH 234. Foundations of Euclidian geometry from the axioms of Hilbert and an introduction to non-Euclidian geometry.
MATH 493: Senior Seminar. 3 hours. Modern topics in mathematics are discussed in a seminar setting. Students integrate their study of mathematics throughout their undergraduate years and explore the connections among mathematics and other courses they have pursued. Departmental assessment of the major is included. This course is designed to be a capstone experience taken during the final semester of the senior year.
MATH 494: Senior Seminar for Secondary Education Math Majors. 3 hours. The history and philosophy of mathematics are discussed in a seminar setting. All students in this course must complete a project wherein familiar questions asked by high school math students are examined and answered in depth. Also, students are required to read and make a presentation on an article from an approved mathematics education journal. Department assessment of the major is included. This course is designed to be a capstone experience taken during the fall semester of the senior year.
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College Algebra
9780495554035
ISBN:
0495554030
Edition: 7 Pub Date: 2008 Publisher: Cengage Learning
Summary: Known for a clear and concise exposition, numerous examples, and plentiful problem sets, Jerome E. Kaufmann and Karen L. Schwitters's COLLEGE ALGEBRA, Seventh Edition, is an easy-to-use book that focuses on building technique and helping students hone their problem-solving skills. The seventh edition focuses on solving equations, inequalities, and problems; and on developing graphing techniques and using the concept ...of a function. Updated with new application problems and examples throughout, the seventh edition is accompanied by a robust collection of teaching and learning resources, including Enhanced WebAssign, an easy-to-use online homework management system for both instructors and
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's Them Engaged. Keep's Them Engaged Blitzer's philosophy: present the full scope of mathematics, while always (1) engaging the student by opening their minds to learning (2) keeping the student engaged on every page (3) explaining ideas directly, simply, and clearly so they don't get "lost" when studying and reviewing. First, he gets students engaged in the study of mathematics by highlighting truly relevant, unique, and engaging applications. He explores math the way it evolved: by describing real problems and how math explains them. In do... MOREing so, it answers the question "When will I ever use this?" Then, Blitzer keeps students engaged by ensuring they don't get lost when studying. Examples are easy to follow because of a three-step learning system - - "See it, Hear it, Try it" embedded into each and every one. He literally "walks" the student through each example by his liberal use of annotations - - the instructor's "voice" that appears throughout.
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Inductive Reasoning, Part 1 of 3 In this video, Sal Khan uses a simple number pattern to help the viewer understand how to use inductive reasoning to figure out what the next number will be. (Because it is a simple number pattern, the viewer will know what the the next number is, but Mr. Khan explains how to answer the mathematical question using inductive reasoning.)Mr. Khan uses the Paint Program (with different colors) to illustrate his points. Sal Khan is the recipient of the 2009 Microsoft Tech Award in Education. (02:01.77 Water Quality Control (MIT) The course material emphasizes mathematical models for predicting distribution and fate of effluents discharged into lakes, reservoirs, rivers, estuaries, and oceans. It also focuses on formulation and structure of models as well as analytical and simple numerical solution techniques. Also discussed are the role of element cycles, such as oxygen, nitrogen, and phosphorus, as water quality indicators; offshore outfalls and diffusion; salinity intrusion in estuaries; and thermal stratification, eu Author(s): AdamsS34 Problem Solving Seminar (MIT) This course, which is geared toward Freshmen, is an undergraduate seminar on mathematical problem solving. It is intended for students who enjoy solving challenging mathematical problems and who are interested in learning various techniques and background information useful for problem solving. Students in this course are expected to compete in a nationwide mathematics contest for undergraduates. Author(s): Rogers, Hartley,Kedlaya, Kiran,Stanley Summing up6.1 Direct proportion Doing and undoingLewis Carroll in Numberland An intriguing biographical exploration of Lewis Carroll, focusing on the author's mathematical career and influences. Author(s): Robin Wilson
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Use Photomerge Exposure in Manual mode Learn how to use the new Photomerge Exposure feature in Photoshop Elements 8 to combine images with different exposures into one perfect image. See why Manual mode is best when combining images that were taken with and without flash.Medieval Rus' Medieval Rus' is a resource designed to support the study of medieval Russian literature, developed by prominent medieval Slavicist David Birnbaum and his graduate students. The site offers: glossaries of church architecture and Christian festivals; external links to historical maps and genealogies; links to relevant associations and mailing lists; but its most useful features are to be found under 'syllabus'. The four unit course (Kievan Rus 988-1240; Mongol Russia 1240-1480; Muscovy 1480-1 Author(s): No creator set
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18523355 Concise Approach to Mathematical Analysis
This text introduces to undergraduates the more abstract concepts of advanced calculus, smoothing the transition from standard calculus to the more rigorous approach of proof writing and a deeper understanding of mathematical analysis. The first part deals with the basic foundation of analysis on the real line; the second part studies more abstract notions in mathematical analysis. Each topic contains a brief introduction and detailed examples
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Mathematics
The field of mathematics encompasses areas such as algebra, geometry,
calculus, probability, statistics, vector analysis, logic, and trigonometry. The
following selected mathematics related resources can be found at the
lower level of the Science Industry and Business Library (SIBL).
Handbooks Generally compendiums of in-depth
essays written by various authors on specific topics in mathematics;
formulas, tables, functions may be included.
Directories Sources for contact information
for societies, associations, government agencies, academic institutions,
and research centers.
Biographies Biographical works specific to
mathematicians can be found in *R-QA28 section as well as in the general
biographical works on scientists *R-Q130 and *R-Q141. Often these works
include historical accounts and developments of mathematical discoveries.
Electronic Databases These databases
are accessible from the Rohatyn Electronic Information Center on the
lower level of SIBL.
Journals SIBL subscribes to English and foreign language
journals in general and specific areas of mathematics. Check CATNYP under
the subject heading "Mathematical—Periodicals" for a comprehensive listing.
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25
Total Time: 3h 37m
Use: Watch Online & Download
Access Period: Unlimited
Created At: 12/02/2008
Last Updated At: 07/20/2010
Taught by Professor Edward Burger, these lessons were really helped me work with Complex Rational Expressions. I was really lost when dealing with these buggers but I watched the video 5 times before I really began to understand these expressions. Everything was explained clearly in the video. I will be looking for a part 2 to this topic.
I was struggling to grasp this concept when reviewing for an exam, as the examples in my text were not sufficient enough. This video was able to articulate the concept in a very clear manner and I can finally exhale and feel confident going into my exam. Thanks!
I love how animated his style is; it makes his teaching style even more engaging. Even when you thought it couldn't get more interesting, his bright orange tie makes it that much more fun to watch his lessons.
But to rate his lesson based on knowledge and ability to teach, I would give it a 10. It's so much easier to learn from a video tutorial than it is for me to read a textbook or sit through a lecture. I can start and stop whenever I want and take breaks without interrupting a class, etc. It's so much easier for me to learn at my own pace! Having these videos on integral algebra online makes me very happy.
Very happy indeed.
Below are the descriptions for each of the lessons included in the
series:
Int Algebra: Solving a Mixture Problem
In this lesson, you will learn how to approach word problems that involve equalities and ratios or fractions percentages, ratios, recipes, mixes, etc an Average
In this lesson, you will learn how to approach word problems that involve averages or means averages or means (many of which include grades or scores Factor Sums and Differences of Cubes
In this lesson, you will learn how to factor the difference of two cubes and how to factor the sum of two cubes. Neither of these are simple factorizations, so Professor Burger will show you what the factorization is and then explain where it comes from. Additionally, he'll show you some ways to remember what these factorizations are. (x^3 - y^3) = (x - y)(x^2+xy+y^2) and (x^3+y^3) = (x+y)(x^2-xy+y^2) Long Division with Polynomials
You know how to use long division to divide two numbers; you can also use long division when dividing polynomials. In this lesson, Professor Burger will review with you how to long divide and then show you how to use long division with polynomials (to evaluate things like (x^4+3x^2-5x-10)/(X^2+3x-5) ). When long dividing, you often end up with a remainder, and this will be the case when using long division on polynomials, as well. This lesson will show you how to find both the quotient and the remainder when dividing two polynomials using long division Synthetic Division with Polynomials
You know how to use long division to divide two numbers; you can also use long division when dividing polynomials. In this lesson, Professor Burger will show you how to eliminate several of the steps in long division by u sing synthetic division. Synthetic division only works when you are dividing by (x + ?) or (x - ?) where ? is a number. In synthetic division, you start by using the coefficients of the polynomial in the numberator with switched signs. In the end, you will end up with the same answer for both the quotient and remainder as you would using long division, but it will be a less harrowing path to get there Multiply-Divide Rational Expressions
In math, a rational expression or rational function is any function which can be written as the ratio of two polynomials. These are inevitably expressed as fractions. In this lesson, you will learn how to multiply and divide rational functions. Rational expressions can be multiplied or divided just like fractions. In walking you through examples of this type of multiplication and division, Professor Burger will highlight things to watch out for and shortcuts that can help you along the way. He will also show you why dividing is the same thing as multiplying by the reciprocal Add-Subtract Rational Expressions
In math, a rational expression or rational function is any function which can be written as the ratio of two polynomials. These are inevitably expressed as fractions. In this lesson, you will learn how to add and subtract rational functions. As with any fractions, to do this, you'll need to find a common denominator. In walking you through examples of this type of addition and subtraction, Professor Burger will highlight things to watch out for and shortcuts that can help you along the way Rewriting Complex Fractions
In math, a rational expression or rational function is any function which can be written as the ratio of two polynomials. These are inevitably expressed as fractions. Dealing with complex rational expressions is the same basic thing as dealing with other rational functions. In a complex rational expression, you generally end up with a variable within a fraction that's with another fraction that also includes another variable. The approaches to addition, subtraction, multiplication and division that are used with simple rational expressions all work the same here. You will need to invert and multiply in order to divide and you will need to find common denominators and least common multiples, etc. In this lesson, you will see a series of examples to see how these complex rational expressions are handled (things like (1/(1/x))/(1/(1/x)^2)). Containing Radicals
When working with equations, you often end up with a radical of some sort (like a square root) on one side of the equation. These type of equations are called radical equations because they contain a square root. To evaluate this type of equation, you'll want to get rid of the radical. This lesson will show you how to approach and solve this type of equation by getting rid of the radical (by isolating the radical alone on one side of the equals sign and then squaring both sides of the equation). When evaluating this type of equation, you will always want to check your solutions in the original equation to make sure that you don't end up with an extraneous root as a solution. Even if the equation solves to give you an extraneous root, it is not a valid solution. An extraneous root is something that is a root to the quadratic but not to the original equation with Two Radicals
In this lesson, Professor Burger will show you how to solve equations that contain two radicals (roots). When you have an equation with two square roots, you'll want to have them on opposite sides of the equal sign. Then, you'll square both sides of the equation. If there is still a radical remaining, you'll have to isolate it on one side of the equation and then square both sides once again. There will be several examples included in this lesson that will show you how to approach this type of problem and then how to check your work Functions and the Vertical Line Test
A function is basically a machine that takes an input value (x) and processes it to produce an output value (y). With a function, if an x value is known, you can find the y value. When graphed, a curve is a function if it passes the vertical line test. In this lesson, Professor Burger will show you how the vertical line test means and how to recognize when a curve does not pass the vertical line test. The vertical line test looks to verify that, for every value of x, only one y value is produced. If something doesn't pass the vertical line test, it is called a relation and not a function Function Notation and Values
In this lesson, Professor Burger will show you how to correctly denote functions and values. By definition, a function has only one value of y for each value of x. A function can always be expressed using the term f(x) instead of y. This lesson will walk through when to use this notation and how to use it correctly to indicate what you want it to be. Additionally, Professor Burger will show you how to verbally say the new notation in addition to how to write it. Last, he'll walk you through a few examples involving functions and their notation and evaluation Domain and Range
While a function always satisfies the vertical line test (for any value of x there is only one value of y), there are functions in which the domain of the function does not include all values of x. In this lesson, we look at the domain of a function (all of the values of x for which we can evaluate the function and find a value of y) and the range of a function (all the values of y that may be generated by evaluating the function for some value of x). In addition to learning about evaluating a function to find the domain and range, Professor Burger will graphically show you how to identify the domain and range Satisfying the Domain of a Function
In this lesson, you will learn how to find all of the allowable x values for a particular function (the function's domain). An allowable x value is one in which you can evaluate the function. There are certain types of numbers which are not allowable, like square roots of negative numbers, numbers with 0 in the denominator, etc. If you evaluate a function and end up with one of these types of numbers, then the x value is deemed to be outside of the domain for the function. Professor Burger will also show you how to correctly denote the domain of a function once you determine what it is Composite Functions
In this lesson, you will learn about a method that you can use to combine functions. The composition of two functions is the way to combine two functions. In this lesson, you will learn how to combine functions (for example, to find f(g(x)) ). There are specific ways to denote these types of composite functions, and you will also learn how to correctly write composite functions (f(g(x)) or (f o g)(x) ). To compose a function (find the composition of functions), you'll have to take the answer of one function and plug it into the other function (to find something like, 'g composed of f of 3'. Professor Burger will also highlight why g(f(3)) is not always equal to f(g(3)). Substitution
In this lesson, Professor Burger will show you how to solve systems of equations using a technique known as substitution. In this approach, you will solve one equation for one of the variables (eg y) and then plug the value (what y is equal to) into into the other equation (anywhere a y appears). This substitution will allow you to solve for x and then in turn solve for y. In order to fully explain how this works, Professor Burger will walk you through several different types of examples Elimination
While you can often solve systems of equations using substitution, you may also find that elimination is a simple approach for some systems of equations. When evaluating a system of linear equations with two linear variables using elimination, you will look for ways to combine the equations (or multiples of the equations) such that the sum of the equations will eliminate one of the variables. Once you eliminate one variable, it should be easy to deduce the value of the other equation. Once you have this, you should be able to plug it in to one of the original equations to solve for the eliminated value by Completing the Square This lesson will teach you how to find solutions by completing the square. In this technique, you'll start by isolating all constants on one side of the equation and all variable terms on the other side. Then, you'll add or subtract something to both sides to complete the square. In this case, you'll end up with x^2+6+9 = 9-1. This equation will be easier to evaluate given that you can simplify it to (x+3)^2 = 8. When you finally get to a solution value for x using this approach, you may need to rationalize a denominator (take radicals out of it), and Professor Burger will review this in the lesson, too Completing the Square: An Example In this lesson, you will learn more advanced techniques to use when solving an equation by completing the square. This lesson will cover what to do when the initial x^2 term contains a coefficient, how to solve problems that involve fractions, how to handle denominators with fractions, etc. This technique is the basis for the quadratic formula, which can always been used to solve quadratic equations Find Vertex by Completing the Square
In this lesson, you will learn how to find the vertex of a parabola given the formula for the parabola. To do this, you will complete the square. By completing the square of the parabola equation, you will be able to get the equation into a standard form that can be more easily evaluated. A parabola is a conic section in which the locus of points constructing it are equidistant from the focus and the directrix. Once we've identified the vertex of a parabola, we can get a good sense for how the parabola is positioned on the Cartesian coordinate plane the Quadratic Formula
The quadratic formula is used to solve for x in quadratic equations, which come in the form ax^2+bx+c=0. This formula is most commonly used when the expression can't be easily factored for evaluation. Oftentimes, this is because the two solutions to the equation are not real numbers. In this lesson, Professor Burger will walk you through when to use the formula, what the alternatives to the formula are and how to apply the formula. He will also explain how and why the formula can give imaginary numbers as solutions and what that means Predict Solution Type by Discriminant
When working with quadratic equations and the quadratic formula, there is a way to determine what type of solutions you will find and how many there will be (2 real solutions or 2 complex solutions or 1 solution) by looking at the coefficients of the quadratic formula. In this lesson, you will learn how to do this by calculating and evaluating the discriminant (d) of the quadratic formula (equal to b^2-4ac, which is a component of the quadratic formula The Pythagorean Theorem
The Pythagorean Theorem describes the relationship between the sides of a right triangle. It asserts that if the hypotenuse is length c and the other two legs are a and b, then a^2+b^2=c^2. This formula has a number of applications, and you will go through many of them in this lesson. Professor Burger will show you how to find one leg of a right triangle if you know the other two or if you know the length of one side and have two polynomials to express the lengths of the other two sides (e.g. if you know the three sides are c=x+2 and a=x and b=x+2 Quadratic Inequalities
In this lesson, Professor Burger will teach you how to solve quadratic (non-linear) inequalities. In a quadratic inequality, there are things like x^2 included. To evaluate these inequalities, we once again start by factoring. Next, you'll find the values for x, for which the quadratic inequality is positive such that you will be able to make a sign chart and then determine the sign (positive or negative) for ech interval delineated on the sign chart. Once you have identified the intervals that satisfy the equation, Professor Burger will show how to properly denote the answer using correct notation Writing an Equation for a Parabola
A parabola is a conic section in which the locus of points constructing it are equidistant from the focus and the directrix. To find the formula of this equation when given the vertex (h,k) and the distance from the focus (p), this lesson will show you how to find the equations for the parabola described by these criteria. There will be two formulas depending on whether p is positive or p is negative (which should indicate whether the parabola opens up or down).
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Book DescriptionI am currently doing a Masters in Games Programming and this is the reccommended text, it assumes no prior knowledge and leads you through the stages with lots of examples and help along the way, definately worth a look!
If, like me, the topic of computer graphics has been frightening because of mathematics then this is the book for you.
I found that this book explained the topic very clearly. One only needs to know the basic school geometry, such as those to do with right-angled triangles, to be able to read this book.
However what is missing is the application of mathematics to create new graphical applications. This book does not cover how to transform mathematical models into screen images or how to code. It does not cover solid models or anti-aliasing. It was not suppose to either. It is a very good introduction to mathematics. You will need to purchase, for example, "Computer Graphics: Principle and Practice" by D. Foley later on.
A rare find these days, a book that just covers the essentials and doesn't bother padding it with useless code snippets and CD offerings. I really appreciated the brevity of style and found that my interest was maintained thoughout the text. There are clear, concise graphical representations that compliment the text very well but if the reader is looking for code examples, I suggest that they look elsewhere.
1 of 1 people found the following review helpful
4.0 out of 5 starsGreat place to start25 April 2000
By A Customer - Published on Amazon.com
Format:Textbook Binding
This is a very good place to start if you are just getting into computer graphics and you need a gentle introduction. I discovered it while reading the book 'Advanced Renderman...' in the section under mathematical preliminaries. The authors recommend it as a good introduction, and I would have to second that. The only gripe I have is that there are several annoying typos (which seems to happen all too often these days).
2 of 3 people found the following review helpful
5.0 out of 5 starsA Must Have If You Are Learning Graphics Programming9 May 2001
By Stephen Rowe - Published on Amazon.com
Format:Textbook Binding
If you are like I was, your math is rusty enough that diving into Foley et al is like reading Greek. This is the best book I've found to teach the mathematical underpinnings of computer graphics. The book starts with basic trig and goes on to linear algebra and some calculus. After this book, you'll be ready to tackle most computer graphics texts. This book is hard to find but well worth it. An acceptable alternative is Mathematics for Computer Graphics Applications.
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Introduction To Algorithms 2nd Edition Solutions
In its new edition, Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity, and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage
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Mathematics is the language and tool of the sciences, a cultural phenomenon with a rich historical traditions, and a
model of abstract reasoning. Historically, mathematical methods
and thinking have proved extraordinarily successful in physics, and engineering. Nowadays, it is used successfully in many new
areas, from computer science to biology and finance. A Mathematics
concentration provides a broad education in various areas of mathematics
in a program flexible enough to accommodate many ranges of interest.
The study of mathematics is an excellent preparation
for many careers; the patterns of careful logical reasoning and
analytical problem solving essential to mathematics are also applicable
in contexts where quantity and measurement play only minor roles.
Thus students of mathematics may go on to excel in medicine, law, politics, or business as well as any of a vast range of scientific
careers. Special programs are offered for those interested in
teaching mathematics at the elementary or high school level or
in actuarial mathematics, the mathematics of insurance. The other
programs split between those which emphasize mathematics as an
independent discipline and those which favor the application of
mathematical tools to problems in other fields. There is considerable
overlap here, and any of these programs may serve as preparation
for either further study in a variety of academic disciplines, including mathematics itself, or intellectually challenging careers
in a wide variety of corporate and governmental settings.
Elementary Mathematics Courses. In order
to accommodate diverse backgrounds and interests, several course
options are available to beginning mathematics students. All courses
require three years of high school mathematics; four years are
strongly recommended and more information is given for some individual
courses below. Students with College Board Advanced Placement
credit and anyone planning to enroll in an upper-level class should
consider one of the Honors sequences and discuss the options with
a mathematics advisor.
Students who need additional preparation for calculus
are tentatively identified by a combination of the math placement
test (given during orientation), college admission test scores
(SAT or ACT), and high school grade point average. Academic advisors
will discuss this placement information with each student and
refer students to a special mathematics advisor when necessary.
Two courses preparatory to the calculus, MATH 105
and 110, are offered. MATH 105 is a course on data analysis, functions
and graphs with an emphasis on problem solving. MATH 110 is a
condensed half-term version of the same material offered as a
self-study course taught through the Math Lab and is only open
to students in MATH 115 who find that they need additional preparation
to successfully complete the course. A maximum total of 4 credits
may be earned in courses numbered 103, 105, and 110. MATH 103
is offered exclusively in the Summer half-term for students in
the Summer Bridge Program.
MATH 127 and 128 are courses containing selected
topics from geometry and number theory, respectively. They are
intended for students who want exposure to mathematical culture
and thinking through a single course. They are neither prerequisite
nor preparation for any further course. No credit will be received
for the election of MATH 127 or 128 if a student already has credit
for a 200-(or higher) level mathematics course.
Each of MATH 115, 185, and 295 is a first course
in calculus and generally credit can be received for only one
course from this list. The Sequence 115-116-215 is appropriate
for most students who want a complete introduction to calculus.
One of MATH 215, 285, or 395 is prerequisite to most more advanced
courses in Mathematics.
The sequences 156-255-256, 175-176-285-286, 185-186-285-286, and 295-296-395-396 are Honors sequences. Students need not be
enrolled in the LS&A Honors Program to enroll in any of these
courses but must have the permission of an Honors advisor. Students
with strong preparation and interest in mathematics are encouraged
to consider these courses.
MATH 185-285 covers much of the material of MATH
115-215 with more attention to the theory in addition to applications.
Most students who take MATH 185 have taken a high school calculus
course, but it is not required. MATH 175-176 assumes a knowledge
of calculus roughly equivalent to MATH 115 and covers a substantial
amount of so-called combinatorial mathematics as well as calculus-related
topics not usually part of the calculus sequence. MATH 175 and
176 are taught by the discovery method: students are presented
with a great variety of problem and encouraged to experiment in
groups using computers. The sequence MATH 295-396 provides a rigorous
introduction to theoretical mathematics. Proofs are stressed over
applications and these courses require a high level of interest
and commitment. Most students electing MATH 295 have completed
a thorough high school calculus. MATH 295-396 is excellent preparation
for mathematics at the advanced undergraduate and graduate level.
Students with strong scores on either the AB or
BC version of the College Board Advanced Placement exam may be
granted credit and advanced placement in one of the sequences
described above; a table explaining the possibilities is available
from advisors and the Department. In addition, there is one course
expressly designed and recommended for students with one or two
semesters of AP credit, MATH 156. Math 156 is an Honors course
intended primarily for science and engineering concentrators and
will emphasize both applications and theory. Interested students
should consult a mathematics advisor for more details.
In rare circumstances and with permission of a Mathematics
advisor, reduced credit may be granted for MATH 185 or 295 after
MATH 115. A list of these and other cases of reduced credit for
courses with overlapping material is available from the Department.
To avoid unexpected reduction in credit, student should always
consult an advisor before switching from one sequence to another.
In all cases a maximum total of 16 credits may be earned for calculus
courses MATH 115 through 396, and no credit can be earned for
a prerequisite to a course taken after the course itself.
Students completing MATH 116 who are principally
interested in the application of mathematics to other fields may
continue either to MATH 215 (Analytic Geometry and Calculus III)
or to MATH 216 (Introduction to Differential Equation -- these
two courses may be taken in either order. Students who have greater
interest in theory or who intend to take more advanced courses
in mathematics should continue with MATH 215 followed by the sequence
MATH 217-316 (Linear Algebra-Differential Equations). MATH 217
(or the Honors version, MATH 513) is required for a concentration
in Mathematics; it both serves as a transition to the more theoretical
material of advanced courses and provides the background required
to optimal treatment of differential equations in MATH 316. MATH
216 is not intended for mathematics concentrators.
Special Departmental Policies. All prerequisite
courses must be satisfied with a grade of C- or above. Students
with lower grades in prerequisite courses must receive special
permission of the instructor to enroll in subsequent courses.
MATH 105. Data, Functions, and Graphs.
Instructor(s):
Prerequisites & Distribution: (4). (MSA). (QR/1). May not be repeated for credit. Students with credit for MATH 103 can elect MATH 105 for only 2 credits. No credit granted to those who have completed any Mathematics course numbered 110 or higher. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.
MATH 105 serves both as a preparatory course to the calculus sequences and as a terminal course for students who need only this level of mathematics. Students who complete MATH 105 are fully prepared for MATH 115. ThisMATH 107. Mathematics for the Information Age.
Section 001.
Instructor(s):
Karen Rhea
Prerequisites & Distribution: Three to four years high school mathematics. (3). (MSA). (QR/1). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
From computers and the Internet to playing a CD or running an election, great progress in modern technology and science has come from understanding how information is exchanged, processed and perceived.
MATH 110. Pre-Calculus (Self-Study).
Instructor(s):
Prerequisites & Distribution: See Elementary Courses above. Enrollment in MATH 110 is by recommendation of MATH 115 instructor and override only. (2). (Excl). May not be repeated for credit. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.
The course covers data analysis by means of functions and graphs. MATH 110 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, half-term version of MATH 105 (MATH 105 covers the same material in a traditional classroom setting) designed for students who appear to be prepared to handle calculus but are not able to successfully complete MATH 115. Students who complete MATH 110 are fully prepared for MATH 115. Students may enroll in MATH 110 only on the recommendation of a mathematics instructor after the third week of classes.
ENROLLMENT IN MATH 110 IS BY PERMISSION OF MATH 115 INSTRUCTOR ONLY. COURSE MEETS SECOND HALF OF THE TERM. STUDENTS WORK INDEPENDENTLY WITH GUIDANCE FROM MATH LAB STAFF.
MATH 115. Calculus I.
Instructor(s):
Prerequisites & Distribution: Four years of high school mathematics. See Elementary Courses above. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit usually is granted for only one course from among 115, 185, and 295. No credit granted to those who have completed MATH 175.
The sequence MATH 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills.
Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. MATH 185 is a somewhat more theoretical course which covers some of the same material. MATH 175 includes some of the material of MATH 115 together with some combinatorial mathematics. A student whose preparation is insufficient for MATH 115 should take MATH 105 (Data, Functions, and Graphs). MATH 116 is the natural sequel. A student who has done very well in this course could enter the honors sequence at this point by taking MATH 186. The cost for this course is over $100 since the student will need a text (to be used for MATH 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended).
See MATH 115 for a general description of the sequence MATH 115-116-215.
Topics include the indefinite integral, techniques of integration, introduction to differential equations, and infinite series. MATH 186 is a somewhat more theoretical course which covers much of the same material. MATH 215 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking MATH 285.
MATH 127. Geometry and the Imagination.
Section 001.
Instructor(s):
Emina Alibegovic
Prerequisites & Distribution: Three years of high school mathematics including a geometry course. Only first-year students, including those with sophomore standing, may pre-register for First-Year Seminars. All others need permission of instructor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. No credit granted to those who have completed a 200- (or higher) level mathematics course (except for MATH 385 and 485).
First-Year Seminar
Credits: (4).
Course Homepage: No homepage submitted.
This course introduces students to the ideas and some of the basic results in Euclidean and non-Euclidean geometry. Beginning with geometry in ancient Greece, the course includes the construction of new geometric objects from old ones by projecting and by taking slices. The next topic is non-Euclidean geometry. This section begins with the independence of Euclid's Fifth Postulate and with the construction of spherical and hyperbolic geometries in which the Fifth Postulate fails; how spherical and hyperbolic geometry differs from Euclidean geometry. The last topic is geometry of higher dimensions: coordinatization — the mathematician's tool for studying higher dimensions; construction of higher-dimensional analogues of some familiar objects like spheres and cubes; discussion of the proper higher-dimensional analogues of some geometric notions (length, angle, orthogonality, etc. ) This course is intended for students who want an introduction to mathematical ideas and culture. Emphasis on conceptual thinking — students will do hands-on experimentation with geometric shapes, patterns, and ideas.
This course is designed for students who seek an introduction to the mathematical concepts and techniques employed by financial institutions such as banks, insurance companies, and pension funds. Actuarial students, and other mathematics concentrators should elect MATH 424, which covers the same topics but on a more rigorous basis requiring considerable use of calculus. Topics covered include: various rates of simple and compound interest, present and accumulated values based on these; annuity functions and their application to amortization, sinking funds, and bond values; depreciation methods; introduction to life tables, life annuity, and life insurance values. This course is not part of a sequence. Students should possess financial calculators.
MATH 186. Honors Calculus II.
Instructor(s):
Prerequisites & Distribution: Permission of the Honors advisor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course from among MATH 116, 156, 176, 186, and 296.
Waitlist Code: 5: Students in LSA College Honors may request overrides from the Honors Office; other students may request them from the Math Dept Office, 2084 East Hall.
MATH 214. Linear Algebra and Differential Equations.
Instructor(s):
Prerequisites & Distribution: MATH 115 and 116This course is intended for second-year students who might otherwise take MATH 216 (Introduction to Differential Equations) but who have a greater need or desire to study Linear Algebra. This may include some Engineering students, particularly from Industrial and Operations engineering (IOE), as well as students of Economics and other quantitative social sciences. Students intending to concentrate in Mathematics must continue to elect MATH 217.
While MATH 216 includes 3-4 weeks of Linear Algebra as a tool in the study of Differential Equations, MATH 214 will include roughly 3 weeks of Differential Equations as an application of Linear Algebra.
The sequence MATH 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a midterm and final exam. Topics Maple software. MATH 285 is a somewhat more theoretical course which covers the same material. For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is MATH 217. Students who intend to take only one further mathematics course and need differential equations should take MATH 216.
MATH 216. Introduction to Differential Equations.
Instructor(s):
Prerequisites & Distribution: MATH 116, 119, 156, 176, 186, or 296. Not intended for Mathematics concentrators. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316 engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence MATH 217-316. After There is a weekly computer lab using MATLAB software. This course is not intended for mathematics concentrators, who should elect the sequence MATH 217-316. MATH 286 covers much of the same material in the honors sequence. The sequence MATH 217-316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. MATH 404 covers further material on differential equations. MATH 217 and 417 cover further material on linear algebra. MATH 371 and 471 cover additional material on numerical methods.
MATH 217. Linear Algebra.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285 Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence MATH 217-316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Therefore the student entering MATH 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of Rn; linear dependence, bases, and dimension; linear transformations; eigenvalues and eigenvectors; diagonalization; and inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. MATH 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. MATH 513 covers more in a much more sophisticated way. The intended course to follow MATH 217 is 316. MATH 217 is also prerequisite for MATH 412 and all more advanced courses in mathematics.
Instructor(s):
Hendrikus Gerardus Derksen
One of the best ways to develop mathematical abilities is by solving problems using a variety of methods. Familiarity with numerous methods is a great asset to the developing student of mathematics. Methods learned in attacking a specific problem frequently find application in many other areas of mathematics. In many instances an interest in and appreciation of mathematics is better developed by solving problems than by hearing formal lectures on specific topics. The student has an opportunity to participate more actively in his/her education and development. This course is intended for superior students who have exhibited both ability and interest in doing mathematics, but it is not restricted to honors students. This course is excellent preparation for the Putnam exam. Students and one or more faculty and graduate student assistants will meet in small groups to explore problems in many different areas of mathematics. Problems will be selected according to the interests and background of the students.
MATH 296. Honors Mathematics II.
Section 001.
Instructor(s):
Brian D Conrad
Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. (4). (Excl). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course from among MATH 156, 176, 186, and 296.
Credits: (4).
Course Homepage: No homepage submitted.
The sequence MATH 295-296-395-396 is a more intensive honors sequence than MATH 185-186-285-286. The material includes all of that of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to learn to understand and construct proofs as well as do calculations and solve problems. The expected background is a thorough understanding of high school algebra and trigonometry. No previous calculus is required, although many students in this course have had some calculus. Students completing this sequence will be ready to take advanced undergraduate and beginning graduate courses. This sequence is not restricted to students enrolled in the LS&A Honors Program. The precise content depends on material covered in MATH 295 but will generally include topics such as infinite series, power series, Taylor expansion, metric spaces. Other topics may include applications of analysis, Weierstrass Approximation theorem, elements of topology, introduction to linear algebra, complex numbers.
Waitlist Code: 5: Students in LSA College Honors may request overrides from the Honors Office; other students may request them from the Math Dept Office, 2084 East Hall.
MATH 310. Elementary Topics in Mathematics.
Section 001 — Math Games & Theory of Games.
Instructor(s):
Morton Brown
Prerequisites & Distribution: Two years of high school mathematics. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
The current offering of the course focuses on game theory. Students study the strategy of several games where mathematical ideas and concepts can play a role. Most of the course will be occupied with the strructure of a variety of two person games of strategy: tic-tac-toe, tic-tac-toe misere, the French military game, hex, nim, the penny dime game, and many others. If there is sufficient interest students can study: dots and boxes, go moku, and some aspects of checkers and chess. There will also be a brief introduction to the classical Von Neuman/Morgenstern theory of mixed strategy games.
One of the main goals of the course (along with every course in the algebra sequence) is to expose students to rigorous, proof-oriented mathematics. Students are required to have taken MATH 217, which should provide a first exposure to this style of mathematics. A distinguishing feature of this course is that the abstract concepts are not studied in isolation. Instead, each topic is studied with the ultimate goal being a real-world application. As currently organized, the course is broken into four parts: the integers "mod n" and linear algebra over the integers mod p, with applications to error correcting codes; some number theory, with applications to public-key cryptography; polynomial algebra, with an emphasis on factoring algorithms over various fields, and permutation groups, with applications to enumeration of discrete structures "up to automorphisms" (a.k.a. Pólya Theory). MATH 412 is a more abstract and proof-oriented course with less emphasis on applications. EECS 303 (Algebraic Foundations of Computer Engineering) covers many of the same topics with a more applied approach. Another good follow-up course is MATH 475 (Number Theory). MATH 312 is one of the alternative prerequisites for MATH 416, and several advanced EECS courses make substantial use of the material of MATH 312. MATH 412 is better preparation for most subsequent mathematics courses.
MATH 316. Differential Equations.
Section 001.
Instructor(s):
Arthur G Wasserman
Prerequisites & Distribution: MATH 215 and 217. (3). (Excl). (BS). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316.
Credits: (3).
Course Homepage: No homepage submitted.
This is an introduction to differential equations for students who have studied linear algebra (MATH 217). It treats techniques of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs are given in class; homework problems include both computational and more conceptually oriented problems. First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout. MATH 216 covers somewhat less material without the use of linear algebra and with less emphasis on theory. MATH 286 is the Honors version of MATH 316. MATH 471 and/or MATH 572 are natural sequels in the area of differential equations, but MATH 316 is also preparation for more theoretical courses such as MATH 451.
MATH 333. Directed Tutoring.
Instructor(s):
Prerequisites & Distribution: Enrollment in the secondary teaching certificate program with concentration in mathematics. Permission of instructor required. (1-3). (Excl). (EXPERIENTIAL). May be repeated for credit for a maximum of 3 credits. Offered mandatory credit/no credit.
Credits: (1-3).
Course Homepage: No homepage submitted.
An experiential mathematics course for exceptional upper-level students in the elementary teacher certification program. Students tutor needy beginners enrolled in the introductory courses (MATH 385 and MATH 489) required of all elementary teachers.
MATH 351. Principles of Analysis.
Section 001.
Instructor(s):
Morton Brown
Prerequisites & Distribution: MATH 215 and 217. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 451.
Credits: (3).
Course Homepage: No homepage submitted.
The content of this course is similar to that of MATH 451 but MATH 351 assumes less background. This course covers topics that might be of greater use to students considering a Mathematical Sciences concentration or a minor in Math. Course content includes: analysis of the real line, rational and irrational numbers, infinity — large and small, limits, convergence, infinite sequences and series, continuous functions, power series and differentiation.
MATH 354. Fourier Analysis and its Applications.
Section 001.
Instructor(s):
Mahdi Asgari
Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 450 or 454.
Credits: (3).
Course Homepage: No homepage submitted.
This course is an introduction to Fourier analysis at an elementary level, emphasizing applications. The main topics are Fourier series, discrete Fourier transforms, and continuous Fourier transforms. A substantial portion of the time is spent on both scientific/technological applications (e.g., signal processing, Fourier optics), and applications in other branches of mathematics (e.g., partial differential equations, probability theory, number theory). Students will do some computer work, using MATLAB, an interactive programming tool that is easy to use, but no previous experience with computers is necessary.
MATH 371 / ENGR 371. Numerical Methods for Engineers and Scientists.
Section 001.
Instructor(s):
David Gammack
Prerequisites & Distribution: ENGR 101; one of MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in Math 471. CAEN lab access fee required for non-Engineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for non-Engineering students.
Course Homepage: No homepage submitted.
This is a survey course of the basic numerical methods which are used to solve practical scientific problems.
Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction
to MATLAB, an interactive program for numerical linear algebra. Convergence theorems are discussed and
applied, but the proofs are not emphasized.
Objectives of the course
Develop numerical methods for approximately solving problems from continuous mathematics on the
computer
Implement these methods in a computer language (MATLAB)
Apply these methods to application problems
Computer language:
In this course, we will make extensive use of Matlab, a technical computing environment for numerical
computation and visualization produced by The MathWorks, Inc. A Matlab manual is available in the MSCC Lab.
Also available is a MATLAB tutorial written by Peter Blossey.
MATH 396. Honors Analysis II.
Section 001.
Instructor(s):
Mario Bonk
This course is a continuation of MATH 395 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Differential and integral calculus of functions on Euclidean spaces. Students who have successfully completed the sequence MATH 295-396 are generally prepared to take a range of advanced undergraduate and graduate courses such as MATH 512, 513, 525, 590, and many others.
MATH 412. Introduction to Modern Algebra.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285; and 217. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 512. Students with credit for MATH 312 should take MATH 512 rather than 412. One credit granted to those who have completed MATH 312.
Credits: (3).
Course Homepage: No homepage submitted.
This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions or the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc.) and their proofs. MATH 217 or equivalent required as background. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, and complex numbers). These are then applied to the study of particular types of mathematical structures such as groups, rings, and fields. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied.
MATH 312 is a somewhat less abstract course which substitutes material on finite automata and other topics for some of the material on rings and fields of MATH 412. MATH 512 is an Honors version of MATH 412 which treats more material in a deeper way. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: MATH 416, 451, 475, 575, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.
MATH 417. Matrix Algebra I.
Instructor(s):
Prerequisites & Distribution: Three courses MATH 513.
Credits: (3).
Course Homepage: No homepage submitted.
Many problems in science, engineering, and mathematics are best formulated in terms of matrices — rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect MATH 217 or 513 (Honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.
MATH 419 is an enriched version of MATH 417 with a somewhat more theoretical emphasis. MATH 217 (despite its lower number) is also a more theoretical course which covers much of the material of MATH 417 at a deeper level. MATH 513 is an Honors version of this course, which is also taken by some mathematics graduate students. MATH 420 is the natural sequel, but this course serves as prerequisite to several courses: MATH 452, 462, 561, and 571.
MATH 419. Linear Spaces and Matrix Theory.
Instructor(s):
Prerequisites & Distribution: Four terms of college mathematics in MATH 513.
Credits: (3).
Course Homepage: No homepage submitted.
MATH 419 covers much of the same ground as MATH 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, and applications to differential and difference equations.
MATH 417 is less rigorous and theoretical and more oriented to applications. MATH 217 is similar to MATH 419 but slightly more proof-oriented. MATH 513 is much more abstract and sophisticated. MATH 420 is the natural sequel, but this course serves as prerequisite to several courses: MATH 452, 462, 561, and 571.
MATH 425 / STATS 425. Introduction to Probability.
Instructor(s):
Mathematics faculty
This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use of the material of MATH 116 and 215. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis. STATS 426 is a natural sequel for students interested in statistics. MATH 523 includes many applications of probability theory.
This course introduces students to the theory of probability and to a number of applications. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances.
There will be approximately 10 problem sets. Grade will be based on two 1-hour midterm exams, 20% each; 20% homework; 40% final exam. pText (required): Sheldon Ross, A First Course in Probability, 6th edition, Prentice-Hall, 2002.
MATH 450. Advanced Mathematics for Engineers I.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. (4). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 354 or 454.
Credits: (4).
Course Homepage: No homepage submitted.
Although this course is designed principally to develop mathematics for application to problems of science and engineering, it also serves as an important bridge for students between the calculus courses and the more demanding advanced courses. Students are expected to learn to read and write mathematics at a more sophisticated level and to combine several techniques to solve problems. Some proofs are given, and students are responsible for a thorough understanding of definitions and theorems. Students should have a good command of the material from MATH 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g. MATH 217, is highly desirable, as is familiarity with Maple software. Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3-space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems; and derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. MATH 450 is an alternative to MATH 451 as a prerequisite for several more advanced courses. MATH 454 and 555 are the natural sequels for students with primary interest in engineering applications.
MATH 451. Advanced Calculus I.
Instructor(s):
Prerequisites & Distribution: MATH 215 and one course beyond MATH 215; or MATH 255 or 285. Intended for concentrators; other students should elect MATH 450. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 351.
Credits: (3).
Course Homepage: No homepage submitted.
This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.) and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as MATH 412) be taken before MATH 451. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits, and continuity; differentiation; integration, and the Fundamental Theorem of Calculus; infinite series; and sequences and series of functions.
There is really no other course which covers the material of MATH 451. Although MATH 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. The natural sequel to MATH 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, MATH 451 treats the theory behind MATH 115-116, while MATH 452 does the same for MATH 215 and a part of MATH 216. MATH 551 is a more advanced version of Math 452. MATH 451 is also a prerequisite for several other courses: MATH 575, 590, 596, and 597.
MATH 452. Advanced Calculus II.
Section 001 — Multivariable Calculus and Elementary Function Theory.
Instructor(s):
Lukas I Geyer
Prerequisites & Distribution: MATH 217, 417, or 419; and MATH 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course does a rigorous development of multivariable calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation. Topics include:
MATH 551 is a higher-level course covering much of the same material with greater emphasis on differential geometry. Math 450 covers the same material and a bit more with more emphasis on applications, and no emphasis on proofs. MATH 452 is prerequisite to MATH 572 and is good general background for any of the more advanced courses in analysis (MATH 596, 597) or differential geometry or topology (MATH 537, 635).
MATH 454. Boundary Value Problems for Partial Differential Equations.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450.
Credits: (3).
Course Homepage: No homepage submitted.
This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; and applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis. Both MATH 455 and 554 cover many of the same topics but are very seldom offered. MATH 454 is prerequisite to MATH 571 and 572, although it is not a formal prerequisite, it is good background for MATH 556.
MATH 462. Mathematical Models.
Section 001.
Instructor(s):
David Bortz
Prerequisites & Distribution: MATH 216, 256, 286, or 316; and MATH 217, 417, or 419. (3). (Excl). (BS). May not be repeated for credit. Students with credit for MATH 362 must have department permission to elect MATH 462.
Credits: (3).
Course Homepage: No homepage submitted.
This course will cover biological models constructed from difference equations and ordinary differential equations. Applications will be drawn from population biology, population genetics, the theory of epidemics, biochemical kinetics, and physiology. Both exact solutions and simple qualitative methods for understanding dynamical systems will be stressed.
MATH 471. Introduction to Numerical Methods.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 371 or 472.
Credits: (3).
Course Homepage: No homepage submitted.
This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proven. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2-point boundary value problems, and the Dirichlet problem for the Laplace equation. MATH 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence MATH 571-572 is mainly taken by graduate students, but should be considered by strong undergraduates. MATH 471 is good preparation for MATH 571 and 572, although it is not prerequisite to these courses.
MATH 475. Elementary Number Theory.
Section 001.
Instructor(s):
Muthukrishnan Krishnamurthy
Prerequisites & Distribution: At least three terms of college mathematics are recommended. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This is an elementary introduction to number theory, especially congruence arithmetic. Number theory is one of the few areas of mathematics in which problems easily describable to a layman (is every even number the sum of two primes?) have remained unsolved for centuries. Recently some of these fascinating but seemingly useless questions have come to be of central importance in the design of codes and cyphers. The methods of number theory are often elementary in requiring little formal background. In addition to strictly number-theoretic questions, concrete examples of structures such as rings and fields from abstract algebra are discussed. Concepts and proofs are emphasized, but there is some discussion of algorithms which permit efficient calculation. Students are expected to do simple proofs and may be asked to perform computer experiments. Although there are no special prerequisites and the course is essentially self-contained, most students have some experience in abstract mathematics and problem solving and are interested in learning proofs. A Computational Laboratory (Math 476, 1 credit) will usually be offered as an optional supplement to this course. Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity, and quadratic fields. MATH 575 moves much faster, covers more material, and requires more difficult exercises. There is some overlap with MATH 412 which stresses the algebraic content. MATH 475 may be followed by Math 575 and is good preparation for MATH 412. All of the advanced number theory courses, MATH 675, 676, 677, 678, and 679, presuppose the material of MATH 575, although a good student may get by with MATH 475. Each of these is devoted to a special subarea of number theory.
MATH 476. Computational Laboratory in Number Theory.
Section 001.
Instructor(s):
Muthukrishnan Krishnamurthy
Prerequisites & Distribution: Prior or concurrent enrollment in MATH 475 or 575. (1). (Excl). (BS). May not be repeated for credit.
Credits: (1).
Course Homepage: No homepage submitted.
Students will be provided software with which to conduct numerical explorations. Students will submit reports of their findings weekly. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Participation in the laboratory should boost the student's performance in MATH 475 or MATH 575. Students in the lab will see mathematics as an exploratory science (as mathematicians do). Students will gain a knowledge of algorithms which have been developed (some quite recently) for number-theoretic purposes, e.g., for factoring. No exams.
Instructor(s):
This course is designed for students who intend to teach junior high or high school mathematics. It is advised that the course be taken relatively early in the program to help the student decide whether or not this is an appropriate goal. Concepts and proofs are emphasized over calculation. The course is conducted in a discussion format. Class participation is expected and constitutes a significant part of the course grade. Topics covered have included problem solving; sets, relations and functions; the real number system and its subsystems; number theory; probability and statistics; difference sequences and equations; interest and annuities; algebra; and logic. This material is covered in the course pack and scattered points in the text book. There is no real alternative, but the requirement of MATH 486 may be waived for strong students who intend to do graduate work in mathematics. Prior completion of MATH 486 may be of use for some students planning to take MATH 312, 412, or 425.
MATH 489. Mathematics for Elementary and Middle School Teachers.
Instructor(s):
Prerequisites & Distribution: MATH 385 or 485. (3). (Excl). May not be repeated for credit. May not be used in any graduate program in mathematics.
Credits: (3).
Course Homepage: No homepage submitted.
This course, together with its predecessor MATH 385, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course.
MATH 490. Introduction to Topology.
Section 001 — An Introduction to Point-Set and Algebraic Topology.
Instructor(s):
Elizabeth A Burslem
Prerequisites & Distribution: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course in an introduction to both point-set and algebraic topology. Although much of the presentation is theoretical and proof-oriented, the material is well-suited for developing intuition and giving convincing proofs which are pictorial or geometric rather than completely rigorous. There are many interesting examples of topologies and manifolds, some from common experience (combing a hairy ball, the utilities problem). In addition to the stated prerequisites, courses containing some group theory (MATH 412 or 512) and advanced calculus (MATH 451) are desirable although not absolutely necessary. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor. These include point-set topology, examples of topological spaces, orientable and non-orientable surfaces, fundamental groups, homotopy, and covering spaces. Metric and Euclidean spaces are emphasized. Math 590 is a deeper and more difficult presentation of much of the same material which is taken mainly by mathematics graduate students. MATH 433 is a related course at about the same level. MATH 490 is not prerequisite for any later course but provides good background for MATH 590 or any of the other courses in geometry or topology.
MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.
Section 001.
Prerequisites & Distribution: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). May be repeated for credit for a maximum of 6 credits. Offered mandatory credit/no credit.
Credits: (1).
Course Homepage: No homepage submitted.
The Applied and Interdisciplinary Mathematics (AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from U-M and from elsewhere). The other central aspect of the course will be a seminar to prepare students with appropriate introductory background material. The seminar will also focus on effective communication methods for interdisciplinary research. MATH 501 is primarily intended for graduate students in the Applied & Interdisciplinary Mathematics M.S. and Ph.D. programs. It is also intended for mathematically curious graduate students from other areas. Qualified undergraduates are welcome to elect the course with the instructor's permission.
Student attendance and participation at all seminar sessions is required. Students will develop and make a short presentation on some aspect of applied and interdisciplinary mathematics.
MATH 512. Algebraic Structures.
Section 001 — Basic Structures of Modern Abstract Algebra.
Instructor(s):
Robert L Griess Jr
Prerequisites & Distribution: MATH 451 or 513. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Student Body: mainly undergrad math concentrators with a few grad students from other fields
Background and Goals: This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students who have completed the 295--396 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students should have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from Math 285 are strongly advised to take some 400-500 level course first, for example, Math 513. Some background in linear algebra is strongly recommended
Content: The course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity. Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitely-generated Abelian groups, permutation representations, the symmetric and alternating groups (2) Ring Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals, and (3) Field Theory: statement of the Fundamental Theorem of Galois Theory, Nullstellensatz, subfields of the complex numbers and the integers mod p.
Alternatives: Math 412 (Introduction to Modern Algebra) is a substantially lower level course covering about half of the material of Math 512. The sequence Math 593--594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412.
Subsequent Courses: Together with Math 513, this course is excellent preparation for the sequence Math 593 — 594.
Text Book: Abstract Algebra, Second Edition by David Dummit and Richard Foote.
MATH 513. Introduction to Linear Algebra.
Instructor(s):
William E Fulton
Prerequisites & Distribution: MATH 412. (3). (Excl). (BS). May not be repeated for credit. Two credits granted to those who have completed MATH 214, 217, 417, or 419.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisites: Math 412 or Math 451 or permission of the instructor
Background and Goals: This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory.
Content: Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; eigenvalues and eigenvectors; applications to linear and linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form. This corresponds to most of the first text with the omission of some starred sections and all but Chapters 8 and 10 of the second text.
Alternatives: Math 419 (Lin. Spaces and Matrix Thy) covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 (Linear Algebra) is similarly proof-oriented but significantly less demanding than Math 513. Math 417 (Matrix Algebra I) is much less abstract and more concerned with applications.
Subsequent Courses: The natural sequel to Math 513 is Math 593 (Algebra I). Math 513 is also prerequisite to several other courses: Math 537, 551, 571, and 575, and may always be substituted for Math 417 or 419.
Section 001.
This course is a continuation of MATH520 (a year-long sequence). It covers the topics of reserving models for life insurance; multiple-life models including joint life and last survivor contingent insurances; multiple-decrement models including disability, retirement and withdrawal; insurance models including expenses; and business and regulatory considerations.
MATH 523. Risk Theory.
Section 001 — Risk Management.
Instructor(s):
Conlon
Required Text: "Loss Models-from Data to Decisions", by Klugman, Panjer and
Willmot, Wiley 1998.
Background and Goals: Risk management is of major concern to all
financial institutions and is an active area of modern finance. This course is
relevant for students with interests in finance, risk management, or insurance.
It provides background for the professional exams in Risk Theory offered by the
Society of Actuaries and the Casualty Actuary Society. Contents: Standard distributions used for claim frequency models and for loss
variables, theory of aggregate claims, compound Poisson claims model, discrete time and continuous time models for the aggregate claims variable, the Chapman-Kolmogorov equation for expectations of aggregate claims variables, the
Poisson process, estimating the probability of ruin, reinsurance schemes
and their implications for profit and risk.
Credibility theory, classical theory for independent events, least
squares theory for correlated events, examples of random variables where the
least squares theory is exact.
Grading: The grade for the course will be determined from
performances on homeworks, a midterm and a final exam.
MATH 525 / STATS 525. Probability Theory.
Section 001.
Instructor(s):
Gautam Bharali
Prerequisites & Distribution: MATH 451 (strongly recommended) or 450. MATH 425 would be helpful. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Background: This course is a fairly rigorous study of the mathematical basis of probability theory. There is some overlap of topics with Math 425, but in Math 525, there is a greater emphasis on the proofs of major results in probability theory. This course and its sequel - Math 526 - are core
courses for the Applied and Interdisciplinary Mathematics (AIM) program.
Content: The notion of a probability space and a random variable, discrete and continuous random variables, independence and expectation, conditional probability and conditional expectations, generating functions and moment generating functions, the Law of Large Numbers, and the Central Limit Theorem comprise the essential core of this course. Further topics, to be decided later (and, if feasible, selected according to audience interest), will be covered in the last month of the semester.
Alternatives: EECS 501 covers some of the above material at a lower level of mathematical rigor. Math 425 (Introduction to Probability) is recommended for students with substantially less mathematical preparation.
Instructor(s):
Virginia R Young
Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in insurance, risk management, or finance. We will cover the following topics: advanced topics in credibility theory, risk measures and premium principles, optimal (re)insurance, reinsurance products, and reinsurance pricing.
I assume that you have taken MATH 523, Risk Theory. In fact, one can think of this course as a continuation of MATH 523 with emphasis on applying the material learned in Risk Theory to more practical settings.
The official text for the course is a set of notes available at UM.CourseTools. In addition, an excellent book concerning modern reinsurance products is Integrating Corporate Risk Management by Prakash Shimpi, published by Texere. I suggest that you buy this book, but I do not require that you do so.
MATH 531. Transformation Groups in Geometry.
Section 001.
Instructor(s):
Emina Alibegovic
Prerequisites: MATH 412 or 512 would be helpful, but neither is necessary.
Text required: None.
Text recommended: Armstrong, Groups and Symmetry; Lyndon: Groups and Geometry.
textbook comment: Your class notes and my handouts will be sufficient. The books
I listed contain some of the material we will cover, but not all of it.
Course description:
The purpose of this course is to explore the close ties between geometry and
algebra. We will study Euclidean and hyperbolic spaces and groups of their
isometries. Our discussions will include, but will not be limited to, free
groups, triangle groups, and Coxeter groups. We will talk about group actions
on spaces, and in particular group actions on trees.
MATH 555. Introduction to Functions of a Complex Variable with Applications.
Section 001.
Instructor(s):
Divakar Viswanath
Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Recent Texts: Complex Variables and Applications, 6th ed. (Churchill and Brown);
Student Body: largely engineering and physics graduate students with some math and engineering undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program
Background and Goals: This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program. Content: Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. This corresponds to Chapters 1--9 of Churchill. Alternatives: Math 596 (Analysis I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.
MATH 557. Methods of Applied Mathematics II.
Section 001.
Prerequisites & Distribution: MATH 217, 419, or 513; 451 and 555. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisites: (1) one of the following: Math 217, 419, or 513 (i.e. a
course in linear algebra); (2) one of the following: Math 216, 256, 286, 316, or 404 (i.e. a course in differential equations); (3) Math 451
(or an equivalent course in
advanced calculus); (4) Math 555 (or an equivalent course in complex
variables).
Text: There is no required text. Lecture notes will be made available
to students from the instructor's website. Recommended texts will be
announced in class.
Audience: Graduate students and advanced undergraduates in applied
mathematics, engineering, or the natural sciences.
Background and Goals: In applied mathematics, we often try to
understand a physical process by formulating and analyzing mathematical
models which in many cases consist of differential equations with
initial and/or boundary conditions. Most of the time, especially if the
equation is nonlinear, an explicit formula for the solution is not
available. Even if we are clever or lucky enough to find an explicit
formula, it may be difficult to extract useful information from it and
in practice, we must settle for a sufficiently accurate approximate
solution obtained by numerical or asymptotic analysis (or a combination
of the two). This course is an introduction to the latter of these two
approximation methods. The material covered in the textbook includes
the nature of asymptotic approximations, asymptotic expansions of
integrals and applications to transform theory (Fourier and Laplace), regular and singular perturbation theory for differential equations
including transition point analysis, the use of matched expansions, and
multiple scale methods. The time remaining after studying these topics
will be devoted to the derivation of several famous canonical model
equations of applied mathematics (e.g. the Korteweg-de Vries equation
and the nonlinear Schroedinger equation) using multiscale asymptotics.
Students will come to understand how these equations arise again and
again from fields of study as diverse as water wave theory, molecular
dynamics, and nonlinear optics.
Grading: Students will be evaluated on the basis of homework
assignments and also participation and lecture attendance.
MATH 558. Ordinary Differential Equations.
Section 001.
Instructor(s):
Andrew J Christlieb
Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisites: Basic Linear Algebra, Ordinary Differential Equations (math 216), Multivariable Calculus (215) and Either Advanced Calculus (math 451) or an advanced mathematical methods course (e.g. Math 454); preferably both.
Course Objective:
This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric techniques and visualization. Much of the motivation for this approach comes from applications. Examples of applications of differential equations to science and engineering are a significant part of the course. There are relatively few proofs.
Course Description: Nonlinear differential equations and iterative maps arise in the mathematical description of numerous systems throughout science and engineering. Such systems may display complicated and rich dynamical behavior. In this course we will focus on the theory of dynamical systems and how it is used in the study of complex systems. The goal of this course is to provide a broad overview of the subject as well as an in-depth analysis of specific examples. The course is intended for students in mathematics, engineering, and the natural sciences. Topics covered will include aspects of autonomous and driven two variable systems including the geometry of phase plane trajectories, periodic solutions, forced oscillations, stability, bifurcations and chaos. Applications to problems from physics, engineering and the natural sciences will arise in the course by way of examples in lecture ad through the homework problems. We will cover material from Chapters 1-5 and 8-13 of the text.
Textbook
Nonlinear Ordinary Differetial Equations, Oxford Press. by: D.W. Jordan and P. Smith
References
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer. John Guckenheimer and Philip Holmes
Nonlinear Differential Equations and Dynamical Systems, Springer. Ferdinand Verhulst
Applications of Centre Manifold Theory, Springer. J. Carr
Nonlinear Systems, Chambridge. P.G. Drazin
Course Objectives:
To provide first-year graduate students with basic understanding of linear programming, its importance, and applications. To discuss algorithms for linear programming, available software and how to use it intelligently.
Section 001.
Instructor(s):
John R Stembridge
Prerequisite: MATH 512 or an equivalent level of mathematical maturity.
This course will be an introduction to algebraic combinatorics.
Previous exposure to combinatorics will not be necessary, but
experience with proof-oriented mathematics at the introductory
graduate or advanced undergraduate level, and linear algebra, will be needed.
Most of the topics we cover will be centered around enumeration and
generating functions. But this is not to say that the course is only
about enumeration — questions about counting are a good starting point
for gaining a deeper understanding of combinatorial structure.
Some of the topics to be covered include sieve methods, the matrix-tree
theorem, Lagrange inversion, the permanent-determinant method, the transfer matrix method, and ordinary and exponential generating
functions.
Recommended text: R. Stanley, Enumerative Combinatorics, Vol. I
Cambridge Univ. Press, 1997.
MATH 567. Introduction to Coding Theory.
Section 001.
Instructor(s):
Hendrikus Gerardus Derksen
Prerequisites & Distribution: One of MATH 217, 419, 513. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Student Body: Undergraduate math majors and EECS graduate students
Background and Goals: This course is designed to introduce math majors to an important area of applications in the communications industry. From a background in linear algebra it will cover the foundations of the theory of error-correcting codes and prepare a student to take further EECS courses or gain employment in this area. For EECS students it will provide a mathematical setting for their study of communications technology.
Content: Introduction to coding theory focusing on the mathematical background for error-correcting codes. Shannon's Theorem and channel capacity. Review of tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes such and Hamming, BCH, cyclic, Melas, Reed-Muller, and Reed-Solomon. Introduction to decoding starting with syndrome decoding and covering weight enumerator polynomials and the Mac-Williams Sloane identity. Further topics range from asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form.
MATH 571. Numerical Methods for Scientific Computing I.
Section 001.
Instructor(s):
James F EppersonThis course is an introduction to numerical linear algebra, a core subject in scientific computing. Three general problems are considered: (1) solving a system of linear equations, (2) computing the eigenvalues and eigenvectors of a matrix, and (3) least squares problems. These problems often arise in applications in science and engineering, and many algorithms have been developed for their solution. However, standard approaches may fail if the size of the problem becomes large or if the problem is ill-conditioned, e.g. the operation count may be prohibitive or computer roundoff error may ruin the answer. We'll investigate these issues and study some of the accurate, efficient, and stable algorithms that have been devised to overcome these difficulties.
The course grade will be based on homework assignments, a midterm exam, and a final exam. Some homework exercises will require computing, for which Matlab is recommended.
MATH 572. Numerical Methods for Scientific Computing II.
Section 001.
Instructor(s):
Divakar ViswanathMath 572 is an introduction to numerical methods for solving
differential equations. These methods are widely used in science
and engineering. The four main segments of the course will
cover the following topics:
MATH 592. Introduction to Algebraic Topology.
Section 001.
Instructor(s):
Igor Kriz
The purpose of this course is to introduce basic concepts
of algebraic topology, in particular fundamental group, covering spaces and homology. These methods provide the
first tools for proving that two topological spaces are
not topologically equivalent (example: the bowling ball
is topologically different from the teacup).
Other simple applications of the methods will
also be given, for example fixed point theorems for
continuous maps.
Prerequisites: basic knowledge of point set topology, such as
from 590 or 591.
Books: There is no ideal text covering all this material on
exactly the level needed (basic but rigorous). Recommended texts
include
Munkres: Elements of Algebraic topology (for homology)
and
J.P.May: A concise course in algebraic topology (for fundamental
group and covering spaces).
Both texts include topics which will not be covered in 592, and are
also suitable textbooks for the next course in algebraic topology, 695.
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Math Department
Introduction
The Mathematics Department of Loyola Academy offers courses in Algebra, Geometry, Pre-Calculus, Calculus and Statistics at all levels from introductory to Honors and Advanced Placement. While only three years of math are required for graduation, the majority of our students take courses for four years. The Mathematics Department provides a challenging, thought-provoking curriculum that is intellectually stimulating and college preparatory in nature, regardless of track level. In addition to traditional course work, students can apply to participate in the Clavius Scholars program. The Mathematics Department furthers its commitment to high quality education through the selection of first-rate textbooks and integration of technology. Students are required to purchase a TI-83 or TI-84 graphing calculator, which is incorporated into instruction. Teachers utilize computer labs and classroom computers for exploration and demonstration, further integrating technology into instruction.
Course Tracking
Loyola Academy requires that students complete a minimum of three credits of mathematics for graduation, while four years of study is strongly recommended. Incoming freshmen are placed in a track using a combination of the results of the STS test and grade school records. Focused attention is given to mathematics and quantitative sub-scores, while also considering the verbal and composite score. Incoming freshmen who have taken Algebra 1 or beyond during middle school will be placed in a course commensurate with their ability. The goal of the Mathematics Department is to find the best possible placement for each of our students. Multiple tracks exist in order to best meet varying student needs. While the majority of students stay in their assigned track for the duration of their Loyola Academy career, track placement is reevaluated on an annual basis. Minimum grade requirements must be met in order to maintain current track placement. Student performance and teacher recommendation can result in a change in a student's track placement for subsequent academic years.
Math Lab
The Mathematics Department recognizes that students sometimes need additional assistance in order to be successful. The Math Lab, located on the first floor of the library, is open daily and is staffed by two part-time math teachers in addition to members of the Mathematics Department. The Math Lab serves a variety of functions by offering homework help, assisting students who have been absent and providing help to students preparing for quizzes and tests.
Loyola Academy's Program for 7th or 8th grade Talented Math Students (LAPTMS)
Loyola Academy is offering a program for talented junior high math students. The Loyola Academy Program for Talented Math Students (LAPTMS) will seek to develop the talents of the area's brightest math students by offering an accelerated math curriculum in which participants will have the opportunity to complete Advanced Placement Calculus and post-Calculus courses by the end of high school. The Loyola Program for Talented Math Students is based on a similar program that has been offered by Johns Hopkins University since 1979, and is currently offered by several other leading universities across the country (i.e.; Northwestern University and Duke University).
Loyola Academy's Program for Talented Math Students is open to students entering at the Algebra 1 or Geometry level of math. Students may begin the program in 7th or 8th grade. LAPTMS 7th or 8th Grade Registration Form.
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Secondary Mathematics I [2011]
The fundamental purpose of Mathematics I is to formalize and extend the mathematics that students learned in the
middle grades. The critical areas, organized into units, deepen and extend understanding of linear relationships, in
part by contrasting them with exponential phenomena, and in part by applying linear models to data that exhibit a
linear trend. Mathematics 1 uses properties and theorems involving congruent figures to deepen and extend understanding of geometric knowledge from prior grades. The final unit in the course ties together the algebraic and
geometric ideas studied. The Mathematical Practice Standards apply throughout each course and, together with the
content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that
makes use of their ability to make sense of problem situations.
Critical Area 1: By the end of eighth grade students have had a variety of experiences working with expressions and
creating equations. In this first unit, students continue this work by using quantities to model and analyze situations,
to interpret expressions, and by creating equations to describe situations.
Critical Area 2: In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain
and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing
functions as objects in their own right. They explore many examples of functions, including sequences; they interpret
functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand
the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that,
depending upon the context, these representations are likely to be approximate and incomplete. Their work includes
functions that can be described or approximated by formulas as well as those that cannot. When functions describe
relationships between quantities arising from a context, students reason with the units in which those quantities are
measured. Students build on and informally extend their understanding of integer exponents to consider exponential
functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential
functions.
Critical Area 3: By the end of eighth grade, students have learned to solve linear equations in one variable and have
applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit
builds on these earlier experiences by asking students to analyze and explain the process of solving an equation
and to justify the process used in solving a system of equations. Students develop fluency writing, interpreting, and
translating between various forms of linear equations and inequalities, and using them to solve problems. They master
the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and
solution of simple exponential equations. Students explore systems of equations and inequalities, and they find and
interpret their solutions. All of this work is grounded on understanding quantities and on relationships between them.
Critical Area 4: This unit builds upon prior students' prior experiences with data, providing students with more formal
means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments
about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.
Critical Area 5: In previous grades, students were asked to draw triangles based on given measurements. They also
have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria,
based on analyses of rigid motions and formal constructions. They solve problems about triangles, quadrilaterals, and
other polygons. They apply reasoning to complete geometric constructions and explain why they work.
Critical Area 6: Building on their work with the Pythagorean Theorem in eighth
grade to find distances, students use a
rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines.Core Standards of the Course
Unit 1: Relationships Between Quantities By the end of eighth grade students have had a variety of experiences working with expressions and creating equations. In this first unit, students continue this work by using quantities to model and analyze situations, to interpret expressions, and by creating equations to describe situations. (ViewSecondary One Textbook.)
Reason quantitatively and use units to solve problems.
Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions.
N.Q.1
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N.Q.2
Define appropriate quantities for the purpose of descriptive modeling.
N.Q.3
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Interpret the structure of expressions.
Limit to linear expressions and to exponential expressions with integer exponents.
A.SSE.1
Interpret expressions that represent a quantity in terms of its context.*Create equations that describe numbers or relationships.
Limit A.CED.1 and A.CED.2 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Limit A.CED.3 to linear equations and inequalities. Limit A.CED.4 to formulas with a linear focus.
A.CED.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.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.CED.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
A.CED.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
SKILLS TO MAINTAIN Reinforce understanding of the properties of integer exponents. The initial experience with exponential expressions, equations, and functions involves integer exponents and builds on this understanding.
Unit 2: Linear and Exponential Relationships In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.
Represent and solve equations and inequalities graphically.
For A.REI.10 focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses. For A.REI.11, focus on cases where f(x) and g(x) are linear or exponential.
A.REI.10
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
A.REI.11
Explain
A.REI.12
Graph
Understand the concept of a function and use function notation.
Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.
Draw examples from linear and exponential functions. In F.IF.3, draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions.
F.IF.1
.IF.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Interpret functions that arise in applications in terms of a context.
For F.IF.4 and 5, focus on linear and exponential functions. For F.IF.6, focus on linear functions and intervals for exponential functions whose domain is a subset of the integers. Mathematics II and III will address other function types. N.RN.1 and N.RN.2 will need to be referenced here before discussing exponential models with continuous domains.
F.IF.4
For.*
F.IF.5
Rel.*
F.IF.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*
Analyze functions using different representations.
For F.IF.7a, 7e, and 9 focus on linear and exponential functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y=3n and y=100·2n.
F.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.9
Compare
Build a function that models a relationship between two quantities.
Limit F.BF.1a, 1b, and 2 to linear and exponential functions. In F.BF.2 connect arithmetic sequences to linear functions and geometric sequences to exponential functions.
F.BF.1
Write a function that describes a relationship between two quantities.*F.BF.2
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*
Build new functions from existing functions.
Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept.
While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard.
F.BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f
Construct and compare linear, quadratic, and exponential models and solve problems.
For F.LE.3, limit to comparisons between exponential and linear models.
F.LE.1
Distinguish between situations that can be modeled with linear functions and with exponential functions.
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F.LE.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.LE.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Interpret expressions for functions in terms of the situation they model.
Limit exponential functions to those of the form f(x) = bx + k .
F.LE.5
Interpret the parameters in a linear or exponential function in terms of a context.
Unit 3: Reasoning with Equations By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation and to justify the process used in solving a system of equations. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. Students explore systems of equations and inequalities, and they find and interpret their solutions. All of this work is grounded on understanding quantities and on relationships between them.
Understand solving equations as a process of reasoning and explain the reasoning.
Students should focus on and master A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in Mathematics III.
A.REI.1
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Solve equations and inequalities in one variable.
Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1/16.
A.REI.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Solve systems of equations.
Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to GPE.5, which requires students to prove the slope criteria for parallel lines.
A.REI.5
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
A.REI.6
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Unit 4: Descriptive Statistics Experience with descriptive statistics began as early as Grade 6. Students were expected to display numerical data and summarize it using measures of center and variability. By the end of middle school they were creating scatterplots and recognizing linear trends in data. This unit builds upon that prior experience, providing students with more formal means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.
Summarize, represent, and interpret data on a single count or measurement variable.
In grades 6 - 8, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.
S.ID.1
Represent data with plots on the real number line (dot plots, histograms, and box plots).
S.ID.2
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
S.ID.3
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
Summarize, represent, and interpret data on two categorical and quantitative variables.
Students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals.
S.ID.6b should be focused on situations for which linear models are appropriate.
S.ID.5
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
S.ID.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
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 and exponential models.
Informally assess the fit of a function by plotting and analyzing residuals.
Fit a linear function for scatter plots that suggest a linear association.
Interpret linear models
Build7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
S.ID.8
Compute (using technology) and interpret the correlation coefficient of a linear fit.
S.ID.9
Distinguish between correlation and causation.
Unit 5: Congruence, Proof, and Constructions In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.
Experiment with transformations in the plane.
Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle.
G.CO.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.CO.2
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G.CO.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G.CO.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.CO.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another
Understand congruence in terms of rigid motions.
Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems.
G.CO.6
Use
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.8
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Make geometric constructions.
Build on prior student experience with simple constructions. Emphasize the ability to formalize and defend how these constructions result in the desired objects.
Some of these constructions are closely related to previous standards and can be introduced in conjunction with them.
G.CO.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G.CO.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Unit 6: Connecting Algebra and Geometry Through Coordinates Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines.
Use coordinates to prove simple geometric theorems algebraically.
This unit has a close connection with Unit 5. Reasoning with triangles in this unit is limited to right triangles; e.g., derive the equation for a line through two points using similar right triangles.
Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in Mathematics I involving systems of equations having no solution or infinitely many solutions.
G.GPE.7 provides practice with the distance formula and its connection with the Pythagorean theorem.
G.GPE.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
G.GPE.5
Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G.GPE.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. *
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Keywords:
Background Tutorials
Numerical and Algebraic Expressions
There is a bunch of vocabulary that you just need to know when it comes to algebra, and coefficient is one of the key words that you have to feel 100% comfortable with. Check out the tutorial and let us know if you want to learn more about coefficients!
Definitions of Linear Systems
A system of equations is a set of equations with the same variables. If the equations are all linear, then you have a system of linear equations! To solve a system of equations, you need to figure out the variable values that solve all the equations involved. This tutorial will introduce you to these systems.
Matrix Definitions
Matrices can help solve all sorts of problems! This tutorial explains what a matrix is and how to find the dimensions of a matrix
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Ultimate Math Refresher for the Gre, Gmat & Sat
9780967759401
ISBN:
0967759404
Pub Date: 1999 Publisher: Lighthouse Review, Incorporated
Summary: A comprehensive math review for the GRE, GMAT, and SAT. This math refresher workbook is designed to clearly and concisely state the basic math rules and principles of arithmetic, algebra, and geometry which a student needs to master. This is accomplished through a series of carefully sequenced practice sets designed to build a student's math skills step-by-step. The workbook emphasizes basic concepts and problem solv...ing skills. Strategies for specific question types on the GRE, GMAT, and SAT are the focus of the Lighthouse Review self study programs
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I would like my students to access through the internet or as a software package
that can be installed on a computer a program that acts like a TI83 or TI84
graphing calculator. I am using Thomson Learning iLrn in an experimental class
that will emphasize using the graphing calculator. Does anyone know of such a
program
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The Economist Numbers Guide - Richard Stutely
Business analysis is a subject of many concepts: time-value of money, discounted cash flow valuation, amortization, depreciation, etc. The Economist Numbers Guide is a handy reference that will help you use these concepts appropriately and explain them clearly to your colleagues.
The Economist Numbers Guide is a short, but comprehensive book that contains everything you need to know about business mathematics. Inside, you'll find information on arithmetic, notation, financial structures, investment analysis, inflation, interest, distributions, graphing, forecasting, sampling, testing, decision trees, Markov chains, and even advanced subjects like linear programming. You'll be hard pressed to find a business situation that requires math this book doesn't cover well.
The Economist Numbers Guide is the reference book I rely on to ensure I'm using business analysis techniques properly and explaining them clearly to other people. If business analysis is part of your job, I highly recommend keeping this book close at hand
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An online testing system developed for entry-skills testing of first-year university students in algebra and calculus is described. The system combines the open-source computer algebra system Maxima with computer scripts to parse student answers, which are entered using standard mathematical notation and conventions. The answers can involve structures such as: lists, variable-precision-floating-point numbers, integers and algebra. This flexibility allows more sophisticated testing designs than the multiple choice, or exact match, paradigms common in other systems, and the implementation is entirely vendor neutral. Experience using the system and ideas for further development are discussed.
Introduction
Computer-aided assessment (CAA) has been widely adopted in higher education and many systems have been developed which have overlapping functionality. For example, Dopper and Sjoer (2004) report that seventeen different systems have been built over fifteen years at Delft University of Technology alone. Our experience at the University of Southern Queensland has involved several systems over a similar period. CAA systems are becoming more sophisticated in terms of the variety of input they will accept, for example, scanning semantic similarities in student C programs (Wang, Su, Wang & Ma, 2007), or grading text of open-ended problem solutions in earth science (Wang, Chang & Li, 2008). Over a similar period, mathematics educators have been evaluating computer algebra systems (CAS) as tools in the learning of mathematics (Artigue, 2002; Nicaud, Bouhineau & Chaachoua, 2004; Klasa, 2010; Berger, 2010). In the assessment of mathematics, in particular, the use of computer algebra systems (CAS) as part of CAA has become commonplace (Sangwin, 2006) and it has been predicted that all CAA will eventually link to CAS (Sangwin, 2003). Computer algebra offers the capacity to accept student algebraic input directly and either evaluate its equivalence to a known answer, or manipulate the input further to determine its properties. This allows the question author to avoid the shortcomings of multiple choice formats where the student knows that one of the given distracters is correct. For example, solving an equation is significantly different from verifying whether one of a number of alternatives is a solution; the student can avoid answering the question as set by checking each answer by substitution.
The first system to make use of CAS in this way was AiM (Klai, Kolokolnikov & Van den Bergh, 2000) and its descendant AiM-TtH (Strickland, 2002; Sangwin, 2003) which used the commercial computer algebra system Maple, but it required students to know Maple syntax and its question authoring system required programming expertise (Sangwin & Grove, 2006). Maple has its own proprietary system, MapleTA, while other systems have been developed for various CAS: CalMath uses Mathematica, CABLE (Naismith & Sangwin, 2004) uses Axiom and STACK (Sangwin & Grove, 2006) which uses Maxima. Each of these systems provided the student with an interactive mathematical experience within a virtual learning environment, that is they were tutorial systems as well as testing systems. Tutorial assessment systems have been used to prepare students for standardised testing (Feng, Heffernan & Koedinger, 2009) and in remedial teaching (Wang, 2010).
At the University of Southern Queensland we wanted to use CAA for entry-level testing (Engelbrecht & Harding, 2004) - assessing the core prerequisite, entry-level skills of students undertaking first-year algebra and calculus. First-year cohorts were large in number, up to 900 students, making manual testing and marking impractical. A wide variation in the skill levels of commencing students had been reported (Jourdan, Cretchley & Passmore, 2007) and early detection and intervention was desirable for the weaker students. It was essential that the entry-level testing be done early in the semester, with minimal expense to the student; that immediate feedback be available to identify those who may be at risk without early remediation; and that the test be delivered online, to make it available to off campus students (i.e. distance students). An existing CAA system was deemed unsuitable because it only allowed questions which were either multiple choice, or exact character-match to a given answer. It was also slow to use as LATEX-coded mathematical objects were first rendered as bitmap images that were subsequently re-embedded into HTML to generate the on screen questions. The quality of these images was thus hardware dependent, and often quite poor. A better system was needed, and we developed the Online Testing System (OTS) described below.
Past experience indicated that the most likely areas of difficulty were algebra, functions, trigonometry and inequalities, which suggested immediately the use of a CAS. However, we did not want to burden students with learning any specialised input syntax so early in their university experience, opting instead for an "informal syntax", along the lines proposed by Sangwin and Ramsden (2007), which was as close as possible on a keyboard to standard mathematical notation and convention. Lastly, we had no budget for proprietary software or licence fees, so we needed a vendor neutral system which used existing open source software.
Features of the Online Testing System
Over the past few years, we have been using an OTS which combines the open source CAS Maxima with PHP scripts and XML configuration files to parse, evaluate and mark student answers which are submitted online. Since it employs standard web technology, the system is accessible to students with only minimal resources, namely a web browser (Java is not required) and a PDF document viewer. It is a testing system rather than a tutorial system and has a number of features which address the difficulties outlined earlier.
Separate question delivery and answer collection
The OTS separates question delivery from answer collection. Students must login to download the question paper as a PDF document. They can either print this or view it on screen; since it is scalable it can be viewed at any desired resolution (even on mobile devices), without loss of readability. Students can work on the paper, away from the computer if that is the way they prefer, before logging in again to submit their answers via a standard web form. Access to the online submission form is unlimited, until the designated closing date, which allows distance students in different time zones to work when it suits them.
Another advantage of PDF question delivery is that it makes the OTS independent of the word processing or authoring system used by question authors. No programming expertise is required and the questions are displayed exactly as the author intended.
Disadvantages are that students cannot work interactively and check their syntax as with AiM and STACK. Also the question paper is printable and therefore not secure, though it would be possible to encrypt the PDF so that it was not printable. For our particular application to entry-level testing, this was not considered a serious problem as the intention is to eventually build up question banks and randomly generate question papers. More attention would need to be given to these issues if the OTS were applied for more security-sensitive purposes such as examinations.
Two-stage Check and Submit
When students enter their answers there are two stages to completing submission. The first stage is to select a Check button, after which any blank fields are highlighted and any specific reminder information is fed back for each question. Reminders are specified by the question author, some examples are:
You should supply a negative answer.
A simple solution of the form x/y is expected. If you don't supply a simple solution it will be marked incorrect.
Give numerical values only, accurate to 4 decimal places. Do not give angles in degrees. Separate multiple answers with a comma; the order of the answers is unimportant.
No syntax checking is performed at this stage, though this would be a desirable enhancement in future. The reminders are intended to help the students check their work before final submission. Students can change their answers at this stage, if they wish, by using the Back button in their browser. They can check and revise as many times as they like, though they only get the same reminders each time.
When checking is complete, the student uploads his or her answers by selecting the Submit button on the form. Their answers are evaluated, marked, the marks totalled and returned to the student's screen with specific feedback. The submitted answers and resulting marks are stored for future processing (e.g. uploading to central grading database) by the OTS. Question authors can optionally specify two types of feedback, one if the question is correct and the other if the question is incorrect. This functionality may be extended in the future.
Informal input syntax
As stated earlier, we did not require students to learn the syntax for Maxima, or commercial CAS systems like Maple or Mathematica, before sitting the test. Instead, we wanted them to be able to input their answers using an informal linear syntax, as natural as calculator entry, and as close to standard mathematical notation and convention as is possible on a keyboard. The implementation should be browser independent and allow for that fact that weaker students are often careless with syntax. For example, (2x-1)(x+3) could be entered either as (2x-1)(x+3), or (x+3)(2x-1), or as (2*x-1)*(x+3), the latter being the full Maxima syntax. If the student entered either of the first two variants, they are prefiltered using a PHP script to "add stars" before being passed to the CAS Maxima, which expands the brackets before checking the answer, thus making the order of the factors irrelevant. Usually, student input was not passed directly to Maxima, but was prefiltered and parsed by one or more PHP scripts. These scripts have been tested to ensure that they do not add any errors to student input, but the raw input is also captured in case of any doubt. Thus, the raw input can be reviewed if the student feels that a mistake has been made.
Sometimes, it is not necessary to call Maxima at all, resulting in faster processing performance. Consider the question:
Express
as a simple fraction involving no negative powers.
Student input was parsed using XML code as shown in the Appendix Example 1 (AE1).
The student input is matched to a regular expression test (regexp) which is handled directly by PHP. But first the string is prefiltered to remove leading, trailing and embedded white space, convert alpha characters to lower case, remove any explicit multiplication symbols *, substitute any grouping done with [ ] or { } pairs by ( ) and finally, remove all ( ) pairs. The filtered string is then compared to the regexp between the @ characters in the <value> element (see AE1). In this case, the @ character functions as a regexp delimiter, but more generally delimits any string to be passed to another script (@ was chosen as it is unlikely to appear in student input). The correct answer 16b/a is matched if the filtered string is either 16b/a or b16/a, the second form being necessary if the student entered the correct string b*16/a.
A second example is the following question.
Expand (x+1)(-2x+1)(x-3)
(Exponents or powers must be typed using the caret character ^. For example, type x2 as x^2)
The XML parsing instructions are given in Appendix Example 2 (AE2).
Two tests are applied to the student input and they must both return a Boolean true value for the question to be marked correct. The two tests are enclosed in the logical <and> container (AE2 lines 2 and 21). The first test is a regular expression match <regexp> (AE2 lines 4-9), which has been negated by the logical <not> tag (AE2 lines 3 and 10). This checks that the student input does not contain three pairs of parentheses i.e. ( )( )( ) or ( )*( )*( ), which would mean that the student had not done any expansion.
The second test (AE2 lines 11-20) specifies a script to be run by the CAS Maxima. The script is specified in the <script> container (AE2 lines 16-18) and expands the difference between the given expression and the prefiltered student input (the @ symbol). The result is compared to the value specified in the <value> element (AE2 line 19) and if it matches, the result of the test is Boolean true. Since the two tests are contained in an <and> element, the final result is the logical AND of the two tests. Notice that any necessary prefiltering must be done in each test separately, as each test is passed a copy of the student's raw input. Also notice, that this code would allow a partial expansion to be marked correct; if the student has correctly removed only one pair of parentheses, then the answer would be marked correct. Of course, this can be changed, if desired, to award only part marks, or no marks, for a partial expansion.
The capacity to prefilter input allows OTS to handle informal input syntax, which can be defined separately for each answer, if necessary, and the ability to combine different tests together allows sophisticated control over how an input is evaluated. Thus partial credit can be awarded for an answer that is not correct, but contains some of the correct working (Ashton, Beevers, Korabinski & Youngson, 2006; Darrah, Fuller & Miller, 2010).
Modular and self-validating
Using XML to specify both answer format and system configuration means that the OTS configuration is modular and self-validating. Invalid specifications or missing fields can be quickly detected and an invalid specification will not be parsed. Validation is done against a RELAXNG schema, chosen because RELAXNG has the capacity to specify data types, but in a way that is easier to use than other schema languages like XML Schema. A typical question specification is given in Appendix Example 3 (AE3).
Each question must have a <title> container, a <solution> container, which specifies how to determine the correct answer (see AE1 and AE2), and optionally, a <reminder> container, the contents of which are returned when the student selects Check, and lastly <feedback> containers which are returned when Submit is selected and the question is marked. Feedback may differ depending upon whether the question is correct or not. Since title, reminders and feedback are provided via a web page, they must be formatted as valid XHTML.
As an authoring language though, XML is rather daunting for some authors. A future plan is to develop an authoring front end to the OTS which would make it accessible to more users.
Data entry types and structures
So far OTS has developed the following data types and structures: integer and variable-precision floating point numbers; inequalities; regular expressions; an algebraic data type, for passing scripts to the CAS and giving access to all the functionality of Maxima; a choice data type, for multiple choice questions; and a general list data type, comprising comma-separated values which can be any of the other data types.
The list data type is very useful for questions such as:
Solve the following cubic equation for x: x3 - 4x2 + x + 6 = 0
(Your answers should be a list separated by commas.)
where one does not want to disclose to students that three answers are expected. Lists also give partial credit for some correct answers.
These data types and structures, combined with the suite of prefilter actions discussed earlier, provide fine-grained control over how student input is processed and evaluated. Thus, allowing the informal input syntax to be implemented. For example, equivalences such as 2.5 = 2 + 1/2 = 5/2 can be handled seamlessly without troubling the student with format directions.
Multistage decision tree
The implementation of the logical combinators <and>, <or> and <not> allows arbitrary combinations of tests to be performed on student input for any question. Since most of the data types are implemented directly in PHP, processing is very efficient and the CAS Maxima is only called when necessary.
Server-side processing
On the server, we have opted simply for tab-separated text files to capture the student input and log system activity. Reporting is currently web-based, however, back-end processing could easily be adapted to other formats, like SQL databases, which would allow more sophisticated query and reporting features.
When starting processes remotely via web interfaces, care must be taken to ensure that the server does not become overloaded, either with running processes, or processes that fail to terminate correctly. To ensure that the server performance is not compromised the following control measures were adopted.
A time limit of 30 seconds was placed on all running PHP scripts. A script cannot run longer than this preset time.
An adjustable maximum number of simultaneous submissions (initially set at 20) are allowed at any one time.
External processes, such as Maxima, are run via a small program that monitors the executable and will terminate the process if it exceeds a preset time limit. For Maxima we have found a time limit of 15 seconds adequate for our purposes. This "monitoring program" is essential as Maxima can be locked into an infinite loop when asked to process strange input.
With these measures in place, server bottlenecks have been avoided and the OTS availability has been high.
Implementation and experience so far
The OTS has been used to process entry-level mathematics students since 2006, enabling hundreds of students to be assessed and counseled appropriately. Students find interacting with the system over the web very natural; it can be can be accessed from anywhere in the world, and is available at night and over weekends, for students to use whenever convenient.
Using CAA has resulted in a huge time saving for staff and provides very early feedback to staff and students alike. During initial testing, student feedback was sought via anonymous exit questionnaires, from which a number of modifications were made to the student interface. The main issues identified were to do with question design and errors of logic in the marking scheme. There were no problems with the underlying engine. Subsequently, the OTS was deployed to deliver the first assignment in seven entry-level mathematics courses, for cohorts in Business, Engineering, Education and Sciences. Student feedback is routinely collected in all courses at USQ via anonymous online questionnaires at the end of each semester. This was one source of information about student reaction to the OTS; the other was comments and questions posted directly by students, either by email or on course online discussion groups, while they were using the system to complete their assignments. We emphasised that fair comment would always be listened to and that electronic marks would be reviewed if requested. With experience, we were better able to anticipate the variations that would arise in student solutions to each question, and to tailor the marking scheme and question feedback appropriately.
The majority of students accept the system well and those who do badly on the test later report that it served as a wake-up call to get them to revise their skills. To date there have been no complaints about accessing and using the system, but some students mistrust the reliability of the marking. Sangwin and Ramsden (2007) report similar fears among students, that they may be marked wrong because of a syntax error. Indeed, some still feel that this was the case even when investigation shows that their error was mathematical. Nonetheless, where a student is still not happy, we either let the student take the test again or mark the test manually, which currently occurs in about 6% of papers.
The main difficulty is developing and testing good questions and specifying the informal syntax for their solution. We learned from student responses and no system can handle perfectly all the possible answers that students may give to any question. Student input was analysed to detect common error patterns and this information was fed back both to question design and solution parsing. This process involved some trial and error, though as we gained more experience using the system, problems became less frequent.
As can be seen from the example questions in this paper, we have tried to help students by giving specific syntax hints where necessary. However, Sangwin and Ramsden (2007) report that around 18% of students still make errors of this kind despite explicit syntax instructions. Their students were expected to enter answers directly in CAS syntax, but in our experience with OTS the incidence of such errors is much lower, around 5%. We believe this is due to the effort we have made to implement an informal input syntax.
Future development
The OTS has shown the feasibility of linking CAA with CAS and implementing an informal linear syntax for student input. As an authoring language though, XML is rather daunting for some authors. It is planned to develop an authoring front-end to the OTS, which would allow questions and answers to be written within one document. The document could be parsed to generate the necessary XML configuration files, the question paper as PDF, and the associated web pages as XHTML. Such a system would make the OTS accessible to many more users.
Currently, the back-end administrative reports and functions use a purpose-built web interface, but capturing the data in an SQL database would integrate better with other data systems at the University. We also plan to investigate more elaborate user verification and access control to enable summative testing online.
We want to develop better-targeted feedback to the student about particular areas of weakness: algebra, functions, trigonometry, inequalities etc and possibly do some syntax checking of student input during the Check phase before submission. This would alert the student to possible syntax errors before they are assessed. One possibility is to use Maxima to render student input as TEX and pass this to a display engine like MathJax (see which could render and display the input as properly formatted mathematics. The student could then be asked to verify, during the Check phase, that the input is what they intended.
The modular design of OTS was intended to allow for automated testing, randomly generating tests from question banks, or targeting tests to areas of weakness. These functions will develop over time and provide web-based assessment, remedial and support materials to students. Another intended improvement is to paginate the Answer Form and allow students to save partially completed answers, which will allow for longer tests to be delivered.
Recommendations
We suggest that the following principles may be helpful to designers and developers of computerised assessment systems.
Separate question delivery from answer collection. Forcing students to sit at the computer to both read the question and enter the answer in the same session is unnecessarily stressful, especially if the student is unfamiliar with the technology. Allowing the students to work off line using their standard tools reduces this stress.
Give instructions about how the answer is to be formatted. Students fear being marked wrong simply because of a formatting error. Hence, the author must think about how the answers will be formatted when writing the question.
Develop a robust informal input syntax. In order to move away from fixed format or multiple choice questions, we need to be able to parse algebraic input and numeric input in a variety of forms. Data typing of input helps the different processing required, as does allowing more than one correct answer to a question.
Anticipate the variety of input expected on any question. For example, will a number be an integer, fraction or decimal, or possibly all three. If rounding is required, how will this be assessed? Where algebraic input is expected how will it be parsed? This kind of flexibility is invaluable in being able to ask more complex questions. With experience, one can predict the most likely student answers and give credit, or partial credit, to those which are correct.
Provide feedback and allow the student to check their answers before final submission. This reduces student anxiety about being marked wrong and increases the value of the test as a learning opportunity.
Allow a computer mark to be manually overridden. No automated system is perfect and knowing that there is a right of appeal greatly reduces student anxiety. Our early tests contained multiple errors, but we fixed and refined them over time. As we gained experience, the error rate declined, as did the number of requests for manual marking.
References
Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245-274. http:/dx.doi.org/10.1023/A:1022103903080
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I've been working through the math essentials book before my course begins this Saturday. There are some areas where I feel I need additional drills/exercises (like inequalities and roots). I'm not sure if I missed it somewhere on TrueTrack, but does Veritas Prep provide additional drills/exercises on their
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Product Synopsis
"The Barnett, Ziegler, Byleen, and Sobecki College MathZone site featuring algorithmic exercises, videos, and other resources accompanies the text
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In continuation with the topics studied in Algebra I, it will develop the real number system and will include a study of the complex numbers as a mathematical system. Students will study the ideas of relations and functions and expand the concept of functions to include quadratic, square root, exponential and logarithmic functions, and rational numbers. Emphasis will also be placed on the analysis of conic concepts with labs and the development of additional real life problem solving skills and applications. Emphasis will be placed on the application of concepts and skills introduced in Algebra II. The level of instruction/curriculum will focus on preparing the student for further advanced placement courses.
I will be available for tutoring Tuesdays and Thursdays after school from 3:00 to 4:00.Any changes to tutoring will be announced in class and on the class website.If the posted tutorial times do not work in your schedule be sure to discuss alternatives with me as soon as possible.Other math teachers are also available at other times which are posted in the math hallway.
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This quantitative reasoning text is written expressly for those students, providing them with the mathematical reasoning and quantitative literacy skills they'll need to make good decisions throughout their lives.
Common-sense applications of mathematics engage students while underscoring the practical, essential uses of math..
For more information about the title Using and Understanding Mathematics: A Quantitative Reasoning Approach (4th
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This textbook provides a comprehensive introduction to the theory and practice of validated numerics, an emerging new field that combines the strengths of scientific computing and pure mathematics. In numerous fields ranging from pharmaceutics and engineering to weather prediction and robotics, fast and precise computations are essential. Based onRuskeepaa gives a general introduction to the most recent versions of Mathematica, the symbolic computation software from Wolfram. The book emphasizes graphics, methods of applied mathematics and statistics, and programming. Mathematica Navigator can be used both as a tutorial and as a handbook. While no previous experience with Mathematica is required,... more...
Tough Test Questions? Missed Lectures? Not Enough Time?Introduces the reader to symbolic computations using Mathematica and enables readers to understand, perform, and optimize sophisticated symbolic computations. This work includes discussions of the symbolic operations such as equation solving, differentiation, series expansion, and integration with more than 200 worked examples
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I am one of those who believe that Calculus is among our species' deepest, richest, farthest-reaching and most beautiful intellectual achievements. This course provides an opportunity for you to discover and appreciate some of the jewels of Calculus. It is my privilege to be in a position to assist you in making those discoveries.
...also learn how to estimate functions by polynomials, so that the numbers such as , , , , , etc. can be estimated using simple addition and multiplication only;
...model problems from geometry and other disciplines using calculus;
...further improve their ability to communicate mathematical ideas and solutions to problems.
...improve their problem-solving ability.
From a most general perspective, the student should see growth in his/her mathematical maturity. The three-semester sequence of calculus courses form the foundation of any serious study of mathematics or other mathematically-oriented disciplines.
Communication Skills: Students will ...
(a)
...collect a portfolio during the course and write a reflection paper.
(b)
...do group work (labs and practice exams) throughout the course, which will involve both written and oral communication.
...improve their ability to write logically valid and precise mathematical proofs and solutions.
Life Value Skills: Students will ...
(a)
...develop an appreciation for the intellectual honesty of deductive reasoning.
(b)
...listen with an open mind and respond with respect.
(c)
...understand the need to do one's own work, to honestly challenge oneself to master the material.
Cultural Skills: Students will ...
(a)
...develop an appreciation of the history of Calculus and the role it has played in mathematics and in other disciplines.
(b)
...learn to use symbolical notation correctly and appropriately.
Aesthetic Skills: Students will ...
(a)
...develop an appreciation for the austere intellectual beauty of deductive reasoning.
(b)
...develop an appreciation for mathematical elegance.
Content:
We plan to cover Chapters 1 through 4 of the textbook.
Course Philosophy and Procedure
Two key components of a success in the course are regular attendance and a fair amount of constant, everyday study. You should try to make sure that your total study time per week at least triples the time spent in class. Working every day on calculus problems is a must. Also, an active class participation, working in small groups, not hesitating to ask me for help both in class and in my office can greatly enhance the success and quality of your learning. You should also use the Learning Center facilities (MC 320) as much as possible.
Grading will be based on three in-class exams ( points each), a cumulative final exam ( points), class participation, take-home problems, group practice exams ( points each), and a ( points worth) portfolio. You will be required to work hard, and will have every opportunity to show what you have learned.
Some of the homework assignments will be graded in two parts - the second part will require you to come to my office and explain your reasoning, answer some questions. Some of those assignments will be ``group assignments''.
In all your work, written and oral, it is essential to provide explanations, justify your reasoning. My grading scale is
A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%.
The following exceptions to that scale are possible:
An A on the final exam (more than points) will raise your grade up, one letter, i.e., a B will turn into an A, a BC will become AB, ....
An outstanding presentation, or an outstanding portfolio can raise your grade up a half letter, i.e., a C will turn into a BC, ....
If one is failing the course by the end of the semester, but has over average on exams, and earns at least points on the final, he/she can get a D for the final grade.
If one is passing the course by the time of the final exam, but earns less than points (a score less than ), that will result in an F for the final grade.
I am looking forward to explore this fascinating subject with you, and for all of us to have an interesting and enjoyable semester.
The Learning Center
provides a number of ways to assist you. In particular, there are drop in hours MTWRF 11:00-11:50 and 3:10-4:00.
If
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